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Title: A Class Room Logic - Deductive and Inductive with Special Application to the - Science and Art of Teaching
Author: McNair, George Hastings
Language: English
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                          A CLASS ROOM LOGIC


                        DEDUCTIVE AND INDUCTIVE

                    WITH SPECIAL APPLICATION TO THE
                      SCIENCE AND ART OF TEACHING


                                  BY


                    GEORGE HASTINGS McNAIR, PH. D.

             HEAD OF DEPARTMENT OF LOGIC AND MATHEMATICS,
                  CITY TRAINING SCHOOL FOR TEACHERS.
                        JAMAICA. NEW YORK CITY


                           THE ETHLAS PRESS
                 FIVE NORTH BROADWAY. NYACK. NEW YORK



                          COPYRIGHT, 1914, BY
                        GEORGE HASTINGS McNAIR



                                  To
                               MY WIFE.



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                               PREFACE.


This treatise is an outgrowth of our class room work in logic.

It has been published in the hope of removing some of the difficulties
which handicap the average student.

We trust that the language is simple and definite and that the
illustrative exercises and diagrams may be helpful in making clear some
of the more abstruse topics.

If a speedy review for examination is necessary, it is recommended
that the briefer course as outlined on page 493 be followed and that
the summaries closing each chapter be carefully read.

Only the fundamentals of deductive and inductive logic have received
attention. Moreover emphasis has been given to those phases which
appear to commend themselves because of their practical value.

Further than this we trust that the book may fulfill in some small way
the larger mission of inspiring better thinking and, in consequence, of
leading to a more serviceable citizenship.

Surely as civilization advances it is with the expectation of giving
greater significance to the assumption “that man is a _rational_
animal.”

I am indebted to a number of writers on logic, notably to Mill,
Lotze, Keynes, Hibben, Fowler, Aikins, Hyslop, Creighton and Jevons.
I am likewise under obligation to that large body of students who, by
frankly revealing their difficulties, have given me a different point
of view.

For constructive criticism and definite encouragement I owe a personal
debt of gratitude to Prof. Charles Gray Shaw of New York University,
to Prof. Frank D. Blodgett of the Oneonta Normal School and to Prin.
A. C. MacLachlan of the Jamaica Training School for Teachers.

                                                            G. H. McN.

  City Training School for Teachers,
         Jamaica, N. Y. City.
           October 3, 1914.



                           TABLE OF CONTENTS

  CHAPTER 1.――THE SCOPE AND NATURE OF LOGIC.
      1. The Mind.
      2. Logic Related to Other Subjects.
      3. Logic Defined.
      4. The Value of Logic to the Student.
      5. Outline.
      6. Summary.
      7. Review Questions.
      8. Questions for Original Thought and Investigation.

  CHAPTER 2.――THOUGHT AND ITS OPERATION.
      1. The Knowing Mind Compared with the Thinking Mind.
      2. Knowing by Intuition.
      3. The Thinking Process.
      4. Notions, Individual and General.
      5. Knowledge and Idea as Related to the Notion.
      6. The Logic of the Psychological Terms Involved in the Notion.
      7. Thought in the Sensation and Percept.
      8. Evolution and the Thinking Mind.
      9. The Concept as a Thought Product.
     10. The Judgment as a Thought Product.
     11. Inference as a Thought Product.
     12. Thinking and Apprehension.
     13. Stages in Thinking.
     14. Outline.
     15. Summary.
     16. Review Questions.
     17. Questions for Original Thought and Investigation.

  CHAPTER 3.――THE PRIMARY LAWS OF THOUGHT.
      1. Two Fundamental Laws.
      2. The Law of Identity.
      3. The Law of Contradiction.
      4. The Law of Excluded Middle.
      5. The Law of Sufficient Reason.
      6. Unity of Primary Laws of Thought.
      7. Outline.
      8. Summary.
      9. Illustrative Exercises.
     10. Review Questions.
     11. Questions for Original Thought and Investigation.

  CHAPTER 4.――LOGICAL TERMS.
      1. Logical Thought and Language Inseparable.
      2. Meaning of Term.
      3. Categorematic and Syncategorematic Words.
      4. Singular Terms.
      5. General Terms.
      6. Collective and Distributive Terms.
      7. Concrete and Abstract Terms.
      8. Connotative and Non-connotative Terms.
      9. Positive and Negative Terms.
     10. Contradictory and Opposite Terms.
     11. Privative and Nego-positive Terms.
     12. Absolute and Relative Terms.
     13. Outline.
     14. Summary.
     15. Illustrative Exercises.
     16. Review Questions.
     17. Questions for Original Thought and Investigation.

  CHAPTER 5.――THE EXTENSION AND INTENSION OF TERMS.
      1. Two-fold Function of Connotative Terms.
      2. Extension and Intension Defined.
      3. Extended Comparison of Extension and Intension.
      4. A List of Connotative Terms Used in Extension and Intension.
      5. Other Forms of Expression for Extension and Intension.
      6. Law of Variation in Extension and Intension.
     6a. Important Facts in Law of Variation.
     6b. Law of Variation Diagrammatically Illustrated.
      7. Outline.
      8. Summary.
      9. Illustrative Exercises.
     10. Review Questions.
     11. Questions for Original Thought and Investigation.

  CHAPTER 6.――DEFINITION.
      1. Importance.
      2. The Predicables.
      3. The Nature of a Definition.
      4. Definition and Division Compared.
      5. The Kinds of Definitions.
      6. When the Three Kinds of Definitions are Serviceable.
      7. The Rules of Logical Definition.
      8. Terms Which Cannot be Defined Logically.
      9. Definitions of Common Educational Terms.
     10. Outline.
     11. Summary.
     12. Illustrative Exercises.
     13. Review Questions.
     14. Questions for Original Thought and Investigation.

  CHAPTER 7.――LOGICAL DIVISION AND CLASSIFICATION.
      1. Nature of Logical Division.
      2. Logical Division Distinguished from Enumeration.
      3. Logical Division as Partition.
      4. Four Rules of Logical Division.
      5. Dichotomy.
      6. Classification Compared with Division.
      7. Kinds of Classification.
      8. Two Rules of Classification.
      9. Use of Division and Classification.
     10. Outline.
     11. Summary.
     12. Review Questions.
     13. Questions for Original Thought and Investigation.

  CHAPTER 8.――LOGICAL PROPOSITIONS.
      1. The Nature of Logical Propositions.
      2. Kinds of Logical Propositions.
      3. The Four Elements of a Categorical Proposition.
      4. Logical and Grammatical Subject and Predicate Distinguished.
      5. The Four Kinds of Categorical Propositions.
      6. Propositions which do not Conform to Logical Type.
      7. Propositions not Necessarily Illogical.
      8. The Relation between Subject and Predicate.
      9. Outline.
     10. Summary.
     11. Illustrative Exercises.
     12. Review Questions.
     13. Questions for Original Thought and Investigation.

  CHAPTER 9.――IMMEDIATE INFERENCE――OPPOSITION.
      1. The Nature of Inference.
      2. Immediate and Mediate Inference.
      3. The Forms of Immediate Inference.
           (1) Opposition.

  CHAPTER 10.――IMMEDIATE INFERENCE (Continued).
           (2) Immediate Inference by Obversion.
           (3) Immediate Inference by Conversion.
           (4) Immediate Inference by Contraversion.
      4. Epitome of the Four Processes of Immediate Inference.
             ◆ Inference by Inversion.
      5. Outline.
      6. Summary.
      7. Illustrative Exercises.
      8. Review Questions.
      9. Problems for Original Thought and Investigation.

  CHAPTER 11.――MEDIATE INFERENCE――THE SYLLOGISM.
      1. Inference and Reasoning.
      2. The Syllogism.
      3. The Rules of the Syllogism.
      4. Rules of Syllogism Explained.
      5. Aristotle’s Dictum.
      6. Canons of the Syllogism.
      7. Mathematical Axioms.
      8. Outline.
      9. Summary.
     10. Illustrative Exercises.
     11. Review Questions.
     12. Questions for Original Thought and Investigation.

  CHAPTER 12.――FIGURES AND MOODS OF THE SYLLOGISM.
      1. The Four Figures of the Syllogism.
      2. The Moods of the Syllogism.
      3. Testing the Validity of the Moods.
      4. Special Canons of the Four Figures.
      5. Special Canons Related.
      6. Mnemonic Lines.
      7. Relative Value of the Four Figures.
      8. Outline.
      9. Summary.
     10. Illustrative Exercises.
     11. Review Questions.
     12. Questions for Original Thought and Investigation.

  CHAPTER 13.――INCOMPLETE SYLLOGISMS AND IRREGULAR ARGUMENTS.
      1. Enthymeme.
      2. Epicheirema.
      3. Polysyllogisms. Prosyllogism――Episyllogism.
      4. Sorites.
      5. Irregular Arguments.
      6. Outline.
      7. Summary.
      8. Review Questions.
      9. Questions for Original Thought and Investigation.

  CHAPTER 14.――CATEGORICAL ARGUMENTS TESTED ACCORDING TO FORM.
      1. Arguments of Form and Matter.
      2. Order of Procedure in a Formal Testing of Arguments.
      3. Illustrative Exercise in Testing Arguments which are Complete
           and whose Premises are Logical.
      4. Illustrative Exercise in Testing Completed Arguments, one or
           both of whose Premises are Illogical.
      5. Incomplete and Irregular Arguments.
      6. Common Mistakes of the Student.
      7. Outline.
      8. Summary.
      9. Review Questions.
     10. Questions for Original Thought and Investigation.

  CHAPTER 15.――HYPOTHETICAL AND DISJUNCTIVE ARGUMENTS INCLUDING THE
          DILEMMA.
      1. Three Kinds of Arguments.
      2. Hypothetical Arguments.
      3. Antecedent and Consequent.
      4. Two Kinds of Hypothetical Arguments.
      5. Rule and Two Fallacies of Hypothetical Argument.
      6. Hypothetical Arguments Reduced to Categorical Form.
      7. Illustrative Exercises Testing Hypothetical Arguments of All
           Kinds.
      8. Disjunctive Arguments.
      9. Two Kinds of Disjunctive Arguments.
     10. First Disjunctive Rule.
     11. Second Disjunctive Rule.
     12. Reduction of Disjunctive Argument.
     13. The Dilemma.
     14. Four Forms of Dilemmatic Arguments.
     15. The Rule of Dilemma.
     16. Illustrative Exercises Testing Disjunctive and Dilemmatic
           Argument.
     17. Ordinary Experiences Related to Disjunctive Proposition and
           Hypothetical Argument.
     18. Outline.
     19. Summary.
     20. Review Questions.
     21. Questions for Original Thought and Investigation.

  CHAPTER 16.――THE LOGICAL FALLACIES OF DEDUCTIVE REASONING.
      1. A Negative Aspect.
      2. Paralogism and Sophism.
      3. A Division of the Deductive Fallacies.
      4. General Divisions Explained.
      5. Fallacies of Immediate Inference.
      6. Fallacies in Language (Equivocation).
      7. Fallacies in Thought (Assumption).
      8. Outline.
      9. Summary.
     10. Illustrative Exercises in Testing Arguments in Both Form and
           Meaning.
     11. Review Questions.
     12. Questions for Original Thought and Investigation.

  CHAPTER 17.――INDUCTIVE REASONING.
      1. Inductive and Deductive Reasoning Distinguished.
      2. The “Inductive Hazard.”
      3. Complexity of the Problem of Induction.
      4. Various Conceptions of Induction.
      5. Induction and Deduction Contiguous Processes.
      6. Induction an Assumption.
      7. Universal Causation.
      8. Uniformity of Nature.
      9. Inductive Assumptions Justified.
     10. Three Forms of Inductive Research.
     11. Induction by Simple Enumeration.
     12. Induction by Analogy.
     13. Induction by Analysis.
     14. Perfect Induction.
     15. Traduction.
     16. Outline.
     17. Summary.
     18. Review Questions.
     19. Questions for Original Thought and Investigation.

  CHAPTER 18.――MILL’S FIVE SPECIAL METHODS OF OBSERVATION AND
          EXPERIMENT.
      1. Aim of Five Methods.
      2. Method of Agreement.
      3. Method of Difference.
      4. The Joint Method of Agreement and Difference.
      5. The Method of Concomitant Variations.
      6. The Method of Residues.
      7. General Purpose and Unity of Five Methods.
      8. Outline.
      9. Summary.
     10. Review Questions.
     11. Questions for Original Thought and Investigation.

  CHAPTER 19.――AUXILIARY ELEMENTS IN INDUCTION.
          OBSERVATION――EXPERIMENT――HYPOTHESIS.
      1. Foundation of Inductive Generalizations.
      2. Observation.
      3. Experiment.
      4. Rules for Logical Observation and Experiment.
      5. Common Errors of Observation and Experiment.
      6. The Hypothesis.
      7. Induction and Hypothesis Distinguished.
      8. Hypothesis and Theory Distinguished.
      9. The Requirements of a Permissible Hypothesis.
     10. Uses of Hypothesis.
     11. Characteristics Needed by Scientific Investigators.
     12. Outline.
     13. Summary.
     14. Review Questions.
     15. Questions for Original Thought and Investigation.

  CHAPTER 20.――LOGIC IN THE CLASS ROOM.
      1. Thought is King.
      2. Special Functions of Induction and Deduction.
      3. Two Types of Minds.
      4. Conservatism in School.
      5. The Method of the Discoverer.
      6. Real Inductive Method not in Vogue in Class Room Work.
      7. As a Method of Instruction, Deduction Superior.
      8. Conquest, not Knowledge, the Desideratum.
      9. Motivation as Related to Spirit of Discovery.
     10. Discoverer’s Method Adapted to Class Room Work.
     11. Question and Answer Method not Necessarily One of Discovery.
     12. Outline.
     13. Summary.
     14. Review Questions.
     15. Questions for Original Thought and Investigation.

  CHAPTER 21.――LOGIC AND LIFE.
      1. Logic Given a Place in a Secondary Course.
      2. Man’s Supremacy Due to Power of Thought.
      3. Importance of Progressive Thinking.
      4. Necessity of Right Thinking.
      5. Indifferent and Careless Thought.
      6. The Rationalization of the World of Chance.
      7. The Rationalization of Business and Political Sophistries.
      8. The Rationalization of the Spirit of Progress.
      9. A Rationalization of the Attitude Toward Work.
     10. The Logic of Success.
     11. Outline.
     12. Summary.
     13. Review Questions.
     14. Questions for Original Thought and Investigation.

  GENERAL EXERCISES IN TESTING CATEGORICAL ARGUMENTS.

  GENERAL EXERCISES IN TESTING HYPOTHETICAL, DISJUNCTIVE AND
          DILEMMATIC ARGUMENTS.

  EXAMINATION QUESTIONS FOR TRAINING SCHOOLS AND COLLEGES.

  BIBLIOGRAPHY.

  OUTLINE OF BRIEFER COURSE.

  INDEX.



                              CHAPTER 1.

                    THE SCOPE AND NATURE OF LOGIC.


=1. THE MIND.=

As to the true conception of matter the world is ignorant. Yet when
asked, “What does matter do?” the reply is, “Matter moves, matter
vibrates.” Moreover, relative to the exact nature of mind, the world
is likewise ignorant. But to the question, “What does mind do?” the
response comes, “The Mind _knows_, the mind _feels_, the mind _wills_.”
The mind has ever manifested itself in these three ways. Because of
this three-fold function it is easy to think of the mind as being
separated into distinct compartments, each constituting an independent
activity. This is erroneous. The mind is a living unit having three
sides but never acting one side at a time. When the mind knows it also
feels in some way and wills to some extent. To illustrate: Music is
heard and one _knows_ it to be Rubinstein’s Melody in F. The execution
being good one _feels_ pleasure. That the pleasurable state may be
augmented one _wills_ a listening attitude. For analytical purposes
the psychologists have a way of naming the state of mind from the
predominating manifestation.


=2. LOGIC RELATED TO OTHER SUBJECTS.=

What the mind _is_ may in time be answered satisfactorily by
philosophy; what the mind _does_ is described by psychology; what the
mind _knows_ is treated by logic. Again: the mind as a whole furnishes
the subject matter for psychology, whereas logic is concerned with the
mind knowing, aesthetics with the mind feeling, and ethics with the
mind willing. Ethics attempts to answer the question, “What is right?”
aesthetics, “What is beautiful?” and logic, “What is true?”

Though both psychology and logic treat of the knowing aspect of the
mind, yet the fields are not identical. The former deals with the
_process_ of the knowing mind as a _whole_, while the latter is
concerned mainly with the _product_ of the knowing mind when it
_thinks_. To be specific: The mind knows when it becomes aware of
anything, moreover, this condition of awareness appears in two ways:
first, immediately or by _intuition_; second, after deliberation or by
_thinking_. For example, one may know immediately or by intuition that
the object in the hand is a lead pencil, but when requested to state
the length of the pencil there is deliberation involving a comparison
of the unknown length with a definite measure. It may now finally
be asserted that the pencil is six inches long. When we know without
hesitation the process involved is intuition, whereas when the
knowledge comes after some sort of comparison the mental act is called
thinking. It, therefore, becomes the business of psychology to deal
with both intuition and thinking while logic devotes its attention
to thinking only, and even in this field the work of logic is more or
less indirect. The specific scope of logic is the product of thinking
or thought.[1] What are the forms of thought? What are the laws of
thought? Are the several thoughts true? These are the questions which
logic is supposed to answer.

For the logician thought has two sources, his own mind and the mind
of others. In the latter case thought becomes accessible through the
medium of language. There is in consequence a close connection between
logic, the science of thought, and grammar, the science of language.
Because of this near relation logic is sometimes called the “grammar
of thought.”

To study any science properly one must have thoughts and since logic is
the science of all thought the subject may be regarded as the _science
of sciences_.


=3. LOGIC DEFINED.=

“Logic is the science of thought.” This definition commonly given
is too brief to be helpful. Should not a definition of any subject
represent a working basis upon which one may build with some knowledge
of what the structure is to be? The following, a little out of the
ordinary, seems to supply this condition: _Logic as a science makes
known the laws and forms of thought and as an art suggests conditions
which must be fulfilled to think rightly._

In justification of the latter definition it may be argued that it
covers the topics usually treated by logicians. It is said that a
science teaches us to know while an art teaches us to do. As a science
logic teaches us to know certain laws which underlie right thinking.
For example, the law of identity which makes possible all affirmative
judgments, such as “Some men are wise,” “All metals are elements,” etc.
Likewise as a science logic acquaints us with certain universal _forms_
to which thought shapes itself, such as definitions, classifications,
inductions, deductions. Further, logic lays down definite _rules_ which
lead to right thinking. To wit: Because it is true of a _part_ of a
class it should not be assumed that it is true of the _whole_ of that
class: or, in short, do not distribute an undistributed term.

A possible profit to the student may result from a study of certain
authentic definitions herewith subjoined:

  (1) “Logic is the science of the laws of thought.” Jevons.

  (2) “Logic is the science which investigates the process of
      thinking.” Creighton.

  (3) “Logic as a science aims to ascertain what are the laws of
      thought; as an art it aims to apply these laws to the detection
      of fallacies or for the determination of correct reasoning.”
      Hyslop.

  (4) “Logic is the art of thinking.” Watts.

  (5) “Logic is the science and also the art of thinking.” Whateley.

  (6) “Logic is the science of the formal and necessary laws of
      thought.” Hamilton.

  (7) “Logic is the science of the regulative laws of the human
      understanding.” Ueberweg.

  (8) “Logic treats of the nature and of the laws of thought.” Hibben.

  (9) “Logic may be defined as the science of the conditions on which
      correct thoughts depend, and the art of attaining to correct and
      avoiding incorrect thoughts.” Fowler.

 (10) “Logic is the science of the operations of the understanding
      which are subservient to the estimation of evidence.” Mill.

 (11) “Logic may be briefly described as a body of doctrines and rules
      having reference to truth.” Bain.

It would seem as if there were as many different definitions as there
are books on the subject. This is due partly to the disposition of
the older logicians to ignore the _art_ of logic and partly to the
difficulty of giving in a few words a satisfactory description of a
broad subject. In the fundamentals of logical doctrine present-day
authorities virtually agree.


=4. THE VALUE OF LOGIC TO THE STUDENT.=

Logic is rapidly coming into favor as a major subject in institutions
devoted to educational theory. Some of the reasons for this change of
attitude are herewith subjoined:

(1) _Logic should stimulate the thought powers._ This is the age of
the survival of the thinker. The fact that the man who thinks _best_
is the man who thinks _much_ and _carefully_ will be accepted by those
who believe that practice makes perfect. “One needs only to observe
the average commuter to conclude that a large percent. of our business
men read too much and think too little.” “Much readee and no thinkee”
was the reply of a Chinaman when asked his opinion of the doings of
the average American. “We as a people are newspaper mad, reading for
entertainment, seldom for mental improvement.”

(2) _Logic aims to secure correct thought._ Are not many of the sins
and most of the failures in this world due to incorrect thinking?

(3) _Logic should train to clear thinking._ It would be difficult to
estimate the loss of energy to the brain worker because he has not
the power to think clearly. Maximum efficiency is impossible with a
befogged brain. How discouraging it is to the student to attempt to
get from the paragraph the thought of the author, who in trying to be
profound succeeds in being profoundly abstruse. There is a probable
need for broad, deep thoughts, but these when placed in a text book
should be sharpened to a point.

(4) _Logic should aid one to estimate aright the statements and
arguments of others._ This is of especial value to the teacher who is
constrained to teach largely from text books. Because it is found in a
book is not proof positive that it is true. Why should we assume that
the book is infallible when we know that the man behind the book is
fallible?

(5) _Logic insists on definite, systematic procedure._ To be logical
is to be businesslike. A study of logic would, no doubt, benefit our
churches and parliamentary orders as well as our schools.

(6) _Logic demands lucid, pointed, accurate expression._ How we would
increase our working efficiency could we but express our thoughts in
an attractive and interesting manner. To listen to the speeches of some
of our great and good men who are concerned in directing the “ship of
state” is sufficient argument that the American schools need more logic.

(7) _Logic is especially adapted to a general mental training._ Despite
the swing of the pendulum of public opinion toward the bread-and-butter
side of life, there are many of high repute who claim that for the
sake of that mental acumen which distinguished the Greek from his
contemporaries we cannot afford to sacrifice everything on the altar
of commercialism.

(8) _Logic worships at the shrine of truth and adds to our store of
knowledge._ What has aided the world more in its march onward than this
deep-seated passion for truth and what has impeded it more than that
vain and wanton indifference to truth which brought to the world its
darkest age?


=5. OUTLINE――=

  THE SCOPE AND NATURE OF LOGIC.

  (1) The Mind.
        Three aspects.
        Unity of.

  (2) Logic Related to Other Subjects.
        Mental philosophy, psychology, logic.
        Psychology, logic, aesthetics, ethics.
        Two ways of knowing.
        Special province of logic.
        Logic and language.
        A science of sciences.

  (3) Logic Defined.
        A general definition.
        A more satisfactory definition.
        A list of authentic definitions.

  (4) The Value of Logic to the Student.
        Eight reasons for its study.


=6. SUMMARY.=

(1) The aspects of the mind are knowing, feeling and willing.

The mind is a living unit and never knows without feeling in some way
and willing to some extent.

(2) What the mind _is_ must be answered by philosophy; what the mind
_does_ by psychology and what the mind _knows_ by logic.

Psychology treats of the mind as a whole, logic of the mind knowing,
aesthetics of the mind feeling and ethics of the mind willing. Ethics
answers the question, What is right? Aesthetics, What is beautiful?
Logic, What is true?

The standpoint of logic is not identical with any particular portion of
psychology.

The mind knows in two ways: (a) by intuition, (b) by thinking. Thinking
is a process――thought a product. Logic deals indirectly with the former
and directly with the latter.

Generally speaking, logic is a systematic study of thought. For the
logician thought has two sources: (a) his own mind and (b) spoken or
written language.

Because of the ambiguity of language logic has much to do with it as a
faulty vehicle of thought.

(3) Logic as a science makes known the laws and forms of thought and
as an art suggests conditions which must be fulfilled to think rightly.
Author.

“Logic may be defined as the science of the conditions on which correct
thoughts depend, and the art of attaining to correct and avoiding
incorrect thoughts.” Fowler.

In the fundamentals of logical doctrine present day logicians virtually
agree.

(4) Logic should stimulate the thought powers; secure correct and clear
thinking; aid in the estimation of arguments; inspire definite,
systematic procedure; demand lucid, pointed, accurate expression and be
especially adapted to general mental discipline.

Logic adds to our store of knowledge and develops a passion for the
truth.


=7. REVIEW QUESTIONS.=

  (1) Explain and illustrate the three ways in which the mind may
      manifest itself.

  (2) Illustrate the fact that the mind acts in unity.

  (3) Show briefly how logic is related to mental philosophy,
      psychology, aesthetics, ethics and grammar.

  (4) Illustrate the two ways of knowing.

  (5) Distinguish between thinking and thought.

  (6) Give a general definition of logic. Why is this definition
      unsatisfactory?

  (7) What are the two sources of thought?

  (8) Why are logic and language so closely related?

  (9) Give that definition of logic which best satisfies you.

 (10) Summarize the benefits which you hope to derive from your study
      of logic.

 (11) Why should teachers be clear thinkers?

 (12) Why should teachers be especially on guard against incorrect
      statements of all kinds?

 (13) Show how logic might be of assistance to the business man.


=8. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Prove that there is nothing real in the world save the mind
      itself.

  (2) “Logic is concerned primarily with how we _ought_ to think and
      only in a secondary way with how we actually think.” Explain this
      quotation.

  (3) Prove that there is no such thing as intuitive knowing.

  (4) Is there any difference between knowledge and thoughts?
      Illustrate.

  (5) Show by illustrations that the English language is ambiguous.

  (6) Prove by concrete illustration that this is the age of the
      survival of the thinker.

  (7) Which is the more harmful: falsehood mixed with truth or
      unadulterated falsehood? Give reasons.

  (8) Give a concrete example of incorrect thinking.

  (9) Show that wrong thinking leads to wrong doing.

 (10) To be worth while must every subject have a practical value?

 (11) “The 20th century virtue is a passion for truth.” Prove the truth
      of this.



                              CHAPTER 2.

                      THOUGHT AND ITS OPERATION.


=1. THE KNOWING MIND COMPARED WITH THE THINKING MIND.=

In the preceding chapter we were told that the mind may know in two
ways (1) by intuition and (2) by thinking. It is thus implied that
the knowing mind includes the thinking mind plus intuition. Thinking
always involves knowing, but knowing need not involve thinking, and
when some logicians maintain that to know a thing one must think it,
there is danger of being misled. They mean by this that in order to
know anything in a permanent and highly serviceable way one must think
it. All animals know, even such a stupid one as the oyster, and yet one
would hardly give an oyster credit for thinking. Only the higher orders
of animal life think. Some argue that the power is confined exclusively
to the human family. This opinion is debatable. If the claimant means
by thinking, reasoning then his ground is well taken. But if he is
willing to give to thinking a broader content, then he has little
defense for his stand. However, attach as broad a meaning to thinking
as the derivation of the word will permit and even then it is a
narrower term than knowing. Thinking plus intuition equals knowing,
and in intuition there is probably no thinking.


=2. KNOWING BY INTUITION.=

It has been affirmed that intuition is the process involved when the
mind knows _instantly_.[2]


  ILLUSTRATIONS:

(1) As I raise my eyes a figure comes to view. My mind knows
_instantly_ that it is the figure three. (2) The ear catches
_immediately_ a tune which is being sung in the room below. Without
deliberation the mind recognizes the tune as America. The mind may
thus know by intuition through any one of the five senses. These are
the wires of connection between the outer world and the mind within
and transmission over these wires may be instantaneous or intuitive.
This is not all. (3) My mind may center its attention on itself and
may recognize there a mental picture or image of a pet dog. Since
this activity is without any apparent deliberation the process must
be intuitive. To define intuitive knowledge as that which comes to the
mind through the senses only is incorrect, as it leaves out altogether
the knowledge the mind may obtain of its own activity as in
illustration “(3).”

Knowledge is anything known. _Intuitive knowledge is knowledge which
comes to the mind immediately by direct observation._ The field for
intuitive knowledge may be the external world or the internal world
though, of course, the former is the more common ground. It is here
that the mind by intuition secures the most of its raw material which,
through the process of thinking, is worked over into a connected,
unified system of lasting value.

The intuitions are the beginning and the basis of all knowledge, and
knowledge gained by intuition is the basis of all thinking.


=3. THE THINKING PROCESS.=

It is claimed that _think_ comes from the same root as _thick_. From
this one would conclude that the process of thinking is virtually a
process of thickening. Surely as one thinks he enriches or thickens
his knowledge. As one thinks percepts into concepts and concepts into
judgments he makes richer in meaning the various notions concerned.
Thinking is largely a matter of pressing many into one: of linking
together the disconnected fragments of the conscious field.


  DEFINITION:

_Thinking is the deliberative process of affirming or denying
connections._

The same idea may be expressed in a variety of ways as the following
indicate.

  (1) “Thinking is the conscious adjustment of a means to an end in
      problematic situations.” Miller.

  (2) “To think is to designate an object through a mark or attribute
      or what is the same thing, to determine a subject through a
      predicate.” Bowen.

  (3) “Thought is the comprehension of a thing under a general notion
      or attribute.” Wm. Hamilton.

  (4) “To think is to make clear through concepts the perceived
      objects.” Dressler.

In the foregoing definitions it is implied that thinking is a
connecting or _thickening_ process. In all forms of thinking from the
simplest to the most complex the knowing mind hunts for some basis of
connection and having found it _thinks_ the relationship into a unified
whole.

The thinking process is the digestive process of the mind. Much as
the digestive organs assimilate the food stuff of the physical world,
so the thinking organ assimilates the food stuff of the mental world.


  ILLUSTRATIONS OF THE THINKING PROCESS:

(1) The child is unable to explain the meaning of “hocus-pocus” as
it occurs in the question, “What hocus-pocus is this?” The child mind
is unable to establish any connection between the word and its real
meaning. In short, is unable to _think into it a meaning_; it therefore
becomes necessary for the teacher to establish some basis of connection
and this he does by suggesting _nonsense_ as a _synonym_.

(2) The teacher holds before the class an Egyptian house god and asks,
“What is it?” After a moment of hesitation some child who has seen
pictures of “his satanic majesty” avers that the object is a “little
devil.” Thus has a connection been established between the idol and
pictures of satan.

(3) John is unable to solve the following problem as he can discern no
connection between the data given and the data required. Problem. 3/4
of my salary is $900, what is my salary?

  Data. Given: 3/4 of salary = $900.
  Required: 4/4 of salary = ?

In order that John may _think_ a solution the teacher must lead him to
see some connection between 3/4 and 4/4. With this in mind the form of
the data is changed to

  Given: 3-fourths = $900
  Required: 4-fourths = ?

or

  Given: 3 parts = $900
  Required: 4 parts = ?

John now notes that 4 parts is 4/3 times 3 parts and consequently
writes 4/3 of $900, which is $1,200 as the answer. Or he may find the
value of 1 part and then of 4 parts.


=4. NOTIONS, INDIVIDUAL AND GENERAL.=

_A notion is any product of the knowing mind――anything which the mind
notes or becomes aware of._

But the mind knows in two ways, by intuition and by thinking. In
consequence the mind has two kinds of notions, those which are
intuitive or _individual notions_ and those which originally result
from thinking or _general notions_.

_An individual notion is a notion of one thing. A general notion is a
notion of a class of things._

_Note._ Here it is necessary to distinguish between a thing and an
object. _An object is a thing which occupies space_ such as a pencil
or a book. “_Thing_” is, therefore, a broader term than “_object_.”
“_A thing is that which has individual existence._” From the viewpoint
of logic “thing” includes objects, qualities, relations, spiritual
entities. Gravitation is a thing but not an object. A tree is both an
object and a thing.


  ILLUSTRATIONS OF NOTIONS.

My notion of the pencil with which I am writing is an individual
notion, but my notion of pencil as a _class name_ is general. My
yellow dog, the honesty of Lincoln, Albert White, New York City, are
individual notions, while dog, honesty, man, city, are general notions.

A sure way to determine whether the notion is individual or general
is to attempt to divide it into its kinds. Only general notions may be
subdivided.


=5. KNOWLEDGE AND IDEA AS RELATED TO THE NOTION.=

Knowledge is anything known, while anything of which the mind becomes
aware is a notion. Notions are always bits of knowledge, but knowledge
is not always a notion. Notions are mental products belonging to the
mind which thinks them, while knowledge, though it must first be a
mental product of someone’s mind, may not necessarily be a product of
yours or mine. Notions are always found in the mind, while knowledge
may be found in books, but not necessarily in some individual mind.
Knowledge stands for everything _known_, the notion, for everything
_noted_. The Egyptians may have possessed much knowledge of which we
may never become aware. Much of their _knowledge_ may never become
_notions_ of the American people. A notion is an existing state of
consciousness. Said notion may be committed to paper, and then it
may give way to another notion. It now ceases to be your notion, but
remains on the printed page, as a bit of knowledge.

“Idea,” because of its ambiguity, really has no place in logic. The
term is frequently restricted to a _reproduced percept_. To illustrate:
When the pencil is before me the mental product is a percept, but when
the pencil is withdrawn and I try to think of it, then have I an _idea_
of “pencil.” Probably _idea_ is most commonly associated with _meaning_
and _belief_. To illustrate: What is your _idea_ as to the meaning of
homogeny? or What are your _ideas_ on the tariff?


=6. THE LOGIC OF THE PSYCHOLOGICAL TERMS INVOLVED IN THE NOTION.=

Concerning the knowing mind the psychologist classifies its activities
and their products as follows:

      _Activity_            _Product_
  (1) Presentative
        (1) Sensation       Sensation
        (2) Perception      Percept

  (2) Representative
        (1) Imagination }   Image
        (2) Memory      }

  (3) Thinking
        (1) Conception      Concept
        (2) Judging         Judgment
        (3) Reasoning       Inference

The notion as _any_ product of the knowing mind includes the _six_
products as indicated by the psychologist.

The individual notion which is intuitive includes the sensation,
percept and image; the general notion which is a thought product stands
for the concept, judgment and inference. To put it mathematically――

                      { sensation }                       }
  Individual notion = { percept   } = intuitive products  }
                      { image     }                       } _notion_
                   { concept   }                          }
  General notion = { judgment  } = thought products       }
                   { inference }                          }

As we shall have occasion frequently to refer to these psychological
terms it may be well to define them.

   _Psychological Definition._         _Logical Definition._

  A sensation is the first and       A sensation is a vague,
  simplest mental result of the      unlocalized mental product of
  stimulation of an incarrying       the knowing mind.
  nerve.

  A percept is a mental              A percept is a consciously
  product which results from a       localized group of sensations.
  consciousness of particular
  material things present to the
  sense.

  An image is a mental product       An image is a reproduced
  which results from particular      percept.
  material things not present to
  the sense.

  A concept is a representation      A concept is a mental product
  in our minds answering to a        arising from thinking many
  general name.                      notions into one class.

  A judgment is the result of        A judgment is the mental
  asserting an agreement or          product arising from conjoining
  disagreement between two ideas.    or disjoining notions.

  An inference is a judgment         An inference is a judgment
  derived from perceiving            derived from antecedent
  relations between other            judgments.
  judgments.

It is seen that the sensations furnish the raw material. Ignoring the
few exceptions we may then say that a percept is a made-over group of
sensations; a concept a thought-made group of percepts; a judgment a
thought-made group of concepts; an inference a judgment derived from
other judgments.

_Developed thinking_ is first found in the concept, and as we study the
thought products, “concept,” “judgment” and “inference,” the truth is
forced upon us that _thinking as a process aims to group the many into
one_. From many percepts is built the one concept, from two concepts
is built the one judgment and from two judgments is built the one
inference.[3]

Speaking figuratively, thinking is a matter of picking up the fragments
along the shore of consciousness and tying them into bundles.


  =7. THOUGHT IN THE SENSATION AND PERCEPT.=

  So far in this discussion it has been assumed that there is
  no thinking involved in the sensation or the percept. There
  are good authorities, however, who insist on dignifying the
  sensation, even with a crude form of thinking. To illustrate:
  One may be reading an interesting novel. The mind is being
  entertained and ignores the activities of the objective world,
  yet we cannot say that the mind is dead to the world outside.
  There is a dim consciousness of certain noises without. These
  unlocalized sounds are sensations; but how is the mind able to
  recognize them as sounds or noises? To interpret the noises is
  it not necessary for the mind to affirm a connection between
  them and some past mental experience? Is it possible for
  the mind to know anything without establishing some kind
  of connection between the outside occurrence and an inner
  situation? If this is granted then in sensation there must be
  implicit thinking.

  As the percept is a localized group of sensations then
  there must be involved in perception a more complex form of
  thinking, since in grouping sensations there is a recognition
  of connections.

  If there is thinking in the sensation which is the simplest and
  lowest form of the knowing-mind then thinking conditions all
  knowledge and really is the basic elemental cell of all knowing.

  On the other hand there are those who maintain that the
  sensation and percept are mere reflections of consciousness; the
  sensation being a reflected quality and the percept a reflected
  object. These mental situations come into being instantly――there
  is no time for thought and we all know that thought requires
  time. (“As quick as thought” is misleading, since light travels
  more rapidly by many times than the agencies of thought.)

  It will probably never be settled to the satisfaction of all
  just when thinking commences. The question is as difficult
  as some others which have never been solved. For example:
  Where does life commence? When does the plant merge into the
  animal? Which was first the egg or the hen? Does the objective
  world really exist or is it only a mental interpretation of
  vibrations? etc.

  Logically considered the question is immaterial. All will agree
  that developed thought is involved in the concept, judgment
  and inference, while, if it appears at all in the percept and
  sensation, it is more or less undeveloped and consequently lies
  quite without the province of the logical field.


=8. EVOLUTION AND THE THINKING MIND.=

Speaking in general terms evolution is a development from a lower to
a higher state. Thus have come the various species of the vegetable
and animal world. The lower orders of life are simple in structure and
function. In the one-celled animate form a single organ performs all of
the work needed to maintain life and perpetuate the species. If these
simple life-forms are cut in two, life continues in the two parts as
if nothing had happened. Aside from their simplicity there is little
of interdependence of function and little of co-ordination of organs
in the lowest life-forms. In short there is no division of labor; “each
cell is a world unto itself.”

An analogous development is seen in the thinking mind. The little
child thinks in _lumps_, and these lumps are only faultily linked
together, but the adult thinks in terms of the _grains_ of the lump,
each grain having its place, which it must occupy for the sake of all
the other grains as well as the entire lump. The child’s thinking is
vague, general and inaccurate, while the adult’s thinking should be
definite, specialized and accurate. Thinking in the lump means little
discrimination and very faulty integration or unity, while thinking
in terms of the grains means detailed discrimination and perfect
integration. To illustrate: The child sees a dog trotting along the
side walk which, according to the suggestion of his mother, he learns
to call “bow-wow.” Later he observes a cat and at once says “bow-wow,”
because all that the child notes is that something with legs, ears
and a tail is trotting along the side walk. Anything which fits these
general marks is a “bow-wow.” Similarly when a child first observes
a robin perched on a gate post he fails to distinguish between the
two――it is _all_ bird from the top of the robin’s head to the bottom
of the gate post.

Progress in thinking is measured by progress in discrimination. The
skilled thinker divides the large unit into very small units, compares
these with each other and then reunites them into a more perfect and
unified whole. First there is an analysis and then a synthesis. Like a
shuttle the power of thought works in and out; it goes in to separate,
it comes out to unify.

There is another aspect in the analogy between the life of the
physical and mental worlds. Somewhere in the order of progress there is
a connecting link between the mineral and vegetable kingdoms, likewise
between the vegetable and animal kingdoms. The sensation is as much
a state of feeling as an act of knowing and consequently is the
connecting link between the _feeling_ mind and the _knowing_ mind. If
the percept is the result of thinking as well as intuition then it may
stand for the dividing line between the _knowing_[4] mind and the
_thinking_ mind.


  =9. THE CONCEPT AS A THOUGHT PRODUCT.=

  _Conception is the process of thinking many notions into one
  class._ The product of such a process is called a concept.
  (1) The concept may stand for a group of concrete _general_
  notions――as the concept _man_, which stands for the five general
  notions: Caucasian, Mongolian, Ethiopian, Malay and American
  Indian. (2) The concept may stand for a group of concrete
  _individual_ notions. For example, the same concept _man_
  represents all of the individual men of the world. (3) The
  concept may stand for a group of _abstract general_ notions. To
  wit: Virtue represents such general notions as honesty, justice,
  industry, purity, etc. (These are general notions because they
  admit of a subdivision into kinds. Industry, for instance,
  may be divided into two kinds: mental industry and physical
  industry.) (4) The concept may stand for a group of _abstract
  individual_ notions. To illustrate: Blueness stands for the
  various shades of blue, as sky blue, bird’s egg blue, navy blue,
  etc.

  Thus does the concept stand for a group of all kinds of notions,
  individual and general, abstract and concrete.


  THE PROCESS OF CONCEPTION ILLUSTRATED.

  I see for the first time in my life a pencil. In other words I
  become conscious of a localized group of sensations――this is a
  percept. I am told that the name of that which I see is _pencil_.
  I note that this particular pencil has a thread of black lead
  encased in a cylindrical strip of wood which is brown in color.
  A second object is presented which I recognize as a pencil
  though the shape is prismatic rather than cylindrical and the
  color green rather than brown. But I call it a pencil because
  it has a thread of black lead encased in a strip of wood. The
  notion which I now have in mind stands for two pencils and
  is therefore represented by a class name. As I observe other
  pencils of various shapes, made of wood and paper with threads
  of different colored lead, my notion of pencil broadens till
  finally it stands for all pencils. This is the process of
  conception according to the definition, namely: “The process of
  thinking many notions into one class.” In this case the notions
  are individual.

  An examination of conception makes evident two distinct
  characteristics. First, I may be able to recognize each
  individual pencil because of the two common qualities, a
  thread of lead and an encasement of some kind. This process of
  the knowing mind whereby it recognizes and affirms connections
  is called thinking as we have already learned. Here is the
  _thinking_ aspect of conception. Second, as the instances of
  the observed objects are multiplied, my notion of pencil is
  _broadened_. It is a building process where many are cemented
  into one; like the blocks of a cement wall. Here we find the
  characteristic which enables us to call the process _conception_.
  This is the mark which distinguishes conception from all the
  other thought processes.


  =10. THE JUDGMENT AS A THOUGHT PRODUCT.=

  Judging is the process of conjoining and disjoining notions.
  The product of judging is the judgment and all judgments are
  expressed by means of propositions. A proposition consists of
  one subject and one predicate connected by some form of the verb
  _be_ or its equivalent.

  (1) A judgment may conjoin or disjoin two individual notions.
        To wit: Conjoined――This pencil belongs to Albert White.
        Disjoined――This pencil does not belong to Mary Smith.

  (2) A judgment may conjoin or disjoin two general notions.
        Conjoined――Some men are virtuous.
        Disjoined――Some men are not virtuous.

  (3) A judgment may conjoin or disjoin a general and an individual
      notion.
        Conjoined――Abraham Lincoln was virtuous.
        Disjoined――Edgar Allen Poe was not temperate.

  In order that the knowing mind may conjoin notions it must
  recognize some mark of similarity or connection. This is the
  _thinking_ aspect of the judgment. It is likewise to a certain
  degree the judging aspect as the latter is simply a matter of
  affirming or denying connections between notions. But thinking
  is a broader term than judging. There may be connections
  established between a name and a notion. For example in the case
  of the dog which the child sees trotting along the sidewalk and
  which the mother refers to as a “bow-wow”; the term “bow-wow” is
  not a percept and has no meaning independent of its association
  with the dog, hence it is not a notion, yet some connection has
  been made in the child’s mind between “bow-wow” and his notion
  of dog. This is a simple form of thinking, but not of judging,
  as the latter affirms or denies connections between _notions_
  only.

  The fact that judging and thinking so closely resemble each
  other has given just cause for some logicians to designate
  judging as the most fundamental element in all thinking. “The
  simplest form of thinking,” says Creighton, “is judging.” In
  order to think many notions into one class it is necessary to
  conjoin notions. To illustrate: The child who has a general
  notion of man sees for the first time a negro. If he recognizes
  the negro as a colored man he must conjoin his general notion of
  man with this individual notion. In short, a concept is built by
  means of a series of judgments. It may be said further that an
  inference is simply a _made-over judgment_. It is thus evident
  that judging appears in both the thought processes of conception
  and inference and, therefore, as a final conclusion it may be
  affirmed that judging, though perhaps not the simplest form of
  thinking, is the basic element of _developed_ thought.


  =11. INFERENCE AS A THOUGHT PRODUCT.=

  Reasoning is the process of deriving a new judgment from a
  consideration of other judgments. The product of any reasoning
  process may be called an inference, although, as will appear in
  a later chapter, inference is commonly used as indicating the
  process as well as the product.

  Often reasoning may assume a syllogistic form with the inference
  as its conclusion. A syllogism is an arrangement of three
  propositions using three different terms. The following are
  syllogisms:

  (1) All children should play.
        Mary is a child.
        Hence, Mary should play.

  (2) No teacher should judge hastily.
        You are a teacher.
        Hence, you should not judge hastily.

  In the second syllogism the inference, “you should not judge
  hastily,” is derived from the other two judgments by merely
  eliminating the common term teacher and disjoining the remaining
  two terms. The inference is consequently a _new_ judgment.
  Therefore, reasoning is only a matter of judging carried to a
  more complex stage.

  To summarize――_conception_ is largely a matter of conjoining
  a general notion with an individual notion, _judging_ of
  conjoining and disjoining all kinds of notions and _inference_
  of conjoining and disjoining judgments. All three processes
  go to form the larger process of thinking. _The concept, the
  judgment and the inference are products arising from conjoining
  and disjoining notions._


=12. THINKING AND APPREHENSION.=

Says Jevons: “Simple apprehension is the act of the mind by which we
merely become aware of something, or have an idea or impression of
it brought into the mind;” while Hyslop states that “The process of
knowledge which gives us percepts is apprehension.” It is obvious that
the idea of the latter is that _apprehension_ yields individual notions
only, while Jevons, in citing the term _iron_ as an illustration of
his definition, would infer that the general notion is the product
of _apprehension_. The term is strikingly ambiguous and will not be
referred to often in this treatise. If the student desires a definition
this will cover the concensus of opinion on the meaning of apprehension.
_Apprehension is that process of the knowing mind which yields the
percept and concept._ Some logicians give to the thinking mind the
three aspects of apprehension, judging and reasoning.


=13. STAGES IN THINKING.=

In all thinking there are three steps or stages which may be termed
_discrimination_, _comparison_, _integration_.

In the case of the two pencils held in the hand, it is noted that
one is longer than the other. Let us analyze the process which
made possible this conclusion. Step one――Attention is given first
to one pencil and then to the other. In each case the pencils are
distinguished from the hand and the other surrounding objects. This is
discrimination. Step two――The pencils are _compared_ in length. Step
three――The two notions are united in the judgment, “Pencil number _one_
is longer than pencil number _two_.” This is integration.

Another illustration. The child is requested to solve this problem: If
8 tons of hay cost $165, what will 16 tons cost?

  Statement:  Given: 8 tons cost $165.
              Required: 16 tons cost?

Discrimination. The child notes that 8 tons cost $165 and at this rate
he is required to find the cost of 16 tons.

Comparison. The child perceives that 16 tons is twice 8 tons.

Integration. The child concludes that the cost of 16 tons will be twice
the cost of 8 tons or $330.

When we think, we first tear to pieces that we may become acquainted
with every part. This may be called analysis. Then we put the related
pieces together again. This may be called synthesis. Before, however,
the parts are re-united a certain amount of comparison is necessary.
The three stages of thought might thus be denominated: (1) analysis,
(2) comparison, (3) synthesis.

After the synthesis or integration it is necessary to name the result,
consequently a fourth step is sometimes given, namely denomination.


=14. OUTLINE.=

  THOUGHT AND ITS OPERATION.

  (1) The Knowing Mind Compared with the Thinking Mind.

  (2) Knowing by Intuition.

  (3) The Thinking Process.
        Defined.
        Other definitions.

  (4) Notions.
        Individual.
        General.
        Thing and object distinguished.

  (5) Knowledge and Idea as Related to the Notion.

  (6) The Logic of Psychological Terms Involved in the Notion.

        The sensation }
        The percept   }  Individual notions.
        The image     }

        The concept   }
        The judgment  }  General notions.
        The inference }

        Terms defined.

  (7) Thought and the Sensation and Percept.

  (8) Evolution and the Thinking Mind.

  (9) The Concept as a Thought Product.

 (10) The Judgment as a Thought Product.
        The simplest form of thinking.

 (11) Inference as a Thought Product.

 (12) Thinking and Apprehension.

 (13) Stages in Thinking.
        Discrimination.
        Comparison.
        Integration.
        (Denomination.)


=15. SUMMARY.=

(1) Knowing is a broader term than thinking as the former equals the
latter plus intuition.

(2) Intuitive knowledge is that which comes to the mind immediately by
direct observation.

Although intuitive knowledge comes to the mind without thought, yet
such knowledge is _essential_ to all thinking. Intuitive knowledge is
the foundation upon which the thinking mind builds.

(3) Thinking is the deliberative process of affirming and denying
connections. Thinking is a “thickening process,” the smaller units
being pressed together to make a larger. Thinking is chiefly a matter
of reducing plurality to unity.

(4) A notion is any product of the knowing mind.

An individual notion is the notion of one thing.

A general notion is a notion of a class of things.

A thing includes objects, qualities, relations or any existing entity.
A thing is that which has _individual existence_.

(5) A bit of knowledge must have been a notion of some one’s mind, but
may not necessarily be a notion of _your_ mind. Knowledge may be found
in books, but a notion is a mental product found only in the mind. Idea
is ambiguous, though its meaning is usually restricted to an image, a
meaning or a belief.

(6) The products of the knowing mind are the sensation, the image,
percept, concept, judgment, inference.

The sensation, image and percept are individual notions, while the
concept, judgment and inference are general notions.

A sensation is a vague, unlocalized product of the knowing mind.

A percept is a consciously localized group of sensations.

An image is a reproduced percept.

A concept is a mental product arising from thinking many notions into
one class.

A judgment is a mental product arising from conjoining and disjoining
notions.

An inference is a judgment derived from antecedent judgments.

The developed thought processes are the concept, the judgment and the
inference.

(7) Just where the simplest form of thinking appears in the various
activities of the knowing mind is still an undecided question. It is
agreed that thinking in its developed and more complex form is found
in conception, judging and reasoning.

(8) Thinking evolves from the simple to the more complex, just as life
has evolved.

The child thinks in vague, indefinite wholes, while the adult thinks in
clear, definite parts. The child discriminates very imperfectly while
the adult discriminates accurately.

The sensation seems to be the connecting link between the feeling mind
and the knowing mind, while the percept links together the knowing mind
and the thinking mind.

(9) Conception is the process of thinking many notions into one class.
The product of such a process is a concept. The concept stands for
groups of all kinds of objects.

Conception has the two aspects of _affirming connections_ and of
_building many into one_. The first is the thinking side of the process
and the second is the mark which distinguishes conception from the
other thought processes.

(10) Judging is the process of conjoining or disjoining notions.
Judgment is the product of judging.

Judgments conjoin and disjoin all kinds of notions.

Judging and thinking, though they closely resemble each other, are not
synonomous terms. Thinking is a broader term in that connections may be
established between a notion and a name for that notion.

Judging is the most fundamental of all thinking, as the concept is
built from a series of judgments and an inference is simply a made-over
judgment.

(11) Inference.

Reasoning is the process of deriving a new judgment from a
consideration of antecedent judgments. This derived judgment may be
called an inference. Sometimes the term inference denotes the process
of reasoning as well as the product.

Reasoning often takes the form of a syllogism.

The concept, the judgment and the inference are _products_ arising from
conjoining and disjoining notions.

(12) Some give to the thinking mind the three aspects, apprehension,
judging and reasoning. Apprehension is another word for the two
processes, perception and conception.

(13) The three important stages in thinking are discrimination,
comparison, integration; or analysis, comparison and synthesis.


=16. REVIEW QUESTIONS.=

  (1) Show the difference between the knowing mind and the thinking
      mind.

  (2) Describe the process known as intuition.

  (3) What is intuitive knowledge?

  (4) Is the assumption that _think_ comes from the same root as
      _thick_ a feasible one? Explain.

  (5) Define thinking in at least two ways.

  (6) “Inability to think is due to inability to note connections.”
      Show this by making use of some problem in arithmetic.

  (7) Distinguish between individual and general notions.

  (8) Which is the broader term, object or thing? Explain.

  (9) What kind of notions only admit of subdivisions? Illustrate.

 (10) What is the difference between knowledge and notions? Explain.

 (11) Explain and illustrate the meaning of idea.

 (12) Classify the various activities of the knowing mind and define
      each.

 (13) Explain by definition and illustration the products of the
      knowing mind.

 (14) Relate the general notion to the psychological products of the
      knowing mind.

 (15) “The thinking mind is a unit.” Explain fully.

 (16) Trace the analogy between the evolution of the physical world and
      the evolution of thought.

 (17) Show that the sensation and the percept may be regarded as
      connecting links between lower and higher states.

 (18) Define and illustrate conception.

 (19) Show that the concept stands for all kinds of notions.

 (20) Point out the thinking aspect of conception as distinguished from
      the activity which gives the process its name.

 (21) Define the judgment. Illustrate two kinds.

 (22) Show that the concept is built by means of a series of judgments.

 (23) Show that judging is the fundamental element in the thought
      products.

 (24) Define and illustrate reasoning.

 (25) Describe the syllogism.

 (26) Explain the use of apprehension.

 (27) What are the stages in thinking? Illustrate fully.

 (28) Show that thinking is a matter of analysis and synthesis.


=17. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Give your argument in favor of the statement, “Dogs think, but do
      not reason.”

  (2) Show by illustration that thinking would be impossible without
      intuition.

  (3) “Thinking is the conscious adjustment of a means to an end in
      problematic situations.” Illustrate this.

  (4) The class is unable to solve the following problem: “I sell
      my house for $12,000, which is a gain of 25% on the cost. Find
      the cost.” What is the trouble? State the problem so that some
      connection is apparent.

  (5) “Two-thirds of my salary is $2,400. What is my salary?” A child
      solves this by dividing $2,400 by two and multiplying this result
      by three. Illustrate a plan for establishing right connections.

  (6) May a judgment express a general notion? Illustrate.

  (7) Is a thought a thing? Illustrate.

  (8) Show the illogic of dividing notions into individual, general and
      abstract.

  (9) Show that goodness is a general notion.

 (10) Is the concept an idea? Explain.

 (11) Prove that a mental image is always an individual notion.

 (12) “In sensation is there implicit thinking?” Argue both sides of
      the question.

 (13) Show that the concept, the judgment and the inference are
      products of the thinking mind.

 (14) Show by illustration where perception ceases and conception
      begins.

 (15) Is there actually any difference between thinking and judging?
      Illustrate.

 (16) “Reasoning is controlled thought.” Explain.

 (17) Of the three stages in thinking which one most concerns the
      teacher? Illustrate.



                              CHAPTER 3.

                     THE PRIMARY LAWS OF THOUGHT.


=1. TWO FUNDAMENTAL LAWS.=

The elemental form of evolved thought is the judgment. The laws
or axioms of thought may, therefore, be discovered by studying the
judgment.

Judging is the process of conjoining and disjoining notions.
When these notions are conjoined the judgment is affirmative; when
disjoined the judgment is negative. To illustrate: “Some men are
wise,” is an affirmative judgment, while “Some men are not wise,” is
a negative judgment. All judgments are either affirmative or negative
and this suggests that there may be but two fundamental laws or axioms
underlying judging or all forms of developed thinking. One law would
condition the affirmative judgment; the other the negative. Such is
actually the case. The law which permits the affirmative judgment is
called the _law of identity_, while the law which allows a negative
judgment is known as the _law of contradiction_. There is a third law
termed the _law of excluded middle_, which is in reality a combination
of the other two.


=2. THE LAW OF IDENTITY.=

In general the law of identity implies a certain permanency throughout
the material world. That door is a door and always will be a door till
the conditions change. If it were not for this law, that everything is
permanently identical with itself, it would be impossible to think at
all. For example: Take away the notion of permanency from the door and
thought becomes at once ridiculous. Suppose that while we are asserting
that the object is a door, it changes to a tree, and while we insist
that the object is now a tree, it changes to a cow, etc. We can readily
see that it would hardly be worth while to think at all.

The law of identity may be stated in three ways: (1) Whatever is, is;
(2) Everything remains identical with itself; (3) The same is the same.


  ABSOLUTE IDENTITY――COMPLETE AND INCOMPLETE.

Applying the law of identity to the affirmative judgment expressed in
the form of a proposition, we find two kinds of identity, absolute and
relative. In the propositions, “Socrates is Socrates,” “dogs are dogs,”
“honesty is honesty,” the subject is _absolutely_ identical with the
predicate――the same in form and meaning. If we were to illustrate the
subject and predicate by two circles they would be of the same size and
shape, the one coinciding with the other point to point.

This kind of absolute identity which makes possible all truisms we
may term, for want of a better name, complete absolute identity. This
would imply that there is an incomplete absolute identity and such
seems to be the case. Examining the definition, “A man is a rational
animal,” we observe that the notion _man_ has the same content or
meaning as the notion _rational animal_. In meaning, then, the two
notions are absolutely identical. The one includes just as many
objects or qualities as the other, and if we were to draw two circles
representing them, they would be of the same size. In form, in mode
of expression, however, the notions differ and the circles, though
coinciding, would need to differ in form, the boundary of one might be
a solid line, the other a dotted. This we may call incomplete absolute
identity. All logical definitions illustrate identities of this kind.


  RELATIVE IDENTITY.

Relative identity is best understood by thinking of it as _partial_
identity, just as we may think of absolute identity as _total_ identity.
In relative identity the _whole_ of one notion may be affirmed of a
_part_ of another notion; or a _part_ of one notion may be affirmed
of a _part_ of another notion. To illustrate: (1) All men are mortal;
(2) Some men are wise. These and their like are made possible because
of the law of relative identity. In the first proposition all of the
“_men_” class is identical with a part of the “_mortal_” class. If
we were to represent this relation by circles, the “men” circle would
be made smaller than the “mortal” circle and placed inside it, as in
Fig. 1.

  Illustration: Fig. 1.

Be it remembered that circles are surfaces, and in Fig. 1 the men
circle is identical with that portion of the mortal circle which is
immediately underneath it.

The same relation may be indicated by a small pad being placed on top
of a larger pad. Then the whole of the smaller pad could be thought
of as being identical with that part of the larger pad which is
immediately underneath.

In the case of the second proposition a part of the “men” class
is identical with a portion of the “wise” class. The two circles
indicating this relation must intersect each other so that a portion
of each may be common ground, as in Fig. 2 where the shaded part
represents the identity.

  Illustration: Fig. 2.

Thus we see that the law of identity underlies all affirmative
propositions. Absolute identity making possible the truism and
definition, and relative identity conditioning all the universal and
particular affirmative propositions which are neither truisms nor
definitions.

The three forms may be symbolized as follows:

  (1) A is A――Absolute complete

  (2) _A_ is A――Absolute incomplete

  (3) A is B――Relative.

The student will note that the “A’s” of absolute incomplete differ in
form.


=3. LAW OF CONTRADICTION.=

The law of contradiction underlies all negative propositions. It is the
mission of this law to tear down or to be destructive in nature; while
the law of identity builds up or is constructive in nature.

The law of contradiction may be stated in this way: It is impossible
for the same thing to be and not to be at the same time and in the same
place. Or better, _it is impossible for the same thing to be itself and
its contradictory at the same time_. Bringing out a further aspect, no
thing can have and not have the same attributes at the same time.

The little word _not_ bisects the universe. All the people in the world
are either honest or not honest, virtuous or not virtuous. These are
contradictory statements and what is comprehended by the one cannot
be comprehended by the other at the same time, any more than a man can
shake his head and nod his head at the same time.

If we assert the identity between two notions then we cannot in the
same breath deny their identity.


  ILLUSTRATIONS:

    (1) A red flower cannot be a red flower and not a red flower at
        the same time.

    (2) No man can be guilty and not guilty at the same time.

    (3) A boy cannot be working and not working at the same time.

If I assert that the flower is red, then I cannot affirm in the same
breath that the flower is not red.


  TWO USES OF NOT.

The word _not_ when used with the copula of a given proposition makes
that _proposition_ negative, as (1) “Some men are not wise.” But when
_not_ is attached to the predicate by a hyphen, the _predicate_ is made
negative, not the proposition, as (2) “Some men are not-wise.” Here
the predicate _not-wise_ is negative, but the proposition in which it
appears is affirmative. It is obvious that the proposition “Some men
are not wise” illustrates the law of contradiction, since the _some
men_ referred to are contradicted of all which is wise. Whereas the
proposition “Some men are not-wise” illustrates relative identity,
since the subject “some men” is affirmed of a part of the predicate
“not-wise.” The student may be led to see these relations by drawing
circles, the one to represent the subject, the other the predicate.
(See page 141.)


  FURTHER ILLUSTRATIONS:

  Some teachers are wise          }
  Some teachers are not-wise      } Illustrate the law of identity.
  Some teachers are unwise        }

  Some teachers are not wise      }
  Some teachers are not not-wise  } Illustrate the law of contradiction.
  Some teachers are not unwise    }

The student must understand that a term and its contradictory destroy
each other. If we affirm something of the one, then we must deny it
of the other, or we undermine the integrity of both. If it is affirmed
of teachers A, B and C that they are wise, then it must be denied that
they are not-wise.


  ILLUSTRATIONS:

    A, B and C are wise.          } These are mutually
    A, B and C are not-wise.      }   destructive.

    A, B and C are wise.          } These are not mutually destructive,
    A, B and C are not not-wise.  }   but virtually mean the same thing.


  SYMBOLIZATION OF THE LAW OF CONTRADICTION.

    A is not not-A.                           A is not B.
    (As A is always A it would        or      or
      be absurd to say that A is              A is not not-B.
      not A.)


  CONTRADICTORY AND OPPOSITE TERMS.

It is easy to use opposite terms in a contradictory sense. This
leads to serious error. “Not-guilty” is the _contradictory_ of
“guilty,” while “innocent” is the _opposite_ of “guilty.” We could
hardly say that the water must either be cold or hot, as it might be
warm. “Not-hot” is the only term which contradicts “hot.” The law of
contradiction has nothing to do with opposites.

Further, it is dangerous to regard words with the negative prefix as
being contradictory of the affirmative form. For example: Valuable and
invaluable are not contradictory. There is likewise some doubt as to
the contradictory nature of such words as agreeable and disagreeable,
though we are sure that agreeable and not-agreeable contradict each
other. To use the “not” with a hyphen is safer than to depend upon some
prefix which is supposed to mean “_not_.”


  ILLUSTRATIONS OF CONTRADICTORY AND OPPOSITE TERMS.

      _Opposite._          _Contradictory._
    ────────────────      ───────────────────
    bad       good        bad       not-bad
    soft      hard        soft      not-soft
    cold      hot         cold      not-cold
    rough     smooth      rough     not-rough
    good      evil        good      not-good
    warm      cool        warm      not-warm
    weak      strong      weak      not-weak


=4. THE LAW OF EXCLUDED MIDDLE.=

The law of excluded middle may be considered as a combination of
identity and contradiction. Identity gives the proposition, “John Doe
is honest.” Contradiction, “John Doe is not honest.” Combine the two
using _either_ and _or_ and we have the excluded middle proposition,
“Either John Doe is honest or he is not honest.”

Excluded middle explains itself. Of the two contradictory notions it
must be either the one or the other. There is no “go-between” notion.

The law may be stated in many ways, as will be seen by the following:
(1) Everything must either be or not be. (2) Either a given judgment
is true or its contradictory is true; there is no middle ground. (3) Of
two contradictory judgments one must be true. (4) Every predicate may
be affirmed or denied of every subject.


  ILLUSTRATIONS:

(1) A man is either mortal or he is not mortal. (2) John Doe is either
honest or not-honest. (3) Either you are going or you are not going.


  SYMBOLIZATION OF EXCLUDED MIDDLE.

    A is either A or not-A
              or
    A is either B or not-B.


=5. THE LAW OF SUFFICIENT REASON.=

The law may be stated in this wise. Every phenomenon, event or
relation must have a sufficient reason for being what it is. To
illustrate: (1) If Venus is the evening star, there must be a
sufficient reason. (2) If the ground is wet, there must be a cause.
Many logicians argue that this law has no place in logic, its
field being that of the physical sciences. The laws of identity,
contradiction and excluded middle are, however, universally regarded
as the Primary Laws of thought.


=6. UNITY OF PRIMARY LAWS OF THOUGHT ILLUSTRATED BY SYMBOLS.=

      (1) Absolute Symbols          Relative Symbols.

    Excluded middle.
      A is either A or not-A.       A is either B or not-B.

    Contradiction.
      A is not not-A.               A is not B or A is not not-B.

    Identity.
      A is A.                       A is not-B or A is B.

      (2) Propositions made to fit symbols.

    Excluded middle.
      A man is either a man or      A man is either honest or
        a not-man.                    not-honest.

    Contradiction.
      A man is not a not-man.       A man is not honest, or a man
                                      is not not-honest.

    Identity.
      A man is a man.               A man is not-honest, or a man
                                      is honest.

The “excluded middle” propositions of the foregoing express
alternatives which are mutually contradictory. There is no middle
ground. The “contradictory propositions” contradict the identity of
the subject with one alternative, while the “identity” propositions
affirm the identity of the subject with the other alternative. This is
made possible because of the principle, “Of two mutually contradictory
terms, if one is true the other must be false.” The foregoing scheme
shows how closely “contradictory” and “identity” propositions are
related to “excluded middle” propositions. Expressed mathematically:
excluded middle = contradiction + identity.


=7. OUTLINE.=

  PRIMARY LAWS OF THOUGHT.

  (1) Two fundamental laws.
        Identity, contradiction.

  (2) Law of identity.
        Absolute――complete, incomplete.
        Relative.

  (3) Law of contradiction.
        Two uses of _not_.
        Contradictory and opposite terms.

  (4) Law of excluded middle.

  (5) Law of sufficient reason.

  (6) Unity of primary laws of thought.


=8. SUMMARY.=

(1) The elemental forms of evolved thought are the affirmative and
negative judgments. This suggests two fundamental laws of thought, the
law of identity and the law of contradiction. The former conditions the
affirmative judgment, the latter the negative.

(2) The law of identity implies a permanency of being. “Everything
remains identical with itself,” is a statement of identity.

Absolute identity may be divided into complete and incomplete identity.

In complete absolute identity the subject is the same as the predicate
in both form and meaning. Truisms illustrate this.

In incomplete absolute identity the subject is identical with the
predicate in meaning only. Illustrated by definitions.

In relative identity the whole of the subject may be affirmed of a part
of the predicate or a part of the subject may be affirmed of a part of
the predicate.

(3) “It is impossible for the same thing to be itself and its
contradictory at the same time,” is a statement of the law of
contradiction. _Identity_ is _con_structive while _contradiction_
is _de_structive in nature. To make the proposition negative the word
_not_ must be used with the copula. “_Not_” attached to the predicate
with a hyphen makes the _predicate_ negative, but not the _proposition_.

To use opposite terms in a contradictory sense leads to serious error.

The safest way of making a positive term a contradictory negative term
is to prefix “_not_” with a hyphen or use “non.”

(4) The law of excluded middle is virtually a combination of identity
and contradiction. It may be stated as follows: “A thing must either be
itself or its contradictory.”

(5) “Every condition must have a sufficient reason for its existence,”
is the law of sufficient reason. Its distinct province is physical
science rather than logic.

(6) The laws may be expressed mathematically: excluded middle =
identity + contradiction.

  SCHEMATIC STATEMENT OF PRIMARY LAWS.

 ┌───────────┬───────────────────┬────────────────┬────────────────────┐
 │   Name    │      Stated       │   Symbolized   │    Illustrated     │
 ├───────────┼───────────────────┼────────────────┼────────────────────┤
 │ Absolute  │ Whatever is, is   │ A is A         │ Work is work       │
 │  identity │                   │                │                    │
 ├───────────┼───────────────────┼────────────────┼────────────────────┤
 │ Relative  │ The whole is      │ All A is B     │ Work is a blessing │
 │  identity │  identical with   │ Some A is B    │ Some play is a     │
 │           │  a part or a part │                │  blessing          │
 │           │  is identical     │                │                    │
 │           │  with a part      │                │                    │
 ├───────────┼───────────────────┼────────────────┼────────────────────┤
 │ Contra-   │ Nothing can both  │ A is not not-A │ Work is not        │
 │  diction  │  be and not be at │     or         │  not-work          │
 │           │  the same time    │ A is not B     │ John is not honest │
 │           │                   │     or         │ Albert is not      │
 │           │                   │ A is not not-B │  not-honest        │
 ├───────────┼───────────────────┼────────────────┼────────────────────┤
 │ Excluded  │ Everything must   │ A is either A  │ Fair play is       │
 │  middle   │  either be or not │  or not-A      │  either fair play  │
 │           │  be               │     or         │  or not-fair play  │
 │           │                   │ A is either B  │ This man is either │
 │           │                   │  or not-B      │  educated or       │
 │           │                   │                │  not-educated      │
 └───────────┴───────────────────┴────────────────┴────────────────────┘


=9. ILLUSTRATIVE EXERCISES.=

(1a) Each of the following propositions is made possible because of the
existence of which law of thought?

In answering this question I summarize in my mind the meaning of each
law of thought. Viz.:

  (1) In complete absolute identity the subject and predicate are
      the same in form and meaning.

  (2) In incomplete absolute identity the subject and predicate
      are the same in meaning, but not in form.

  (3) In relative identity either the whole or a part of the
      subject is identical with a part of the predicate.

  (4) The law of contradiction always denies the identity between
      subject and predicate.

  (5) Excluded middle conditions all alternative expressions.


  THE PROPOSITIONS.

  (1) “A thief is a thief.” Complete absolute identity.

  (2) “Thinking is the process of affirming or denying connections.”
      Incomplete absolute identity.

  (3) “All good men are wise.” Relative identity.

  (4) “No triangle has interior angles whose sum is greater than
      two right angles.” Contradiction.

  (5) “A stitch in time saves nine.” Relative identity.

  (6) “Judging is the process of conjoining and disjoining notions.”
      Incomplete absolute identity.

  (7) “You are either a voter in this district or you are not a
      voter in this district.” Excluded middle.

  (8) “Some people do not know how to live.” Contradiction.

  (9) “All is well that ends well.” Incomplete absolute identity.

 (10) “Some men teach school.” Relative identity.

 (11) “None of the planets are as large as the sun.” Contradictory.

 (12) “All the trees in this grove are maple.” Relative identity.

(1b) Indicate the law which conditions each of the following
propositions:

  (1) “He who laughs last laughs best.”

  (2) “Perfect is perfect.”

  (3) “He is a wolf in sheep’s clothing.”

  (4) “Either your memory is poor or you are telling a deliberate
      falsehood.”

  (5) “Some of our greatest teachers thought they were failures.”

  (6) “No man of sense would ever try to get something for nothing.”

  (7) “Failure is _not to try_.”

  (8) “Success is the right man in the right place doing his best.”

  (9) “Every man is insane on some topic.”

 (10) “Some pupils are not industrious.”

 (11) “You are either a genius or a successful fakir.”

 (12) “Honesty is the best policy.”


=10. REVIEW QUESTIONS.=

  (1) How many kinds of judgments are there? Illustrate.

  (2) Name the fundamental laws of thought and explain how they are
      related to the kinds of judgments.

  (3) Show that it would be impossible to think at all were it not for
      the law of identity.

  (4) State the law of identity in three ways.

  (5) Explain the kinds of absolute identity. Illustrate by
      propositions and by circles.

  (6) Explain by word and by diagrammatical illustration relative
      identity.

  (7) Symbolize the three forms of identity. Fit words to these
      symbols.

  (8) State in three ways the law of contradiction.

  (9) Show by illustration that _not_ bisects the world.

 (10) Explain the uses of _not_.

 (11) Prove that “John Doe is not-honest,” illustrates identity and
      not contradiction.

 (12) Symbolize in three ways contradiction. Fit words to these
      symbols.

 (13) Illustrate contradictory and opposite terms.

 (14) Show that words with negative prefixes are not necessarily the
      contradictory of the corresponding affirmative forms.

 (15) State and explain the law of excluded middle.

 (16) Symbolize the law of excluded middle.

 (17) State the law of sufficient reason. Illustrate.

 (18) Illustrate the unity of the three primary laws of thought.


=11. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Prove that the judgment is the elemental form of evolved thought.

  (2) What is meant by evolved thought?

  (3) Show that “Whatever is, is” is a statement of complete absolute
      identity only.

  (4) State incomplete absolute identity.

  (5) By means of _one_ proposition state relative identity.

  (6) Show that incomplete absolute identity is a term more or less
      illogical.

  (7) Show that these statements are exact expressions of relative
      identity:
        All men are some wise.
        Some men are some wise.

  (8) Why is the law of contradiction so named?

  (9) Show that space may be bisected by drawing a circle upon the
      black board.

 (10) Show that there is a difference in meaning between “You are not
      honest” and “You are not-honest.”

 (11) Is there any difference in meaning between disagreeable and not
      agreeable?

 (12) Which is the stronger term not-just or unjust? Why?

 (13) Give a list of words in which the contradictory forms are
      expressed by the ordinary prefixes.

 (14) Illustrate by circles the law of excluded middle.

 (15) Illustrate by a line-diagram the difference between contradictory
      and opposite terms.

 (16) Show that the province of the law of sufficient reason is
      physical science.



                              CHAPTER 4.

                            LOGICAL TERMS.


=1. LOGICAL THOUGHT AND LANGUAGE INSEPARABLE.=

Any impression upon the mind tends to manifest itself in some form of
expression. Impression which arouses thought tends to expression in the
form of symbols. Thought and symbol go hand in hand. Expression, taking
the form of word-symbols, constitutes a word-language.

It is commonly supposed that language is serviceable mainly in
communicating one’s thoughts to others, but language does service
in another way which is quite as important. It tends to clarify and
make definite all thought. Without a word-language thinking would
lack continuity; would be vague, loose, illogical. The right use of a
word-language, therefore, is a necessary adjunct to logical thought.
The basic element of a word-language is the logical term.


=2. MEANING OF LOGICAL TERM.=

A notion has been referred to as any product of the knowing mind.
When we express these notions in words such expressions may be called
logical terms.

Definition. _A logical term is a word or a group of words denoting
a definite notion._ Illustrations: Honesty, Chicago, tree, walking,
the man who was ill, beautiful roses. This is a list of logical terms,
because each word or group of words denotes a _notion_ of some kind.
It is now evident that any subject or predicate with its modifiers
constitutes a logical term. In the proposition, “The beautiful red
house on the hill, owned by Mr. Jones, has burned,” the term used
as the subject consists of eleven words. The reader must not confuse
logical terms with grammatical parts of speech. “Of” is a preposition
but not a logical term, as no definite notion is indicated.


=3. CATEGOREMATIC AND SYNCATEGOREMATIC WORDS.=

There are some words which, when used alone, denote definite notions,
such as man, tree, dog, justice. On the other hand there are other
words which, when used alone, do not stand for a definite notion, such
as up, beautifully, a, and.

Words like those in the first list are called categorematic words,
while those in the second list illustrate syncategorematic words.


  DEFINITION.

_A categorematic word is one which forms a logical term unaided by
other words. A syncategorematic word is one which must be used with
other words to form a logical term._

Any word or group of words which can be used as either subject or
predicate of a proposition is a logical term. If the _one_ word in
question can be used as either subject or predicate of a proposition
then it must be a categorematic word. If it is impossible to use the
one word as either subject or predicate of a proposition then this is
a sure indication that such a word is syncategorematic. For example,
there is no sense in the expressions, “_And_ is honest,” “_Of_ is not
true”; hence _and_ and _of_ are syncategorematic.

We may conclude from this that nouns, descriptive adjectives and
verbs may be categorematic words, while adverbs, prepositions and
conjunctions are syncategorematic words.


=4. SINGULAR TERMS.=

_A singular term is a term which denotes one object or one attribute._

Proper nouns, when they stand for individuals, are singular terms, such
as John Adams, Mississippi River, Socrates. Some proper names stand for
a class of objects, as the Caesars, the Mephistopheles, the Napoleons.
But when thus used they lose their character as proper names. Such
names, therefore, are general terms, not singular.

Common nouns may be made singular by some modifying word, as the first
man, the pole star, the highest good, my pet dog, etc.

Certain attributes which imply a _oneness_ or a distinct individuality
are singular, such as absolute justice, birds-egg blue, perfect
happiness, etc.

Some claim that terms like water, air, salt, etc., are singular, as
they stand for one thing. This, however, cannot be if such terms admit
the possibility of classification as: hard water, soft water, mineral
water.


=5. GENERAL TERMS.=

_A general term is one which denotes an indefinite number of objects or
attributes._

Class-names are general terms, such as men, chair, tree, army, nation.
Words like redness, sweetness, justice, are probably general in that
they denote a combination of qualities or may be subdivided into kinds.

The way the term is employed in the proposition should determine its
singular or general nature.


=6. COLLECTIVE AND DISTRIBUTIVE TERMS.=

_A collective term is a general term which indicates an indefinite
number of objects as one whole._ Such words as class, crowd, army,
forest, nation, are collective.

_A distributive term is a general term which indicates an indefinite
number of objects as a whole, and also may be used to refer to each one
of the group separately._ Such as man, pupil, tree, book.

It is easy to distinguish collective from distributive terms when we
attempt to use them in the designation of individuals. Pointing to
a body of troops, one may remark, “There is the regiment.” But when
pointing to _one man_ in the regiment, he could hardly say, “There is
the regiment.” “Regiment” is therefore collective because it may be
used with reference to the whole body of troops but cannot be used in
connection with any individual of that body. On the other hand in the
sentence, “Man is mortal,” “man” refers to the whole family of men. It
also indicates any one of them. As, “This man, John Doe, is mortal.”
Thus “man” is distributive. The distributive term, therefore, can be
used in a two-fold sense; namely, to denote the whole or to denote each.

It must be noted that, viewed from a different standpoint, some
collective terms become distributive in nature. As for example in the
proposition, “The army of the world is composed of able bodied men,”
_army_ is used with reference to all armies. While it may be used to
designate some particular army, as The American army.

Collective terms have been classified as general terms. It must
be borne in mind, however, that such may be made singular by some
modifying word. For example, _people_ is a general term, but _American
people_ is a singular term in that it refers to one people, being thus
limited by the word American.


=7. CONCRETE AND ABSTRACT TERMS.=

_A concrete term is a term which denotes a thing_; e. g., this man,
that tree, John Doe, denote in each case a thing. Man and tree, denote
many things. All are concrete.

_An abstract term is a term which denotes an attribute of a thing_;
e. g., whiteness, patience, squareness, are abstract terms.

Such words as red, honest, just, are concrete; while redness, honesty,
justice, are abstract.

On first thought it might be inferred that “red” is the name of
an attribute just as much as “redness.” This is a mistaken thought,
however, as when we use the word red we mean red something――an _object_
which is red in color, not the color itself. For example, in saying
the house is red, we refer to the thing that is red, not to the color
redness.

Descriptive adjectives, because they describe things, are concrete.
They do not alone name qualities of things, hence they are not abstract.


=8. CONNOTATIVE AND NON-CONNOTATIVE TERMS.=

_A connotative term is one which denotes a subject and at the same
time implies an attribute._ (A subject is anything which possesses
attributes.)

All concrete general terms are connotative because they denote subjects
and at the same time stand for certain attributes; e. g., “man” denotes
many subjects; in fact, it stands for all the men in the world; it also
implies rationality, the power of speech, power of locomotion, etc.
“Triangle” stands for all plane figures of three sides; it likewise
stands for the qualities, three-sided, three-cornered, etc. Both “man”
and “triangle” are connotative.

_A non-connotative term is one which denotes a subject only, or implies
an attribute only._ Such words as Boston, Columbus, The Elizabeth
White, denote a subject only. “Blueness,” “justice,” “width,” imply an
attribute only. All these terms are non-connotative. The words blue,
just, wide, are connotative. “Blue,” for example, denotes all blue
things, as the blue sky, the blue sea; at the same time “blue” implies
that something possesses the quality, _blueness_.

Generally speaking, proper and abstract nouns are non-connotative;
though such proper nouns as Mount Washington, Mississippi River, are,
no doubt, connotative, as they denote an object and imply at least one
attribute. In the case of Mount Washington an object is surely denoted,
and the attribute mountainous is implied. Any proper noun which conveys
definite information is connotative. It may be claimed that all proper
nouns give information. For example, to many _Boston_ indicates not
only an object, but the qualities common to a city. In reply it may
be said that “Boston” might indicate a boat, or a dog, or almost any
individual object.


=9. POSITIVE AND NEGATIVE TERMS.=

_A positive term is one which signifies the possession of certain
attributes_; e. g., metal, man, teacher, happy, honest.

_A negative term is one which signifies the absence of certain
attributes_; e. g., inorganic, unhappy, non-metallic.

Terms which have the prefix not, non, un, in, dis, etc., or the affix
less, are usually considered negative. The fact that there are some
exceptions to this must not be overlooked. For example, unloosed,
invaluable, are positive terms.

In theory every positive term has its corresponding negative; as pure,
impure; organic, inorganic; metal, non-metal; good, not-good.

In some instances the language does not supply the word with the
negative prefix because no need of it has been felt. The only way to
express the negative of such words as good, table, etc., is to prefix
“not” or “non.”


=10. CONTRADICTORY AND OPPOSITE TERMS. (See page 38).=

Positive terms with their negatives have contradictory meanings and
therefore are referred to as contradictory terms. For example, honest
and not-honest, metallic and non-metallic, perfect and imperfect,
are contradictory terms. Such terms are mutually destructive. When we
assert the truth of one we also imply the falsity of the other. If, for
example, we assert that Abraham Lincoln was honest, we carry with this
assertion the implication that Lincoln was not not-honest, or that any
statement to the effect that he was not honest is false.

Contradictory terms, when used in a sentence, illustrate the law of
excluded middle, as in the statements: “John’s recitation is either
perfect or imperfect.” “This teacher is either just or not-just.”
There is no middle ground in such propositions.

When contradictory terms are used in classification the whole is
divided into but two classes; e. g.:

    honest        not-honest
    agreeable     not-agreeable
    metallic      non-metallic
    perfect       imperfect
    pure          impure
    organic       inorganic

All the men in the world are either honest or not-honest. All the
substances in existence are either organic or inorganic, etc.

It will also be seen from this list that the contradictory of the
positive form is not always indicated by using the prefix. Honest
and dishonest, or agreeable and disagreeable, are not contradictory
terms. In the case of agreeable and disagreeable, there seems to be
the middle ground of absolute indifference. For example: the music of
the orchestra is agreeable while the humming of the enthusiast back
of me is decidedly disagreeable; but as to the noise upon the street,
it is neither agreeable nor disagreeable as long practice has made me
indifferent to it.

When there is any doubt as to the terms being contradictory, the safest
plan is to prefix “not” or “non” to the positive form.

Terms which oppose each other but do not contradict are said to be
opposite or contrary terms. The following list illustrate opposite
terms:

    hot           cold
    cool          warm
    less          greater
    wise          foolish
    bitter        sweet
    soft          hard
    tall          short
    agreeable     disagreeable

All these terms admit of a medium. In the case of hot or cold, for
example, a substance need not necessarily be either. It may be warm or
cool.

Terms seem to be contradictory when it is a matter of quality, but
opposite when it is a question of quantity or degree.


=11. PRIVATIVE AND NEGO-POSITIVE TERMS.=

_A privative term is one which is positive in form but negative in
meaning._ Such words as blind, deaf, dumb, dead, maimed, orphaned, are
privative terms, in that there is no negative prefix or suffix and yet
they denote the absence of certain qualities. “Blind,” for example, is
positive in form, but denotes absence of sight.

_A nego-positive term is one which is negative in form but positive
in meaning._ Such terms as invaluable, unloosed, immoral, indwell, are
nego-positive because, though they have negative prefixes, yet they
possess a certain positive meaning. “Invaluable,” for instance, does
not mean not-valuable, but very valuable.


=12. ABSOLUTE AND RELATIVE TERMS.=

_An absolute term is one whose meaning becomes intelligible without
reference to other terms._ Automobile, water, tree, house, book, are
absolute terms. Any of them may be made clear to a child or a foreigner
without special reference to other terms. For example, the child will
recognize from certain common marks the automobile every time he sees
it. The marks of tree, house, flower, are apparent to every one.

_A relative term is one which derives its meaning from its relation
to some other term._ Parent, teacher, shepherd, monarch, eldest, cause,
commander, are relative terms. For example, in explaining the meaning
of “_parent_” to a foreigner, reference must be made to “_child_.”
The pairs of terms thus associated are spoken of as correlatives.
Parent and child, teacher and pupil, shepherd and flock, monarch and
subject, eldest and youngest, cause and effect, commander and army,
are correlative terms. Either one of each pair is the correlate to the
other, and every relative term needs its correlate to make its meaning
clear. To say that a relative term denotes an object which cannot be
thought of without reference to some other object, is confusing, as it
is quite impossible to think of any object without calling to mind some
other object or notion. Fire calls to mind water; tree suggests shade,
etc.


=13. OUTLINE.=

  LOGICAL TERMS.

  (1) Meaning of term.

  (2) Categorematic and syncategorematic words.

  (3) Kinds of terms.
        Singular terms.
        General terms.
          (a) Collective terms.
          (b) Distributive terms.
        Concrete and abstract terms.
        Connotative and non-connotative terms.
        Positive and negative terms.
        Contradictory and opposite terms.
        Privative and nego-positive terms.
        Absolute and relative terms.


=14. SUMMARY.=

A logical term is a word or group of words denoting a definite notion.

A singular term is a term which denotes one object or one attribute.

A general term is a term which denotes an indefinite number of objects
or attributes.

General terms are collective or distributive.

A collective term is a general term which indicates an indefinite
number of objects considered as _one whole_.

A distributive term is a general term which indicates an indefinite
number of objects as a whole and also may be used to refer to each one
of the group separately.

A concrete term is a term which denotes a thing.

An abstract term is a term which denotes the attribute of a thing.

A connotative term is one which denotes a subject and at the same time
implies an attribute.

A non-connotative term is one which denotes a subject only or implies
an attribute only.

A positive term is one which signifies the possession of certain
attributes.

A negative term is one which signifies the absence of certain
attributes.

In theory every positive term has its negative. As related to each
other positive and negative terms are said to be contradictory. If
one denotes a true notion then the other denotes a false notion.

Some terms oppose each other but do not flatly contradict. As related
to each other such terms are said to be opposite.

A privative term is one which is positive in form but negative in
meaning.

A nego-positive term is one which is negative in form but positive in
meaning.

An absolute term is one whose meaning becomes intelligible without
reference to other terms.

A relative term is one which derives its meaning from its relation to
some other term.


=15. ILLUSTRATIVE EXERCISES.=

(1a) The words in italics are categorematic.

  (1) “_Honesty_ is the _best policy._”

  (2) “_A wise teacher_ never _scolds._”

  (3) “The _woodcock_ has a _long bill_ and _eyes high_ up on the
      _head._”

NOTE――If there is any doubt as to such words as never, on, etc., being
syncategorematic, attempt to use them as subject or predicate of a
proposition; e. g., John is _never_.

(1b) Underscore the categorematic words in the following:

  (1) “Socrates was the greatest teacher of pagan times.”

  (2) “Play is nature’s way of teaching a child how to work.”

  (3) “A man may be what he chooses if he is willing to pay the price.”

(2a) In the following, words enclosed in parentheses are logical terms:

  (1) (“All men) are (mortal.”)

  (2) (“The law of identity) is (one of the primary laws of thought.”)

  (3) (“Judging) is (the process of conjoining and disjoining
      notions.”)

(2b) Indicate the logical terms in the sentences under 1b.

(3a) The logical characteristics of the term _teacher_ are

  (1) general term,

  (2) distributive term,

  (3) concrete term,

  (4) connotative term,

  (5) positive term,

  (6) relative term.

(3b) The logical characteristics of other terms are as follows:

  (1) Goodness――general, abstract, non-connotative, positive, abstract.

  (2) Soft――general, concrete, non-connotative, positive, “hard” is its
      opposite, “not-soft” is its contradictory, absolute.

  (3) Disagreeable――general, concrete, non-connotative, “agreeable”
      is its opposite, “not-disagreeable” is its contradictory,
      nego-positive, absolute.

  (4) Aristotle――singular, concrete, non-connotative, positive,
      absolute.

  (5) Class――general, collective, concrete, connotative, positive,
      relative.

(3c) Give the logical characteristics of the following terms: justice,
Abraham Lincoln, tree, library, America, president, principle, sympathy,
dumb, nation.


=16. REVIEW QUESTIONS.=

  (1) What is the connection between logical thinking and language?

  (2) Why is _man_ a categorematic word?

  (3) Why is _beautifully_ syncategorematic?

  (4) Distinguish between singular and general terms.

  (5) Show how a collective term may be used in a distributive sense.

  (6) Why are the words _tree_ and _book_ distributive?

  (7) Distinguish between concrete and abstract terms.

  (8) Define and illustrate a non-connotative term.

  (9) Why are concrete general terms connotative?

 (10) Distinguish between positive and privative terms.

 (11) Why is not the word _immoral_ negative?

 (12) Give the opposite of “hot.” What is the contradictory of “hot”?

 (13) Distinguish by definition and illustration between relative and
      absolute terms.

 (14) What is the correlate of the word _effect_?


=17. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Is it possible to think independent of language?

  (2) May words be spoken or written without thought? Illustrate.

  (3) Are categorematic words always logical terms?

  (4) Must all the words of a logical term be categorematic?

  (5) Are pronouns and auxiliary verbs categorematic?

  (6) Indicate the logical connection between the terms of a
      proposition and the termini of a railroad.

  (7) Show that attribute is a broader term than quality.

  (8) Is the word _Washington_ general or singular? Give reasons.

  (9) Make the word _dog_ a singular term.

 (10) Give an illustration where the word _class_ would not be
      collective.

 (11) “All the members of the baseball team are star players.”
      How has the term _star players_ been used, collectively or
      distributively?

 (12) Why may the term _New York City_ be connotative to a New Yorker
      and non-connotative to a Patagonian?

 (13) So far as your present knowledge of the martyred president
      Abraham Lincoln is concerned, is the term _Abraham Lincoln_
      connotative or non-connotative?

 (14) Are non-connotative terms always singular? Illustrate.

 (15) Are singular terms always non-connotative?

 (16) What is the difference in meaning between immoral and unmoral,
      disagreeable and not-agreeable?

 (17) Why is immoral a nego-positive term while unmoral is negative?

 (18) What is the contradictory of the opposite of _wise_?

 (19) Show that there is some ground for believing all terms to be
      relative.

 (20) Is _army_ a relative term? If “army” were used so as to be
      distributive in nature would it then be general or collective?

 (21) Why should the pronoun be ignored by the logician?

 (22) Show the difference between thing and subject.

 (23) Argue to the effect that no term can be non-connotative.



                              CHAPTER 5.

                 THE EXTENSION AND INTENSION OF TERMS.


=1. TWO-FOLD FUNCTION OF CONNOTATIVE TERMS. (See page 52.)=

It has been indicated that a connotative term is one which possesses
the double function of signifying a subject as well as an attribute.
It may be observed here that an attribute of a notion is any mark,
property or characteristic of that notion. Attribute, then, represents
quality, relation or quantity. By a subject is meant anything which
possesses attributes. Most subjects stand for objects and most
attributes are qualities; consequently, for the sake of simplicity,
we may use subject and object interchangeably; likewise, attribute and
quality.

A connotative term, therefore, denotes an object at the same time
it implies a quality. To illustrate: The symbol _man_ stands for the
various individual men of the world, such as Lincoln, Washington,
Alfred the Great, etc., or for certain qualities like rationality,
power of speech and power of locomotion. The connotative term _teacher_
may be used to _denote_ Socrates, Pestalozzi, Thomas Arnold, or
_connote_ such qualities as ability to instruct, sympathy, and
scholarship. The term _planet_ stands for such objects as Venus, Earth,
and Mars, and for such qualities as rotation upon axis, revolution
about sun, and opaque or semi-opaque bodies. In each of the three
illustrations the term is employed in the two-fold sense of denoting
objects and of implying qualities.


=2. EXTENSION AND INTENSION DEFINED.=

This double function of connotative terms furnishes an important
topic for the student of logic――the Extension and Intension of Terms.
In short, some authorities claim that to master the extension and
intension of terms is virtually to master the entire subject of
logic. Though this position may be an exaggerated one, yet it tends
to emphasize the importance of the topic.

_A term is used in extension when it is employed with reference to the
objects for which the term stands._

When the term triangle is used to refer to the objects isosceles
triangle, scalene triangle, right triangle, it is employed in extension.

_A term is used in intension when it is employed with reference to the
attributes for which the term stands._

The term triangle is employed in intension when we use it to refer to
the qualities, three sided and three angled.


=3. EXTENDED COMPARISON OF EXTENSION AND INTENSION.=

A connotative term seems to be two dimensional――it has extent or length
and intent or depth.

“Extension consists of the things to which the term _ap_plies,” while
“intension consists of the properties which the term _im_plies.”

Extension is quantitative, while intension is qualitative. An
extensional use means to point out or number objects, while an
intensional use means to describe by naming qualities. To name is to
use a term in extension――to describe is to use a term in intension.

To divide a term into its kinds we must regard it in an extensional
sense; e. g., the term _man_ may be divided into Caucasian, Mongolian,
Malay, Ethiopian, American Indian.

To define a term we must regard it in an intensional sense; e. g., man
is a rational animal.

Etymologically considered extension means to _stretch out_, intension,
to _stretch within_. To use a term extensionally one must _look out_.
To use a term intensionally one must _look in_.

In attempting to use a term in extension we may ask ourselves the
question, “What are the kinds?” or “To what objects may the term be
applied?” While if we would use a term in intension the question should
be, “What does it mean?” or “What are the qualities?” Let us, for
example, use the term _metal_ in the two senses, first in extension,
second in intension. Question: To what _objects_ may the term _metal_
be applied? Answer: Metal may be applied to the objects silver, gold
and iron. Thus has metal been employed in extension.

Question: What are the _qualities_ of _metal_? Answer: The qualities
are element, metallic lustre, good conductor of heat and electricity.
Thus has metal been used in _intension_.

NOTE. Since an attribute is anything which _belongs_ to a subject, then
the _parts_ of a subject must be classed as attributes. Hence, a term
is used intensionally when reference is made to its parts.


=4. A LIST OF CONNOTATIVE TERMS USED IN EXTENSION AND INTENSION.=

  _The Term._     _Extensional Use._        _Intensional Use._

                                            { roots, branches, trunk.
  tree.           maple, oak, beech.        {           or
                                            { woody-fiber, sap, bark.

  house.          stone, brick, cement.     foundation, frame-work,
                                              roof.

  dog.            shepherd, fox terrier,    carnivorous, quadruped,
                    bull.                     propensity to bark.

  book.           textbook, dictionary,     cover, leaves, binding.
                    encyclopaedia.

  quadrilateral.  trapezium, trapezoid,     four sides, four angles,
                    parallelogram.            limited plane.

  logic.          theoretical logic,        science of thinking, art of
                    applied logic,            right thinking, treats of
                    educational logic.        laws of thought.

  star.           Sirius, Arcturus, Vega.   heavenly body, gives light
                                              and heat, twinkles.

  force.          gravitation, molecular,   { produces motion
                    atomic.                 { changes motion
                                            { destroys motion.

  term.           general, singular,        word or group of words,
                    non-connotative.          definite idea.

  government.     monarchy, aristocracy,    body of people,
                    democracy.                established form of law,
                                              banded together for
                                              mutual protection.

  bird.           crow, robin, pigeon.      biped, feathered, winged.


=5. OTHER FORMS OF EXPRESSION FOR EXTENSION AND INTENSION.=

    _Extension._      _Intension._
    comprehension     content
    extent            intent
    breadth           depth
    denotation        connotation
    application       implication

Formerly the words extension and intension were applied to concepts
while denotation and connotation were applied to terms representing the
concepts, but now the words are interchangeable. Denotation, the noun,
and denote, the verb, signify, etymologically, a _marking off_. To
denote is to mark off or indicate the objects or classes of objects for
which the term stands. Connotation, the noun, and connote, the verb,
signify _to mark along with_. To connote is to mark along with the
object, its attributes.

The terms which should be remembered are

    extension }     { intension
        or    } and {     or
   denotation }     { connotation


=6. LAW OF VARIATION IN EXTENSION AND INTENSION.=

It has been noted that the intension of a term has reference to its
qualities while extension considers its application to various objects.
It may be wise to experiment with the extension and the intension of
certain terms as types with a view of ascertaining how the two ideas
are related to each other. For the sake of definiteness let us make use
of the following scheme:

                                  I.

      _Intensional_                           _Extensional_

  (1) four sides      }
  (2) parallel sides  } common qualities of { (1) squares
  (3) equal sides     }
  (4) right angles    }

  (1) four sides      }
  (2) parallel sides  } common qualities of { (1) squares
  (3) equal sides     }                     { (2) rhombs

                                            { (1) squares
  (1) four sides      } common qualities of { (2) rhombs
  (2) parallel sides  }                     { (3) rectangles
                                            { (4) rhomboids

                                            { (1) squares
                                            { (2) rhombs
  (1) four sides      } common qualities of { (3) rectangles
                                            { (4) rhomboids
                                            { (5) trapezoids
                                            { (6) trapeziums


                                  II.

                                            { (1) nebulae
                                            { (2) fixed stars
  (1) heavenly body   } common qualities of { (3) sun
                                            { (4) comets
                                            { (5) meteors
                                            { (6) moon

                                            { (1) nebulae
  (1) heavenly body   } common qualities of { (2) fixed stars
  (2) self-luminous   }                     { (3) sun
                                            { (4) comets

  (1) heavenly body   }                     { (1) nebulae
  (2) self-luminous   } common qualities of { (2) fixed stars
  (3) fixed           }                     { (3) sun

  (1) heavenly body   }
  (2) self-luminous   } common qualities of { (1) nebulae
  (3) fixed           }                     { (2) fixed stars
  (4) twinkle         }

  (1) heavenly body   }
  (2) self-luminous   }
  (3) fixed           } common qualities of { (1) nebulae
  (4) twinkle         }
  (5) foggy           }

In considering the first illustration we observe that as the number of
qualities is decreased, the number of objects increases. While in the
second example as the qualities are increased, the number of objects
decreases. It would appear from this that the intension and extension
of a term are _inversely_ related to each other. As the one increases
the other decreases and _vice versa_. It is customary to state this
relation in the form of a law known as the law of variation. “_As the
intension of a term is increased its extension is decreased and vice
versa_,” or the extension and intension of a term vary in an inverse
ratio to each other. To further illustrate: this _book_ refers to a
large number of objects; add to the qualities of book those of _text
book_ and the application is much reduced. In other words as we
increase the intension, the extension is diminished. Increase the
intension further by adding the quality _English_ text book and the
extension becomes still less.


=6a. TWO IMPORTANT FACTS IN THE LAW OF VARIATION.=

In studying the law of variation two facts are especially evident.
(1) The law applies only to a series of terms representing notions
of the same family. The extension and intension of “text book,” for
example, could not be compared with the extension and intension of
“house” as they belong to a different class of words, the genus of
_text book_ being _book_, while the genus of _house_ is _building_.

To illustrate the law of variation, determine upon any class name,
then think of its proximate genus (the next _higher-up_ class to which
it belongs). Continue this till the series is sufficiently complete
to illustrate the law. Or proceed in the opposite direction. That
is, after selecting the class name think of the next lower term in
the class and thus continue till series is complete. Illustration:
The class name _man_ is determined upon; the proximate genus of man
is _biped_, the proximate genus of biped is _animal_, and so on. Or
thinking downward: a proximate species of man is _white man_, of white
man. _European_, etc.

Thus the series:

                                animal
                                 biped
                                 _man_
                               white man
                               European

(2) As a second fact: the increase and decrease is not a mathematical
one. That is, by doubling the extension the intension is not halved.
Or if the intension is decreased by one quality the extension is not
necessarily increased by one object. Thus “man” stands for one billion
seven hundred million beings or objects. Decrease the intension of
“man” by the one quality of rationality and the extension would include
all bipeds――many billion objects.


=6b. THE LAW OF VARIATION DIAGRAMMATICALLY ILLUSTRATED.=

In a general way _lines_ may be used to represent the variation
in extension and intension. For example: we may let a line an inch
long represent the extension of _man_, one two inches long represent
the extension of _biped_, three inches long represent the extension
of _animal_, etc. While on the other hand, if a line an inch long
represents the intension of _man_, a line one-half inch long may
be used to represent the intension of _biped_, one a quarter of an
inch long to represent the intension of _animal_, etc. The following
illustrates this scheme in connection with another series of words:

                  _Extension_             _Intension_
                      ――         barn       ――――――――
                     ――――      building      ――――――
                    ――――――     structure      ――――
                   ――――――――     object         ――

In the foregoing scheme _building_ refers to a greater number of
objects than _barn_, hence the line under _extension_ representing
_building_ should be longer than the line for _barn_. Likewise
_structure_, referring to a greater number of objects than _building_,
is represented by a longer line. Thus when the series is viewed
from top to bottom a gradual increase in extension is noted. Giving
attention to the intensional use of the series we note that _building_
has fewer qualities than _barn_, _structure_ fewer than _building_ and
_object_ fewer than _structure_. Therefore, from top to bottom, the
intension of the terms gradually decreases.

The variation may be made still more apparent if triangles are used,
one triangle being placed upon the other, vertex to base, like the
following:

  Illustration: ( ‡ Intension and Extension )

“Biped” is written near the base or in the broadest part of the
extension triangle because it denotes the greatest number of objects,
and is, therefore, broadest in extension. “Man” occupies a narrower
part of the extension triangle because it refers to fewer objects or
is narrower in extension than “biped.” “Arnold” occupies the narrowest
part of the extension triangle because it is the narrowest in extension.
On the other hand “Arnold” occupies the broadest part of the intension
triangle because intensionally it possesses more qualities than
the others, while “biped,” having the least depth in intension or
possessing the fewest qualities, occupies the narrowest portion of the
intension triangle.


=7. OUTLINE.=

  THE EXTENSION AND INTENSION OF TERMS.

  1. Two-fold Function of Connotative Terms.

  2. Extension and Intension Defined.

  3. Extended Comparison of Extension and Intension.

  4. A List of Connotative Terms used in Extension and Intension.

  5. Other Forms of Expression for Extension and Intension.

  6. Law of Variation in Extension and Intension.

 6a. Two Important Facts in the Law of Variation.

 6b. The Law of Variation Diagrammatically Illustrated.


=8. SUMMARY.=

1. Connotative terms are used in a two-fold sense: first, to denote
objects; second, to imply qualities.

2. A term is used in extension when it is employed with reference
to the objects for which the term stands. A term is used in intension
when it is employed with reference to the qualities for which the term
stands.

3. The answer to either of the following questions will lead one to use
any term in extension: First, what are the kinds? or second, to what
objects may the term be applied?

The answer to either of the following questions will lead to the use
of any term in intension: First, what does it mean? or second, what are
the qualities?

4. To illustrate extension and intension it is best to use the
class-names in every day speech.

5. The word denotation is commonly used for extension and connotation
for intension.

6. “As the intension of a term is increased its extension is decreased
and _vice versa_,” is a statement of the Law of Variation in the
extension and intension of terms.

6a. The law of variation applies only to a series of terms representing
notions of the same class or family, the words being arranged in a
species-genus order. The increase and decrease of the extension and
intension of a series is not proportional.

6b. The law of variation is best explained by using two triangles, one
super-imposed upon the other vertex to base and base to vertex.


=9. ILLUSTRATIVE EXERCISES.=

1a. Employ the following terms in extension――European, flower, term,
truth.

            { Russian
  European  { Englishman
            { Scotchman

            { lily
    flower  { rose
            { pansy

            { singular
      term  { distributive
            { collective

            { Truth has no extension.
     truth  {   Since it refers to a
            {   quality only, it is
            {   non-connotative.

1b. Employ the following in extension――grain, rock, soil, precious
stone.

2a. Use intensionally bird, quadruped, letter, John.

            { two feet
      bird  { ability to fly
            { feathers

            { four feet
 quadruped  { back bone
            { hairy covering

            { heading
    letter  { body
            { complimentary close

            { John has no intension.
      John  {   Since it refers to
            {   an object only, it is
            {   non-connotative.

2b. Use the following in intension――word, table, purity, government.

3a. The use of a term in extension follows when attempting to answer
two questions: First, what are the kinds? Second, to what objects may
the term be applied? Make application of this with reference to the
term _man_.

1. What are the kinds of men? Caucasian, Malay, Mongolian, Ethiopian,
Redman.

2. To what objects does the term _man_ refer? George Washington, Chas.
Hughes, John Smith.

In both 1 and 2 the word _man_ is used to denote objects, hence it is
employed in extension.

3b. Use the term _vegetable_ in extension by answering the two
questions in 3a.

4a. Decrease one by one the qualities of some common object with a
view of noting how when the intension is decreased the extension is
increased.

    _Intension_                   _Extension_

  binding                   } {
  leaves                    } {
  cover                     } { school arithmetic
  printed matter            } {
  designed for instruction  } {
  instruction in arithmetic } {

  binding                   } {
  leaves                    } { school arithmetic
  cover                     } { school grammar
  printed matter            } { school speller, etc.
  designed for instruction  } {

  binding                   } { school arithmetic
  leaves                    } { school grammar
  cover                     } { school speller, etc.
  printed matter            } { encyclopaedia
                            } { novel

                            } { school arithmetic
  binding                   } { school grammar
  leaves                    } { school speller, etc.
  cover                     } { encyclopaedia
                            } { novel
                            } { note book

4b. With a view of noting how when the intension is decreased the
extension is increased, decrease one by one the common qualities of
_peach tree_.

5a. In the following series what word could be substituted for “mammal”
and why? Being, organized being, animal, vertebrate, mammal. Answer:
Fish, reptile, or bird; because there are at least seven classes of
animals which belong to the vertebrate family, any one of which could
be used to complete the series.

5b. Form a series of which “Baldwin apple” has the narrowest extension.
What terms may be substituted for “Baldwin apple?”

6a. In a series of which “pupil” is a member show that the increase
and decrease is not proportional. The series: logic pupil, pupil, youth,
human being, being. In decreasing the intension of “logic pupil” by
dropping the one quality, logic, the extension is made larger by many
more than one, as “pupil” represents many more objects than “logic
pupil.” Therefore, the increase is not in proportion to the decrease.

6b. In a series in which “ruler” appears, show that the increase and
decrease is not proportional.

7. From the following list select the proper words of the series;
arrange them; draw and name the triangles: Caesar, brute, man, Roman,
American, biped, sensuous being, animal, individual.


=10. REVIEW QUESTIONS.=

  1. What is a connotative term? Illustrate.

  2. Which is the broader term, quality or attribute? Why?

  3. When is a term used in extension?

  4. Use the term triangle in intension.

  5. As an aid to using a term in extension or intension what questions
     may one ask himself?

  6. By asking these questions use the term _clock_ in both extension
     and intension.

  7. By experimenting with the qualities of a rectangle show that as
     the intension is decreased the extension is increased.

  8. Write a list of five connotative terms. Prove that they are
     connotative by illustrating their extension and intension.

  9. The term metal { denotes  } such qualities as element, metallic
                    { connotes }
     lustre, conductor of heat and electricity. In the foregoing which
     of the two words following the brace should be used? Give reasons.

 10. State the law of variation in two ways.

 11. As one studies the law of variation what two facts are especially
     evident? Explain fully.

 12. For the purpose of illustrating the law of variation form a series
     of which _desk_ is a member. Draw and name the triangles.


=11. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  1. Which is the broader term, subject or object? Prove it.

  2. If a term like _Caesar_ is given extension does it become a
     general term? Why?

  3. Using “man” as a member of each, arrange at least three different
     series.

  4. Why may it be said that a connotative term is _two_ dimensional?

  5. Is there a word which has a broader extension than “being”? Why?

  6. Prove that _youth_ has less intension than _human being_.

  7. Devise a series of words in which the variation is proportional.

  8. Advance arguments supporting the hypothesis that the term _John_
     has neither extension nor intension.

  9. Suggest arguments to prove that “George Washington” has both
     extension and intension.



                              CHAPTER 6.

                              DEFINITION.


=1. IMPORTANCE.=

To be clear, cogent, concise and consistent is to be logical.
Reference has been made to a striking tendency on the part of writers
and speakers to use words loosely. It is a noticeable fact that
scholars generally aim to be profound rather than clear, philosophical
rather than pointed.

In the use of text books more or less pedagogical these are the common
complaints: “I don’t understand what he means” or “You have to read so
much to get so little.” This condition gives to the topic of definition
a prominence which cannot be overlooked by those who are seeking the
truth; because the definition is the clearest, briefest and altogether
the most satisfactory way of describing an idea. Likewise the habit
of defining any doubtful term reduces to a minimum the possibility of
misunderstanding.

The subject must appeal strongly to the instructor, as he, above all
others, should make his work stand for clearness, pointedness and
continuity.


=2. THE PREDICABLES.=

_A predicable is a term which can be affirmed or predicated of any
subject._ In the proposition, “A man is a rational animal,” the term
“rational animal” is a predicable, because it can be affirmed of the
subject _man_.

To gain a clear knowledge of the definition it is quite necessary
to understand the _five predicables_ which we shall consider in the
following order:

  1. Genus.

  2. Species.

  3. Differentia (difference).

  4. Property.

  5. Accident.

(1) _Genus_ and (2) _Species_.

Genus and species are relative terms and can best be defined together.

_A genus is a term which stands for two or more subordinate classes._

_A species is a term which represents one of the subordinate classes._

The genus may be subdivided into species; the species together form the
genus.

To illustrate: The term _man_ stands for five subordinate classes or
species, as white, black, brown, yellow and red. “Man” is, therefore,
a genus, while “white man” and “black man,” etc., are species. The
term “polygon” is a genus with reference to “trigon,” “tetragon,”
“pentagon,” etc., while “trigon” is a species of “polygon.”

Any given genus may be a species of some higher class. That is, “man,”
which is a genus with reference to the kinds of men, is a species
of the higher class “biped,” while “biped” is a species of “animal,”
“animal” a species of “organized being,” “organized being” of “material
being,” “material being” of “being.” But here we stop, as there is no
higher grade to which “being” can be referred. This highest genus takes
the name of _summum genus_.

Similarly any given species may be a genus of some lower class.
“White man,” for example, which is a species of “man,” is a genus of
“American,” “Englishman,” “German,” “Frenchman,” etc. “American” is
a genus of “New Yorker,” “Californian,” etc., while “New Yorker” is
a genus of “Smith of Jamaica.” This last term is an individual and
cannot be subdivided. It represents the lowest possible species and
is referred to in logic as _infima species_.

It is obvious that the highest genus cannot become a species, neither
can the lowest species become a genus.


  PROXIMATE GENUS.

The proximate genus is the next class above. To illustrate: “Animal”
is a genus of “man,” but “biped” is the proximate genus of “man.”
“Quadrilateral” is the genus of “square,” but “rectangle” is the
proximate genus. The next class above “trigon” is _polygon_ not
_figure_. Hence “polygon” is the proximate genus of “trigon.”


  GENUS AND SPECIES OF NATURAL HISTORY.

In natural history the following terms are used to denote the
various grades of kinship in any scheme of classification: (1) kingdom,
(2) class, (3) order, (4) family, (5) genus, (6) species, (7) variety,
(8) the individual thing. Here “genus” and “species” are absolute not
relative and occupy a _fixed_ place in the scheme, while from a logical
viewpoint any of the grades indicated between the lowest and highest
would be the species of the next higher grade or a genus of the next
lower; e. g., _order_ is a species of “class,” while it is the genus
of “family.”


  GENUS, A DOUBLE MEANING.

We recall that any class name or genus has a double use, extensional
and intensional. When considered from the standpoint of its extension,
a genus represents a group of objects or is _mathematical_ in its
application, but when used in an intensional sense it represents a
group of qualities or is _logical_ in its application.

Considered extensionally the genus refers to a larger number of objects
than the species. But when viewed intensionally the species refers to
more qualities than the genus. This was made clear when discussing the
law of variation in the extension and intension of terms.

(3) _Differentia._

_The differentia of a term is that attribute which distinguishes a
given species from all the other species of the genus._

It has been observed that the species refers to more qualities than
the genus. In fact, it represents all the attributes of the genus plus
those which distinguish the particular species from the other species
of the genus. These additional qualities are the differentiæ of the
particular species.


  TO ILLUSTRATE:

The attribute which distinguishes man from the other bipeds of the
world is his rationality. That which distinguishes the rectangle from
the other parallelograms is its four right angles. The attributes
_rationality_ and _right angles_ are differentiae.

(4) _Property._

_A property of a term is any attribute which helps to make the term
what it is._ Thus “consciousness” is a property of man, “binding” a
property of book, “angles” a property of triangle. Deprive the terms
of these attributes and their true nature is altered.

A differentia is a property according to the foregoing definition.
However, Jevons defines “property” as “Any quality which is common to
the whole of a class, but is not necessary to mark out the class from
other classes.” This viewpoint excludes “differentia” from the notion
of property. The difference in opinion is of slight importance.

(5) _Accident._

_An accident of a term is any attribute which does not help to make
the term what it is._ It may indifferently belong or not belong to the
term. Deprive a term of an accident and the nature of the term remains
unchanged. Thus, a teacher’s position, a man’s watch, the fact that the
angle is one of 80° are all accidents.

It is obvious that a property is a _constant_ attribute while an
accident is variable. This gives to the former a universal validity
while the latter is more or less shifting and uncertain. All triangles
must have _three angles_ (property) while the _value of each angle in
degrees_ (accident) admits of unlimited variation.

Some logicians divide accidents into separable and inseparable. A man’s
_hat_ would be a separable accident while his _birthplace_ would be an
inseparable accident.


  FIVE PREDICABLES ILLUSTRATED.

In the following brief descriptions the five predicables are
designated:

           (species)         (genus)           (differentia)
  (1) This ――――――――― is a ――――――――――――― with ―――――――――――――――――
           rectangle      parallelogram      four right angles
                             (accident)
      its longer sides being ――――――――――.
                             ten inches

           (species)      (differentia) (prox. genus)
  (2) This ――――――――― is a ――――――――――――― ――――――――――――― with the
              man           rational        biped
          (property)               (accident)
      ――――――――――――――――――― and a ――――――――――――――――.
      power of locomotion       ruddy complexion

        (species)      (genus)    (differentia)      (property)
  (3) A ――――――――― is a ――――――― of ――――――――――――― and ――――――――――――,
         trigon        polygon     three sides      three angles
                            (accident)
      ―――――――――――――――――――――――――――――――――――――――――――――――――――――.
      the sum of the angles being equal to two right angles


=3. THE NATURE OF A DEFINITION.=

It will be remembered that an individual notion is a notion of a single
thing or attribute, while a general notion is a notion of a class of
things or a group of attributes. A term which represents an individual
notion is known as a _singular term_, while a term which stands for a
general notion is referred to as a _general term_.

One may explain the meaning of a singular term which stands
for one thing by enumerating its various attributes. For example,
such attributes as a piercing bark, a yellow color, intelligent,
companionable, a strong liking for sweetmeats, explain the meaning
of the singular term “Fido.” Likewise we may explain the meaning of
a general term by enumerating its attributes. To illustrate: power of
speech, rationality, ability to laugh, etc., explain the meaning of
the general term _man_. The explanation of the singular term fits only
Fido. There is probably no other dog in the world just like Fido. But
the explanation of the general term _man_ may be applied to _all_ men.

A brief enumeration of attributes which may be applied to a _class of
things_ often takes the form of a definition. The word definition comes
from the word _definire_, meaning to limit or fix the bounds of.

A definition, then, consists of the enumeration of such attributes
as distinguish a term from all other terms. In this sense it would
seem that the singular term _Fido_, as well as the general term
_man_, admits of definition, but it is usual for logicians to confine
definition to the _general term_. Singular terms may be _described_;
general terms, _defined_.


  A DEFINITION OF DEFINITION.

_A definition of a term is a statement of its meaning by enumerating
its characteristic attributes._

That the enumeration must be in terms of its distinguishing or
characteristic attributes is implied in the derivation of the term
_definition_. The attributes must establish limits or bounds, just as a
line fence limits a land owner’s possessions. To indicate that man is a
creature possessing the power of locomotion, sense of sight and ability
to eat, is surely not a definition, as the marks are not characteristic
of men only. These attributes set no boundary between man and horse,
consequently the statement is a faulty _description_ of man, not
a _definition_. But when the enumeration includes such attributes
as power of speech, rationality and ability to laugh, then does the
description become a definition. To put it differently: A definition is
a description of a term by means of its distinguishing attributes. This
statement may be considered a definition of man, though somewhat faulty:
“A man is a creature who is rational and who possesses the power of
speech and ability to laugh.”


=4. DEFINITION AND DIVISION COMPARED.=

We have learned that general terms when connotative may be used
extensionally or intensionally.

A definition indicates the _intensional_ nature of a term, while a
statement which points out the _extensional_ nature of a term is known
as logical division. More briefly: A definition is an _intensional_
statement of the nature of the term, while logical division is an
_extensional_ statement of the nature of the term.

To illustrate: The following statements are definitions:

  (1) A dog is a domesticated quadruped of the genus canis and given
      to barking.

  (2) A quadrilateral is a rectilinear figure of four sides.

  (3) Soil is a substance composed of pulverized rock and decayed
      vegetable matter in which plants will grow.

The following represent Logical Division:

  (1) Dogs are divided into hounds, terriers, bull, etc.

  (2) The kinds of quadrilaterals are trapeziums, trapezoids and
      parallelograms.

  (3) The various soils are loam, sand, clay, muck, etc.


=5. THE KINDS OF DEFINITIONS.=

Generally speaking there are three kinds of definitions, namely,
(1) Etymological, (2) Descriptive, (3) Logical.[5]

(1) _An etymological definition is one based upon the derivation of the
term._

This kind of a definition, which gives merely the meaning of the
symbol, is sometimes called a _nominal_ or verbal definition; while
a _real_ definition is regarded as one which gives the _meaning
of the notion_ for which the symbol stands. The modern logician is
inclined to ignore this classification on the argument that to make a
distinction between a symbol and the notion it symbolizes is simply to
misunderstand the relation which exists between them. If the definition
does not agree with the thing then it cannot correctly explain the
term which represents the thing. Define correctly the term and one has
defined correctly the notion signified by the term.

The attributes of a term may be separated into three classes:
differentia, property and accident. It would appear possible,
therefore, to define a term by enumerating the accidents only or by
enumerating the properties, or, finally, by stating the differentiae.
But if the enumeration is confined to accidents the chances are that
the statement will be a description, not a definition, as accidents
are seldom sufficiently characteristic to determine the boundaries of
a term. This leaves open two distinct ways of defining a term: First,
by naming the properties or properties and accidents only; second,
by stating the differentiæ only. The former kind is the so-called
descriptive definition, while the latter is the logical.

(2) _A descriptive definition of a term is a description of its nature
by means of its properties and accidents._

(3) _A logical definition of a term is a description of its nature by
means of its differentiæ._


  THE THREE KINDS OF DEFINITIONS ILLUSTRATED AND COMPARED.

Etymological Definition of Trigon.

  A trigon is a figure of three corners.

Descriptive:

  A trigon is a figure which has three sides and three angles, the sum
    of the latter being equal to two right angles.

Logical:

  A trigon is a polygon of three angles.

It is seen that an etymological definition is simply a root-word
analysis. In the case of _trigon_, the prefix comes from the Greek,
meaning three, while the root-word comes from the Greek meaning corner.

The descriptive definition of trigon names the properties, “three
sides and three angles” (differentiæ) and the accident, “the sum of
the angles of which equals two right angles.”

The logical definition of trigon simply states the proximate genus,
“polygon,” and the differentia, “three angles.”


=6. WHEN THE THREE KINDS OF DEFINITIONS ARE SERVICEABLE.=

The etymological definition is helpful in furnishing a cue for
remembering the descriptive and logical definitions. It also leads
to precision of expression――the right word in the right place. Here
is where the knowledge of a foreign language, particularly Latin, is
helpful.

The descriptive definition is best adapted to the child-mind. Children
think in the large; are not given to hair-splitting discriminations,
and, therefore, _many_ characteristic marks must be mentioned in
order to insure a mastery of the content. With children the logical
definition is often too brief to be clear. For example, it is easy
to see which of the following definitions would be better adapted
to the child-mind. _Logical_: A square is an equilateral rectangle.
_Descriptive_: A square is a figure of four equal sides and four right
angles.

The logical definition may be introduced to the student of the
secondary school.

Few exercises are better adapted to the development of powers of
discrimination and precision than practice in defining logically the
common terms of every-day life. For example: “A book is a pack of
paper-sheets bound together.” “A chair is a piece of furniture with
back and seat, designed for the seating of one person.” “A lead pencil
is a cylindrical writing implement with lead through the center.”
“A door is an obstacle designed to swing in and out to open and close
an entrance.” “An eraser is an implement made to rub out written or
printed characters.”

These definitions, coming from training school students, are not above
criticism, yet they illustrate the point in hand.


=7. THE RULES OF LOGICAL DEFINITION.=

Five rules summarize the requirements to which a logical definition
must conform.


  FIRST RULE.

_A logical definition should state the essential attributes of the
species defined._

This means that a logical definition should contain the species, the
proximate genus and the differentia. As the terms species, genus and
differentia have been explained, it will be sufficient to briefly
illustrate this rule.

_Logical According to the First Rule._

        species      genus      differentia
  (1) A ――――――― is a ――――― with ―――――――――――.
         bird        biped       feathers

        species      genus          differentia
  (2) A ――――――― is a ―――――― ―――――――――――――――――――――――――――.
        mascot       person supposed to bring good luck

      species       genus        differentia
  (3) ―――――――― is a ―――――― of ―――――――――――――――――.
      Religion      system    faith and worship

        species          genus      differentia
  (4) A ―――――――― is a ―――――――――――― ―――――――――――――.
        moonbeam      ray of light from the moon

_Illogical According to the First Rule._

  (1) A man is a rational animal.
        (“Biped” is the proximate genus, not “animal.”)

  (2) A Connotative term always denotes both an object and an attribute.
        (No genus.)

  (3) A trigon is a polygon.
        (No differentia.)

  (4) It is a term which denotes an indefinite number of objects or
      attributes.
        (No species.)

_The Foregoing Illogical Definitions Made Logical._

  (1) A man is a rational _biped_.

  (2) A Connotative term is a _term_ which denotes both an object and
      an attribute.

  (3) A trigon is a polygon of _three angles_.

  (4) A _general term_ is a term which denotes an indefinite number of
      objects or attributes.


  SECOND RULE.

_A logical definition should be exactly equivalent to the species
defined._

This means that the species must equal the genus plus the differentia
or the subject and predicate of the definition must be co-extensive――of
the same bigness. The subject must refer to the same number of objects
as the predicate.

A man upon the witness stand makes the declaration that he will testify
to the truth, the whole truth and nothing but the truth. A logical
definition must contain _the species, the whole species_ and _nothing
but the species_. If the definition does not include all the species,
it is too narrow; while on the other hand, if it includes other species
of the genus it is too broad.

An excellent test of this second requirement is to interchange subject
and predicate. If the interchanged proposition means the same as the
original then the conditions have been met. To illustrate: Original――A
trigon is a polygon of three angles. Interchanged――A polygon of three
angles is a trigon.

The very best way of making the definition conform to this rule is to
put to oneself these three questions: 1. Does it include all of the
species? 2. Does it exclude all other species of the genus? 3. Has it
any unnecessary marks?

To exemplify: Let us ask the three questions relative to the following
logical definitions:

  (1) A parallelogram is a quadrilateral whose opposite sides are
      parallel.

  (2) A bird is a biped with feathers.

_Questions_:

  (1) Does the definition include all the parallelograms? Yes. Does it
      exclude all other quadrilaterals? Yes. Are there any unnecessary
      marks? No.

  (2) Does it include all birds? Yes. Does it exclude all other bipeds?
      Yes. Any unnecessary marks? No.

_Illogical According to the Second Rule._

  (1) A man is a vertebrate animal.

      (Too broad. Does not exclude other species of the genus, such as
      horses, dogs, etc.)

  (2) A barn is a building where horses are kept.

      (Too narrow. Does not include all of the species, such as cow
      barn.)

  (3) An equilateral triangle is a triangle all of whose sides and
      angles are equal.

      (Equal angles is an unnecessary mark.)

_The Foregoing Definitions Made Logical._

  (1) A man is a rational biped. (Proximate genus.)

  (2) A barn is a building where horses and cattle are kept and hay and
      grain are stored.

  (3) An equilateral triangle is a triangle all of whose sides are
      equal.


  THIRD RULE.

_A definition must not repeat the name to be defined nor contain any
synonym of it._

A violation of this rule is known as “a circle in defining” (_circulus
in definiendo_).

There are some exceptions to this rule, as in the case of compound
words and a species which takes its name from its proximate genus. To
say that a hobby-horse is a horse, or that an equilateral triangle is a
triangle, is not only allowable but necessary, that the proximate genus
may be used.

_The following definitions are illogical according to the third rule_:

  (1) A teacher is one who _teaches_.

  (2) Life is the sum of the _vital_ functions.

  (3) A sensation is that which comes to the mind through the _senses_.


  FOURTH RULE.

_A definition must not be expressed in obscure, figurative or ambiguous
language._

A violation of this rule is referred to in logic as “defining the
unknown by the still more unknown” (_ignotum per ignotius_).

It is known that the purpose of definition is to make clear some
obscure term, consequently unless every word used is understood the
chief aim of the definition has been defeated.

From this it must not be inferred that all definitions should be free
from technical terms. Such a restriction would make the defining of
many terms unsatisfactory and in a few cases practically impossible.
To the student of evolution the following definition by Spencer
is intelligible while to the uninitiated it would appear obscure:
“Evolution is a continuous change from an indefinite, incoherent
homogeneity to a definite coherent heterogeneity through successive
differentiations and integrations.”

This rule insists upon simple language when it is possible to use such
in giving an accurate and comprehensive meaning to the term defined.

_Illogical Definitions According to the Fourth Rule._

  (1) “A net is something which is reticulated and decussated, with
      interstices between the intersections.” Dr. Johnson.

  (2) “Thought is only a cognition of the necessary relations of our
      concepts.”

  (3) “The soul is the entelechy, or first form of an organized body
      which has potential life.” Aristotle.


  FIFTH RULE.

_When possible the definition must be affirmative rather than negative._

The fact that there are a considerable number of terms which admit of a
negative definition only, takes from the force of this rule. Such terms
as deafness, inexpressible, infidel and the like can best be defined
negatively.

It likewise happens that when words are used in pairs it is expedient
to define one affirmatively and the other negatively. Recall, for
example, the definitions of relative and absolute terms: “A relative
term is one which needs another term to make its meaning clear.”
“An absolute term is one which does not need another term to make its
meaning clear.”

_Illogical Definitions According to the Fifth Rule._

  (1) A gentleman is a man who is not rude.

  (2) An element is a substance which is not a compound.

  (3) An univocal term is a term which does not have more than one
      meaning.


=8. TERMS WHICH CANNOT BE DEFINED LOGICALLY.=

A logical definition insists upon a proximate genus and differentia.
But as there is no genus higher than the highest genus (_summum genus_)
then surely such cannot be defined logically. The words _being_ and
_thing_ illustrate terms of this class. Moreover, it is impossible
to give a satisfactory definition of an individual (_infima species_)
as no attributes can be mentioned which will distinguish definitely
and permanently the individual from others of the class. We may
perceive the attributes but not those that are possessed solely
by the individual. To say that Abraham Lincoln was a man who was
simple and honest is not a definition, as other men have had the
same characteristics.

Again there are a few terms such as life, death, time and space which
cannot be defined satisfactorily. These terms seem to be in a class by
themselves or of their own genus (_sui generis_).

Since a definition of a term is a brief explanation of it by means of
its attributes, it follows that collective terms and terms standing for
a single attribute are incapable of definition. Such terms as group,
pain, attribute, belong to this class.

We may say, then, that there are some terms too high, some too low and
some too peculiar to come within the province of logical definition.
In short, “_summum genus_,” “_infima species_” and “_sui generis_” are
incapable of definition.


=9. DEFINITIONS OF COMMON EDUCATIONAL TERMS.=

  (1) _Development_ is the process whereby the latent possibilities
      of an individual are unfolded or the invisible conditions of a
      situation are made apparent.

      Development means expansion according to principle, while
      unfolding may or may not involve a principle.

  (2) _Education_ is the process employed in developing systematically,
      symmetrically and progressively all of the capabilities of a
      single life; or

  (3) _Education_ is the process of modifying experience in order to
      make the life as valuable as it ought to be.

  (4) _Teaching_ is the art of occasioning those activities which
      result in knowledge, power and skill.

      It is the duty of the true teacher to inspire the child to
      activity along right lines. Through his own activity the child
      shapes his inner world which is sometimes termed _character_.

      Knowledge is anything known, power is ability to act, skill is a
      readiness of action.

  (5) _Instruction_ is the art of occasioning those activities which
      result in knowledge.

      Instruction develops the understanding; teaching develops
      character.

  (6) _Training_ is the occasioning of those activities which, by means
      of directed exercise, result in power and skill.

      Training and education are not interchangeable. Training implies
      an outside authority, while education, which involves inner
      development, may proceed without supervision.

  (7) _Knowledge_ is anything acquired by the act of knowing.

  (8) _Learning_ is the act of acquiring knowledge or skill.

  (9) Instruction, training, teaching, learning and education all
      involve activity.

      Instruction arouses activity which results in knowledge;
      training directs activity which produces power and skill;
      teaching includes both instruction and training. Learning is an
      activity which results in knowledge and skill, while education
      is a developing process which involves all the others.

 (10) A _science_ is knowledge classified for the purpose of
      discovering general truths.

 (11) _An art_ is a skillful application of knowledge and power to
      practice.

      “A science teaches us to know, an art to do.”

 (12) A _fact_ is a single, individual, particular thing made or done.

      A _truth_ is general knowledge which exactly conforms to the
      facts.

      A truth may be a definition, rule, law, or principle.

 (13) A _fact_ as opposed to hypothesis is an occurrence which is true
      beyond doubt.

      An _hypothesis_ is a supposition advanced to explain an
      occurrence or a group of occurrences.

      A _theory_ is a general hypothesis which has been partly verified.

 (14) _Theory_ as opposed to practice means _general knowledge_, while
      practice involves the putting into operation one’s theories.

 (15) A _fact_ as opposed to phenomenon is something accomplished.
      A phenomenon is something shown.

 (16) A _method-whole_ is any subdivision of the matter for instruction
      which leads to a generalization.

 (17) _Method_ is an orderly procedure according to a recognized system
      of rules and principles.

      As the term is commonly used it includes not only the arrangement
      of the subject matter for instruction but the mode of presenting
      the same to the mind.

 (18) _Induction_ is the process of proceeding from the less general to
      the more general.

      _Deduction_ is the process of proceeding from the more general to
      the less general.

 (19) The terms induction and deduction may have reference to forms of
      reasoning or to methods of teaching.

      The _inductive method_ is the method of deriving a general truth
      from individual instances.

      The _deductive method_ is the method of applying a general truth
      to individual instances.

      The inductive method is objective, while the deductive method is
      subjective. Induction is the method of discovery; deduction is
      the method of instruction.

 (20) _Analysis_ is the process of separating a whole into its related
      parts.

      _Synthesis_ is the process of uniting the related parts to form
      the whole.

 (21) The _analytic method_ is the method of proceeding from the whole
      to the related parts.

      The _synthetic method_ is the method of proceeding from the
      related parts to the completed whole.

 (22) Analysis and synthesis deal with _single things_, while induction
      and deduction are concerned with _classes of things_.

 (23) The _complete method_ consists of three elements: (1) induction,
      (2) deduction, (3) verification or proof.

      When the emphasis is placed on the inductive phase, the complete
      method is sometimes termed the development method.


=10. OUTLINE.=

DEFINITION.

  (1) Importance.

  (2) The Predicables.
        Genus――species――summum genus――infima species.
        Proximate Genus.
        Genus and Species of Natural History.
        Genus, Double meaning of
          Differentia.
          Property.
          Accident.
            Separable, Inseparable.

  (3) Nature of Definition.

  (4) Definition and Division Compared.

  (5) The Kinds of Definitions.
        (1) Etymological.
        (2) Descriptive.
        (3) Logical.


  THREE KINDS ILLUSTRATED AND COMPARED.

  (6) When the Three Kinds are Serviceable.

  (7) The Rules of Logical Definition.
        (1) _Essentials._
        (2) Same _size_.
        (3) Do not _repeat_.
        (4) _Unambiguous._
        (5) _Language_ affirmative.

  (8) Terms Which Cannot be Defined Logically.
        Summum genus.
        Infima species.
        Sui generis.
        Collective terms.
        A single attribute.


=11. SUMMARY.=

(1) To be logical one must acquire the habit of accurate definition.

This topic ought to appeal strongly to the school teacher, who should
above all others make his work stand for clearness, pointedness and
continuity.

(2) A predicable is a term which can be affirmed or predicated of any
subject.

The five predicables are Genus, Species, Differentia, Property and
Accident.

  (1) A Genus is a term which stands for two or more subordinate
      classes.

  (2) A Species is a term which represents one of the subordinate
      classes.

      The _proximate genus_ of a species is the next class above the
      species, while the _summum genus_ is the highest possible class
      in any graded series of terms. The lowest class is the _infima
      species_ of that series. The lowest class may be individual.

      In natural history genus and species are not relative terms, but
      absolute, having a fixed place in the series of gradations.

      The term genus possesses a double meaning: it may be used to
      represent objects (extensionally) or qualities (intensionally).

  (3) The differentia is that attribute which distinguishes a given
      species from all the other species of the genus.

  (4) A property of a term is any attribute which helps to make that
      term what it is.

      Differentia is a property according to definition. Some logicians
      would not include the differentia in the content of the term
      property.

  (5) An accident of a term is any attribute which does not help
      to make it what it is. Some authorities divide accidents into
      separable and inseparable.

(3) A definition of a term is a statement of its meaning by enumerating
its characteristic attributes.

(4) Definitions explain a term intensionally, while logical division
explains a term extensionally.

(5) There are three kinds of definitions: (1) etymological,
(2) descriptive, (3) logical.

An etymological definition is based upon the derivation of the term;
a descriptive definition states the characteristic properties and
accidents of a term, while a logical definition is simply a statement
of the differentia of a term.

(6) The etymological definition leads to precision of expression, the
descriptive definition is best adapted to the child-mind, while the
logical definition belongs to the realm of secondary education.

(7) Five rules summarize the requirements to which a logical definition
must conform. In a word or two these five rules are: Every logical
definition must (1) state the genus and differentia, (2) be equivalent
to the species defined, (3) not repeat the name to be defined, (4) not
be expressed in obscure language, (5) commonly be affirmative.

(8) Some terms are too high (summum genus), some too low (infima
species), some too peculiar (sui generis) to come within the province
of logical definition.


=12. ILLUSTRATIVE EXERCISES.=

1a. The italicized words in the following propositions are predicables
because they are affirmed of the subject:

  (1) “This man weighs _one hundred fifty pounds_.”

  (2) “A bird is a _feathered biped_.”

  (3) “The earnest teacher is an _indefatigable worker_.”

  (4) “Walking is the _most beneficial outdoor exercise_.”

1b. Underscore the predicables in the following:

  (1) “All men are rational.”

  (2) “Teachers must be just.”

  (3) “Every form of unhappiness springs from a wrong condition of the
      mind.”

  (4) “Calmness of mind is one of the beautiful jewels of wisdom.”

2a. To clarify our ideas it is an excellent plan to select a group
of words belonging to the same genus with a view of defining them as
simply and expeditiously as possible. As an illustration _building_
may be selected as a genus. The word _kind_ will suggest to us the
species, such as dwelling, church, theatre, school, barn, bird-house,
granary and smoke-house. Next it is necessary to discover the basis of
distinction. This seems to be the use to which the building is put. Now
we are ready for the definitions:

  _Species_         _Genus_         _Differentia_

  A dwelling      is a building   where people live.
  A church        is a building   where people worship.
  A theatre       is a building   where people act.
  A school        is a building   where children are taught.
  A barn          is a building   where domestic animals, hay and
                                    grain are kept.
  A bird-house    is a building   designed for birds.
  A granary       is a building   where grain is stored.
  A smoke-house   is a building   where meat is smoked.

2b. By selecting _man_ as the genus, define the terms Caucasian,
Mongolian, Ethiopian, Malay and American Indian. Treat the term _chair_
in the same manner.

3a. One may easily distinguish a property from an accident by asking
himself the question, “Would subtracting the attribute from the term
alter its identity”? For example in the following, I find that the
words italicized are properties because subtracting each from the term
changes its identity:

   _Term_   _Attributes_

  man       age, _rationality_, possessions.
  book      _binding_, _leaves_, size, color, contents.
  radium    _emits intense light and heat_, costs a million dollars
                a pound.
  snail     _air-breathing mollusk_, moves slowly.
  slush     _soft mud and snow_, six inches deep.

3b. Indicate the common attributes of the following terms, underscoring
the properties: Tree, teacher, garden, house, river.

4. The rules summarize well the essentials of the subject matter of the
logical definition. Therefore, it is highly important for the student
to have these rules at the “tip of the tongue.” With this in view a
device of this nature may be helpful. Make each letter of the word
rules stand for the initial letter of a suggestive word in each of the
five _rules_. For example: r (repeat), u (unambiguous), l (language
affirmative), e (essential), s (same size).

With a little study “r and repeat,” “u and unambiguous,” “l and
language affirmative,” “e and essential,” “s and same size” may be
firmly linked together in the memory. _Repeat_ suggests the third
rule, do not repeat the name, etc.; _unambiguous_, the fourth rule,
not ambiguous language, etc.; _language affirmative_, the fifth rule;
_essentials_, the first rule; _same size_, the second rule, subject
and predicate must be of same size. The fact that the rules are not
recalled in order of treatment is inconsequential.

It is the writer’s experience that fifteen minutes of concentrated
study upon this device or one similar to it will indelibly stamp upon
the mind these troublesome rules.

The student may be able to devise a more helpful keyword.


=13. REVIEW QUESTIONS.=

  (1) Why should the subject of definition appeal strongly to the
      school teacher?

  (2) Define a predicable.

  (3) Name in order the five predicables.

  (4) Define and illustrate the terms genus and species.

  (5) Explain the terms summum genus, infima species, sui generis.

  (6) Illustrate proximate genus.

  (7) Explain the terms genus and species as used in natural history.

  (8) Exemplify the double meaning of the genus man.

  (9) Define and illustrate differentia.

 (10) In what sense is the species a richer term than the genus?

 (11) Distinguish between property and accident.

 (12) Illustrate separable and inseparable accidents.

 (13) Give descriptive definitions of the following, indicating the
      five predicables: logic, general term, non-connotative term,
      obversion.

 (14) Define definition; illustrate.

 (15) Distinguish between definition and division.

 (16) Name, define and illustrate the three kinds of definitions.

 (17) Distinguish between real and verbal definitions.

 (18) Define in three ways the following: king, government, city, metal.

 (19) State the rules of logical definition.

 (20) What words may be used as _cues_ to aid in recalling the rules
      for logical definition?

 (21) Under what circumstances will the wise teacher make use of each
      of three kinds of definitions?

 (22) Relative to the second rule for logical definition what are the
      _three_ questions that one should ask himself?

 (23) Explain the exceptions to the third rule.

 (24) In connection with the fourth rule what may be said as to the use
      of technical terms?

 (25) What facts take from the force of the fifth rule?

 (26) What classes of words do not admit of logical definition?
      Illustrate.

 (27) Define education, teaching, instruction, training.

 (28) Distinguish by illustration between induction and synthesis;
      deduction and analysis.


=14. QUESTIONS FOR ORIGINAL THOUGHT
AND INVESTIGATION.=

  (1) Why should the _scholar_ be tempted to speak and write
      illogically?

  (2) Name the parts of speech that may be classed as predicables.

  (3) Explain the ten categories as given by Aristotle.

  (4) Show that genus and species are relative terms.

  (5) Why should the definition be needed most in the abstract sciences,
      such as theology, ethics, political economy, juris-prudence and
      psychology?

  (6) Define sin, life, wrong, personality, habit, character.

  (7) From the viewpoint of natural history find the species in the
      series of terms of which _polygon_ is a member.

  (8) What is the plural of differentia?

  (9) Why should logic insist upon the proximate genus?

 (10) (a) Man is a rational animal.
      (b) Man is a rational biped (proximate genus).

      In the case of the immature mind the first definition would be
      clearer. Why?

 (11) “A property of a term is any mark or characteristic which belongs
      to that term.” Is this definition logical? Give reasons.

 (12) What is the difference between the logical and the popular
      conception of _property_?

 (13) Is there any difference between the logical and popular
      conception of accidents?

 (14) “The term _conferentia_ might be used to stand for the essence of
      the genus, as the term differentia represents the essence of the
      species.”[6] Explain this.

 (15) John Stuart Mill affirms that there is no such thing as a real
      definition. Discuss this.

 (16) In your opinion, of the five rules of logical definition what one
      is violated most by the average teacher? Give reasons.

 (17) Distinguish between symbol and content.

 (18) Why are descriptive definitions best for young children? What
      educational principle is involved?

 (19) From the standpoint of the five rules for logical definition
      criticise the following:

      (1) A man is a reasonable vertebrate.

      (2) A gentleman is a man with no visible means of support.

      (3) A man is an organized entity whose cognitive powers function
          rationally.

      (4) A metal is an element with a metallic luster.

      (5) A triangle is a figure of three sides.

      (6) A teacher is one who imparts knowledge.

      (7) Education is the process of drawing out all that is beautiful
          in the body and noble in the soul.

      (8) A democrat is a man who believes in free trade.

      (9) A government is a commonwealth controlled by direct vote of
          the people.

 (20) Write the foregoing definitions in logical form.

 (21) Since man is the only animal given to laughter, why is not the
      following a logical definition: “Man is a laughing animal.”

 (22) “A logical definition should contain the species, the genus and
      the appropriate differentia.” Is there any reason for using the
      term appropriate?

 (23) In connection with genus and species explain subaltern.

 (24) Is laughter a property of human being or an accident?

 (25) Show how a pedagogue may be an instructor but not a teacher.

 (26) Illustrate the complete method.

 (27) Show that induction may consist of a series of analyses; also a
      series of syntheses.



                              CHAPTER 7.

                 LOGICAL DIVISION AND CLASSIFICATION.


=1. NATURE OF LOGICAL DIVISION.=

The term _genus_ is used for any class name which stands for two or
more subordinate classes while the term _species_ is made to stand for
any one of the subordinate classes.

The proximate genus of any species is the _next_ class above. For
example the proximate genus of man is biped, not animal.

_Logical division is the process of separating a proximate genus into
its co-ordinate species._


  ILLUSTRATIONS:

          _Genus_           _Species_

                        { Fixed stars
                        { Planets
  (1) Heavenly bodies   { Satellites
                        { Comets
                        { Meteors
                        { Nebulae

                        { Leptocardians
                        { Fishes
  (2) Vertebrates       { Amphibians
                        { Reptiles
                        { Birds
                        { Mammals


                        { Caucasian
                        { Mongolian
  (3) Man               { Malay
                        { Ethiopian
                        { American Indian

                        { Monarchy
  (4) Government        { Aristocracy
                        { Democracy


=2. LOGICAL DIVISION DISTINGUISHED FROM ENUMERATION.=

When the genus is separated at once into _individual objects_ the
process is not logical division, but simple enumeration. Logical
division implies a separating into smaller class terms, each term being
a genus of still smaller subdivisions. This process may be continued
till the last division gives individuals as species. Enumeration takes
place when the first subdivision results in a list of individuals. To
illustrate:

                          _Logical Division._

                              { Science teacher
                    Teacher   { Mathematics teacher
                              { English teacher
                              { Modern language teacher

                            _Enumeration._

                              { John J. Brown
                    Teacher   { H. G. White
                              { Mary Jones
                              { Alice Smith


=3. LOGICAL DIVISION AS PARTITION.=

_Partition is the process of separating an individual thing into its
parts._

The partition is quantitative or mathematical when the separation is
in terms of space or time, but when otherwise the partition becomes
qualitative or logical. Or to put it in another way, the partition
is mathematical when the separation gives parts and logical when the
separation gives ingredients.

To illustrate:

            {                  { branches
            { quantitative     { leaves
            {     or           { roots
            { (mathematical)   { trunk
  (1) Tree  {
            { qualitative      { woody fibre
            {     or           { capillary attraction
            { (logical)        { sap
            {                  { chlorophyll

            { quantitative     { roof
            {     or           { frame-work
            { (mathematical)   { foundation
  (2) House {
            { qualitative      { wood
            {     or           { iron
            { (logical)        { stone
            {                  { plaster

An easy way to determine that the separation involves logical division
proper and not partition is to affirm the connection between a class
and a sub-class. To wit: A man is a biped; a square is a rectangle;
a Caucasian is a man, etc. If such an affirmation cannot be made then
the separation involved is not properly logical division but probably
partition. For example it cannot be said that a _roof is a house_, or
that _sap is a tree_. It is seen, then, that a logical division of any
genus may be summarized in the form of a series of judgments of which
a species is the subject and the genus is the predicate. For example,
by a logical division quadrilaterals may be divided into trapeziums,
trapezoids and parallelograms; this process may then be summarized in
a series of three judgments: (1) A trapezium is a quadrilateral; (2) A
trapezoid is a quadrilateral; (3) A parallelogram is a quadrilateral.


=4. RULES OF LOGICAL DIVISION.=

When the logical division of a genus is under consideration there are
four rules which should be observed.

FIRST RULE. _There must be but one principle of division (fundamentum
divisionis)._ To divide mankind into white man, Australian, yellow man,
African and red man is a violation of this rule as the _two_ principles
of color and geographical location are involved. A division in which
more than one principle is used is sometimes referred to as _cross
division_ because the various species cross each other. For example in
the foregoing there are many white men who are Australians.

This rule applies only to one division. Where there is a series
of divisions a new principle may be employed in each division. For
example, in dividing triangles into scalene, isosceles and equilateral,
the equality of sides is the principle involved, but, in subdividing
isosceles triangles into right angled and oblique angled, the principle
employed concerns the nature of the angle.

SECOND RULE. _The co-ordinate species must be mutually exclusive._
There must be no overlapping. The illustration given in the first
rule is likewise a violation of this rule. Another example in which
this second rule is not obeyed may be found in most geometries where
triangles are divided into scalene, isosceles and equilateral. Here
the second and third classes are not mutually exclusive since all
equilateral triangles are isosceles according to the usual definition,
“An isosceles triangle is a triangle having two equal sides.” All
equilateral triangles have _two_ equal sides.

THIRD RULE. _The division must be exhaustive._ That is, the species
taken together must equal the whole genus. The sum of the species must
be co-extensive with the genus.

Dividing man into Caucasian, Ethiopian and Mongolian would be a
violation of this rule, as there are at least two other species of man,
Malay and American Indian.

A distinction should be made between an exhaustive division and
a complete division as the latter is not a logical requirement.
To divide government into monarchy, aristocracy and democracy is
exhaustive but incomplete. Exhaustive because there is no other kind
of government, all the species are included; but incomplete in that
monarchy may be divided into absolute and limited; democracy into pure
and representative.

FOURTH RULE. _The division must proceed from the proximate genus to the
immediate species._ There should be no sudden jumps from a high genus
to a low species. The division must be gradual and continuous; step by
step. To divide government into limited monarchy, absolute monarchy,
pure democracy and representative democracy would be a violation of
this rule, as government is the proximate genus of monarchy, not of
limited monarchy, therefore one step has been omitted. Such an omission
involves a step from grandfather to grandchild, so to speak, the
generation of father having been left out.

A violation of this rule is most insidious when some of the species
of a subdivision are immediate while others are not. To wit: dividing
government into monarchy, aristocracy, pure democracy and republic,
or dividing quadrilaterals into trapeziums, trapezoids, rectangles,
squares, rhomboids and rhombs.


=5. DICHOTOMY.=

Dichotomy comes from the Greek, meaning _to cut in two_. _Dichotomy
is a continual division of a genus into two species which are
contradictory in nature._

Contradictory terms are such as admit of no middle ground. They divide
the whole universe of thought into two classes. For example, honest and
not-honest, pure and impure, perfect and imperfect, are contradictory
terms. Dichotomy thus affords an easy opportunity for an exhaustive
division as in the use of contradictories nothing in the universe need
be omitted.

An historical illustration of dichotomy is the “Tree of Porphyry” named
after Porphyrius, a Neo-Platonic philosopher of the third century.

                        _Tree of Porphyry._

                            Substance.
                               /  \
                              /    \
                      Corporeal    Incorporeal
                              \
                               \
                               Body
                               /  \
                              /    \
                        Animate    Inanimate
                              \
                               \
                           Living Being
                               /  \
                              /    \
                       Sensible    Insensible
                              \
                               \
                              Animal
                               /  \
                              /    \
                       Rational    Irrational
                               \
                                \
                                Man
                               / | \
                              /  |  \
                             /   |   \
                     Socrates  Plato  Other Men

This kind of division is not altogether satisfactory as the negative
side is too indefinite. On the other hand, if both subdivisions
are made positive then there is danger of making the opposing terms
contrary rather than contradictory. This, of course, would be a
serious logical fallacy, as contrary terms admit of middle ground while
contradictory terms give no choice, it is either the one or the other.

The use of dichotomy becomes evident in situations where new
and unexpected discoveries may be made. Without disturbing the
classification the new species may be appended to the negative side
of the division. The following illustrates:

                Vertebrates
             ┌───────┴──────────┐
             │                  │
       Leptocardians    Not-leptocardians
                     ┌──────────┴───┐
                     │              │
                   Fish           Not-fish
                        ┌───────────┴────┐
                        │                │
                     Amphibians    Not-amphibians
                          ┌──────────────┴────┐
                          │                   │
                       Reptiles        Not-reptiles
                             ┌────────────────┴───┐
                             │                    │
                          Birds              Not-birds
                               ┌──────────────────┴──┐
                               │                     │
                             Mammals            Not-mammals
                                                     │
                                             _The New Species_


=6. CLASSIFICATION――COMPARED WITH DIVISION.=

_Classification is the process of grouping notions according to their
resemblances or connections._

So far as results are concerned there is no difference between
logical division and classification. Both processes may give us the
same orderly scheme of heads and subheads. The difference lies in
the process itself. Division is _deductive_ in nature as it proceeds
from the more general genus to the less general species. While
classification is _inductive_ as it groups the less general species
under the more general genus. Division differentiates unity into
multiplicity, while classification reduces multiplicity to unity. It
follows that the one is the inverse of the other. The difference in the
mode of procedure may be illustrated by using the common classification
or division of triangles. For example:

Without any knowledge of the kinds of triangles the student discovers
by examining the various shapes of many triangles that there is a
group in which _none_ of the sides are equal. For the lack of a better
name he terms these non-equilateral (scalene). Further observation
discloses another group in which two of the sides are equal. These he
names bi-equilateral (isosceles). Finally a third group is designated
as tri-equilateral (equilateral). This process is classification.
Division would consist in separating the genus triangle into the three
kinds――scalene, isosceles, equilateral.


=7. KINDS OF CLASSIFICATION――ARTIFICIAL AND NATURAL.=

_An artificial classification is one in which the grouping is made on
the basis of some arbitrary connection._ Cataloguing alphabetically the
books in a library illustrates this kind of classification. Likewise
the arrangement of the names in a directory or a telephone book. The
connecting mark being the initial letter of the title or name. The
reason why Mills and Meyers are put in the same group is that both
names happen to commence with the letter _M_.

Artificial classifications are resorted to for some special purpose,
designed by man, not by nature. Consequently artificial classifications
are sometimes called _special_ or _working_ classifications.

_A natural classification is one in which the grouping is made on the
basis of some inherent mark of resemblance._

Classifications in animal and plant life are the best illustrations
of this kind. Such classifications are suggested by nature and not by
man, and may, therefore, be called _general_ or scientific. The main
aim of natural classification is to derive general truths and arrange
knowledge so that it may be easily remembered.


=8. TWO RULES OF CLASSIFICATION.=

The rules of logical division are applicable in the making of a logical
classification. In addition to these an artificial classification
should be made to conform to the one rule: _The classification must be
appropriate to the purpose in hand._ Likewise a natural classification
should be made to conform to the rule: _Every classification should
afford opportunity for the greatest possible number of general
assertions._


=9. USE OF DIVISION AND CLASSIFICATION IN THE SCHOOL ROOM.=

It has been stated that classification and division aim at the
same result. Classification reduces multiplicity to unity while
division differentiates unity into multiplicity. In short, division is
_deductive_ while classification is _inductive_ in mode of procedure.
Therefore, classification should be used in those situations which
call for induction and division in cases where deduction is the better
method.

Speaking generally, classification should be used with small children
when the essential thing is to present the concrete facts with a view
of leading the children to discover for themselves the truths contained
therein.

With older pupils division may be used, if the purpose is to set in
order facts which are already known.


=10. TOPICAL OUTLINE.=

  LOGICAL DIVISION AND CLASSIFICATION.

  (1) Nature of Logical Division.
        Genus――species.
          Illustrations.

  (2) Logical Division Distinguished from Enumeration.
        Illustrations.

  (3) Logical Division and Partition.
        Quantitative――qualitative.
        How summarized.

  (4) Four Rules of Logical Division.
        (1) One principle――cross division.
        (2) Mutually exclusive.
        (3) Exhaustive――complete.
        (4) Immediate species.

  (5) Dichotomy.
        Contradictory terms.
        Tree of Porphyry.
        Use illustrated.

  (6) Classification Compared with Division.

  (7) Kinds. Artificial――Natural.

  (8) Two Rules of Classification.
        (1) Appropriate.
        (2) Afford many Assertions.

  (9) Use of Division and Classification.


=11. SUMMARY.=

(1) Logical division is the process of separating a proximate genus
into its co-ordinate species.

(2) The first subdivision in a logical division gives class terms,
while the first subdivision in an enumeration gives individual objects.

(3) Partition is the process of separating an individual thing into
its parts. These parts may be either quantitative or qualitative.

A logical division of any genus may be summarized in a series of
judgments of which a species is the subject and the genus is the
predicate.

(4) The four rules of logical division are: the division must (1) be
based on one principle, (2) have species mutually exclusive, (3) be
exhaustive and (4) proceed from proximate genus to immediate species.

A violation of the first rule gives a cross division.

Exhaustive division is easily confused with a complete or finished
division.

(5) Dichotomy is a continual division of a genus into two species which
are contradictory in nature.

An historical illustration of dichotomy is the Tree of Porphyry.

Dichotomy is of service in the field of new and unexpected discoveries.

(6) Classification is the process of grouping notions according to
their resemblances or connections.

Classification is inductive in nature, division deductive.
Classification unifies, division differentiates.

(7) An artificial classification is made on the basis of some
arbitrary connection; a natural classification, on some inherent mark
of resemblance.

(8) The rules of logical division are applicable in any classification.
In addition to these a classification should (1) be appropriate and
(2) afford opportunity for the greatest possible number of assertions.

(9) Classification should be the mode of procedure in the lower grades,
division in the higher grades.


=12. REVIEW QUESTIONS.=

  (1) Define and illustrate logical division.

  (2) What is the meaning of proximate genus?

  (3) How does logical division differ from enumeration? Illustrate.

  (4) Distinguish between logical division, and physical division or
      partition.

  (5) Illustrate a quantitative partition; a qualitative partition.

  (6) Illustrate how a logical division may be summarized in the form
      of a series of judgments.

  (7) State and explain the rules of logical division.

  (8) State the rules violated in the following divisions, explaining
      in full:

                      { Primary
                      { Secondary
        (1) Education { Collegiate
                      { Technical
                      { Scientific
                      { Professional

                      { Infancy
        (2) Life      { Childhood
                      { Youth
                      { Old age

                      { Caucasian
                      { Ethiopian
        (3) Man       { Malay
                      { Mongolian
                      { American

                      { Cement
                      { Frame
        (4) Buildings { Stone
                      { Dwellings
                      { Barns
                      { Churches

  (9) Show the difference between contradictory and opposite terms.

 (10) Define dichotomy.

 (11) Illustrate the Tree of Porphyry and indicate its use to
      scientists.

 (12) Illustrate the difference between classification and division.

 (13) Why should classification be the mode of procedure when dealing
      with immature minds?

 (14) Illustrate the difference between an artificial and a natural
      classification.

 (15) State and explain the two rules of classification.

 (16) Show which of the following divisions are logical and which are
      not:
        (1) The manifestations of the mind into knowing, thinking and
            feeling.
        (2) Books into mathematical and non-mathematical.
        (3) Students into those who are industrious, athletic and
            shiftless.
        (4) Flowers into roses, carnations and lilies.
        (5) Planets into those which are larger than the earth and
            those which are smaller.


=13. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Analyze by mathematical partition the terms world, plant, book.

  (2) Analyze by logical partition the terms granite, water, air.

  (3) What rule is violated if the logical division is applied to the
      genus rather than the proximate genus?

  (4) Divide logically the following terms: school, religion, book,
      vegetable, life.

  (5) “Each new subdivision may adopt a new principle of division.”
      Illustrate this.

  (6) Explain and illustrate the meaning of the terms super-ordinate,
      sub-ordinate and co-ordinate.

  (7) Define and illustrate metaphysical division and verbal division.

  (8) Give a definition of an isosceles triangle which will make
      logical the division of triangles into scalene, isosceles and
      equilateral.

  (9) “The evolution of all truth develops progressively through three
      stages.” “The first is the thesis; the second is the antithesis;
      the third is the synthesis.” Explain this in terms of trichotomy.

 (10) Illustrate the difference between a complete division and an
      exhaustive division.

 (11) Show in what ways, if any, the following divisions violate the
      rules of logical division.


                        { 1 Memory (History)
        Human Learning  { 2 Imagination (Poetry)
          (by Bacon)    { 3 Reason (Philosophy)
                        {     or (the Sciences)

                        { 1 Mathematics
                        { 2 Astronomy
        Sciences        { 3 Physics
          (by Comte)    { 4 Chemistry
                        { 5 Biology
                        { 6 Sociology
                        { 7 Morals



                              CHAPTER 8.

                         LOGICAL PROPOSITIONS


=1. THE NATURE OF LOGICAL PROPOSITIONS.=

Judging has been defined as the process of conjoining or disjoining
notions. This may be put in another way: “Judging is the process of
asserting or denying the agreement between two notions.” The product
of the act of judging is a judgment and when judgments are put in
word-form such expressions are called logical propositions.

Definition: _A logical proposition is a judgment expressed in words._
Just as percept and concept notions are expressed by means of logical
terms so judgment notions may be expressed by logical propositions.

To illustrate: The terms _the squirrel_ and _cracking a nut_ express
two notions, and when an agreement between them is asserted and the
product is expressed in word form, then such an expression becomes the
logical proposition, “The squirrel is cracking a nut.”

The following being _expressed_ judgments are logical propositions:

  (1) All men are mortal.

  (2) Some men are wise.

  (3) No men are immortal.

  (4) Some men are not wise.

  (5) No sane person is a lover of vice.

  (6) Some good orators are not good statesmen.

  (7) Every man is fallible.

  (8) If it rains, I shall not go.

  (9) He is either sane or insane.


=2. KINDS OF LOGICAL PROPOSITIONS.=

There are three kinds of logical propositions; namely, categorical,
hypothetical and disjunctive.

_A categorical proposition_ is one in which the assertion is made
unconditionally. _An hypothetical proposition_ is one in which the
assertion depends upon a condition. _A disjunctive proposition_ is
one which asserts an alternative.


  THE THREE KINDS ILLUSTRATED:

  (1) “Every dog has his day.” Categorical.

  (2) “If you do your best, success will reward you.” Hypothetical.

  (3) “He is either stupid or indolent.” Disjunctive.

  (4) “All vices are reprehensible.” Categorical.

  (5) “Either you are very talented or very industrious.” Disjunctive.

  (6) “If capital punishment does not aid society, it should be
      abolished.” Hypothetical.

  (7) “You may go provided your teacher is willing.” Hypothetical.

  (8) “No intelligent man can ignore the practice of temperance.”
      Categorical.

By studying the illustrations it will be observed that the categorical
propositions are direct, bold, assertive statements, whereas the
hypothetical are limited by conditions which make them less forceful.
In the second proposition, for example, “success will reward you,”
is limited by the condition, “If you do your best.” The disjunctive
may be regarded as categorical in form, but hypothetical in meaning,
because in such a proposition as, “He is either, stupid or indolent,”
a direct assertion is made which suggests the categorical, and yet it
may be implied that, if he is stupid then he is not indolent; this is
indicative of the hypothetical.

Some logicians classify propositions as categorical and conditional,
the conditional being subdivided into hypothetical and disjunctive. The
first classification seems preferable, however, as it conforms to the
three modes of reasoning.

The common word-signs of the categorical proposition are _all_,
_every_, _each_, _any_, _no_ and _some_, while those of the
hypothetical are _if_, _even if_, _unless_, _although_, _though_,
_provided that_, _when_, or any word or group of words denoting a
condition. The disjunctive symbols are _either――or_.


=3. THE FOUR ELEMENTS OF A CATEGORICAL PROPOSITION.=

Every categorical proposition should have four elements; namely,
the quantity sign, the logical subject, the copula and the logical
predicate. In the foregoing categorical propositions the quantity signs
are respectively, _every_, _all_ and _no_. In any case the quantity
sign is always attached to the subject and indicates its breadth or
extension. For example, in the two propositions, “All men are mortal”
and “Some men are wise,” the quantity sign _all_ makes the term _man_
much broader than does the quantity sign _some_.

The logical subject of a categorical proposition is the term of which
something is affirmed or denied, whereas the logical predicate of a
categorical proposition is the term which is affirmed or denied of
the subject. In the two propositions, “All men are mortal” and “No men
are immortal,” the term about which something is affirmed or denied is
_men_, while the terms which are affirmed and denied of the subject are
respectively _mortal_ and _immortal_. “Men” is, therefore, the logical
subject of each proposition, while “mortal” is the logical predicate
of the first and “immortal” the logical predicate of the second. The
copula is the connecting word between the logical subject and predicate
and denotes whether or not the latter is affirmed or denied of the
former. The copula is always some form of “_to be_” or its equivalent.
When the predicate is _denied_ of the subject, “_not_” may be used with
the copula and considered a part of it. To illustrate: in the logical
proposition, “Some men are not wise,” “_are not_” may be regarded as
the copula.

The four elements are indicated in the following categorical
propositions:

    _Quantity
      sign_     _Logical subject_   _Copula_  _Logical predicate_

      All       fixed stars         are       self-luminous

      No        wise man            is        going to steal

      Some      quadrupeds          are       domestic animals

      Some      glittering things   are not   gold

      Some      boys                are not   discreet

      A few     men                 are       multi-millionaires

      Every     citizen             is        duty-bound to vote

The student must ever keep in mind the fact that to be absolutely
logical all categorical propositions must be expressed in terms of
the _four elements_. However, life is too short and man is too busy to
speak always in terms of the four elements. Moreover, to be logical may
often compel an awkwardness of expression and a lack of euphony which
could hardly be tolerated. For these reasons the utterances in ordinary
conversation are frequently illogical so far as the four elements are
concerned, though not necessarily illogical in meaning. When it is
desired to test the validity of any series of statements leading up to
some generalization, it may become necessary to express the statement
in terms of the four elements. The student should gain some facility
in this, otherwise he may be readily led into fallacious reasoning.

The following statements taken at random from newspapers are given in
the original and then expressed in terms of the four elements:

      _The Original_                _In Terms of the Four Elements_

  (1) You came too late.        (1) The person is one who came too
                                    late.

  (2) I saw the swell turnout   (2) The man was one who saw the swell
      coming along.                 turnout coming along.

  (3) All of the men walked.    (3) All of the men were those who
                                    walked.

  (4) The robbers cut a hole    (4) All the robbers were the ones who
      in this floor.                cut a hole in this floor.

  (5) Some of these flew away.  (5) Some birds were those which
                                    flew away.

  (6) The rain interfered with  (6) The rain was that which interfered
      the attendance.               with the attendance.

  (7) Our habits make or        (7) All our habits are forces which
      unmake us.                    make or unmake us.

  (8) We all had a fine time.   (8) All the party were those who had
                                    a fine time.

In argumentative discourse it is often sufficient to “think the
proposition” in terms of the four elements without taking the time to
actually express it.


=4. LOGICAL AND GRAMMATICAL SUBJECT AND PREDICATE DISTINGUISHED.=

The grammatical subject is _one_ word while the logical subject is the
grammatical subject with all its modifiers except the quantity sign.
For example: in the proposition, “All white men are Caucasians,” _men_
is the grammatical subject, while _white men_ is the logical subject.
_All_ being the quantity sign simply indicates the extension of _men_
and is not a part of the logical subject.

The grammatical predicate is the verb-form together with any predicate
noun or adjective, while the logical predicate is the predicate word
or words and all its modifiers. The grammatical predicate includes
the copula, but the logical predicate never includes the copula. The
grammatical predicate does not include the object, while the logical
predicate always includes what is equivalent to the object and all
its modifiers. To illustrate: in the proposition, “Some men are wise,”
_are wise_ is the grammatical predicate, while _wise_ is the logical
predicate. And in the proposition, “He burned the red house on the
hill,” _burned_ is the grammatical predicate, while _the one who burned
the red house on the hill_ is the logical predicate.


=5. THE FOUR KINDS OF CATEGORICAL PROPOSITIONS.=

Categorical propositions are divided according to their _quantity_
into _Universal_ and _Particular_ and according to their _quality_ into
_Affirmative_ and _Negative_.

_A universal proposition is one in which the predicate refers to the
whole of the logical subject._


  ILLUSTRATIONS:

    (1) All men are mortal.

    (2) All civilized men cook their food.

    (3) No dogs are immortal.

    (4) Every man was once a boy.

Considering the first proposition, “_mortal_,” the logical predicate,
refers to the whole of the logical subject “_men_.” Similarly “_cook
their food_” refers to the whole of the term “_civilized men_”;
“_immortal_” to the whole of the term “_dogs_,” and “_once a boy_”
to the whole of the term “_man_.”

In considering the definition of a universal proposition it is
necessary to keep in mind the distinction between a logical and a
grammatical subject, as in the second proposition the logical predicate,
“cook their food,” refers to only a part of the grammatical subject,
_men_, and, therefore, the proposition might fallaciously be termed a
particular proposition rather than a universal.

_A particular proposition is one in which the predicate refers to only
a part of the logical subject._


  ILLUSTRATIONS:

  (1) Some men are wise.

  (2) Some animals are not quadrupeds.

  (3) Most elements are metals.

  (4) Many children are mischievous.

In the foregoing propositions _some_, _most_ and _many_ are quantity
signs and, therefore, must not be considered as a part of the logical
subjects. Considering the logical subjects and predicates in order,
the term _wise_ refers to only a part of the _men_ in the world,
_quadrupeds_ to only a part of the _animals_, _metals_ to only a part
of the _elements_ and _mischievous_ to only a part of the _children_.

_An affirmative proposition is one which expresses an agreement between
subject and predicate._

_A negative proposition is one which expresses a disagreement between
subject and predicate._

Affirmative propositions conjoin terms, negative propositions disjoin
terms. In the first the agreement is affirmed; in the second the
agreement is denied.


  ILLUSTRATIONS:

    None of the captives escaped. Negative.

    Some teachers are just. Affirmative.

    All trees grow towards heaven. Affirmative.

    Some people are not companionable. Negative.

    No person is above criticism. Negative.

Dividing both universal and particular propositions as to quality,
gives four kinds; namely, universal affirmative, universal negative,
particular affirmative and particular negative. No topic in logic
demands greater familiarity than these four types, as every proposition
must be reduced to one of the four before it can be used as a basis of
reasoning.

For the sake of brevity the symbols A, E, I and O are used to
designate respectively the universal affirmative, the universal
negative, the particular affirmative and the particular negative. A and
I, symbolizing the affirmative propositions, are the first and second
vowels in _Affirmo_, while E and O, symbolizing the negatives, are the
vowels in _Nego_. The common sign of the universal affirmative, or the
A proposition is _all_; of the universal negative, or E proposition
_no_; of the particular affirmative, or I proposition _some_; of the
particular negative, or O proposition _some_ with _not_ as a part of
the copula. The accompanying classification summarizes these facts,
S and P being used to symbolize the terms “_subject_” and “_predicate_.”

                                                _Illustrations_

                {             { Affirmative-A   All S is P
  Categorical   { Universal   { Negative-E      No S is P
  Propositions  {
                {             { Affirmative-I   Some S is P
                { Particular  { Negative-O      Some S is not P

Henceforth the symbols A, E, I, O will be used to designate the four
kinds of categorical propositions. The propositions have other quantity
signs aside from the ones used above. These may be summarized:

                    { A――all, every, each, any, whole.
                    { E――no, none, all-not.
  Quantity signs of { I――some, certain, most, a few, many, the
                    {     greatest part, any number.
                    { O――some - - not, few.


=6. PROPOSITIONS WHICH DO NOT CONFORM TO THE LOGICAL TYPE.=

It has been observed that all expressed judgments must be reduced to
one of the four logical types A, E, I or O, before they can be used
argumentatively. Logic insists upon definiteness and clearness――there
must be no ambiguity, no opportunity for a wrong interpretation. From
this viewpoint the four types fulfill every requirement. Their meaning
cannot be misunderstood. To any one with normal intelligence their
significance may be made perfectly clear. Any argument when once put in
terms of the four types may be spelled out with mathematical precision.
In consequence it is of prime importance that the four types not only
be well understood, but that a certain facility be gained in reducing
ordinary conversation to some _one_ of these types.

(1) Indefinite and Elliptical Propositions.

It is known that every logical proposition must be expressed in terms
of the four elements――_quantity sign_, _logical subject_, _copula_ and
_logical predicate_, consequently the four types A, E, I and O which
epitomize every form of logical proposition, are expressed in terms
of these four elements. But in common conversation often the quantity
sign, as well as the copula, is omitted. See section 3.

Propositions without the quantity sign are called _indefinite_,
while those with the suppressed copula may be termed _elliptical_
propositions. Both may be made logical as the attending illustrations
will indicate:

          _Illogical_                           _Logical_

  _Indefinite_
  Men are fighting animals.         _All_ men are fighting animals.
                                      (A)

  Lilies are not roses.             _No_ lilies are roses. (E)

  Good is the object of moral       _All_ good is the object of moral
    approbation.                      approbation. (A)

  Perfect happiness is              _In all cases_ perfect happiness
    impossible.                       is impossible. (A)

  _Elliptical_
  Fashion rules the world.          _All_ fashions _are_ ruling the
                                      world. (A)

  Trees grow.                       _All_ trees _are plants which_
                                      grow. (A)

  Children play.                    _All_ children _are_ playful. (A)

  Some men cheat.                   Some men _are persons who_ cheat.
                                      (I)

Here it is noted that the logical form of some propositions is not
always the most forceful. Often the logical form gives an awkward
construction and should be resorted to only for purposes of logical
argument.

The reduction of either kind to the logical form must be determined
by the meaning of the proposition. As a usual thing the indefinite is
_universal_ (either an A or an E) in meaning, while the problem of the
elliptical is to give it in terms of the _copula_, expressed with as
little awkwardness as possible.

General truths, because attended with no quantity sign, might be
classed as indefinite propositions, though their universality is so
apparent that they may be unhesitatingly classed as universals.


  ILLUSTRATIONS:

    “Things equal to the same thing are equal to each other.”

    “Trees grow in direct opposition to gravity.”

    “Honesty is the best policy.”

    “A stitch in time saves nine.”

Because the indefinite proposition is so frequently of a general
nature, it is sometimes classed as _general_ rather than _indefinite_.

Sir William Hamilton would class the indefinite as an _indesignate
proposition_.

(2) Grammatical Sentences.

The grammarian divides sentences into five kinds; namely, declarative,
interrogative, imperative, optative, exclamatory. But logic recognizes
only the declarative, as it has already been seen that the four logical
types are declarative in nature. A logical proposition, then, is always
a sentence, but all sentences are not logical propositions. The four
kinds of sentences which are not logical propositions may be usually
reduced to one of the four types as the attending illustrations will
indicate:

  _Illogical_                       _Logical_

  Interrogative. Do men have        The question is asked, Do men have
    the power of reason?              the power of reason?[7] (A)

  Imperative. “Thou shalt not       All men are commanded not to
    steal.”                           steal, or you are one who should
                                      not steal. (E)

  Optative. “I would I had a        I am one who desires a million
    million.”                         dollars. (A)

  Exclamatory. “Oh, how you         You are one who frightened me. (A)
    frightened me!”

(3) Individual Propositions.

_An individual proposition is one which has a singular subject_; e. g.,
_Abraham Lincoln_ was an honest man. _Peter the Great_ was Russia’s
greatest ruler. _The maple tree in my yard_ is dying of old age. These
propositions, having a singular term as subject, are individual or
singular in nature. As the predicate refers to the whole of the logical
subject, individual propositions are classed as _universal_.

(4) Plurative Propositions.

Plurative propositions are those introduced by “most,” “few,” “a few,”
or equivalent quantity signs. For example, “_Most_ birds are useful to
man”; “_Few_ men know how to live”; “_A few_ of the prisoners escaped,”
are plurative propositions. “Most” means more than half, while “few”
and “a few” mean less than half. In either case the proposition is
particular. Stated logically, the illustrative propositions would take
the form of “Some birds are useful to man”; “Some men do not know how
to live”; “Some of the prisoners escaped.”

The reader will observe the difference in significance between _few_
and _a few_. The former is negative in character and when introducing
a proposition makes it a particular negative (O). The latter always
introduces a particular affirmative (I).

(5) Partitive Propositions.

Partitive propositions are particulars which imply a complementary
opposite. These arise through the ambiguous use of _all-not_, _some_
and _few_. _All-not_ may sometimes be interpreted as _not all_ and
sometimes as _no_. To illustrate: The proposition, “All men are
not mortal,” is distinctly a universal negative or an E, while the
proposition, “All that glitters is not gold,” is a particular negative
or an O. The logical form of the first is, “No men are mortal,” and
of the second, “Some glittering things are not gold.” When used in the
“not-all” sense, the proposition is partitive because if the O-meaning
is intended the I is implied. For example, “All that glitters is not
gold,” is partitive because the statement implies that some glittering
things _are_ gold (I) as well as the complement, “Some glittering
things _are not_ gold” (O). A knowledge of both the affirmative and
negative aspects is taken for granted in the statement of either the
one or the other.

“All-not,” then, is negative in any case, but universal when it
means _no_ and particular when it means _not all_. Any proposition is
partitive in nature when the quantity sign is _not all_, or _all-not_
interpreted as the equivalent of _not all_.

It may be observed here that _all_ has two distinct uses. First, it
may be used in a collective sense; second, in a distributive sense. For
example: _All_ is used in the _collective_ sense in such propositions
as, “All the members of the football team weighed exactly one ton,” or
“All the angles of the triangle are equal to two right angles.” Using
_all_ in the distributive sense would make true these: “All the members
of the football team weigh more than 140 pounds”; “All the angles of
a triangle are less than two right angles.” _All_ is used collectively
when reference is made to an aggregate, but distributively when
reference is made to each.

The quantity sign _some_ is likewise ambiguous, as it may mean (1) some
only――some, but not all, or (2) some at least――some, it may be all or
not all. When “some” is used as the quantity sign of any particular
proposition which has been accepted as logical, the second meaning,
“some at least,” is always implied. This interpretation of “_some_”
will be explained more in detail in a succeeding section.

When _some_ is used in the sense of _some only_, the partitive nature
of the proposition is apparent, as both I and O are implied. For
example, with reference to the human family, to say that “some only are
wise” necessitates an investigation, which leads to the discovery that
some are wise, while others are not wise. If the proposition be an I,
then its complementary O is implied, or if it be an O, the I is implied.

_Few_ given as a sign of a plurative proposition also serves as a sign
of the partitive. The plurative aspect is prominent when it is said
that “Few men can be millionaires” and emphasis is placed upon the
meaning that “Most men cannot be millionaires.” But when emphasis is
given to “few,” as meaning _few only_ rather than the _most are not_,
then the I and the O are both implied; e. g., _Some men become
millionaires, but the most do not_.

To put it in a word, “all-not,” “some” and “few” introduce
partitive propositions when the meaning implies both an I and an O.
When treating such in logic the meaning which seems to be given the
greater prominence must be accepted. Surely in the statement, “All that
glitters is not gold,” the O-interpretation is the one intended; namely,
“Some things which glitter are not gold.”


  ILLUSTRATIONS:

  (1) “_All_ men are not honest.”

  (2) “_Few_ men live to be a hundred.”

  (3) “_Some_ men are consistent.”

The first proposition with the emphasis placed upon _all_ suggesting
that _some men are not honest_, is the intended proposition while _some
men are honest_ is the implied. In reducing it to the logical form the
intended proposition is the one which should be used.

With the emphasis upon _few_ and _some_, the second and third
propositions may be interpreted as follows: (2) Intended proposition,
_Some men do not live to be a hundred_. Implied proposition, _Some
men do live to be a hundred_. (3) Intended proposition, _Some men are
consistent_. Implied proposition, _Some men are not consistent_.

(6) Exceptive Propositions.

These are introduced by such signs as _all except_, _all but_,
_all save_. To wit: (1) “_All except_ James and John may be excused”;
(2) “_All but_ a few of the culprits have been arrested”; (3) “_All_
birds _save_ the English sparrow are serviceable to man” are exceptive
propositions.

Exceptive propositions are universal when the exceptions are mentioned.
Universal propositions necessitate a subject more or less definite, as
the predicate of such must refer to the whole of a _definite_ subject.
It follows that in exceptive statements definiteness is secured when
the exceptions are mentioned, therefore it becomes clear how all
such propositions must be universal. Of the illustrations, the first
and third propositions are universal. Any exceptive proposition is
particular when the exceptions are referred to in general terms or
when the subject is followed by _et cetera_. The second illustrative
proposition is particular.

(7) Exclusive Propositions.

Of all propositions which vary from the logical form the exclusive
is the most misleading. Exclusives are accompanied by such words as
“only,” “alone,” “none but,” and “except.” Their peculiarity rests
in the fact that reference is made to the _whole_ of the predicate,
but only to a _part_ of the subject. For example, in the exclusive
proposition, “Only elements are metals,” _metals_ is referred to as
a whole while _elements_ is considered only in part. The true meaning
is “Some elements are all metals,” or to put it in logical form, “All
metals are elements.” _The easiest way to deal with an exclusive is
to interchange subject and predicate (convert simply) and call the
proposition an A._


  PROCESS ILLUSTRATED:

  _Exclusive Proposition_           _Reduced to Logical Form_

  1. None but high school           All who enter Training School
      graduates may enter             must be high school graduates.
      Training School.

  2. Only first-class               All parlor cars are for
      passengers are allowed in       first-class passengers.
      parlor cars.

  3. Residents alone are            All who are licensed to teach are
      licensed to teach.              residents.

  4. No admittance except on        All who have business may be
      business.                       admitted.

  5. Only bad men are not-wise.     All who are not-wise are bad men.

  6. Only some men are wise.        All who are wise are men.

It is claimed by good authority that the real nature of the exclusive
is best expressed by _negating_ the subject and calling the proposition
an E; e. g., exclusive: “Only elements are metals”; logical form: “No
not-elements are metals” (E). In a succeeding chapter it is explained
how an E admits of first simple conversion and then obversion. The
following illustrate these two processes:

    _Original E_: “No not-elements are metals.”

    _Simple conversion_: “No metals are not-elements.”

    _Obversion_: “All metals are elements.”

From this it may be seen that the statement, “The easiest way to deal
with an exclusive is to interchange subject and predicate and call the
proposition an A,” is substantially correct.

(8) Inverted Propositions.

The poet often employs the inverted proposition, illustrated by
the following: “Blessed are the merciful;” “Great is this man of
war.” An interchanging of subject and predicate makes these poetical
constructions logical; e. g., “All the merciful are blessed;” “This man
of war is great.”

NOTE.――The student should not be misled by the relative clause. Often
it may be interpreted as a part of the predicate rather than the
subject. To wit: “No man is a friend who betrays a confidence”; clearly
the logical subject is _no man who betrays a confidence_.


=7. PROPOSITIONS WHICH ARE NOT NECESSARILY ILLOGICAL.=

(1) _Analytic and Synthetic Propositions._

_An analytic proposition is one in which the predicate gives
information already implied in the subject._ Thus, “_Fire burns_,”
“_Water is wet_,” “_A triangle has three angles_” are analytic
propositions because the predicates do not give added information
to one who has any conception of the subjects. Because the attribute
mentioned by the predicate is an essential one, analytic propositions
are sometimes termed _essential_ propositions. Other names for the same
kind of proposition are _verbal_ and _explicative_.

_A synthetic proposition is one in which the predicate gives
information not necessarily implied in the subject. “Fire protects men
from the wild animal.” “A cubic foot of water weighs 62½ lbs.” “The sum
of the interior angles of a triangle is equal to two right angles.”_
These are synthetic because a common conception of the meaning of
the subject would not need to include the information given by the
predicate. Other names for synthetic propositions are _accidental_,
_real_ and _ampliative_.

The distinction between analytic and synthetic propositions is not
so clear as would on first thought appear. “_Fire burns_” might give
added information to the child or savage who knows only of the light
emitted by fire. To them, then, the proposition would be synthetic. The
distinction must be based upon the assumption that the same words mean
about the same thing to people in general.

This analytic-synthetic division of propositions finds a significance
in the domain of philosophy. To the logician the distinction is of
slight importance save in the so-called verbal disputes, viz.: disputes
which turn on the meaning of words.

(2) _Modal and Pure Propositions._

_A modal proposition states the mode or manner in which the predicate
belongs to the subject._ The signs of modal propositions are the
adverbs of time, place, degree, manner. Illustrations: “James is
walking _rapidly_.” “Honesty is _always_ the best policy.” “Aristotle
was _probably_ the greatest thinker of ancient times.”

_A pure proposition simply states that the predicate belongs, or
does not belong, to the subject._ Illustrations: “James is walking.”
“Honesty is the best policy.” “Aristotle was the greatest thinker of
ancient times.”

Some logicians refer to modal propositions as being such as indicate
_degrees_ of belief. Such words as “probably,” “certainly,” etc., would
indicate their modality.

As logic has to do with the pure proposition and not the modal, the
difference of opinion is of little import.

(3) _Truistic Propositions._

_A truistic proposition is one in which the predicate repeats the words
and the meaning of the subject._ Illustrations: “A man is a man,” “A
beast is a beast,” “A traitor is a traitor,” “What I have done I have
done.”

The truistic proposition is of little importance except in cases
where the subject is used extensionally while the predicate is used
intensionally. In the illustration, “A man is a man,” the subject
merely stands for a member of the man family, while the predicate may
indicate certain manly qualities. Against such ambiguities the logician
must be on guard.


=8. THE RELATION BETWEEN SUBJECT AND PREDICATE.=

In Chapter 5 the extension and intension of terms was explained. The
student recalls, for instance, that the term “man” may be used to
denote objects, as “white man,” “black man,” “red man,” etc. In this
sense the term “man” is used _extensionally_. But when made to stand
for the attributes “rationality,” “power of speech,” etc., the term
“man” is used intensionally.

In considering the relation between subject and predicate it is
customary to employ the terms in an extensional sense only, since such
a restriction serves the purpose of syllogistic reasoning and
conversion.

Let us, then, give attention to the _extension_ of the subject and
predicate of the categorical propositions A, E, I, O.

(1) _The Universal Affirmative or A Proposition._

_All S is P symbolizes the A proposition._ This may be interpreted as
meaning that all of the subject belongs to a part of the predicate, or
that all of the subject belongs to all of the predicate. The first
interpretation is the usual one and may be illustrated by the following
propositions:

  1. “All men are mortal.”

  2. “All trees grow.”

  3. “All metals are elements.”

It is obvious that the subjects of these propositions include every
specimen of the particular class mentioned. For example: The subject
_all men_ includes every specimen of the human family; _all trees_
includes every object of that class; _all metals_ covers everything
which the scientist classes as such. In the three propositions, then,
reference is made to the _whole_ subject but to only a _part_ of the
predicate, as other beings beside men, such as the horse, are mortal;
and other plants aside from trees, such as the sun flower, grow; other
substances, namely oxygen, are elements.

For the sake of making the logical meaning of the four propositions
clearer, recourse may be made to Euler’s diagrams, so named because
the Swiss mathematician and logician, Leonhard Euler, first used them.

The first illustration of the A proposition, “All men are mortal,”
may be represented by two circles, a larger circle standing for the
predicate, _mortal_, and a smaller circle entirely inside the larger
representing the subject, _men_. Thus:

  Illustration:   FIG. 1.

It is evident that all of the smaller circle belongs to the larger.
This diagram will then fit any proposition where it may be said that
all of the subject belongs to a part of the predicate, or which may
be symbolized as “All S is some P.” (All the subject is some of the
predicate.)

The student knows that circles are plane surfaces and when such a
statement as “All men are mortal” is given, reference is made to only
that part of the “mortal” circle which is _directly underneath_ the
“men” circle. Nothing has been said relative to the remaining part of
the “mortal” circle.

“_A_” propositions which may be interpreted as meaning “All S is
all P” are called co-extensive A’s because the subject and predicate
are exactly equal in extension. Such propositions are best illustrated
by definitions; e. g.:

  1. “A man is a rational biped.”

  2. “A trigon is a polygon of three sides.”

  3. “Teaching is the art of occasioning those activities which result
     in knowledge, power and skill.”

To represent the meaning of the co-extensive A by the Euler diagram,
two circles of the same size may be drawn, one coinciding at every
point with the other. If the first circle is drawn heavily in black and
the second dotted in red, it will make clear to the eye that there are
two circles.

(2) _The Universal Negative or E Proposition._

_“No S is P” best symbolizes the E proposition_, though sometimes the
universal negative is written “All S is not P.” This latter form, as
has been explained, is ambiguous and therefore illogical.

“No S is P” surely means that no part of the subject belongs to any
part of the predicate and no part of the predicate belongs to any part
of the subject. The subject and predicate are mutually exclusive.

The following illustrate the E proposition:

  1. “No man is immortal.”

  2. “No true teacher works for money.”

  3. “No thorough student can remain unwise.”

 The E proposition may be represented by two circles, the
one entirely without the other as in Fig. 2:

  Illustration:   FIG. 2.

(3) _The Particular Affirmative or I Proposition._

_This may be symbolized as “Some S is P,”_ and considered as meaning
that a _part_ of the subject belongs to a _part_ of the predicate. It
has already been noted that “some” is ambiguous and that its logical
signification is “some at least_.” (It may be all or it may not be
all.) For example, the only logical interpretation which can be placed
on “Some men are wise” is, that the investigation has resulted in
finding only a _part_ of the man family wise. Whether or not all are
wise is unknown as the entire field has not received attention. In no
case can it be assumed that all the others are _not_ wise.

The I proposition illustrated:

  1. “Some men are wise.”

  2. “Some animals are vertebrates.”

  3. “Some teachers are inspiring.”

The meaning of the I proposition may be represented by two circles
intersecting each other:

  Illustration:   FIG. 3.

The significant feature of the diagram is the shaded part which
represents a part of the “men” circle as belonging to a part of the
“wise” circle. The unshaded part of each circle is the unknown field.

(4) _The Particular Negative or O Proposition._

_The common symbolization of the O is “Some S is not P.”_ Put in
statement form: Some of the subject is excluded from the whole of the
predicate. Here, as in the I, the same logical import must be given to
_some_; e. g., in the proposition, “Some men are not wise,” our
knowledge is confined to the group who are not wise. Whether or not the
others are wise or not-wise is unknown.

Illustrations of the O proposition:

  1. “Some men are not wise.”

  2. “Some laws are not just.”

  3. “Some novels are not helpful.”

The significance of the O proposition may be shown by two intersecting
circles as in Fig. 4:

  Illustration:   FIG. 4.

A similar diagram represents the I proposition, the only difference
being in the _part shaded_. In the O proposition the investigated field
is all of the “men” circle _outside_ of the “wise” circle, while in the
I proposition the known field is that part of the “men” circle _inside_
the “wise” circle.

In comparing the four diagrams the student will note that the
affirmative propositions are _inclusive_, while the negative
propositions are _exclusive_.

(5) _The Distribution of Subject and Predicate._

_A term is said to be distributed when it is referred to as a definite
whole._

In the proposition, “All men are mortal,” the subject _all men_ is
considered as a whole. “_All_ men” stands for every specimen of the
human race; not a single one has been left out. Again, _the whole_ is
definite; any one, if he were given the time and opportunity, could
ascertain by actual count just how many “all men” represented.

It should be observed that if the word _definite_ is not incorporated
in the definition of a distributed term, there is afforded an
opportunity for error. The attending illustrations will make this clear:

  1. “All the students except John and James are dismissed.”

  2. “All the students except John, James, etc., are dismissed.”

The subject of the first proposition is distributed, while the subject
of the second is undistributed. Reasons: The first subject, “All the
students except John and James,” is referred to as a whole and that
whole is definite, therefore, it is distributed; the second subject,
“All the students except John, James, etc.,” is referred to as a whole,
but as the whole is not definite, the term is not distributed. Because
_all_ is the quantity sign of the second subject the casual observer
might easily be misled in designating it as a distributed term.

Here it may be well to explain that when reference is made to subject
or predicate the _logical_ subject or predicate is meant. Unless this
is constantly kept in mind error results; e. g., in the proposition,
“All white men are Caucasians,” the logical subject is “white men,”
not “men.” If the subject were “men,” it would be undistributed, as
the whole of the man family is not considered, but the actual subject,
being “white men,” _is_ distributed because the predicate refers to
_all_ white men.

Recurring to the illustration, “All men are mortal,” we have concluded
that the subject “all men” is distributed. The predicate, “mortal,”
however, is undistributed, as reference is made to it only in part;
i. e., there are other beings aside from men that are mortal, such as
“trees,” “horses,” “dogs,” etc. _In all A propositions of the type of
“all men are mortal,” the subject is distributed while the predicate
is undistributed._ This relation is clearly shown by the diagrammatical
illustration, Fig. 1. Here _all_ of the “men” circle is identical with
only a part of the “mortal” circle. In other words, the _whole_ of the
“men” circle is considered, while reference is made to only a _part_ of
the “mortal” circle.

_In the case of the co-extensive A both subject and predicate are
distributed._ Relative to the co-extensive “All men are rational
animals,” it could likewise be said that “all rational animals are
men,” or that “all men are all of the rational animals.” Reference is
thus made to _all_ of the definite predicate as well as to all of the
definite subject.

In the E propositions, such as “No men are immortal,” the whole of the
subject is excluded from the whole of the predicate. This makes evident
the fact that _both terms are distributed_. See Fig. 2.

The I proposition, such as “Some men are wise,” concerns itself with
only a part of the subject and only a part of the predicate,
_consequently neither subject nor predicate is distributed_. This
relation is verified by the representation, Fig. 3.

_In the O proposition the subject is undistributed, while the predicate
is distributed._ For example, in the proposition, “Some men are not
wise,” “some men” would indicate that only a part of the logical
subject is under consideration. But the predicate is distributed
because “some men” is denied of the _whole_ of the predicate, “wise.”
This may become clear by studying Fig. 4. Here all of the shaded part
which stands for the subject, “some men,” is excluded from the whole
of the “wise” circle. But all of the shaded part is only a part of the
entire “men” circle, consequently the subject which the shaded part
represents (some men) is undistributed. The predicate, “wise,” however,
is distributed, as the subject is excluded from every part of it. It is
well to remember that _not_, when used with the copula, distributes the
predicate which follows it.

If the student is to succeed in testing the value of arguments, he must
ever have “at the tip of his tongue” his knowledge of the distribution
of the terms of the four logical propositions. With this in view the
following schemes are offered:

                                  I.

                          _Subject_      _Predicate_

                    A   distributed     undistributed

                    E   distributed     distributed

                    I   undistributed   undistributed

                    O   undistributed   distributed

                                  II.

                    A   distributed     undistributed

                    O   undistributed   distributed

                    E   distributed     distributed

                    I   undistributed   undistributed

                                 III.

                    A   All S is P
                           └─┘
                    E   No S is P
                          └─┘  └─┘
                    I   Some S is P

                    O   Some S is not P
                                     └─┘

Referring to scheme II it may be observed that A and O contradict each
other; i. e., where A is distributed O is undistributed and vice versa.
A similar relation exists between E and I.

In scheme III the bracket └─┘ under the symbol indicates the term which
is distributed.

IV. As a fourth scheme a “key word” might be adopted. Any of these
three might be used: (1) saepeo, or (2) asebinop, or (3) uaesneop.
The significance of “saepeo” is this: “_s_” stands for _subject
distributed_, “_p_” for _predicate distributed_, “_a_” “_e_” “_o_”
for the _logical propositions_ where any distribution occurs. Putting
the letters together gives this: subject distributed of propositions
A and E, predicate distributed of propositions E and O.

Similarly, “asebinop” stands for this: “_as_,” _a_ distributes its
_s_ubject; “_eb_” _e_ distributes _b_oth; “_in_,” _i_ distributes
_n_either; “_op_,” _o_ distributes the _p_redicate.

In the coined word “_uaesneop_” appear six letters which compose
“saepeo,” and the letters have the same significance. The two
additional letters, u and n, stand for universal and negative. The
interpretation of the entire word, therefore, is this: “_uaes_,”
the _u_niversals _a_ and _e_ distribute their _s_ubjects; _neop_,
the _n_egatives _e_ and _o_ distribute their _p_ redicates.

It seems to me that the last word is the most helpful as it emphasizes
the two facts which are the most used; namely, (1) Only the universals
distribute their subjects; (2) Only the negatives distribute their
predicates.

If the student will visualize “_uaesneop_” so thoroughly as never
to forget it, he will not experience difficulty in determining the
distribution of the terms of the four logical propositions.


=9. OUTLINE.=

  LOGICAL PROPOSITIONS.

  (1) The nature of logical propositions.

  (2) Kinds of logical propositions.
        Categorical
        Hypothetical
        Disjunctive

  (3) The four elements of a categorical proposition.

  (4) Logical and grammatical subject and predicate distinguished.

  (5) The four kinds of categorical propositions.
        Universal affirmative   A
        Universal negative      E
        Particular affirmative  I
        Particular negative     O

  (6) Propositions which do not conform to the logical type.
        Indefinite and elliptical
        Grammatical sentences
        Individual
        Plurative
        Partitive
        Exceptive
        Exclusive
        Inverted

  (7) Propositions not necessarily illogical.
        Analytic and synthetic
        Modal and pure
        Truistic

  (8) The relation between subject and predicate of the four logical
      propositions.
        Euler’s diagrams
        Distribution of subject and predicate
        Uaesneop
        Asebinop
        Saepeo


=10. SUMMARY.=

(1) A logical proposition is a judgment expressed in words.

(2) The three kinds of logical propositions are categorical,
hypothetical, disjunctive.

  A categorical proposition is one in which the assertion is made
  unconditionally.

  A hypothetical proposition is one in which the assertion depends
  upon a condition.

  A disjunctive proposition is one which asserts an alternative.

The most common word-signs of the categorical proposition are “all,”
“no,” “some” and “some-not,” of the hypothetical, “if” and of the
disjunctive, “either-or.”

(3) Every logical categorical proposition has the four elements:
quantity sign, subject, copula and predicate.

The quantity sign indicates the extension of the proposition; the
logical subject is that of which something is affirmed or denied;
the logical predicate is the term which is affirmed or denied of the
subject; the copula is always some form of “to be” and is used to
connect subject and predicate. “_Not_” is sometimes used with the
copula.

The statements of ordinary conversation are usually not expressed in
terms of the four elements, but must be, before they can be used in
testing arguments.

(4) One word usually constitutes the grammatical subject while
a word with all its modifiers goes to make up the logical subject.
The verb with any predicate word is the grammatical predicate. The
logical predicate is all which follows the copula――it may include the
predicate-word and all its modifiers as well as the modified object.

(5) Categorical propositions are divided into four kinds; universal
affirmative (A), universal negative (E), particular affirmative (I),
particular negative (O). For the sake of brevity these four are
respectively denoted by the vowels A, E, I, O.

  An A proposition is one in which the predicate affirms something of
  _all_ of the logical subject.

  An E proposition is one in which the predicate denies something of
  _all_ of the logical subject.

  An I proposition is one in which the predicate affirms something of
  a part of the logical subject.

  An O proposition is one in which the predicate denies something of
  a part of the logical subject.

Every proposition must be reduced to one of the four types before it
can be used as a basis of argumentation.

It is incumbent on the student to recognize these four types with
precision and accuracy.

(6) There are a few proposition types which are recognized as being
illogical in form. These may be defined as follows:

  (1) An indefinite proposition is one without the quantity sign. It
      usually may be classed as universal.

  (2) An elliptical proposition is one in which the copula is
      suppressed.

  (3) An individual proposition is one which has a singular subject.
      It is universal in content.

  (4) Plurative propositions are those introduced by “most,” “a few”
      or some equivalent quantity sign. These are particular in meaning.

  (5) Partitive propositions are particulars which imply a
      complementary opposite. These arise through the ambiguous use of
      “all-not,” “some” and “few.”

“All-not” sometimes means “_no_,” while at other times it may mean
“_not-all_.” If the quantity sign means the latter, then it introduces
a partitive proposition.

“_Some_” may mean “_some only_,” or “_some at least_.” The latter is
the logical meaning. The former interpretation makes the proposition
partitive. When “few” means “few only,” it is partitive in nature.

  (6) Exceptive propositions are those introduced by such signs as
      “all except,” “all but,” “all save,” etc. They are universal
      only when the exceptions are mentioned.

  (7) Exclusive propositions are those introduced by such words as
      “only,” “alone,” “none but” and “except.” In an exclusive the
      predicate and not the subject is distributed. Consequently
      the easiest way to make an exclusive logical is to interchange
      subject and predicate and call it an A.

  (8) An inverted proposition is one where the predicate precedes the
      subject. Interchanging them gives the logical form.

Of the grammatical sentences only the declarative is logical.

The relative clause, though out of place, must be used with the word it
modifies.

(7) There are other propositions, though not illogical, to which the
logician usually gives some attention. These may be defined as follows:

  (1) An analytical proposition is one in which the predicate gives
      information already implied in the subject.

  (2) A synthetic proposition is one in which the predicate gives
      information not implied in the subject.

  (3) A modal proposition is one which states the manner in which the
      predicate belongs to the subject. The adverbs of time, place,
      degree and manner are the signs of the modal proposition.

  (4) A pure proposition simply states that the predicate belongs or
      does not belong to the subject.

  (5) A truistic or tautologous proposition is one in which the
      predicate repeats the words and meaning of the subject.

(8) In considering the relation which may exist between subject and
predicate, the two terms are employed in extension only, as this use
best serves the interests of inference.

The extensional relation between subject and predicate of the four
logical propositions may be stated as follows:

  Ordinary A――All of the subject belongs to a part of the predicate.

  Co-extensive A――All of the subject belongs to all of the predicate.

  E――None of the subject belongs to any part of the predicate.

  I――Some of the subject belongs to some of the predicate.

  O――Some of the subject is excluded from the whole of the predicate.

In general it may be said that the affirmative propositions are
_in_clusive while the negatives are _ex_clusive.

A term is said to be distributed when it is referred to as a definite
whole.

“A” distributes the logical subject only, “E” both logical subject and
logical predicate, “I” neither logical subject nor logical predicate,
“O” the logical predicate only. The co-extensive “A” distributes both
subject and predicate.

It is essential that the student know by heart the distribution of
the terms of the logical propositions. Some keyword like _uaesneop_
may be used as an aid to the memory. This means the _u_niversals _A_
and _E_ distribute their _s_ubjects, while the _n_egatives _E_ and _O_
distribute their _p_redicates.


=11. ILLUSTRATIVE EXERCISES.=

(1a) Examine the following list of propositions with a view to
classifying them as “A’s,” “E’s,” “I’s” or “O’s.”

  _E_ 1. “None of the inmates voted.”

  _A_ 2. “Benj. Franklin was the best educated American.”

  _I_ 3. “Some doctors deem it right to lie to their patients.”

  _A_ 4. “All earnest teachers need to observe the teaching of others.”

  _I_ 5. “Some politicians are honest.”

  _A_ 6. “Fools rush in where angels fear to tread.”

  _O_ 7. “Some proverbs are not true to life.”

  _E_ 8. “No man should infringe upon the rights of others.”

I recall that an affirmative proposition in which the predicate refers
to the whole of the subject is an A, while one where the predicate
refers to only a part of the subject is an I. Further, a negative
proposition where the predicate refers to the whole of the subject is
an E, while one in which the predicate refers to only a part of the
subject is an O. With these facts in mind, I classify the propositions
as indicated.

(1b) In a similar manner classify as to quantity and quality the
following:

  (1) “All worthy workers grow to look like their work.”

  (2) “Every dog has his day.”

  (3) “Some of the presidents were not popular.”

  (4) “No unskilled laborer can afford to own an automobile.”

  (5) “Some of the ‘election prophets’ were sadly mistaken.”

(2a) Classify the following propositions and make the illogical,
logical:

  (1) “Only first-class passengers may ride in parlor cars.”

  (2) “Haste makes waste.”

  (3) “Few men know how to act under stress.”

  (4) “All which seems to ring true is not true.”

  (5) “Members alone are admitted.”

  (6) “None but men of integrity need apply.”

  (7) “Horses trot.”

  (8) “Blessed are they which are persecuted for righteousness sake.”

The first proposition is an exclusive and may be made logical by
converting and calling it an A, viz.: “All who ride in parlor cars are
first-class passengers.” (A)

The second is indefinite and elliptical and is made logical by
prefixing the universal quantity sign and expressing in terms of the
four elements. The logical form is, “All who make haste are those who
are wasteful.” (A)

The third is plurative in nature and means, “Most men do not know how
to act under stress.” It would be classed as an O.

The fourth is partitive in nature because of the ambiguous use of
“all――not.” It means, “Some who seem to ring true are not true.” (O)

The fifth is an exclusive. By converting and changing to an A the
proposition takes the logical form, “All who are admitted are members.”

The sixth is likewise an exclusive, the logical form being, “All who
apply must be men of integrity.”

The seventh is an elliptical proposition. Logical form: “All horses are
trotting animals.”

The eighth is an inverted or poetical proposition. It is made logical
by interchanging subject and predicate. Logical form: “Those who are
persecuted for righteousness sake are blessed.”

(2b) Classify the attending propositions and change to the logical form,
if necessary:

  (1) “Only truthful men are honest.”

  (2) “The stokers alone were saved.”

  (3) “All who run do not think.”

  (4) “Honesty is the best policy.”

  (5) “They laugh that win.”

  (6) “The good alone are happy.”

  (7) “Knowledge is power.”

  (8) “Only the actions of the just smell sweet and blossom in the
      dust.”


=12. REVIEW QUESTIONS.=

  (1) Define and illustrate logical propositions.

  (2) Define and exemplify the three kinds of logical propositions.

  (3) What are the usual quantity signs of the four kinds of
      propositions?

  (4) Name and define the four elements of a logical proposition.

  (5) Select from the printed page five propositions which are not
      expressed in terms of the four elements, and so express them.

  (6) Distinguish between logical and grammatical subject; likewise
      between logical and grammatical predicate.

  (7) Define and illustrate the four kinds of categorical propositions.

  (8) What makes an understanding of the four logical propositions so
      important?

  (9) Give the unusual quantity signs of the logical propositions.

 (10) What should guide one in making an indefinite proposition logical?

 (11) How are general truths usually classified?

 (12) Change _birds fly_ to the logical form.

 (13) How many and what kinds of grammatical sentences are logical?

 (14) How would the logician deal with interrogative sentences?

 (15) Give illustrations of individual propositions. How are they
      usually classified?

 (16) Explain the logical mode of dealing with the plurative
      proposition.

 (17) Exemplify the ambiguity of “all-not,” “some” and “few.”

 (18) Why are propositions introduced by “all-not,” “some” and “few”
      called partitive?

 (19) Use “_all_” in both a partitive and collective sense. Which
      signification has logic adopted?

 (20) When are exceptive propositions universal and when particular?

 (21) What is an exclusive proposition?

 (22) Explain by circles the exclusive.

 (23) Tell in full how to change an exclusive to logical form.

 (24) Tell how the logician would deal with such poetical expressions
      as “Blessed are the pure in heart,” “Tell me not in mournful
      numbers,” “Strenuous is the man of state.”

 (25) What distinction does the logician make between analytic and
      synthetic propositions?

 (26) Illustrate the difference between the so-called modal and pure
      propositions.

 (27) Explain and illustrate the truistic proposition.

 (28) Show by circles the relation existing between the subject and
      predicate of all the logical propositions.

 (29) State in good English the relation between the subject and
      predicate of all the logical propositions.

 (30) Relative to the distribution of terms apply the words “uaesneop”
      and “asebinop.” Which one is the more serviceable?

 (31) Distinguish between the grammatical and logical subject.

 (32) Explain by circles the distribution of the terms of the four
      logical propositions.

 (33) The statement, “A part of the subject is excluded from the whole
      of the predicate,” describes which proposition? Explain how it
      indicates that the predicate is distributed.


=13. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Show that a judgment may be an individual notion as well as a
      general notion.

  (2) Many logicians classify logical propositions in this wise:

                      { Categorical
        Proposition   {
                      { Conditional { Hypothetical
                                    { Disjunctive

      Give arguments for and against such a classification.

  (3) “All men are bipeds” is a judgment of extension, while “Man is
      wise” is a judgment of intension. Explain.

  (4) “To be logical is to be pedantic.” Discuss this.

  (5) Why is the proposition, “He runs,” illogical? Make it logical.

  (6) Point out the reasons for calling, “White men are Caucasians,”
      a particular proposition.

  (7) What makes it necessary to change the propositions of ordinary
      conversation to those of the four logical types?

  (8) Some would call the individual proposition particular. Argue the
      question.

  (9) Make a list of five propositions in common speech and show how
      their partitive implication may mislead.

 (10) Explain by circles _some only_ and _some at least_.

 (11) Explain how “_et cetera_” may change a universal to a particular
      proposition.

 (12) “The real nature of an exclusive is best shown by negating the
      subject and calling the proposition an E.” Give arguments for
      and against this statement.

 (13) Show that with the immature mind all propositions must be
      synthetical.

 (14) Explain how a proposition may be truistic in form but not in
      meaning.

 (15) Show by the Euler diagram how easy it is for the careless student
      to think that an “O” does not distribute its predicate.

 (16) Explain by the use of two pads (a small yellow one and a large
      white one) the distribution of terms.

 (17) When the logician makes reference to the subject of a
      proposition, show that he should exercise care in designating it
      as the _logical_ subject.



                              CHAPTER 9.

                   IMMEDIATE INFERENCE――OPPOSITION.


=1. THE NATURE OF INFERENCE.=

_Inference is the thought process of deriving a judgment from one or
two antecedent judgments._

The process is simply a matter of expressing explicitly in a final
judgment, a truth that was implied in one or two previous judgments. To
exemplify: From the antecedent truth, that “All teachers should be fair
minded,” one may derive a consequent truth that “This teacher, Albert
White, should be fair minded.” Or from the statement, “All men are
mortal,” one may derive the judgment, “No men are immortal.” Because
the ground is wet we conclude that it has rained. If _all_ dogs are
quadrupeds then surely _some_ dogs are quadrupeds. Finally from the two
propositions, “All training school students are high school graduates,”
and “Mary Jones is a training school student,” we are led to conclude
that “Mary Jones is a high school graduate.”


=2. IMMEDIATE AND MEDIATE INFERENCE.=

It has been noted that a truth may be derived from a consideration
of _one_ or _two_ antecedent judgments. To illustrate further: From
the judgment, “All men are fallible,” we may derive the conclusion
that “No men are infallible”; or, from the two judgments, “All men
are fallible,” and “Socrates was a man,” we may readily infer that
“Socrates was fallible.” These two modes of inference take the names
of _immediate_ inference and _mediate_ inference. Let us express these
two kinds in equation form:


                                  I.

                                                  _Equation Form,
      _Ordinary Form._                            Using Initial
                                                  Letters._

      Antecedent judgment: All men are fallible.  All m are f
                                                  ―――――――――――
      Conclusion: No men are infallible.          No m are i


                                      II.

      First antecedent judgment: All men are
        fallible.                                 All m are f

      Second antecedent judgment: Socrates was
        a man.                                        S was m
                                                      ―――――――
      Conclusion: Socrates was fallible.          ∴   S was f

Giving attention to the antecedent judgments of the second argument
it is noted that the terms “_f_” and “_S_” are referred to the common
term “_m_.” In logic this common term is known as the _middle term_.
As there is but one antecedent judgment in the first argument there can
be no common or middle term. The first argument is an illustration of
_immediate_ inference; the second of _mediate_ inference. This suggests
the definitions:

_Immediate inference is inference without the use of a middle term._

_Mediate inference is inference by means of a middle term._


=3. THE FORMS OF IMMEDIATE INFERENCE.=

Many logicians recognize four forms of immediate inference. These
four forms are (1) _opposition_, (2) _obversion_, (3) _conversion_,
(4) _contraversion._[8]

(1) _IMMEDIATE INFERENCE BY OPPOSITION._

We have learned that to be logical all categorical assertions must
be reduced to some one of the four propositions, A, E, I, O. If these
four logical propositions be given the _same subject and predicate_,
certain definite relations will become evident; therefore, _Opposition
is said to exist between propositions which are given the same subject
and predicate, but differ in quality, or in quantity, or in both_.

The following illustrative outline will make this clear:

                  1.
       _Original Proposition._
    I. All men are mortal. (A)
   II. No men are immortal. (E)
  III. Some men are wise. (I)
   IV. Some men are mortal. (I)
    V. Some men are not wise. (O)
   VI. Some men are not immortal. (O)

                  2.
       _Opposite in Quantity._
    I. Some men are mortal. (I)
   II. Some men are not immortal. (O)
  III. All men are wise. (A)
   IV. All men are mortal. (A)
    V. No men are wise. (E)
   VI. No men are immortal. (E)

                  3.
       _Opposite in Quality._
    I. No men are mortal. (E)
   II. All men are immortal. (A)
  III. Some men are not wise. (O)
   IV. Some men are not mortal. (O)
    V. Some men are wise. (I)
   VI. Some men are immortal. (I)

                  4.
       _Opposite in Both._
    I. Some men are not mortal. (O)
   II. Some men are immortal. (I)
  III. No men are wise. (E)
   IV. No men are mortal. (E)
    V. All men are wise. (A)
   VI. All men are immortal. (A)

Granting the truth of the propositions in the first column, it follows
that those in the second column differ in quantity. That is, in “Some
men are mortal” a _smaller_ number of men is referred to than in “All
men are mortal.” A similar variation in quantity obtains with the
other propositions in the second column. Moreover, the propositions
in the third column are the negative of the corresponding ones in the
first; while the fourth column propositions differ from the first in
_both_ quantity and quality. Thus opposition exists to a greater or
less degree between all. We may now ask ourselves the question, “When
the propositions are related to each other in opposition which ones are
true and which ones are false?” Giving attention to the propositions
in row “I,” we note that if the universal affirmative, “All men are
mortal,” is true, then the particular affirmative, “Some men are
mortal,” is likewise true; because of the principle, “What is true
of the whole of the class is true of a part of that class.” But the
universal negative, “No men are mortal,” and the particular negative,
“Some men are not mortal,” are both false. Briefly stated: If A is
true, then I is true, but, both E and O are false.

Regarding row “II” we may conclude that if E is true, then O is
likewise true, but both A and I are false.

As to rows “III” and “IV,” granting the truth of the I propositions,
“Some men are wise” and “Some men are mortal,” we are able to assert
that of the two A propositions, “All men are wise,” and “All men are
mortal,” the first is false while the second is true. A is, therefore,
indeterminate, or doubtful. Of the O propositions, “Some men are not
wise,” is true while, “Some men are not mortal,” is false. Therefore,
O is doubtful. Both of the E propositions are false. Hence, the
conclusion relative to rows “III” and “IV” is: If I is true, A and O
are doubtful, while E is false.

Concerning rows “V” and “VI” it will be seen without further
explanation that if O is true, then E and I are doubtful and A is false.


  THE SCHEME OF OPPOSITION.

The conditions of opposition are easily comprehended and remembered
when recourse is made to the following scheme:

                      A      E      I      O
    If A be true    true   false  true   false
    If E be true    false  true   false  true
    If I be true    doubt  false  true   doubt
    If O be true    false  doubt  doubt  true

To use the above scheme, read horizontally from left to right. For
example: If A be true, then all in the row opposite obtains; that
is, A is true, E is false, I is true, and O is false. (We take it for
granted that the student will see that the first column belongs to A,
the second to E, the third to I, and the fourth to O.) If E be true,
then A is false, E is true, I is false, O is true, etc.

The whole of opposition is comprehended in _two facts_ which are based
upon _one principle_. This is the principle: _Whatever may be said of
the entire class may be said of a part of that class._ To put it in
another way: Whatever is affirmed of all may be affirmed of some, or,
Whatever is denied of all may be denied of some. To illustrate:

    Accepted truth: All planets rotate. (A)
    Accepted inference: Some planets rotate. (I)
                      or
    Accepted truth: No planet is a sun. (E)
    Accepted inference: Some planets are not suns. (O)

These are the two facts: First, _a particular affirmative may be
derived from a universal affirmative_. Second, _a particular negative
may be derived from a universal negative_. Or, more briefly: _An I may
be derived from an A, and an O from an E_.


  SQUARE OF OPPOSITION.

  Illustration:   ( ‡ Square of Opposition)

Aristotle represented the relations of the four logical propositions by
what is termed the _square of opposition_. Viewed from the standpoint
of the square, the relations may be summed up as follows:

1. _Contrary Propositions._

_Why so named._

As related to each other, A and E are said to be contrary because they
seem to express _contrariety_ to the greatest degree.

_Relation stated._

If one is true, the other must be false, but both may be false.

_Illustrations._

(1) If one is true, the other must be false; e. g., if A is true, as
“All metals are elements,” then E is false, as “No metals are elements.”
Or, if E is true, as “No birds are quadrupeds,” then A is false, as
“All birds are quadrupeds.”

(2) Both may be false. If A is false, as “All men are wise,” then E may
be false, as “No men are wise.”

2. _Subcontrary Propositions._

_Why so named._

Propositions I and O are said to be related to each other in a
subcontrary manner because they are contrary as to each other and
“_under_” their universals A and E.

_Relation stated._

If one is false, the other must be true, or, both may be true.

_Illustrations._

(1) If one is false, the other must be true.

If I is false, as “Some metals are compounds,” then, O is true, as
“Some metals (at least) are not compounds.” Or, if O is false, as “Some
metals are not elements,” then I is true, as “Some metals are elements.”

(2) Both may be true.

If I is true, as “Some men are wise,” then O also may be true, as “Some
men are not wise.”

3. _Subalterns._

_Why so named._

Etymologically considered subaltern means _under the one_, thus
proposition I is under A, and O is under E.

_Relation stated._

_First Relation._

Subalterns are related to each other as are the universals and
particulars; hence,

(1) If the universal is true, the particular under it is also true;
while if the particular is true, the corresponding universal may, or,
may not, be true.

_Illustrations._

(a) If the universal is true, the particular under it is true.

If A is true, as “All metals are elements,” then I is true, as “Some
metals are elements.” Or, if E is true, as “No metals are compounds,”
then, O is also true, as “Some metals (at least) are not compounds.”

(b) If the particular is true, the corresponding universal may, or, may
not, be true.

If I is true, as “Some men are wise,” or, “Some men are mortal,” then A
may be false, as “All men are wise,” or, A may be true, as “All men are
mortal.” Or, if O is true, as “Some men are not wise,” or, “Some men
are not immortal,” then E may be false, as “No men are wise”; or, true,
as “No men are immortal.”

_Second Relation._

(2) If the universal is false, the particular under it may or may not
be true, but, if the particular is false, the universal above it must
be false.

_Illustrations._

(a) If the universal is false, the particular under it may or may not
be true.

If A is false, as “All metals are compounds,” or “All men are wise,”
then I may be false, as “Some metals are compounds,” or, I may be true,
as “Some men are wise.” Or, if E is false, as “No men are mortal,” or,
“No men are wise,” then O may be false, as “Some men are not mortal,”
or, O may be true, as “Some men are not wise.”

(b) If the particular is false, the universal above it must be false.

If I is false, as “Some men are trees,” then A is false, as “All men
are trees.” Or, if O is false, as “Some men are not bipeds,” then E is
also false, as “No men are bipeds.”

4. _Contradictory Propositions._

_Why so named._

The propositions A and O, likewise E and I, are called contradictory
propositions because they oppose each other in both quantity and
quality. They are mutually opposed to each other or _absolutely_
contradictory.

_Relation stated._

If one is true the other must be false.

_Illustrations._

(1) A and O compared.

If A is true, as “All metals are elements,” then, O is false, as “Some
metals are not elements.” Or, if O is true, as “Some metals are not
compounds,” then A is false, as “All metals are compounds.”

(2) E and I compared.

If E is true, as “No birds are quadrupeds,” then I is false, as “Some
birds are quadrupeds.” Or, if I is true, as “Some birds are bipeds,”
then E is false, as “No birds are bipeds.”

The chief value of the square of opposition springs from the
contradictory propositions. The square shows conclusively that any
universal affirmative assertion (an A) may best be contradicted by
proving a particular negative (an O). For example: To satisfactorily
refute the statement that, in this section, all birds migrate to the
south in winter, it would be sufficient to prove that the English
sparrow and starling do _not_ migrate to the south. The square likewise
makes evident that any universal negative (an E) may be conclusively
denied by establishing the truth of a particular affirmative (an I).
To illustrate: The easiest way to prove the falsity of “No trusts are
honest” is to present facts showing that at least trusts A and B _are_
honest.

_The Individual Proposition._

An individual proposition is one with an individual subject such
as “Aristotle was wise.” In logic, the individual proposition is
classed as a universal. This seems to be a bit irregular, as with the
individual proposition there is no particular, while, the strictly
_logical_ universal _always_ implies a particular. Because of this
variation from the true logical form the relations, as indicated by
the square of opposition, do _not_ apply to the individual proposition.
For example: According to the square A and E are contrary, but, when
individual, A and E contradict each other, as “Aristotle was wise”
(A)――“Aristotle was not wise” (E).



                              CHAPTER 10.

              IMMEDIATE INFERENCE (CONTINUED)――OBVERSION,
               CONVERSION, CONTRAVERSION AND INVERSION.


(2) _IMMEDIATE INFERENCE BY OBVERSION._

_Obversion is the process of changing a proposition from the
affirmative form to its equivalent negative or from the negative form
to its equivalent affirmative._

Some authorities refer to this process as “Inference by Privitive
Conception,” but Obversion seems to be a better term.

Obversion is based upon the principle that _two negatives are
equivalent to one affirmative_. With this double negative principle in
mind let us experiment with the four logical propositions, A, E, I, O.

_The A Proposition._

Example: “All thoughtful men are wise.” Insert the double negative and
the proposition reads: “All thoughtful men are not not-wise.” Changed
to the logical form this becomes: “No thoughtful men are not-wise.”
Simplified and we have, finally: “No thoughtful men are unwise.”
Thus by the process of obversion we have passed from the original
proposition, “All thoughtful men are wise,” to “No thoughtful men
are unwise.” In the first proposition the subject “_thoughtful men_”
is denied of the predicate “_unwise_.” Assuming that “unwise” is the
contradictory of “wise,” then: “What is affirmed of a predicate may
be denied of its contradictory.” Recourse to circles will make this
clearer. In the previous chapter it has been suggested that _not_
bisects the world. For example: What can _not_ be included in the
_wise_ class may be placed under the not-wise or _unwise_ class.
Likewise a circle bisects space――there is the space inside the circle
and the space outside the circle. Let the space inside the circle
represent all wise beings, then the space outside the circle would
represent all not-wise or unwise beings; e. g.,

  Illustration:   FIG. 5.

Now representing _thoughtful men_ by a smaller circle and placing it
inside the larger we have,

  Illustration:   FIG. 6.

Referring to Fig. 6 we note that all of the smaller circle belongs
to the larger or that none of the smaller circle belongs to the space
outside of the larger. Hence the two propositions: “All thoughtful men
are wise” (A), and “No thoughtful men are unwise” (E) have virtually
the same meaning though the same subject is related to different
predicates.

The use of the positive or negative form depends upon circumstances.
Often the negative puts the thought in a more forceful way.

In passing from, “All thoughtful men are wise,” to “No thoughtful
men are unwise,” it was necessary to prefix _not_ to the predicate
_wise_ and substitute for _not_ its equivalent _un_. If the original
predicate were unwise or not-wise, then the reverse order of dropping
the _un_ or _not_ could be followed. This process of prefixing the
_not_ to an affirmative predicate or of dropping the _not_ from a
negative predicate is referred to as _negating the predicate_. Before
substituting _in_, _im_, _un_, etc., for _not_, one must make sure
that the substitution really gives the contradictory; there are some
logicians who claim that unwise, for instance, is not the contradictory
of wise.

In comparing the first proposition with the second it is observed that
the first is an A, while the second is an E, also that the predicate of
the first was _negated_ to form the predicate of the second. Thus the
rule: Negate the predicate and change A to E.

_To sum up_:

  The obversion of an A proposition.

1. Principle:

  Two negatives are equivalent to one affirmative.

2. Rule:

  Negate the predicate and change the A to an E by using the sign
  _no_ instead of _all_.

3. Process illustrated.

    _The Original Proposition (A)_      _The Obverse (E)_

    All men are mortal.                 No men are immortal.

    All maples are trees.               No maples are not-trees.

    All teachers should be              No teacher should be
      sympathetic.                        un-sympathetic.

    All pain is unpleasant.             No pain is pleasant.

    All men are imperfect.              No men are perfect.

    All birds are feathered             No birds are non-feathered
      animals.                            animals.

    All men are not-trees.              No men are trees.

    All scalene triangles are           No scalene triangles are
      non-equilateral.                    equilateral.

_The E Proposition._

It is obvious that the process of obverting an E is simply the reverse
of obverting an A. Consequently, the same principle obtains; whereas
the _process_ may be illustrated by reading the foregoing illustrations
reversely.

The rule for obverting E is: _Negate the predicate and change the E to
an A by changing the sign no to all._

_The I Proposition._

Let us note the result when the double negative principle is applied to
the I proposition.

Original: “Some men are wise.”

Adding two negatives: “Some men are not not-wise.”

The foregoing simplified: “Some men are not unwise.”

In comparing the first proposition with the last it is observed that
the first is an I while the last is an O; it is also observed that the
predicate of the first was _negated_ in order to form the predicate
of the last. Thus the rule: “Negate the predicate and change the I to
an O.”

The use of circles may make this clearer:

  Illustration:   FIG. 7.

The significant part of Fig. 7 is that which is inked. Here we have
represented the part of the “men” circle which is common to the “wise”
circle. Thus the inked part represents “Some men are wise.” If the
inked part is entirely inside of the “wise” circle, no part of it can
belong to the “unwise” space without. Thus the obverse, “Some men are
not unwise.”

_Summary._

The obversion of an I proposition.

1. Principle:

  Same as with A.

2. Rule:

  Negate the predicate and change the I to an O.

3. Process illustrated.

  _The Original Proposition (I)_    _The Obverse (O)_

  Some water is pure.               Some water is not impure.

  Some curves are perfect.          Some curves are not imperfect.

  Some friends are loyal.           Some friends are not disloyal.

  Some men are true.                Some men are not not-true.

  Some precious stones are          Some precious stones are not
    imperfect.                        perfect.

  Some plants are not-trees.        Some plants are not trees.

  Some boys are not-honest.         Some boys are not honest.

It must be borne in mind that when “_not_” is used without the hyphen
it makes the proposition negative, because when “_unhyphened_,” “_not_”
must be thought of in connection with the copula and not in connection
with the predicate; while “_not_” attached to the predicate with
a hyphen simply makes the predicate negative without affecting the
quality of the proposition; e. g., “Some plants are not trees” is
a negative proposition, while “Some plants are not-trees” is an
affirmative proposition with a negative predicate.

It may not be clearly seen how it is possible, by following the rule
given, to pass from such a proposition as “Some plants are _not-trees_,”
to “Some plants are _not trees_.” Let us illustrate the steps:

  1. The original: “Some plants are not-trees.”

  2. Negating predicate: “Some plants are trees.”

  3. Changing to an O: “Some plants are not trees.”

Dropping the not from “1” and then adding it again to “2” is simply
putting into operation the double negative idea, so that there is no
violation of the principle.

_The O Proposition._

O bears the same relation to I that E bears to A. The principle
involved is the same. The process is illustrated by reading reversely
the scheme of illustrations under I. The rule is as follows: _To obvert
an O negate the predicate and change the O to an I by eliminating the
not._

_Summary of Obverting the Four Logical Propositions._

1. Principle:

  Two negatives are equivalent to one affirmative.

2. Rules:

                          { (1) A to E
    Negate the predicate  { (2) E to A
      and change          { (3) I to O
                          { (4) O to I


(3) _IMMEDIATE INFERENCE BY CONVERSION._

_Conversion is the process of inferring from a given proposition
another which has, as its subject, the predicate of the given
proposition, and, as its predicate, the subject of the given
proposition._ It is simply a matter of transposing subject and
predicate. The original proposition is called the _convertend_ while
the derived proposition is named the _converse_.

The process of conversion is limited by two rules. First rule. No term
must be distributed in the converse which is not distributed in the
convertend. Second rule. The quality of the converse must be the same
as that of the convertend. More briefly: (1) _Do not distribute an
undistributed term._ (2) _Do not change the quality._

We recall that a term is distributed when it is referred to as a
_definite whole_. An undistributed term is referred to only _in part_.
The principle underlying rule “1,” therefore, is the one which forms
the basis of inference by opposition; namely, “_Whatever may be said of
the entire class may be said of a part of that class_.” The converse of
this is _not true_, that is, “What is said of part of a class cannot be
said of the whole of that class.” When we distribute an undistributed
term we are saying of the _whole_ class what was said only of a _part_
of that class. This is fallacious. On the other hand, we may say of a
part what was said of the whole, or “undistribute” a distributed term.

We recall that the conclusion of the whole matter of inference by
opposition was, that only an I could be inferred from an A and only an
O from an E, or to put it in another way: Only an affirmative from an
affirmative and only a negative from a negative. This establishes the
truth of the second rule in conversion: “Do not change the quality.”

Let us apply the two rules to the four logical propositions.

_Converting an A proposition._

Take as a type, “All horses are quadrupeds.” Here the subject
“_horses_” is distributed, but the predicate “_quadrupeds_” is
undistributed. In transposing subject and predicate we cannot
distribute the term “quadrupeds,” according to the rule which says, “Do
not distribute an undistributed term.” Hence in interchanging subject
and predicate we cannot say, “All quadrupeds are horses,” but must
limit the assertion to, “_Some_ quadrupeds are horses.” Logicians call
this process _Conversion_ by _Limitation_.

_Conversion by Limitation Exemplified Further._

    _Convertend._                 _Converse._

    All metals are elements.      Some elements are metals.

    All bees buzz.                Some buzzing insects are bees.

    All men are fallible.         Some fallible beings are men.

    All good teachers are         Some sympathetic persons are good
      sympathetic.                  teachers.

The conclusions from the foregoing are these: First, the usual mode of
converting an A is to interchange subject and predicate, limiting the
latter by the word “_some_” or a word of similar significance. Second,
this mode is called _conversion by limitation_. Third, the converse of
an _A_ is an _I._

_The Co-extensive A._

In the conversion of A propositions there is the one exception of
“co-extensive A’s,” such as truisms and definitions. It will be
remembered that with these both subject and predicate are distributed;
hence, they may be interchanged without limiting the predicate by
“some.” To illustrate: The converse of the truism, “A man is a _man_.”
is “A _man_ is a man,” while the converse of the definition, “A man
is a rational animal,” is “A rational animal is a man.” This mode
of interchanging subject and predicate without limiting the latter
is called _Simple Conversion_. The ordinary A proposition is thus
_converted_ by _limitation_, while the co-extensive A is _converted
simply_.

_Converting an E proposition._

As both terms of the E proposition are distributed it is not possible
to violate the rule of distribution. It is to be remembered that no
fallacy is committed by “undistributing” a term which is already
distributed.

_Illustrations._


    _Convertend._                 _Converse._

    No men are immortal.          No immortals are men. Simply.

    No birds are quadrupeds.      No quadrupeds are birds. Simply.

    No metals are compounds.      No compounds are metals. Simply.

    No men are immortal.          Some immortals (at least) are not
                                    men. Limitation.

    No birds are quadrupeds.      Some quadrupeds are not birds.
                                    Limitation.

    No metals are compounds.      Some compounds are not metals.
                                    Limitation.

Three facts are evident relative to the converting of an E. First: An E
proposition may be converted either _simply_ or by _limitation_. Second:
E may be converted into either _E_ or _O_. Third: If the converse is
an O then is the inference a _weakened_ one, being _particular_ when it
could just as well be _universal_.

_Converting an I proposition._

With an I proposition neither term is distributed. Thus care must be
used lest an undistributed term in the convertend be distributed in the
converse. _Illustrations_:

    _Convertend._                 _Converse._

    Some men are wise.            Some wise beings are men.

    Some teachers scold.          Some who scold are teachers.

    Some high school graduates    Some who enter college are high
      enter college.                school graduates.

    Some Americans live simply.   Some who live simply are Americans.

From the foregoing we conclude first, that I is _converted simply_;
second, that I is converted _into I_.

_The O Proposition._

With an O proposition the subject is _undistributed_ while the
predicate is distributed. This condition presents a peculiar
difficulty. Consider, for example, the O proposition, “Some men are
not wise.” Convert this into, “Some wise beings are not men,” and the
_undistributed_ subject of the convertend, which is “men,” becomes the
_distributed_ predicate of the converse. _Thus the O proposition cannot
be converted without violating the rule for distribution._

_A Summary of How the Four Logical Propositions May be Converted._

  1. _A._ The ordinary A proposition may be converted by _limitation_
          only. The co-extensive A may be converted _simply_.

  2. _E._ The E proposition is converted _simply_. The E may also be
          converted by _limitation_, but the inference thus obtained is
          _weakened_.

  3. _I._ The I proposition may be converted _simply_ only.

  4. _O._ The O proposition _cannot be converted_.


(4) _INFERENCE BY CONTRAVERSION._ (Contraposition).

This mode of inference is usually referred to as inference by
_contraposition_, but contraversion, indicating more definitely the
nature of the process, is a better term. Contraversion involves _two_
steps: First, _obversion_; second, _conversion_. The same principles
and rules evident in these _two_ processes obtain in inference by
contraversion. The following scheme, therefore, ought to be sufficient
to make the matter clear:

_Inference by Contraversion._

    1. _The Given Proposition._    2. _Obverted._

    A. All men are mortal.            No men are immortal.

       All trees are plants.          No trees are not-plants.

    E. No men are infallible.         All men are fallible.

       No men are trees.              All men are not-trees.

    I. Some men are wise.             Some men are not not-wise.

    O. Some water is not pure.        Some water is impure.

       Some houses are not white.     Some houses are not-white.

3. _Converted; giving the contraverse of the original proposition._

      No immortals are men.

      No not-plants are trees.

      Some fallible beings are men.

      Some not-trees are men.

      An O cannot be converted, consequently the contraversion of an
        I is impossible.

      Some impure liquids are water.

      Some not-white buildings are houses.

It is indicated in the foregoing scheme that “I” cannot be contraverted.
This is due to the fact that the obverse of an I is an O, and it will
be remembered that “O” cannot be converted. All the other propositions
admit of contraversion.


=4. EPITOME OF THE FOUR PROCESSES OF IMMEDIATE INFERENCE IN CONNECTION
    WITH THE FOUR LOGICAL PROPOSITIONS.=

  [*]│Proposition│Name of      │Inference  │Principle
     │symbolized │Process      │symbolized │involved
  ───┼───────────┼─────────────┼───────────┼─────────────────────────
   A │All S is   │Opposition   │Some S is  │What is said of _all_ may
     │  P[†]     │             │  P (I)    │  be said of _some_.
     │           │             │           │
     │           │Obversion    │No S is    │Two negatives are
     │           │             │  not-P (E)│  equivalent to one
     │           │             │           │  affirmative.
     │           │             │           │
     │           │Conversion by│Some P is  │An undistributed term
     │           │  Limitation │  S (I)    │  cannot be distributed.
     │           │             │           │
     │           │Contraversion│No not-P   │Same principles which
     │           │             │  is S (E) │  obtain in obverting
     │           │             │           │  A and converting E.
  ───┼───────────┼─────────────┼───────────┼─────────────────────────
   E │No S is P  │Opposition   │Some S is  │What is said of _all_ may
     │           │             │  not P (O)│  be said of _some_.
     │           │             │           │
     │           │Obversion    │All S is   │Two negatives are
     │           │             │  not-P (A)│  equivalent to one
     │           │             │           │  affirmative.
     │           │             │           │
     │           │Simple       │No P is S  │Distribution not
     │           │ Conversion  │  (E)      │  affected.
     │           │             │           │
     │           │Contraversion│Some not-P │An undistributed term
     │           │             │  is S (I) │  cannot be distributed.
  ───┼───────────┼─────────────┼───────────┼─────────────────────────
   I │Some S is P│Opposition   │Doubtful   │None.
     │           │             │           │
     │           │Obversion    │Some S     │Two negatives are
     │           │             │  is not   │  equivalent to one
     │           │             │  not-P (O)│  affirmative.
     │           │             │           │
     │           │Conversion   │Some P is  │Distribution not
     │           │             │  S (I)    │  affected.
     │           │             │           │
     │           │Contraversion│Impossible │None.
  ───┼───────────┼─────────────┼───────────┼─────────────────────────
   O │Some S is  │Opposition   │Doubtful   │None.
     │  not P    │             │           │
     │           │             │           │
     │           │Obversion    │Some S is  │Two negatives are
     │           │             │  not-P (I)│  equivalent to one
     │           │             │           │  affirmative.
     │           │             │           │
     │           │Conversion   │Impossible │None.
     │           │             │           │
     │           │Contraversion│Some not-P │Same as in obversion of
     │           │             │  is S (I) │  O and conversion of I.

                        * – Name of proposition
         † – “S” represents any subject and “P” any predicate.


  INFERENCE BY INVERSION.

Some logicians treat of a form of immediate inference known as
inversion though it is of small importance and of little practical
value.

The process can be applied only to propositions A and E. In the one
case the _contradictory subject_ is limited by “_some_” and then denied
of the predicate, whereas, in the other case, the contradictory subject
is merely affirmed of the predicate.

_Illustrations._

        _The Given Proposition._  _The Inverse._

    I.  All S is P. (A)           Some not-S is not P. (O)
        All planets rotate.       Some not-planets do not rotate.

   II.  No S is P. (E)            Some not-S is P. (I)
        No men are immortal.      Some not-men are immortal.


From the foregoing we are able to conclude _that the inverse of “A” is
found by negating the subject and changing to an “O”; while the inverse
of “E” is found by negating the subject and changing to an “I.”_


=5. OUTLINE.=

  IMMEDIATE INFERENCE――OPPOSITION――OBVERSION, CONVERSION,
  CONTRAVERSION AND INVERSION.

  1. The Nature of Inference.

  2. Immediate and Mediate Inference.

  3. The Forms of Immediate Inference.
      (1) Opposition.
            (a) Scheme of Opposition.
            (b) Square of Opposition.
      (2) Obversion.
      (3) Conversion.
            (a) Simply.
            (b) By Limitation.
      (4) Contraversion.
          Inversion.


=6. SUMMARY.=

1. Inference is the thought process of deriving a judgment from one or
two antecedent judgments.

2. Immediate inference is inference without the use of a middle term.
Mediate inference is inference by means of a middle term.

3. The four common forms of immediate inference are (1) opposition,
(2) obversion, (3) conversion, (4) contraversion.

(1) The name _opposition_ stands for certain definite relations which
exist between the logical propositions when they are given the same
subject and predicate. The one principle underlying opposition is:
Whatever is said of the entire class may be said of a part of that
class. The two statements which sum up opposition are first, an I may
be derived from an A; and second, an O may be derived from an E.

The crucial fact made obvious by the square of opposition is that A and
O are mutually contradictory; likewise E and I.

(2) Obversion is the process of passing from an affirmative to its
equivalent negative or from a negative to its equivalent affirmative.
“Two negatives are equivalent to one affirmative,” is the basic
principle of obversion.

The proposition A may be obverted by negating the predicate and
changing to an E. “E” is obverted by negating the predicate and
changing to an A. “I” is obverted by negating the predicate and
changing to an O. “O” is obverted by negating the predicate and
changing to an I.

(3) Conversion is the process of inferring from a given proposition
another which has as its subject the _predicate of the given
proposition_ and as its predicate the _subject of the given
proposition_.

Conversion is limited by the two rules, (1) do not distribute an
undistributed term; (2) do not change the quality.

To convert an A interchange subject and predicate, limiting the latter
by _some_, or a word of like significance. This is called conversion by
limitation.

The co-extensive A may be converted without limiting the predicate.
This is called simple conversion.

An E proposition may be converted either simply or by limitation. When
converted by limitation the inference is a _weakened_ one.

An I proposition is converted simply only.

The O proposition does not admit of conversion.

(4) Immediate inference by contraversion is a process involving first
_obversion_ and then _conversion_.

“A,” “E” and “O” may be contraverted; “I” cannot be contraverted.


=7. ILLUSTRATIVE EXERCISES.=

(1a) From the antecedent judgment, “All weeds are plants,” I am able
to derive by immediate inference these judgments: (1) “All weeds are
not not-plants,” or “No weeds are not plants.” (2) “No not-plants are
weeds.” (3) “Some plants are weeds.” (4) “Some weeds are plants.”

(1b) “All vertebrates have a backbone.” From the foregoing judgment
derive immediately five different conclusions.

(2a) “All good citizens try to vote,”

  “Albert White is a _good citizen_,”
  Hence, “Albert White will try to vote.”

I know that the above is an example of mediate inference because the
two antecedent judgments make use of the middle term, “_good citizen_.”

(2b) Why is the following illustrative of mediate inference?

  “All wise men are close observers,”
  “All wise men are thoughtful,”
  Hence, “Some thoughtful men are close observers.”

(3a) Derive immediate inferences by opposition from the following:

  (1) “Good men are wise.”

  (2) “No teacher can afford to be unjust.”

  (3) “All birds fly.”

  (4) “None of the inner planets are as large as the earth.”

I first determine that “1” and “3” are A propositions, while “2” and
“4” are E’s. Then I recall that by opposition an I may be derived from
an A and an O from an E. Hence, the inferences are:

  (1) “Some good men are wise.”

  (2) “Some teachers cannot afford to be unjust.”

  (3) “Some birds fly.”

  (4) “Some of the inner planets are not so large as the earth.”

(3b) Derive by opposition inferences from the following:

  (1) “No true woman will neglect her home for society.”

  (2) “All patriotic men love the flag.”

  (3) “Fools rush in where angels fear to tread.”

(4a) Obvert the following:

  (1) “All earnest teachers are diligent students.”

  (2) “No self-respecting man can afford to be careless in his personal
      appearance.”

  (3) “Some of the great teachers of the past did not practice what
      they preached.”

  (4) “Some weeds are beautiful.”

I determine first the logical character of each proposition, finding
the first to be an A, the second an E, the third an O and the fourth an
I. Then I recall that in obversion the predicate must always be
_negated_ and an A must be changed to an E or an E to an A; also an I
must be changed to an O or an O to an I. Hence, the obverse of each
proposition is:

  (1) “No earnest teacher is a not-diligent student.”

  (2) “All self-respecting men can afford to be not-careless (careful)
      in their personal appearance.”

  (3) “Some of the great teachers of the past did not-practice (failed
      to practice) what they preached.”

  (4) “Some weeds are not not-beautiful.”

(4b) Infer by obversion from the following:

  (1) “All roses are beautiful.”

  (2) “None of the members of the stock exchange are dishonest.”

  (3) “Some pupils are not industrious.”

  (4) “Some teachers are tactful.”

(5a) Convert the following:

  (1) “All that glitters is not gold.”

  (2) “All good men are wise.”

  (3) “Some books are to be chewed and digested.”

  (4) “No man is perfectly happy.”

It is first necessary to determine the logical character of each
proposition. Carelessness might lead one to call the first proposition
an A because it is introduced by the quantity sign “_all_.” But on
second thought we note that the meaning is to the effect that _some
glittering things are not gold_; this is an O. It is clear that
the second is an A, the third an I and the fourth an E. It is now
expedient to recall the rules regarding conversion. These are, (1) do
not distribute an undistributed term; (2) do not change the quality.
We may now attempt to interchange the subject and predicate of each
proposition, with the following results:

  (1) Conversion impossible.

  (2) “Some wise men are good men.”

  (3) “Some things to be chewed and digested are books.”

  (4) “No perfectly happy being is a man.”

When attempting to convert proposition (1), I find that the subject
which is undistributed becomes distributed, hence the rule pertaining
to distribution is violated. This conclusion is verified by recalling
the fact that an O proposition cannot be converted. The second
proposition, being an A, is converted by limitation; while the third
and fourth are converted simply, as is the natural procedure with all
I’s and E’s.

(5b) Convert these propositions:

  (1) “Blessed are the meek.” (All the meek are blessed.)

  (2) “None but material bodies gravitate.” (All gravitating bodies are
      material.)

  (3) “Gold is not a compound substance.”

  (4) “Usually cruel men are cowards.”

NOTE.――_The first proposition is poetical while the second is an
exclusive._

(6a) Contravert the following propositions:

  (1) “All virtue is praiseworthy.”

  (2) “Some teachers are not tactful.”

  (3) “A man who lies is not to be trusted.”

Contraversion consists in obverting first, and then converting;
consequently, the contraverse of the three propositions is as follows:

  (1) “No unpraiseworthy deed is virtue.”

  (2) “Some not-tactful persons are teachers.”

  (3) “Some untrustworthy men are those who lie.”

(6b) Write the contraverse of the following:

  (1) “All honest men pay their debts.”

  (2) “All men are rational.”

  (3) “Nearly all the troops have left the town.”

  (4) “Some teachers are not patient.”

(7a) The attending scheme indicates the logical process and rule
involved in passing from one proposition to another:

  A. “All men are imperfect.”
        Process: Obversion.
        Rule: Negate predicate and change to E.

  E. “No men are perfect.”
        Process: Simple Conversion.
        Rule: Interchange subject and predicate.

  E. “No perfect beings are men.”
        Process: Contraversion.
        Rule: Obvert and then convert.

  I. “Some not-men are perfect beings.”

(7b) Treat in a manner similar to the above the proposition, “All
horses are quadrupeds.”


=8. REVIEW QUESTIONS.=

  (1) What is inference?

  (2) What is the meaning of antecedent?

  (3) Define (1) judging, (2) a judgment.

  (4) All roses are beautiful,
        This flower is a rose,
        This flower is beautiful.

      Write this example of mediate inference in equation form. Name
      the middle term.

  (5) Define immediate inference. Illustrate.

  (6) Define mediate inference. Illustrate.

  (7) Name the five forms of immediate inference.

  (8) What principle is involved in inference by opposition?

  (9) Draw the scheme of opposition.

 (10) Make use of this scheme in deriving inferences from the following
      propositions:
        (a) “Good men are wise.”
        (b) “No king is infallible.”
        (c) “Cattle are ruminants.”
        (d) “All who cheat the railroads are not honest.”

 (11) What are contradictory propositions? Illustrate.

 (12) What would be the simplest way of disproving the statement that
      “No great religious teacher has been consistent?”

 (13) Why are A and E said to be contrary propositions?

 (14) Define obversion.

 (15) By what other name is obversion known?

 (16) State the basic principle of obversion.

 (17) Illustrate the process known as _negating the predicate_.

 (18) State the rule for obverting an A proposition.

 (19) Obvert the following:
        (1) “All the boys in my room are industrious.”
        (2) “Honesty is the best policy.”
        (3) “Only the industrious are truly successful.”

 (20) First state the rule and then obvert the following:
        (1) “Some plants are biennial.”
        (2) “Planets are not suns.”
        (3) “Blessed are the merciful.”
        (4) “These samples are not perfect.”

 (21) Define conversion.

 (22) State and illustrate the rules which condition the process of
      conversion.

 (23) Convert, if possible, the following:
        (1) “Some men practice sophistry.”
        (2) “Few men know how to live.”
        (3) “Some of the inhabitants are not civilized.”
        (4) “All the world is a stage.”
        (5) “None of my pupils failed.”
        (6) “Experience is a hard taskmaster.”

 (24) Why may co-extensive propositions be converted simply?

 (25) Describe the process of inference by contraversion.


=9. PROBLEMS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) What ground is there for the belief that immediate inference, so
      called, is merely a matter of the interpretation of propositions?

  (2) Is there any difference between reasoning and inference?

  (3) When the conclusion is reached that two rooms are of the same
      width, because each is five yards wide, what is the middle term?

  (4) Put in equation form:
        All teachers instruct,
        John Jones is a teacher,
        John Jones instructs.

      Show that the equations are not _absolutely true_.

  (5) Indicate the true relation between the subjects and predicates
      of the foregoing by using the algebraic signs > and <.

  (6) Why cannot an A be derived from an I?

  (7) Why cannot an O be derived from an A?

  (8) The basic principle of obversion is “Two negatives are equivalent
      to one affirmative.” Show by means of circles that this is not
      absolutely true; take as an illustrative proposition, “No men are
      not mortal.”

  (9) Show that agreeable and disagreeable are not contradictory terms.

 (10) Why should the logician class individual propositions as
      universal?

 (11) Show by circles that there is a difference in signification
      between, “Some men are _not wise_” and “Some men are _not-wise_.”

 (12) Show by circles that the O proposition cannot be converted.

 (13) “The I proposition cannot be contraverted.” Make this clear.

 (14) Is there any difference in meaning between, “All illogical work
      is unscholarly” and “No illogical work is scholarly?” Explain by
      circles.

 (15) State the logical process involved in passing from each
      proposition to its succeeding one:

      (1) “All men are imperfect.”

      (2) “No men are perfect.”

      (3) “No perfect beings are men.”

      (4) “Some not-men are perfect beings.”

      (5) “Some perfect beings are not-men.”

      (6) “Some perfect beings are not men.”

 (16) It is sometimes said that in sub-contraries there is really no
      opposition. Do you agree? Give arguments.



                              CHAPTER 11.

                   MEDIATE INFERENCE. THE SYLLOGISM.


=1. INFERENCE AND REASONING.=

Inference has been defined as both a product and a process. When
used to indicate a process the term inference becomes synonomous with
reasoning. If logicians could agree to confine inference to the product
and reasoning to the process, it would remove an ambiguity which is
more or less misleading. But since this has not become the custom, we
shall use inference as indicating the process as well as the product.

_Definitions――Middle Term Explained._

Inference is the thought process of deriving a judgment from one or two
antecedent judgments.

_Mediate inference is inference by means of a middle term._

Reasoning of this nature involves three terms, two of which are
compared with a third or middle term, and then related to each other
to form a new judgment. The middle term is the common unit, or the
_standard_ by which the other terms are measured. To illustrate: If
John and James are each six feet tall, then plainly, they are of the
same height. The standard, or _middle_ term, is “_six feet tall_.”


=2. THE SYLLOGISM.=

Just as the judgment is expressed by means of the proposition, so
mediate inference is best expressed by means of the syllogism.[9] The
following are syllogisms:

  (1) James is six feet tall,
      John is six feet tall,
      Hence James is as tall as John.

  (2) All true teachers are just,
      You are a true teacher,
      Hence you are just.

  (3) All men are mortal,
      You are a man,
      Hence you are mortal.


=3. THE RULES OF THE SYLLOGISM.=

All syllogistic reasoning is conditioned by the following eight rules:

  (1) A syllogism must have three, and only three, different terms.

  (2) A syllogism must have three, and only three, propositions.

  (3) The middle term must be distributed at least once.

  (4) No term must be distributed in the conclusion which is not also
      distributed in a premise.

  (5) No conclusion can be drawn from two negative premises.

  (6) If one premise be negative, the conclusion must be negative; and
      conversely, to prove a negative conclusion, one of the premises
      must be negative.

  (7) No conclusion can be drawn from two particular premises.

  (8) If one premise be particular, the conclusion must be particular.

These rules are exceedingly important, as their observance is necessary
in all mediate reasoning. The student needs, not only to understand
the meaning of these rules, but he needs to commit them to memory so
thoroughly that they may be recalled without hesitation or mistake. To
aid the memory, the eight rules may be divided into these four groups:

    I. Rules one and two relate to the _composition of the syllogism_.

   II. Rules three and four pertain to the _distribution of terms_.

  III. Rules five and six have reference to _negative premises_.

   IV. Rules seven and eight concern _particular premises_.


=4. RULES OF THE SYLLOGISM EXPLAINED.=

(1) _A syllogism must have three and only three terms._

It is common to represent the various syllogistic forms by symbols,
the same symbols always standing for the same terms. In this treatment
we shall let capital G stand for the major term, as “major” means
_greater_; capital S for the minor term, as “minor” means _smaller_,
and capital M for the middle term. G, S and M, the initial letters
of greater (major), smaller (minor) and middle, will be the constant
symbols for these terms; just as A, E, I and O are used as the constant
symbols for the four logical propositions.

_Illustration._

  Syllogism written in full:
                All men are mortal,
                Socrates is a man,
                ――――――――――――――――――
    (Therefore) Socrates is mortal.

  Syllogism symbolized:
                All M is G
                    S is M
                    ――――――
                  ∴ S is G

_The major term is always the predicate and the minor term the
subject of the conclusion._ The conclusion of the foregoing syllogism
is, “Socrates is mortal.” Since G stands for the predicate of every
conclusion, then it stands for “_mortal_,” the predicate of the above
conclusion. For a similar reason, S stands for the subject, namely,
“_Socrates_”; while M represents the middle term, “_man_.”

Since every syllogism must have three propositions, and since it takes
two terms to form a proposition, then it follows that every syllogism
must contain _six terms_. But, as no syllogism can have _more_ than
three _different_ terms, we conclude that each term of the syllogism
must be used _twice_. In the foregoing example, G thus appears,
not only in the last proposition, or conclusion, but in the first
proposition also. Similarly, both S and M occur twice. Every logical
syllogism, then, contains first, _a major term_, which is always the
_predicate_ of the conclusion and appears _once_ in the premises;
second, a _minor term_, which is always the _subject_ of the conclusion
and appears _once_ in the premises; and third, a _middle term_ to which
the other two terms are referred.

There are two ways of locating the middle term; first, it is the term
which is used in _both_ the premises; second, it is the term which
_never appears in the conclusion_. Likewise, there are two ways of
locating the major and minor terms; first, the major term is always
the _predicate_ and the minor term the _subject_ of the conclusion;
second, the major term is usually the _broader_ and the minor term the
_narrower_ of the two. If the major and minor terms seem to be of about
the same extension or breadth, then the term in the first proposition,
which is not the middle term, is the major.

In the attending syllogisms the three terms are designated:

           (middle)           (major)
              |                  |
  (1) All true teachers are sympathetic,

      (minor)      (middle)
         |            |
        You are a true teacher,

      (minor)     (major)
         |           |
      ∴ You are sympathetic.

           (major)        (middle)
              |              |
  (2) No shell fish are vertebrates,

           (minor)      (middle)
              |            |
        All trout are vertebrates,

          (minor)      (major)
             |            |
      ∴ No trout are shell fish.

The necessity of having but _three_ different terms in any syllogism
may be understood by supposing that there are four different terms;
then it would follow that there could be _no standard_ or common link.
In the axiom, “Things equal to the same thing are equal to each other,”
the _same thing_ is the common standard or link. Two things which equal
two different things are not equal to each other.

The impossibility of reasoning from four terms may be shown by circles.

  All men are mortal.
  All trees grow.

  Illustration:   FIG. 8.

These circles show that no connection can be established between either
group. Using four terms in any syllogism is known as the _fallacy of
four terms_.

(2) _A syllogism must have three and only three propositions._ The
proposition containing the major term is called the _major premise_,
while the one containing the minor term is called the _minor premise_.
In a strictly logical syllogism the major premise is written first,
the minor premise second and the conclusion third. In common parlance,
however, the minor premise or even the conclusion may appear first.

The conclusion of a syllogism is always preceded by _therefore_, or
its equivalent, which may be written or understood. The premises always
answer the question, Why is the conclusion true? The premises are often
preceded by such words as _for_ and _because_.

The attending irregular syllogisms are arranged logically and the
premises and conclusions indicated:

(1a) _Illogical._

“You must take an examination because all who enter the school are
examined and you, as I understand it, are planning to enter.”

(2a) “Some of these books are not well bound, for they are going to
pieces as no well bound book would do.”

(1b) _Logical._

  All who enter this school are examined, Major premise.
  You are planning to enter this school, Minor premise.
  You must be examined. Conclusion.

(2b)

  No well bound book goes to pieces, Major premise.
  Some of these books are going to pieces, Minor premise.
  Some of these books are not well bound. Conclusion.

The fact that all syllogisms must have three and only three premises
follows from rule “1.” One premise must compare the middle term with
the “major”; another premise must compare the middle term with the
“minor”; while the conclusion links together the “major” and the
“minor.”

(3) _The middle term must be distributed at least once._ The rule is
usually given in this way, “The middle term must be distributed once at
least, and must not be ambiguous.” In this treatment the last part of
the rule has been omitted because it must be apparent to the student
that a middle term used in two senses is virtually equivalent to _two
different terms_; such an “ambiguous middle” would, in consequence,
give a syllogism of _four terms_.

Rules 3 and 4 are of greater importance than the others because they
are more frequently violated. If the middle term is not distributed
at least once, the fallacy is referred to as “_undistributed middle_.”
If the distributed major term of the conclusion is not distributed
in the major premise, then the fallacy is called, “_illicit process
of the major term_”; and finally, if the distributed minor term of
the conclusion is not distributed in the minor premise the fallacy is
denominated an “_illicit process of the minor term_.” These two illicit
processes may be abbreviated to illicit major and illicit minor.

Recall that any term is distributed when it is referred to as a
definite whole. Unless the whole of the middle term is considered it
fails to become a common standard of comparison. This becomes clear
when recourse is made to the circles.

_Illustration._

Syllogism in which the middle term is not distributed:

    All men are mortal,
    All trees are mortal,
  ∴ All trees are men.

All the propositions are A’s and consequently the predicates of each
are undistributed, as A distributes the subject _only_. Therefore the
middle term, “_mortal_,” is not distributed in either of the premises
and thus the fallacy.

Fallacy shown by circles:

  Illustration:   FIG. 9.

These circles indicate the correct meaning of the two premises. By them
it is seen that all of the “men” circle belongs to the “mortal” circle
and all of the “tree” circle belongs to the “mortal” circle, but in
this case there is no connection between the “men” and “tree” circles.
Thus, to say that “_All trees are men_,” is fallacious. We have no
right to either affirm or deny the connection between men and trees.
If “mortal” were distributed we would have this right as the following
will make clear:

    All men are mortal,
    No stones are mortal,
  ∴ No stones are men.

  Illustration:   FIG. 10.

Here the middle term _mortal_ is distributed in the second premise
as in it the subject “_stones_” is _excluded_ from the entire mortal
territory. This conclusion is verified by the formal statement that “E”
distributes both subject and predicate. Since all of the “men” circle
belongs to the “mortal” circle and none of the “stones” circle belongs
to the “mortal” circle then none of the “stones” circle can belong to
the “men” circle.

(4) _No term must be distributed in the conclusion which is not also
distributed in its premise._

It has been affirmed that a term is distributed when it is referred to
as a definite whole. To put it in another way, a term is distributed
when it is employed in its fullest sense. It is obvious that we should
not employ a term in its fullest sense in the conclusion when it has
been used only in a partial sense in its premise. What is said of the
part cannot necessarily be said of the whole. For example: Because
_some_ men are honest it does not follow that _all_ men are honest. Of
course the converse of this is true, namely, if it could be proved that
all men are honest then surely it would follow that some of the men are
honest. To put it briefly: What is true of _all_ is true of _some_ but
what is true of _some_ is not necessarily true of _all_.

To distribute a term in the conclusion when it is not distributed in
the premise where it occurs is equivalent to saying, “what is true of
some is true of all.” This error which violates rule “4” leads to the
two fallacies of illicit process of the major and minor terms. The
following illustrate the two fallacies.

Syllogism illustrating illicit major:

    All trees grow,
    No men are trees,
  ∴ No men grow.

The first premise is an A and consequently its subject is distributed.
The second premise and conclusion being E’s have both subject and
predicate distributed. Thus _grow_, as used in the conclusion, is
distributed, but, as used in the major premise, it is not distributed.
Fallacy shown by circles:

  Illustration:   FIG. 11.

Here all of the “tree” circle belongs to the “grow” circle and none
of the “men” circle belongs to the “tree” circle, hence the diagram
correctly represents the meaning of the two premises and shows the
fallacy of concluding that _no men grow_. The “men” circle, being
entirely within the “grow” circle, indicates that _all men grow_.
Syllogism illustrating illicit minor:

    All true teachers are just,
    All true teachers are sympathetic,
  ∴ All the sympathetic are just.

Each proposition being an A distributes its subject. But the subject of
the conclusion which is “_the sympathetic_” is not distributed in the
minor premise, as an A proposition distributes its subject only. Hence
the fallacy of illicit minor.

Fallacy shown by circles:

  Illustration:   FIG. 12.

The diagram correctly represents the two premises since all of the
“true teacher” circle belongs to both the “just” and “sympathetic”
circles. But all of the “sympathetic” circle does not belong to the
“just” circle. Hence the fallacy.

(5) _No conclusion can be drawn from two negative premises._

When two terms are both denied of a third term, it is quite impossible
to draw any conclusion relative to the two terms, as the absolute
exclusion of the third term eliminates any possibility of a common link
or standard.

The circles will make this apparent:

    No men are immortal,
    No trees are immortal,

  Illustration:   FIG. 13.

“No trees are men” is the conclusion represented by Fig. 13.

Other possible conclusions are, “_All trees are men_,” “_All men are
trees_” and “_Some men are trees_.”

It is thus seen that no definite conclusion can be drawn. It may now
be said that when the major and minor terms are used in two negative
premises the connection between them is _indeterminate_. This violation
of rule “5” may be termed the fallacy of _two negatives_.

(6) _If one premise be negative the conclusion must be negative; and
conversely, to prove a negative conclusion one of the premises must be
negative._

Referring to the first part of this rule, it may be said of two terms
that if one is affirmed and the other denied of a third term, then the
two terms must be denied of each other. The attending syllogism and its
“circled” representation will throw light upon this:

    No men are immortal,
    All Americans are men,
  ∴ No Americans are immortal.

  Illustration:   FIG. 14.

Since none of the “men” circle belongs to the “immortal” circle and all
of the “American” circle is inside the “men” circle, it is evident that
none of the “American” circle can belong to any part of the “immortal”
circle. Thus it is manifest that an affirmative conclusion like, “All
Americans are immortal,” is invalid.

The converse of rule 6, “To prove a negative conclusion, one of the
premises must be negative,” may be explained by the general principle
in logic that when two terms are known to disagree, one must agree with
a third term while the other must disagree. If both agreed with a third,
then the conclusion would of necessity be affirmative. If both
disagreed no conclusion could be drawn. A violation of rule 6 may be
called the fallacy of _negative conclusion_.

(7) _No conclusion can be drawn from two particular premises._ Proof:

  (1) All the possible combinations of the two particular premises
      I and O are, (1) IO, (2) OI, (3) II, (4) OO.

  “IO” considered.

  (2) Since O is a negative premise the conclusion would have to be
      negative according to rule 6. (If one premise is negative, the
      conclusion must be negative.)

  (3) If the conclusion is negative, then its predicate, which is
      the major term, must be distributed. (All negative propositions
      distribute their predicates.)

  (4) If the major term is distributed in the conclusion, it must
      be distributed in the major premise, rule 4. (No term must be
      distributed in the conclusion, which is not also distributed in
      one of the premises.)

  (5) Hence two terms must be distributed in the premises, the major
      term according to (4) and the middle term according to rule 3.

  (6) But I distributes neither term and O distributes its predicate
      only; I and O together, then, distribute but _one_ term.

  (7) To draw a negative conclusion the premises must distribute two
      terms, the middle and the major, according to the foregoing.

  (8) Hence a conclusion from I and O is untenable. The same may be
      said of “OI.”

  “II” considered.

  (1) The I proposition distributes neither subject nor predicate,
      hence the premises “II” would distribute no term.

  (2) But the middle term must be distributed at least once according
      to rule 3.

  (3) Therefore no conclusion can be drawn from “II.”

  A valid conclusion from “OO” is impossible according to rule 5.

(8) _If one premise be particular the conclusion must be particular._
Proof: The possible combinations conditioned by rule 8 are AI, AO, EI,
EO, IO, II, OO.

  “AI” considered.

  (1) Proposition A distributes its subject, proposition I neither;
      hence “AI” together distribute but one term.

  (2) According to rule 3 this one term must be the middle term.

  (3) The minor term must, therefore, be undistributed in the minor
      premise, and in consequence undistributed in the conclusion.

  (4) But this undistributed minor term is the subject of the
      conclusion; hence said conclusion must be particular, as only
      particulars have an undistributed subject.

  “AO” and “EI” considered.

  Proof:

  (1) “AO” distribute two terms; so do “EI.”

  (2) Both “AO” and “EI” must have negative conclusions according to
      rule 6.

  (3) A negative conclusion distributes its predicate which is the
      major term.

  (4) The major term and the middle term must be distributed in the
      premises. Rules 4 and 3.

  (5) Thus the third term, which is the minor, cannot be distributed
      in the minor premise and, consequently, the minor cannot be
      distributed in the conclusion.

  (6) This necessitates a particular conclusion.

Premises EO and OO, being negative, cannot yield a conclusion according
to rule 5; similarly, neither can the particulars IO and II because of
rule 7.


=5. THE DICTUM OF ARISTOTLE.=

Aristotle gives an axiom on which all syllogistic inference is
based. Indeed from this fundamental principle the significant rules
of the syllogism could be derived. The dictum is stated in this wise:
“Whatever is predicated, whether affirmatively or negatively, of
a term distributed may be predicated in the manner of everything
contained under it.” The following statements represent various ways
of explaining this dictum:

  (1) Whatever is said of a term used in its fullest sense may likewise
      be said of that term when used only in a partial sense.

  (2) What is true of the whole is true of the part.

  (3) “What pertains to the higher class pertains also to the lower.”
      Since this dictum is the basic principle underlying the important
      rules of the syllogism, it is unnecessary to dwell longer
      upon it; because an explanation of the rules is, virtually, an
      explanation of the dictum.


=6. CANONS OF THE SYLLOGISM.=

The dictum of Aristotle is ostensibly a self-evident truth, and some
logicians have put this truth in the form of three axiomatic statements
which are known as the _canons of the syllogism_. These are as follows:

  (1) “Two terms agreeing with one and the same third term agree with
      each other.”

  (2) “Two terms of which one agrees and the other does not agree with
      one and the same third term, do not agree with each other.”

  (3) “Two terms both disagreeing with one and the same third term may
      or may not agree with each other.”

Making use of the symbols as explained on a previous page of this
chapter, it will be seen that the first canon conforms to this
syllogistic type:

    All M is G
    All S is M
    ――――――――――
  ∴ All S is G

The two terms are S and G, while M is the third term.

The attending symbolizations illustrate, respectively, the second and
third canons:

    No M is G
    All S is M
    ――――――――――
  ∴ No S is G

    No M is G
    No S is M
    Conclusion indeterminate.


=7. THREE MATHEMATICAL AXIOMS.=

Analogous to the three canons treated in “6,” there are certain
mathematical axioms which are here stated:

  (1) “Things equal to the same thing are equal to each other.”

  (2) “One thing equal to and the other thing not equal to the same
      third thing are not equal to each other.”

  (3) “Things not equal to the same thing may or may not equal each
      other.”

Illustrations of the three axioms:

  (1) If x equals 5, and y equals 5, then x equals y.

  (2) If x equals 5, and y does not equal 5, then x does not equal y.

  (3) If x does not equal 5, and y does not equal 5, then x may or may
      not equal y.


=8. OUTLINE.=

  MEDIATE INFERENCE.

  (1) Inference and reasoning.
      Definitions. Middle term explained.

  (2) The analogy between the judgment and the syllogism.

  (3) Rules of the syllogism given. Eight in number.

  (4) Rules of the syllogism explained:

      Rule 1. Syllogistic symbols.
              Major, minor, and middle terms; how found.
              Fallacy of four terms.

      Rule 2. Major and minor premises and conclusion, how determined.
              Logical arrangement.
              Reason for three propositions.

      Rule 3. Reason for omitting “ambiguous middle” from rule.
              Undistributed and distributed middle explained.

      Rule 4. Illicit major and minor explained and illustrated.

      Rule 5. Fallacy of two negatives.

      Rule 6. Fallacy of negative conclusion.

      Rule 7. Fallacy of two particulars.

      Rule 8. Fallacy of particular conclusion.

  (5) Aristotle’s dictum.

  (6) Canons of the syllogism.

  (7) Mathematical axioms.


=9. SUMMARY.=

(1) Inference is a term used to denote a process as well as a product.
As a process reasoning and inference are in reality synonomous terms.

Inference is a thought process of deriving a judgment from one or two
antecedent judgments.

Mediate inference is inference by means of a middle term. Mediate
inference makes use of three terms, two of which are compared with a
third term as a standard. This third term is called the middle term.

(2) The syllogism is the common mode of expression for mediate
inference.

(3) Valid syllogistic reasoning is conditioned by eight rules. The
first and second relate to the composition of the syllogism; the
third and fourth to the distribution of terms; the fifth and sixth
to negative premises; the seventh and eighth to particular premises.

(4) All syllogisms must have three terms: the major, the minor, and the
middle. The middle term occurs twice in the premises but never appears
in the conclusion. The minor term is always the subject, and the major
term the predicate of the conclusion. The major term is usually broader
than the minor.

No conclusion can be drawn from four terms. To attempt this gives rise
to the fallacy of four terms.

All syllogisms must have three propositions, the major and the
minor premises, and the conclusion. The major premise first and the
minor second is the more logical arrangement, although the common
conversational form is to use the minor premise first.

Ambiguous middle amounts to the fallacy of four terms.

Unless the middle term is distributed at least once in the syllogism,
it fails to become a common standard.

Distributing a term in the conclusion, without its being distributed in
its premise, is equivalent to asserting that, “What is true of a part
is true of the whole.” This error results in the fallacies of illicit
major and minor.

A conclusion from two negatives is impossible, because of the total
exclusion of the middle term.

Of two terms, if one is affirmed and the other denied of a third term,
then they must be denied of each other; and, conversely, if two terms
are to be denied of each other, one must be affirmed and the other
denied of a given third term. This fundamental principle necessitates
deriving a negative conclusion from two premises when one is negative.
It, likewise, compels the converse of this.

A valid conclusion from two particulars is untenable because of the
two negative fallacies, or some fallacy relative to the distribution
of terms.

One particular premise forces a particular conclusion because of the
fallacies of two negatives, two particulars, and illicit minor.

(5) Aristotle’s dictum simplified means, “What is true of the whole is
true of the part.”

(6) The canons of the syllogism, three in number, are:

  (1) “Two terms agreeing with one and the same third term agree with
      each other.”

  (2) “Two terms of which one agrees and the other does not agree with
      one and the same third term do not agree with each other.”

  (3) “Two terms both disagreeing with one and the same third term may
      or may not agree with each other.”

(7) The foregoing canons may be stated as mathematical axioms.


=10. ILLUSTRATIVE EXERCISES.=

(1a) Make use of the proper symbols and indicate the three terms of
each of the attending syllogisms:

  (1) All fixed stars twinkle,
      Vega is a fixed star,
      ――――――――――――――――――――――――
    ∴ Vega twinkles.

  (2) All men are rational beings,
      No tree is a rational being,
      ――――――――――――――――――――――――――――
    ∴ No trees are men.

  (3) All good citizens are law abiding,
      All good citizens vote,
      ――――――――――――――――――――――――――――――――――
    ∴ Some who vote are law abiding.

I recall that the three terms are the middle, the major and the minor,
and that the “middle” does not occur in the conclusion, whereas the
“major” is always the predicate and the “minor” the subject of the
conclusion. The symbols M, G and S being the initial letters of middle,
greater and smaller, I make use of these in designating the three terms,
as the following will illustrate:

               M         G
  (1) All fixed stars twinkle,

        S            M
      Vega is a fixed star,
      ――――――――――――――――――――――――
        S     G
    ∴ Vega twinkles.

“Twinkles” being the predicate of the conclusion is designated as being
the major term by putting the letter G above it. Then “G” is placed
above the term “twinkle” in the first premise.

“S” is placed above the subject of the conclusion to indicate that it
is the minor term. “S” is also placed above “Vega,” the minor term, as
found in the second premise.

The remaining term, “fixed stars,” must be the middle term, therefore
I place “M” above it. The fact that “fixed star” does not occur in the
conclusion verifies this.

Using only the symbols, the syllogism takes this form:

    All M is G
        S is M
        ――――――
      ∴ S is G

Using the symbols to represent the other syllogisms, we have

  (2) All G is M
       No S is M
       ―――――――――
     ∴ No S is G

  (3) All M is G
      All M is S
          ――――――
   ∴ Some S is G

(1b) Indicate by symbols the three terms of the following syllogisms:

  (1) No trees are men,
      All rational beings are men,
      ――――――――――――――――――――――――――――
    ∴ No rational being is a tree.

  (2) All men have the power of speech,
      You are a man,
      ―――――――――――――――――――――――――――――――――
    ∴ You have the power of speech.

  (3) Some men are wise,
      All men are rational,
      ―――――――――――――――――――――
    ∴ Some rational beings are wise.

(2a) Illustrate by syllogism the fallacy of undistributed middle. An
easy way is to use _the middle term as the predicate of two A premises_.
This yields the fallacy because an A proposition does not distribute
the predicate.

The illustration: distributed terms underscored.

    All true teachers are students,
        ―――――――――――――
    All scholars are students,
        ――――――――
    ――――――――――――――――――――――――――――――――
  ∴ All scholars are true teachers.
        ――――――――

(2b) Give two illustrations of undistributed middle.

(3a) Give syllogistic illustrations of the fallacies of illicit major
and minor.

_Illicit Major._

_Use the middle term as the subject of an A proposition, and then as
the predicate of an E proposition._ This would necessitate a negative
conclusion in which the major term is distributed. But the major term
is not distributed in the major premise, hence the fallacy.

Illustration in which the distributed terms are underscored:

    All men are mortal,
        ―――
    No trees are men,
       ―――――     ―――
    ―――――――――――――――――――
  ∴ No trees are mortal.
       ―――――     ――――――

_Illicit Minor._

To illustrate this fallacy one may use _the middle term as the subject
of two A premises_. This would give an A conclusion in which the
subject is distributed. But this same term is not distributed in its
premise because here it is used as the predicate of an A. Illustration:

    All earnest students study,
        ――――――――――――――――
    All earnest students desire to succeed,
        ――――――――――――――――
    ―――――――――――――――――――――――――――――――――――――――
  ∴ All who desire to succeed study.
    ―――――――――――――――――――――――――


=11. REVIEW QUESTIONS.=

  (1) Distinguish between inference and reasoning.

  (2) Define inference. Mediate inference.

  (3) Illustrate the difference between mediate and immediate
      inference.

  (4) Explain by illustration the use of the middle term.

  (5) Exemplify the syllogism.

  (6) State the rules of the syllogism.

  (7) From the attending syllogisms select the three terms:

      (1) All patriotic citizens vote,
          You are a patriotic citizen,
        ∴ You should vote.

      (2) No honest man would misrepresent,
          (but) John Smith did misrepresent,
        ∴ John Smith is not honest.

  (8) Symbolize the foregoing syllogisms.

  (9) Illustrate by syllogisms the fallacy of four terms.

 (10) Indicate by circles that a valid conclusion cannot be drawn from
      four terms.

 (11) Why must a syllogism have three and only three propositions?

 (12) Indicate how the three propositions of an argument may be
      designated. What is the logical arrangement?

 (13) Show that an ambiguous middle amounts to a fallacy of four terms.

 (14) Explain and illustrate undistributed middle, illicit major,
      illicit minor.

 (15) Exemplify the fallacies of question “14” by using circles.

 (16) Explain by circles why a conclusion cannot be drawn from two
      negatives.

 (17) Make clear that a negative conclusion must follow, if one premise
      be negative.

 (18) State and explain the principle which underlies the rule, “If the
      conclusion is negative one premise must be negative.”

 (19) Prove by the process of elimination that no conclusion can be
      drawn from two particulars.

 (20) In a way similar to that of question “19” show that if one
      premise be particular the conclusion must be particular.

 (21) State and explain Aristotle’s dictum.

 (22) State the canons of the syllogism.

 (23) Symbolize and explain by circles the three canons.

 (24) Illustrate the three mathematical axioms which the canons
      suggest.


=12. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Give an illustration of a valid conclusion being drawn from four
      terms.

  (2) Explain by circles the foregoing.

  (3) From three different business transactions, select the middle
      term of comparison.

  (4) Why should not those who are given to much which is argumentative,
      speak in syllogistic terms?

  (5) “He is a man of high ideals, and you know him to be strictly
      honest, therefore you have no excuse for not voting for him.”
      Recast this quotation with a view of making a logical syllogism.

  (6) Show by circles that there may be a vital difference between a
      _syllogism_ of three terms and an _equation_ of three terms.

  (7) Indicate by illustration that in conversational argumentation the
      minor premise naturally comes first.

  (8) Show by circles the meaning of “indeterminate conclusion.”

  (9) Rule five states that no conclusion can be drawn from two
      negatives. Defend this rule in connection with the following
      syllogism, which seems to contain a valid conclusion:

          Any statement which is not true cannot be accepted,
          This statement is not true,
        ∴ It cannot be accepted.

 (10) If the conclusion is particular, must one premise be particular?
      Explain.



                              CHAPTER 12.

                  FIGURES AND MOODS OF THE SYLLOGISM.


=1. THE FOUR FIGURES OF THE SYLLOGISM.=

By a figure of a syllogism is meant some particular arrangement of the
three terms in the two premises. The conclusion is eliminated from this
discussion, because in it the arrangement of the terms is constant, the
major term always being used as the predicate of the conclusion and the
minor as the subject. Using the symbols M, G and S, we find that there
are four possible arrangements and, therefore, but _four figures_.
These may be represented as follows:

    First     Second    Third     Fourth
    figure    figure    figure    figure
    M ― G      G ― M     M ― G     G ― M
    S ― M      S ― M     M ― S     M ― S
    ―――――      ―――――     ―――――     ―――――
    S ― G      S ― G     S ― G     S ― G

No matter what the syllogism, if it is to be proved “_logical_,” it
should be made to fit one of the four figure-types. To be sure, it may
fit the figure without being logical, but it cannot be strictly logical
without fitting the figure. The following valid syllogisms conform to
the four figures as will be seen by the symbolized terms:

                     M        G
  First figure: All men are mortal,

                   S           M
                Socrates is a man,

                   S          G
              ∴ Socrates is mortal.

                M ― G
                S ― M
                ―――――
                S ― G

                         G                       M
  Second figure: All good citizens love their country,

                       S                  M
                 No criminal loves his country,

                       S              G
               ∴ No criminal is a good citizen.

                 G ― M
                 S ― M
                 ―――――
                 S ― G

                        M                G
  Third figure: All good citizens are law abiding,

                        M          S
                All good citizens vote,

                          S          G
              ∴ Some who vote are law abiding.

                M ― G
                M ― S
                ―――――
                S ― G

                         G             M
  Fourth figure: Some teachers are fair minded,

                                 M            S
                 All who are fair minded are just,

                          S               G
               ∴ Some just persons are teachers.

                 G ― M
                 M ― S
                 ―――――
                 S ― G

Here, then, are the types that represent all the syllogisms which
mediate inference may use. _Logic recognizes no other._ Since every
successful student of logic must be familiar with the four figures, the
following may be used as a suggestive aid to reproducing the figures at
will:

First. It is easy for any one to remember this syllogism:

    All men are mortal,
    Socrates is a man,
  ∴ Socrates is mortal.

In fact, it comes down to us from the time of Aristotle, and is
therefore a patriot of many generations to whom the faithful should
touch their hats. Let us, then, be ready to reproduce this syllogism
with automatic precision, since it will enable us to know at once the
_position of the terms_ in the first figure. Second. Converting the
terms of the _major premise_ of the first figure gives the second
figure, as, e. g.:

    First figure.             Second figure.
       M ― G       (Convert)      G ― M
       S ― M                      S ― M
       ―――――                      ―――――
       S ― G                      S ― G

Third. Converting the terms of the _minor premise_ of the first figure
gives the third figure, as, e. g.:

    First figure.             Third figure.
       M ― G                      M ― G
       S ― M       (Convert)      M ― S
       ―――――                      ―――――
       S ― G                      S ― M

Fourth. Converting the terms of _both the major and minor premises_ of
the first figure gives the fourth, as, e. g.:

    First figure.             Fourth figure.
       M ― G       (Convert)      G ― M
       S ― M       (Convert)      M ― S
       ―――――                      ―――――
       S ― G                      S ― G

To summarize: _The second, third and fourth figures may be derived from
the first. Converting the major premise of the first figure gives the
second figure; converting the minor premise gives the third figure; and
converting both premises gives the fourth figure._


=2. THE MOODS OF THE SYLLOGISM.=

By the mood of a syllogism is meant some particular arrangement of
the propositions which compose the syllogisms. “_Mood_” stands for
an arrangement of the _propositions_, while “_figure_” represents an
arrangement of the _terms_ in any syllogism.

Combining any three of the four logical propositions gives a mood, as,
e. g.,

    (1) E   (2) A   (3) E
        A       I       I
        E       I       O

are moods. The first one has an E proposition for the major premise, an
A for the minor and an E for the conclusion. This syllogism represents
the first mood given above:

    E   No men are trees,
    A   All Americans are men,
    E ∴ No Americans are trees.

It would not be difficult to determine by actual experiment, just how
many moods could be formed, and of these, how many would admit of valid
conclusions. It may be seen that there are sixty-four permutations of
the four logical propositions, taken three at a time. These are in part:

    (1)   (2)   (3)   (4)   (5)   (6)   (7)   (8)
     A     A     A     A     A     A     A     A
     A     A     A     A     E     E     E     E
     A     E     I     O     A     E     I     O

    (9)  (10)  (11)  (12)  (13)  (14)  (15)  (16)
     A     A     A     A     A     A     A     A
     I     I     I     I     O     O     O     O
     A     E     I     O     A     E     I     O

And so the permutations could be continued. Substituting E for the
major premise of the above group would give another group of sixteen,
while a like substitution of I and O would result in two more groups,
sixteen in each. This gives sixty-four in all.[10]


=3. TESTING THE VALIDITY OF THE MOODS.=

In order to put the moods to good use, it is necessary to ascertain
which ones yield a valid conclusion in any figure. If each were valid
in all of the four figures, there would be 256. But it is obvious that
such is not the case.

Referring to the sixteen permutations given above, we find that the
_“negative-conclusion” rule_ makes invalid 2, 4, 5, 7, 10, 12, 13 and
15; whereas the rule for particulars throws out 9 and 14. This leaves
the following as the probable valid moods in one or more of the figures:
1, 3, 6, 8, 11, 16. But to be certain of this the investigation must be
                    A
continued. The mood A has stood the test of the rules for negative and
                    A
particular conclusions; now let us test this mood from the standpoint
of the distribution of terms, using it in all four figures:


        _First_   _Second_  _Third_   _Fourth_
    A    M ― G     G ― M     M ― G     G ― M
         ―         ―         ―         ―
    A    S ― M     S ― M     M ― S     M ― S
         ―         ―         ―         ―
        ―――――――   ―――――――   ―――――――   ―――――――
    A    S ― G     S ― G     S ― G     S ― G
         ―         ―         ―         ―

As an A proposition distributes its subject only, we underscore the
subject of each proposition in all the figures. (This underscoring is
a simple way to indicate distribution.)

We now find that the mood is valid in the first figure, because the
middle term is distributed at least once; namely, in the major premise,
and there is no term distributed in the conclusion which is not already
distributed in the premise where it occurs. On the other hand, the mood
A
A is invalid in the second, because of “_undistributed middle_,” and
A
invalid in the third and fourth, because S is distributed in the
conclusion but not distributed in the premise where it occurs (illicit
minor).

Let us try AII in the four figures:

    A   M ― G     G ― M     M ― G     G ― M
        ―         ―         ―         ―
    I   S ― M     S ― M     M ― S     M ― S
        ―――――     ―――――     ―――――     ―――――
    I   S ― G     S ― G     S ― G     S ― G

We underscore the subject of the A proposition in each of the four
figures. As I distributes neither subject nor predicate, no other term
                                              A
should be underscored. It is now evident that I is not valid in figures
                                              I
two and four, because in both figures the middle term is undistributed
(undistributed middle).

In a like manner all the other moods might be tested. Logicians, who
have done this, have found 24 to be valid. Five of these have weakened
conclusions; i. e., a particular conclusion when it could just as well
              A
be universal. E illustrates this as the conclusion _could be E_. This
              O

syllogism exemplifies the weakened conclusion:

  A   All trees grow,
  E   No sticks are trees,
  O ∴ Some sticks do not grow.

This conclusion is true, since “some” means “some at least.” Yet the
conclusion is weak, because there is nothing to interfere with the
broader and stronger conclusion that, “No sticks grow.” There are,
therefore, only 19 valid and serviceable moods. These are as follows:

                    (1)   (2)   (3)   (4)   (5)   (6)
                  {  A     E     A     E     ―     ―  }
  First figure    {  A     A     I     I     ―     ―  }   4
                  {  A     E     I     O     ―     ―  }

                  {  E     A     A     E     ―     ―  }
  Second figure   {  A     E     O     I     ―     ―  }   8
                  {  E     E     O     O     ―     ―  }

                  {  A     I     A     E     O     E  }
  Third figure    {  A     A     I     A     A     I  }  14
                  {  I     I     I     O     O     O  }

                  {  A     A     I     E     E     ―  }
  Fourth figure   {  A     E     A     A     I     ―  }  19
                  {  I     E     I     O     O     ―  }

                                                                 A
Of these nineteen moods it is not much of a tax to remember that A is
                                                                 A
                                        E
valid only in the first figure; whereas A is valid in the first and
                                        E
                A                               E
second figures; I in the first and third; while I is valid in all. This
                I                               O
knowledge, however, should be used only as one would employ the answers
in arithmetic. Testing the validity of a mood in the four figures is an
exceedingly valuable thought-exercise, which a knowledge of the final
result might easily vitiate. It is, no doubt, best to test the value
of any mood without such knowledge, and then compare the result by
referring to the foregoing list of valid moods. It is not always wise
to work with the answer in mind, yet it is most satisfying to know of
a _certainty_ that one’s reasoning has led to a truth which others have
verified.


=4. SPECIAL CANONS OF THE FOUR FIGURES.=

As a deductive exercise in clear, logical thought, the indirect proof
involved in establishing certain principles underlying the four figures,
is of immense value. On no account should this section be omitted. The
mere fact that it appears to be a difficult section is proof positive
that the student is in need of just such exercises.

_Canons of the first figure._

  (1) The minor premise must be affirmative.

  (2) The major premise must be universal.

_Problem: The minor premise must be affirmative._

_Data_: Given the form of the first figure, which is,

    M ― G
    S ― M
    ―――――
    S ― G

_Proof_: (1) If the minor premise is not affirmative then it must
be negative; because affirmative and negative propositions, being
contradictory in nature, admit of no middle ground.

(2) If the minor premise is negative, the conclusion must be negative;
for the reason that a negative premise necessitates a negative
conclusion.

(3) If the conclusion is negative then its predicate, G, must be
distributed; since all negatives distribute their predicates.

(4) If the predicate of the conclusion, which is the major term, is
distributed, then it must be distributed in the premise where it occurs,
which is the major premise; for any term which is distributed in the
conclusion must be distributed in the premise where it occurs.

(5) If the major term, which is the predicate of the major premise,
is distributed, then the major premise must be negative; because only
negatives distribute their predicates.

(6) The result of this argument, then, gives _two_ negative premises,
and we know from rule 3 that a conclusion from two negatives is
untenable.

(7) Since the minor premise cannot be negative, it must be affirmative.

_Problem: To prove that the major premise must be universal._

_Data_: Given the form of the first figure:

    M ― G
    S ― M
    ―――――
    S ― G

_Proof_: (1) The predicate of the minor premise, M, which is the middle
term, is undistributed; because no affirmative proposition distributes
its predicate.

(2) The middle term must be distributed in the major premise; since in
any syllogism the middle term must be distributed at least once.

(3) As the middle term, M, used as the subject of the major premise,
must be distributed, then the major premise must be universal; because
only universals distribute their subjects.

_Epitome._

_In the first figure, the minor premise must be affirmative, since
making it negative necessitates making the major premise negative also;
the major premise must be universal in order to distribute the middle
term at least once._

_Special canons of the second figure._

(1) One premise must be negative.

(2) The major premise must be universal.

_Problem: To prove that one premise must be negative._

_Data_: Given the form of the second figure:

    G ― M
    S ― M
    ―――――
    S ― G

_Proof_: (1) The middle term, M, is the predicate of both premises.

(2) The middle term must be distributed at least once, according to
rule 3.

(3) Hence one premise must be negative; since only negatives distribute
their predicates.

_Problem: To prove that the major premise must be universal._

_Data_: Given the form of the second figure:

    G ― M
    S ― M
    ―――――
    S ― G

_Proof_: (1) As one premise must be negative, it follows that the
conclusion must be negative according to rule 6.

(2) If the conclusion is negative, then its predicate, G, the major
term, must be distributed; since all negatives distribute their
predicates.

(3) When distributed in the conclusion, the major term, G, must also be
distributed in the major premise, where it is used as the subject. See
rule 4.

(4) Hence the major premise must be universal; for only universals
distribute their subjects.

_Epitome._

_In the second figure one premise must be negative in order to
distribute the middle term at least once; and the major premise must be
universal that the major term, which is distributed in the conclusion,
may be distributed in the premise where it occurs._

_Canons of the third figure._

(1) The minor premise must be affirmative.

(2) The conclusion must be particular.

_Problem: To prove that the minor premise must be affirmative._

_Data_: Given the form of the third figure, which is,

    M ― G
    M ― S
    ―――――
    S ― G

_Proof_: (1) Suppose the minor premise were negative, then the
conclusion would have to be negative, and this would distribute the
predicate G.

(2) A distributed predicate would necessitate its being distributed in
the major premise.

(3) But G, being the conclusion of the major premise, could be
distributed only by a negative proposition.

(4) This would result in two negatives; therefore no conclusion could
be drawn, if the minor premise were negative.

_Problem: To prove that the conclusion must be particular._

_Data_: Given the form of the third figure:

    M ― G
    M ― S
    ―――――
    S ― G

_Proof_: (1) The minor term, which is the predicate of the affirmative
minor premise, is undistributed; because no affirmative distributes its
predicate.

(2) If undistributed in the premise, then the minor term must remain
undistributed in the conclusion, where it is used as the subject.

(3) The conclusion must, then, be particular; since all universals
distribute their subjects.

_Epitome._

_In the third figure, unless the minor premise be affirmative, there
can be no conclusion; since a negative minor would necessitate a
negative major. An affirmative minor compels a particular conclusion,
in order that the minor term, in the conclusion, may remain
undistributed._

_Canons of the fourth figure._

(1) If the major premise is affirmative, the minor premise must be
universal.

(2) If the minor premise is affirmative, the conclusion must be
particular.

(3) If either premise is negative, the major must be universal.

_Problem: To prove that if the major is affirmative, the minor must be
universal._

_Data_: Given the form of the fourth figure:

    G ― M
    M ― S
    ―――――
    S ― G

_Proof_: (1) If the major premise is affirmative, then its predicate
which is the middle term, M, is undistributed; for no affirmative
distributes its predicate.

(2) The middle term must then be distributed in the “minor” according
to rule 3.

(3) Then the “minor” must be universal; since only universals
distribute their subjects.

_Problem: To prove that if the minor is affirmative, the conclusion
must be particular._

_Data_: Given the form of the fourth figure:

    G ― M
    M ― S
    ―――――
    S ― G

_Proof_: (1) If the minor premise be affirmative, then S, its
predicate, must be undistributed; because no affirmative distributes
its predicate.

(2) Since S is undistributed in the minor premise, it must remain
undistributed in the conclusion where it is used as the subject.

_Problem: To prove that if either premise is negative, the major must
be universal._

_Data_: Given the form of the fourth figure:

    G ― M
    M ― S
    ―――――
    S ― G

_Proof_: (1) If one of the premises is negative, then the conclusion
must be negative according to rule 6.

(2) If the conclusion is negative, then the predicate, G, must be
distributed.

(3) If G is distributed in the conclusion, it must be distributed in
the major premise.

(4) The major premise must be universal; as G is used as its subject,
and only universals distribute their subjects.

_Epitome._

_In the fourth figure, if the “major” is affirmative, the “minor” must
be universal in order to distribute the middle term. If the minor is
affirmative, the conclusion must be particular; otherwise the fallacy
of illicit minor would result. If either premise is negative, the major
must be universal to avoid the fallacy of illicit major._


=5. SPECIAL CANONS RELATED.=

After a particular mood has been tested in the regular way, it has been
intimated that the student may refer to the tabulated list of valid
moods to ascertain, with a certainty, the validity of his reasoning.
This is equivalent to referring to the answers in arithmetic; for
if the student is unable to find the mood in the figure in which he
has proved it valid, then he knows that he has made some mistake in
his reasoning. A second check, though not absolute, is to recall the
special canons of section four. If, for example, our reasoning has led
                   A
us to believe that E is valid in the first figure, we may recall that
                   E
the minor premise of the first figure must be affirmative and therefore
AEE cannot be valid.

A few suggestions relative to memorizing the special canons may not
be out of place. The two canons of the first figure must be committed,
and then it may be remembered that the _second figure is the negative
figure of logic_. Other figures _may_ yield a negative conclusion,
but the second _must_ yield a negative conclusion. Since a negative
conclusion necessitates a negative premise, it follows that the second
figure must always appear with one premise negative. The other canon
which pertains to the major premise is the same as the “major premise”
canon of the first figure.

_The third figure is the particular figure of logic._ Other figures may
yield particular conclusions, but the third _must_ do so. This helps us
to remember the canon that the conclusion of the third figure must be
particular. The other canon which relates to the minor premise is the
same as the “minor premise” canon of the first figure. The canons of
the fourth figure are in reality a summary of the canons of the other
three figures.


=6. MNEMONIC LINES.=

As a device for remembering the 19 valid moods, the logicians of an
earlier day originated a combination of coined words which, though
rather unscientific, may be easily committed to memory. Since, however,
it is of much more value to test the moods by means of the general
rules of the syllogism than it is to try to remember these moods, the
mnemonic lines are of slight value. They are treated here merely as an
item of historical interest.

(1) _Barbara_, _Celarent_, _Darii_, _Ferio_que prioris;

(2) _Cesare_, _Camestres_, _Festino_, _Baroko_, secundæ;

(3) Tertia, _Darapti_, _Disamis_, _Datisi_, _Felapton_, _Bokardo_,
_Ferison_, habet; Quarta insuper addit

(4) _Bramantip_, _Camenes_, _Dimaris_, _Fesapo_, _Fresison_.

The only letters in these lines which mean nothing are l, n, r, t and
small b and d; all the others have a signification. For example, the
_vowels_ of the italicized words signify the various valid moods, as
e. g., the first line indicates the moods AAA, EAE, AII, EIO. The Latin
words, printed in ordinary type, are intended to make evident that the
moods indicated by the artificial italicized words of the first line,
belong to the _first_ figure; that the moods of the next four words,
belong to the _second_ figure; while the _third_ figure includes the
next six, and the _fourth_ figure the last five. It is now seen that
Festino, for example, stands for that mood of the second figure which
has an E for its major premise, an I for its minor premise, and an O
for its conclusion.

The first figure was called by Aristotle the _perfect figure_, whereas
the second and third were the _imperfect figures_. The fourth figure
was given no place in the works of Aristotle; its discovery is credited
to Galen, a celebrated teacher of medicine of the second century.
According to Aristotle, the first figure is the most serviceable and
the most convincing and, therefore, as a final test of their validity,
the moods of the other figures should be changed to the _first_.
This process in logic is termed _Reduction_. In this reduction of
the imperfect figures to the perfect, the _capital letters_ of the
artificial words, together with s, p, m, and k, have a definite meaning.
The capital letters indicate that certain moods of the imperfect
figures can be reduced to the corresponding moods of the first figure;
e. g., Festino (eio) of the second figure, Felapton (eao) of the third
figure, and Fesapo (eao) of the fourth figure may all be reduced to
Ferio (eio) of the first figure. This is known because _F_ is the
initial letter of each word. _s_ signifies that the proposition denoted
by the preceding vowel is to be converted simply. To illustrate: _s_
                                                     E
in Fesapo means that the major premise E of the mood A of the fourth
                                                     O
figure must be converted _simply_ in order to change the mood to Ferio
of the first figure. _p_ indicates that the proposition represented
by the vowel which precedes p must be converted by _limitation_ (per
accidens). _m_ (mutare) makes evident that the premises are to be
interchanged, the major of the old becoming the minor of the new,
and the minor of the old becoming the major of the new. _k_ denotes
that the mood, such as Baroko, must be reduced by a special process
known as _indirect reduction_. These directions may now be followed
as illustrative of the process of reduction.

                                    A
(1) _Given: A syllogism in Darapti_ A
                                    I
                 M              G
    A    All true teachers are just,
                 M                  S
    A    All true teachers are sympathetic,
                         S             G
    I  ∴ Some sympathetic persons are just.

                                      A
The symbols indicate that the mood is A or is in Darapti and that this
                                      I
mood is used in the third figure.

                     A
_Problem_: To reduce A of the third figure to some mood of the first
                     I
figure.

_Process_: D, being the initial letter of Darapti, suggests that its
mood must be reduced to one indicated by a word of the first figure
                                                        A
whose initial letter is D. This mood is in Darii, or is I.
                                                        I

The _p_ in Darapti indicates that the proposition represented by the
preceding vowel must be converted by limitation. This proposition is
the minor premise; converting it by limitation gives: “Some sympathetic
persons are true teachers.” As there are no other significant letters
the reduction is complete and we have this:

               M              G
  A    All true teachers are just,

                       S                M
  I    Some sympathetic persons are true teachers,

                       S             G
  I  ∴ Some sympathetic persons are just.

                                             A
The symbolization indicates that the mood is I of the first figure, or
                                             I
is in Darii.

                                      A
(2) _Given: A syllogism in Camestres_ E
                                      E
               G              M
  A    All true teachers are just,

           S                           M
  E    No one who shows partiality is just,

           S                                G
  E  ∴ No one who shows partiality is a true teacher.

The symbols show that the mood is AEE of the second figure or in
Camestres. Judging from the initial letter C, the mood in Camestres
                                        E
must be reduced to the mood in Celarent A.
                                        E


The letter m between a and e indicates that the major and minor
premises of the given syllogism must be interchanged. The letters
following both e’s suggest that the minor premise and the conclusion
of the syllogism must be converted simply.

This is the resulting syllogism:

              M                 G
  E    No just person shows partiality,

               S                 M
  A    All true teachers are just persons,

              S                  G
  E  ∴ No true teacher shows partiality.

                   E
Here, then, is the A of the first figure or the mood in Celarent.
                   E

According to the ancient theory, reduction is necessary as a matter of
_final_ and _absolute proof_ that the conclusion follows from the given
premises. But, as this claim has been satisfactorily refuted by modern
logicians, we need not give more space to the process. The meaning of
k, as related to “_indirect reduction_,” is explained in most of the
earlier works on logic. See Hyslop, page 193.


=7. RELATIVE VALUE OF THE FOUR FIGURES.=

_The first figure._

The first figure is known as the _perfect_ figure; because it is the
only one which proves _all_ of the four logical propositions. Recalling
the moods of the first figure makes this evident:

     A    E    A    E
     A    A    I    I
    _A_  _E_  _I_  _O_

It is likewise the more natural figure; because it is the only one
which uses both the subject and predicate of the conclusion in the same
relative places as they appear in the premises. Symbolizing the figure
makes this apparent:

    M  ― _G_
   _S_ ―  M
   _S_ ― _G_

The first figure, being the only figure which proves a “universal
affirmation” (A), is used most by the _scientist_; as the object of
science is to establish _universal affirmative_ truths.

_The second figure._

As the second figure conditions negative conclusions only, it is called
the figure of _disproof_, or the exclusive figure. It is easy to see
how negative conclusions may be used to narrow the inquiry down to one
definite theory. For example, suppose it is desired to ascertain which
boy of the five broke the window; by a series of deductions the teacher
may be able to prove that the culprit is not A, not B, not C and not D;
hence the guilty one must be E. This figure is virtually the one used
in diagnosing most diseases.

_The third figure._

The third figure admits of particular conclusions only, and in
consequence is of little value to the scientist. Since, however, the
easiest way to contradict a universal affirmative (A) or a universal
negative (E), is to prove the truth, respectively, of a particular
negative (O) and a particular affirmative (I), it follows that the
third figure serves a purpose.

_The fourth figure._

This figure is so nearly like the first that it is of little value; in
fact, it may be changed to the first by simply interchanging the major
and minor premises. Some authorities refuse to recognize the fourth
figure.


=8. OUTLINE.=

  FIGURES AND MOODS OF THE SYLLOGISM.

  (1) The four figures of the syllogism.
        Definition――symbolization.
        Illustrations――device for remembering.

  (2) The moods of the syllogism.
        _Twenty-four_ valid.

  (3) Testing the validity of the moods.
        Application of the general rules of the syllogism.
        Weakened conclusion――five.
        _Nineteen_ useful moods.
        A thought exercise.

  (4) Special canons of the four figures.
        Proof of the two canons of the first figure.
        Proof of the two canons of the second figure.
        Proof of the two canons of the third figure.
        Proof of the three canons of the fourth figure.

  (5) Special canons related.
        Used as checks.

  (6) Mnemonic lines.
        Their use explained.
        Reduction.

  (7) Relative value of the four figures.


=9. SUMMARY.=

(1) By a syllogistic figure is meant some particular arrangement of the
three terms in the two premises.

This arrangement yields four figures which are designated by the
position of the middle term.

To be logical, any syllogism must conform to _one_ of the four figures.
The first figure is suggested by the position of the terms of the
“Socrates is mortal” syllogism. The second is derived by converting
the _major premise_ of the first; while the third figure results
from converting the _minor premise_ of the first, and the fourth by
converting _both_ major and minor of the first.

(2) By a mood of a syllogism is meant some particular arrangement of
the propositions which compose it.

There are 64 moods but only 24 are valid.

(3) The validity of the various moods may be tested by applying to them
the rules of the syllogism. No mood is valid if it violates any one of
the eight rules.

A “weakened conclusion” is a particular conclusion which could just as
well be universal.

Of the 24 valid moods five have weakened conclusions. This leaves but
19 useful moods.

Testing the validity of the various moods in the four figures is a most
valuable thought exercise.

(4) The deductive exercise involved in establishing certain special
canons of the four figures is of immense value and should not be
omitted.

In the first figure it may be proved (1) that the minor premise must
be affirmative; since making it negative necessitates making the major
premise negative, and no conclusion can be drawn from two negatives;
(2) that the major premise must be universal in order to distribute the
middle term at least once.

In the second figure it may be proved (1) that one premise must be
negative in order to distribute the middle term; (2) that the major
premise must be universal in order to distribute its subject, which
is distributed in the negative conclusion where it appears as the
predicate.

In the third figure it may be proved (1) that the minor premise must
be affirmative in order to prevent the “two negative” fallacy; (2) that
an affirmative minor necessitates a particular conclusion, because the
minor term in the conclusion must remain undistributed.

In the fourth figure it may be proved (1) that if the major is
affirmative, the minor must be universal in order to distribute the
middle term; (2) that if the minor is affirmative, the conclusion must
be particular in order to avoid committing the fallacy of illicit minor;
(3) that if either premise is negative, the major must be universal to
avoid the fallacy of illicit major.

(5) A knowledge of the special canons is helpful in that it may be used
to check fallacious reasoning.

(6) Certain mnemonic lines were used by the Schoolmen as an aid in
recalling the nineteen valid moods, and also as a suggestive device to
aid in the process known as Reduction.

The process of reduction is merely a matter of changing to the first
figure the moods of the other figures. This process is no longer
thought to be necessary.

(7) The first figure, called the perfect figure, is the one used
most by scientists, as it is the only figure which proves a universal
affirmative truth. The second figure is the negative, or figure of
disproof, and is used mainly for the purpose of eliminating all the
conditions of the inquiry save _one_. The third figure serves a purpose
in affording an easy way to contradict a universal assertion; this is
the figure of particulars. The fourth figure, because it so closely
resembles the first, is of little value.


=10. ILLUSTRATIVE EXERCISES.=

Question 1a. By making use of the rules for negatives and particulars,
                                          O   A   A
test the validity of the following moods: I   I   A
                                          A   A   I

Answer: The first mood has the negative O as its major premise, and
the affirmative A as its conclusion; the mood is thus invalid; because
a negative premise necessitates a negative conclusion according to
rule 6.

The second mood contains the particular proposition I as its minor
premise, and thus should have a particular conclusion according to
rule 8. But the conclusion A is universal and, therefore, the mood is
invalid.

The premises of the third mood are universal and the conclusion
particular. The mood, however, is valid, because rule 8 does not work
both ways, as does rule 6. When a universal can just as well be drawn,
then the particular becomes a weakened conclusion.

(1b) Using the rules for negatives and particulars, test the validity
                  A  E  E
of the following: A  O  A
                  E  O  O

(2a) Paying no regard to “figure,” derive as many conclusions as
possible from the following sets of premises: E  A
                                              I  E

Answer: _E_. The major premise of this mood, being negative,
        _I_

necessitates a negative conclusion, according to rule 6, and the minor
premise, being particular, compels a particular conclusion, according
to rule 8. Since the conclusion must be negative and particular, then
                                                            E
O is the only one which can be drawn. The completed mood is I.
                                                            O

_A_. This mood must have a negative conclusion, because the minor
_E_
premise is negative; this would necessitate either E or O; but O
as a conclusion would be, in this case, a _weakened_ one; since E
distributing both terms would necessarily distribute the minor; which
fact would permit the minor to be distributed in the conclusion. Thus
the conclusion could just as well be universal as particular. The
                  A
completed mood is E.
                  E

(2b) From the following sets of premises derive as many conclusions as
possible paying no attention to figure: E   A   O
                                        A   A   A

(3a) Making use of all the general rules of the syllogism, test the
                                                   A
validity of the following mood in all the figures: A.
                                                   I

Answer:       1       2       3       4
        A   M ― G   G ― M   M ― G   G ― M
            ―       ―       ―       ―
        A   S ― M   S ― M   M ― S   M ― S
            ―       ―       ―       ―
            ―――――   ―――――   ―――――   ―――――
        I   S ― G   S ― G   S ― G   S ― G

An underscored symbol indicates a distributed term. Since A distributes
its subject, the subjects of both premises are underscored in all the
figures. No term is underscored in the conclusions; since I distributes
neither term. In the first figure the middle term is distributed in the
major premise, and no term is distributed in the conclusion. Since both
premises are affirmative, the rules for negatives are not applicable;
and as a particular may be drawn from two universals, if there is no
violation of the rules for distribution, this mood seems to be valid in
the first figure. It is, however, a _weakened_ conclusion; since an A
could just as well be drawn. The mood is invalid in the second figure
because of undistributed middle, but valid in both the third and fourth;
since in both cases the middle term is distributed at least once.

(3b) Determine the validity of the attending moods in all the figures
                I  A  E
giving reasons: A  O  A.
                I  O  O


=11. REVIEW QUESTIONS.=

  (1) Define a logical figure and illustrate by means of some ordinary
      syllogistic argument.

  (2) Symbolize the four figures and give suggestions for remembering
      them.

  (3) Write syllogisms which illustrate each of the four figures.

  (4) Define mood as it is used in logic. Illustrate.

  (5) How many moods are valid?

  (6) Explain by illustration a “weakened conclusion.”

                           A                         E
  (7) Test the validity of E in the third figure; of I in the third.
                           E                         O

  (8) Independent of all helps, prove the truth of the canons of the
      first figure.

  (9) In a similar way prove the canons of the second, third and fourth
      figures.

 (10) So far as testing arguments is concerned, what use may be made
      of the special canons of the syllogism?

 (11) Offer a few suggestions for remembering the special canons.

 (12) Why did Aristotle attach so much importance to reduction in
      logic?

 (13) Justify calling the first figure the “perfect figure,” and the
      others the “imperfect figures.”

 (14) Treat of the relative value of the four figures.

 (15) Show by illustration that the second figure is the exclusive
      figure.

                                                   E  O  I
 (16) Test the following moods in all the figures: I  A  A
                                                   A  O  I

                      A  E  E  A  A  E  A  A  A
                      E  I  A  E  I  E  O  A  I.
                      O  O  O  O  E  I  I  I  I


=12. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Give an illustration of a syllogism in the fourth figure which
      might just as well be written in the first figure.

  (2) May a syllogism, which is invalid in the fourth figure, be made
      valid by writing it in the form of the first figure? Prove it.

  (3) Show why it is impossible to apply all the rules of the ***

  (4) Show the difference between a direct and an indirect proof.

                A
  (5) Show that A is valid in the first figure when the major premise
                O
      (A) is co-extensive.

  (6) The third figure is known as the figure of particular
      conclusions. Why should not the second canon of that figure be,
      “_One premise_ must be particular” rather than “The conclusion
      must be particular?”

  (7) Show that there is some ground for thinking that, as a final
      test, moods in the other figures should be reduced to the first.

  (8) Illustrate the fact that the second figure is the figure of
      disproof; whereas the third is the figure of contradictions.

  (9) “To be logical a syllogism must conform to one of the four
      figures, but this does not mean, necessarily, that _all
      arguments_ must conform to some figure.” Explain this.



                              CHAPTER 13.

            INCOMPLETE SYLLOGISMS AND IRREGULAR ARGUMENTS.


=1. ENTHYMEME.=

_An enthymeme is a syllogism in which one of the three propositions is
omitted._

Suppressing the major premise gives an enthymeme of the _first order_;
whereas if the minor premise be suppressed, the enthymeme becomes one
of the _second order_; while omitting the conclusions gives an
enthymeme of the _third order_.

_Illustrations_:

  Complete syllogism.
          All true teachers are just,
          You are a true teacher,
  (Hence) You are just.

  Enthymeme of _first order_; major premise omitted.
          ..........................
          You are a true teacher,
  (Hence) You are just.

  Enthymeme of _second order_; minor premise omitted.
          All true teachers are just,
          ..........................
  (Hence) You are just.

  Enthymeme of the _third order_; conclusion omitted.
          All true teachers are just,
    (And) You are a true teacher,
          ..........................

To argue in terms of the complete syllogism is the _unusual_, not the
_usual_ method. We have a way of abbreviating our remarks; expressing
only the necessary and leaving the obvious to be taken for granted.
Thus the enthymeme becomes the natural form of expression. But the mere
fact that a part of the argument is omitted, makes it more essential
for the student to think clearly and with careful continuity, that no
error may intrude itself.

Probably the most common enthymemes are those of the first order.
This may be explained by the fact that the major premise is usually the
most universal of the three propositions, and, in consequence, the one
which would be the most generally understood. The following represent
enthymemes of this order, gleaned from the ordinary conversation of
ordinary people:

  (1) “Your beets won’t grow, because you are planting them in the
      wrong time of the moon.”

  (2) “You, being a member of the Sunday School, should be ashamed of
      such language.”

  (3) “Being the son of your father, you ought to have some pride in
      this matter.”

  (4) “We are going to have an open winter, because I have observed
      that the hornets’ nests are near the ground.”

  (5) “You had better put in lots of coal, for I have noticed that the
      squirrels have gathered in more nuts than usual.”

Judging from these enthymemes, it would seem to be more natural to
assert the conclusion and follow this by a reason in the form of a
minor premise, leaving the _major_ to the intelligence of the auditor.

The enthymeme of the second order occurs only infrequently, since
it seems to be an unnatural mode of expression, though sometimes it
appears to lend emphasis to the conclusion; e. g., “All untrustworthy
boys come to a bad end, and I predict that you will come to a bad end.”

Enthymemes of the third order are commonly used for the sake of
emphasis, as the following make evident:

  (1) “No business man wants an indolent boy, and you are indolent.”

  (2) “All successful teachers are interested in their work, and you
      plan to be a successful teacher.”

  (3) “Humility is a sign of greatness, and Lincoln possessed this
      quality.”


=2. EPICHEIREMA.=

_An epicheirema is a syllogism in which one or both of the premises is
an enthymeme._ To put it in another way: An epicheirema is a syllogism
in which one or both of the premises is supported by a reason.

When one premise is an enthymeme the syllogism is termed a _single_
epicheirema; whereas when both premises are enthymemes it becomes a
_double_ epicheirema.

_Single epicheirema._

  All men are mortal, _because all men die_,
      Socrates was a man,
    ∴ Socrates was mortal.

_Double epicheirema._

  All men are mortal, _because all men die_,
      Socrates was a man, _because he was a rational animal_,
    ∴ Socrates was mortal.

It is obvious that supporting each premise with a reason lends strength
to the argument. This justifies the use of the epicheirema.


=3. POLYSYLLOGISM.=

_A polysyllogism is a series of syllogisms in which the conclusion of
a preceding syllogism becomes a premise of a succeeding one._

The syllogism in the series whose conclusion becomes a premise of the
succeeding syllogism is termed a _prosyllogism_; while the syllogism
which uses as one of its premises the conclusion of the preceding
syllogism is called an _episyllogism_.

_Illustrations._

                  {  A quadruped is an animal,       }
                  {  A dog is a quadruped,           } _Prosyllogism_
  _Polysyllogism_ {∴ A dog is an animal.           } }
                  {  Fido is a dog,                }   _Episyllogism_
                  {∴ Fido is an animal.            }

                  {  All who libel an associate      }
                  {    are unprofessional,           }
                  {  This teacher has libelled       } _Prosyllogism_
                  {    her associate,                }
  _Polysyllogism_ {∴ This teacher is               } }
                  {    unprofessional.             } }
                  {  All who are unprofessional    }   _Episyllogism_
                  {    should be disciplined,      }
                  {∴ This teacher should be        }
                  {    disciplined.                }


=4. SORITES.=

_A sorites is a series of syllogisms in which all of the conclusions
are omitted except the last one._

Just as the epicheirema is a combination of enthymemes of the _first_
and _second orders_, so the sorites is a combination of enthymemes of
the _third order_. If each conclusion were written, the sorites would
take the form of prosyllogisms and episyllogisms. Two forms of the
sorites are recognized by logicians. These are the _progressive_ or
Aristotelian, and the _regressive_ or Goclenian.

_Illustrations._

                _Progressive_

  _Symbolized._       _Put in Word Form._
        All A is B    Thomas Arnold was a teacher,
        All B is C    A teacher is a man,
        All C is D    A man is a biped,
        All D is E    A biped is an animal,
  Hence all A is E    Hence Thomas Arnold was an animal.

                _Regressive_

        All A is B    A biped is an animal,
        All C is A    A man is a biped,
        All D is C    A teacher is a man,
        All E is D    Thomas Arnold was a teacher,
  Hence all E is B    Hence Thomas Arnold was an animal.

When regarded from the viewpoint of extension, the progressive sorites
proceeds from the smaller to the larger while the regressive is the
converse of this. The point may be illustrated by circles:

  Illustration:   FIG. 15.

  Circle 1 stands for Thomas Arnold.
  Circle 2 stands for teacher.
  Circle 3 stands for man.
  Circle 4 stands for biped.
  Circle 5 stands for animal.

The progressive sorites proceeds from the smaller circle to the larger,
thus:

          All of circle 1 belongs to 2
          All of circle 2 belongs to 3
          All of circle 3 belongs to 4
          All of circle 4 belongs to 5
   Hence, All of circle 1 belongs to 5

The regressive sorites proceeds from the larger to the smaller; i. e.:

          All of circle 4 belongs to 5
          All of circle 3 belongs to 4
          All of circle 2 belongs to 3
          All of circle 1 belongs to 2
   Hence, All of circle 1 belongs to 5

Other differences become apparent when the omitted conclusions are
expressed.

            _Progressive_

  _Symbolized_    _Word Form_
    All A is B    T. Arnold was a teacher, (_A_)
    All B is C    A teacher is a man, (_A_)
  ∴ All A is C  ∴ T. Arnold was a man. (_A_)
    All C is D    A man is a biped, (_A_)
  ∴ All A is D  ∴ T. Arnold was a biped. (_A_)
    All D is E    A biped is an animal, (_A_)
  ∴ All A is E  ∴ T. Arnold was an animal. (_A_)

In the three completed syllogisms it becomes evident that the
progressive sorites uses the minor as its first premise and in
consequence takes the form of the fourth figure, though the reasoning
is according to the first figure.

The progressive sorites must conform to the following rules:

  (1) The first premise may be universal or particular, all the others
      _must_ be universal.

  (2) The last premise may be affirmative or negative; all the others
      _must_ be affirmative.

A violation of the first rule would result in undistributed middle;
whereas a violation of the second rule would give illicit major. These
rules may be illustrated by giving attention to the symbols of the
foregoing completed syllogisms.

The first completed syllogism of the sorites is:

    All A is B
    All B is C
  ∴ All A is C

Securing a logical arrangement by interchanging the major and minor
premises gives:

           (M)  (G)
  (A)   All B is C (First premise universal)
            ―

           (S)  (M)
  (A)   All A is B

           (S)  (G)
  (A) ∴ All A is C
            ―

Applying the rules we find this syllogism valid, or we may recall that
A
A is valid in the first figure.
A

Let us now make the first premise of the sorites _particular_ and test.

    Some A is B
    All B is C
  ∴ Some A is C

_Arranged logically_:

           (M)  (G)
  (A)   All B is C
            ―

            (S)  (M)
  (I)   Some A is B

            (S)  (G)
  (I) ∴ Some A is C

_Proof_:

Since one premise is particular the conclusion must be particular.
(Rule 7) As there are no negatives in the argument, only one conclusion
is possible; namely, a particular affirmative (I). Thus, instead of
the conclusion, “All A is C,” which is an (A), it must be, “Some A
is C,” or an (I). Underscoring the distributed term, it is seen that
the middle term is distributed in the major premise and that no term
is distributed in the conclusion. Thus the mood is valid. This is
“checked” when we recall that AII is always valid in the first figure.
We have now shown that the first premise of a progressive sorites may
be _universal_ or _particular_. Let us further proceed to prove that
all the other premises must be universal.

_Data_: Given the first completed syllogism of the sorites:

    All A is B
    All B is C
  ∴ All A is C

_Proof_: Let any other premise, such as the second, be particular; this
gives the following:

    All A is B
    Some B is C
  ∴ Some A is C

_Arranged logically_: Mood, figure, and distribution indicated.

            (M)  (G)
  (I)   Some B is C

           (S)  (M)
  (A)   All A is B
            ―

            (S)  (G)
  (I) ∴ Some A is C

We note at once that the middle term is undistributed, hence the
     I
mood A is invalid in the first figure; reference to the valid moods in
     I
figure _one_ “checks” this conclusion. Since no premise, other than the
first, can be particular, then all save the first must be universal.

The truth of the first rule has been demonstrated, and now we may
follow a similar plan to prove the truth of the second rule.

_Problem_: To prove that the last premise may be negative.[11]

_Data_: Given the _last_ completed syllogism:

  {   All A is D
  {   All D is E
  { ∴ All A is E

Let us make the last premise negative (E) and test the result. (As all
but the first must be universal we cannot use an O.)

      All A is D
      No D is E
    ∴ No A is E

_Arranged logically and symbolized_:

          (M)  (G)
  (E)   No D is E
           ―    ―

           (S)  (M)
  (A)   All A is D
            ―    ―

          (S)  (G)
  (E) ∴ No A is E
           ―    ―

_Proof_: Negative premise; negative conclusion. No particulars. Middle
term distributed in major premise. No term distributed in conclusion
which is not distributed in premise where it occurs. Syllogism valid.
We must now prove that all the other premises must be affirmative.

_Problem_: To prove that no other premise can be negative, or that all
others must be affirmative.

_Data_: Given last syllogism of sorites with the first premise negative.
(Any other may be taken.)

    No A is D
    All D is E
  ∴ No A is E

_Arranged logically and symbolized_:

           (M)  (G)
  (A)   All D is E
            ―

          (S)  (M)
  (E)   No A is D

          (S)  (G)
  (E) ∴ No A is E
           ―    ―

_Proof_: “G” is distributed in the conclusion but not in the major
premise. Fallacy of illicit major. Hence no other premise can be
negative.

We may now consider the completed syllogisms of the _regressive_
sorites.

    All A is B
    All C is A
  ∴ All C is B
    All D is C
  ∴ All D is B
    All E is D
  ∴ All E is B

By examining the foregoing it becomes apparent that the regressive
sorites, both in form and in the reasoning, adapts itself to the first
figure.

The rules of the regressive sorites are just the reverse of the
progressive. These are:

  (1) The first premise may be negative; all the others must be
      affirmative.

  (2) The last premise may be particular; all the others must be
      universal.

It would be a valuable exercise for the student to test these rules
according to the plan pursued in treating the progressive sorites.


=5. IRREGULAR ARGUMENTS.=

It has been intimated that a syllogistic argument, in order to be
logical, should be made to conform to the _rules of the syllogism_. It
must not be inferred from this, however, that all deductive reasoning
is included by the logical forms here treated. There seem to be
arguments which yield valid conclusions, and yet which are not logical
in the strict sense of the word. The following illustrate some of these
forms:

(1) _ Quantitative Arguments._

    John is taller than James,
    Albert is taller than John,
  ∴ Albert is taller than James.

Here, apparently, is a fallacy of four terms: these four terms are
(1) John, (2) taller than James, (3) Albert, (4) taller than John. Yet
we know that the argument is valid. There is not a particle of doubt in
the mind relative to the truth of the conclusion that “Albert is taller
than James.” We are consequently forced to the inference that such
quantitative arguments lie outside the field of syllogistic reasoning.
The argument involves this new principle, “Whatever is greater than a
_second thing_ which is greater than a _third thing_ is itself greater
than a third thing.”

There are many other arguments similar to this which are not
syllogistic in nature. To wit: A equals B, B equals C, C equals D;
A equals D. A is a brother of B, B is a brother of C, C is a brother
of D; A is a brother of D. A is west of B, B is west of C, C is west
of D; A is west of D.

(2) _Plurative Arguments._

These are arguments in which the propositions are introduced by _more_
or _most_; e. g.:

      Most (more than half) of the team are seniors,
      Most (at least half) of the team are under twenty,
    ∴ Some students under twenty are seniors.

                I
Here we have an I which is evidently valid. No term distributed and yet
                I
the conclusion is unquestionably true. This is due to the fact that the
propositions are so worded as to force an overlapping of the major and
minor terms. The student may illustrate this relation by circles.


=6. OUTLINE.=

  INCOMPLETE SYLLOGISMS AND IRREGULAR ARGUMENTS.

  (1) Enthymeme.
        First, second and third orders.
        Natural form.

  (2) Epicheirema.
        Single, double.

  (3) Polysyllogism.
        Prosyllogism, episyllogism.

  (4) Sorites.
        Progressive, regressive.
        Two rules of each.

  (5) Irregular Arguments.
        Quantitative, plurative.


=7. SUMMARY.=

(1) An enthymeme is a syllogism in which one of the three propositions
is omitted. Suppressing the major premise gives an enthymeme of the
_first_ order; omitting the minor gives one of the _second_ order;
while omitting the conclusion gives one of the _third_ order.

The enthymeme is really the natural form of expression. Enthymemes of
the first order are the most _common_ while those of the third order
are the most _emphatic_.

(2) An epicheirema is a syllogism in which one or more of the premises
is an enthymeme. An epicheirema is said to be _single_ when but one
premise is an enthymeme, and _double_ when both premises are enthymemes.

(3) A polysyllogism is a series of syllogisms in which the conclusion
of the _preceding_ syllogism becomes a premise of the _succeeding_ one.
The one of the series whose conclusion becomes a premise is termed a
prosyllogism; while the one which uses the conclusion as a premise is
called an episyllogism.

(4) A sorites is a series of syllogisms in which all the conclusions
are omitted except the last one.

The two kinds of sorites are the progressive and regressive. The
progressive uses the “minor” as its first premise and adopts the form
of the _fourth_ figure, whereas the regressive uses the “major” as its
first premise and adopts the form of the _first_ figure.

The two rules of the progressive sorites are, (1) “The first premise
may be particular, all the others must be universal”; (2) “The last
premise may be negative, all the others must be affirmative.”

The two rules of the regressive are, (1) “The first premise may be
negative, all the others must be affirmative”; (2) “The last premise
may be particular, all the others must be universal”.

(5) Irregular arguments are such as yield valid conclusions and yet do
not conform to the syllogistic rules.

The quantitative argument expresses quantity and contains four terms.
This argument is based on the principle, “What ever is greater than a
second thing which is greater than a third thing is itself greater than
a third thing.”

Plurative arguments are introduced by “more” or “most” and give in
consequence a valid conclusion from two particulars. This is due to the
overlapping of the major and minor terms.


=8. REVIEW QUESTIONS.=

  (1) Define and illustrate an enthymeme.

  (2) Illustrate the enthymemes of the three orders and point out their
      distinct uses.

  (3) Why should the enthymeme demand closer thought than the ordinary
      syllogism?

  (4) Define and illustrate the epicheirema.

  (5) Of what use is the epicheirema? Illustrate.

  (6) Define and illustrate a prosyllogism and an episyllogism.

  (7) Why are polysyllogisms so called?

  (8) Define and illustrate the sorites.

  (9) Relate the sorites and the epicheirema to the enthymeme.

 (10) Illustrate the two forms of sorites.

 (11) Explain the two forms of sorites by means of a diagram.

 (12) Prove the truth of the two rules of the progressive sorites.

 (13) Illustrate two kinds of irregular arguments and show that they
      are valid.

 (14) Complete the five enthymemes of page 248 and indicate their mood
      and figure.


=9. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Why should enthymemes of the second order be less common than
      those of the first?

  (2) You desire to make it evident to a child that a small beginning
      often leads to a momentous ending; do so in terms of the
      enthymeme of the first order.

  (3) Show that prosyllogism and episyllogism are relative terms.

  (4) When the common premise of the “pro” and “epi” syllogism is
      omitted what abbreviated form results?

  (5) From the viewpoint of your definition criticise this: “A sorites
      is a series of prosyllogisms and episyllogisms in which all of
      the conclusions are suppressed except the last.”

  (6) Prove the truth of the two rules of the regressive sorites.

  (7) Show that the prosyllogism and the episyllogism may be
      progressive or regressive.

  (8) “Reasoning from cause to effect”――is such progressive or
      regressive? Explain.

  (9) Which is inductive in nature, the progressive form of reasoning
      or the regressive? Explain.

 (10) Test the validity of the enthymemes on pages 248 and 249.

 (11) “A sorites is at least as immediately convincing as the chain of
      syllogisms into which it can be decomposed.” Discuss this.



                              CHAPTER 14.

            CATEGORICAL ARGUMENTS TESTED ACCORDING TO FORM.


=1. ARGUMENTS OF FORM AND MATTER.=

The matter relative to the syllogism treated in chapters 11, 12
and 13 is given primarily to enable the reader to test the validity
of categorical arguments. Such arguments must be viewed from the
two standpoints of _form_ and _matter_, since it is one of the chief
purposes of logic to enable the student to detect fallacious reasoning,
no matter how subtly it may be concealed. Therefore, that one may gain
marked facility in this kind of work, it becomes necessary to proceed
with _thoroughness_ and _confidence_. The _meaning_ of arguments and
the various _material_ fallacies may be treated later; but we are now
equipped with sufficient knowledge and experience to test the validity
of arguments from the viewpoint of _form_.


=2. ORDER OF PROCEDURE IN THE FORMAL TESTING OF ARGUMENTS.=

In testing categorical arguments _three_ things are essential; first,
_to follow a definite plan_; second, _to give reasons_; third, _to give
the author the benefit of the doubt_. In view of these essentials, we
suggest this outline which may be helpful to the inexperienced:

  (1) Arrange logically and complete the syllogism.

  (2) Determine the figure and mood by using symbols.

  (3) Apply the rules for negatives and particulars.

  (4) Indicate the distribution by underscoring the terms distributed.

  (5) Apply the rules for distribution.

  (6) Name fallacies, if any, giving reasons.

We recall that to be strictly logical any categorical argument must
take this form: first, major premise; second, minor premise; third,
conclusion. Often in common conversation either the minor premise or
conclusion is given first. Illustrations of this: (1) “He cannot be a
gentleman (conclusion); for no gentleman would do such a thing (major
premise), and there is no doubt but that he did it” (minor premise).
(2) “He has the making of a good teacher (conclusion); because he not
only knows, but he knows how to impart what he knows (minor premise),
and this is a sure sign of a good teacher” (major premise). When the
argument appears in this illogical form, the first duty of the student
is to arrange it logically. To do this he must be able to recognize
readily the premises and the conclusion. To this end these facts may
be of assistance:

  (1) A premise always answers the question “Why”, and is often
      introduced by such words as “_for_,” “_because_,” “_since_,”
      and the like.

  (2) The conclusion is usually introduced by “_therefore_,” “_hence_,”
      “_it follows_,” etc.

  (3) When there are no word-signs those mentioned in the foregoing may
      be inserted with a view of determining which is the conclusion,
      and which are the premises.

_Suggestions relative to completing abbreviated arguments:_

(1) If the conclusion is to be supplied, select the term used twice in
the premises; this, the middle term, must not appear in the conclusion.
The other two terms may now be connected (copulated) to form the
conclusion, the _narrower_ term (minor) being used as the subject,
unless it occurs in what clearly seems to be the major premise. (2) If
either premise is to be supplied, unite the middle term with the
_subject_ of the conclusion for the minor premise, and with the
_predicate_ of the conclusion for the major premise. (3) In supplying
any missing proposition, care should be taken to make the argument
_valid_, if this can be done in conformity with good English, good
sense, and the rules of logic.

As regards the determination of the figure it is well to locate the
middle term first, placing above it the symbol M. Then “G” (greater)
may be placed above the major term and “S” (smaller) above the minor.


=3. ILLUSTRATIVE EXERCISES IN TESTING ARGUMENTS WHICH ARE ALREADY
    COMPLETE, REGULAR, AND LOGICALLY ARRANGED.=

               M           G
  (1) A   All dogs are quadrupeds,
              ――――

                  S           M
      A   All greyhounds are dogs,
              ――――――――――

                  S              G
      A ∴ All greyhounds are quadrupeds.
              ――――――――――

                                                     { A
This argument is in the first figure, the mood being { A. All
                                                     { A
the propositions are affirmative and universal, consequently the
rules pertaining to negatives and particulars are inapplicable. “A”
distributes the subject only, hence all the subjects are underscored.
The middle term “_dog_” is distributed in the major premise, and the
minor term “_greyhound_,” which is distributed in the conclusion, is
likewise distributed in the minor premise. The argument is, therefore,
valid in _form_. This may be verified by referring to a list of valid
moods in the first figure.

                    G                    M
  (2) E   No prejudiced person is open to conviction,
             ―――――――――――――――――    ――――――――――――――――――

                       S                     M
      A   All fair minded persons are open to conviction,
              ―――――――――――――――――――

                      S                 G
      E ∴ No fair minded person is prejudiced.
             ――――――――――――――――――

                                           { E
The argument is in the second figure; mood { A. There is one
                                           { E
negative premise and the conclusion is negative; no particulars.
“E” distributes both terms, “A” the subject only. The middle term
is distributed in the major premise. Both major and minor terms are
distributed in the conclusion, but they are likewise distributed in
the premises where they are used. The argument is, therefore, valid.
Reference to the valid moods of the second figure confirms this
conclusion.

                    M        G
  (3) A   All good citizens vote,
              ―――――――――――――

                    M            S
      A   All good citizens obey the law,
              ―――――――――――――

                     S          G
      A ∴ All who obey the law vote.
              ――――――――――――――――

            { A
The mood is { A used in the third figure. All the propositions are
            { A
A’s, hence the negative and particular rules are inapplicable. “A”
distributes its subject. The middle term is distributed in both
premises. “All who obey the law” is distributed in the conclusion but
not in the premise where it is used. Therefore the argument is invalid.
                                   { A
The fallacy being _illicit minor_. { A is not found in the third
                                   { A
figure’s list of valid moods.

                    M        G
  (4) A   All good citizens vote,
              ―――――――――――――

                S                M
      E   No criminal is a good citizen,
             ――――――――      ――――――――――――

                S       G
      E ∴ No criminal votes.

                             { A
The mood of this argument is { E used in the first figure. One premise
                             { E

negative; conclusion negative; no particulars. “A” distributes the
subject only; “E” both subject and predicate. The middle term, “good
citizens,” is distributed in both premises. The major term, “votes,” is
distributed in the conclusion but not in the premise where it is used.
                                                            { A
The argument is invalid, the fallacy being _illicit major_. { E is not
                                                            { E
found in the first figure’s list of valid moods.

                    G                M
  (5) A   All true teachers are sympathetic,
              ―――――――――――――

                      S                   M
      A   All lovers of children are sympathetic,
              ――――――――――――――――――

                      S                    G
      A ∴ All lovers of children are true teachers.
              ――――――――――――――――――

                             { A
The mood of this argument is { A used in the second figure. There are
                             { A
no negatives and no particulars. “A” distributes its subject only. The
middle term, “sympathetic,” is distributed in neither premise, hence
the argument is invalid. Fallacy of _undistributed middle_. Referring
                                           { A
to the list of valid moods, we do not find { A in the second figure.
                                           { A

                    M              G
  (6) A   All thoughtful men are humane,
              ――――――――――――――

                    S                 M
      A   All good citizens are thoughtful men,
              ―――――――――――――

                     S             G
      I ∴ Some good citizens are humane.

            { A
The mood is { A in the first figure. No negatives; no particulars. “A”
            { I
distributes its subject only; “I” distributes neither term. Middle term,
distributed in the major premise; no term distributed in the conclusion.
The argument is, therefore, valid. The conclusion is _weakened_ as it
                                     { A
could just as well be an A. The mood { A in the first figure is valid,
                                     { I
but of little value because of the weakened conclusion.


=4. ILLUSTRATIVE EXERCISE IN TESTING COMPLETED ARGUMENTS, ONE OR BOTH
    PREMISES BEING ILLOGICAL.=

_Arguments containing exclusive propositions._

  (1)   Only first class passengers may ride in the parlor car,
        All these are first class passengers,
      ∴ They may ride in the parlor car.

Propositions introduced by such words as _only_, _none but_,
_alone_ and their equivalents are _exclusive_ propositions. Since these
distribute their predicates, but do not distribute their subjects, the
most convenient way of dealing with them is to _interchange subject
and predicate_ and then regard them as _“A” propositions_. As the
first proposition of the argument is an exclusive, we must deal with
it accordingly. Interchanging subject and predicate and introducing it
with _all_ places the argument in this form:

                    G                                M
  A   (All) The parlor car is reserved for first class passengers,
                ――――――――――

            S                 M
  A   All these are first class passengers,
          ―――――

            S                       G
  A ∴ All these may ride in the parlor car.
          ―――――

                             { A
The mood of this argument is { A in the second figure. No negatives;
                             { A
no particulars. “A” distributes its subject only; the middle term is
thus undistributed. The argument is invalid, the fallacy being that of
_undistributed middle_.

(2) “No one but a thief would take these books without asking for them,
and it has been proved that you took the books; that is the reason I
have called you a thief.”

It is clear that “_no one but_” is equivalent to “_only_.” Thus the
first proposition of the argument is an exclusive, and may be made
logical by interchanging subject and predicate and calling it an “A.”
As a result of this the argument takes the following form:

                    M                     G
  A   (All) These books were taken by a thief,
                  ―――――

       S               M
  A   You took these books,
      ―――

       S          G
  A ∴ You are a thief.
      ―――

We have now had sufficient experience to recognize the validity of mood
AAA in the first figure.

  (3) “None but the brave deserve the fair,
       And you are not fair.”

Making the exclusive logical and completing gives:

                 M                 G
  A   (All) The fair deserve the brave,
                ――――

       S           M
  E   You are not fair,
      ―――

       S                       G
  E ∴ You do not deserve the brave.
      ―――

                             { A
The mood of this argument is { E used in the first figure. There
                             { E
is a negative premise, also a negative conclusion; no particulars.
The middle term is distributed twice. The major term “_brave_” is
distributed in the conclusion but not in the major premise; hence
the argument is invalid, the fallacy being _illicit major_.

NOTE.――There may be some doubt in the student’s mind as to the
proposition “None but the brave deserve the fair,” really meaning
“All the fair deserve the brave.” This doubt may be better satisfied by
treating the exclusive in the second way as indicated on page 137, to
wit: Negate the subject of the exclusive, then give it the form of the
regular “E.” This results in “No not-brave persons deserve the fair,”
which, after first converting and then obverting becomes, “All the fair
deserve the brave.”

_Arguments Containing Individual Propositions._

  (4) “George Washington never told a lie, but you, when tempted,
      yielded with no qualms of conscience.”

Completing, and arranging logically gives:

  E   George Washington never told a lie,
      ―――――――――――――――――

  A   You _did_ tell a lie,
      ―――

  E ∴ You (in this respect) are not like George Washington.
      ―――                                ―――――――――――――――――

Treated properly this argument proves to be valid; the student, however,
is apt to deal with such in this wise:

  O   George Washington never told a lie,
                                     ―――

  I   You did tell a lie,
      ―――

  O ∴ You (in this respect) are not like George Washington.
      ―――                                ―――――――――――――――――

When placed in this mood the argument is invalid; since the major
term, which is distributed in the conclusion, is not distributed in
the premise where it occurs (_illicit major_). It is the tendency on
the part of students to classify as particular, a proposition which
has as its subject a _singular term_. Such propositions we have learned
to call _individual_. The cause of this tendency is easily explained:
Consider the propositions, (1) “This man is mortal”; (2) “Some men are
mortal”; (3) “All men are mortal.” In the first instance “_mortal_”
refers to the subject “_man_” which is narrower in significance than
“_some men_” to which “mortal” of the second proposition refers.
In consequence, it is very natural to infer that if, “_Some men are
mortal_,” is particular, then, “_This man is mortal_,” is likewise
particular. The error springs from a wrong conception of particular
as used in logic; the content of the term has little to do with
_extension_, but is chiefly concerned with _indefiniteness_. A
particular proposition is one in which the predicate refers to only
a part of an _indefinite_ subject. If the subject is referred to as
a whole, and this whole is more or less definite, then the proposition
is universal. Since “mortal” refers to the _whole_ of the definite term
“_this man_,” as positively as it refers to the whole of “_all men_,”
there is as much justification in calling the _first_ proposition
universal as there is in calling the _third_ universal. _It may be
remembered, then, that logicians class as universal all individual
propositions._

_Arguments Containing Partitive Propositions._

  (5)   All that glitters is not gold,
        Tinsel glitters,
      ∴ Tinsel is not gold.

The quantity sign “_all_” when used with “_not_” is ambiguous; it may
mean “_no_” or “_some-not_.” The only way to determine which meaning is
intended is to try both these quantity signs, selecting the one which
seems to fit best the author’s meaning. When “all-not” means “some-not”
the proposition which it introduces is called a _partitive_ proposition;
since such always suggests a complementary proposition. (See page 133.)
For example, “Some glittering things are _not_ gold,” suggests its
complement, “Some glittering things _are_ gold.” In testing the
foregoing argument it is clear that “_All that glitters is not gold_”
does not mean “_No glittering thing is gold_,” so much as it implies
“_Some glittering things are not gold_.” Thus the argument takes this
form:

                  M                   G
  O   Some glittering things are not gold,
                                     ――――

        S       M
  A   Tinsel glitters,
      ――――――

        S            G
  E ∴ Tinsel is not gold.
      ――――――        ――――

            { O
The mood is { A in the first figure. There is one negative premise
            { E
(O), and the conclusion is negative. There is one particular premise
(O), but the conclusion is _not_ particular. This makes the argument
invalid according to rule 8; viz.: “A particular premise necessitates
a particular conclusion.” Carrying the test still further it will be
seen that there is likewise the fallacy of _undistributed middle_.

_Other arguments where one of the premises is partitive._

“All scholars are not wise and, therefore, Aristotle was not wise.”
“All democrats are not free-traders, but most of the men of this
particular club are democrats, and hence they are of a different faith
(not free-traders).”

“All the members of the club are not good players, and James belongs to
the club.”

“All educated men do not write good English; therefore, you ought not
to express surprise when informed that X, though an educated man, uses
poor English.”

The major premise in each of the foregoing is partitive in nature and
should be changed to the following form before the argument is tested;
taking these in order we have: “Some scholars are not wise”; “Some
democrats are not free-traders”; “Some of the members of the club are
not good players”; “Some educated men do not write good English.” Let
us test the validity of the last one:

  (6) O   Some educated men do not write good English,
                                   ——————————————————

      A   X is an educated man,
          —

      E ∴ X does not write good English (uses poor English).
          —          ——————————————————

Like the first one of the list, this is invalid inasmuch as a
particular premise should yield a particular conclusion, not one which
is universal. The argument also contains the fallacy of _undistributed
middle_.

_Arguments Containing Inverted Propositions._

(7) “Blessed are the merciful: for they shall obtain mercy.” The first
proposition, being poetical in construction, is typical of the inverted
form. These are usually made logical by _simple conversion_. Since
premises usually follow “_for_,” or equivalent word-signs, it is easy
to see that “for they shall obtain mercy” is one of the premises; while
the other, the broader of the two, is understood.

Arranged logically the argument assumes this form:

                     M              G
  A   Those who obtain mercy are blessed,
                ————————————

             S                    M
  A   The merciful shall obtain mercy,
          ————————

             S            G
  A ∴ The merciful are blessed.
          ————————

                      { A
Here we have the mood { A in the first figure, which we know to be
                      { A
valid.

_Other arguments where one of the propositions is inverted._

“Blessed are the pure in heart: for they shall see God.”

“To thine own self be true, and it must follow, as the night the day,
thou canst not then be false to any man.”

“A king thou art and, therefore, thy commands shall be, yea, _must_ be
obeyed.”

Taking the inverted propositions in order and making each logical, the
following is the result: “The pure in heart are blessed”; “You be true
to yourself, and....”; “You are a king, therefore....”


=5. ARGUMENTS WHICH ARE INCOMPLETE AND MORE OR LESS IRREGULAR.=

  (1) “He must be a star player; for he played fullback on the team
      which won the championship.”

  (2) “The man is not to be trusted; because he served a term of
      90 days in jail.”

  (3) “Only material bodies gravitate, and ether does not gravitate.”

  (4) “If only fools despise knowledge, this man cannot be a fool.”

  (5) “A charitable man has no merit in relieving distress; because he
      merely does what is pleasing to himself.”

  (6) “It is evident that all who get justice buy it; since only the
      rich get it.”

The above arguments thrown into logical form and validity or invalidity
stated: (The student should test these in detail.)

                                     M
  (1) A   All belonging to the team which won the championship were
              ――――――――――――――――――――――――――――――――――――――――――――――――

                  G
          star players,
               ―――――――

          S                   M
      A   He played with the team which won the championship,
          ――

          S              G
      A ∴ He is a star player. _Valid in form._
          ――

                             M                                     G
  (2) E   One who serves a term of 90 days in jail is not to be trusted,
          ――――――――――――――――――――――――――――――――――――――――              ―――――――

                S            M
      A   This man served a term of 90 days in jail,
               ―――

                S                  G
      E ∴ This man is not to be trusted. _Valid in form._
               ―――              ―――――――

                      M                 G
  (3) A   All gravitating bodies are material,
              ――――――――――――――――――

            S                M
      E   Ether does not gravitate,
          ―――――          ―――――――――

            S             G
      E ∴ Ether is not material. _Illicit major._
          ―――――        ――――――――

                        M                 G
  (4) A   All who despise knowledge are fools,
              ―――――――――――――――――――――

                S                   M
      E   This man does not despise knowledge,
               ―――          ―――――――――――――――――

                S            G
      E ∴ This man is not a fool. _Illicit major._
               ―――          ――――

                                    M
  (5) E   No one who merely does what is pleasing to himself has
             ―――――――――――――――――――――――――――――――――――――――――――――――

                       G
          merit in relieving distress,
          ―――――――――――――――――――――――――――

                   S                                    M
      A   A charitable man merely does what is pleasing to himself,
            ――――――――――――――

                    S                        G
      E ∴ No charitable man has merit in relieving distress. _Valid
             ――――――――――――――     ―――――――――――――――――――――――――――
          in form._

                   M          G
  (6) A   All the rich buy justice,

                     S             M
      A   All who get justice are rich,
              ―――――――――――――――

                     S            G
      A ∴ All who get justice buy it. _Valid in form._
              ―――――――――――――――

In supplying suppressed premises the critic is duty bound to give
the author the benefit of the doubt, if by so doing no principle in
logic is violated and the proposition conforms to good English and good
sense. Often it is not easy to perceive in the abbreviated argument
the meaning intended; in such instances all legitimate effort should be
directed to making the argument valid. To illustrate: In supplying the
major premise of argument “6” it would be easy to make it, “All justice
is bought by the rich”; in consequence the critic could pronounce the
argument invalid as the middle term would be undistributed.

Before asserting that an argument is fallacious because it has four
terms rather than three, the student must make sure that there are
no synonyms or equivalents used. In argument “4,” for instance, there
are apparently the four terms: (1) “foolish,” (2) “despise knowledge,”
(3) “man,” (4) “fool”; but to regard “_foolish_” and “_fool_” as
synonyms does not seem like undue liberty. The following arguments
further illustrate this need of _recognizing synonyms_:

“Human beings are accountable for their conduct; brutes, not being
human, are therefore _free from responsibility_.” (Not accountable for
their conduct.)

“Not all educated men spell correctly; because one often finds mistakes
in the writings of _college graduates_.” (Educated men.)

“Modern education is not popular in this state; for it increases the
tax rate, and the popularity of everything, which _touches the pocket_
of these frugal Yankees, (increases the tax rate) is very short lived.”
(Not popular.) In common parlance the use of synonyms is so prevalent
that ready ability to substitute equivalents in word, phrase, and
clause form is needed by him who would be skillful in testing all kinds
of arguments.

It has already become apparent to the student that the _number_ of the
noun or the _tense_ of the verb is of small logical consequence. A very
large proportion of the formal fallacies in argumentation concern the
rules of distribution which are summarized in the dictum “What may be
said of the whole may be said of part of that whole.”


=6. COMMON MISTAKES OF STUDENTS IN TESTING ARGUMENTS.=

The most common mistakes made by the student when testing arguments
are as follows: (1) Using the exclusive as an “A” without interchanging
subject and predicate; e. g., interpreting the proposition, “Only
high school graduates may enter the training school,” as meaning “All
high school graduates may enter the training school.” (2) _Calling
individual propositions particular_; e. g., interpreting “Socrates is
mortal” as an “I” rather than an “A.” (3) Signifying that partitive
propositions are “A’s” rather than “O’s”; e. g., “All that glitters is
not gold” interpreted as meaning that “All glittering things are gold,”
rather than “Some glittering things are not gold.” (A). (4) Concluding
that a fallacy of four terms has been committed when two terms are
synonomous. (5) Failing to interchange the subject and predicate of
inverted propositions.


=7. OUTLINE.=

  CATEGORICAL ARGUMENTS TESTED ACCORDING TO FORM.

  (1) Arguments of form and matter.

  (2) Order of procedure in the formal testing of arguments.
        The outline.
        Determining premises and conclusion.
        Completing abbreviated arguments.

  (3) Illustrative exercises in testing arguments which are complete
      and whose premises are logical.

  (4) Illustrative exercises in testing completed arguments, one or
      both of whose premises are illogical.
        Exclusive premises, individual premises, partitive premises,
          inverted premises.

  (5) Incomplete and irregular arguments.

  (6) Common mistakes of the student.


=8. SUMMARY.=

(1) In determining their validity, arguments must be tested from the
two viewpoints of _form_ and _matter_.

(2) In testing categorical arguments it is quite necessary to be
definite, to give reasons, and to give the author the benefit of the
doubt.

With this in view the attending outline is suggestive:

  1. Arrange logically and complete.
  2. Determine the figure and mood.
  3. Apply rules for negatives and particulars.
  4. Indicate distribution.
  5. Apply rules for distribution.
  6. Name fallacies, if any, giving reasons.

The logical arrangement of syllogistic arguments is

  1. Major premise.
  2. Minor premise.
  3. Conclusion.

Any proposition in a syllogism which answers the question “_Why?_”
is a premise, whereas the conclusion follows “_therefore_”, or its
equivalent either written or understood. If a conclusion is to be
supplied, unite the two terms which are used but _once_ in the premises,
using the “minor premise term” as the subject. If a premise is to be
supplied, unite the middle term with the “minor” to form the minor
premise and with the “major” to form the major premise.

(3) Arguments which are regular, complete, and logically arranged,
may be tested by symbolizing the mood and figure, underscoring the
distributed terms, and then applying the general rules of the syllogism.

(4) Arguments with illogical premises may not be tested with impunity
till the faulty premises are made logical. The exclusive, an illogical
proposition introduced by only, alone, none but, and the like, may be
made logical by interchanging subject and predicate and calling the
proposition an A. The individual proposition is one with a singular
subject. In testing, individual propositions are classed as universal.
Propositions introduced by “all-not” are usually given the significance
of “some-not”. These are called partitive propositions, which in the
testing, should be denominated “O’s”.

Inverted propositions when subjected to the test for validity must be
converted _simply_ and then classified. (Usually as A’s.)

(5) In supplying propositions which are taken for granted, the aim
should be to make the argument valid, provided this can be done without
violating the rules of logic, English, and common sense.

Ability to _substitute_ equivalent words, phrases, or clauses is
demanded of the student of logic, inasmuch as such substitution is
frequently needed in the testing of arguments.

Number and tense have little significance in dealing with arguments.

(6) The common mistakes of students made in testing arguments concern
exclusive, partitive and inverted propositions, and an inability to
recognize expressions equivalent in meaning.


=9. REVIEW QUESTIONS.=

  (1) Name and explain the two standpoints from which all arguments
      must be viewed.

  (2) Give an outline of procedure which may be serviceable in the
      testing of categorical arguments.

  (3) Give illustrations showing that the logical order of categorical
      arguments is not the usual mode of procedure in common parlance.

  (4) Offer suggestions which may aid in designating a premise; a
      conclusion.

  (5) How would you proceed in forming any one of the three
      propositions of a syllogism when the other two are given?

  (6) Designate the premises and the conclusion in the following,
      supplying any proposition which may be omitted, also arrange
      logically and test the validity.

        (1) “The people of this country are suffering from an overdose
            of prosperity; consequently a period of hard times will be
            a valuable lesson.” (The conclusion should be recast so as
            to read, “A period of hard times will cure the people of
            this country.” The minor premise is, “Those who suffer from
            an overdose of prosperity may be cured by a period of hard
            times.”)

        (2) “I am a teacher; you are not what I am; hence you are not
            a teacher.”

        (3) “To kill a man is murder, therefore war is murder.”

        (4) “You have not adopted the best policy since honesty has
            always been and will always be the best policy.”

        (5) “Since the road is criminally mismanaged, why should not
            the authorities be indicted as criminals?”

        (6) “Early to bed and early to rise makes a man healthy,
            wealthy and wise. I am none of these; hence my sleeping
            hours have been wrong.”

  (7) Illustrate a weakened conclusion.

  (8) Explain the exclusive proposition and indicate how the logician
      should treat it.

  (9) Arrange logically and test the following:

        (1) Only weak men become intemperate, and Edgar Allen Poe was
            surely intemperate.

        (2) No admittance except on business; hence you cannot be
            admitted.

        (3) Virtuous acts are praiseworthy, and indiscriminate giving
            is not a virtuous act.

 (10) Explain why individual propositions are classed as universal.

 (11) Write an argument whose major premise is a partitive proposition;
      arrange logically and test validity.

 (12) Arrange and test this argument: “Blessed are the poor in spirit:
      for theirs is the kingdom of heaven.”

 (13) Complete, arrange and test.

        (1) “The object of war is to settle disputes; hence soldiers
            are the best peacemakers.”

        (2) “The various species of brutes being created to prey upon
            one another proves that man is intended to prey upon them.”

        (3) “The end of everything is its perfection; death being the
            end of life is its perfection.”

        (4) “All the trees of the yard make a thick shade and this is
            one of them.”

        (5) “Minds of moderate caliber ordinarily condemn everything
            which is beyond their range, and his is such a mind.”

        (6) “The best of all medicines are fresh air and sleep, and you
            are sorely in need of both.”

        (7) “Every hen comes from an egg; every egg comes from a hen;
            therefore every egg comes from an egg.”

        (8) “He cannot have been there――otherwise I should have seen
            him.”

 (14) “It is fair to give the author the benefit of the doubt when we
      set ourselves up as censors worthy of the name.” Explain this.

 (15) Illustrate by citing arguments the need of detecting terms which
      are equivalents in signification.

 (16) How does the logician look upon number and tense as treated in
      grammar?

 (17) Illustrate and test an argument in which one of the premises is
      elliptical.

 (18) Summarize the most common mistakes made by students in the
      testing of categorical arguments; illustrate these mistakes and
      then write in logical form.


=10. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Give illustrations of arguments which are valid in form but
      invalid in meaning. Explain.

  (2) May an argument be valid in meaning but invalid in form?
      Exemplify.

  (3) Put a simple problem in arithmetic in syllogistic form and show
      that the minor premise naturally comes first.

  (4) In the practice of law is there any custom analogous to giving
      the author the benefit of the doubt in logical argumentation?

  (5) Test in detail the following arguments:

        (1) “All wise presidents strive to give heed to the demands of
            the people, but this president has not done so.”

        (2) “The existence of God is not universally believed, hence it
            cannot be true.”

        (3) “The institution has prospered under the present régime
            therefore why change it?”

        (4) “The man is guilty because seven out of the nine witnesses
            so testified.”

        (5) “I know three men who cleared not less than ten thousand
            dollars in this business; and why cannot I do as much?”

  (6) “Only members may vote and, since you are not a member, you will
      not be allowed to vote.” Change the exclusive in this argument in
      the two ways suggested in Chapter 8, page 126. Test the argument
      in both cases.

  (7) Show by illustration that the quantity sign “_all_” when used
      with “_not_” may in some cases mean “_no_” and in others
      “_some-not_”.

  (8) Make two selections from some poet of authority representing
      arguments with an inverted premise.

  (9) Select from news papers three arguments which seem to illustrate
      the fallacy of four terms but which in reality do not. Explain.

 (10) Wherein could the elliptical proposition lead to error?

 (11) Put the following in syllogistic form and test:

        (1) “That persons may reason without language is proven by the
            circumstances that infants reason and yet have no language.”

        (2) “The scriptures cannot come from God because they contain
            some things which cannot be comprehended by man.”

        (3) “When Columbus was sailing the ocean in search of a new
            world, he fell in with a flock of land birds and concluded
            that he could not be far from land.”

        (4) “Bolingbroke in arguing against the truth of the Christian
            religion shows that the Christian religion has bred
            contentions.” “Burke answered him by showing that civil
            government had bred contentions.”



                              CHAPTER 15.

           HYPOTHETICAL ARGUMENTS, AND DISJUNCTIVE ARGUMENTS
                        INCLUDING THE DILEMMA.


=1. THREE KINDS OF ARGUMENTS.=

The proposition, constituting the basic unit of the argument, would
of necessity be indicative of the nature of said argument; therefore
the three general kinds of propositions, categorical, hypothetical and
disjunctive, suggest the three kinds of arguments which are in turn
categorical, hypothetical and disjunctive. Categorical arguments are
those in which all of the propositions are categorical. Since this kind
has been treated, it remains for us to consider the other two.


=2. HYPOTHETICAL ARGUMENTS.=

We have observed that a hypothetical proposition is one in which the
assertion depends on a condition; for example, in the proposition, “If
it is pleasant, I will call on you to-morrow,” the _calling_ depends
on the state of the weather. “I will call on you to-morrow,” is the
_assertion_ which is limited by the _condition_, “If the weather is
pleasant.” Definition:

_The hypothetical argument or syllogism is one in which the major
premise is hypothetical and the minor premise categorical._

  ILLUSTRATION:

    If the people are right more than half of the time, the world will
      progress;

    And the people _are_ right more than half of the time,

    Hence the world will progress.

In contradistinction to _disjunctives_, hypothetical propositions and
hypothetical syllogisms are frequently referred to as “_conjunctive_.”


=3. THE ANTECEDENT AND CONSEQUENT.=

Facility in detecting the antecedent and consequent of hypotheticals
is required in order to deal intelligently with the argument. The
hypothetical proposition has been defined as one in which the assertion
is limited by a condition. The _consequent_ is the assertion and
usually follows (though not always) the _antecedent_ which is the
limiting condition. _First the antecedent and then the consequent is
the logical order_ as the derivative meaning of the words antecedent
and consequent would indicate. The antecedent is introduced by such
words as “if,” “though,” “unless,” “suppose,” “granted that,” “when,”
etc.


  ILLUSTRATIONS:

          _Antecedent._                  _Consequent._
    1. If you study,                  you will pass.
    2. If it rains,                   it is cloudy.
    3. If two is added to three,      the result is five.
    4. If you are temperate,          you will live to a ripe old age.

          _Consequent._                   _Antecedent._
    5. I will go,                     unless you wire me to the
                                        contrary.
    6. I will pay you,                when you present your bill.
    7. I shall make the trip in       granted that I have no accidents.
        ten hours,
    8. My overcoat would not have     if the door had been locked.
        been stolen,


=4. TWO KINDS OF HYPOTHETICAL ARGUMENTS.=

The two kinds of hypothetical syllogisms are the _constructive_ and
_destructive_.

_A constructive hypothetical syllogism is one in which the minor
premise affirms the antecedent._

_A destructive hypothetical syllogism is one in which the minor premise
denies the consequent._

The constructive hypothetical is sometimes referred to as the “_modus
ponens_”; whereas the destructive hypothetical is called the “_modus
tollens_.”


  ILLUSTRATIONS:

                _Constructive_ Hypothetical Syllogisms.

       _Symbols._                 _Words._

     If A is B, C is D    If you are diligent, you will succeed;
        A is B            And you _are_ diligent,
      ∴ C is D            Therefore you will succeed.

                _Destructive_ Hypothetical Syllogisms.

     If A is B, C is D    If you had been diligent, you would have
                            succeeded;
        C is not D        But you did not succeed,
      ∴ A is not B        Therefore you were not diligent.


=5. THE RULE AND TWO FALLACIES OF THE HYPOTHETICAL ARGUMENT.=

From a given hypothetical proposition it is possible to construct
_four_ different hypothetical syllogisms, as the attending
illustrations make evident:

Consider the hypothetical proposition “If it has rained, the ground is
damp.”

  (1) Minor premise affirms antecedent.
        If it has rained, the ground is damp;
        It has rained,
        Therefore the ground is damp.

  (2) Minor premise denies antecedent.
        If it has rained, the ground is damp;
        It has not rained,
        Therefore the ground is not damp.

  (3) Minor premise affirms consequent.
        If it has rained, the ground is damp;
        The ground is damp,
        Therefore it has rained.

  (4) Minor premise denies the consequent.
        If it has rained, the ground is damp;
        The ground is not damp,
        Therefore it has not rained.

Without any knowledge of the rules of the hypothetical syllogism let
us strive to determine how many of the foregoing are valid. Relative to
the first, it would be impossible for any rain to fall without making
the ground somewhat damp; a few drops would be sufficient. In short,
if the antecedent happens, the consequent _must_ follow. It seems,
therefore, that the first argument is _valid_. Considering the second:
rain is not the only cause for the dampness of the ground, as it might
result from the falling of dew, or from a dense fog; _no rain_ does not
necessarily mean _no dampness_. It is clear that if the antecedent does
not happen, the consequent may or may not follow. Thus it appears that
the second argument is _invalid_. Attention to the third makes evident
a condition similar to the second: the ground may be made damp by
agencies other than rain, such as fog and dew. Thus the third argument
is likewise _invalid_. But in the fourth argument it is obvious that
if the ground is not damp, then there could have been neither rain, nor
fog, nor dew. No dampness shuts out _all_ of the conditions, including
the rain. Therefore the fourth argument is _valid_.

This investigation suggests a rule for hypothetical arguments. Since
only the first and fourth arguments are valid, this is the rule which
must obtain: _The minor premise should either affirm the antecedent or
deny the consequent._

Any violation of this rule would result in the fallacies of _denying
the antecedent_ or _affirming the consequent_.

There is one exception to this rule which must not be overlooked;
viz.: If the antecedent and consequent of the hypothetical proposition
are _co-extensive_ then both may be either affirmed or denied.


  ILLUSTRATIONS:

    (1) If the rectangle is equilateral, then it is a square;
        The rectangle is equilateral,
      ∴ It is a square.

    (2) If the rectangle is equilateral, then it is a square;
        The rectangle is not equilateral,
      ∴ The rectangle is not a square.

    (3) If the rectangle is equilateral, then it is a square;
        It is a square,
      ∴ The rectangle is equilateral.

    (4) If the rectangle is equilateral, then it is a square;
        It is not a square,
      ∴ The rectangle is not equilateral.


=6. HYPOTHETICAL ARGUMENTS REDUCED TO THE CATEGORICAL FORM.=

The hypothetical syllogism so closely resembles the categorical that
it may be changed to it by a slight alteration in the wording. After
testing the hypothetical by its own rule, it may be expedient to reduce
the argument to the categorical form, and subject it to a second test
in which the categorical rules are applied. This reduction usually
necessitates two steps; first, _change the propositions which represent
the antecedent and consequent to a subject term and a predicate term
respectively and then unite them to form the major premise_; second,
_supply a new minor term, if necessary_.

Illustrations of Reduction; and Comparison of Hypothetical and
Categorical Fallacies:

_Hypothetical Form_:

  (1) If it has rained, the ground is damp;
      It has rained,
    ∴ The ground is damp.

_Categorical Form_:

             M                       G
  A   The falling rain makes the ground damp,
      ――――――――――――――――           ―――――――――――

              S               M
  A   In this case rain has fallen,
      ―――――――――――――――――

           S                         G
  A ∴ In this case the ground is damp ground.
      ――――――――――――

It is seen that the argument in the hypothetical form is valid as the
minor premise affirms the antecedent. Reducing to the categorical gives
                         { A
to the argument the mode { A in the first figure which we know to be
                         { A
valid.

_Hypothetical_:

  (2) If one were wise, he would study;
      But you will not study,
    ∴ You are not wise.

_Categorical_:

             G              M
  A   A wise person would study,
        ―――――――――――

       S             M
  E ∴ You will not study,
      ―――          ―――――

       S           G
  E ∴ You are not wise.
      ―――         ――――

In the hypothetical form the argument is valid since the minor premise
                                                              { A
denies the consequent. Reducing to the categorical gives mood { E in
                                                              { E
the second figure. This is valid.

_Hypothetical_:

  (3) If the wind blows from the south, it will rain;
      But the wind is not blowing from the south,
      Hence it is not going to rain.

_Categorical_:

          M              G
  A   South wind brings rain,
      ――――――――――

          S                  M
  E   This wind is not a south wind,
      ―――――――――          ――――――――――

          S                  G
  E ∴ This wind will not bring rain.
      ―――――――――          ――――――――――

Hypothetically considered, the minor premise denies the antecedent and
consequently the argument is invalid. Reducing to the categorical form,
it is found that the major term is distributed in the conclusion, but
is not distributed in the major premise; hence the fallacy of _illicit
major_ is committed.

_Hypothetical_:

  (4) If a man is just, he will obey the golden rule;
      This judge has obeyed the golden rule,
      Hence he is just.

_Categorical_:

           G                        M
  A   A just man will obey the golden rule,
        ――――――――

             S                      M
  A   This judge has obeyed the golden rule,
           ―――――

             S           G
  A ∴ This judge is a just man.
           ―――――

Hypothetically considered, the minor premise affirms the consequent
and thus the argument is fallacious; when changed to the categorical
we find the fallacy of _undistributed middle_. If other examples were
taken, it could be proved that the hypothetical fallacy of _denying
the antecedent_ is usually equivalent to the categorical fallacy of
_illicit major_; whereas the hypothetical fallacy of _affirming the
consequent_ amounts to _undistributed middle_.

In reducing some hypotheticals it is necessary to make use of such
expressions as, “_the case of_” or “_the circumstances that_.” The
attending argument will illustrate this:

        If Jefferson was right, man was created free and equal;
  (but) Man was not created free and equal,
      ∴ Jefferson was not right.

_Reduced to the categorical_:

                      G
    The case of Jefferson being right is the case of man being created
        ―――――――――――――――――――――――――――――
            M
      free and equal;

     S                        M
    Man was not created free and equal,
    ―――                 ――――――――――――――

        S                          G
  ∴ Jefferson (this man) was not right.
    ―――――――――                    ―――――

The argument is valid in both cases.


=7. ILLUSTRATIVE EXERCISE TESTING HYPOTHETICAL ARGUMENTS OF ALL KINDS.=

The following brief outline may be followed in testing hypothetical
arguments:

  1. Arrange logically.
  2. Determine antecedent and consequent.
  3. Apply the hypothetical rule; name fallacies giving reasons.
  4. Reduce to categorical form.
  5. Apply the categorical rules, giving fallacies with reasons.

  (1) If a man is properly educated, he will not despise manual labor;
      therefore I conclude that you have not been properly educated,
      since you dislike to work with your hands.

_Arranged logically and antecedent and consequent indicated_:

    If a man is properly educated (antecedent), he will not despise
        manual labor (consequent);
    You despise manual labor (dislike to work with your hands),
  ∴ You have not been properly educated.

The minor premise denies the consequent, hence the argument is valid
according to the rule, “The minor premise must affirm the antecedent
or deny the consequent.” The student should note that the consequent is
negative and therefore its denial must be an affirmative proposition.

_Reduced to the categorical_:

                  G                             M
  E   A properly educated man will not despise manual labor;
        ―――――――――――――――――――――          ――――――――――――――――――――

       S           M
  A   You despise manual labor,
      ―――

       S                          G
  E ∴ You have not been properly educated.
      ―――               ―――――――――――――――――

Regarded categorically this is valid. Why?

(2) “If one believes in the tenets of the democratic party, then he
should vote for its candidates; and since A does believe in them I have
asked him to vote for me.”

_Arranged, and antecedent and consequent indicated._

    If one believes in the tenets of the democratic party (antecedent),
      then he should vote for its candidates (consequent);
    And A does believe in these tenets,
  ∴ He should vote for its candidates (I have asked him to vote for
      me).

The minor premise affirms the antecedent and thus the argument is valid
according to rule.

_Reduced to the categorical_:

                                M
  A   One who believes in the tenets of the democratic party should
      ――――――――――――――――――――――――――――――――――――――――――――――――――――――
                         G
        vote for its candidates,

      S                     M
  A   A believes in these tenets,
      ―

      S                         G
  A ∴ A should vote for its candidates.
      ―

                                      { A
Reduced to the categorical gives mood { A in the first figure and this
                                      { A
we know to be valid.

(3) “If the weather had not been pleasant, I could not have come; but
as the weather is pleasant, here I am.”

_Arranged and antecedent and consequent indicated_:

    If the weather had not been pleasant (antecedent), I could not have
      come (consequent);
    The weather is pleasant,
  ∴ I have come (here I am).

The minor premise _denies_ the antecedent and consequently the argument
is invalid according to the rule. (An affirmative minor premise denies
a negative antecedent.)

_Reduced to the categorical_:

  E   Unpleasant weather would not permit me to come,
  E   This weather is not unpleasant,
  A ∴ This weather enabled me to come.

Fallacy of two negative premises.

(4) “If one pays his debts, he will not be ‘black-listed’; but since
you are ‘black-listed,’ I conclude that you have not paid your debts.”

_Arranged logically and antecedent and consequent indicated_:

    If one pays his debts (antecedent), he will not be “black-listed”
      (consequent);
    You are “black-listed,”
  ∴ You have not paid your debts.

The minor premise denies the consequent hence the argument is valid.

_Reduced to categorical form_:

                   G                    M
  E   No one who pays his debts is black listed,
         ――――――――――――――――――――――    ――――――――――――

       S           M
  A   You are black listed,
      ―――

       S                  G
  E ∴ You have not paid your debts.
      ―――          ―――――――――――――――

         { E
The mood { A in the second figure is valid.
         { E

(5) “Men would do right for the sake of themselves, if they appreciated
the law of retribution; but they never think of that.”

_Arranged, completed, and tested_:

  If they appreciated the law of retribution (antecedent), men would
      do right for the sake of themselves (consequent);
  But they do not appreciate the law of retribution (never think of
      that),
  Hence they do not do right for the sake of themselves.

Fallacy of denying the antecedent.

_Reduced to the categorical_:

                                   M
  A   The case of men appreciating the law of retribution, is the case
               ――――――――――――――――――――――――――――――――――――――――――
            G
        of men doing right for the sake of themselves;

           S                         M
  E   But men do not appreciate the law of retribution,
          ―――        ―――――――――――――――――――――――――――――――――

       S                           G
  E ∴ Men do not do right for the sake of themselves.
      ―――           ――――――――――――――――――――――――――――――――

Fallacy of illicit major.

(6) “If an animal is a vertebrate, then it must have a backbone; but
the books say that this animal is not a vertebrate, hence it cannot
have a backbone.”

Since the minor premise denies the antecedent it would appear that the
argument is invalid; yet common knowledge and common sense dictate that
the conclusion is true. Surely no invertebrate can have a backbone. As
a matter of fact the antecedent and consequent are _co-extensive_ and
therefore the hypothetical rule is not applicable.

_Reduced to the categorical_:

           M                     G
  A   Vertebrates must have a backbone (Co-extensive),
      ―――――――――――           ――――――――――

             S                  M
  E   This animal is not a vertebrate,
           ――――――          ――――――――――

  E ∴ This animal cannot have a backbone.
           ――――――             ――――――――――

As co-extensive A’s distribute their predicates the possibility of
there being a fallacy of illicit major is forestalled.

Categorically considered the argument is likewise valid.


=8. DISJUNCTIVE ARGUMENTS.=

It has been observed that a disjunctive proposition is one which
expresses an alternative. _A disjunctive syllogism is one in which the
major premise is a disjunctive proposition._


  ILLUSTRATION:

    The boy is either honest or dishonest,
    He is honest,
  ∴ He is not dishonest.


=9. THE TWO KINDS OF DISJUNCTIVE ARGUMENTS.=

The two forms of disjunctive arguments are _the one which by affirming
denies_ and _the one which by denying affirms_. The former is known by
the Latin words “_modus ponendo tollens_”; while the latter is termed
the “_modus tollendo ponens_.”


  ILLUSTRATIONS:

  (1) By affirming denies.

        The defendant is either guilty or innocent,
        He is guilty,
      ∴ He is not innocent.

              or

        The defendant is either guilty or innocent,
        He is innocent,
      ∴ He is not guilty.

  (2) By denying affirms.

        The defendant is either guilty or innocent,
        He is not guilty,
      ∴ He is innocent.

              or

        The defendant is either guilty or innocent,
        He is not innocent,
      ∴ He is guilty.


=10. THE FIRST RULE OF DISJUNCTIVE ARGUMENTS.=

It may be said that disjunctive arguments depend on _two_ rules. This
is the first: _The major premise must assert a logical disjunction._
A logical disjunction involves two requisites; first, the alternatives
must be _mutually exclusive_; second, the _enumeration_ must be
_complete_.

_Illustrations of illogical major premise._

  _Terms not mutually exclusive_:

    This boy is either inattentive or indolent,
    He is not inattentive,
  ∴ He is indolent.

It is obvious that the boy might be _both_ inattentive and indolent.
Experience teaches that the qualities are usually concurrent, and to
assume that the boy must be either one or the other is a clear case of
“begging the question.”

Some logicians maintain that “_either――or_” signify that both
alternatives _cannot be false_, but that both _may be true_. If this
viewpoint were adopted, the major premise of the illustration would
_not_ be a case of begging the question. It is unnecessary to argue the
point, if it is made perfectly clear which view is to obtain in this
discussion. Briefly stated the two points are these. First opinion:
“_Either――or_” when used logically, mean that if the first alternative
is false the second must be true, or if the first alternative is true
the second must be false. Second opinion: “_Either――or_” when used
logically mean that if the first alternative is false, the second must
be true; but if the first alternative is true, the other may or may
not be true. _This treatise adopts the first opinion._ With us all
alternative arguments to be logical must be mutually contradictory;
i. e., _when one is false, the other must be true and when one is true
the other must be false; both_ cannot be false, neither can _both_
be true. When it is intended that this implication should not obtain,
then the expressed alternative will take this form, “The boy is either
inattentive or indolent or _both_.”

Other examples where the terms of the disjunctive may not be mutually
exclusive:

  (1) “Lord Bacon was either exceedingly studious or phenomenally
      bright.” (Undoubtedly he was both.)

  (2) “This teacher is a graduate either of Harvard or of Yale.”
      (Perhaps both.)

  (3) “The defendant is either a liar or a thief.” (The one often
      leads to the other.)

  (4) “To succeed one must either seize the opportunity as it passes
      or make his own.” (The best success results from doing both.)


_Incomplete enumeration_:

      The cause of the disease was either the water or the milk,
      It was not the milk,
    ∴ It was the water.

When such an argument as this is advanced, it must be with the
knowledge that every other alternative has received satisfactory
investigation. Without this assurance one could justly claim that the
disease might have been caused by the _meat_ or _fish_ supply. Complete
enumeration means that the investigation has narrowed the facts to
the boundary of the field covered by the alternatives. The fallacy
of incomplete enumeration is also one of “begging the question.”

Other examples of a possible incomplete enumeration:

  (1) “Jones lives either in Boston or New York.”

  (2) “Mary is studying either algebra or geometry.”

  (3) “He either committed suicide or was lynched.”

  (4) “Either the Giants or the Boston Americans will win the pennant.”


=11. SECOND RULE OF DISJUNCTIVE ARGUMENTS.=

The second rule is made so self evident by the first that there is
little need of a detailed discussion concerning it. The rule is this:
_When the minor premise affirms or denies one of the alternatives of
a logical disjunction, the conclusion must, in order, deny or affirm
all of the others._ To put it differently: When the “minor” affirms,
the conclusion must deny every other alternative, and vice versa.
When there are but two alternatives reference to any of the foregoing
disjunctive arguments will make the rule clear. There may be, however,
_more_ than two alternatives. In such a case, if the first rule is
observed then the second becomes applicable.


  ILLUSTRATIONS:

  (1)   John Doe lives either in Boston, Albany, or New York;
        He lives in New York,
      ∴ He does not live in either Boston or Albany.

                    or

        He does not live in New York,
      ∴ He lives in either Boston or Albany.

  (2)   The season must have been either summer, or autumn, or winter,
          or spring;
        It was neither autumn, nor winter, nor spring,
      ∴ It must have been summer.

                    or

        It was either autumn, or winter, or spring,
      ∴ It could not have been summer.


=12. REDUCTION OF THE DISJUNCTIVE ARGUMENT TO THE HYPOTHETICAL AND
    THEN TO THE CATEGORICAL.=

It would seem that the laws of the disjunctive contradict those of the
categorical syllogism; for we apparently derive from two affirmatives a
negative conclusion, and we also derive an affirmative conclusion when
one premise is negative. This objection is seen to be nugatory when the
disjunctive is reduced to the categorical form. The reduction involves
the two steps of first changing the disjunctive to the hypothetical
form and then to the categorical form. The following illustrations will
suffice to make the matter clear:

  (1) _Disjunctive._
        A is either B or C
        A is B
      ∴ A is not C

      _Hypothetical._
        If A is B, then A is not C
        A is B
      ∴ A is not C

      _Categorical._
        The case of A being B is the case of A not being C
        In this case A is B
      ∴ A is not C

  (2) _Disjunctive._
        The defendant is either guilty or innocent;
        He is not innocent,
      ∴ He is guilty.

      _Hypothetical._
        If the defendant is guilty, then he is not innocent;
        But he is guilty,
      ∴ He is not innocent.

      _Categorical._
        The case of the defendant being guilty is the case of the
          defendant not being innocent,
        In this case the defendant is guilty,
      ∴ In this case the defendant is not innocent.


=13. THE DILEMMA.=

The majority of us are acquainted with the dilemma as related to the
activities of life. One is in a dilemma when there are two courses open
to him but neither is particularly enticing. One is placed in a dilemma
when he is forced to choose the lesser of two evils. For example, one
may, without the proper equipment, be overtaken by a heavy rain storm;
he seeks the shelter of a wayside shed; the rain continues so that
he is forced either to miss his train, or to endure the discomfort of
a drenching. Thus the logical dilemma limits one to a choice between
alternatives, either one of which might well be avoided.

_Definition._

_The dilemma is a syllogism in which the major premise consists of
two or more hypothetical propositions, while the minor premise is a
disjunctive proposition._

It being a combination of hypothetical and disjunctive propositions
the dilemma is sometimes appropriately referred to as the
“hypothetico-disjunctive” argument. The order of the premises is
indifferent, yet it seems to be more natural to use the hypothetical
first; thus the definition.


=14. FOUR FORMS.=

The four forms of the dilemma are the _simple constructive_, the
_simple destructive_, the _complex constructive_, and the _complex
destructive_. The following symbolizations illustrate these four kinds:

_Simple Constructive Dilemma._

  If A is B, W is X; and if C is D, W is X,
  But either A is B or C is D,
  Hence W is X.

This is termed a simple dilemma because there is but _one_
consequent; namely, W is X. The conclusion being affirmative makes it
_constructive_.

_Simple Destructive Dilemma._

  If A is B, W is X; and if A is B, Y is Z,
  But either W is not X or Y is not Z,
  Hence A is not B.

This is simple because there is but one antecedent, A is B, and
destructive because the conclusion is _negative_.

_Complex Constructive Dilemma._

  If A is B, W is X; and if C is D, Y is Z,
  But either A is B or C is D,
  Hence either W is X or Y is Z.

This is complex because there are two antecedents and two consequents;
constructive, inasmuch as the conclusion is affirmative.

_Complex Destructive Dilemma._

  If A is B, W is X; and if C is D, Y is Z,
  But either W is not X or Y is not Z,
  Hence either A is not B or C is not D.

This is complex because there are two antecedents as well as two
consequents, and destructive because the conclusion is negative.
Briefly: (1) A simple dilemma is one where either the antecedent or
consequent is repeated; whereas if neither is repeated the dilemma is
complex. (2) A constructive dilemma contains an affirmative conclusion;
while a destructive dilemma uses a negative conclusion. (3) A simple
dilemma has as its conclusion a categorical proposition; whereas the
conclusion of a complex dilemma is always disjunctive.

If the number of antecedents and consequents be increased, a trilemma,
tetralemma, etc., may result.


  ILLUSTRATION――_Trilemma._

  If A is B, W is X; and if C is D, Y is Z; and if E is F, U is V,
  But either A is B, or C is D, or E is F,
  Hence either W is X, or Y is Z, or U is V.

Some authorities define a dilemma as a syllogism in which the
“major-hypothetical” has _more than one antecedent_ while the “minor”
must be disjunctive. This viewpoint necessarily _excludes_ the second
form or the simple destructive dilemma. The weight of authority,
however, appears to favor the classification here recommended.


=15. THE ONE RULE INVOLVED IN DILEMMATIC ARGUMENTS.=

Since the major premise of the dilemma is hypothetical, the rule for
testing such would of necessity be the hypothetical rule; namely, “The
minor premise must either affirm the antecedent or deny the consequent.”
As this rule and the fallacies incident to it have been treated in
detail, further discussion is unnecessary.


=16. ILLUSTRATIVE EXERCISE TESTING DISJUNCTIVE AND DILEMMATIC
     ARGUMENTS.=

  (1) If the arithmetic contains useful facts, it will help to good
        citizenship; and if it trains the powers of reason, it will
        help to good citizenship,
      But the arithmetic either contains useful facts or trains the
        powers of reason,
      Hence it will help to good citizenship.

This is a simple constructive dilemma in which the minor premise
affirms the antecedents. The argument is, therefore, valid since it
conforms to the rules of the hypothetical syllogism. The fact that the
minor premise may not be a perfect disjunctive does not invalidate the
conclusion, inasmuch as it is perfectly obvious that if the arithmetic
fulfilled both the requirements of the antecedents, the conclusion
would still obtain. It may, therefore, be inferred that if the dilemma
conforms to the rules of the hypothetical argument, it is valid, though
the disjunctive proposition which it contains may not be strictly
logical.

(2) A man is either temperate or intemperate; and, as I have seen you
drunk several times, I conclude that you are intemperate.

_Arranged logically._

    A man is either temperate or intemperate,
    You are not temperate,
  ∴ You are intemperate.

It would seem that the major premise is a logical disjunctive, since
temperate and intemperate indicate that the alternatives are _mutually
exclusive_ and the _enumeration complete_. And since the minor premise
denies one alternative while the conclusion affirms the other, we may
infer that the argument is valid.

(3) If a man is honest, he will either pay his debts or explain; but
this fellow paid no heed to the repeated notifications.

_Arranged logically._

    If a man is honest, he will pay his debts; and if he is honest, he
      will explain in case he cannot pay,
    This man neither paid his debt, nor explained,
  ∴ This man is not honest.

This is a simple destructive dilemma, and since the minor premise
denies the consequents it is valid.

(4) A voter must either favor protection or free trade; and since you
do not favor protection, you must be a free trader. The disjunctive is
not logical as one might believe in universal reciprocity. The argument
is, therefore, invalid. _Why?_

(5) If a man were loyal, he would not be unduly critical; and if he
were wise, he would not be too loquacious; but I find this clerk has
been both unduly critical and too loquacious; hence I consider that he
has been not only unwise but strikingly disloyal.

This complex dilemma is valid since the minor premise denies the two
consequents.


=17. ORDINARY EXPERIENCES RELATED TO THE DISJUNCTIVE PROPOSITION AND
     HYPOTHETICAL ARGUMENT.=

(1) One desires to take a certain trip which involves various routes;
information from time tables reveals the fact that there are three
routes A, B, and C. Concerning the conditions of the journey the most
important factor is the _matter of comfort_. Further investigation
makes evident that route B will be the most comfortable one, and
consequently is the route selected. Putting this ordinary experience
in argumentative form gives the following:

  The route is to be either A, or B, or C;
  I will take route A; if it is the most comfortable; (co-extensive)
  A is not the most comfortable route,
  Hence I will not take route A.
  If B is the most comfortable route, I will take it;
  B is the most comfortable route,
  Hence I will take route B.

(2) The symptoms suggest either malarial or typhoid fever; the
physician is undecided till a blood test makes evident that it is not
typhoid.

_Considered argumentatively._

  This disease is either malarial or typhoid fever;
  If it is typhoid, the blood will reveal certain evidences;
  But the blood does not reveal these evidences,
  Hence the disease is not typhoid.

(3) The natural bent of the youth suggests the profession of either the
ministry or teaching. He finally decides to follow the one in which he
can best serve his fellows. This, after mature deliberation, appears to
him to be the work of the teacher. Thrown into the form of an argument
the following results:

  I am best fitted for either the pulpit or the schoolroom;
  If the schoolroom furnishes the richest field for helping my fellows,
    I will choose that work;
  The schoolroom _does_ appear to furnish such a field,
  Hence I will choose the work of the teacher.

It would appear from these ordinary experiences that frequently we
are brought face to face with a choice of alternatives which are not
unattractive, as in the case of the dilemma. Moreover, some condition
suggests itself which, if proved or disproved, will lead to a choice
of _one_ of these alternatives. Such circumstances when thrown into
the form of an argument present a _disjunctive proposition_ followed
by a _hypothetical argument_. To put it differently: Often in our daily
affairs a most prominent limiting condition induces us to select one
out of several alternatives. These alternatives are _not_ dilemmatic
in nature.


=18. OUTLINE.=

  HYPOTHETICAL ARGUMENTS, AND DISJUNCTIVE ARGUMENTS INCLUDING THE
  DILEMMA.

  (1) Three kinds of arguments
        Categorical, hypothetical, disjunctive.

  (2) Hypothetical arguments
        Defined, illustrated.

  (3) Antecedent and consequent.
        How determined, illustrations.

  (4) Two kinds of hypothetical arguments
        Constructive, destructive, illustrations.

  (5) Rule and two fallacies of the hypothetical argument.
        Illustrations and application of rules.
        Fallacy of denying antecedent.
        Fallacy of affirming consequent.
        Co-extensive hypotheticals.

  (6) Hypothetical arguments reduced to the categorical form.
        Rule, illustrations.
        Hypothetical and categorical arguments compared.

  (7) Illustrative exercises testing hypothetical arguments of all
      kinds.

  (8) Disjunctive arguments.
        Defined, illustrated.

  (9) Two kinds of disjunctive arguments.
        By “affirming denies,” by “denying affirms.” Illustration.

 (10) First rule.
        Stated, illustrated.

 (11) Second rule
        Stated, illustrated.

 (12) Reduction of disjunctive argument
        Two steps.

 (13) The dilemma
        Definition.

 (14) Four forms of dilemmatic arguments
        Simple constructive, simple destructive,
        Complex constructive, complex destructive.
        Illustrations.

 (15) The rule.

 (16) Illustrative exercises testing disjunctive and dilemmatic
      arguments.

 (17) Ordinary experiences related to the disjunctive proposition and
      hypothetical argument.


=19. SUMMARY.=

(1) Just as there are three kinds of propositions so there are three
kinds of arguments; namely, categorical, hypothetical, disjunctive.

(2) Categorical syllogistic arguments are those in which all of the
propositions are categorical.

Hypothetical syllogistic arguments are those in which the major premise
is hypothetical.

In contradistinction to disjunctives, hypothetical arguments may be
referred to as “_conjunctive_”.

(3) The hypothetical proposition is composed of antecedent and
consequent; the former being the limiting condition; while the latter
is the direct assertion. As the words indicate the antecedent usually
precedes the consequent. The signs of the antecedent are “if,” “though,”
“unless,” “suppose,” “granted that,” “when,” etc.

(4) The two kinds of hypothetical syllogisms are the constructive and
destructive; the former is involved when the minor premise affirms the
antecedent; the latter when the minor premise denies the consequent.
These two kinds are sometimes referred to as “_modus ponens_” and
“_modus tollens_” respectively.

(5) Out of the four possible hypothetical syllogisms only _two_ are
valid as investigation proves this rule: _The minor premise must affirm
the antecedent or deny the consequent._ In the case of the hypothetical
proposition being _co-extensive_, the rule does not apply.

(6) Hypothetical arguments may be reduced to the categorical by
contracting the antecedent of the hypothetical proposition to form
the subject-term, and by contracting the consequent of the hypothetical
proposition to form the predicate-term of the major premise of the
categorical syllogism. If it is necessary, supply a new minor term.

Denying the antecedent is a matter of illicit major; whereas affirming
the consequent is equivalent to undistributed middle.

(7) Hypothetical arguments may be tested by following this outline:

  (1) Arrange logically.
  (2) Determine antecedent and consequent.
  (3) Apply hypothetical rule.
  (4) Reduce to categorical form.
  (5) Apply categorical rules.

(8) A disjunctive syllogism is one in which the major premise is a
disjunctive proposition.

(9) The two kinds of disjunctives are those which “_by affirming deny_”
and those which “_by denying affirm_.”

(10) In testing disjunctive arguments there are _two_ rules involved:
_First_, “The major premise must assert a _logical disjunction_.” This
necessitates the two requisites “_the alternatives must be mutually
exclusive_” and the “_enumeration must be complete_.” The two opinions
relative to the nature of an alternative assertion are, first, if one
is false, the other must be true and vice versa; and second, if one
is false, the other must be true, but _both_ may be true. The first is
adopted in this discussion.

_Second._ The second rule involved is “When the minor premise affirms
or denies _one_ of the alternatives of a logical disjunctive the
conclusion must deny or affirm all of the others.”

(11) Subjecting the disjunctive arguments to the categorical test
gives evidence to the close relation existing between the two forms.
A logical disjunctive proves to be logical when reduced to the
categorical. The reduction entails the two steps, first, reduce to
the hypothetical; second, reduce to the categorical.

(12) The logical meaning of the dilemma is suggested by the popular
conception. One is said to be in a dilemma when two courses are open
to him, neither of which is specially attractive.

A logical dilemma presents two alternatives either one of which might
well be avoided.

The major premise of the dilemma is hypothetical; while the minor is
disjunctive.

(13) The four forms of the dilemma are the simple constructive,
the simple destructive, the complex constructive and the complex
destructive.

(14) The dilemma is subject to the hypothetical rule which is, “The
minor premise must either affirm the antecedent or deny the consequent.”

(15) The minor premise need not be a logical disjunctive provided the
major conforms to the hypothetical rule.

(16) Frequently when ordinary experiences are reduced to augmentative
form they present a disjunctive proposition followed by a hypothetical
argument.


=20. REVIEW QUESTIONS.=

  (1) Relate the three kinds of arguments to the three general kinds
      of propositions.

  (2) Define and illustrate the hypothetical argument.

  (3) Explain the term conjunctive with reference to hypothetical
      arguments.

  (4) Explain and illustrate antecedent and consequent in hypothetical
      arguments.

  (5) Select from the following the antecedent and consequent:

        (1) “I usually succeed when I try.”
        (2) “I will not undertake it unless you guarantee half of the
            sum needed.”
        (3) “Though I speak with the tongues of men and of angels,
            and have not charity, I am become as sounding brass or a
            tinkling cymbal.”

  (6) Illustrate the two kinds of hypothetical syllogisms which are
      valid.

  (7) State and explain the rule to which hypothetical arguments must
      conform.

  (8) State and exemplify the one exception to the hypothetical rule.

  (9) Explain how hypothetical arguments may be reduced to the
      categorical form. Illustrate.

 (10) Show by illustration that denying the antecedent is equivalent
      to _illicit major_, while affirming the consequent is equivalent
      to _undistributed middle_.

 (11) Reduce to the categorical form and test:

        “If Napoleon had possessed more of the spirit of Washington, he
        would have been less famous but a better man than he was; but
        he did not possess the spirit of the ‘Father of His Country.’”

 (12) Test according to outline the following hypothetical arguments:

        (1) “If it be a good thing to have faith, then certainly he who
            believes in the bible of a pagan has faith and must have a
            good thing.”
        (2) “If a 10-inch charge burst inside of a tank, there would
            be nothing left of the tank. It would be blown into small
            pieces.”
        (3) “If the plate found had been originally on the outside of
            the ship, I should have judged that there must be green
            paint on it, but I could not find green paint on that part
            of the ship.”
        (4) “If I mistake not, you are the man who did not pay me for
            that pair of shoes. I am sure that you are the man as I
            never forget a face.”
        (5) “If the maxim ‘Early to bed and early to rise makes one
            healthy, wealthy and wise’ were true, I would have been a
            millionaire long ago.”

 (13) Define and illustrate a disjunctive argument.

 (14) Exemplify the two kinds of disjunctive arguments.

 (15) What is meant by a logical disjunction?

 (16) “The alternatives must be mutually exclusive.” Explain this,
      illustrating fully.

 (17) Cite cases where the enumeration is not complete.

 (18) State in complete form both of the rules to which all disjunctive
      arguments must conform.

 (19) Show by illustration how the disjunctive syllogism may be reduced
      to the categorical.

 (20) Define and illustrate the dilemma.

 (21) Give examples, using symbols, of the four dilemmatic forms.
      Explain why these forms are so named.

 (22) Why does the hypothetical rule apply to the dilemmatic syllogism?

 (23) Test the validity of the following: Give reasons.

        (1) “If a substance is solid it possesses elasticity and
            so also it does if it be a liquid or gaseous; but all
            substances are either solid, liquid or gaseous; therefore,
            all substances possess elasticity.”
        (2) “If men were prudent, they would act morally for their own
            good; if benevolent, for the good of others. But many men
            will not act morally, either for their own good or that of
            others; such men, therefore, are not prudent or benevolent.”
        (3) “If the majority of those who use public houses are
            prepared to close them, legislation is unnecessary; but if
            they are not prepared for such a measure, then to force it
            upon them by outside pressure is both dangerous and unjust.”
        (4) “The man is either a liar or a fool and in either case he
            is beneath my attention.”
        (5) “Either he is sincere or else he is the most astute
            impostor the world has ever produced; for me I prefer to
            think him sincere.”

 (24) Explain the relation that many experiences appear to bear toward
      an argument introduced by a disjunctive proposition and followed
      by a hypothetical syllogism. Illustrate.


=21. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) May both premises of a hypothetical argument be hypothetical
      propositions? Explain. See Fowler p. 115.

  (2) Which of the two is valid? Explain.

        (1) If A is B, C is D
            If A is B, E is F
          ∴ If C is D, E is F

        (2) If A is B, C is D
            If C is D, E is F
          ∴ If A is B, E is F

  (3) Show by circles that two of the possible four hypothetical
      arguments are invalid.

  (4) What categorical rules does the hypothetical argument seem to
      violate? Explain.

  (5) Originate a hypothetical syllogism whose antecedent and
      consequent are both negative. Test its validity.

  (6) Originate a co-extensive hypothetical argument and show that four
      valid syllogisms may be derived from it.

  (7) Explain by word and illustration the two meanings which may be
      attached to “either-or.”

  (8) If we accepted the opinion that both alternates of a disjunctive
      may be true, which kind of disjunctive argument would it
      invalidate?

  (9) In a logical disjunction what law of thought is involved?
      Explain.

 (10) Why do the laws of the disjunctive seem to contradict the
      categorical rules? Explain fully.

 (11) Show by drawing on common experience that a logical dilemma is
      closely related to the popular conception of dilemma.

 (12) Illustrate by symbols and then place in good English a pentalemma.

 (13) State a definition of a dilemma which excludes the simple
      destructive form.

 (14) Give a common experience which, when thrown into argumentative
      form, results in a disjunctive proposition followed by a
      hypothetical syllogism. Coin a name for such a combination.



                              CHAPTER 16.

             THE LOGICAL FALLACIES OF DEDUCTIVE REASONING.


=1. A NEGATIVE ASPECT.=

It has been remarked that “Logic as a science makes known the laws
and forms of thought and as an art suggests conditions which must
be fulfilled to think rightly.” In recent chapters we have discussed
the second aspect of the definition; in these we have attempted to
answer the question, “What _rules_ must be followed in order to reason
correctly?” We are now ready to treat the same aspect from a negative
point of view namely, what _errors_ must be avoided in order to reason
correctly? What are the fallacies which we must strive to avoid in our
_own_ thinking, and attempt to correct in the thinking of _others_?


=2. PARALOGISM AND SOPHISM.=

“Fallacy” comes from the Latin _fallacia_, meaning deceptive or
erroneous, and therefore _a fallacy in logic is any error in reasoning
which has an appearance of correctness_. If the writer or speaker is
_himself_ deceived by the fallacy, then such is called a _Paralogism_;
but if the fallacy is committed by him for the expressed purpose of
deceiving others, then such becomes a _Sophism_. During the time of the
Schoolmen the Sophism was in such high repute that it required even a
Socrates to puncture this ignominious bubble of vain trickery. In fact,
Socrates, the greatest of all pagan educators, _led_ the crusade which
has relegated to the “logical dust bin” the notion that _skill in the
art of framing sophisms is a scholarly accomplishment_. Many believe
modern sophistry to be the chief social and commercial evil of the
day, and to Socrates must be given the credit for teaching us to look
upon those who would practice sophism with righteous indignation and
pronounced disgust. However, paralogism and not sophism is the more
legitimate field for the student of logic; his problem being, “What
are the common errors which I, as a writer and speaker, must strive to
avoid?”


=3. A DIVISION OF THE DEDUCTIVE FALLACIES.=

The mistakes of induction will occupy our attention in a later
chapter. We are now concerned with the fallacies of deduction. Any
classification or division of the deductive fallacies must of necessity
be faulty. Even the labors of Aristotle in this regard are now
pronounced crude and unsatisfactory. This is due to the divergence of
opinion as to the signification of some of the fallacies, as well as
to the fact that no division is free from the fault of an _overlapping_
of the species. As a result of this lack of unanimity in definition and
lack of ability in making the species mutually exclusive, any division
of the deductive fallacies must be more or less illogical.

Aristotle divides the fallacies of deduction into two groups:
(1) Fallacies _in dictione_, or formal fallacies. (2) Fallacies _extra
dictionem_, or material fallacies. This division has received universal
approval and though many distinctions made by him have been abandoned,
yet most logicians retain his phraseology. Since many of the technical
terms which Aristotle used have lived through the generations under
the conventional meaning which he assigned to them, it becomes less
confusing to adhere as closely as possible to these terms. Therefore,
in the attending division only those changes have been made which
progress and experience have forced upon us. What remains of this
chapter will be devoted to explaining these fallacies as they appear
in this division. For the sake of clearness and definiteness it is
strongly recommended that the student study the outline extensively
enough to be able to reproduce it.

                             _Fallacies._

                       _Formal_ (_In dictione_)

                              {  1. Opposition
           1. Immediate       {  2. Obversion
                inference     {  3. Conversion
                              {  4. Contraversion

                              {  5. Four terms
                              {  6. Undistributed middle
           2. Categorical     {  7. Illicit major
                arguments     {  8. Illicit minor
                              {  9. Negative premises
                              { 10. Particular premises

           3. Hypothetical    { 11. Denying the antecedent
                arguments     { 12. Affirming the consequent

           4. Disjunctive     { 13. Illogical disjunction
                arguments     {

                      _Material_ (_In dictionem_)

                              {  1. Ambiguous middle
                              {  2. Amphibology
           1. In Language     {  3. Accent
                Equivocation  {  4. Composition
                              {  5. Division
                              {  6. Figure of speech

                              {  1. Accident
                              {  2. Converse accident
           2. In Thought      {  3. Irrelevant conclusion
                Assumption    {  4. Non sequitur
                              {  5. False cause
                              {  6. Complex question
                              {  7. Begging the question


=4. GENERAL DIVISIONS EXPLAINED.=

The _formal fallacies_ are those which concern the _form_ of the
argument rather than the meaning. These fallacies arise from an
improper use of words as arbitrary signs of thought, not from any
inconsistency in the thought itself. To commit a formal fallacy we
must violate one of the specific rules of logic. For this reason the
formal fallacies are easier of comprehension. Moreover, because of this
definiteness logicians are better able to come to some agreement as to
their content and import. Classing the fallacies of immediate inference
as formal is somewhat of an innovation; but since they occur because of
the breaking of certain definite rules, and since immediate inference
is a matter of changing the form without altering the meaning, we
believe there is some justification for this position. Some would class
“immediate inference” fallacies with the material fallacies of language.

The _material fallacies_ are fallacies of meaning and not of form. They
are those arising from inconsistency in thought, and from imperfect
ways of interpreting this thought as it appears in language. No very
specific rules of logic are violated by them and for this reason there
are those who would entirely eliminate the material fallacies from the
field of logic. But since _thought_ is even more subtle than _form_ in
its deceitful machinations, we believe that the material fallacy calls
for special attention on the part of the logician.

Material fallacies are divided into two kinds. First, those which have
reference to wrong thinking, or fallacies _in thought_; and, second,
those which are due mainly to an incorrect interpretation of words,
or fallacies _in language_. The former result from inconsistency and
unreasonableness in thought, whereas the latter come from lack of
precision in expression.


=5. FALLACIES OF IMMEDIATE INFERENCE.=

Fallacies of immediate inference arise from some violation of the rules
which this topic enunciates.

(1) _Opposition._

Among other statements opposition posits these two: (1) When the
particular is true its opposing universal is indeterminate; (2) A
universal negative does not necessarily contradict a universal
affirmative.

These signify that neither an A nor an E must be assumed to be true
when the corresponding I or O is true, and that E may not always
contradict A, nor O contradict I.


  ILLUSTRATIONS OF FALLACIES OF OPPOSITION.

(1) Since some men are wise, then I may conclude that all men are wise.
(2) I have contradicted his statement “all men are honest” by proving
that _no men are honest_.

There is little difference between fallacies like (1) and fallacies of
converse accident. Concerning illustration (2), _both_ statements are
false; but to contradict we know that if one is false, the other must
be true.

(2) _Obversion._

“Two negatives are equivalent to one affirmative,” is the principle
underlying obversion. The most common fallacy in obversion springs from
using one negative instead of two.


  ILLUSTRATIONS OF FALLACIOUS OBVERSION.

(a) Original: Some men are not wise. Obverse: (incorrect) Some men are
wise.

(b) Original: All true teachers are just. Obverse: (incorrect) All true
teachers are not just.

(3) _Conversion._

Conversion involves the interchanging of the subject and predicate
of a proposition without affecting the distribution; in consequence
the usual fallacy incident to this interchange is _distributing_ an
undistributed term.


  ILLUSTRATIONS OF FALLACY OF CONVERSION.

(a) Original: All fixed stars are heavenly bodies. Converted:
(incorrectly) All heavenly bodies are fixed stars.

(b) Original: Some men are not wise. Converted: (incorrectly) Some wise
beings are not men.

(4) _Contraversion._

As this process involves the two steps of obversion and conversion,
fallacies appertaining to contraversion would relate to these two steps.


  ILLUSTRATIONS OF FALLACIES OF CONTRAVERSION.

(a) Original: No honest man fails to pay his debts. Contraverted:
(incorrectly) Some who do not pay their debts are honest men.

(b) Original: Some animals are quadrupeds. Contraverted: (incorrectly)
Some not-quadrupeds are not animals.

The formal fallacies of categorical, hypothetical, and disjunctive
arguments have received detailed treatment in chapters 11, 14 and 15;
we may, therefore, devote our attention to the material fallacies
without further delay.


=6. FALLACIES OF LANGUAGE. (Equivocation.)=

These are the fallacies of _double meaning_. It is known that an
equivocal term is one which permits two or more interpretations;
similarly a _proposition_ which admits of two or more interpretations
may be denominated equivocal. Thus the term equivocation has come to
stand for all errors in language resulting from a possibility of more
than one interpretation. This justifies the position of referring to
all of the six fallacies in language as fallacies also of equivocation.

(1) _Ambiguous middle._

Ambiguous middle explains itself. _It is the fallacy of giving to the
middle term a double meaning._ In form the argument may contain but
three terms, yet in meaning there are in reality four terms. For this
reason ambiguous middle and the fallacy of four terms appear to be
about one and the same thing; but in this treatment we shall regard
them as mutually exclusive, and this is the distinction:

Invalid arguments of “ambiguous middle” have only _three_ terms in form
but _four_ terms in meaning. This signifies that the middle term though
identical in _form_ is given a _double meaning_.

Invalid arguments of “four terms” always have four terms in both _form_
and _meaning_; they are “logical quadrupeds” in every sense of the word.


  ILLUSTRATIONS.

_Ambiguous middle._

  (a) “_Necessity_ is the mother of invention,”
      Bread is a _necessity_,
    ∴ Bread is the mother of invention.

  (b) “_Nothing is_ better than wisdom,”
      Dry bread is better than _nothing_,
    ∴ Dry bread is better than wisdom.

  (c) A _church_ is a force for good in any community,
      A slate roof is good for a _church_,
    ∴ A slate roof is a force for good in any community.

_Fallacies of four terms._

  (a) All _true teachers_ are just,
      John Doe is an _educator_,
    ∴ John Doe is just.

  (b) _Milk_ is nourishing,
      This substance is a _white fluid_,
    ∴ This substance is nourishing.

  (c) _Thieves_ should be imprisoned,
      This man _has taken what does not belong to him_,
    ∴ This man should be imprisoned.

In the “four-term” fallacies, observe that the four terms occur in
the _premises_. When a fourth term is introduced in the conclusion,
the material fallacy of _non sequitur_ has been committed.

(2) _Amphibology_ (_or amphiboly_).

Amphibology is a fallacy resulting from an ambiguous proposition
rather than from the ambiguity of any particular term. _The fallacy of
amphibology is committed when the spoken or written proposition conveys
more than one meaning._ The ancient oracles indulged in this sort of
fallacy, the reason for such indulgence being obvious; the oracles
were not too positive as to the outcome of their prognostications, and
therefore were especially careful to cover _every emergency_.

A careless use of relative clauses and prepositional phrases often
results in the fallacy of amphibology.


  ILLUSTRATIONS OF THE FALLACY OF AMPHIBOLOGY.

  (a) “You the enemy will slay.”

  (b) “The Duke yet lives that Henry shall depose.”

  (c) “Wanted a piano by a young lady made of mahogany.”

  (d) “You your father will punish.”

(3) _Accent._

_This fallacy springs from placing undue emphasis on some word or group
of words._ Naturally such accentuation may convey a meaning entirely
foreign to the author’s intent. Newspapers are guilty of this fallacy
when they select a few words from a speech and use them as headlines
without further explanation. A politician may quote a sentence uttered
by an opponent and fail to relate it to what preceded or followed.
A cartoonist may arouse the prejudice of public opinion by giving
ridiculous emphasis to some idiosyncracy possessed by the subject of
his attack.


  ILLUSTRATIONS OF FALLACIES OF ACCENT.

  (a) “Thou shalt not bear false witness against thy _neighbor_.”

      By giving undue emphasis to _neighbor_, the notion is clearly
      conveyed that one may bear false witness against all who are
      _not_ neighbors.

  (b) “You must not crib when taking _my_ examinations.”

  (c) _What the “Spellbinder” said._

      “I may say, as a side remark, that the labor unions are guilty of
      developing a nation of shirks, when they prohibit a phenomenally
      efficient workman from doing his best.” “I do not wish to be
      misunderstood in this.” “I believe in labor unions but in this
      particular they are dead wrong.”

      _What the newspaper reported._

      (Headline) “The Labor Union Scored as a Training School for
      Shirks.” “―――― said in his speech in ―――― Hall that the Union was
      responsible for the development of a nation of shirks.” “A good
      man,” said he, “is not permitted to do his best work.”

(4) _Composition._

_The fallacy of composition is committed when it is assumed that what
is true distributively is likewise true collectively._ A term is used
in a distributive sense when it is applied to _each individual_ of
the class; whereas a term is used in a collective sense when it is
applied to the class considered as _one whole_. “All” meaning _each
one considered separately_ and “all” meaning _the whole_ furnishes a
frequent pitfall for this fallacy.


  ILLUSTRATIONS OF THE FALLACY OF COMPOSITION.

  (a) “Every member of the team is a star player; hence I expect that
      the entire aggregation will be a winner.”

  (b) “All the men of the jury are fair minded; therefore we have good
      reason for supposing that the jury’s verdict will be in accord
      with the rules of justice.”

  (c) “Thirteen and twenty-three are odd numbers; thirty-six is equal
      to thirteen and twenty-three; hence thirty-six is an odd number.”

  (d) “All the angles of a triangle are less than two right angles;
      hence the angles X, Y and Z are less than two right angles.”

  (e) In governmental affairs the assumption, that a law which benefits
      one section will benefit all, is a fallacy of composition.

(5) _Division._

_The fallacy of division is committed when it is assumed that what is
true collectively is true distributively._ Division is the converse of
composition. Composition is a fallacious procedure from a distributive
to a collective use; while division is a fallacious procedure from
a collective to a distributive use. The fallacy of division may
be illustrated by giving the converse of the illustrations under
composition:

  (a) “The team is a star playing team; and since Smith is the ‘first
      baseman’ of the team, he must be a star player.”

  (b) “The jury rendered a just decision; hence the foreman is a fair
      minded man.”

  (c) Thirty-seven is an odd number,
      Nine and twenty-eight are thirty-seven,
    ∴ Nine and twenty-eight are odd numbers.

  (d) All the angles of a triangle are equal to two right angles,
      A is an angle of a triangle,
    ∴ A is equal to two right angles.

(6) _Figure of Speech._

_This fallacy results from assuming that words of the same root have
the same meaning._ Since the same root-word may be used as a noun,
verb, adjective, etc., it does not follow that in these various forms
it retains a common meaning. “Address” as a noun and “address” as a
verb convey two distinct meanings.

The following are examples of this fallacy:

  (a) No _designing_ person should be trusted,
      This architect is a _designer_,
    ∴ This architect should not be trusted.

  (b) Justifiable _investigation_ is wise,
      This man is a just _investigator_,
    ∴ This man is wise.

These fallacies are not classed as those of “four terms” because two
terms so _closely resemble_ each other in _form_, and yet they are
not fallacies of ambiguous middle; since the middle terms are _not
identical_ in form.


=7. FALLACIES IN THOUGHT.=

The fallacies in thought arise through a tendency to assume as true
that which demands further proof. Any one who is more anxious to
be _right_ than to _win_ will make sure that nothing has been taken
for granted which should receive further investigation, or that no
truth has been given a presumptuous twist in order to make it fit the
particular case under discussion. Because these errors in thought may
be attributed chiefly to undue assumptions, we may denominate them as
the fallacies of assumption.

(1) _Accident._

_The fallacy of accident occurs when one reasons from a general truth
to an accidental case._ Doctrinaires and theoretic enthusiasts are
partial to this fallacy. It is so easy to lay down a general formula or
remedy and then attempt to apply it to every accidental circumstance.
Grandmother with her catnip tea and mustard plaster, however we may
cherish the memory of the dear old soul, was nevertheless guilty of
the fallacy of accident. Applying maxims and proverbs to particular
instances is still another way of committing the fallacy.


  EXAMPLES OF FALLACIES OF ACCIDENT.

  (a) “Honesty is the best policy,” thinks the physician as he reveals
      the cold, hard truth to his patient and thus shortens the
      patient’s life.

  (b) Spirituous liquor in excess acts as a poison, and therefore
      should not be used to resuscitate an extreme case.

  (c) “What is bought in the market is eaten; raw meat is bought in the
      market; therefore it is eaten.”

  (d) “Early to bed and early to rise makes one healthy, wealthy and
      wise.” I shall practice this for ten years and by that time hope
      to be healthy, wealthy and wise.

  (e) John has earned the enviable (?) reputation of being the “worst
      boy in school,” hence he is going to be the worst boy in “my
      grade.”

  (f) Mary is an inveterate whisperer; and since I know that some one
      is whispering, I am sure that that some one is Mary.

  (g) Being a convict, he is not to be trusted.

(2) _Converse Accident._

_As the title implies this is the fallacy of reasoning from an
accidental case to a general truth._ Illustrations:

  (a) “John has been a bad boy to-day; and hence he is going to make
      trouble during the entire term.”

  (b) “This food is good for hens; and hence it is good for all
      domestic fowls.”

  (c) “I know of several men who have been phenomenally serviceable
      to mankind, and none of these men were college trained; hence
      I conclude that college education is not essential to the
      attainment of the highest state of efficiency.”

Relative to both accident and converse accident, it may be said that
they obtain because all general truths, such as rules, principles,
definitions, maxims, etc., have their _exceptions_; and it is through
these exceptions that the two fallacies are made possible.

_Accident and Converse Accident Distinguished from Division and
Composition._

The fallacy of accident, we have learned, occurs when one reasons
from a _general truth_ to an _accidental case_; whereas the fallacy
of division obtains when one reasons from a _collective_ use of a
term to a _distributive_ use; in both cases the procedure is from a
_larger unit_ to a _smaller unit_. Moreover, with converse accident and
composition, the movement is from the _smaller unit_ to the _larger_.
Because of this similarity there is danger of confusing the two kinds
of fallacies. As a matter of distinction between the fallacies of
accident, and composition and division the attending comparative
_résumé_ may be of value:

  (1) _Division_ is similar in movement to _accident_, while
      _composition_ resembles _converse accident_.

  (2) A valuable cue for remembering which way division and accident
      move, is to recall that division in arithmetic is a procedure
      from the larger unit to the smaller, and therefore that division
      in logic would have the same signification.

  (3) Division and composition pertain to _mathematical_ wholes; while
      accident and converse accident relate to _logical_ wholes.

  (4) The aggregates of division and composition may be counted or
      enumerated easily; while the accident and converse accident
      aggregates (or generals) are not easily enumerated.

  (5) Division and composition relate to logical _terms_, whereas
      accident and converse accident relate to general _truths_.

  (6) Division and composition use a _term_ in a _collective_ sense
      and then in a separate or _distributive_ sense, or vice versa;
      accident and converse accident use a _thought_ in a _general_
      and then in an _accidental_ sense, or vice versa.

_Irrelevant Conclusion_ (_Ignoratio Elenchi_).

_The fallacy of irrelevant conclusion results when the argument does
not squarely meet the point at issue._ It is the fallacy of arguing to
the wrong point either purposely or through ignorance. One in defense,
who has a weak case, may be tempted to divert attention from the point
in hand, realizing that a close analysis of the matter in dispute
will tend to his undoing. In such instances (1) the lawyer will abuse
the plaintiff, (2) the demagogue will tell humorous stories, (3) the
teacher will take advantage of the ignorance of the pupil, (4) the
scholar will refer to authority and (5) the magnate will fall back
upon the power of position and wealth. These forms of “_rhetorical
thinking_” are as harmful as they are popular, and furnish one of the
chief reasons for giving to the common people a better understanding
of “how to think” as well as “how _not_ to think.”

Definite names have been given to the various forms of irrelevant
conclusion which may be summarized as follows:

_Argumentum ad populum._

This is the fallacy of appealing to the feelings, passions and
prejudices of an audience rather than to their good sense and powers of
reason. It is probably the most common of the group. To excite sympathy,
the lawyer for the defense may speak feelingly of the _suffering_
that an unfavorable verdict will bring to the wife and children of the
accused.

_Argumentum ad hominem._

Here the character of the opponent is defamed with a view of
discrediting him with the court or audience. “Mud throwing” in times of
political agitation is a good example of this fallacy.

_Argumentum ad ignorantiam._

This fallacy comes from taking advantage of the ignorance of the
opponent; the fallacy assumes that the original supposition has been
proved if one is unable to prove the _contradictory_ of the original.
Illustration: Mars is inhabited because no one is able to prove that
Mars is not inhabited.

_Argumentum ad baculum._

In this all argumentation is made to give way to the forces of personal
opposition and to the power of money. Illustration: A political
committee seating those delegates only, who will vote _their_ way; and,
doing this, not from the merits of the case, but because said committee
happen to have a sufficient number of votes to “put the thing through.”

_Argumentum ad verecundiam._

This fallacy comes from supposing that the whole thing may be settled
by citing some noted authority who apparently substantiates the
argument advanced.

_Epitome of five forms of Irrelevant Conclusion_:

  (1) Appealing to the audience.

  (2) Defaming the character of the opponent.

  (3) Inability to prove the contradictory.

  (4) Gaining the point by force.

  (5) Citing authority.

_Non Sequitur (False Consequent)._

_This is the fallacy of deriving a conclusion which does not follow
from the premises._ The fallacy obtains whenever material appears in
the conclusion, which has no bearing on the case under discussion.
“_Irrelevant conclusion_” pertains to the establishment of the premises
while “_non sequitur_” is concerned with the conclusion only. We know
that a logical thinker constructs the conclusion from material already
presented by the premises; “_Non sequitur_” uses material in the
conclusion which is found in neither premise.

“_Non sequitur_” differs from the fallacy of four terms in that the
latter uses the fourth term in the premises while the former introduces
the fourth term in the conclusion, and in a form so well obscured that
it sometimes escapes notice. Illustration:

    All men are thinking animals,
    Socrates was a man,
  ∴ Socrates was a scholar.

It does not follow that because a man is a thinking animal that he will
become scholarly.

_False Cause._

_This is the fallacy of assuming that because two happenings have
occurred together several times, the one is the cause of the other._
This very common fallacy is due to lack of discrimination, and to the
exaggerations incident to fear and superstition. Illustrations:

  (a) Planting vegetables which grow down, such as the beet, during
      the last two days of the waxing moon in order to have a larger
      yield. So far as we know the moon has no influence over growing
      vegetables.

  (b) Thirteen seated at a table is an indication that one of the
      number will die during the year. This is one of the most absurd
      fallacies that has ever been visited upon an intelligent people.

It is seen that “False Cause” is closely related to “_Non Sequitur_.”

_Complex Question_ (_Double Question_).

_This fallacy obtains when an assumption is put in the form of a
question._


  ILLUSTRATIONS:

  (a) A wise father who did not want to tempt beyond the yielding
      point his three-year-old son, asked, pointing to the scratches on
      the new mahogany piano, “Freddie, did you do that last night or
      this morning?”

  (b) What caused you to desist from slandering your neighbors; New
      Year’s resolutions or the preaching of Dominie X?

  (c) A daily paper anecdote:

      “Charles Bradlaugh, the noted English free-thinker, once engaged
      in a discussion with a dissenting minister. He insisted that
      the minister should answer questions by a simple yes or no,
      asserting that every question should be replied to in that
      manner.” The reverend gentleman arose and said, “Mr. Bradlaugh,
      will you allow me to ask you a question on these terms?”
      “Certainly,” said Mr. Bradlaugh. “Then, may I ask, have you
      given up beating your wife?”

_Begging the Question_ (_Petitio Principii_).

This is a fallacy _of deriving a conclusion from notions which in
themselves demand proof_.

The fallacy is not committed when the assertion is self-evident. It
is easy to claim that our opponent is begging the question as soon as
we see that he is getting the better of us. One may himself beg the
question by being too ready to charge others with begging the question.
When the opponent adopts premises which are commonly accepted, he does
not beg the question. One commits the fallacy when he _seems to prove_
the conclusion more satisfactorily than he really does. This he may
accomplish by covertly taking for granted the truth of notions which
have not the stamp of universal approval. The fallacy of begging the
question assumes three forms:

(1) _The assumption of an unproved premise_ (_assumptio non probata_).

In this either the major or the minor premise, or both may demand
more substantial proof. It must be borne in mind, however, that the
disputant must not ask for further proof after he has once accepted the
premises, or after the opponent has met his demands to the satisfaction
of commonly accepted authority.

_Examples of begging the question by assuming unproved premises_:

  (a)   All patriotic citizens are honest at heart,
        This man charged with graft is a patriotic citizen,
      ∴ This man charged with graft is honest at heart.

      “All patriotic citizens are honest at heart,” is not an accepted
      truth and thus demands proof.

  (b) A famous sophism of the Greek philosopher by which he proved that
      motion was impossible, is an excellent illustration of an assumed
      premise:

        “If motion is possible, a body must move either in the place
          where it is, or in the place where it is not;
        But a body cannot move in the place where it is; and of course
          it cannot move where it is not,
        Therefore, motion is impossible.”

      Referring to this, De Morgan claims “Movement is change, and so a
      body requires _two_ places in order to move.” _A body cannot move
      in the place where it is, but must be moved from place to place._
      The major premise being assumed, this sophism illustrates the
      fallacy of begging the question.

  (c) The most subtle form of begging the question is an enthymeme
      where the suppressed premise is the one assumed; e. g., “You,
      being a teacher, should not do as other people do.”

      Completed and arranged the argument becomes:

        No teacher should do as other people do,
        You are a teacher,
      ∴ You should not do as other people do.

      Surely the major premise demands proof.


(2) _Reasoning in a Circle_ (_Circulus in probando_).

This form of begging the question occurs, “When a conclusion is based
upon a premise which in an earlier stage of the argument was itself
based upon this very conclusion.” To put it in another way: Reasoning
in a circle involves proving the truth of a conclusion by using a
particular premise, and then proving the truth of the particular
premise by using the conclusion. From premise to conclusion and from
conclusion to premise completes the circle.

_Examples of begging the question by reasoning in a circle_:

  (a) It is wrong because my conscience pricks me, and my conscience
      pricks me because it is wrong.

  (b) “The effeminate walk shows a lack of force; because no forceful
      man walks that way.”

  (c) Says Hamilton, “Plato, in his _Phoedo_, demonstrates the
      immortality of the soul from its simplicity; and in the
      _Republic_, he demonstrates its simplicity from its immortality.”

(3) _Question Begging Epithets and Appellations._

This is the fallacy of assuming the point at issue by means of a
carefully selected epithet.

Scientists sometimes assume to clarify an inexplicable phenomenon by
giving it a technical name. Politicians are exceedingly free with their
epithets and appellations, and the records of religious disputes prove
that the theologian often resorted to this device.

_Examples of begging the question by using epithets and appellations_:

  (a) We must attribute the disease to _heredity_.

  (b) The candidate for governor is an _animated feather duster_.

  (c) They call him _Blue Charlie_.

  (d) It is the policy of the _big stick_.

  (e) The _muck-raker_ seldom makes an efficient servant of the people.

It is seen that the use of these epithets and appellations is simply a
rhetorical device for the purpose of creating either a favorable or
unfavorable impression.


=8. OUTLINE.=

  THE LOGICAL FALLACIES OF DEDUCTIVE REASONING.

  (1) A negative aspect of definition of logic.

  (2) Paralogism and sophism.
        Distinguished. Mission of Socrates.

  (3) A division of the deductive fallacies.
        More or less faulty. Aristotle’s phraseology retained.
        Division given.

  (4) General divisions explained.
        Formal and material. Material fallacies in _language_ and in
            _thought_.

  (5) Fallacies of immediate inference.
        Opposition, obversion, conversion, contraversion.

  (6) Fallacies in language (also fallacies of equivocation).
        Ambiguous middle――distinguished from four terms.
        Amphibology.
        Accent.
        Composition――“all” a pitfall.
        Division.
        Figure of speech.

  (7) Fallacies in thought――(also fallacies of assumption).
        Accident.
        Converse accident. Made possible by exceptions.
        Accident and converse accident distinguished from composition
            and division.
        Comparative résumé.
        Irrelevant conclusion (_ignoratio elenchi_).
          _Argumentum ad populum._
          _Argumentum ad hominem._
          _Argumentum ad ignorantiam._
          _Argumentum ad baculum._
          _Argumentum ad verecundiam._
        _Non sequitur_ (false consequent).
        False cause.
        Complex question.
        Begging the question (_petitio principii_).
          Assumption of premise.
          Reasoning in a circle.
          Question begging epithets and appellations.


=9. SUMMARY.=

(1) Logic as a science makes known the laws and forms of thought and as
an art suggests conditions which must be fulfilled in order to think
rightly.

A discussion of the second phase of the definition would be incomplete
without a consideration of the negative aspect as well as the positive.
Such a viewpoint makes evident the question “What errors must be
avoided in order to reason correctly?” An answer to this question is
given under the caption of Logical Fallacies.

(2) A logical fallacy is any error in reasoning which has the
appearance of correctness.

A fallacy which deceives the writer or speaker himself is termed
a paralogism, whereas a fallacy formed for the express purpose of
deceiving another is denominated a sophism.

It was the pagan teacher Socrates who taught modern thought to frown
upon all forms of sophism; these exist to-day much as they did in the
olden time.

(3) Because of disagreement as to definition, and because of inability
to prevent an overlapping of species, any logical division of the
deductive fallacies must be faulty.

In the division of the deductive fallacies, this treatise retains the
phraseology and form worked out by Aristotle, so far as such retention
is consistent with the changes incident to the advances of time.

(4) Formal fallacies occur because of careless and improper use of
words as arbitrary signs. Formal fallacies are definite and easy of
comprehension.

The material fallacies are due to certain inconsistencies in thought
and to imperfect ways of interpreting language. They are more subtle
and thus more difficult of comprehension than the formal fallacies.

There are material fallacies in thought and material fallacies in
language; the former are due to _looseness in thinking_ and the latter
to _lack of precision in expression_.

(5) Fallacies of opposition result most frequently from deriving
universals from their corresponding particulars, and from assuming
to contradict affirmative universals by negative universals and
affirmative particulars by negative particulars.

The common fallacy in the process of obversion consists in using one
negative instead of two, whereas the ordinary error of conversion is a
matter of distributing an undistributed term.

Fallacies of contraversion must involve either those of obversion or
conversion since the process is a combination of the two.

(6) Fallacies in language, because they result from permitting
more than one interpretation, may be also denominated fallacies of
equivocation.

  (1) Ambiguous middle is the fallacy of giving to the middle term a
      double meaning.

      The fallacy of four terms, as the name signifies, exists when
      the argument has four terms in both form and meaning. Ambiguous
      middle is a matter of four terms in meaning but only three in
      form.

  (2) The fallacy of amphibology is committed when the given
      proposition conveys more than one meaning. In order to maintain
      their prestige the ancient oracles made use of this fallacy.

  (3) The fallacy of accent springs from placing undue emphasis on some
      word or group of words. Newspaper and demagogues are prone to
      this error, that they may thus create an unfavorable impression
      towards those whom they oppose.

  (4) The fallacy of composition is committed when it is assumed that
      what is true _distributively_ is likewise true _collectively_.
      “All” meaning _each one_ and “all” meaning the _whole class_
      often leads to the fallacy of composition.

  (5) The fallacy of division is committed when it is assumed that what
      is true _collectively_ is true _distributively_.

      Division is the converse of composition.

  (6) The fallacy of figure of speech is occasioned by assuming that
      words of the same root have the same meaning.

(7) Fallacies in thought are likewise called fallacies of assumption,
because of the tendency to assume as true something which demands
further proof.

  (1) The fallacy of accident occurs when one reasons from a general
      truth to an accident case. It is the favored fallacy of the
      doctrinaire, the reformer and the vender of “cure-alls.”

  (2) The fallacy of converse accident occurs when one reasons from an
      accidental case to a general truth.

Both accident and converse accident are made possible because rules,
definitions, maxims, etc., have _exceptions_. It is easy to confuse
division and composition with the fallacies of accident. Division and
composition are concerned with the _collective and distributive use of
terms_, whereas the fallacies of accident involve the use of notions
in a _general and accidental sense_. The former represent notions which
may be _counted or enumerated_ while the latter concern notions which
are _logical_ rather than numerical. Composition and division involve
“number of,” accident, “meaning of.”

  (3) The fallacy of irrelevant conclusion results when the argument
      does not squarely meet the point at issue. It is the fallacy of
      arguing to the wrong point either purposely or ignorantly. This
      may be accomplished by (1) appealing to sympathy of audience,
      (2) defaming character of opponent, (3) assuming that the fact is
      true because of inability to prove the contradictory, (4) gaining
      point by force, (5) citing authority.

  (4) “_Non sequitur_” is the fallacy of deriving a conclusion which
      does not follow from the premises. It involves introducing new
      material in the conclusion.

  (5) “False cause” is the fallacy of assuming that because two
      happenings have occurred together several times the one is
      the cause of the other. The fallacy is due largely to the
      exaggerations of fear and superstition.

  (6) The fallacy of complex question consists in putting an assumption
      in the form of a question.

  (7) Begging the question is the fallacy of deriving a conclusion from
      notions which in themselves demand proof.

      This fallacy takes the three forms of (1) the assumption of an
      unproved premise, (2) reasoning in a circle, (3) question begging
      epithets and appellations.


=10. ILLUSTRATIVE EXERCISES IN THE TESTING OF ARGUMENTS IN BOTH FORM
     AND MEANING.=

  (1a)  He who wilfully takes the life of another should be
        electrocuted, This sharp shooter has wilfully taken the life
        of another, Hence he should be electrocuted.

                                                              { A
In form we know this argument to be valid since it is in mood { A of
                                                              { A
the first figure. But as the conclusion does not meet with our approval,
we are forced to the belief that there must be a material fallacy.
Such proves to be the case. In the first instance, “Wilfully takes
the life of another” is used in a personal, individual, selfish sense,
whereas in the second instance the expression is used in a general,
“servant-of-the-government” signification. The argument is, therefore,
invalid, the fallacy being ambiguous middle.

(1b) From the viewpoint of both _form_ and _meaning_ test the following:
“Events which are not probable happen almost every day; but what
happens every day are very probable events; therefore events which are
not probable are very probable.”

  (2a)  The planets have those attributes needed in the support of life,
        Mars is a planet,
        Hence Mars has those attributes needed in the support of life.

                      { A
This is valid in form { A in the first figure. The major premise posits
                      { A
a fact which has not been proved; the argument is therefore invalid in
meaning, the fallacy being that of _begging the question_.

(2b) “The end of a thing is its perfection; death is the end of life,
therefore death is the perfection of life.”

Indicate the fallacy in the foregoing, giving reasons.

  (3a)  The countries of Europe abound in beggars,
        France is a country in Europe,
      ∴ France abounds in beggars.

“The countries of Europe” in the major premise is used in a collective
sense, while the same expression in the minor premise is used in a
distributive sense. The argument is, therefore, invalid in meaning;
_fallacy of division_.

(3b) State and explain the material fallacy in the following:

The states believe in the income tax principle; hence Vermont’s vote
will be favorable to this.

(4a) “On general principles I believe that one is better off when he
abstains from both tea and coffee; and this is the reason why I offer
you a cup of hot water.”

The individual to whom the hot water was offered might have been
in great need of a mild stimulant. Here, then, is an exception to
the general principle and the fallacy committed is clearly that of
_accident_.

(4b) “Books are a source both of instruction and amusement; a table of
logarithms is a book; therefore it is a source both of instruction and
amusement.” Jevons.

Designate with explanations the fallacy in the above argument.

(5) “Twice have I started out on Friday and both times I had tire
trouble.” Fallacy of _false cause_.

(6) “Where do you spend your vacation, in Palestine or Rome?” Fallacy
of _complex question_.

(7) “Of all the men of that department he seemed to be the most
trustworthy, and I pride myself on my ability to judge men in this
regard; but now even the police cannot find him.”

The fact that the police cannot find him has nothing to do with the
argument. The fallacy is that of _non sequitur_.

(8) “You must not whisper in _my_ classes.” Fallacy of _accent_.

(9) “I am a Progressive because I believe in progress.” Fallacy of
_figure of speech_.

(10) “I know it is true because I found it in our text book.” Fallacy
of _irrelevant conclusion_.


=11. REVIEW QUESTIONS.=

  (1) Give the negative aspect of the second part of the definition of
      logic.

  (2) Define and illustrate the term fallacy as it is used in logic.

  (3) Distinguish between a paralogism and a sophism.

  (4) Tell of the mission of Socrates.

  (5) What reasons may be given for such a divergence of opinion on a
      proper classification of the fallacies of deduction?

  (6) Give a complete outline, without explanation, of the deductive
      fallacies.

  (7) Distinguish between formal and material fallacies.

  (8) Explain the two kinds of material fallacies.

  (9) Illustrate the fallacies of immediate inference.

 (10) Why should the fallacies in language be likewise termed fallacies
      of equivocation?

 (11) Explain and illustrate ambiguous middle.

 (12) Illustrate the fallacy of amphibology.

 (13) Explain by illustration the fallacy of accent.

 (14) Explain and exemplify the fallacies of composition and division.

 (15) Illustrate the fallacy of figure of speech.

 (16) Give reasons for denominating the fallacies in thought as
      fallacies also of assumption.

 (17) Define and illustrate the fallacies of accident and converse
      accident.

 (18) Distinguish between the fallacies of composition and division and
      the two fallacies of accident.

 (19) “Every rule has its exception,” what has this to do with the
      fallacies of accident?

 (20) Explain and illustrate the fallacy of irrelevant conclusion.

 (21) Name the various ways in which irrelevant conclusion may be
      committed.

 (22) Illustrate the fallacy of _non sequitur_.

 (23) Explain the fallacy of false cause.

 (24) Give examples of the complex question.

 (25) How may the teacher use the complex question to advantage?

 (26) Explain the fallacy of begging the question.

 (27) Illustrate the three forms of begging the question.

 (28) From the viewpoint of form and meaning, test the validity of the
      following:

      (1) “No soldiers should be brought into the field who are not
          well qualified to perform their part; none but veterans are
          well qualified to perform their part, therefore, none but
          veterans should be brought into the field.” Whately.

      (2) “For the proverb is true, ‘That light gains make heavy
          purses;’ for light gains come thick, whereas great gains come
          but now and then.” Bacon.

      (3) “Whatever is given on the evidence of sense may be taken as a
          fact; the existence of God, therefore, is not a fact, for it
          is not evident to sense.” St. Andrew. 1896.

      (4) “All the trees in the park make a thick shade; this is one of
          them, therefore this tree makes a thick shade.” Jevons.

      (5) “What we eat grew in the field; loaves of bread are what we
          eat; therefore loaves of bread grew in the fields.” Jevons.

      (6) “Who is most hungry eats most; who eats least is most hungry;
          therefore who eats least eats most.” Jevons.

      (7) “Great talkers should be cropped, for they have no need of
          ears.” Franklin.

      (8) “Love your enemies, for they tell you your faults.” Franklin.

      (9) “All the works of Shakespeare cannot be read in a day;
          therefore the play of Hamlet, being one of the works of
          Shakespeare, cannot be read in a day.” Jevons.

     (10) “Logic as it was cultivated by the schoolmen proved a
          fruitless study; therefore logic as it is cultivated at the
          present day must be a fruitless study likewise.” Jevons.


=12. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Is there any difference in content between error and fallacy?
      Illustrate.

  (2) In what ways do trusts indulge in sophisms?

  (3) May the sophism be used conscientiously by the country doctor?
      Explain.

  (4) Give in substance Aristotle’s classification of fallacies.

  (5) Select the fallacies which could with justice be called fallacies
      of interpretation. See Creighton.

  (6) Explain in full the popular conception of equivocation.

  (7) Indicate the marks which distinguish the following: Ambiguous
      middle, fallacy of four terms, non sequitur, figure of speech.

  (8) “Why should Jeremy Bentham employ a person to read to him who
      habitually read in a monotonous tone of voice?” Jevons――Hill.

  (9) Originate a sentence of about ten words and through the fallacy
      of accent secure as many different meanings as possible.

 (10) Show that the fallacy of figure of speech might be classed as a
      fallacy of four terms.

 (11) To what fallacies, in your opinion, are teachers especially
      given?

 (12) Show that the fallacy of accident could be classed as one of
      ambiguous middle.

 (13) “When the Puritan settlers in New England passed their three
      famous resolutions――Resolved, _first_, that the earth is the
      Lord’s and the fullness thereof; _secondly_, that he hath given
      it to his Saints; _thirdly_, that we are his Saints. What fallacy
      did the Puritan Fathers commit?” Ryland.

 (14) A Dutchman afflicted with pneumonia arises at midnight and eats
      a large quantity of sauerkraut. The Dutchman gets well, whereat
      his physician writes in his little book on remedies, “_Sauerkraut
      sure cure for pneumonia_.” The physician was guilty of what
      fallacy? Why?

 (15) De Morgan quotes from Boccaccio this: “A servant who was roasting
      a stork for his master was prevailed upon by his sweetheart to
      cut off a leg for her to eat. When the bird came upon the table
      the master desired to know what had become of the other leg. The
      man answered that storks never had more than one leg. The master,
      very angry, but determined to strike his servant dumb before
      he punished him, took him next day into the fields where they
      saw storks, standing each on one leg, as storks do. The servant
      turned triumphantly to his master; on which the latter shouted,
      and the birds put down their other legs and flew away. Ah, sir,
      said the servant, you did not shout to the stork at dinner
      yesterday; if you had done so he would have shown his other leg.”
      What fallacy does this quotation from Boccaccio illustrate?

 (16) Why should begging the question and irrelevant conclusion be
      classed as fallacies of the “forgotten issue?”

 (17) From the standpoint of both form and meaning test the validity of
      the following:

      (1) “Virtue is the child of knowledge and vice of ignorance;
          therefore education, periodical literature, traveling,
          ventilation, drainage and the arts of life, when fully
          carried out, serve to make a population moral and happy.”
          Hibben.

      (2) “The civil power has the right of ecclesiastical jurisdiction
          and administration, therefore parliament may impose articles
          of faith on the church or suppress dioceses.” Hibben.

      (3) “Seeing that abundance of work is a sure sign of industrial
          prosperity, it follows that fire and hurricane benefit
          industry, because they undoubtedly create work.”
          St. Andrews――1895.

      (4) “Riches are for spending, and spending for honor and good
          action; therefore, extraordinary expense must be limited by
          the worth of the occasion.” Bacon.

      (5) “And let a man beware how he keepeth company with choleric
          and quarrelsome persons; for they will engage him into their
          own quarrels.” Bacon.

      (6) “He that resteth upon gains certain, shall hardly grow to
          great riches; and he that puts all upon adventures, doth
          oftentimes break and come to poverty. It is good, therefore,
          to guard adventures with certainties that they may uphold
          losses.” Bacon.



                              CHAPTER 17.

                         INDUCTIVE REASONING.


=1. INDUCTIVE AND DEDUCTIVE REASONING DISTINGUISHED.=

It has been remarked that inference is the process of deriving a
judgment from one or two antecedent judgments, and that mediate
inference is inference by means of a middle term. But to reason by
means of a middle term necessitates two judgments; hence mediate
inference might be defined as the process of deriving a judgment from
_two_ antecedent judgments. In this treatment mediate inference and
reasoning have been used interchangeably. This, then, becomes our
definition for reasoning: _Reasoning is the process of deriving a
judgment from two antecedent judgments._

The syllogism results when the process of reasoning is formally
clothed in words. Moreover, the conclusion of the syllogism may be
_more_ general than the premises or _less_ general. This suggests the
two important kinds of reasoning; namely, inductive and deductive.
Inductive reasoning is reasoning from less general premises to a more
general conclusion. Deductive reasoning is reasoning from more general
premises to a less general conclusion.


  ILLUSTRATION:

    _Inductive Syllogism._            _Deductive Syllogism._

    The robin, crow, sparrow,         All birds have wings,
      etc. have wings,                The robin, crow, sparrow,
    The robin, crow, sparrow,           etc. are birds,
      etc. are birds,               ∴ The robin, crow, sparrow,
  ∴ All birds have wings.               etc. have wings.

    Iron, silver, gold, etc. are      All metals are elements,
      elements,                       Iron, silver, gold, etc. are
    Iron, silver, gold, etc. are        metals,
      metals,                       ∴ Iron, silver, gold, etc. are
  ∴ All metals are elements.            elements.

    Boston, New York, Chicago,        All large cities have fine
      etc. have fine harbors,           harbors.
    Boston, New York, Chicago,        Boston, New York, Chicago, etc.
      etc. are large cities,            are large cities,
  ∴ All large cities have fine      ∴ Boston, New York, Chicago, etc.
      harbors,                          have fine harbors.

The student who is sufficiently familiar with the canons of the
deductive syllogism will at once detect the fallacy of _illicit
minor_ in the foregoing inductive syllogisms; i. e., “_birds_” when
used as the predicate of the minor premise of the first syllogism
is undistributed, but as the subject of the conclusion “_birds_” is
distributed. The same might be said concerning the terms “_metals_”
and “_large cities_.” A portion of this chapter will be devoted to
answering this criticism. At this point it may be stated that _the
inductive syllogism is not supposed to conform perfectly to the canons
of the deductive syllogism_.


=2. THE INDUCTIVE HAZARD.=

Referring to the first inductive syllogism of section _one_, it is
assumed that the robin, crow and sparrow are representative birds, and
that we are thus justified in concluding that if these _type_ birds
have wings, then _all_ birds must have wings. Of course this is more
or less of a conjecture or “a hazard”; since birds without wings may
exist in some undiscovered corner of the globe. However, inasmuch
as the generalization concerns a representative quality, we deem the
assumption fairly well founded. The logical right to take this “leap
into the unknown” will be discussed later. It will profit us at this
time to realize more fully how essential the “inductive hazard” is
to the progress of the world. When the Schoolmen of mediæval time
refused to venture, they failed to progress, and thus came the dark
days. Whenever man has ignored this God given instinct which leads to
discovery, the world has stood still. _This willingness to “take a leap
into the dark” with the hope of finding, in the shadow, truth which
would enhance man’s power and increase his serviceableness, has given
to the world about all that is worth while._ It was the spirit of the
_hazard_ which pushed Columbus to the discovery of a new world; which
gave Newton the secrets of the motions of the universe; which enabled
Edison to harness a multitude of lurking forces; and Morse and Bell to
reduce distance to its lowest terms. In ordinary affairs with ordinary
men those succeed _best_ who manifest _most_ a safe, steady, persistent
spirit of discovery. _Here, then, in the “inductive hazard” have we a
most important phase of school life which, in this day of making the
work easy, is being sadly neglected._ On the other hand, an unregulated
and insane spirit of venture may result in a great waste of energy, and
in the development of low ideals of recklessness and inaccuracy. The
“inductive hazard” must be cultivated; yet it must be regulated as well,
and, as the reader already realizes, logic needs to concern itself
mainly with this _regulative_ aspect.


=3. THE COMPLEXITY OF THE PROBLEM OF INDUCTION.=

The problem of induction is much more complex than that of deduction
because of these reasons: First. Deduction as a process of reasoning
was the only kind discussed by the logicians for _two thousand years_.
Aristotle is called the father of deductive logic and this Intellectual
Giant, the greatest of ancient time and possibly of all time, so
perfected the form of deductive reasoning that, up to the time of
Francis Bacon, no scholar possessed the temerity to gainsay its
supremacy in the field of logical reasoning. For twenty centuries
Aristotle’s Deductive Logic was the _Logicians’ Bible_. On the other
hand, inductive reasoning, though it was briefly discussed by Aristotle,
received little attention till the versatile Francis Bacon placed it
upon the stage of the thinking world. This makes deduction nearly two
thousand years older than induction. Time, by eliminating the personal
equation and exposing in various ways fallacious thinking, tends to
unify and universalize truth. Hence, logicians are agreed so far as the
fundamentals of deductive logic are concerned, but are still at odds
over the true conception and use of inductive logic.

A second reason for this confused status in the field of inductive
logic is the fact of its being more closely related to the events of
every day living. Induction is the natural method of childhood; the
popular method of the school room; and the most used method of common
life. In consequence its ramifications are so varied and multitudinous,
that it will take centuries of thinking to reduce the doctrine of
induction to that uniformity and definiteness which so distinguishes
deduction.


=4. THE VARIOUS CONCEPTIONS OF INDUCTION.=

The attending quotations will give the student a fair idea of the
leading conceptions concerning induction:

(1) “Induction is the process by which we conclude that what is true
of certain individuals of a class is true of the whole class, or that
what is true at certain times will be true under similar circumstances
at all times.” “Induction, as above defined, is a process of inference;
it proceeds from the known to the unknown.” “Any process in which what
seems the conclusion is no wider than the premises from which it is
drawn, does not fall within the meaning of the term.”――J. S. Mill, A
System of Logic, 1892, p. 175.

(2) “An induction is a generalization or an inference based upon
propositions that state observed facts.” “The truth inferred may be
general or particular, but it must be one which we cannot perceive in a
single act of observation.”――Ballentine’s Inductive Logic, 1896, p. 14.

(3) “Induction is the process of inference by which we get at general
truths from particular facts or cases.”――Ryland’s Logic, 1900, p. 148.

(4) “Induction may be defined as the legitimate inference of the
general from the particular, or, of the more general from the less
general.”――Fowler, 1905, p. 10, Vol. 2.

(5) “The term induction has been used by logicians to denote this
leap of the mind from the limitations of its positive knowledge
to belief in universal laws.” “In pedagogy, however, the term
is applied to the whole process of arriving at general truths or
principles.”――Salisbury’s Theory of Teaching, p. 156.


=5. INDUCTION AND DEDUCTION CONTIGUOUS PROCESSES.=

If there is one thing above another which modern logic is emphasizing
it is the _unity of the mind_ and the _contiguity of thinking_.
Induction and deduction are _dove-tailed processes_ which characterize
all thinking worthy of the name. Where induction ceases, deduction
commences, and vice versa. It becomes the function of inductive
thinking to establish a connection between what has been experienced
and what has not been experienced. Therefore, the conclusion of an
induction must always contain more than is implied in the premises. The
premises denote facts which have been observed; whereas the conclusion
denotes the observed facts of the premises plus analogous facts which
have _not been observed_. Inductive thought ventures into the unknown,
and attempts to establish a bond of connection between it and something
already known. Induction seeks new knowledge, and does so by taking
that “leap into the dark” already referred to as the “inductive hazard.”

As soon as the mind reaches a universal truth, it sets to work to
clarify this truth. Such is accomplished by reference to other facts
which the universal is supposed to include; and this application of the
general to the particular is _deduction_. _Induction discovers_ the new
knowledge while _deduction clarifies_ it.


=6. INDUCTION AN ASSUMPTION.=

In this treatment induction as a general process has been subdivided
into induction as a mode of inference and induction as a method.
_Induction as a mode of inference is the process of reasoning from less
general premises to a more general conclusion; whereas induction as a
method is a procedure from the observation of individual facts to a
realization of a universal truth._ In either case the conclusion of an
inductive process always implies more than is contained in the premises.
This gives to the conclusion an uncertainty. No induction is absolutely
free from doubt except the so-called perfect induction, which form will
receive attention in a later section.


  =7. UNIVERSAL CAUSATION.=

  All inductive assumptions are made possible because of two
  laws――_universal causation_ and _uniformity of nature_.

  The law of universal causation may be stated in this wise:
  _Nothing can occur without a cause and every cause has its
  effect._ “It is a universal truth, that every fact which has
  a beginning has a cause.”――Mill.


  SIMPLE ILLUSTRATIONS OF UNIVERSAL CAUSATION.

  The sun rises in the east. The boy throws a stone through the
  window. A democratic wave sweeps the country. Prices of food
  stuff are high. The bullet, shot out into space, finally falls
  to the earth. Each one of these occurrences has a cause.[12]

  That universal causation is a fundamental condition of all
  induction may be further illustrated. The astronomer notes that
  the stars in the vicinity of Vega seem to be moving outward
  _from_ a common center; whereas in the opposite part of the sky
  the stars seem to be moving inward _toward_ a common center.
  Having observed this phenomenon, the astronomer at once looks
  for a cause. Finally he decides that the phenomenon is due to
  the fact that the sun, with his attending family, is moving
  towards Vega. Arranged, the argument may take this form:

  The stars in the vicinity of Vega seem to be moving outward
  from a common center, whereas in the opposite part of the sky
  the stars seem to be moving inward, When descending a mountain
  the trees at the foot seem to move outward from, and those at
  the top inward toward, a common center, When riding on the train
  the ties in front seem to move outward while those in the rear
  seem to move inward. From this we conclude that the sun with
  the Earth and other planets is moving toward a spot in the sky
  near Vega. Were it not for the assumption that the phenomenon
  relative to the stars _had a cause_ there could have been no
  induction. Moreover, any investigation concerning “democratic
  waves,” “prices of food stuffs,” etc., must assume as a starting
  point that these phenomena have causes.

  It would appear that the mind is not satisfied with a mere
  passive observation of the occurrences of the world but is
  inclined to reach out for the “whys and wherefores.” Due partly
  to this reason, “universal causation” is often referred to as
  an _a priori_ law; meaning that it is a law which cannot be
  _proved_, but must be _assumed_ in all thinking.


  =8. THE LAW OF THE UNIFORMITY OF NATURE.=

  Law stated: _The same antecedents are invariably followed by the
  same consequents._ “That the course of nature is uniform is the
  fundamental principle of induction.”――Mill. “It is not enough to
  feel assured that nothing can happen without a cause (causation);
  I must also feel assured that the same cause will invariably be
  followed by the same effect.”――Fowler.

  Referring to the observed phenomenon of the outward movement
  of the stars about Vega, the astronomer might advance as an
  hypothesis the fact of the solar system’s movement toward
  Vega. Having done this he could then experiment with a view of
  verifying this hypothesis. In this experiment he would attempt
  to introduce the same cause surrounded by similar circumstances,
  and then watch for the same effect. To make it concrete: suppose
  the astronomer paints the side of a barn dark blue and bedecks
  this with stars of white. Then taking a position as far removed
  from the blue surface as his eyesight will permit, he runs
  toward the barn watching the apparent movement of the artificial
  stars. A similar experiment could be performed by substituting
  for the _starred_ barn, the stumps on a side hill. In both
  experiments he assumes that like conditions will be followed by
  constant results. That is, in these particular cases, advancing
  toward a group of objects is always followed by an apparent
  separation of said objects.

  This law of uniformity of nature not only underlies inductive
  thinking but it really conditions all thinking. It implies
  that the universe is a rational system functioning in a uniform
  manner. Moreover, it suggests that the interpretations of the
  mind are likewise uniform and whenever the mind proves a fact
  to be a universal truth, this truth will _always_ remain a
  truth unless the conditions change. In fact were it not for the
  uniformity of nature, all activity whatsoever would be rendered
  nugatory. Because of this law we have a right to assume that
  grinding a knife under right conditions will always tend
  to sharpen it; that surrounding a live seed with a proper
  environment will result in growth; that water at the same
  altitude will boil at a constant temperature, etc., etc.

  The student will discern the close connection between these two
  laws and the laws of thought. There is really no distinctive
  mark between the law of causation and the law of sufficient
  reason, while “uniformity of nature” includes identity as
  one of its distinctive features. The laws differ, however,
  in their application, “causation” and “uniformity of nature”
  conditioning inductive thinking, while the others are concerned
  with deductive thinking.

  Because “uniformity of nature” expresses facts of experience, it
  is regarded as an _empirical_ law, as contrasted with the law of
  causation, which is supposed to be based upon an innate mental
  conception or is an _a priori_ law.


=9. INDUCTIVE ASSUMPTION JUSTIFIED.=

The function of induction seems to be to universalize particulars. The
mind of man has ever been engaged in establishing connections among the
concrete experiences of daily life. This ability of his to generalize
his individual experiences has been one of the chief agencies in
elevating him to the position of “King of the animal world.” In this
disposition to generalize man has taken it for granted that nature is
honest; that what she tells him under given conditions, she will tell
him again under identical conditions. To put it in logical terms man
can depend upon the _invariability_ of nature’s activities, or _upon
the uniformity of nature_. Here, then, is one of the most fundamental
laws not only of induction but of all activity. But this law implies a
second quite as fundamental. If every cause is invariably followed by
the same effect under like conditions, then it is thereby implied that
every cause has an affect and every event is due to some cause. This,
too, is invariable. In consequence of these facts man is justified
in thinking that nature is not only honest and therefore “she
gives me _confidence_, but her every activity _means something_ and
therefore she arouses my _curiosity_.” “Uniformity of nature” engenders
confidence, “universal causation” inspires the spirit of discovery
and with these two weapons man is willing to venture into the jungle
of the unknown. _Why is man eager to undertake the “inductive hazard?”
Because, through the laws of universal causation and uniformity
of nature, his curiosity is aroused, and he is given confidence in
nature’s activities._


=10. THREE FORMS OF INDUCTIVE RESEARCH.=

Induction is a matter of universalizing less universal experiences.
In this the process may assume any one of three forms, namely:
(1) Induction by simple enumeration; (_inductio per enumerationem_);
(2) Induction by analogy; (3) Induction by analysis.


  THREE FORMS ILLUSTRATED:

(1) _Simple enumeration._

Having observed a few instances the generalization is, “All birds have
wings.” The certitude of this may now be strengthened by observing more
birds and finding without exception that each has wings.

(2) _Analogy._

By noting on Mars geometric markings which resemble canals, the
generalization is vouchsafed that Mars is inhabited by human beings.
Other similarities in atmospheric conditions, existence of land and
water, etc., tend to make this generalization more plausible.

(3) _Analysis._

By analyzing water taken from a certain spring, it is found to contain
hydrogen and oxygen in the proportion of 1 to 8; in consequence a
generalization to this effect is posited. Analyses of specimens from
other sources yield similar results and thus the generalization is
given greater certitude.

As a usual thing the particular form which the induction assumes
depends on the nature of the topic under investigation and also on
the mental make-up of the investigator. The general statement that
all birds have wings could hardly be derived by means of analogy or
analysis, but is a matter of a casual observation of many instances.
Moreover, that mind given to accurate observation, but not inclined to
note resemblances or to carry on experiments, would naturally follow
the first inductive type. On the other hand, simple enumeration would
be impossible in questions like the habitability of Mars, and would
yield no results in cases requiring definite scientific experimentation
like electrolysis.

It is worthy of note that some topics lend themselves to all three
modes of procedure. To wit: (1) Enumeration. Without being taught the
rule the child is given a list of examples involving the dividing of
a decimal by a decimal and is asked to solve them. By comparing his
answers with those in the book, he somewhat accidentally discovers
what seems to be the correct rule for pointing off in the quotient.
By following this rule and each time comparing answers he establishes
the truth. (2) Analogy. If .24 ÷ .6 is the first example, the child
may resort to the well known process of dividing a common fraction by a
                 (  24    60   24    4  )
common fraction, ( ――― ÷ ――― = ―― = ――, ) then, because of their close
                 ( 100   100   60   10  )
resemblance, he may reason that decimal fractions should yield the same
result. (3) Analysis. Here the child reasons that since division of
decimals is the inverse of multiplication of decimals, the rule for
pointing off might be the inverse of the multiplication rule. By trying
this out and proving his answer in each example, he becomes convinced
of the correctness of his reasoning.


=11. INDUCTION BY SIMPLE ENUMERATION.=

As its name implies this type of inductive research consists
_in observing many instances which may exemplify the particular
uniformity under consideration_. The process is quantitative rather
than qualitative, the certitude of the generalization depending on
the _mass of facts collected_ rather than on any striking resemblance
or any detailed analysis. The aim is to observe, accurately if not
scientifically, instance after instance until all doubt is removed.
The outcome of such observation may be three fold. (1) The enumeration
may be complete. This gives the so-called “perfect induction” which
will receive attention later. (2) The enumeration may be incomplete
and _without exceptions_; generalizing in this way from uncontradicted
experience gives what are termed “_empirical_” truths. (3) The
enumeration may be incomplete _with_ exceptions. It is obvious that
this type of induction could give no valid generalization; but the
result may be put in the form of a ratio between the uniformities and
the exceptions. Such a procedure is a mere “_calculation of chances_”
and the result simply an expressed probability.


  THE THREE KINDS OF SIMPLE ENUMERATION ILLUSTRATED.

The subject to receive investigation is a _school examination_.

  (1) Complete enumeration. Every paper is read and marked; this leads
      to the generalization, “All the class have passed.”

  (2) Incomplete enumeration with no exceptions. Representative
      papers are read and marked in which no failures are found.
      Generalization, “Probably all of the class have passed.”

  (3) Incomplete enumeration with exceptions. Representative papers are
      read and marked in which there are 20 failures out of the hundred
      papers examined. Generalization, “Probably about 80% of the class
      have passed.”

Briefly, simple enumeration may take the form of (1) _a perfect
induction_, (2) _a probable induction_, (3) _a mere calculation of
chances_. The first necessitates completed experience, the second
uncontradicted experience and the third contradicted experience.


=12. INDUCTION BY ANALOGY.=

_Induction by analogy assumes that if two (or more) things resemble
each other in certain respects, they belong to the same type, and,
therefore, any fact known of the one may be affirmed of the other._


  THE TYPE.

As the definition implies, analogy involves an extensive use of types;
let us, therefore, become better acquainted with them as instruments
in analogical inductions. A type is _one_ of a group which embodies the
_essential characteristics_ of that group. How easy and natural it is
to dismiss a complex topic with the citing of an example which may be
regarded as a type; how common is the use of examples in the school
room! On second thought it becomes apparent that analogical induction
by example or type is the most common of all forms of induction either
as a method or a mode of inference. _Analogy by example (or type)
assumes that if two or more things are of the same type, they resemble
each other in every essential property._

Illustrations of analogical inductions by example or type.

(1) _Mathematics._

      Example:  a  + b
                a  + b
                ―――――――――――――
                a² + ab
                   + ab + b²
                ―――――――――――――
                a² + 2ab + b²

Inductive Inference: The square of the sum of two quantities is equal
to the square of the first, plus twice the first by the second, plus
the square of the second.

(2) _Nature._

This corn sent me as a sample produced heavy, full ears, and many of
them; hence (inductive inference), if I plant corn _like this sample_
under like conditions, I will receive in return heavy, full ears, and
many of them.

(3) _Geography._

Cities like New York, located on the coast, possess a larger foreign
element than the inland cities like Philadelphia.

(4) _Grammar._

A noun is the name of anything, as the examples, “George Washington”
and “house” would indicate.

In deriving a generalization from one or two examples the prime
essential is to select types which are _truly representative_.
Often the example used is a special type and in consequence does not
exemplify all of the essential characteristics of the group. To teach
the nature of a parallelogram by using a rectangle only, is an easy way
to commit this error; or one may affirm that the class can easily cover
the work, when the judgment is based entirely on knowledge concerning
the _brightest_ one of the grade.

Type work when judicially used is a positive time saver and a very
present help in times of perplexity. Let the skillful teacher use types
and examples extensively yet cautiously.


  THE MARK OF SIMILARITY.

As opposed to analogy by type there is a second form; namely, _analogy
by one or more similar marks or qualities_. This form is best described
by the definition: _When two things resemble each other in a few marks
or qualities they resemble each other in other marks or qualities._

Illustrations of analogy by marks.

  (1) Noting that two students have the same surname, I infer that they
      are brothers.

  (2) A man with a book under his arm rings the door bell and asks to
      see “the lady of the house.” At once the conclusion is drawn that
      the caller is a book agent.

  (3) Two automobiles, resembling each other in shape of body, force
      one to the conclusion that the machines are of the same make.


  THE ERRORS OF ANALOGY BY MARKS OF SIMILARITY.

It follows that analogy by example gives generalizations of much
greater certitude than analogy by one or two marks of resemblance.
Here is a field bespattered from boundary to boundary with erroneous
thinking. The principle of resemblance being an innate tendency, this
form of error is most common with the immature. The child reasons
by analogy when he invests the poodle with the despised cognomen of
“kitty”; or honors every man who wears glasses with “papa.” In the
childhood of the race natural events were interpreted by means of
analogy. The wind blowing through the trees made sounds much like the
human voice; hence these noises were attributed to spirits. Primeval
man was led to believe by analogy that everything which moved was alive.
We may, therefore, think of our revered forbear as engaged in the
undignified task of running after his shadow, or chasing a leaf around
a stump.


  THE VALUE OF ANALOGY.

Analogy being rich in its suggestions is the favored process of the
scientist and inventor. Newton reasoned by analogy when he tentatively
affirmed of the moon what he positively knew of the apple. Franklin’s
reasoning was analogical when he discovered the identity of the
electric spark and lightning. Because this form of induction so often
leads to error and at best involves a degree of probability far below
induction by analysis, some logicians are inclined to ignore its
generalizations altogether. Others deem this a mistake because of these
reasons: First. Analogy is serviceable to a high degree in suggesting
hypotheses which may be advanced either for the purpose of explanation
or verification. It has already been indicated that analogy is the
common instrument used by the inventor and discoverer. Second. The
principle of analogy, in reality, lies at the basis of classification;
because in this, things are grouped according to their resemblances.
Third. Analogical induction affords valuable training in originality
and initiative. A mind which easily and naturally discerns analogies
is “fertile in new ideas.”


  REQUIREMENTS OF A TRUE ANALOGY.

It has been remarked that the certitude of an induction by simple
enumeration depends upon the number of uncontradicted instances. In
analogy the case is different as the process emphasizes the _weight_
of the points of resemblance rather than the _number_. In substance
the requirements of a logical analogy are three.

First. _The points of resemblance must be representative and not
exceptional._ For example: The argument that Mars is inhabited because
it has two moons is of little worth, since we have no proof that
moonshine is essential to life; this point of resemblance is not
representative. On the other hand, if the basis of argument is the
fact that Mars has an atmosphere, the conclusion carries some weight;
as air seems to be essential to life.

Second. _The points of resemblance must outweigh the points of
difference._ That is, the ratio of probability must always be in
favor of the resembling instances. Since it is not a matter of numbers
but of weight, a numerical proportion like this would be misleading:
Resemblances: Differences = 10:6. It is obvious that the six
differences might more than outweigh the ten resemblances. The safer
way, if it were possible, would be to attach a _value_ to each point of
resemblance or difference, and then express the proportion in terms of
the sums of these values.

Third. _There must be no difference which is absolutely incompatible
with the affirmation which we wish to prove._ For example, the fact
that the moon has no atmosphere renders nugatory any attempt to prove
the habitability of the moon.


=13. INDUCTION BY ANALYSIS.=

This, the third form of inductive research, is by far the most
important. Simple enumeration, because it depends upon the number of
observed instances, consumes much time; while we have already noted how
easy it is for analogy to lead to error. At the best, the conclusion
of these methods must be subjected to analytic investigation, if we are
seeking universal validity. Induction by analysis is superior to the
other forms because it secures a _higher degree of probability_ and is
a _positive time saver_.

Defined. We have learned that analysis is the process of separating
a whole into its related parts. We thus define induction by analysis
as _the process of separating a whole into its parts with a view of
deriving a generalization relative to the nature and causal connection
of these parts_.


  ILLUSTRATIONS:

(1) Concerning the generalization that “all birds have wings,” it
becomes possible to observe in detail the nature of the wings and
advance the hypothesis that these wings are designed for aërial
navigation. This hypothesis may then be strengthened by observing that
the entire structure of the bird is adapted to flying.

(2) If it were possible to analyze the atmosphere, water, and soil of
Mars, and should such analysis reveal a composition similar to that of
the earth, it would illustrate well not only the _method_ of analysis
but also its superiority over the other methods of investigation.

(3) The physician, in diagnosing a “case,” observes that the symptoms
resemble those of typhoid; but to be positive of the truth of his
diagnosis, he takes a blood test. Noting the resemblances is induction
by analogy; but the blood test involves induction by analysis.

Induction by analysis concerns hypothesis, observation, and experiment,
including Mill’s experimental methods. These topics will receive due
attention in the chapters which follow. It will be sufficient to close
this discussion with a brief treatment of perfect induction, and
traduction.


=14. PERFECT INDUCTION.=

As has been indicated under simple enumeration, _a perfect induction
is one in which the premises enumerate all the instances denoted by the
conclusion_.


  ILLUSTRATIONS:

  (1) A, B, C, D, and E are all Reactionaries,
      (All) The members of the committee are A, B, C, D, and E,
    ∴ (All) The members of the committee are Reactionaries.

  (2) John, James, Albert, and Peter all have perfect eyesight,
      John, James, Albert, and Peter are all the boys of my family,
    ∴ All the boys of my family have perfect eyesight.

  (3) The first, second, and third groups are up to grade,
      The first, second, and third groups include all of the children
        in my room,
      Hence all the children in my room are up to grade.

Because the conclusion of a perfect induction gives nothing
new――nothing but what is found in the premises, some claim that
the process is practically valueless. From the viewpoint of the
discoverer this position is well taken; yet to universalize particular
observations puts the knowledge in compact, usable form, and saves
one the trouble of returning each time to the consideration of each
particular. Thus as a process which leads to verified universals,
perfect induction is a _time saver_. In the second place it was the
method used by Socrates when he desired to lead up to a definition or
some other general truth. The Sophists were given to a careless use
of the “inductive hazard”; they were prone to generalize from one
or two particulars, or what is worse, to establish a generalization
and then attempt to fit the particular instances to it. This led to a
superficiality which the Great Pagan Educator abhorred. The fact that
perfect induction was the method used by Socrates to counteract the
teachings of the Sophists, is sufficient vindication for its use in
discouraging the indefensible assumptions of to-day, and in inspiring
warrantable generalizations based on accurate observation.

In the school room with classes addicted to careless, inaccurate work,
to accept nothing but a perfectly induced generalization, when this
is feasible, is a most valuable lesson. For example, the teacher may
not accept the generalization that all of the “first class” cities
of the U. S. are located on navigable waterways, until the pupils
have investigated the waterway conditions of every city belonging
to the class. On the other hand, there may be individual cases of
“cocksureness” which need attention. The teacher can do little for the
“know-it-all youngster” until he pricks the bubble of conceit. This may
be accomplished by allowing the youth to draw a generalization, which
seems to meet all the requirements of truth arrived at by means of
an _imperfect induction_; then without warning let the teacher give
an instance which will show the generalization to be _false_. This
involves what Socrates termed the “torpedo’s shock.” To illustrate:
Consider the “prime number” formula given by Jevons. In deriving this,
direct the class to add 2 to its square, and to this sum add 41. Give
similar directions relative to numbers 3, 4, 7 and 10. Indicating the
work as directed, would give the following:

  (1)  2 +  2² + 41 =  47
  (2)  3 +  3² + 41 =  53
  (3)  4 +  4² + 41 =  61
  (4)  7 +  7² + 41 =  97
  (5) 10 + 10² + 41 = 151

A question or two will make apparent the fact that all the results
are prime numbers, and then the generalization may be drawn; namely,
X + X² + 41 = prime number. Now without warning, but under the
assumption that you desire to test deductively the general formula,
let X = 40. This gives (40 + 40² + 41) 1681, which is the square of
41 and is, therefore, not a prime number.


=15. TRADUCTION.=

It may have been noted by the student that “perfect induction” is
not induction at all according to the definition; viz.: Inductive
reasoning is reasoning from less general premises to a more general
conclusion. Referring to the first illustration of the previous section
it is apparent that the conclusion is no broader than the premises.
Ostensibly, the conclusion is a mere summary, or a generalization
of the facts mentioned in the premises. Moreover perfect induction
does not readily conform to the definition of deductive reasoning,
as in this the movement must be from the more general to the less.
We are thus forced to the conclusion that perfect induction is a form
of a third type of reasoning which is known under the cognomen of
_traduction_. This is from the Latin _trans_, and _ducere_ meaning
to lead across. Definition: _Traductive reasoning is reasoning to a
conclusion which is neither less general nor more general than the
premises._

Aside from the case of perfect induction there are other types
which well illustrate traduction. These are: First. _Reasoning from
particular_ (_or individuals_) to _particular_ (or individuals).


  ILLUSTRATION:

    Highland Street is the longest street in Jamaica,
    Highland Street is not so long as Broadway of New York City,
  ∴ The longest street of Jamaica is not so long as Broadway of
      New York City.

Second. _Reasoning from general to general._


  ILLUSTRATION:

    All growing things die,
    All living things are growing things,
  ∴ All living things die.

It may be observed that all of the propositions in traduction are
co-extensive “A’s” or “E’s”; hence all the terms are distributed. This
eliminates any possibility of committing fallacies of distribution.
Further, the propositions may be interchanged at will, without
invalidating the particular conclusion selected. To illustrate we may
change the last argument to this:

    All growing things are living things,
    All things that die are growing things,
  ∴ All things that die are living things.

From the viewpoint of authenticity traduction is the most, and
induction the least dependable; whereas the certitude of deductive
reasoning lies somewhere between the two. On the other hand, when
looked at from the ground of serviceableness the order is reversed,
induction being the most useful form of inference and traduction the
least.


=16. OUTLINE.=

  INDUCTIVE REASONING.

  (1) Inductive and Deductive Reasoning Distinguished.

  (2) The “Inductive Hazard.”
        Essential in world’s progress.
        Cultivated and regulated in school.

  (3) Complexity of the Problem of Induction.

  (4) Various Conceptions of Induction.
        Quotations from prominent authorities.

  (5) Induction and Deduction Contiguous Processes.

  (6) Induction an Assumption.
        A mode of inference; A method.

  (7) Universal Causation.
        Law stated and illustrated.
        Conditions all induction.

  (8) Uniformity of Nature.
        Defined and illustrated.
        Conditions all induction.
        Empirical.

  (9) Inductive Assumptions Justified.

 (10) Three Forms of Inductive Research.
        (1) Enumeration (2) Analogy (3) Analysis.
        Illustrated.
        Conditions determine form followed.

 (11) Induction by Simple Enumeration.
        Defined and illustrated.
        Outcome threefold――these illustrated.

 (12) Induction by Analogy.
        Two conceptions.
        Analogy by type or example. Illustrations representative.
        Error of analogy.
        Suggestiveness of analogy.
        Value of analogy.
        Requirements of a true analogy. Three.

 (13) Induction by Analysis.
        Importance.
        Defined and illustrated.

 (14) Perfect Induction.
        Defined and illustrated.
        Its use.
        Method of Socrates.

 (15) Traduction.
        Defined and illustrated.
        Three methods compared.


=17. SUMMARY.=

(1) Reasoning is the process of deriving a judgment from two antecedent
judgments. The syllogism is a common form of expressing the process of
reasoning.

Inductive reasoning is reasoning from less general premises to a more
general conclusion.

Deductive reasoning is reasoning from more general premises to a less
general conclusion.

The inductive syllogism is not supposed to conform to the canons of the
deductive syllogism.

(2) Positing in the conclusion more than is indicated in the premises
involves what is known as the “_inductive hazard_.”

The inductive hazard which is another expression for the spirit of
discovery, should be fostered in the school room since it has been one
of the great forces in human progress; but this venturesome spirit must
be regulated by rules, principles, and systematic procedure, or low
ideals of recklessness and inaccuracy will result.

(3) The problem of induction is more complex than that of deduction;
because the former is a comparatively new subject, and also is more
closely related to the activities of life.

(4) The opinion relative to the exact nature of induction, though
varied, may be summed up in the thought of its being the process which
leads to general truths, derived from the observation of individual
facts.

(5) Induction and deduction are contiguous processes which go to
make up the more general process of thinking. Where induction ceases,
deduction naturally commences; induction discovers new knowledge,
deduction clarifies it.

(6) Induction as a general process may be treated as a mode of
inference or as a method. In either case the conclusion comprehends
more than is contained in the premises.

Since no imperfect induction is absolutely free from doubt, on what
ground are we justified in making any inductive assumptions? The answer
follows:

(7 and 8) “Nothing can occur without a cause and every cause has
its effect,” is the law of universal causation; while the law of the
uniformity of nature is “the same antecedents are universally followed
by the same consequents.” These two laws justify inductive assumptions,
and, in a sense, condition all thinking.

(9) Uniformity of nature gives man confidence, while universal
causation arouses his curiosity. With these two weapons he is willing
to “march into the unknown.”

(10) As the process of universalizing individual experiences, induction
assumes the three forms of simple enumeration, analogy and analysis.
The form adopted is not always elective but is controlled largely by
the exigency of the case. Some topics lend themselves to all three
modes.

(11) Induction by simple enumeration consists in observing many
instances which exemplify the uniformity under consideration.
_Complete_ enumeration gives the so called perfect inductive inference;
_incomplete but uncontradicted_ enumeration leads to empirical truths;
while _incomplete_ and _contradictive_ enumeration involves a mere
calculation of chances.

(12) Induction by analogy assumes that if two (or more) things resemble
each other in certain respects, they belong to the same type, and,
therefore, any fact known of the one, may be affirmed of the other.

A most common form of analogy is reasoning by type or example. In
this it is assumed that if two or more things are of the same type,
they resemble each in every essential property. The type must be truly
representative. A second form of analogy is reasoning by marks of
resemblance. This second form often leads to egregious error.

Analogy is especially valuable in suggesting hypotheses and in giving
training in originality and initiative.

A true analogy demands that the points of resemblance be representative;
that they outweigh the points of difference, and that no disagreement
be incompatible.

(13) Induction by analysis is the process of dividing a whole into its
parts with a view of deriving a generalization relative to the nature
and causal connection of these parts.

Induction by analysis makes use of the hypothesis, of observation and
experiment, including Mill’s five methods.

(14) A perfect induction is one in which the premises enumerate all of
the instances denoted by the conclusion. It is serviceable in inspiring
care and accuracy in the establishment of generalizations.

(15) Traduction is the process of reasoning to a conclusion which is
neither less general nor more general than the premises.

Traduction includes reasoning from particular to particular or from
general to general. Perfect induction is in reality a form of
traduction.

Induction, though the most useful form of inference, is the most
untrustworthy; whereas traduction is just the reverse of this.


=18. REVIEW QUESTIONS.=

  (1) Define and illustrate reasoning.

  (2) Distinguish by definition and illustration between inductive and
      deductive reasoning.

  (3) Explain the “inductive hazard” and show its use to man.

  (4) “For twenty centuries Aristotle’s Deductive Logic was the
      logician’s bible.” Explain this.

  (5) Show that induction and deduction are contiguous processes.

  (6) Distinguish between induction as a mode of inference and
      induction as a method.

  (7) State and explain the law of universal causation. Illustrate
      fully.

  (8) Make evident that a cause may involve many antecedents.

  (9) State and explain by illustration the law of uniformity of
      nature.

 (10) Verify by illustration the notion that the “fact of causation”
      conditions all induction.

 (11) Which of the two laws is empirical, “causation” or “uniformity”?
      Why?

 (12) Show that induction is a form of thinking.

 (13) Why should the law of uniformity of nature convince man that
      nature is honest? Illustrate.

 (14) Show that the law of universal causation stirs the spirit of
      discovery.

 (15) Name and illustrate the three forms of induction.

 (16) Why is it that the tendencies of the investigator often determine
      the inductive form which he adopts?

 (17) Explain by illustration the three-fold outcome of induction by
      simple enumeration.

 (18) Selecting some class room experience, illustrate analogy by
      example or type.

 (19) Define and exemplify types as used in logic.

 (20) Remark upon the errors incident to analogy.

 (21) Summarize the advantages which induction by analogy offers.

 (22) State and exemplify the requirements of true analogies.

 (23) Indicate the superiority of induction by analysis over the other
      two forms. Illustrate.

 (24) Define and illustrate perfect induction.

 (25) Under what circumstances is perfect induction justified?

 (26) Define and illustrate traduction.

 (27) Indicate the various forms of traduction.


=19. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Show the connection between illicit minor and the “inductive
      hazard.”

  (2) Show by illustration that time tends to universalize truth.

  (3) “Induction and not deduction is the natural method of the child
      mind.” Prove the correctness of this statement.

  (4) “Induction is the process of inference by which we get at general
      truths from particular facts or cases.” Prove that this is not
      strictly correct according to definition.

  (5) As related to establishing general truths, what are the special
      functions of induction and deduction?

  (6) Show that an inductive inference must of necessity be more or
      less uncertain.

  (7) Is there any distinction between the laws of universal causation
      and sufficient reason? Hyslop’s Elements of Logic, page 329.

  (8) Show that universal causation and uniformity of nature are
      complementary laws. Hyslop, p. 330.

  (9) Relate the “fact of causation” to the laws of thought.

 (10) Distinguish between empirical and “a priori” laws.

 (11) When Harvey discovered the circulation of the blood, what form
      of induction did he use?

 (12) What form of reasoning did Columbus follow in proving that the
      earth is spherical?

 (13) “It is said that the greatness of Darwin was due largely to his
      habit of never ignoring an exception.” Justify by illustration
      the truth of this assertion.

 (14) In analogical reasoning by example, under what conditions would
      _one_ illustration be as convincing as _many_?

 (15) “Considering the similarities and differences, the weight of the
      argument favors Mars’ habitability.” Suppose the proportion of
      probability were something like this――Resemblances: Differences
      = 8:7; wherein might the conclusion be erroneous?

 (16) Mention a mark or characteristic which would make the
      habitability of Mars incompatible?

 (17) Select a topic for investigation which is peculiarly adapted to
      enumeration; to analogy; to analysis.

 (18) “The uniformities we expect to find in the world take two
      main aspects, one of which is indicated by the term _thing_ and
      the other by the term _circumstance_.” Aikin’s Principles of
      Logic, 1905; p. 233. In the light of the two fundamental laws
      of universal causation and uniformity of nature explain and
      illustrate the quotation.

 (19) Explain the principle of teleology as related to analogy. Hibben,
      1908; p. 317.



                              CHAPTER 18.

      THE FIVE SPECIAL METHODS OF OBSERVATION AND EXPERIMENT.[13]


=1. THE AIM OF THE FIVE METHODS.=

The primary forms of induction have been divided into _simple
enumeration_, _analogy_ and _analysis_ . Conditioning these forms are
the two laws, _uniformity of nature_ and _universal causation_. Since
these laws are always concerned with causes, we may refer to them as
together expressing the fundamental “_fact of causation_.” Wherever
there is a causal connection, no matter how slight, these laws obtain.

Though “the fact of causation” probably conditions all forms of
induction, it is most conspicuous in the third form; namely, Analysis.
Here the main aim is to establish a causal connection of some kind; an
aim which may be accomplished through the medium of observation and
experiment. Incident to this notion, John Stewart Mill formulated five
experimental methods of induction. These are known according to the
following distinctive titles:

  1. The Method of Agreement.

  2. The Method of Difference.

  3. The Joint Method of Agreement and Difference.

  4. The Method of Concomitant Variations.

  5. The Method of Residues.


=2. METHOD OF AGREEMENT.=

(1) Principle stated. As stated by Mill the principle of the Method of
Agreement is this: “If two or more instances of the phenomenon under
investigation have only one circumstance in common, the circumstances
in which alone all the instances agree is the cause (or effect) of the
given phenomenon.”

This notion is given in clearer terms by Jevons and Creighton. Viz.:
“_The sole invariable antecedent of a phenomenon is probably its
cause_”; and “_The sole invariable consequent of a phenomenon is
probably its effect_.”

It is known that an antecedent is anything which _precedes_; while a
consequent is anything which _follows_. To be regarded as a cause, an
antecedent must be _invariable_, and to be regarded as an effect, a
consequent must likewise be _invariable_. Antecedents and consequents
which are in no way constant could hardly have any causal connection.

(2) Method symbolized. Let P₁, P₂, P₃, P₄, etc., represent the
phenomenon as it may appear the first, second, third, fourth, etc.,
times, and let A, B, C, etc., stand for the various antecedents, or the
various consequents as the case may demand. These two forms may now be
used to illustrate the two statements which summarize Agreement:

First statement.

   _Antecedents._    _Consequents._
          A B C D ―― P₁
          A D E F ―― P₂
          A L M N ―― P₃
          A O P Q ―― P₄

Second statement.

          P₁ ―― A B C D
          P₂ ―― A D E F
          P₃ ―― A L M N
          P₄ ―― A O P Q

In the first case, the sole invariable antecedent is A, and, therefore,
we infer that A is probably the cause of P. In the second case, the
invariable consequent being A, is probably the effect of P.

(3) Concrete examples illustrating first statement.

_The Problem_: Cause of John’s tardiness.

On investigation the various _antecedents_ are these: (1) John has
his breakfast at _seven_; (2) after breakfast he carries his father’s
_dinner_ to him and (3) feeds the hens; and then (4) goes to school by
the _path through the woods and around the mill pond_.

_Phenomenon as a consequent._ John is tardy. Determining to do away
with the tardiness, the teacher brings about a variation in the
antecedents, varying _one at a time_ taken in the order indicated above.

     To wit: (1) Varying the first antecedent.
                 John breakfasts at 6:30;
                 Other antecedents the same;
  (_Phenomenon_) But John is tardy.

             (2) Varying the second antecedent.
                 The younger brother carries the dinner;
                 Other antecedents the same;
  (_Phenomenon_) John is tardy.

             (3) Varying the third antecedent.
                 Another brother cares for the hens;
                 Other antecedents the same;

  (_Phenomenon_) John is still tardy.

The teacher is now quite certain that the tardiness is due to the route
through the woods and around the pond.

Using, as symbols, the initial letters of the italicized “key-words”
of the antecedents as stated above, the case of tardiness may be
symbolized as follows:

  _Key words_   _Symbols_

    seven           s
    dinner          d
    hens            h
    woods           w
    tardy           t₁, t₂, t₃

  _Antecedents_ _Phenomenon_
     s d h w        t₁
     e d h w        t₂
     s b h w        t₃
     s d a w        t₄

“w” standing for route through the woods, is seen to be the invariable
antecedent.

(4) Concrete example illustrating the second statement.

_The Problem_: To determine the effect of direct primaries.

            _First trial._

  _Antecedent_            _Consequents_

                { 1. Greater expense to candidate,
    Direct      { 2. Greater interest shown,
      primary   { 3. Better men nominated,
                { 4. “Bumper” crops.

            _Second trial._

                { 1. Greater expense to candidate,
    Direct      { 2. Greater interest shown,
      primary   { 3. Better men nominated,
                { 4. Crops below average.

            _Third trial._

                { 1. No greater expense,
    Direct      { 2. Greater interest shown,
      primary   { 3. Better men nominated,
                { 4. Crops average.

            _Fourth trial._

                { 1. No greater expense,
    Direct      { 2. No greater interest,
      primary   { 3. Better men nominated,
                { 4. Crops average.

It is seen that the invariable consequent is, “Better men nominated.”
We may, therefore, conclude that this is a probable effect of “Direct
primaries.”

(5) Distinguishing features of method of agreement. The essential
characteristics of the method of agreement are three:

  First, _The phenomenon always occurs_.
  Second, _There is at least one invariable antecedent_.
  Third, _The other antecedents vary_.

Giving attention to the attending symbolized illustrations it may be
noted that “P,” the phenomenon, always happens; while in the case of
the first symbolization, “D” is the invariable antecedent and “A, B, C,
E, G, L, M, F, I” are the variable antecedents. “K” is the invariable
antecedent of the second and “H, I, L, T, M, W, X, Y, Z, S” are the
variable antecedents.

  _Antecedents_   _Consequents_

  1. A B C D E ―――――――― P₁
     A B C D G ―――――――― P₂
     L B C D M ―――――――― P₃
     A F G D M ―――――――― P₄
     L B C D I ―――――――― P₅

  2. H I K L T ―――――――― P₁
     K L M T W ―――――――― P₂
     M T L K W ―――――――― P₃
     X H K Y Z ―――――――― P₄
     T W L K S ―――――――― P₅

(6) A Matter of Observation and Experiment.

On studying the problem relative to the tardiness of John, it appears
that in obtaining the various antecedents the work would be largely
a matter of _observation_. Carrying the father’s dinner, the route
through the woods, etc., are facts which observation would make evident.
However, when it becomes necessary to vary these antecedents with a
view to finding the invariable one, the procedure is experimental as
well as a matter of casual observation. Moreover, in connection with
the direct primary problem the question would be largely a matter of
experiment; though observation would obtain as a subsidiary condition.
We may conclude from this that the method of agreement involves _both
observation and experiment_; and since the student will discover
that the other methods impose similar demands, we are justified
in designating these five special methods of induction as those of
observation as well as of experiment.

(7) Advantages and Disadvantages of the Method of Agreement.

The concrete cases given to illustrate the method of agreement present
a simple combination of antecedents and consequents. In life, however,
such simplicity does not usually obtain and in consequence the method
of agreement gives rise to a few serious difficulties. These may be
summarized as (a) Plurality of causes; (b) Immaterial antecedents;
(c) Complexity of phenomena; (d) Uncertainty of conclusion.

(a) Plurality of causes is mentioned by Mill as constituting the
“characteristic imperfection” of the method of agreement. As the term
signifies, plurality of causes represents a condition where a given
phenomenon has more than one cause, or where different causes produce
the same effect. For example, “A poor crop” may be due to drought,
neglect, pests, etc.; heat may be caused by friction, electricity,
combustion. Unfavorable home conditions; ill health; dislike for
teacher――any one of these might be followed by irregular attendance.

(b) Immaterial antecedents are those which precede a given phenomenon
and yet, under the most favorable situations, have no causal connection
with said phenomenon. For example, the various antecedents of the
heavy rain may have been a south wind, forgetting to take an umbrella,
missing the car and having to walk, etc. Clearly these antecedents,
with the exception of the first, are immaterial.

(c) The law of agreement demands that _all the material antecedents_
receive consideration, but often the situation is too complex to
make this possible; a fair illustration of such would be an attempt
to ascertain all of the antecedents of “the high cost of living.”

(d) The law of agreement never precludes the possibility of error; as
it is quite impossible to carry the analysis to the point of absolute
certainty. Of all the methods, “agreement” is the least reliable.
Despite the foregoing objections, however, the method is of positive
value because of its suggestiveness; opening the door to plausible
hypotheses it gives the investigators a _working basis_.


=3. METHOD OF DIFFERENCE.=

(1) Principle stated.

Says Mill, “If an instance in which the phenomenon under investigation
occurs, and an instance in which it does not occur, have every
circumstance in common save one, that one occurring only in the former;
the circumstance in which alone the two instances differ is the effect
or the cause of an indispensable part of the cause, of the phenomenon.”

To put this in simple terms: _Whatever is invariably present when the
phenomenon occurs and invariably absent when the phenomenon does not
occur, other circumstances remaining the same, is probably the cause
or the effect of the phenomenon._

(2) Method symbolized.

Using the same symbols as were used in “Agreement.”

  _Antecedents_   _Consequents_
     A B C D            P
     ‒ B C D            ‒

                or

           P         A B C D
           ‒         ‒ B C D

In the first instance A is probably the cause of the phenomenon, since
it is present when the phenomenon occurs and absent when it does not
occur. For a similar reason, A is the effect in the second case.

(3) Concrete illustrations.

(A) A wise teacher in ascertaining the cause of John’s tardiness would
have suggested at once a change of route. Using as symbols the initial
letters of the key-words of the antecedents in the case, the following
results:

  s d h w     t
  s d h −     ‒

(B) _First trial._

_Problem_: Unprepared home work.

          _Antecedents_                 _Consequents_

  1. Length of lesson,              }
  2. Definiteness of lesson,        } Work _not_ properly
  3. Amount of interest shown,      }   prepared.
  4. Physical condition the same.   }

_Second trial._

  1. _Length_ of lesson the same,   }
  2. _Lesson made more definite_,   } Work _properly_ prepared.
  3. _Interest_ the same,           }
  4. Physical _condition_ the same. }

The foregoing symbolized:

  L D I C     W
  L ‒ I C     ‒

It is seen that indefiniteness of lesson assignment is the cause of the
unprepared home work.

(4) Advantages and disadvantages of the Method of Difference.

The main difficulty attending the use of the method of difference is
the _complexity_ of phenomenon. The very nature of the method insists
as an essential requirement that _only one material antecedent shall
be varied at a time_. In life the variations are more or less confused,
and it is often not only impossible to observe cases of a single
variation, but frequently error comes through _overlooking_ antecedents
which are material to the case under investigation. For these reasons
the Method of Difference is more a method of _experiment_ than it is
a method of observation. By controlling the circumstances it becomes
possible to vary but _one_ antecedent at a time, and also to bring into
prominence _all_ of the _material_ antecedents.

Bacon claims that _all “crucial instances”_ are merely applications of
the Method of Difference. By crucial instance he means any fact which
will enable us to determine _at once_ which supposition is the correct
one. For example, the physician may not know whether it is malaria or
typhoid fever till he takes a blood test; such a test typifies “crucial
instances.” The various tests in chemistry are likewise cases of
crucial instances, and, in consequence, this science makes use of
“Difference” more than any other method.

(5) Characteristic features of Method of Difference.

There are three distinguishing marks of the Method of Difference: these
are, (1) The phenomenon does not always happen; (2) _One_ antecedent is
variable; (3) The other antecedents are more or less invariable.

The following symbolizations will make these three characteristics
evident:

      _Antecedents_   _Consequents_

  (1)     A B C             P
          A ‒ C             ‒
  (2)     ‒ B C             ‒
          X B C             P
  (3)     L M T K           P
          L M ‒ K           ‒


  AGREEMENT AND DIFFERENCE COMPARED.

(a) The methods of Agreement and Difference are complementary as may
be discerned by comparing their characteristic features: In Agreement
the phenomenon _always occurs_; in Difference the phenomenon _does
not always occur_: In Agreement there is one _invariable_ antecedent;
whereas in Difference there is one _variable_ antecedent: In Agreement
the other antecedents are more or less _variable_; but in Difference
the other antecedents are more or less _invariable_.

(b) According to Mill the Method of Agreement insists that what _can
be eliminated_ is _not_ connected; whereas the Method of Difference
implies that what _cannot be eliminated is_ connected.

(c) The Method of Agreement is more a method of observation, since
it is chiefly concerned with the _discovery of causes_. The Method of
Difference is distinctly a method of experiment, because its usual aim
is to _discover effects_.

(d) The Method of Agreement is so called because the object is to
compare several instances to determine in what respect they _agree_;
but in the case of Difference instances are compared to determine in
what respects they _differ_.

(e) The conclusions of the Method of Difference involve greater
certainty than those of Agreement and, therefore, the former method
should be adopted when there is a choice.


=4. THE JOINT METHOD OF AGREEMENT AND DIFFERENCE.=

(1) Principle stated.

The uncertainty of the conclusions of Agreement and the impossibility
at times of employing directly the Method of Difference, give rise to
the use of the combination of Agreement and Difference known as the
Joint Method. As stated by Mill, the principle conditioning the Joint
Method is this: “If two or more instances in which the phenomenon
occurs have only one circumstance in common, while two or more
instances in which it does not occur have nothing in common save the
absence of that circumstance, the circumstance in which alone the two
sets of instances differ is the effect or the cause or an indispensable
part of the cause, of the phenomenon.” More briefly the notion may
be stated in this wise: _Among many instances, if one circumstance is
invariably present when the phenomenon occurs, and invariably absent
when the phenomenon does not occur this circumstance is probably the
cause or the effect of the phenomenon._

This principle differs from the one underlying the Method of Difference
in that the instances considered are more varied and more numerous. The
principle of Difference requires but _two_ sets of instances, while the
Joint Method demands at least three; _two_ when the phenomenon occurs
and _one_ when it does not occur. A study of the symbolizations and
illustrations will clarify this distinction.

(2) Joint Method symbolized.

If we use circumstances and phenomenon in place of antecedents and
consequent, then one symbolization may be made to stand for
ascertaining either the invariable antecedent, or the invariable
consequent.

     _Circumstances_  _Phenomenon_

  1.     A B C D            P₁
  2.     A D E F            P₂
  3.     A L M N            P₃
  4.     A O P Q            P₄
  5.       O P Q            ―
  6.       L M N            ―
  7.       D E F            ―
  8.       B C D            ―

It is obvious that the first, second, third and fourth groups of
instances illustrate the principle of Agreement; whereas the first and
eighth, the second and seventh, the third and sixth, and the fourth and
fifth illustrate in each case, the principle of Difference.

(3) Concrete Examples illustrating Joint Method.

_The problem_: Too much whispering.

     _Antecedents_                  _Consequent_

  1. Insufficient work,           }
     Lack of interest,            } Much whispering.
     _Seated near a friend._      }

  2. More work,                   }
     Lack of interest,            } Much whispering.
     _Seated near a friend._      }

  3. More work,                   }
     More interest,               } Much whispering.
     _Seated near a friend._      }

  4. More work,                   }
     More interest,               } Not much whispering.
     _Not seated near a friend._  }

  5. More work,                   }
     Lack of interest,            } Not much whispering.
     _Not seated near a friend._  }

  6. Insufficient work,           }
     Lack of interest,            } Not much whispering.
     _Not seated near a friend._  }

From this it may be concluded that the undue amount of whispering is
caused by seating particular friends near each other.

_The problem_: Poor recitations.

   _Antecedents_    _Consequent_

  1. Long lesson,                   }
     Faulty assignment of lesson,   } Poor recitation.
     _Fear of teacher._             }

  2. Lesson made shorter,           }
     Faulty assignment of lesson,   } Poor recitation.
     _Fear of teacher._             }

  3. Lesson made shorter,           }
     A more careful assignment,     } Poor recitation.
     _Fear of teacher._             }

  4. Lesson made shorter,           }
     A more careful assignment,     } Good recitation.
     _Removal of fear of teacher._  }

  5. Lesson made shorter,           }
     Faulty assignment,             } Good recitation.
     _No fear of teacher._          }

  6. Lesson long,                   }
     Faulty assignment,             } Good recitation.
     _No fear of teacher._          }

Fear of teacher is the cause of the poor recitation.

(4) Distinguishing features.

Being a combination of Agreement and Difference the Joint Method
possesses the characteristics of _each_, though more or less modified.
The distinguishing marks may be summarized as follows:

  (1) Of the first group of instances:
        (1) The phenomenon must always occur,
        (2) One antecedent must be invariable,
        (3) The other antecedents must be more or less variable.

  (2) Of the second group of instances:
        (1) The phenomenon must never occur,
        (2) One antecedent must be variable,
        (3) The other antecedents must be more or less invariable.

Briefly, the one principle concerned is this: There must be an
invariable conjunction between the phenomenon involved and the
antecedent suspected of being the cause.

(5) Advantages and Disadvantages of the Joint Method.

Since the Joint Method permits a consideration of the negative aspect
of the question as well as the affirmative, the opportunities for
testing the many instances concerned are doubled. In consequence, the
conclusions of the Joint Method are more positive than those of the
other methods. It follows that this same opportunity to multiply the
instances would tend to lessen the other objections raised against the
Method of Agreement; viz., plurality of causes, immaterial antecedents,
complexity of phenomenon.

The student must regard the given illustrative symbolizations and
concrete examples as being of the simplest form; in life such are the
exceptions rather than the rule. When investigating questions, like
the cause of the high cost of living, the effect of high tariff, the
reason for the typhoid epidemic, etc., there is often a confusion of
circumstances which makes the Joint Method unsatisfactory, even though
it furnishes a larger opportunity for the multiplication of instances.

The strongest case which the Joint Method is able to present is when
the negative instances repeat the positive in every detail, with the
one exception of the variable antecedent. To wit:

  _Strong Argument_:

    _Circumstances_  _Phenomenon_
        A B C             P₁
        A L M             P₂
        ‒ L M             ―
        ‒ B C             ―

  _Weak Argument_:
        A B C             P₁
        A L M             P₂
        ‒ R S             ―
        ‒ T K             ―

Despite the disadvantages, the conditions of the Joint Method are more
or less ideal; since the positive branch of the argument _suggests_
the hypothesis, while the negative branch _proves_ the accuracy or
inaccuracy of such.


=5. METHOD OF CONCOMITANT VARIATIONS.=

(1) Principle stated.

Mill’s statement is this: “Whatever phenomenon varies in any manner
whenever another phenomenon varies in a particular manner, is either a
cause or an effect of that phenomenon, or is connected with it through
some fact of causation.”

To put it differently: _If when one phenomenon varies alone, another
also varies alone, the one is either the cause or the effect of the
other._

(2) Concomitant Variations symbolized.

  Circumstances   Phenomenon
        A              P
      A + a          P + p
   (A + a) − a    (P + p) − p

It is evident from this that little “a” is the cause or the effect of
little “p.” To put it in concrete form:

Let A = X number of calories of heat,

And P = 68° F., the original temperature of room,

And a = candle burning in room for ½ hour,

And p = 2° F.

Then

              _Antecedents_                     _Consequent_

  X no. of cal. of heat in room             = 68° F. temp. of room
  X no. of cal. of heat + burning candle    = 68° + 2° = 70°
  X (no. of cal. of heat + burning candle)  = (68° + 2°) − 2° = 68°
      − burning candle

As large “A” is increased and decreased by little “a” so large “P”
appears to be increased and decreased by little “p.” This strongly
suggests a causal connection between little “a” and little “p.”

(3) Other concrete illustrations.

_Problem_: To ascertain nature of sound.

              _Antecedent_                      _Consequent_

  Bell rung when within a glass jar filled    Loud sound.
    with air,
  Some of the air pumped out of the jar,      Sound not so loud.
  More air pumped into jar again,             Sound louder again.

The conclusion must be that air has something to do with the production
of sound.

_Problem_: To find best feed for egg production.

  100 lbs. beef scraps,   }
  100 lbs. wheat,         } 30 doz. eggs.
  100 lbs. oats,          }
  100 lbs. corn,          }

   50 lbs. beef scraps,   }
  100 lbs. wheat,         } 27 doz. eggs.
  100 lbs. oats,          }
  100 lbs. corn,          }

   90 lbs. beef scraps,   }
  100 lbs. wheat,         } 28 doz. eggs.
  100 lbs. oats,          }
  100 lbs. corn,          }

Since the variation in the amount of beef scraps is accompanied by a
like variation in the number of eggs produced, it may be assumed that
beef scraps are essential to large egg production.

(4) Distinguishing features.
      The phenomenon always occurs but in varying degrees;
      One antecedent varies in degree;
      The other antecedents are invariable.

(5) Advantages and disadvantages.

Concomitant Variations is applicable in cases when it is impossible
to use Difference. Recourse is made to the latter when the phenomenon
can be made to appear or disappear at will, but there are times when
it is impossible to cause the phenomenon to disappear altogether. For
example, in the case of the varying degrees of heat in the room it
would be scientifically impossible to take _all_ of the heat out of
the room; or in experimenting with gravitation, to do away with its
influence entirely, is beyond the power of man. It is thus evident that
Concomitant Variations may be used in cases where the conditions forbid
doing away entirely with the phenomenon.

The special function of Concomitant Variations seems to be to establish
the exact quantitative relation between the varying cause and the
varying effect. To illustrate: As a general law it is known that bodies
attract each other in varying degrees according to their distances
apart and according to their relative sizes; by Concomitant Variations
this law has been given definite quantitative value and reads like
this: “Bodies attract each other directly as the product of their
masses, and inversely as the square of the distance between them.”
This illustration suggests that the variation between antecedent and
consequent may be _direct_ or _inverse_.

The error most common in this method is the assumption that the
quantitative relation between two varying phenomena will always be
according to a _constant ratio_. For example, when being reduced
from a high temperature to 39⅕° F., water steadily contracts; but at
39⅕° F. it commences to expand until it becomes ice. Thus the ratio
of contraction of water is constant only within certain limits. In any
event the established ratio of variation can with absolute safety be
applied only to the _instances investigated_. Another disadvantage
incident to this method, is the situation of two elements varying
together constantly, and yet having no causal connection whatever.


=6. THE METHOD OF RESIDUES.=

(1) Principle stated.

As stated by Mill the principle of residue is this: “Subtract from
any phenomenon such part as is known by previous inductions to be the
effect of certain antecedents, and the residue of the phenomenon is the
effect of the remaining antecedents.”

In simpler form the notion is this: _Subtract from any phenomenon those
parts of it which are known to be the effect of certain antecedents,
and what is left of the phenomenon is the effect of the remaining
antecedents_.

(2) Principle symbolized.

  _Antecedent_   _Consequent_

       A              x
       B              y
       C              z

The total cause of the phenomenon xyz is ABC. But it is known that
the cause of x is the antecedent A; whereas the cause of y is the
antecedent B; hence it is concluded that the cause of z is the
antecedent C.

(3) Concrete illustrations.

_Problem_: To find the weight of coal.

  _Antecedents_ _Consequents_

  Weight of driver,   }
  Weight of wagon,    } = 4200 lbs.
  Weight of coal.     }

  Weight of driver,   } = { 200 lbs.  } = 2200 lbs.
  Weight of wagon,    }   { 2000 lbs. }

Hence we may conclude that the weight of coal is 4200 lbs. − 2200 lbs.,
or 2000 lbs.

Perhaps the most noted instance in history of the application of
this method, was the one which resulted in the discovery of Neptune.
In calculating the orbit of Uranus, it was found that the combined
attractions of the sun and the known planets did not account for the
path which Uranus took. There was some unknown influence at work.
Assuming that this unaccountable attraction was due to the presence of
another planet beyond the orbit of Uranus, an Englishman by the name
of Adams, and later the Frenchman Leverrier, were able to indicate by
the principle of Residues, the spot where this planet should be. By
directing the telescope toward this point, Neptune was discovered.

(4) Distinguishing features:
      The phenomenon always occurs,
      The antecedents are usually invariable,
      Some of the antecedents are known to be the cause of a part of
        the phenomenon.

(5) Advantages and disadvantages.

The Method of Residues gives three distinct results: First, it tells
_what_ is left over after all the other parts of the phenomenon have
been explained. Second, it tells _how much_ is left over, and third,
it calls attention to the _unexplained parts_ of the phenomenon. For
example, in the first concrete illustration, by subtracting the known
quantities from the total quantity, what is left over is found to be
coal; not only so but we are able to calculate the exact amount of
coal. This illustrates the first and second results of the Method of
Residues. (Like concomitant variations it is seen that residues is
serviceable in given _definite quantitative values_.) The discovery of
Neptune illustrates well the third result of this method; i. e., after
accounting for every other force, it was found that there was yet a
force at work which had never been explained. It is this third feature
of _unexplained residues_ which has placed “Science in its present
advanced state.” “Most of the phenomena which nature presents are
complicated; and when the effects of all known causes are estimated
with exactness, and subducted, the residual facts are constantly
appearing in the form of phenomena altogether new, and leading to the
most important conclusions.” So says John Herschel. Almost all of the
discoveries in astronomy have come about in this way. If a heavenly
body does not behave as it should according to the established theory,
then either the theory is wrong or there is some _residual phenomenon_
which needs to be explained. Its suggestiveness is, therefore, the most
important function of this method, though this very feature is the one
which makes evident its greatest disadvantage. The unexplained residual
phenomenon may be very complex and, therefore, a careless observer is
apt to overlook a lurking element which in reality is the true cause.


=7. THE GENERAL PURPOSE AND UNITY OF THE FIVE METHODS.=

Thinking has been defined as the deliberative process of affirming
and denying connections. It is obvious that these five methods are a
matter of affirming and denying connections between antecedents and
consequents. As soon as the looked for connections are established,
the antecedents and consequents are known to be related to each other
as causes and effects. In this attempt to find and prove connections
the Method of Agreement is chiefly valuable in suggesting workable
hypotheses, and the method of difference in verifying, through
experiment, the correctness or incorrectness of these hypotheses.

In substance the principle conditioning both methods is this: “_If
a single antecedent is invariably present when the phenomenon is
present and invariably absent when the phenomenon is absent then this
antecedent is the cause of the phenomenon._” To put it still more
briefly: _Between two phenomena there is a causal connection, if the
conjunction between the two is invariable._ It is the business of
Agreement to _single out the one antecedent_ and of Difference to
show, by presenting the negative as well as the affirmative side of the
case, that the _conjunction of the one antecedent and the particular
phenomenon is invariable_. The Joint Method is merely a combination
of Agreement and Difference carried into more varied and complex
situations. The methods of Concomitant Variations and Residues are
merely modifications of Difference; the former being used when the
_chief feature is the fluctuation_ of the phenomenon, and the latter
when it is desired to find _what is left over_.

_Agreement suggests the hypothesis, “difference” proves it; the
joint method is “difference” more or less complicated, concomitant
variations is “difference” applied to fluctuating phenomena, residues
is “difference” used to find what and how much is left over._

Agreement is the method of observation and belongs to the physician
and nature student. Difference and the Joint Method are experimental
devices which are used by the physicist and chemist. Concomitant
Variations is the method of unstable phenomena and naturally attaches
itself to the economist and statistician. Residues is the method of
“lurking exceptions” and is favored by the astronomer and mathematician.
Residues, being the method of “what is left over,” is the most common
in daily affairs.[14]

_All the five methods are forms of inductive thinking which lead to the
establishment of causal connections by means of the principle of the
invariable conjunction of phenomena._


=8. OUTLINE.=

  THE FIVE SPECIAL METHODS OF OBSERVATION AND EXPERIMENT.

  (1) Aim of Five Methods.
        Fundamental fact of causation.

      Aim of analysis.
                    { agreement
                    { difference
        Methods of  { joint
                    { concomitant variations
                    { residues

  (2) Method of Agreement.
        Principle stated
        Method symbolized
        Method illustrated
        Distinguishing features of method
        A matter of observation and experiment
        Advantages and disadvantages

  (3) Method of Difference.
        Principle stated
        Method symbolized
        Method illustrated
        Advantages and disadvantages
        Characteristic features
        Agreement and Difference compared

  (4) The Joint Method of Agreement and Difference
        Principle stated
        Method symbolized
        Concrete illustrations
        Distinguishing features
        Advantages and disadvantages

  (5) Method of Concomitant Variations
        Principle stated
        Method symbolized
        Concrete illustrations
        Distinguishing features
        Advantages and disadvantages

  (6) The Method of Residues
        Principle stated
        Method symbolized
        Concrete illustrations
        Distinguishing features
        Advantages and disadvantages

  (7) General Purpose and Unity of Five Methods
        One fundamental principle


=9. SUMMARY.=

(1) The fundamental fact of causation underlies the three forms of
induction, but is most conspicuous in the method of analysis and may
be ascertained by recourse to one of the experimental methods.

(2) The principle of the method of agreement may be summed up in the
two statements: The sole invariable antecedent of a phenomenon is
probably its cause and the sole invariable consequent of a phenomenon
is probably its effect. These two statements may be symbolized and
illustrated.

The essential characteristics of the method of agreement are the
phenomenon always occurs; there is at least one invariable antecedent;
the other antecedents vary.

The method of agreement together with the other four methods may justly
be termed methods of experiment as well as methods of observation.

The difficulties of the method of agreement are in the main plurality
of causes, immaterial antecedents, complexity of phenomenon and
uncertainty of conclusion. These difficulties may be summarized as
involving a phenomenon which may have several causes; may be preceded
by conditions of no causal consequence; may be so involved as to
prevent exhaustive examination; and may give unreliable conclusions.

Agreement is valuable chiefly in furnishing to the investigator
plausible hypotheses.

(3) The principle of difference is this: “Whatever is invariably
present when the phenomenon occurs and invariably absent when the
phenomenon does not occur, other circumstances remaining the same,
is probably the cause or the effect of the phenomenon.”

Like agreement, difference admits of symbolization and illustration by
concrete examples.

The chief difficulties attending difference are: in nature varying
_one_ antecedent at a time is infrequent, and it is easy to overlook
antecedents which are closely related to the case under investigation.

Difference is the most common method of the experimental sciences.
The characteristic features of difference are, the phenomenon does
not always occur, one antecedent is variable, while the others are
invariable.

The methods of agreement and difference are complementary processes.
Agreement attempts to eliminate all the antecedents but one, while
difference aims to eliminate one only. Agreement is a method of
observation, while difference is a method of experiment. The conclusion
of the method of difference gives greater certainty than that of the
method of agreement.

(4) The joint method may be stated in this way: Among many instances if
one circumstance is invariably present when the phenomenon occurs and
invariably absent when the phenomenon does not occur, this circumstance
is probably the cause or the effect of the phenomenon.

The instances of the joint method are more numerous and more varied
than those of either agreement or difference.

The joint method has the distinguishing characteristics of both
agreement and difference.

Because it furnishes greater opportunities for multiplying and varying
the instances involved, the joint method presents fewer objections than
either of the two separate methods.

The positive branch of the joint method suggests the hypothesis, while
the negative branch proves it. This makes the method somewhat ideal.

(5) The principle of concomitant variations may be stated as follows:
If when one phenomenon varies alone, and another also varies alone,
the one is either the cause or the effect of the other. This is the
method of fluctuation, and is used when it is impossible to make
the phenomenon disappear altogether, as in the case of difference.

The chief function of concomitant variations is to establish exact
quantitative relations between cause and effect.

(6) The principle of residues is this: Subtract from any phenomenon
those parts of it which are known to be the effect of certain
antecedents, and what is left of the phenomenon is the effect of the
remaining antecedent.

The most valuable feature of residues is its suggestiveness; an
attempt to explain the “residual phenomenon” has led to many important
scientific discoveries.

(7) The five methods are concerned with the establishment of causal
connections between phenomena. Agreement _suggests_ the connection
while difference _proves it_. The other methods are modified
applications of difference, necessitated by some peculiar form which
the phenomenon may take. A statement of the one principle involved is:
“If the conjunction between two phenomena is _invariable_ then there is
a causal connection.”

All of the methods are forms of inductive thinking.


=10. REVIEW QUESTIONS.=

  (1) Explain “the fundamental fact of causation.”

  (2) Show that the fact of causation is most conspicuous in induction
      by analysis.

  (3) Name the five special inductive methods of observation and
      experiment.

  (4) State, symbolize, and illustrate the method of agreement.

  (5) Give examples of antecedents which do not function as causes.

  (6) Show that the “special methods” are a matter of both observation
      and experiment.

  (7) Give the distinguishing features of the method of agreement;
      illustrate by reference to the symbols.

  (8) Exemplify the plurality of causes; immaterial antecedents;
      complexity of phenomenon.

  (9) Show that the conclusions of agreement are largely hypothetical.

 (10) State, symbolize, and illustrate the method of difference.

 (11) Show by illustration that, in the method of difference, only one
      antecedent should be varied at a time.

 (12) Show that difference is naturally a method of experiment.

 (13) Explain Bacon’s use of the term “crucial instances.”

 (14) Name and explain the characteristic features of the method of
      difference.

 (15) Show that agreement and difference are complementary.

 (16) Explain and illustrate the joint method.

 (17) What inference may be drawn from the following instances:

        _Antecedents_    _Consequents_

           A L M T          p q r g
           B L M E          z q r x
           B C M E          r z x y
           A M T H          p q g o

 (18) “Mr. Darwin, in his experiment on cross and self fertilization
      in the vegetable kingdom, placed a net about one hundred flower
      heads, thus protecting them from the bees. He at the same time
      placed one hundred other flower heads of the same variety of
      plant where they would be exposed to the bees. He obtained
      the following result: The protected flowers failed to yield
      a single seed. The others yielded about 2,720 seeds. Thus
      cross-fertilization was proved.” (Hibben).

      What method did Darwin employ? Symbolize the experiment.

 (19) Summarize the distinguishing marks of the joint method.

 (20) Show that the joint method is more ideal than either agreement
      or difference.

 (21) State and give concrete illustrations of the law of concomitant
      variations.

 (22) What is the chief function of concomitant variations? Illustrate.

 (23) Give instances where it would be impossible to use difference,
      but easy to use concomitant variations.

 (24) Explain this: “The quantitative variation between antecedent and
      consequent may be either direct or inverse.”

 (25) State and explain by illustration the method of residues.

 (26) What are the advantages and disadvantages of the principle of
      residues?

 (27) State the principle which virtually sums up the five methods.

 (28) Write briefly on the practical applications of the five methods
      to the ordinary walks of life.


=11. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) Trace the connection between the method of agreement and
      induction by simple enumeration.

  (2) Show that Mill’s methods may properly be termed “Inductive
      Methods of Scientific Investigation.”

  (3) How may it be shown by “agreement” that the high cost of living
      is due to the tendency to spend more than we earn?

  (4) Assume that you are a member of the Board of Health, and that you
      desire to ascertain the cause of the diphtheria epidemic by means
      of the principle of agreement.

  (5) What is the error involved in coming to the conclusion that to
      sit at table where there are thirteen, may mean the death of
      _one_ of the thirteen before the end of the year.

  (6) Indicate how it could be shown, by the method of difference, that
      the mosquito is responsible for the propagation of yellow fever.

  (7) “Another experiment similar to this was tried by Plateau, who put
      some food of which cockroaches are fond on a table and surrounded
      it with a low circular wall of cardboard. He then put some
      cockroaches on the table; they evidently scented the food, and
      made straight for it. He then removed their antennae.” (Hibben).
      Complete and give with explanations the method used.

  (8) “In some cases it is impossible to remove an element which is
      supposed to be the cause of an effect under investigation.”
      Explain and illustrate.

  (9) “Extreme care must be taken that, in the withdrawing of any
      element, no other element is inadvertently introduced.” Tyndale
      supposed he had proved _spontaneous generation_, when, after
      sealing in a jar of boiled water a wisp of baked hay, he found,
      after many days, indications of life within the bottle. In
      transferring the hay to the bottle, he carried the former across
      the room. What element was inadvertently introduced?

 (10) “The attempt to determine the numerical relations according to
      which two phenomena vary, requires the utmost caution as soon as
      our inference outsteps the limits of our observations.” (Fowler).
      Explain this in connection with the law of concomitant variations.

 (11) “When the effects of all known causes are estimated with
      exactness and subducted, the _residual facts_ are constantly
      appearing in the form of phenomena altogether new, and leading
      to the most important conclusions.” Make clear by illustration
      this quotation which has reference to the principle of residues.

 (12) Explain “invariable conjunction of phenomena.”

 (13) Investigate by means of one of the five methods the following
      problems:

        (1) “All vegetables which grow to root should be planted during
            the last two days of the waxing moon.”

        (2) “In this section the _south_ wind is the _storm_ wind.”

        (3) “Mischief is the outcome of misdirected energy.”

        (4) “Bad boys usually receive unjust treatment.”

        (5) “An ounce of prevention is worth a pound of cure.”



                              CHAPTER 19.

                 THE AUXILIARY ELEMENTS OF INDUCTION.

                 OBSERVATION――EXPERIMENT――HYPOTHESIS.


=1. THE FOUNDATION OF INDUCTIVE GENERALIZATIONS.=

Induction is the process of universalizing particular facts. _The
starting point is the fact._ Through observation the investigator
gathers _facts_, and then works them over with a view of finding
_uniformities_. The mind cannot build inductive generalizations without
facts any more than a mason can build a brick wall without the bricks.

A fact is any particular thing _made_ or _done_ or is that which may
be acquired by means of the presentative (perception and imagination)
powers of the mind. The state of awareness which results from the
observation of facts is an individual notion. This presents another
aspect of the inductive process; namely, “It is a matter of building
general notions from individual notions, acquired by the observation
of facts.” To illustrate: I note that A, B, C and D are honest in their
dealings with me, hence I come to the conclusion that some men are
honest. A fact is something done, consequently the actual doing of the
honest things by A, B, C and D are facts. Each state of awareness of
each fact is an individual notion. The mind now discerns a uniformity
in these facts and derives the general notion that “some men are
honest.”


=2. OBSERVATION.=

Facts are acquired by means of observation. When the mind fixes the
attention upon any phenomenon it _observes_ it. The term observation
means “to watch for” and may be defined _as the act of watching for
phenomena as they may occur_. The observation may be only casual,
or it may be _willed_ or rational. It is the latter aspect which
most concerns the logician. In this sense observation means careful,
painstaking, systematic perception. It involves the concentration of
consciousness upon the case in hand, or the actual giving of attention.
The thing observed may be external, when the observation takes the
form of _sense-perception_; or it may be internal, when the observation
becomes a matter of _introspection_.


=3. EXPERIMENT.=

In observation we simply _watch_ the phenomenon; in experiment we
_make_ it. In experiment we not only observe, but we _manipulate_ the
circumstances so as to present the phenomenon under the most favorable
conditions for observation. “In observation,” says Mill, “we _find_ an
instance in nature suited to our purposes”; whilst in experiment, by an
artificial arrangement of circumstances, we _make_ an instance suited
to our purpose. In observation we watch for causes; in experiment we
work for effects. We may thus define experiment as _the act of making
phenomena occur for the purpose of watching for effects_. In experiment
there is much which is merely observation. In fact experiment _is_
observation in which the phenomenon is artificially produced. For
the sake of definiteness, however, any observation which involves a
_manipulation of circumstances_, may be designated as _experimental_.


=4. RULES FOR LOGICAL OBSERVATION AND EXPERIMENT.=

To the uninitiated, the matter of observation seems an easy task, and
yet when one hears two honest men swear to diametrically opposite facts
which have come to them from observing the same phenomenon, his faith
is shaken. “Eyes have they but see not” is a logical truth as well as
a moral one. Only the observation of the trained can be depended upon;
and yet this should not discourage the layman, for even he, by a little
conscientious effort towards careful observation, may greatly increase
his store of accurate knowledge and add to the joy of living.

The attending rules are usually heeded by the trained scientist in
matters of observation and experiment:

_First Rule. The observations should be precise._ The time, the place,
the surrounding conditions _must_ be accurately noted. Many artificial
contrivances have been devised because of the desire of the scientist
to be precise. Instruments like the balance, the thermometer, the
microscope, etc., has he invented, and various devices and methods
has he adopted for the sake of _precision_. A common method is to take
an _average of observations_. For example, to estimate justly the
class work of a student, the teacher should not be content with the
ratings of one or two recitations, but must average the ratings of many
recitations. Again, a child may be led to discover approximately the
value of the sum of the interior angles of a triangle by measuring the
angles of many triangles and then striking an average; assume that the
following results are obtained by such a procedure: (1) 178, (2) 181,
(3) 179, (4) 180, (5) 182; adding these and dividing by 5 gives 180.

_Second Rule. The observations should concern only the material
circumstances of the case in hand._ All the non essentials may be
ignored, as they serve only to distract attention. For example, (1) in
order to get the “right count” all other sounds must be ignored save
that of the _fire gong_; (2) in finding the depth of the water for the
building of a dam, soundings ten miles away from the objective point
could be of little value. On the other hand, it is easy to overlook
certain lurking essentials. To observe such, it is necessary to resort
to what the psychologist terms a “_preadjustment of attention_.” We
must know with exactness what we are looking for. We must have a mental
image of what we wish to see. The astronomer in the discovery of a
new planet must know the exact spot where it ought to be, and have
a clear mental image of its appearance. This _expectant attention_
is a necessity in the case of the physician who is anxious to make no
mistakes in his diagnosis. If he is looking for pneumonia, he must have
a very distinct auditory image of the sound of an affected lung. It
should be remarked, however, that this very preadjustment of attention,
with the untrained, frequently leads to illusion. We are so anxious to
see what we are looking for that nine-tenths of what we believe we see
is only inference. How easy it is to read into a phenomenon something
that is entirely foreign to it; to read between the lines; to see only
the reflection of our own ideas. “Verily the mental picture of what we
wish to see becomes so vivid that we are positive of the thing being
external.” Thus the drunkard sees snakes and the superstitious see
ghosts. _Reading into the external what is only vividly internal is
probably the most common error with the untrained observer._

_Third Rule. The observed circumstances should be varied as much as
possible._ To observe a fact from a different viewpoint may not only
broaden the original notion, but it may change it entirely. In order to
gain a true notion of the effect of a particular nostrum on the human
organism, it becomes necessary to experiment with persons of different
ages, living under different environments, and inheriting different
constitutions. Those who are noted for pronouncing broad, safe and
sane judgments upon momentous questions are those who are “all-angled
observers.”

_Fourth Rule. The observed phenomenon should, if possible, be isolated
from all interfering phenomena._

In studying the action of a drug or a food, all other drugs or foods
must be eliminated. The effect of gravitation on a body cannot be
recorded accurately unless the experiments are made in a vacuum. When
studying the deflections of the compass, all magnetic substances must
be removed from the field.


=5. COMMON ERRORS OF OBSERVATION AND EXPERIMENT.=

The rules for scientific observation have suggested certain common
errors which may now be considered.

(1) Preconceived ideas.

There is not an unholy belief nor an unwholesome theory which cannot be
bolstered up by means of apparent facts. For example, that monstrosity
of Puritan thought known as “Salem Witchcraft” was substantiated by
facts honestly observed. Again, having made up his mind that it is
going to be “so and so,” the statistician goes out into the highways
and byways and gathers the facts which vindicate his judgment. Further,
the democrat finds that the _majority_ of the voters are democrats;
while the republican is confident that _two-thirds_ of the voters
are for republicanism. Here again is the fallacy springing from
a preadjustment of attention. _We see what we want to see._ Only
the highly trained observer is able, with impunity, to make use of
preadjusted attention, and even with him, it is not easy to remove
from the situation _belief_ and _prejudice_.

The true observer undertakes his work with his mind open to _anything_
which the eye may bring him, though it may topple into the dust his
dearest theory and most cherished belief; he proceeds――_the mind a
“clean white page.”_

(2) The “observed” and the “inferred” confused.

This error has already received some attention. It may be remarked
further, however, that, psychologically considered, observation is a
matter of interpreting the new by means of the old. Of necessity the
interpretation, whatever it may be, will assume the complexion of the
particular “old knowledge” which the mind is able to use. In short, a
man _will_ see what his previous environment has _trained_ him to see;
the conscientious gardener sees the weeds, whilst the artist may see
nothing but the flowers. It follows, therefore, that all observation
must be largely a matter of inference based on experience. In looking
at the moon, for example, all I actually see is a patch of color; the
form and distant location of the moon being a matter of experience.

The inference referred to in this heading is not that which is
_necessary_ for perception, but that which is _suggested_ by perception.
To illustrate: It is icy; three men are running for a car; Smith raises
his arm; Jones slips to the ground; and Brown testifies, that “Smith
knocked Jones down.” Brown _observed_, that Smith assumed the proper
attitude and that Jones conveniently went down at the right time; and
then _inferred_ the rest.

(3) Ignoring the exceptions.

This comes through an over anxiety to prove our theory. With this
mental attitude, the observations which are corroborative will so
completely fill the mental field, that the exceptions are made to
seem of no consequence. This accounts for the superstition attached
to _thirteen_. As a coincidence some one at some time died who had
previously eaten at a table where there were thirteen. Perhaps during
the life of the superstitious one this happened on two or three
occasions, but the fact so impresses the subject that he ignores
the _dozen_ times when death did not follow. Other generalizations
belonging to this class are (1) people never die at flood tide;
(2) there must be three accidents in succession; (3) the first sight
of the new moon over the right shoulder is a good omen; (4) seeds which
grow to root do best when planted during the last days of the waxing
moon; (5) horse chestnut in pocket guards against sore throat, etc.

(4) Sympathy and undue interest.

The influence of the heart over the brain is well known. A physician
is liable to this error when he attempts to prescribe for one of his
own family. Sympathy not only warps the judgment but it may actually
interfere with the accuracy of an honest observer’s perceptive powers.

(5) Inattention and a fallible memory.

These short comings are too apparent to demand discussion.


=6. THE HYPOTHESIS.=

Having observed the facts, the mind naturally seeks for explanation
of the same. Hence taking the facts as a cue and bringing into play a
constructive imagination, a plausible supposition is advanced, which is
then proved or disproved. Such a supposition is known as an hypothesis.

Definition. _An hypothesis is a supposition advanced for purposes of
explanation and proof._

First illustration. The facts are known that light travels from the
sun to the earth, and at the rate of 186 thousand miles per second.
These facts suggest the problems: (1) How does the light reach the
earth? (2) Why this rate of speed; why so much faster than the rate at
which sound travels? To solve these problems, or to explain the facts,
the “ether” hypothesis is advanced: viz., “A rare medium called ether
pervades space and transmits the light and heat of the sun.” This
hypothesis has never been conclusively proved.

Second illustration. Fact: The child leans forward and squints his eyes,
when attempting to read work which has been placed on the black board;
hypothesis: The child is near sighted.


=7. INDUCTION AND HYPOTHESIS DISTINGUISHED.=

Induction is a matter of realizing generalizations from the observation
of facts. The product of such is an induction, but we know that an
hypothesis is likewise a generalization based upon facts. What is the
difference? An induction, as such, is a _broader_ term than hypothesis.
As soon as the hypothesis is proved or disproved, it ceases to be
an hypothesis, but still remains an induction. An hypothesis, being
advanced for purposes of explanation ceases to be an hypothesis
when, in the last analysis, it fails to explain. Moreover, as soon as
the hypothesis is shown to be an undoubted truth, it also loses its
distinctive hypothetic marks. _An hypothesis is merely a tentative
induction._


  ILLUSTRATIONS:

(1) The hypothesis is advanced that the fire started from the coal
range in the kitchen. After the incendiary is caught, this supposition
ceases to be an hypothesis.

(2) It is suspected, that my insomnia is due to the three cups
of strong coffee indulged in at the evening meal. As soon as this
supposition is proved by experimental means (law of difference), it
ceases to be an hypothesis and becomes an unpopular inductive truth.


=8. HYPOTHESIS AND THEORY.=

In common parlance hypothesis and theory are used interchangeably.
We refer to the “nebular hypothesis” or the “nebular theory”; to the
“hypothesis of the sun’s heat” or “the theory of the sun’s heat.” On
the other hand, we say “the theory of gravitation,” “the theory of
evolution,” etc., with certain uniformity. From these observations we
may infer that hypothesis and theory may be used interchangeably when
the facts are of a _low probability_; but when the facts have undergone
_cogent verification_, it is more correct to use theory in their
designation rather than hypothesis. “_A theory is a partially
verified hypothesis._” It has been remarked that theory has a second
signification of being a term which stands for “any body of acquired
truth.” It is unfortunate that its use could not be confined to this
latter conception.


=9. THE REQUIREMENTS OF A PERMISSIBLE HYPOTHESIS.=

Any hypothesis should be made to conform to the following
requisites: (1) _The hypothesis must be conceivable._ The hypothetic
generalizations of primeval days were mere fancy. For example, the
loud noise from the clouds on dark days was the angry voice of the
God of the skies. Even in this day when a complex situation cannot
be explained there comes the temptation to draw entirely upon the
imagination, and advance an hypothesis which is absurd in every sense
of the word. The permissible hypothesis demands that there be some
ground for the conjecture. A fact or two at least must be used as the
foundation for whatever the constructive imagination may build. On
the other hand the past has taught us that we cannot afford to be too
exacting in the enforcement of this rule. The ideas of Copernicus,
Newton, Harvey, Darwin, and many another of the world’s best thinkers,
were looked upon at first as being ridiculous. There is always a bare
possibility of a “lurking truth” in the conjecture, _and no broad
minded and sanely educated man can afford to scoff blindly at something
which may seem to him mere fancy_. Prejudice and a willful blindness to
truth, have ever been imminent stumbling blocks in the path of progress.

(2) _The hypothesis must be capable of proof or disproof._ This
means, that where it is possible the hypothesis should touch, in one
form or another, our experience. If the hypothesis is wholly unlike
any experience we may have had, it becomes _impossible_ to ascertain,
whether it agrees or disagrees with the facts, which it is supposed
to explain. A legitimate hypothesis must furnish some opportunity
for securing facts to prove or disprove it. For example, to advance
an hypothesis relative to the conditions of the next world is hardly
permissible, as “spirit-facts” are entirely without our field of
experience. Surely, one returning from Heaven could give us no
conception of it; because there is nothing in the carnal mind that
may be used to interpret the experiences that must function in the
Celestial City.

(3) _The hypothesis must be adequate._ It should take into
consideration all the known facts. It stands to reason that, if _one_
known fact is ignored, the entire procedure is thus vitiated. It would
be absurd to suppose the moon to be inhabited without giving heed to
the fact of its having no atmosphere.

(4) _The hypothesis must be as simple as possible._ We must, of
course, recognize situations which in themselves are too complex to
admit of simple conjectures. The purport of the fourth rule is, that
the hypothesis should not be made _unnecessarily_ complex.

(5) _The hypothesis should not contradict any verified truth._ Any
conjecture which opposed the law of gravitation would be out of place.
Of course it is possible to have only apparent conflicts between the
new hypothesis and the old law. Further observation should show that
no such clash exists.


=10. THE USES OF HYPOTHESES.=

The hypothesis is serviceable mainly in these particulars:

(1) As a working basis.

When one is confronted with a huge mass of facts it becomes necessary
to start somewhere, and with as little waste of time and energy as
possible. Almost anything is better than a haphazard floundering which
reaches “nowhere.” So the investigator _hazards_ a tentative theory,
which he at once proceeds to verify. If verification fails, then he may
discard this first hypothesis for a better one.

(2) As a guide to ultimate truth.

Much might be said relative to the use of rejected hypotheses. By
means of these, science has advanced step by step towards the full
light of perfect knowledge. As has been remarked, no true scientist
cares to overlook the opportunity for suggestive inspiration which
some forsaken hypothesis may afford him. Just as the individual attains
the best success by _using his failures as stepping stones_, so the
true scientific discoverer _climbs up to the light on the stairway of
discarded hypotheses_. By testing and rejecting the false hypotheses,
the situation becomes more definite and the problem more accurately
defined. “Kepler himself tried no less than nineteen different
hypotheses before he hit upon the right one, and his ultimate success
was doubtless in no slight degree due to his unsuccessful efforts.”

(3) As a discoverer of immediate truth.

Often, moreover, the hypothesis leads directly to positive verification.
The supposition advanced may hit the truth squarely; and may be of such
peculiar nature as to lead easily to clear and conclusive proof.

(4) As affording a probable explanation of a problem which will not
lend itself to an entirely satisfactory solution.

The theory of evolution may illustrate this fourth use; while the
history of the discovery of Neptune illustrates the third.


=11. CHARACTERISTICS NEEDED BY SCIENTIFIC INVESTIGATORS.=

The hypothesis is referred to “as the great instrument of science.” The
greatest thinkers of time have possessed the courage and the conscience
to step from the known to the unknown; to hazard a guess as to the
meaning of what they saw, and then subject their guess to a rigorous
test. This procedure involves three elements on the part of the
investigator: (1) Power of accurate observation. (2) Constructive
imagination. (3) A passion for truth.

(1) An hypothesis formed without an accurate knowledge of facts
is not only useless, but often it may work positive harm. To
advance serviceable suppositions which are not grounded on fact,
is as impossible, as it is to build a house without a foundation.
The hypothesis is an image of the constructive imagination, but
the pedestal of this image must rest on the ground of fact. The
investigator who would be scientific must exercise scrupulous care in
securing his facts through observation and experiment. The rules and
errors involved in such a procedure have received sufficient attention.

(2) After the investigator has his facts to build upon; and these may
be few or many――sometimes even a single fact is sufficient――then may he
theorize as to a possible explanation of them. Here is where the real
work of the born genius tells. To some the facts are nothing but words,
to others they mean universal laws and great inventions. Who but a
Newton could have seen the law of gravitation in the falling apple?
Who but an Edison could have seen the phonograph in the sound wave and
wax? It must be recognized that this remarkable imaginative insight is
inborn in some cases; and yet this does not preclude the necessity for
_cultivating_ this power, though it may be only in a rudimentary state.
Here is another opportunity for the school teacher; namely, to train in
every legitimate way the _constructive imagination_.

(3) Having once constructed the hypothesis, the honest scientific
investigator at once proceeds to subject it to a series of most
rigorous tests. It is well to see big things in a little fact; to
have a mind as fertile in new ideas as a watered garden――this is
genius! But is it not more incumbent to have a conscience so keen,
that nothing will be allowed to pass for truth which has not received
ample verification? Intellectual dishonesty is quite as common as moral
dishonesty. Moreover, one must maintain an _open mind, absolute candor,
and a willingness to abandon the most cherished theory_. Often it is
much easier to _explain away_ contradictory facts than it is to forsake
a _pet theory_.


=12. OUTLINE.=

  THE AUXILIARY ELEMENTS IN INDUCTION――OBSERVATION――EXPERIMENT――
  HYPOTHESIS.

  (1) The Foundation of Inductive Generalizations.

  (2) Observation. Defined.

  (3) Experiment. Defined.
        Compared with Observation.

  (4) Rules for Logical Observation and Experiment.
        Their need.
        First Rule.
        Second Rule.
        Third Rule.
        Fourth Rule.

  (5) Common Errors of Observation and Experiment.
        (1) Preconceived Ideas.
        (2) Confusing the Observed with the Inferred.
        (3) Ignoring the Exceptions.
        (4) Sympathy and Undue Interest.
        (5) Inattention and a Fallible Memory.

  (6) The Hypothesis.
        Defined and Illustrated.

  (7) Induction and Hypothesis Distinguished.

  (8) Hypothesis and Theory Distinguished.

  (9) The Requirements of a Permissible Hypothesis.
        (1) Conceivable,
        (2) Capable of proof or disproof,
        (3) Adequate,
        (4) Simple,
        (5) Not contradictory.

 (10) Uses of Hypothesis.
        (1) A working basis,
        (2) Guide to ultimate truth,
        (3) Discoverer of immediate truth,
        (4) Probable explanation.

 (11) Characteristics Required by Scientific Investigators.
        (1) Accurate observer,
        (2) Constructive imagination,
        (3) Passion for truth.


=13. SUMMARY.=

(1) Facts are the foundation of all inductive generalizations.
Induction is largely a matter of building general notions from
individual notions derived from the observation of facts.

(2) Observation is the act of watching the phenomena as they may occur.
It involves the voluntary concentration of consciousness on the case in
hand.

(3) Experiment is the act of making phenomena occur for the purpose
of watching for effects. It is in reality a form of observation which
necessitates a manipulation of circumstances.

(4) The average man is not given to careful observation. The rules
adopted by scientific observers are: (1) The observation should
be _precise_; (2) should concern only the material circumstances;
(3) should be varied; (4) should be isolated.

For the sake of _precision_ many instruments have been invented and
methods devised; notably instruments for accurate measurements, such as
the balance and thermometer, and methods like the method of averages.

Frequently a situation may be so complicated as to demand a
“preadjustment of attention.” With the untrained this very
preadjustment may lead to serious error.

An “all-angled observer” is the most trustworthy.

(5) Errors in observation come from preconceived ideas; confusing
perception with inference; ignoring the exceptions; sympathy;
inattention; and a fallible memory.

(6) An hypothesis is a supposition advanced for purposes of explanation
and proof.

(7) An hypothesis is a tentative induction. As soon as it is deprived
of its tentative nature it ceases to be an hypothesis.

(8) Hypothesis and theory are often used interchangeably when reference
is made to phenomena of low probability. Theory should be used only in
instances of high probability.

(9) A permissible hypothesis must be (1) conceivable; (2) capable of
proof or disproof; (3) adequate; (4) simple; (5) must not contradict
any verified truth.

(10) The hypothesis is especially serviceable in these four particulars:
(1) as a working basis; (2) as a guide to ultimate truth; (3) as a
discoverer of immediate truth; (4) as affording probable explanations.

(11) There are certain characteristics which an honest and courageous
investigator needs to possess. These are: (1) undoubted ability as
an accurate observer of facts, (2) a constructive imagination, (3) a
passion for truth.

To build an acceptable hypothesis without fact is as impossible as it
is to build a house without a foundation.

The genius, because of his imaginative insight, transforms the simple
fact into a complex invention or law.

A prevailing “_intellectual dishonesty_” suggests the need of “_a
greater passion for truth_.”


=14. REVIEW QUESTIONS.=

  (1) Show that facts are the raw material of induction.

  (2) Define and illustrate a fact.

  (3) Define induction in terms of the notion.

  (4) Define and illustrate observation.

  (5) Define and illustrate experiment.

  (6) Show the difference between observation and experiment.

  (7) State and exemplify the rules for logical observation and
      experiment.

  (8) Illustrate the method of averaging observations.

  (9) Explain “preadjustment of attention.”

 (10) What is the most common error with the untrained observer?
      Explain and illustrate.

 (11) Explain the expression “all-angled observer.”

 (12) State and exemplify the errors of observation and experiment.

 (13) To what error in observation are superstitions generally due?

 (14) Define and illustrate hypothesis.

 (15) Indicate the difference between an hypothesis and an ordinary
      induction.

 (16) When may theory and hypothesis be used interchangeably?
      Illustrate.

 (17) Show by illustration that the term theory is ambiguous.

 (18) Summarize the requirements of a permissible hypothesis.
      Illustrate.

 (19) Select some school room experience with a view of making it
      conform to the requirements of a permissible hypothesis.

 (20) Explain and illustrate the uses of hypothesis.

 (21) “The scientific discoverer climbs up to the light on the stairway
      of discarded hypotheses.” Explain.

 (22) Write a short theme on “Characteristics Required by Scientific
      Investigators.”


=15. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) “Land and sea breezes are due to a difference in temperature.”
      Is this a fact or a law? Explain your position.

  (2) Give three different definitions of induction. Which one have you
      adopted? Defend your position.

  (3) Define and illustrate observation.

  (4) Distinguish between observation and attention.

  (5) “In observation we _find_, in experiment we _make_.” What is
      meant by this?

  (6) Give illustrations of falsehood due to careless observation.

  (7) Argue for and against the use of “expectant attention” in
      observation.

  (8) “Nine-tenths of what we see comes from within.” Do you believe
      this? Labor the question.

  (9) Offer suggestions which, if followed, should lead to scientific
      observation.

 (10) “One must be just before he is sympathetic.” Relate this to the
      fine art of accurate observation.

 (11) Is an hypothesis a generalization? Explain.

 (12) Give school room examples of hypotheses which lead to injustice.

 (13) “An hypothesis is merely a tentative induction.” Make clear this
      assertion.

 (14) Illustrate inconceivable hypotheses by drawing on your knowledge
      of ancient history.

 (15) “Prejudice and willful blindness to truth have ever been imminent
      stumbling blocks in the path of progress.” Expatiate upon this.

 (16) Are the hypotheses advanced concerning communications from the
      spiritual world capable of proof or disproof? Give reasons.

 (17) Show by historical examples the use of discarded hypotheses.

 (18) “Genius is another name for hard work.” Do you agree? Defend your
      position.

 (19) “The man to whom nothing ever occurs is unlikely to make any
      important discoveries.” Discuss this.



                              CHAPTER 20.

                       LOGIC IN THE CLASS ROOM.


=1. THOUGHT IS KING.=

“Our habits make or unmake us.” “In a thoughtless hour a groove is
imbedded in the nerve substance, and thereafter, nine-tenths of the
life flows through that groove.” Habit is, indeed, a most powerful and
a most tyrannical master; and yet it has come within the observation
and even the experience of many, that _thought is even more masterful
than habit_. Appearing at the psychological moment and in a pedagogical
way, a thought may be made to possess the mind with force sufficient to
break almost any habit. From an ethical point of view, the exceptions
to this are due to an inability to arouse thought of sufficient
strength. Moreover, mental reactions which result in habit are
originally brought about through some thought process. Speaking in
general terms, it may be affirmed that thought _makes habit_ and if
sufficiently strong _breaks habit_. That our habits make or unmake
us may be true, but is it not likewise true that our thoughts make or
unmake our habits?

Thought is king; thought has made man king of the animal kingdom, and
if thought has figured so largely in the evolution of the human animal
in past ages, may we not assume that it will sway the future ages in
like manner? Thought is a product of the class room. Here thoughts
which _make_ habits, and thoughts which _break_ habits have full sway.
As the children of the American schools think to-day, so will the men
of American life think on the morrow; and as America _thinks_ so will
she ultimately _do_. This lends vital import to any object which may
either inspire or regulate thought.


=2. SPECIAL FUNCTION OF INDUCTION AND DEDUCTION.=

As commonly treated logic is a _regulative_ subject. This implies the
two aspects of _direction_ and _correction_. Logic directs by means
of the _laws_ and _forms_ of thought, and corrects by means of the
_rules_ of right thinking. To a certain degree both departments of
logic are directive as well as corrective; but it is worthy of remark
that inductive logic emphasizes the former, while deductive logic
lays stress upon the latter. It is inductive logic which shows how
man has acquired _new_ knowledge; inductive logic explains the mode
of procedure adopted by the discoverer and the inventor. On the other
hand, deductive logic is distinctly a science of criticism. Induction
_directs_ to new truth; deduction aims to _modify_ and _correct_ new
truth.


=3. TWO TYPES OF MIND.=

Though there are many _special_ forms of thought, yet there are but
two _general_ forms; namely, induction and deduction. Inductive thought
seeks the new; deductive thought corrects the old. Similarly, there
are two types of mind: the inductive type and the deductive type.
The former _reaches out for new things_, the latter is satisfied with
_ordering the old_. In politics the man with the inductive type of
mind becomes a “Liberal” or a “Progressive”; while the man with the
deductive type of mind becomes a “Conservative” or a “Standpatter.” It
must be conceded that _both are needed in the development of the best
form of Democracy_. We need an _unfettered_ freedom as advocated by
Jefferson; but we also need an _ordered_ freedom as taught by Hamilton.


=4. TOO MUCH CONSERVATISM IN SCHOOL ROOM.=

Since the beginning these two mental types have been in evidence――the
liberal who wants to _do_ things, and the conservative who wants to
_weigh_ things. With the liberal, it is fight whether or no; with the
conservative, it is fight provided the enemy is not too formidable. The
one _dares_; the other _cautions_: both are needed to balance the world.

Liberalism and conservatism may be fostered in the school room, and
to maintain a true balance each must receive its share of attention.
Is such the case? The passing of “district-school-individualism” and
the coming of “graded-school-collectism” has transferred the emphasis
from liberalism to conservatism――from the inductive type to the
deductive type. In this day it seems to be more important to have
the child’s work _orderly_, than to have it _original_. In the main,
examination papers call for correct knowledge and not for thought;
in the main, promotions are based on _accuracy_, not on _initiative_.
The conservative type being in control, the schools are sending out
too many “Deductives,” not enough “Inductives.” The world needs more
Columbuses and Edisons.


=5. THE METHOD OF THE DISCOVERER.=

A change must come. The methods of instruction are too didactic and
not sufficiently inspirational. Greater attention must be given to the
spirit of discovery and less to the spirit of correction. _The teacher
must lead less and follow more; must correct less and suggest more;
must tell less and direct more._ If we are to give greater attention to
the training of discoverers, logic may aid in this crusade by calling
attention to the common mode of procedure which the discoverers of the
past have adopted. This is a legitimate topic for the logician, since
_induction_, _deduction_, _hypothesis_, and _proof_ have ever been
common tools in the discoverer’s workshop. With a view to becoming
better acquainted with the common mode of procedure of the man who
seeks for new truth, let us study two typical instances:

  (1) The Discovery of Neptune.

  The discovery of Neptune was a double one. Early in the present
  century it was found that Uranus was straying widely from his
  theoretic positions, and the cause of this deviation was for
  a long time unsuspected. Two astronomers, Adams in England and
  Leverrier in France, the former in 1843 and the latter in 1845,
  undertook to find out the cause of this perturbation, on the
  supposition of an undiscovered planet beyond Uranus. Adams
  reached his result first, and the English astronomers began
  to search for the suspected planet with their telescopes, by
  first making a careful map of all the stars in that part of the
  sky. But Leverrier, on reaching the conclusion of his search,
  sent his result to the Berlin observatory, where it chanced
  that an accurate map had just been formed of all the stars
  in the suspected region. On comparing this with the sky, the
  new planet, afterward called Neptune, was at once discovered,
  23d September, 1846.

  (2) Bees are guided in their flight by a knowledge of their
  surroundings, not by a general sense of direction.

  “M. Romanes took a score of bees in a box out to sea, where
  there could be no landmarks to guide the insects home. None of
  them returned home. Then he liberated a second lot of bees on
  the seashore and none of these returning, he liberated another
  lot on the lawn between the shore and the house. None of these
  returned, although the distance from the lawn to the hive was
  not more than two hundred yards. Lastly he liberated bees in
  different parts of the flower garden on either side of the house,
  and these at once returned to the hive.” (Hibben.)

A multiplication of instances would only give stronger evidence
to the fact that the mode of procedure adopted by the discoverer
and inventor conforms to these three general steps: (1) antecedent
facts, (2) forming an hypothesis, (3) verification. It will be to our
advantage to study somewhat in detail these three steps.

(1) Antecedent facts.

In the discovery of Neptune the _decisive_ or _crucial fact_ was the
knowledge that Uranus deviated from his true path about the sun. This
knowledge was obtained through observation and mathematical calculation.
But the hypothesis of the existence of another planet could not
have been formed had it not been for the _more fundamental facts_ of
inertia, gravitation, falling bodies, etc. For the sake of definiteness
antecedent facts may thus be divided into _foundation_ facts and
_crucial or decisive_ facts. The latter are an outgrowth of the former.
The _foundation_ fact of the second illustration is Romanes’ knowledge
of animal instinct; while the _crucial fact_ is, no doubt, the
observation that bees fly in a circle before starting for home. In
the case of Newton’s discovery of the law of gravitation, the falling
of the apple was the crucial fact; while his knowledge of terrestrial
gravity proved to be the vital foundation fact.

_A crucial fact is one which leads immediately to the formation of
a reasonable hypothesis._ It is not to be inferred from this that
the same fact becomes a crucial one to _all_ alike. The falling of
the apple was only crucial to a genius like Newton. With the average
only _extraordinary_ facts become crucial; but with the genius any
_ordinary_ fact may become crucial. Both the scholar and the genius may
have the foundation facts, but only the latter may be able to read into
a dry fact or event, a new world of truth.

(2) Forming an Hypothesis.

From the viewpoint of logical correctness, the matter of hypothesis
has received due attention in an antecedent chapter; we need now
to look at the subject through the eyes of the discoverer, not the
logician. The crucial fact at first creates an intellectual perplexity
which is accompanied with an uneasy, dissatisfied state of mind. This
unsatisfied feeling drives the intellect to protracted thought. As a
final result some hypothesis is constructed which seems to explain the
crucial fact. Here is where analogy functions in a most vital manner.
No hypothesis is forthcoming unless it resembles the crucial fact. It
has been remarked elsewhere that analogy is the basic element in the
forming of hypotheses. So it transpires, that the protracted thought
referred to, is virtually a mental effort to detect significant
resemblances between the well known crucial fact, and some hypothetical
fact which the imagination may picture. To put it differently: The
crucial fact arouses a mental state of unrest which in turn drives
the mind to a “still hunt” for _relations_. The establishment of the
hypothesis is simply a makeshift, designed to satisfy this “mental
urge.” In the discovery of Neptune the crucial fact, the deviation of
Uranus, produced a state of uneasiness in the minds of the astronomers.
Surely something was wrong. This urged them to further meditation,
which finally resulted in the hypothesis that there must be an unknown
planet beyond the orbit of Uranus. They assumed, of course, that the
relation between this supposed planet and Uranus was analogous to the
relation between any two of the known planets. In the case of Newton
the falling apple stirred his astute mind to the assumption that the
same force which pulled the apple, likewise pulled the moon towards the
earth. Here we have again (1) the crucial fact, (2) the mental urge,
(3) the analogous hypothesis.

(3) Verification.

Forming an hypothesis only partly fulfills the demands of an
unsatisfied intellect. The true discoverer, being possessed with a
passion for truth, seeks for “the truth, the whole truth, and nothing
but the truth.” In consequence the hypothesis is subjected to tests
which may lead to its confirmation, its rejection, or its modification.

The two possible modes of verification are recourse to experience
and appeal to reason; or _empirical_ proofs and _rational_ proofs. In
the former the hypothesis is compared with facts by means of further
observation and experiment. M. Romanes’ experience with the bees is
a fair illustration of this form. Possibly the student has already
noted that Romanes’ mode of procedure conforms to the “joint method
of agreement and difference.” In the case of rational proofs the
hypothesis is subjected to deductive demonstration, either of the
form of syllogistic argument or mathematical calculation. A fair
sample of this kind of verification is Newton’s discovery of universal
gravitation. When he decided that the moon and the apple might be
controlled by the same universal force, he undertook to establish his
hypothesis by mathematical calculation. At first his figures seemed to
disprove his theory, but after a wait of ten years, new data relative
to the diameter of the earth, removed the apparent discrepancy.
In the case of the discovery of Neptune, the verification was both
rational and empirical. Mathematical (rational) calculation led to
the assumption that the new planet must be at such a point. With this
knowledge the observer was enabled to turn his telescope to the spot
indicated and there, true to the calculations, was Neptune (empirical).

To summarize: The method of the discoverer involves a knowledge
of certain fundamental facts; the observation of crucial facts; a
mental unrest; the constructing of an hypothesis through analogy; and
finally verification by either appeal to experience, or mathematical
demonstration.


=6. THE REAL INDUCTIVE METHOD OR DISCOVERER’S METHOD NOT IN VOGUE
    IN CLASS ROOM WORK.=

It has been remarked elsewhere that there are two general mind types,
the liberal and the conservative. Also that the natural method of
thought animating the former is inductive; while the natural method
of thought of the latter is deductive. The “liberal” is the apostle of
_new_ truth; the “conservative” an apostle of _safe_ truth. Both types
are needed; the one to balance the other. In consequence both methods
are of service in the class room; here each should be given its
proportionate place. That this condition does not obtain may not be
apparent, since much attention is being given to certain inductive
forms, such as “proceeding from the concrete to the abstract,” “from
the particular to the general,” “from the known to the related unknown,”
etc. Likewise the courses of study and the various text books, claim
to advocate the use of the inductive process. Seemingly these facts
point toward a very general observance of the inductive tenets. This
is true with one vital exception: Induction is the natural method of
the discoverer. With it he _acquires_ knowledge; but in the class room
induction is used to _impart_ knowledge. In life the discoverer takes
the initiative, thinks his own thoughts first hand; but in the school
room, above the kindergarten, the child is not allowed to take the
initiative, not even in his play. All is planned for him, all doled
out, not in the raw, but partially made over. The teacher must impart
a certain amount of knowledge in a given time, and consequently she
must “set the pace” in this race for _second hand facts_. To allow the
child to lead; to give him the benefit of his own individuality; to
permit him to use the God given spirit of discovery which clamors for
recognition; would be suicidal according to our present standards. If
the plan of the discoverer were followed, the course of study could not
be covered; children would fail of promotion; and criticism would be
forthcoming from both principal and parent.

_In the average class room of the day the inductive FORM prevails
but the SPIRIT is not in evidence._ Like a wolf in sheep’s clothing
induction has entered the class room to devour that primal force in the
child’s make-up, which has raised his race above his simian ancestors.
Our class room methods, being inductive in form but deductive in spirit,
may train the youngster to be a _camp follower_ but never a _leader_
in thought and action. The call of the day is for more initiative; for
more originality; for more individuality; for more enthusiasm. There
is too much form without the spirit; so much that bespeaks system and
refinement without those native impulses and native abilities which
mark one child from another. Like the books of a library our children
are classified and labeled, and when more come in the others are dusted
and placed on the next higher shelf. How many more centuries must we
wait before the schools will adopt, in spirit as well as in form, the
pedagogical principles of life? Will the time ever come when it may be
said _that all our leaders in thought and action are college graduates_?


=7. AS A METHOD OF INSTRUCTION DEDUCTION IS SUPERIOR TO INDUCTION.=

The inductive method is pre-eminently the method of the discoverer only
when it involves both the _form_, which he follows, and the _spirit_,
which he evinces. The so-called method of the school room is _inductive
in form_, as the procedure is from particular facts to general truths;
but _deductive in spirit_, as it is used to _impart_ knowledge. If
it were inductive in spirit, the child would be allowed to _acquire
knowledge entirely through his own initiative_. Deduction is the method
of _instruction_, whereas induction is the method of _discovery_. That
the child of the school is _instructed_ or better “deducted” and _not
generally allowed to discover_, is a situation so apparent that we need
not labor the point further.

Because the inductive process has been made a method of instruction,
it has been robbed of its chief advantage over deduction. Indeed, as
a method of instruction, deduction is really the superior method. It
requires less time, demands greater concentration, often arouses more
interest, and creates situations which are less involved.


=8. CONQUEST NOT KNOWLEDGE THE DESIDERATUM.=

In all great inventions, man has taken his cue from nature. In
inventing the telescope, his model was the eye; in building his house,
his inspiration was the cave. In reality man has accepted nature’s
suggestions, and then attempted to improve upon them. In this he
has met with success. From the crab apple tree, he has developed the
northern spy; from the _wild_ hen which laid 25 eggs a year, he has
evolved the modern hen which produces 225 eggs a year. Moreover, man
has attained his greatest successes by _enlarging_ upon the thoughts
of nature and not by _unmixed substitutions_. Burbank, through a
long process of years, has changed the color of a flower, but in
accomplishing this did he not use some hidden tendency of nature?
Burbank, with all his wisdom, cannot give a flower color unaided by
“Dame Nature.”

When man commenced to study nature’s mode of education, he saw that
fearful sacrifices were entailed, both in time and in energy as well
as in life itself; and so he evolved a _more economical way_ of leading
the child through the experiences of the race. In consequence, he has
developed the present splendid system of education.

In the evolution of all great institutions, there are in evidence
crucial weaknesses, and in the evolution of man’s educational system
it appears that he has erred in adopting nature’s _form of education_
without her _spirit of education_. In his anxiety to have the young
acquire as much as possible, man has overshot nature’s true purpose.
For example, the big word in man’s educational system is _knowledge_;
but the big word in nature’s educational system is _conquest_. Nature
gives man knowledge simply to reward him _for his effort_; but man
would give to his fellow the reward _without the effort_. According
to nature, the strongest men are those who _overcome_ most; according
to man, the strongest men are those who _know_ most. The common
educational principles, such as, “From the concrete to the abstract
and from the known to the related unknown,” etc., are interpreted
by man from the viewpoint of _knowledge_; whereas nature would teach
that these are a feasible way to develop _power――to grow manhood_.
It is seen that nature uses knowledge only as a means to an end, and
therefore when man uses knowledge as an end only, he is trying to
substitute a plan of his own for nature’s plan. _The best results can
be secured only when man co-operates with nature in developing, and at
the same time regulating, the spirit of conquest._


=9. MOTIVATION AS RELATED TO THE SPIRIT OF DISCOVERY.=

It has been remarked in this chapter, that the “crucial fact” serves to
stir the mind of the natural born discoverer to an activity raised to
the nth power of effectiveness. Naturally, the intent of such activity
is to _solve the mysteries_ which the crucial fact may suggest. This
passion of the mind to “know more about it” is appropriately termed
“the mental urge.” From the viewpoint of the pedagogue, the “mental
urge” is simply an intrinsic interest in the situation at hand; an
interest born of an innate or acquired passion to know the truth.

With the average child, the “mental urge” is strong only when the
situations appeal to some immediate need or vital experience. The
attempt to make the school work concrete and vital; to make it answer
the child’s natural curiosities and real necessities, is dignified
with the name “_motivation_.” It is obvious that this is a new term
for an old condition. To motivate the work, means to give to it an
attractiveness which _any_ situation might have for the true born
discoverer and inventor. _If we would use the discoverer’s method
successfully, we must learn the art of motivating the work._ This may
be accomplished by appealing to the play instincts, to the business
instincts, and to the vital interests of every day life.


=10. DISCOVERER’S METHOD OR THE REAL INDUCTIVE METHOD ADAPTED TO
     CLASS ROOM WORK.=

A revolt has already set in against this insatiate desire to teach
_knowledge_, rather than to teach the _child_. Many schools are
permitting a study of those topics which vitally concern every day
life. Less attention is being given to formal discipline, and more
attention to self activity. Gradually will the scheme of education
be directed toward fitting the school work to the child, rather
than fitting the child to the school work. When this new thought
in education is fully upon us, then will every device and method be
directed toward giving full scope to the _spirit of inquiry_, which
so completely possesses every normal child.

It now remains for us to indicate ways in which the spirit of inquiry,
or the “discoverer’s method,” may be adapted to school room work. In
the first illustration, we shall outline the topic as it is generally
given in the average school where attention is paid to development work.
This will then be followed by a second outline which may be suggestive
of the discoverer’s mode of procedure.

First illustration. _School Room Method._

I. Aim: To teach addition of business fractions.

II. Preparation: (Only type examples given).

       (1)            (2)             (3)
    3 bushels       3 parts   Rule: Only like numbers
  + 5 bushels     + 5 parts         can be added.
  ―――――――――――     ―――――――――
    8 bushels       8 parts

III. Presentation:

  (1) (2)  (3) (4)
    3 ninths      3/9    2/3 =   4/6    2/3 =   8/12
  + 5 ninths    + 5/9    1/6 = + 1/6    3/4 = + 9/12
  ――――――――――    ―――――    ―――――――――――    ――――――――――――
    8 ninths      8/9            5/6           17/12

IV. Summary:

  (1) Only like fractions can be added.

  (2) Change unlike fractions to like fractions.

  (3) Add the numerators, placing the sum over the common denominator.

V. Application:

Examples and problems involving similar and dissimilar fractions.

Before undertaking to illustrate the discoverer’s method, it may be
well to designate in order the evident steps as they would appear to
the pedagogue:

  (1) Motivate the topic to be presented.

  (2) Bring to mind appropriate “foundation facts.”

  (3) Make evident the “crucial fact.”

  (4) Lead to the forming of an hypothesis through analogy.

  (5) Afford ample opportunity to prove the hypothesis.

_Discoverer’s Method Adapted._

Lesson Plan.

I. Aim: (1) _By playing upon the curiosity or by exposing a vital need,
create a strong desire to know how to add business fractions._
(Motivate the topic.)

Curiosity: “We all know what a fraction is and we know, too, how to
change fractions to higher or lower terms.” “Now I wonder how many know
how to _add_ fractions, such as 2/5 and 1/5?” “Don’t you tell any one,
Mary, but just write your answer on a piece of paper and show it to
me.” (Mary’s answer shows that she has thought correctly, but figured
incorrectly. John, after having raised his hand, shows his answer to
the teacher.) “John has the right answer.” “That’s fine, but let us
keep the secret, John.” “I wonder how many others there are in this
class who will find the right way?” etc., _or_

Vital need: Discuss with the class the various occupations of life
and secure expressions of preference. Some may plan to be real estate
agents, others contractors or book keepers, etc. “George, you plan to
be a book keeper.” “Let us suppose that I have given you the job of
book keeper in my factory.” “Show that you are worth your wages by
adding these numbers: 124¾, 647⅔.” “What! can’t do it?” “Then I don’t
want you!” etc.

II. Preparation:

(2) Bring to the foreground the necessary _foundational knowledge_.
Suggestions:

    4 bushels       8 parts
  + 3 bushels     + 2 parts
  ―――――――――――     ―――――――――
    7 bushels      10 parts

III. Presentation:

(3) Make evident the _crucial fact_. Suggestions:

  Add     2 fifths      3 eighths       3/8
        + 1 fifth     + 1 eighth      + 1/8
        ――――――――――    ―――――――――――     ―――――
          3 fifths      4 eighths

(4) Without further suggestion, give the young discoverer suitable
opportunity for finding the sum of 3/8 and 1/8. In the act of
discovering, an implicit _hypothesis_ takes form in the mind through
analogous reasoning. This point marks the climax of the lesson――the
supreme moment, when the skill and tact of the teacher is assessed
to the limit. Just here the child must have a comfortable environment
where perfect concentration is possible. Nothing must be forced;
and there should be nothing suggestive of disgrace or shame, if the
youthful Columbus is unsuccessful. The first attempt should be without
books. If more help is needed, access to books may be given. If the
investigation is still without definite result, then _as a last resort_
the teacher may, in the presence of the child, _add fractions_, solving
_with deliberation_ example after example, until the child believes he
has discovered the process.

(5) Stimulate a desire to _verify the facts discovered_.

Suggestions leading to verification: Afford opportunity for
mathematical demonstration. Illustrations: The fractions 1/4 and 3/8
have been added in this way――

    1/4 =   2/8
    3/8 = + 3/8
            ―――
            5/8

Use is now made of the crucial fact, when the example assumes this
form――

    2 eighths
  + 3 eighths
  ―――――――――――
    5 eighths

_Or_ if the class has been trained in the use of the diagram the
following may be the form of proof:

          { ━━━━━━━━━━
    1/4   { ──────────
          { ━━━━━━━━━━
            ──────────
            ━━━━━━━━━━
          { ──────────
    3/8   { ━━━━━━━━━━
          { ──────────
          { ━━━━━━━━━━

Explanation from diagram. I see that 1/4 equals 2 parts and 3/8 equals
3 parts; the sum of 3 parts and 2 parts are 5 parts. But the name of
the part is eighths; hence the answer 5 parts may be written 5 eighths,
or 5/8. Thus the final form is

    2 parts
  + 3 parts
  ─────────
    5 parts = 5/8

Give opportunity to consult answers in text books as further
verification.

The _summary_ and _application_ of adding fractions according to the
“discoverer’s method,” are virtually the same as the corresponding
steps in the “school room method.”

_Second Illustration of Discoverer’s Method._

David P. Page in his Theory and Practice of Teaching well illustrates
the discoverer’s method in conducting a general exercise in nature
study. We cannot do better than to quote from him:

  “It is the purpose of the following remarks to give a specimen
  of the manner of conducting exercises with reference to _waking
  up the mind_ in the school and also in the district. Let us
  suppose that the teacher has promised that on the next day,
  at ten minutes past ten o’clock, he shall request the whole
  school to give their attention five minutes to something that
  he may have to show them. This very announcement will excite an
  interest both in school and at home (playing upon the curiosity);
  and when the children come in the morning they will be more
  wakeful than usual till the fixed time arrives. At the precise
  time, the teacher gives the signal agreed upon, and all the
  pupils drop their studies and sit erect. When there is perfect
  silence and strict attention by all, he takes from his pocket
  an ear of corn and in silence holds it up before the school.
  The children smile, for it is a familiar object (foundational
  knowledge already in hand); and they probably did not suspect
  they were to be fed with corn.”

  Teacher. “Now, children,” addressing himself to the youngest,
  “I am going to ask you only one question about this ear of corn.
  If you can answer it, I shall be very glad. As soon as I ask the
  question, those who are under seven years old, and think they
  can give an answer, may raise their hand. _What is this ear of
  corn for?_”

  Several of the children raise their hands, and the teacher
  points to one after another in order, and they rise and give
  their answers.

  Mary. It is to feed the geese with.

  John. Yes, and the hens, too, and the pigs.

  Sarah. My father gives corn to the cows.

  Laura. It is good to eat. They shell it from the cobs and send
  it to the mill, and it is ground into meal. They make bread of
  the meal and we eat it.

  “I am sorry to tell you that none of you have mentioned the use
  I was thinking of, though, I confess, I expected it every minute.
  I shall now put the ear of corn in my desk, and no one of you
  must speak to me about it till to-morrow. You may now take your
  studies.”

  The consequence of this would be that various families, father,
  mother and older brothers and sisters, would resolve themselves
  into a committee of the whole on the ear of corn: and by the
  next morning several children would have something further to
  communicate on the subject. The hour would this day be awaited
  with great interest and the first signal would produce perfect
  silence.

  The teacher now takes the ear of corn from the desk and displays
  it before the school; and quite a number of hands are instantly
  raised as if eager to be the first to tell what other use they
  have discovered for it.

  The teacher now says pleasantly, “The use I am thinking of you
  have all observed, I have no doubt; it is a very important use,
  indeed; but as it is a little out of the common course (crucial
  facts) I shall not be surprised if you cannot give it. However,
  you may try.”

  “It is good to boil,” says little Susan, almost springing from
  the floor as she speaks. “And it is for squirrels to eat,” says
  little Samuel. “I saw one carry away a whole mouthful yesterday
  from the cornfield.”

  Others still mention other uses. Perhaps, however, none
  will name the one the teacher has in his own mind; he should
  cordially welcome the answer if perchance it is given.
  (Supposing that it has not been given.) “I have told you
  that the answer I was thinking of was a very simple one; it
  is something you have all observed and you may be a little
  disappointed when I tell you. The use I have been thinking of
  for the ear of corn is this: _It is to plant._ _It is for seed_,
  to propagate that species of plant called corn.” (Verification.)
  Here the children may look disappointed as much as to say, We
  knew that before. The teacher continues: “And this is a very
  important use for the corn; for if for one year none should be
  planted, and all the ears that grew the year before should be
  consumed, we should have no more corn. The other uses you have
  named were merely secondary. But I mean to make something more
  of my ear of corn. My next question is: _Do other plants have
  seed?_” Here is a new field of inquiry, etc., etc.

From the standpoint of “the greatest amount of knowledge in the
shortest possible time,” this mode of presentation consumes an
inexcusable amount of time and is, therefore, “impracticable.” But
when viewed from the ground of interest, originality, initiative, and
conquest――the watchwords of the “new thought in education”; there is
no real waste in either time or energy. _The spirit and method of the
discoverer will no doubt be the educational slogan of the future age._

Epitome of Discoverer’s Method, adapted to the class room:

  (1) Motivate the topic to be presented.

  (2) Bring to mind, if necessary, the “foundational facts.”

  (3) Make evident the “crucial fact.”

  (4) Furnish every opportunity for a first-hand discovery of the
      “lesson-point” (establishing hypothesis through analogy).

  (5) Let the hypothesis be verified.

_The entire situation must be one of freedom, zeal, originality, and
initiative._


=11. THE QUESTION AND ANSWER METHOD NOT NECESSARILY ONE OF DISCOVERY.=

No one mode of presentation is more universally used than the “question
and answer.” The advantages of this mode are many and the teacher who
is an adept in the art of questioning, from the standpoint of knowledge,
is generally efficient. The common error, however, incident to much
questioning, is that of asking “telling questions.” By the use of such,
the class is forced along the desired channel of thought so rigorously
as to have a condition _where the spirit of inquiry is entirely
wanting_. It is possible to conform to the rules of good questioning,
and yet rob the class of all originality and initiative. From the
teacher’s viewpoint, the discoverer’s method demands few questions;
it is the method of _suggestion_ rather than one of questions.
Avowedly in this method, the children should ask and answer their own
questions. Viewed from the ground of discovery there are three modes
of presentation which may represent a progressive series. These are
(1) the lecture mode, (2) the question and answer mode, (3) the mode
by suggestion. In the first there is _little_ of the spirit of the
discoverer; in the second there is a trifle more of the spirit of
the discoverer; while in the third there is _much_ of this spirit.
The student is advised to select some class room topic with a view
to illustrating these three modes of presentation.


=12. OUTLINE.=

  LOGIC IN THE CLASS ROOM.

  (1) Thought is king.

  (2) Special functions of induction and deduction.

  (3) Two types of mind.
        Inductive or liberal.
        Deductive or conservative.

  (4) Too much conservatism in school.

  (5) The method of the discoverer.
        Three steps

        1. Antecedent     { 1. Foundational
            facts         { 2. Crucial

        2. Forming        { 1. “Mental urge”
            hypothesis    { 2. Analogy

        3. Verification   { 1. Empirical
                          { 2. Rational

  (6) The real inductive method or Discoverer’s Method not in vogue in
      class room work.

  (7) As a method of instruction, deduction is superior to induction.

  (8) Conquest, not knowledge, the desideratum.

  (9) Motivation as related to the spirit of discovery.

 (10) Discoverer’s method or the real inductive method adapted to
      class room work.
        School room method.
        Discoverer’s method.
        Epitome.

 (11) Question and answer method not necessarily one of discovery.


=13. SUMMARY.=

(1) Thought is king in that it is the ruling factor in the making and
breaking of habit. This lends import to logic, which is the science of
thought.

(2) The chief function of induction is to discover new truth; whereas
deduction aims at clarifying and correcting new truth. Inductive logic
makes known the special forms of thought which the discoverer uses;
while deductive logic tends to show how he verifies the truth thus
obtained.

(3) Just as there are two general forms of thinking, inductive and
deductive; so there are two general types of mind, the inductive
and the deductive; the former leads to liberalism, the latter to
conservatism. Both types are needed to maintain a safe balance.

(4) The schools of the day are emphasizing the deductive phase of
work to the sacrifice of the inductive. They are neglecting the
Columbuses and the Edisons of the class. The course of study makes for
a conservatism, which “nips in the bud” any marked tendency to discover
and invent.

(5) Logic may aid in the crusade against the ultra conservative
tendency of class method, by giving emphasis to the method of the
discoverer and inventor. An analysis of this method reveals these
three steps: antecedent facts, forming an hypothesis and verification.
Antecedent facts may be divided into foundational and crucial.
A crucial fact leads immediately to the formation of the hypothesis;
whereas the foundational facts represent that body of knowledge
which makes it possible to interpret the crucial fact. The crucial
fact creates an unsatisfied state of mind, which, in turn, urges the
discoverer to construct some satisfactory hypothesis. Inference by
analogy is the process used in such a construction. The two modes of
verification are recourse to experience, or empirical; and appeal to
reason, or rational.

(6) In the class room, induction is used in form, not in spirit; in
consequence we are neglecting the generals for the camp followers.

(7) The inductive method is logically the method of discovery, while
the deductive method is the method of instruction. In the class room,
both methods have been devoted to the matter of instruction. Because of
this, induction has been robbed of its chief advantage over deduction.

(8) Man has attained his greatest success by _enlarging_ upon the
thoughts of nature and not by an _absolute substitution_. In enlarging
upon nature’s way of educating the child, man has adopted her form of
procedure, but has lost her spirit of work. In his scheme of education,
man’s watchword is _knowledge_, while nature’s is _conquest_. To
seek knowledge without inspiring the spirit of conquest is man’s way;
whereas nature’s way is to encourage the spirit of conquest by using
knowledge as a reward. Man must co-operate with nature, if the best
results are to be secured.

(9) In the case of the true discoverer, it is not necessary to endow
the object of his thought with added attractiveness; but with the child
enthusiasm may need to be stimulated by “motivating” the subject in
hand. This may be accomplished by appealing directly to the vital needs,
worldly necessities, and innate cravings of the child mind.

(10) A revolt is in evidence against that insatiate desire to teach
knowledge, which has been so marked in the past. Already schools are
introducing departments of work which look toward _conquest_ rather
than _knowledge_.

When adapted to the school room the discoverer’s method naturally
resolves itself into these five steps:

  (1) “Motivate” the topic for presentation.

  (2) Bring to mind “foundational facts.”

  (3) Vividly make evident the “crucial fact.”

  (4) Lead to discovery of “lesson-point.”

  (5) Afford opportunity for verification.

(11) The question and answer method of presenting work, does not
necessarily give full scope to the spirit of inquiry as emulated by the
true born discoverer.

As a matter of affording opportunity for the development of the spirit
of discovery, there are three modes of presentation which may be
arranged in a progressive series:

  (1) The lecture mode in which there is little opportunity for
      discovery.

  (2) The question and answer mode which permits some opportunity for
      discovery.

  (3) The mode by suggestion which permits ample opportunity for
      discovery.


=14. REVIEW QUESTIONS.=

  (1) Show that thought may be made to make and break habit.

  (2) “Induction directs to new truth, deduction aims to modify and
      correct new truth.” Explain and illustrate this.

  (3) Relate radicalism and conservatism to induction and deduction.

  (4) Show that in the present day school situations, the spirit of
      deduction prevails.

  (5) Describe a discovery which is a typical illustration of the
      _discoverer’s method_.

  (6) Indicate with explanation the general steps in the discoverer’s
      method.

  (7) Show by illustration the difference between “foundational facts”
      and “crucial facts.”

  (8) Explain how the “crucial fact” leads to the construction of an
      hypothesis.

  (9) Explain and illustrate the two ways of verification.

 (10) Distinguish between the inductive method as it is used in the
      class room, and the inductive method as used by the discoverer.

 (11) Show that in his inventions, man enlarges upon the thoughts of
      nature.

 (12) Explain “motivation” and show that it is a new name for an old
      situation.

 (13) In adapting the discoverer’s method to class work, what are the
      successive steps to be followed?

 (14) Show by illustration that the question and answer method is not
      necessarily one which encourages the spirit of discovery.


=15. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) “Our pet thoughts control us.” Discuss this.

  (2) Select some class room experience for the purpose of showing that
      induction is especially _directive_ in nature, whereas deduction
      is more or less _corrective_ in nature.

  (3) “There are just two kinds of people in the world, the
      _Inductives_ and the _Deductives_.” Explain.

  (4) Are the schools sending out too many _Deductives_? Argue the
      question.

  (5) “It is the business of the teacher to teach himself out of the
      business.” Explain.

  (6) Look up the discovery of the laws of the pendulum, with a view
      of showing that the event well illustrates the fact of the three
      general steps in the discoverer’s method.

  (7) “With the average, only extraordinary facts become crucial; but
      with the genius any ordinary fact may become crucial.” Make this
      clear.

  (8) Explain “mental urge.” Illustrate.

  (9) Illustrate “empirical proof,” also “rational proof.”

 (10) Show by illustration that the inductive method as used in the
      class room, falls far short of being the method of the discoverer.

 (11) Indicate by citing historical examples, that conquest rather than
      knowledge makes for manhood.

 (12) Show how you would motivate a topic in geography.

 (13) Outline a plan for teaching some topic in nature according to the
      discoverer’s method.

 (14) Select a topic in arithmetic, for the purpose of giving a
      comparative illustration of the “question and answer mode” of
      presentation, and the “mode by suggestion.”



                              CHAPTER 21.

                            LOGIC AND LIFE.


=1. LOGIC GIVEN A PLACE IN A SECONDARY COURSE.=

“To prepare for complete living” seems to be the ultimate aim of
education, and any school subject which does not aid to this end must
be eliminated from the courses of study. “Knowledge for the sake of
knowledge” will not do in this age of practical efficiency. _A subject
in order to survive must show indications of doing its share in this
larger business of man building._ If it can be made evident that
logic lends itself in no undecided terms to such an aim, then may its
incorporation in a secondary course of study be not only justified but
more highly appreciated.


=2. MAN’S SUPREMACY DUE TO POWER OF THOUGHT.=

That man is the supreme agent of intelligent progress is due to three
factors: First, to the existence of the natural world; second, to the
existence of man himself; third, to man’s ability to think. Given life
and the world as a place to evolve that life, and it is barely possible
that man might have _survived_, but without _thought_ he could never
have become _supreme_. _Man is king of the animal kingdom because of
his power of thought._ Let us illustrate:

Ages ago when England was a part of the main land; when there was no
North Sea nor English Channel; we are told that there lived in the
forest tracts there about many large and ferocious animals; such as
the elephant, the lion, and the tiger. There lived also in the region a
_smaller_ and apparently a _weaker_ animal. This creature had no tusks
to hook with, no great jaws to crunch with, and no claws to tear with;
and an eye witness would have said “Such a weakling has no possible
chance against these enemies of his; he and his descendants will
succumb and the species will become extinct.” The region was tropical;
but, of a sudden, a cataclysmic twist changed the temperature from
a torrid to a frigid state. What happened? The large and ferocious
animals either migrated to the south or froze to death; _but this
weakling put on furs, built fires, and remained in the jungle as its
king_. His name was man, and though he had no horns to hook with, he
possessed a _brain to think with_; this gave him supremacy over the
forces of nature.

From the beginning the adaptation of the lower animals has been
physical; whereas man’s has been more or less intellectual. By means
of deliberative thought man made the bow and arrow which could kill
at a distance of 200 yards; then he invented the repeating rifle which
may kill a mile away. _Thought_ has taught man to harness the forces
of nature in the form of all kinds of invention. _Thought_ has given
man the power to build bridges and palaces, to paint pictures, to
chisel angels. _Thought_ has pierced the fog of ignorance and brought
light to the dark spots of the globe. _Thought_ has build nations and
established the spirit of good will on earth. Through the long years,
_thought_ has been the _one tool of conquest_ which has enabled man
to build for himself, out of the furnishings of nature, _a heaven on
earth_.

Can you recall a department of life which thought has not embellished?
Can you recall a single factor that has been raised to the nth power
of efficiency without thought? Steam and electricity _plus thought_
lights the world, unites the world, feeds and clothes the world. To-day,
as in the olden time, men who think are ever at a premium. This holds
true from the Shopkeeper to the Magnate of Wall Street; from Basil,
the Blacksmith, to Edison, the King Inventor; from Reuben, the Farmer,
to Burbank, the Wizard.


=3. IMPORTANCE OF PROGRESSIVE THOUGHT.=

Man not only thinks but he thinks _progressively_. The _average horse_
of to-day, for example, is probably no more intelligent than was the
average equine of the time of Alexander the Great, whose war horse,
Bucephalus, attained historical fame. Yet, intellectually, the _average
man_ of to-day is _far_ above the average man of Alexander’s time.
“Horse-knowledge” is more or less stationary. Through instinct
each generation makes use of the knowledge of its ancestors without
any noticeable accretions. But “man-knowledge” is a growing product
of progressive thought. Man appropriates all the knowledge of his
forbears, and then adds to this a bit of his own. By being able to
think progressively, man is enabled to _stand upon the shoulders of
his ancestors_ and thus to take advantage of a broader vision.

We are now led to the conclusion that man’s supremacy is due not only
to his ability to think, but to his power of _progressive_ thought.


=4. NECESSITY OF RIGHT THINKING.=

In the main, man’s thinking has been for his good, that is, in the
long run, it has contributed to his general progress. If this had not
been so, long since would he have dropped back to the level of the
non-thinking animals.

Thinking has been defined as the process of affirming or denying
connections. Right thinking is, therefore, a matter of affirming
the right connections or denying the wrong connections. To put it
differently: right thinking is the process of adjusting the best means
to a right end; whereas wrong thinking is a matter of overlooking the
best means, or directing improper means to a wrong end. Right thinking
involves _proper adjustment_; wrong thinking _improper adjustment_. In
the intellectual world as in the physical, _improper adjustment_ means
extinction. Illustrations of this:

(1) A contractor undertakes to build a skyscraper. In the excavation
an old wall is discovered. The thought of the contractor is, “I must
make a pot of money out of this job, and since this old wall is in the
right spot I will build on it, and thus save me ‘five hundred’.” In
the course of ten years, without warning, the building topples over and
fifty women and children are killed. The contractor is convicted and
sent to prison for life. If the builder had thought the right thought;
namely, “I want to put up a building that will stand for generations,”
he would have survived the competition of his fellows and entered his
long home with success etched upon his soul.

(2) Two school teachers, A and B, are working in the same system. A’s
ambition is to be promoted and she uses “pull” as the means. For a time
she succeeds in pulling the wires, and likewise in pulling the “wool”
over the eyes of the Board of Education. B aspires to professional
growth, using as the means every opportunity for genuine improvement.
In time both are known as they really are, not as they seem to be. A
is denominated a “shirk,” a politician, a _mere school keeper_; whereas
B is looked up to as the best equipped worker in the building, a _real
school teacher_.

There may seem to be many exceptions to this point of view, and yet
in the last analysis we find that these exceptions are only apparent.
When we maintain that right thinking means survival and wrong thinking
extinction, we assume that the standard adopted is _genuine efficiency_
and not a _certain money basis_. High positions may be _secured_
through wrong thinking, but these cannot be _filled creditably_ without
the preponderance of right thinking.


=5. INDIFFERENT AND CARELESS THOUGHT.=

It may be advanced as a plausible hypothesis that man, especially if he
is an American, finds much trouble for himself, and makes much trouble
for the world because of his indifference to thought. To leap first and
look afterwards is the spirit of youth, and America is young. _Think
twice before you look and look twice before you leap_ is sound logical
doctrine. A logically minded man _rationalizes_ every new proposition
before he adopts it. He marshals before the mind the favorable points
and then bombards them with every conceivable objection. With the
steady eye of an honest, earnest, open minded critic, he weighs the
unfavorable against the favorable and then accepts the indications of
the balance unequivocally. If logic did nothing else save to inspire
young people to thus rationalize every doubtful undertaking, it would
do its share toward world betterment.


=6. THE RATIONALIZATION OF THE WORLD OF CHANCE.=

Man seems to be a natural born gambler. He loves to “take a chance”
and herein lies much of his unhappiness. Without discussing the
evils of the stock exchange, the horrors of the gambling den, and the
unbusiness like procedure of the race track, we may merely attempt here
to show how _the rationalization of the conception of chance_ may be
instrumental in dimming the glare of gambling to the average youth.

(1) The meaning of the term chance.

The term chance implies an inability to find a cause for any
particular event. Whenever we trust to luck, we do so through ignorance.
In reality every thing in this world is ordered _according to law_,
and if we possessed infinite knowledge concerning these laws, then,
for us, the word “chance” would have no meaning. One accomplishment
of knowledge has been to rationalize superstition and chance. “Not a
grain of sand lies upon the beach, but infinite knowledge would account
for its lying there; and the cause of every falling leaf is guided by
the same principles of mechanics as rule the motions of the heavenly
bodies.”――Jevon’s Prin. of Science, vol. I, p. 225.

That chance is a literal confession of ignorance, is a wholesome truth
for all to bear in mind. If we were not so ignorant of atmospheric
conditions, we would never be caught in the rain without an umbrella;
if we knew perfectly the laws of mechanics, we would not speed our car
and trust to luck that the car would hold together.

(2) Chance mathematically considered.

The principle of the “calculation of chances” has been discussed
elsewhere. It will be sufficient here to illustrate the principle from
a mathematical point of view.

Suppose that a jeweller desires to dispose of a ten-dollar watch
by a raffle. He may place a hundred numbers in a box, one of which
corresponds to the number on the watch. My chance of drawing the right
number is one out of a hundred and may be expressed by the fraction
1/100. The fact that I may draw the right number on the first trial or
on the last trial is immaterial. The real meaning of the ratio “one out
of a hundred” is, that in the long run, I shall _lose 99 times_ where
I gain but once. This implies, that if I pay 25 cents for each draw,
I shall in the end pay 99 times 25 cents for the watch, or I will have
paid $24.75 for a ten dollar watch.

(3) Chance and gambling.

In all forms of gambling no wealth is produced. What one man gains the
other man loses. In addition to this the institution which projects the
gambling scheme must be supported. In consequence, _more money must be
lost than can possibly be gained_. This leads to the conclusion that
on the basis of averages he who would gamble must terminate his career
“behind the game.” Statistics verify this conclusion.

(4) Chance and investments.

Interest, which is money paid for the use of money, is high when the
demand for money exceeds the supply and low when the supply equals or
exceeds the demand. The fact that the supply is short is largely due
to the lack of confidence on the part of the investor. This means that
he is unwilling to take the risk. Thus the principle: “_High rate of
interest, great risk; low rate of interest, little risk._”


=7. THE RATIONALIZATION OF POLITICAL AND BUSINESS SOPHISTRIES.=

“Win right or wrong” is a nut shell statement of modern sophistry.
Corollaries to this are such aphorisms as “Of two evils choose the
lesser”; “Do evil that good may come,” etc. Armed with these platitudes
the modern business and political octopus will play the bully and
squeeze the life out of the little fellow in the name of economy;
will pay for editorials to elect the “right man”; will evade bad laws
so-called; institute lobbies; buy votes; and perpetrate a thousand
other immoral deeds in the name of “good business” or of “party
loyalty.”

Half truths are the most atrocious of all kinds of fallacies in that
they are the most misleading. “Do evil that good may come” is but half
of the whole truth “Do evil that good may come, _provided there is
no other way open_.” Again, “Of two evils choose the lesser, _if a
complete enumeration has shown that there is not a third course_.”

A development of a finer ability of discernment under right influence
should lead the common citizen to see _through_ these various
sophistries practiced by corporate greed, and should enable him by
means of the ballot to “blaze a better way.”

The “public is a blunderbuss” because the average man either cannot,
or will not, think his own thoughts. By developing greater skill and
arousing greater interest in the _thinking process_, the crowd of camp
followers will be reduced; selfish bossism will die; and a truer and
more efficient democracy will reign supreme.


=8. THE RATIONALIZATION OF THE SPIRIT OF PROGRESS.=

Genuine progress comes through a happy combination of the old with the
new. A love for the old only, means _ultra conservatism_; whereas a
love for the new only, means _ultra radicalism_; a love for both means
_rational liberalism_.

That people love the old way may be attributed to two forces which will
receive attention here.

(1) Race instinct.

It may be said that “life is a brief space between two eternities――a
path between infinity and infinitude.” “Man is a pedestrian who
perambulates along the way.” The eternities concern him not so much as
the _path_ which stretches between them. In a former day, one of the
striking characteristics of the western plain was the _beaten path_
stretching out along the table-land like an elongated, dust colored
serpent; and often following this path would be a herd of buffalo
winding its way in single file around boulder and ant hill till shut
from view by the distant horizon. Thus has man travelled along the
beaten path, following the “foot prints of the ages.” Here and there
and everywhere do we see signs of those who have gone on before; father,
grandfather, great grandfather; yes, even to the toe marks of those
primeval ancestors of ours who shambled along the way, nobody knows how
many years ago. From the dark recesses of the cave, have our forbears
thrown a lasso of blood about our necks, and it seems as if we _must_
follow the old, old way. “Being acorns of the ancestral oak,” we
grow similar oak tree tendencies, living over again the life of our
progenitors. “There lies in every soul the history of the universe.”

(2) Imitation.

But there is another reason for this ultra conservative spirit and it
is that nature’s chief mode of instruction is by means of _imitation_.
To every living thing of wood or field nature seems to say, “Your
parents are always right, do as they do for this is the best way
to learn the lessons of life.” A man thinks, feels and wills his
way through life in a certain manner largely because his father did
likewise. Moreover, we not only imitate those who have gone on before,
but we counterfeit each other; fashion is another name for world wide
mimicry. We imitate our friends and those whom we admire; we talk like
them, we walk like them, we live like them.

It now appears that we are held to the path of the past by means of
_race instinct_ and the _power of imitation_, and we are thus prone
to believe that the old way is good enough. It is evident that to get
out of the beaten path is dangerous. The wild animal that deserts the
habits of the race dies a premature death, and the man who possesses
the temerity to struggle through the thicket of new things must, of
necessity, shorten his span of life. To follow the “same old rut”
is easiest for the teacher; to be loyal to the “grand old party” is
safest for the politician. But to the contrary, if every man of every
generation had followed the beaten path blindly――without deviation, the
human race would now be a horde of simians. Because man has possessed
the power of progressive thought, he has developed the spirit of
radicalism and has thereby made himself supreme.

“The old way anyway――the old way right or wrong” has been the world’s
biggest stumbling block. Every innovation must fight for its life.
Every good thing has to be condemned in its day and generation. It is
Huxley who suggests three stages for the course of a new idea: First,
it is revolutionary; second, it will make little difference; third, _I
have always believed in it_. On the other hand, the new way anyway; “we
must have a change whether or no”; “we must have something different
despite the cost,” have ever been the slogans of waste and destitution.
The wars which have not resulted from the prejudice of ultra
conservatism have been brought about through the thoughtlessness of
ultra radicalism. The revolutionist, the freak and the anarchist,
products of impulse and the spirit of discontent, spring from an unwise
love of change.

The world needs conservatism and radicalism not so much as it needs
_rationalism_. It needs men who can hold to the _good of the old_ and
adopt the _best of the new_; men who neither “rust out” nor “waste
out”; but _wear out_. That rational progress may obtain, there must
be a perfect dovetailing of the old with the new. Man must leave the
beaten path not altogether, but at times. He needs to blaze out a _new_
way not so much as he needs to straighten the bends, tunnel through
the mountains, and fill in the swamps of the _old_ way. _A rational
“liberalism” implies a willingness to follow the old path with a view
to improving the imperfections thereof._


=9. A RATIONALIZATION OF THE ATTITUDE TOWARD WORK.=

On the assumption that true happiness is the ultimate aim of life,
we may conclude that anything which does not contribute to this
end functions as a curse and not as a blessing. Happiness involves
physical comfort and mental joy. To have comfort of the body implies
moderate means. The poor cannot be happy because of bodily want. When
“physical-man” is not given proper nourishment for healthy growth,
then does he _goad_ “spiritual-man” with the pricks of appetite and
pain till his wants are appeased. This is a law of nature. On the
other hand happiness is not attained through acquisition; neither
the millionaires, nor the scholars, nor the famous are the happiest.
This is a fact apparent to all. Over worry and over excitement follow
closely the heels of much money and high position. Too little brings
unhappiness through want; too much brings unhappiness through worry.
Therefore man is _cursed by his work_ when the remuneration is not
enough for comfort of body, or when the income is too much for poise
of mind.

Unless the organs of the body are used they atrophy. Every cell of
the physical makeup demands exercise. Work which is not drudgery;
work which causes the organs of the body and the powers of the mind to
function normally; work which gives comfort without luxury; work which
forces one to the highest actualization of his physical and spiritual
powers is _man’s greatest blessing_. In and through such work will man
attain his highest state of happiness.


=10. THE LOGIC OF SUCCESS.=

We may now hope to show that material aggrandizement, the adopted
standard of success, is one of the illogical factors of modern life.

The tree of the forest always grows toward the light. It pushes its way
through the darkness of the soil into the shadow of the underbrush and
finally out into the unobstructed light of the sun. This parallels the
progress of the race. From the darkness of _savagery_ into the shadow
of _barbarism_, and finally out into the full light of _civilization_.
Thus has man grown steadily and continually toward better things. But
“better things” is a relative term and has changed with the development
of the race. “A good healthy idea may not live longer than twenty
years.” In consequence growth toward the light has been in accordance
with man’s _conception_ of a higher and a better life; which conception
is ever changing.

Moreover, growth toward the best is always rewarded by real happiness.
It therefore follows that the _right road to real happiness_ extends
along the way of _better things_ as conceived by the traveller, _man_.

Any force which tends to lift the world up toward more light is a
blessing, and any personality which contributes to this end is a
success. When one drops a pin it falls _down_ toward the earth, at
the same time the earth comes _up_ to meet the pin. This is according
to the universal law of gravitation. It is true that the earth moves
the pin through a much greater space than the pin moves the earth, and
yet the fact remains that the pin _does move the earth_. The extent to
which the smaller body is able to move the larger, depends on the two
factors of _weight_ and _relative position_. If the pin were lighter
or farther away it would influence the earth so much the less. In like
manner does the “pin-man” influence the “human-world.” The extent of
this influence is controlled by man’s _weight_, or his “lifting power,”
and the _position_ which he occupies; just as the attraction of the pin
for the earth is controlled by _weight_ and _position_.

The facts of history have proved that man’s power to lift depends not
so much upon what he _has_ as upon what he _is_. In short, lifting
power is directly in proportion to _personal worth_. Moreover, man’s
ability to draw humanity up may be increased or decreased by the
position which he occupies. Such a position must function for the
best good of the world, and at the same time must elicit the highest
development of the man.


  TO SUMMARIZE:

Individual success involves these three elements:

First――_A man of personal worth._

Second――_A position which draws out the best in the man._

Third――_A work which definitely contributes to the uplift of the world._

A definition is now in order:

_Success is the right man in the right place doing his best for the
highest good of the world._


=11. OUTLINE.=

  LOGIC AND LIFE.

  (1) Logic given a place in a secondary course.

  (2) Man’s supremacy due to power of thought.

  (3) Importance of progressive thinking.

  (4) Necessity of right thinking.

  (5) Indifferent and careless thought.

  (6) The rationalization of the world of chance.
        (1) Meaning of the term chance.
        (2) Chance mathematically considered.
        (3) Chance and gambling.
        (4) Chance and investments.

  (7) The rationalization of business and political sophistries.

  (8) The rationalization of the spirit of progress.

  (9) A rationalization of the attitude toward work.

 (10) The logic of success.


=12. SUMMARY.=

(1) To justify its having a place in any course of study, logic must
lend itself to character building.

(2) Man is king of the animal kingdom because of his power of thought.
From the beginning his adaption has been more or less intellectual and
his chief weapon of conquest has ever been his thinking brain.

(3) Man’s supremacy has been due not only to his ability to think, but
also to his power of _progressive_ thought.

(4) Right thinking is the process of adjusting the best means to a
right end. Wrong thinking involves improper adjustment, which in turn
results in extinction.

(5) A “logically-minded” man _rationalizes_ every new proposition
before he adopts it. That is, he analyzes with the utmost care and with
unprejudiced frankness all the favorable and unfavorable situations;
he then throws them into the balance of honest judgment and adopts the
indications of said balance, unequivocally.

(6) Chance is a confession of ignorance. If man possessed infinite
knowledge, the term chance would have no place in his vocabulary.

The games of chance are money making schemes supported by the gambling
fraternity. On the basis of averages, the gambler, in the long run,
must terminate his career “behind the game.”

High rate of interest implies great risk; low rate of interest little
risk.

(7) “Win right or wrong” epitomizes the teachings of modern sophistry.
With the coming of better thinking, a more efficient democracy will
obtain.

(8) Rational progress combines the best of the old with what seems to
be the best of the new.

Blind love for the old, or ultra conservatism, is due to the two forces
of race instinct and power of imitation.

An adherence to the “old way anyway” may mean retrogression; whereas
following the new way, simply because of its newness, may involve
unnecessary waste.

(9) Work which is not drudgery; work which causes the organs of the
body and the powers of the mind to function normally; work which
gives comfort without luxury; work which forces one to the highest
actualization of his physical and spiritual powers is man’s greatest
blessing.

(10) Logically considered personal aggrandizement is not a true
standard of success. Success involves personal worth rather than
personal holding.

Success is measured by man’s ability to help the world on toward better
things.

Success is the right man in the right place doing his best for the
highest good of the world.


=13. REVIEW QUESTIONS.=

  (1) What is the ultimate aim of education? Show that logic
      contributes to this end.

  (2) Prove that man’s power of thought has ever been his best weapon
      of conquest.

  (3) Exemplify the distinction between non-progressive and progressive
      thinking.

  (4) Define right thinking. Illustrate.

  (5) “A logically-minded man rationalizes every new project before
      undertaking it.” Give a concrete instance in explanation of this.

  (6) “Chance is a literal confession of ignorance.” Demonstrate this.

  (7) Give a mathematical illustration proving that schemes of chance
      are simply money making devices for the benefit of those who
      project them.

  (8) The average gambler must terminate his career behind the game.
      Prove this.

  (9) Why should high rate of interest imply great risk?

 (10) Show that a half truth is a most misleading fallacy.

 (11) Illustrate a business sophistry. Explain.

 (12) Write a brief theme on “The Rationalization of the Spirit of
      Progress.”

 (13) Under what conditions may work become man’s greatest blessing?

 (14) Define success. Illustrate.

 (15) In the light of your definition of success discuss the following:
      “The only failure is not to try.”


=14. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

  (1) “To prepare for complete living” is the end of education.
      Interpret and discuss this quotation from Spencer.

  (2) Mention some discovery or invention which represents the power
      of progressive thought.

  (3) “Man’s adaptation has been largely intellectual while the
      adaptation of the camel has been physical.” Explain.

  (4) Interpret the expression, “The son stands upon the shoulders of
      the father.”

  (5) Illustrate instances where man’s thinking has not been for his
      best interests.

  (6) Indicate how wrong thinking led to the Civil War.

  (7) Distinguish between legitimate speculation and gambling.

  (8) Name and explain the logical elements involved in a low rate of
      interest.

  (9) How may training in right thinking lead to more efficient
      citizenship?

 (10) “There lies in every soul a history of the universe.” Show the
      truth of this.

 (11) Show by illustration that imitation is one of nature’s chief
      modes of instruction.

 (12) Explain the meaning of drudgery.

 (13) Mention instances where work is a curse.

 (14) Is success possible when the right man is found doing his best
      in the wrong place?

 (15) Whom do you consider the most successful American? Give reasons.



                   GENERAL EXERCISES IN TESTING THE
                  VALIDITY OF CATEGORICAL ARGUMENTS.


_Let the student give attention to the fallacies in meaning as well as
to the fallacies in form._

   1. None but those who are contented with their lot in life can
      justly be considered happy. But the truly wise man will always
      make himself contented with his lot in life, and, therefore he
      may justly be considered happy. Keynes.

   2. Suffering is a title to an excellent inheritance; for God
      chastens every son whom he receives. Keynes.

   3. No young man is wise; for only experience can give wisdom, and
      experience comes only with age. Keynes.

   4. Dr. Johnson remarked that “a man who sold a penknife was not
      necessarily an iron-monger.” Against what logical fallacy was
      this remark directed? Explain. Keynes.

   5. This pamphlet contains seditious doctrines, the spread of
      which may be dangerous to the state; hence the pamphlet must
      be suppressed. Keynes.

   6. Good workmen do not complain of their tools: my pupils do not
      complain of their tools; therefore, my pupils are probably good
      workmen. Keynes.

   7. Knowledge gives power; consequently, since power is desirable,
      knowledge is desirable. Keynes.

   8. Some who are truly wise are not learned; but the virtuous alone
      are truly wise; the learned, therefore, are not always virtuous.
      Keynes.

   9. The spread of education among the lower orders will make them
      unfit for their work; for it has always had that effect on those
      among them who happen to have acquired it in previous times.
      Keynes.

  10. Slavery is a natural institution and therefore ought not to be
      abolished. Russell.

  11. The yardstick of success is the dollar, and you have made your
      millions.

  12. “All who talk well are not necessarily intelligent, and A is
      certainly a spell-binder.”

  13. Gold and silver are the wealth of a country; consequently,
      the diminution of gold and silver by exportation must mean the
      diminution of the wealth of a country. Russell.

  14. A miracle is unbelievable, because it fails to conform to known
      laws of nature.

  15. Improbable events happen every day; now, what happens every day
      is a probable event; therefore, improbable events are probable
      events.

  16. What fallacy did Columbus commit when he made the egg stand on
      end by breaking one end?

  17. Some holder of a ticket is sure to draw the prize; and, as I am
      a ticket holder, I am sure to draw the prize. Russell.

  18. All the members of the jury are just men, hence you may trust the
      foreman.

  19. Select the star players of the country and you will have a team
      which cannot be beaten.

  20. All the houses on this street present a pretty picture; this
      house, therefore, which is on this street, will make a fine
      picture.

  21. What is the good of all your teaching, for every day we hear of
      wrong doing made possible by education.

  22. You are not what I am; I am a teacher; hence you are not a
      teacher.

  23. The student of history is compelled to admit the law of progress,
      for he finds that society has never stood still. Russell.

  24. This bill must have been designed to bleed the people because it
      is supported by the grafters of the state.

  25. “To close the saloons on Sunday is contrary to the wishes of the
      people of the city; hence those ‘farmer legislators’ should keep
      hands off.”

  26. Success is the right man in the right place doing his best, and
      you are working to the limit.

  27. Early to bed and early to rise, makes one healthy, wealthy and
      wise. It is, therefore, easy enough to get rich.

  28. Honesty being the best policy, I must tell the truth to my
      patient, though to tell him that he cannot live will shorten his
      life many days.

  29. A stitch in times saves nine, hence an ounce of prevention is
      worth a pound of cure.

  30. The richest man I know used to sweep his office every morning,
      hence it pays to commence at the bottom.

  31. Cramming is an injurious habit, since it makes the building of
      logical memories practically impossible.

  32. A strong will means a trained will; struggle is an indication of
      weakness.

  33. There is no such thing as a national or state conscience;
      therefore, no judgments can fall upon a sinful nation. Hibben.

  34. The principles of justice are variable; the appointments of
      nature are invariable; therefore, the principles of justice are
      no appointment of nature. Aristotle.

  35. Intelligence and not sex should be the standard; therefore, let
      the women have their way.

  36. “War by killing off the men of the country gives the living a
      better opportunity to succeed because of reduced competition.”

  37. Since you deem yourself a misfit, in the name of common sense,
      why do you not change your occupation?

  38. The conquest of America by Europeans has been a good thing for
      the world; since no eminent historian doubts it.



      GENERAL EXERCISES IN TESTING THE VALIDITY OF HYPOTHETICAL,
                 DISJUNCTIVE AND DILEMMATIC ARGUMENTS.


_The student must remember to give attention to the fallacies in
meaning as well as to the fallacies in form._

   1. If I speak at length, he is bored; if I speak briefly, he is
      offended; therefore I will not speak at all.

   2. If virtue is involuntary, vice is also involuntary; but vice is
      voluntary, hence virtue is also.

   3. If a man cannot make progress toward perfection, he must either
      be a brute or a divinity; but no man is either; therefore every
      man is capable of such progress. Fowler.

   4. If education is popular, compulsion is unnecessary; if unpopular,
      compulsion will not be tolerated. Fowler.

   5. If you are to recover from this illness, then you will. If you
      are not to recover, then you will not, hence what is the use of
      calling in a physician?

   6. If your act was right, your conscience will approve it; if wrong,
      your conscience will prick you. Either your act was right or
      wrong, so you can depend upon your conscience.

   7. If he is intoxicated then he is not responsible, but he acts like
      a sober man.

   8. If the Elixir of Life is of any value, those who take it will
      improve in health; now my friend who has been taking it has
      improved in health, and therefore the elixir is of value as a
      curative agent. Hyslop.

   9. If you will settle down to business, you may still win out,
      because I am confident it is not too late for hard work to be
      effective.

  10. If the end justifies the means then money used for any object of
      charity may be secured in any way.

  11. If might is right then money talks, but I find that occasionally
      money proves ineffective.

  12. If the majority of those who use public houses are prepared
      to close them, legislation is unnecessary, but if they are not
      prepared for such a measure, then to force it on them by outside
      pressure is both dangerous and unjust. Hyslop.


  13. If the conscience is infallible in matters of right and wrong,
      then sin is just one thing; namely, doing that which is contrary
      to one’s conscience. We believe that an educated conscience is
      infallible.

  14. If the earth were of equal density throughout, it would be about
      2½ times as dense as water; but it is about 5½ times as dense;
      therefore the earth must be of unequal density. Hyslop.

  15. The end of human life is either perfection or happiness; death
      is the end of human life, therefore death is either perfection
      or happiness. Creighton.

  16. That chauffeur either lost his head or was drunk because no sane
      man would deliberately run down an innocent child.

  17. If you argue on a subject which you do not understand, you will
      prove yourself a fool; for this is a mistake which fools always
      make. Keynes.

  18. If you are a man of your word, you will live up to your
      agreement, or if you have any self respect, you will do the manly
      thing. Now your neighbors tell me that you are a man in the habit
      of making good your promises.



                   SETS OF EXAMINATION QUESTIONS FOR
                    TRAINING SCHOOLS AND COLLEGES.


_Answer ten questions._                                 Time, 2 hours.

                                Set I.

   1. Define and illustrate obversion and state the principle which
      conditions the process.

   2. Give directions for making the following propositions logical:
        (1) Only first class passengers may ride in parlor cars.
        (2) All who claim to be pious are not pious.
        (3) “Blessed are the merciful.”

   3. Write a theme of 200 words on “Logic and Life.”

   4. Put into syllogistic form and test the validity of this argument.
      “We are going to have an open winter because the hornets’ nests
      are near the ground.”

   5. Justify the teaching of logic in an institution which offers
      courses in Educational Theory.

   6. Correct the following definitions, stating the rules violated:
        (1) A man is an organized entity whose cognitive powers
            function rationally.
        (2) A bird is an animal that flies.
        (3) A scholar is an educated man with scholarly attainments.

   7. Prove that in the first figure the minor premise must be
      affirmative.

   8. Investigate a case of habitual tardiness by making use of the
      canon of difference.

   9. Describe with illustrations the various ways of begging the
      question.

  10. Why should classification rather than logical division be the
      mode of procedure in the case of small children? Illustrate.

  11. Illustrate the following:
        (1) non connotative term,
        (2) undistributed middle,
        (3) fallacy of accident.


                                Set II.

_Answer ten questions._                                 Time, 2 hours.

Throw the following into the form of a syllogism and criticise, giving
reasons:

   1. “I do not know how to teach school as I have had no experience.”

   2. “Only the honest should be in business and you are not honest.”

   3. Why should all teachers study logic? Give arguments in full.

   4. Describe Mill’s methods of induction and illustrate one.

   5. Give and explain the rules of logical definition.

   6. Explain the distribution of terms and illustrate by circles the
      meaning of the four logical propositions.

   7. Define the following:
        (1) teaching,
        (2) extension of terms,
        (3) obversion,
        (4) hypothesis,
        (5) relative term.

   8. Give a class room illustration of the Complete Method.

   9. Distinguish between
        (1) distributive and collective terms,
        (2) analysis and deduction,
        (3) logical division and classification.

  10. Illustrate the following:
        (1) contradictory proposition,
        (2) analogy,
        (3) law of identity,
        (4) singular term,
        (5) univocal term.

  11. Convert, if possible, the following:
        (1) Some men are honest.
        (2) All that glitters is not gold.
        (3) All kings are fallible.


                               Set III.

_Answer ten questions._                                 Time, 2 hours.

   1. Investigate by the Joint Method of Induction this question: “Why
      is John absent so often?”

   2. Explain and illustrate:
        (1) contradictory propositions,
        (2) illicit middle,
        (3) obversion,
        (4) contraversion,
        (5) synthesis.

   3. State and exemplify the rules of logical division.

   4. Write a theme of at least 150 words on one of the following:
        (1) Induction as the Discoverer’s Method.
        (2) A Rational View of Success.

   5. Define logically:
        (1) teaching,
        (2) deduction,
        (3) education,
        (4) analysis,
        (5) money.

   6. Distinguish between the extension and intension of terms.

   7. Exemplify:
        (1) an absolute term,
        (2) the complete method,
        (3) non connotative terms,
        (4) fallacy of accident,
        (5) hypothesis.

   8. “Educated among savages, he could not be expected to know the
      customs of polite society.” Is this valid? Reasons.

   9. “The signs indicate that you are either stupid or unprepared; but
      the past proves that you are not the former.” Test the validity.

  10. Discuss comprehensively one of the following topics:
        (1) The Fallacies.
        (2) Thinking.
        (3) Abbreviated Arguments.


                                Set IV.

_Answer ten questions._                                 Time, 2 hours.

   1. Exemplify:
        (1) the law of variation in the extension and intension of
            terms,
        (2) a distributed predicate.

   2. Indicate with explanation the logical errors:
        (1) A teacher assumes that the “bad boy of the school” is going
            to cause trouble in her room.
        (2) All the men of the Commission are fair minded men, hence
            they will render a fair decision.

   3. What experimental method of induction is the most positive in its
      conclusion? Illustrate this method.

   4. State and illustrate the rules of logical definition.

   5. Obvert each of the four logical propositions. Explain the
      principle involved.

      Test the validity of the following arguments:

   6. “Horses, not being human, cannot reason.”

   7. “Only the industrious deserve to succeed and you have never done
      a hard day’s work in your life.”

   8. “If you had been wise, you would have refused to stoop to the
      methods of the firm, but you were not wise.”

   9. From this premise construct a valid syllogism: “All large cities
      owe their size to some commercial advantage.”

  10. Define and illustrate the following: analogy, hypothesis,
      thinking, connotative term, relative term.

  11. Distinguish between:
        (1) Analysis and deduction.
        (2) Logical division and classification.
        (3) Relative and absolute identity.


                                Set V.

_Answer ten questions._                                 Time, 2 hours.

Test the validity, giving reasons:

   1. All successful teachers are industrious, but you are not
      industrious because you are not successful.

   2. John was a troublesome boy in the first and second grades,
      therefore he is going to make trouble for the third grade
      teacher.

   3. Teaching is the art of imparting knowledge. Criticise, giving
      reasons. Define correctly, pointing out the essentials.

   4. Explain the extensional and intensional use of terms and
      illustrate the law of variation.

   5. Describe Mill’s experimental methods of induction. Symbolize the
      joint method.

   6. Define the following: analysis, law of identity, obversion.

   7. Illustrate the laws of thought.

   8. Write on one of the following topics:
        (1) Complete Method,
        (2) Right Thinking.

   9. “The science of logic never made a man reason rightly.” Discuss
      this question.

  10. Explain and illustrate the enthymeme.


                                Set VI.

_Answer ten questions._                                 Time, 2 hours.

   1. Exemplify the following:
        (1) illicit minor,
        (2) begging the question,
        (3) law of excluded middle,
        (4) inductive method.

   2. Write a short theme on one of these topics:
        (1) Thinking.
        (2) Logical Terms.

      Test the validity of the attending arguments, giving reasons:

   3. “He who talks much usually says little and you are certainly a
      great talker.”

   4. “You must be industrious, since only such truly succeed.”

   5. Illustrate and give the characteristic marks of the joint method
      of induction.

   6. Summarize the benefits to be derived from a study of logic.

   7. State and illustrate the rules of logical definition.

   8. Distinguish between
        (1) extension and intension,
        (2) opposite and contradictory terms,
        (3) analysis and synthesis.

   9. Define and illustrate hypothesis, obversion, sorites,
      hypothetical argument.

  10. Explain and illustrate the three forms of induction.

  11. Distinguish logically between a teacher and an instructor.



                             BIBLIOGRAPHY.

Aikins. The Principles of Logic. Henry Holt and Co., New York. 1905.

Bain. Logic, Inductive and Deductive. Longmans, Green and Co. 1902.

Bosanquet. The Essentials of Logic. The MacMillan Co., London. 1910.

Bradley. The Principles of Logic. London. 1886.

Creighton. Introductory Logic. The MacMillan Co., New York. 1905.

Dewey. Studies in Logical Theory. The University of Chicago Press.
  1903.

Fowler. The Elements of Deductive and Inductive Logic. Oxford. 1905.

Hibben. Logic, Deductive and Inductive. Chas. Scribner’s Sons, New
  York. 1906.

Hyslop. Elements of Logic. Chas. Scribner’s Sons, New York. 1905.

Jevons-Hill. Elements of Logic. American Book Co., New York. 1883.

Keynes. Formal Logic. The MacMillan Co., London. 1906.

Lotze. Logic. Translated by B. Bosanquet, 2 vols. Oxford. 1888.

McCosh. Laws of Discursive Thought. Chas. Scribner’s Sons. 1906.

Mill. A System of Logic, 2 vols. Longmans, Green and Co., London. 1904.

Russell. Elementary Logic. The MacMillan Co., New York. 1908.

Ryland. Logic. George Bell and Sons, London. 1900.

Sigwart. Logic. Translated by Helen Dendy, 2 vols. The MacMillan Co.
  1895.

Swinburne. Picture Logic. Longmans, Green and Co., London. 1904.

Taylor. Elementary Logic. Chas. Scribner’s Sons, New York. 1911.

Venn. The Logic of Chance. The MacMillan Co., New York.



                      OUTLINE OF BRIEFER COURSE.


      Subject

    =I. THOUGHT AND ITS LAWS=
          ⭘ Logic Defined
          ⭘ The Thinking Process.
          ⭘ Stages in Thinking
          ⭘ The Law of Identity
          ⭘ The Law of Contradiction
          ⭘ The Law of Excluded Middle

   =II. LOGICAL TERMS=
          ⭘ All of Chapter 4

  =III. EXTENTSION AND INTENSION OF TERMS=
          ⭘ All of Chapter 5

   =IV. DEFINITION=
          ⭘ All of Chapter 6

    =V. LOGICAL DIVISION AND CLASSIFICATION=
          ⭘ All of Chapter 7

   =VI. LOGICAL PROPOSITIONS=
          ⭘ All of Chapter 8 Except Section 7

  =VII. IMMEDIATE INFERENCE=
          ⭘ All of Chapter 10

 =VIII. MEDIATE INFERENCE=
          ⭘ All of Chapter 11 Except Section 8

   =IX. FIGURES AND MOODS=
          ⭘ The Four Figures of the Syllogism
          ⭘ The Moods of the Syllogism
          ⭘ Testing the Validity of the Moods

    =X. INCOMPLETE SYLLOGISMS=
          ⭘ Enthymeme
          ⭘ Polysyllogisms
          ⭘ Sorites

   =XI. CATEGORICAL ARGUMENTS TESTED=
          ⭘ All of Chapter 14

  =XII. HYPOTHETICAL AND DISJUNCTIVE ARGUMENTS=
          ⭘ All of Chapter 15 except Sections 13,14, 15 and 17

 =XIII. LOGICAL FALLACIES=
          ⭘ All of Chapter 16

  =XIV. INDUCTIVE REASONING=
          ⭘ All of Chapter 17 Except Sections 3, 4, 7, 8 and 9

   =XV. MILL’S METHODS OF OBSERVATION AND EXPERIMENT=
          ⭘ All of Chapter 18

  =XVI. OBSERVATION, EXPERIMENT AND HYPOTHESIS=
          ⭘ All of Chapter 19



                                 INDEX


                                   A

  Absolute Terms, 56.

  Abstract Terms, 51.

  Accent, Fallacy of, 330.

  Accident, 81;
    Fallacy of, 334.

  Affirmative Proposition, 127.

  Agreement, Method of, 387.

  All――not, Some, Few, Logical Significance of, 133.

  Ambiguous Middle, 328.

  Amphibology, 329.

  Analogy, 368.

  Analysis, Definition of, 97;
    As a Method, 97;
    Induction by, 373.

  Analytic Propositions, 138;
    Method, 97.

  Antecedent, 289.

  Apprehension and Thinking, 24.

  Arguments, Irregular, 258;
    Testing of Categorical, 263;
    Incomplete, 247;
    General Exercises, 481;
    Mistakes of Students in Connection with, 281;
    Hypothetical, 288;
    Disjunctive, 302;
    Dilemmatic, 308.

  Argumentum ad populum, 338;
    ad hominem, 338;
    ad ignorantiam, 338;
    ad baculum, 338;
    ad verecundiam, 339.

  Aristotle’s Dictum, 208.

  Art, Definition of, 96.

  Auxiliary Elements of Induction, 418.


                                   B

  Bain Quoted, 12.

  Ballentine Quoted, 359.

  Barbara, Celarent, etc., 234.

  Begging the Question, 341.

  Bibliography, 492.

  Bowen Quoted, 12.

  Briefer Course, Outline of, 493.


                                   C

  Canons of Syllogism, 209;
    of Four Figures, 226.

  Categorematic Words, 48.

  Categorical Arguments, 263;
    Tested, 263;
    General Exercises, 481.

  Categorical Propositions Defined, 121;
    Four Elements, 122;
    Four Kinds, 126;
    Classification of, 128.

  Cause, Fallacy of False, 340.

  Chance, Rationalization of, 468.

  Child, Thinking of, 20.

  Circulus in Probando, 343.

  Classification Compared with Division, 112;
    Kinds, 113;
    Rules of, 114;
    Use, 114.

  Co-extensive Propositions, 142.

  Collective Terms, 50.

  Comparison, Stages in Thinking, 25.

  Complete Method, Three Elements, 97.

  Composition, Fallacy of, 331.

  Concept, Definition of, 17;
    a Thought Product, 21.

  Concomitant Variations, 402.

  Concrete Terms, 51.

  Connotative Terms, Two-fold Function of, 62;
    Definition of, 52;
    a List of, 65.

  Conquest the Desideratum, 447.

  Consequent, 289;
    Fallacy of False, 339.

  Contradiction, Law of, 35.

  Contradictory Terms, 53;
    Propositions, 167.

  Contrary Propositions, 165.

  Contraversion, 181;
    Fallacies of, 327.

  Converse Accident, Fallacy of, 335.

  Conversion, 176;
    by Limitation, 178;
    Simply, 179;
    Fallacies of, 327.

  Copula, 123.

  Creighton Quoted, 4, 387, 485.


                                   D

  Deduction, Defined, 96;
    as a Method, 97;
    Special Function of, 438.

  Definition, Importance of, 77;
    the Predicables, 77;
    Nature of, 82;
    Definition of, 83;
    Compared with Division, 84;
    Kinds of, 85;
    When Serviceable, 87;
    Rules of, 88;
    Terms which Cannot be Defined, 93;
    of Common Educational Terms, 94.

  Denomination, Stages in Thought, 26.

  Denotation and Connotation of Terms, 66.

  Descriptive Definition, 86.

  Development, Definition of, 94.

  Dichotomy, 110.

  Difference, Method of, 393.

  Differentia, 80.

  Dilemma, 308.

  Discoverer’s Method, 440.

  Disjunctive Arguments, 302;
    Rules of, 303;
    Logical Disjunction, 303;
    Reduction of, 307.

  Distribution of Subject and Predicate of Propositions, 145;
    Schemes for Remembering, 148.

  Division, Definition of Logical, 105;
    As Partition, 107;
    Compared with Definition, 84;
    Distinguished from Enumeration, 106;
    Rules of, 108;
    Compared with Classification, 112;
    Use of, 114;
    Fallacy of, 332.

  Dressler Quoted, 12.


                                   E

  Education, Defined, 94;
    Compared with Instruction, 95.

  Educational Terms Defined, 94.

  Elements of the Logical Proposition, 123.

  Elliptical Propositions, 129.

  Enthymeme, 247.

  Epicheirema, 249.

  Episyllogism, 250.

  Epithets, Question Begging, 343.

  Essential Attributes of Definition, 88.

  Etymological Definition, 85.

  Euler’s Diagrams, 141.

  Evolution and the Thinking Mind, 19.

  Examination Questions, 486.

  Exceptive Propositions, 135.

  Excluded Middle, Law of, 39.

  Exclusive Propositions, 136.

  Exercises, Testing Arguments, 481.

  Experiment as an Element in Induction, 419.

  Extension and Intension of Terms, Defined, 63;
    Compared, 63;
    Used in Comparison, 65;
    Other Forms of Expression for, 66;
    Law of Variation in, 66.


                                   F

  Fact, Defined, 96.

  Fallacies, of Deductive Reasoning, 322;
    Paralogism and Sophism, 322;
    Division of, 323;
    of Immediate Inference, 326;
    in Form, 194, 199;
    Hypothetical, 291;
    Disjunctive, 303;
    of Language, 328;
    in Thought, 334.

  False Cause, 340.

  False Consequent, Fallacy of, 339.

  Figure of Speech, Fallacy of, 333.

  Figures of Syllogism, 218;
    Special Canons of, 226;
    Perfect and Imperfect, 235;
    Reduction, 235;
    Relative Value of, 239.

  Formal Fallacies, 197, 324.

  Four Terms, Fallacy of, 329.

  Fowler Quoted, 4, 360.

  Fundamentum Divisionis, 108.


                                   G

  General Exercises in Testing Arguments, 481.

  General Terms, 49.

  Genus and Species, 78.

  Grammatical Subject and Predicate, 125.

  Grammatical Sentences, 131.


                                   H

  Hamilton Quoted, 4, 12, 131.

  Hibben Quoted, 4, 441.

  Huxley Quoted, 473.

  Hypothetical Arguments, 288;
    Kinds, 290;
    Rules and Fallacies, 291;
    Reduced to Categorical, 293;
    Illustrative Exercise in Testing, 297;
    General Exercises, 484.

  Hypothesis, Defined, 96, 425;
    and Theory, 427;
    Requirements of, 427;
    Uses of, 429.


                                   I

  Identity, Law of, 32;
    Absolute, 33;
    Complete and Incomplete, 33;
    Relative, 34.

  Illicit Major and Minor, 199;
    Illustration of, 215.

  Image, Definition of, 17.

  Immediate Inference, 159;
    by Obversion, 170;
    by Opposition, 161;
    by Conversion, 176;
    by Contraversion, 181;
    Epitome of Four Processes, 182;
    by Inversion, 183;
    Fallacies of, 326.

  Imperfect Induction, 361.

  Indefinite Propositions, 129.

  Individual Proposition, Nature of, 132;
    in Opposition, 168.

  Induction, Defined, 96;
    as a Method, 97;
    Reasoning, 355;
    and the Hazard, 356;
    the Three Forms of, 365;
    Perfect, 375;
    Special Function of, 438.

  Inference, Definition of, 18;
    a Thought Product, 24;
    Immediate, 159;
    Mediate, 192.

  Infima Species, 79.

  Instruction Defined, 95.

  Intension of Terms, 63.

  Integration, a Stage in Thought, 26.

  Inversion, 183.

  Inverted Proposition, 137.

  Irregular Arguments, 258.

  Irrelevant Conclusion, 337.


                                   J

  Jevons Quoted, 4, 25, 387, 468.

  Joint Method of Agreement and Difference, 397.

  Judgment, Definition of, 17;
    a Thought Product, 22;
    Most Fundamental Element in Thinking, 23.


                                   K

  Keynes Quoted, 481, 485.

  Kinds of Definitions, 85.

  Knowing, by Intuition and by Thinking, 2;
    Knowing and Thinking Compared, 10;
    by Intuition, 11;
    Habitual, 11.

  Knowledge, Defined, 95;
    Intuitive, 11.


                                   L

  Language and Thought Inseparable, 47.

  Law of Variation in Extension and Intension, Stated, 66;
    Two Important Facts in, 69;
    Diagrammatically Illustrated, 70, 71.

  Laws of Sufficient Reason, 40;
    of Universal Causation, 361;
    of Uniformity of Nature, 362.

  Laws of Thought, 32;
    Unity of, 40;
    Schematic Statement of, 43.

  Learning, Defined, 95.

  Logic, Defined, 3;
    Authentic Definitions of, 4;
    Grammar of Thought, 3;
    Science of Sciences, 3;
    the Value of to the Student, 5;
    Related to Other Subjects, 1;
    Specific Scope, 2.

  Logic in the Class Room, 437.

  Logic and Life, 463.

  Logic of Success, 475.

  Logical Definition, 85, 88.

  Logical Disjunction, 303.

  Logical Subject and Predicate, 125.


                                   M

  Major Term, 196.

  Material Fallacies, 323, 324, 325, 328.

  Mediate Inference, 192;
    the Syllogism, 192;
    Rules of Syllogism, 193.

  Method Defined, 96;
    Inductive and Deductive, 97;
    Complete, 97.

  Method-Whole Defined, 96.

  Middle Term, 192, 193, 196.

  Mill Quoted, 5, 359, 361, 387, 393, 397, 402, 406.

  Mill’s Experimental Methods, 386.

  Miller Quoted, 12.

  Mind, the Unity of, 1;
    Knowing and Thinking Compared, 10.

  Minor Term, 196.

  Mnemonic Lines, 234.

  Modal Proposition, 139.

  Modus Ponendo Tollens, etc., 302.

  Moods of Syllogism, 221;
    Testing Validity of, 223.

  Motivation as Related to Spirit of Discovery, 449.


                                   N

  Negative Proposition, 127.

  Negative Terms, 53.

  Nego-positive Terms, 55.

  Non-connotative Terms, 52.

  Non Sequitur, Fallacy of, 339.

  Not, Bisects the World, 36;
    Two Uses of, 36.

  Notion, Definition, 14;
    Individual, 14;
    General, 14;
    Distinguished from Knowledge, 15;
    Distinguished from Idea, 16;
    Psychological Terms Involved in, 16.


                                   O

  Observation, 419;
    Rules of, 420;
    Errors of, 423.

  Obversion, Definition of, 170;
    Fallacies of, 326.

  Opposite Terms, 53.

  Opposition, Nature of, 161;
    Scheme of, 163;
    Square of, 164.

  Outline of Briefer Course, 493.


                                   P

  Page Quoted, 453.

  Particular Propositions, 126;
    Affirmative, 143;
    Negative, 144.

  Partition, 107.

  Partitive Propositions, 133.

  Percept, Definition of, 17;
    Related to Thought, 18.

  Perfect Induction, 375.

  Petitio Principii, 341.

  Plurative Propositions, 132.

  Polysyllogism, 250.

  Porphyry, Tree of, 111.

  Positive Terms, 53.

  Predicables. Defined, 77;
    Named, 78;
    Illustrated, 82.

  Predicate, Grammatical and Logical, 125;
    Distribution of, 145.

  Primary Laws of Thought, 32.

  Privative Terms, 55.

  Progressive Thought, 465.

  Property, 81.

  Propositions, Definition of Logical, 120.

  Prosyllogism, 250.

  Proximate Genus, 79.

  Pure Proposition, 139.


                                   Q

  Quantity Signs, 123.

  Quantity and Quality of Propositions, 126.

  Question and Answer, not a Method of Discovery, 457.

  Question Begging Epithets, 343.

  Question, Complex, 340.


                                   R

  Rationalization, of Chance, 468;
    of Political and Business Sophistries, 470;
    of the Spirit of Progress, 471;
    of the Attitude toward Work, 474.

  Reasoning, Defined, 24, 355;
    Inductive, 355;
    Deductive, 355.

  Reduction of Figures, 235.

  Relation between Subject and Predicate, 140.

  Relative Terms, 56.

  Residues, Method of, 406.

  Right Thinking, 466.

  Rules, of Logical Definition, 88;
    of Logical Division, 108;
    of Classification, 114;
    of the Syllogism, 193;
    of the Hypothetical Argument, 291;
    of the Disjunctive, 303.

  Russell Quoted, 481, 482.

  Ryland Quoted, 481, 482.


                                   S

  Salisbury Quoted, 360.

  Science, Defined, 95.

  Sensation, Defined, 17;
    Related to Thought, 18.

  Simple Conversion, 179.

  Simple Enumeration, 367.

  Singular Terms, 49.

  Socrates, 322.

  Sorites, 251.

  Species, 78.

  Square of Opposition, 164.

  Subaltern Propositions, 164.

  Subcontrary Propositions, 164.

  Subject, Logical, 123;
    Grammatical and Logical Distinguished, 125;
    Distribution of, 145.

  Success, Logic of, 475.

  Sufficient Reason, Law of, 40.

  Summum Genus, 79.

  Syllogism, a Product of Inference, 24;
    Nature of, 192;
    Rules of, 193;
    Undistributed Middle, 199;
    Illicit Major, 199;
    Illicit Minor, 199;
    Aristotle’s Dictum, 208;
    Canons of, 209;
    Mathematical Axioms of, 210;
    Four Figures of, 218;
    Moods of, 221;
    Incomplete, 247.

  Syncategorematic Words, 48.

  Synthesis, Defined, 97;
    as a Method, 97.

  Synthetic Proposition, 138.


                                   T

  Teaching, Defined, 94;
    Compared with Instruction and Education, 95.

  Terms, Extension and Intension of, 63;
    Used in Extension and Intension, 65;
    which Cannot be Defined, 93;
    Contradictory and Opposite, 38;
    Logical, 47;
    Singular and General, 49;
    Collective and Distributive, 50;
    Concrete and Abstract, 51;
    Connotative and Non-connotative, 52;
    Positive and Negative, 53;
    Contradictory and Opposite, 53;
    Privative and Nego-positive, 55;
    Absolute and Relative, 56.

  Theory Defined, 96.

  Thinking, Definition of, 12;
    Illustration of Process, 13;
    Compared with Knowing, 10;
    Compared with Intuition, 2;
    the Process, 12;
    Groups Many Into One, 18;
    in the Sensation and Percept, 18;
    Evolution and Thinking Mind, 19;
    of the Child, 20;
    of the Adult, 20;
    and the Concept, 21;
    and the Judgment, 22;
    and Apprehension, 24;
    Stages in, 25;
    in the Inference, 24;
    Laws of, 32;
    Unity of Laws, 40;
    Progressive, 465;
    Right, Necessity of, 466;
    Indifferent and Careless, 467.

  Thought and Language, 47.

  Thought is King, 437.

  Traduction, 377.

  Training, Definition of, 95.

  Tree of Porphyry, 111.

  Truistic Proposition, 139.

  Truth Defined, 96.


                                   U

  Uberweg Quoted, 4.

  Undistributed Middle, 199;
    Illustration of, 214.

  Uniformity of Nature, 362.

  Universal Affirmative Proposition, 140.

  Universal Causation, 361.

  Universal Negative Proposition, 142.

  Universal Propositions, 126.


                                   V


  Variations, Method of Concomitant, 402.


                                   W

  Watts Quoted, 4.

  Weakened Conclusion, 224.

  Whately Quoted, 4.

  Word-signs of Categorical Propositions, 122.



                              Footnotes.


    1 – NOTE. Sometimes thinking and thought are used
        interchangeably. This is confusing. Properly, “thinking”
        is always a process of the knowing mind while “thought”
        is the product of this process, just as the flour of the
        gristmill is the _product_ of the grinding _process_.


    2 – Intuitive knowing might be termed habitual knowing.

    3 – Mediate Inference.

    4 – Intuitive Knowing.

    5 – Hyslop’s Elements of Logic (1901), page 100.

    6 – Hyslop.

    7 – Men do have the power of reason.

    8 – Sometimes called contraposition.

    9 – From the Greek meaning _to reason with_.

   10 – The student may be sufficiently interested to complete the
        list.

   11 – The student should prove that the last premise may be
        affirmative.

   12 – This cause, however, need not be a _single_ antecedent, in
        fact it seldom is. “This cause, philosophically speaking,
        is the sum total of the conditions, positive and negative,
        taken together.”――Mill. The cause of the price of food stuff
        being high, involves many conditions, or antecedents, so
        interwoven that it is impossible to designate any one as
        being the chief factor concerned.

   13 – Those might be named the Five Special Methods of Induction
        by Analysis.

   14 – All cases of finding the net proceeds are examples of the
        law of residue.



                         Transcriber’s Notes.


  The following corrections have been made in the text:

  Page 22:
    Sentence starting: First, I may be....
      – ‘recignize’ replaced with ‘recognize’
        (able to recognize each)

  Page 61, Question 16:
    Sentence starting: What is the difference....
      – ‘differenece’ replaced with ‘difference’
        (the difference in meaning)

  Page 61, Question 23:
    Sentence starting: Argue to the effect....
      – ‘non-contotative’ replaced with ‘non-connotative’
        (can be non-connotative.)

  Page 86:
    Sentence starting: A logical definition....
      – ‘digerentiæ’ replaced with ‘differentiæ’
        (by means of its differentiæ.)

  Page 144:
    Sentence starting: Here, as in the I,...
      – ‘comfined’ replaced with ‘confined’
        (knowledge is confined to)

  Page 204:
    Sentence starting: (6) _If one premise....
      – ‘nagtive’ replaced with ‘negative’
        (one premise be negative)

  Page 240:
    Sentence starting: Since, however, the easiest....
      – ‘affirmavite’ replaced with ‘affirmative’
        (and a particular affirmative)

  Page 246:
    Sentence starting: (3) Show why it is....
      – Printer error: lost end of point (3) and duplicated part of
          point (4)

  Page 276:
    Sentence starting: Other arguments where....
      – ‘agruments’ replaced with ‘arguments’
        (Other arguments where)

  Page 286:
    Sentence starting: (4) In the practice....
      – ‘analagous’ replaced with ‘analogous’
        (custom analogous to giving)

  Page 286:
    Sentence starting: (4) “The man is guilty....
      – ‘(4)’ missing from text
        ((4) “The man is guilty)

  Page 325:
    Sentence starting: These fallacies arise from....
      – ‘abitrary’ replaced with ‘arbitrary’
        (words as arbitrary signs)

  Page 326:
    Sentence starting: There is little difference....
      – ‘fallicies’ replaced with ‘fallacies’
        (and fallacies of converse)

  Page 371:
    Sentence starting: (1) Noting that two....
      – ‘sirname’ replaced with ‘surname’
        (have the same surname,)

  Page 376:
    Sentence starting: In the school room....
      – ‘generalziation’ replaced with ‘generalization’
        (perfectly induced generalization,)

  Page 382:
    Sentence starting: _Complete_ enumeration gives....
      – ‘uncontradiced’ replaced with ‘uncontradicted’
        (_incomplete but uncontradicted_)

  Page 405:
    Sentence starting: It is thus evident that....
      – ‘prenomenon’ replaced with ‘phenomenon’
        (entirely with the phenomenon.)

  Page 422:
    Sentence starting: _Reading into the external....
      – ‘vivdly’ replaced with ‘vividly’
        (what is only vividly internal)





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