The principles of logic

Preface

The deductive part of this book may require a few words of explanation.

In most text-books on logic deduction and induction are treated from two very different standpoints. Induction is regarded as a means for the investigation of nature, and its canons are frankly objective or ' material '. They tell us how things must be related to each other in one respect if they are also related in some other respect {e.g., how two events must be related in time if one is the cause of the other) ; and they do not say anything about the relation of our thoughts to each other as mere thoughts, or give any rules for the arrangement and manipulation of the words in which these thoughts are expressed.

Deduction, on the other hand, is usually defined as the ** science of the laws of thought'' — as one writer says, its *' object-matter is thought'', — and it is treated altogether from the subjective or ' formal ' standpoint, as though mere thoughts as such could be consistent or inconsistent, wholly regardless of the nature of the object to which they refer; and then, when it comes to working out the details of the subject from this standpoint, the ' laws of thought ' are treated practically as though they were laws for the right arrangement and manipulation of words. Hence we have rules of the syllogism and allied rules for conversion and obversion which say nothing whatever about the things under discussion and their relations to each other, but tell us only how we must or must not arrange our words in discussing them.

iii

IV PREFACE.

In this book I have tried to treat deduction from the objective standpoint that everybody assumes in the treatment of induction. Consequently I have omitted the traditional rules of the syllogism and put in their place a direct statement of the principles on which we reason in the different figures, adding certain ' cautions ' that must be observed if the principles are not to be misapplied; and I have treated conversion and obversion in much the same way.

The fourth figure of the syllogism seemed to me not to represent any distinct principle of reasoning, and therefore to have no proper place in the objective treatment of logic; but I have explained the traditional way of dealing with it. The * algebra ' of logic I have omitted altogether. Readers interested in it are referred to the ** Johns Hopkins Studies in Logic'' (Little, Brown & Co., Boston).

However imperfect my own treatment of deduction from the objective standpoint may be, I believe that the standpoint itself is not only more correct philosophically than the subjective, but also better pedagogically; for we do far more to make a student clear-headed by teaching him to look a situation in the face and analyze it than by giving him any amount of dexterity in the reduction of arguments to a given verbal form.

My indebtedness to other authors is apparent. I am no less indebted to individuals — particularly to colleagues in other departments who have given me valuable suggestions on matters related to their special subjects; but most of all to Dr. W. T. Marvin, to whom I read the whole book while it was still in manuscript.

H. A. A.

CHAPTER I.— Introduction

Two kinds of thinking, i — ^Judgments and propositions, 2 — A different view of judgments, 4 — Logical thinking is objective, 6 — Scope of logic^ 6 — Truth impersonal, 7 — Corollary, 9 — Fact and feeling, 10.

CHAPTER n.— The Meanings of Words.

Blunders of interpretation, 13 — How words become ambiguous, 16 — What ambiguities are most dangerous, 17 — Blunders of inference, 22 — Paronymous terms, 28 — How to deal with ambiguities, 29 — Definition, 30 — How to frame definitions, 31 — Illustrations, 32 — Where precision is needed most, 34.

CHAPTER in. — The Meaning of Statements as a

Whole.

Conventional phrases, 36 — Oblique senses, 37 — Exaggeration, 38 — Amphibology, 39 — Accident, 39 — Accent, 42.

CHAPTER IV. — Division and Classification.

Relation to definition, 45— Principles of division and subdivision, 45— Cross-division, 47 — Some technical names, 50 — Scientific classification and its purposes, 50 — Naming, 55.

CHAPTER v.— The Uses of Single Words and Phrases.

Terms, 56 — Demonstrative and descriptive, 58 — Connotative and non-connotative, 60 — Singular and general, 61 — Epithets, 62 — Collective and distributive, 63— The danger, 64 — Abstract and con-

V

VI CONTENTS.

Crete, 65 — Hypostatising abstractions, 66 — Secondary meaning of 'abstract*, 70 — Positive and negative, 71 — Relative and absolute : first sense, 73 — Second sense, 74— Dangers, 75.

CHAPTER VI. — The Relations Expressed in Propositions.

Five fundamental relations, 77 — Certain combined relations, 84.

CHAPTER VH. — What Propositions Imply about

Existence.

The two * subjects ', 87 — Denials of relations and of existence, 89 — The conception of reality, 91.

CHAPTER Vni. — The Formal Characteristics of

Propositions.

Quantity and quality, 96 — Ambiguities of quantity and quality, 98 — Undesignated quantity or quality, 99 — Double quantity, 102 — Exclusives and exceptives, 103 — Disjunctives and hypotheticals, 106.

CHAPTER IX. — The Opposition of Propositions.

With common propositions, 11 1 — With exclusives and exceptives, 115 — Symbols, 117.

CHAPTER X. — Inference and the so-called Laws of

Thought.

Inference — what, 121 — Deduction, 122 — The three 'Laws of Thought', 124— Obversion, 130.

CHAPTER XL — Immediate Inference, or Inference FROM A Single Premise.

Conversion, 131 — The traditional treatment, 132 — The treatment by diagrams, 140 — A broader treatment, 142.

CHAPTER XII. — Mediate Inference and Syllogism.

Limitations of deduction, 145 — 'Figures', 148.

CONTENTS. vii

CHAPTER XIII. — The First Figure of the Syllogism.

General function, 150 — Is there real inference? 152 — Principle and cautions, 155.

CHAPTER XIV. — The Second Figure of the Syllogism.

Function and general cautions, 162 — Quantity in the second figure, 167.

CHAPTER XV. — The Third Figure of the Syllogism.

Purpose, principle, and general caution, 171 — Negative relations, 174 — Quantity of the premises, 175.

CHAPTER XVI.— The Alleged Fourth Figure.

Origin of the figure, 176 — Three ways of dealing with it, 176 — Formal reduction, 178.

CHAPTER XVII. — Other Deductive Arguments.

Hypothetical syllogisms, 183 — Disjunctive syllogisms, 186 — Dilemmas, 186 — Three forms of abbreviated argument, 188.

CHAPTER XVIIL— Blunders in Word and Blunders

IN Thought.

'Purely logical ', 193 — Two kinds of material fallacies, 195.

CHAPTER XIX.— The Forgotten Issue.

Petitio principii, 196 — Includes epithets, 197 — And circle, 197 — Ignoratio Elenchi, 198 — Includes 'adhominem', 201.

CHAPTER XX. — The Ill-conceived Universe.

The assumed universe, 204 — The confusion of universes, 208 — The neglected aspect, 211 — The neglected member, 215 — The neglected whole, 219.

CHAPTER XXI. — The Difference between Induction

AND Deduction.

; Difference in limitations, 221 — Difference in certainty, 226,

Vlil CONTENTS.

CHAPTER XXIL— The Uniformity of Nature.

How we come to believe in it, 228 — Its two main aspects, 232 — Law, 237 — Precision in uniformity, 238 — The thing-aspect of this precision, 241— Proof of uniformity, 243.

CHAPTER XXni. — Science and the Peculiarities of

THE Relations that it Traces.

Twofold work of science, 244 — Peculiarities of individual identity and causal interaction, 245.

CHAPTER XXIV.— The Method of Exhaustion and

THE Search for Particular Uniformities.

* Perfect induction', 250— Why we accept less, 251 — Justification practical, 252 — When may we guess? 252 — How problems and relations are interwoven, 253.

CHAPTER XXV. — Induction by Simple Enumeration AND THE Search for Causes.

Analysis, inference, and explanation, 257 — Analysis and clear thinking, 259.

CHAPTER XXVI.— The Methods of Difference and

Agreement.

Different ways of exhausting the universe, 261 — The method of difference, 262 — The method of agreement, 264— Does either method really exhaust ? 265 — Advantages of each, 267 — Plurality of possible causes, 267 — Two kinds of data compared, 270 — Post hoc ergo propter hoc^ 272— Advantages of experiment, 272— Assumption in experiment, 273.

CHAPTER XXVIL— The Joint Method of Agreement

AND Difference.

Its function, 275 — Compared with simple method of difference, 278,

CHAPTER XXVIIL— Counteracting and Complex

Causes.

Counteracting causes, 281— Causes •compounded* or ^combined', 283,

CONTENTS. IX

CHAPTER XXIX. — The Methods of Residues and Concomitant Variations.

Quantitative treatment of causes, 287— Method of residues, 288— Method of concomitant variations, 290 — Cautions, 293.

CHAPTER XXX.— Group Comparisons, or the Method

OF Statistics.

Principle and uses, 296 — Number of data, 300— Personal equation, etc.. 300— Preconceptions and interests, 302 — Accidental selection, 304 — Misplaced accuracy, 306.

CHAPTER XXXL— Means, or Averages.

General conception, 312— Various kinds, 313 — First use of average, 317 — Second use of average, 320 — Third use of average, 324 — Measures of error, 327.

CHAPTER XXXn.— Probability.

Why needed, 330 — What it is not, 331— What it is, 333 — What value? 334 — Why hard to estimate, 336— Caution, 337 — Mathematical principles, 338 — Four more cautions, 340.

CHAPTER XXXIII.— Observation and Memory.

Observation and inference, 343 — Credulity, 345 — Errors cumulative, 347 — Unprejudiced observers, 347 — Two classes of errors, 348— Memory and its dangers, 350 — What we remember, 350— Honest lies, 352 — Resulting commonplaces, 354.

CHAPTER XXXIV.— The Discovery of Past and Future Events in General.

The starting-point, 357 — The limit, 358— ' Monuments ', 359— Circumstantial evidence, 361.

CHAPTER XXXV.— Testimony.

Its importance, 363— Its unique value and danger, 364 — Accepting, rejecting, and weighing, 364— Expert evidence ; caution. 367— Inferring without trusting, 371— More refined methods^ 373— Who is

CONTENTS.

the witness ? 376 — Joint authorship, 378 — How does he know ? 384 — Hearsay, 384— Is he truthful? 386 — Arbitrary tests, 386 — Character, 387 — Circumstances, 388 — Interests, 389 — Consistency and general probability, 392 — Confirmation and contradiction, 393 — The judge must judge impartially, 397 — According to evidence, 398 — Must judge for himself, 398 — Must have a problem and make it definite, 400 — Issue before evidence, 402 — Must not be managed by the witnesses, 403 — Must not decide in a hurry, 404.

CHAPTER XXXVL— The Three Ultimate Tests of

Truth.

Consistency, 405 — Conceivability, 406 — The third test, 409 — Uniformity in the mass, 409 — Analogy, 413 — Absurdity, 414 — Simplicity, 416 — The right to assume these principles, 421 — The limits of proof, 423.

Index 4^5

LOGIC.

CHAPTER I.

Introduction

The business of Logic is to help us to think clearly and objectively, express ourselves plainly and accurately, reason correctly, and estimate aright the statements and arguments of others.

We cannot fully understand what is meant by the statement that logic helps us to think clearly unless we distinguish carefully between two kinds of thinking.

The first kind makes some statement or asks Two kinds of

tninkiiig:.

some question; the second consists in a mere play of mental images, such as takes place when we follow quite passively the successive notes of a piece of music, hearing each as it passes but doing nothing more, or when the music runs through our head afterwards in the same way, or when half-dozing we watch the pictures that float before us, or when in the same passive way we feel yet do not note the sensations that come from our limbs as we walk or row or take some other such mechanical exercise. But just as soon as we recognize that the music is or is not beautiful, that one note is following another, that some one is playing, or that we are hearing, feeling, or imagining something, then we get back for the moment at least to the first kind of thinking; for to the mere passive images we have added something else and there is now an active affirmation or denial.

The second kind of thinking — that which consists in the mere play of sensations or images — serves no purpose beyond the pleasure or recreation of the thinker; and logic has nothing to do with it. Consequently, w^hen w^e speak of clear thinking we do not refer to the vividness or continuity of such passive images, but rather to the definiteness and consistency of active affirmations, denials, or questions.

When we affirm or deny anything we are said to form a belief, to come to a conclusion, or to pass a judgment; and

the sentence in which the judo^ment is expressed Judgments . ^^ , . . V> ,• ^

andpropo- is called a proposition. Froposiiions can thus sitions. 1 1 r> 1 . . ,

be denned as sentences expressing judgments.

According to this definition interrogative sentences are not propositions, for a question implies the absence of a judgment. On the other hand negative sentences are propositions; for it is quite as much an act of judging to deny that something is the case, e.g., to say or think ' The day is not hot', as to affirm it — to say or think 'The day is hot'. There is thus a vast difference between saying that a thing is not so and not saying that it is so, in spite of the fact that we often say ' I do not think so ' when we mean ' I think not '.

Judgments always have reference to something other than themselves. Like the eye which always looks outward and never sees itself, they are always concerned with some object or other which lies beyond them, and never with themselves. This is most important. Perhaps it can be made clear by an illustration from grammar. When I say " The word ' good ' is an adjective ", I am speaking of that word as it occurred somewhere else; for as I use it here it is a noun. W^ords never refer to themselves, but to something else which they name or ' mean '. The same is true of judgments. When I say " The house is on fire ", I do not mean to state anything about myself or about my judgment with

JUDGMENTS AND VRDPOSTTIONS. 3

reference to the house. I do not even mean to say that I judge that the house is on fire. I do judge it; and the statement expresses my judgment. But to express a judgment is to tell something about the object thought about or judged of — in this case the house — not about the thought or judgment itself, or the person who passes it, or the words in which it is expressed. When we wish to see our own eyes we do not look at them directly, but at their image in a mirror. So when we wish to know our own thought we must get a new thought with the thought in question as its object, e.g,, ' I thought that the house was on fire', or * I recognized that the house was on fire '. In these propositions I am talking about my own thought or judgment; in the former I was not. But even here the thought of which I am speaking is not the thought that I am expressing, for I am speaking of the past thought and expressing the present thought about it.

That about which a judgment is passed, i.e., that about which something is asserted (affirmed or denied), is called the Subject or Object of our thought. If it is a real thing or person, it is also called the subject of the relations that are affirmed of it, and the w^ord used to point it out is usually the subject of the sentence or proposition in which the judgment is expressed. Thus, in the example just given, the real house is the subject of the state we call being on fire, for it is the house — not something else — that is blazing; the real house is the subject of the judgment, for it is the real house that I am thinking and speaking about; but it is the word ' house ' that is the subject of the sentence. It is not the word ' house ' that is on fire or of which I am thinking or speaking, and it is not the real house that is part of a sentence. Thus the subject of a sentence is not that about which something is stated in the sentence, but it is the name ^that about which something is stated.

We should never confuse these three different meanings of the word Subject — what we are thinking about, or the subject of thought; the thing that is in a certain state, or the subject of that state; and the name of what we are thinking and speaking about, or the subject of a sentence.

There are plenty of logicians and other writers who have not made or recognized this distinction. Locke, for example, who wrote more than two hundred years A different ,, ... . , ■, ^ ^ ^

view of ago, tells us m his celebrated Lssay Concernmg

judgments. ^-t tti t- y> ^ ^ 11 -HI

Human Understandmg that knowledge is the

perception of the agreement or disagreement of our ideas '\

This definition takes no account whatever of the things

beyond our ideas to which they are supposed to refer» So

Jevons tells us in his " Lessons in Logic ", a text-book that

is very widely used, that an act of judgment " consists in

comparing together two notions or ideas of objects derived

from simple apprehension, so as to ascertain whether they

agree or differ ". But when we judge that the house is on

fire we do not compare our idea of the house with our idea

of fire, find that in some way or other the two ideas ' agree ',

and then use the copula ' is ' to indicate the agreement and

to fasten the two agreeing ideas together. On the contrary

we are usually wholly absorbed in the house and its fate and

do not think about our own ideas at all. When we say that

a person can tell ' only about his own ideas of things ' it

would be more accurate to say that he can express only his

own opinions. But to express an opinion about a thing

is to tell not about the opinion itself — about one's own

thoughts and their relations to each other — but about the

thing as one believes it to be, its states and its relations to

other things. If we really agreed with Locke when he says

or implies that everything we know is our own idea instead

of something beyond it, we should have to agree also w4th

Hume when he says that we " can form ideas which shall be

no greater \i,e., larger] than the smallest atom of the animal

spirits of an insect or a thousand times less than a mite'',

merely because we can think of things so small; when he

speaks of perceptions or sensations as " composed of parts "

A DIFFERENT VIEW OK JUDGMENTS. 5

because the things that \ve perceive or feel are so composed; and when he seriously discusses " the infinite divisibility of our ideas of space and time. '' *

If there is no distinction between thought and the object thought of, Flume is right enough in talking of large and small ideas; but if there is a distinction, we must not assume that a thought of a large or complex object is any larger than the thought of one that is small or simple. A good photograph of a single brick involves just as complex a chemical process on the sensitive plate as that of a whole house; to pick a single grape is just as complex an act as to pick a whole bunch; and in the same way to think of a mob is not to have a mob of thoughts. Some acts of thinking are undoubtedly more complex than others; but the complexity of the thought in no way corresponds with that of the object thought about, f

* See his <' Treatise of Human Nature", Book I, Part II, Sec. I. David Hume lived from 17 ii to 1776. He was probably the greatest of all British philosophers.

+ As for the ' simple apprehension ' spoken of by Jevons and so many other writers, it has little or nothing to do with an act of judgment. We can divide a sentence into subject and predicate, or, if we like, into subject, predicate, and copula ; but we cannot make any such division in the judgment which the sentence expresses. " Simple apprehension ", says Jevons, " is the act of mind by which we merely become aware of something, or have a notion, idea, or impression of it brought into the mind. The adjective simple means apart from other things, and apprehefision the taking hold by the mind. Thus the name or term /r<?« instantaneously makes the mind think of a strong and very useful metal, but does not tell us anything about it, or compare it with anything else." {^Eleinentary Lessons in Logic ^ p. ii.) But let the reader spend five minutes trying to think of such a metal without making some statement about it. He can say ^ Iron is strong,' or ' Iron is useful,' or ' Iron is a metal,' or 'This is iron,' and during the five minutes he will doubtless make an immense number of other statements; but when he tries to think of it without discovering some relation or passing some judgment he will probably find himself mechanically repeating some word applicable to iron and really thinking about something else; or it may be that with the word upon his lips or some visual image in his phantasy

Since all thinking has reference to some reality beyond

itself, we think clearly when we discern the object that we

are thinking about without confusion, and we

thinking: is reason correctly when we see how one relation

objective. r ^ • - ^ 1 T ^ 1 . ,

ot a thing involves another, in order to think clearly and reason correctly it is therefore necessary to look outwards continually beyond ourselves and beyond the words used by others towards the things that we or they are thinking about, in order to see these things and all their essential relations as they are. Unless we do this we cannot succeed either in expressing ourselves plainly and accurately or in forming a right estimate of the reasoning of others. Thus the habit of closely examining the reality beyond us and of testing all our thoughts and words with reference to that reality is necessary for all the aims of logic.

Logic is often defined as the science of the laws of thought; but if what we have been saying is correct, it would Scope of be far more appropriate to say that it points out

logic. ^j^^ i^^^.g ^^ things which all thought should

respect, or that it deals with the mutual implications of the relations of things. The special sciences and metaphysics also study relations of things and the way in which one involves another, but with a somewhat different purpose. Each of the special sciences is concerned with some one group of things and relations, and when it inquires how one relation involves another it is for the sake of gaining more knowledge about the particular things and relations in question. Its aim is thus the attainment of wider or more exact knowledge in some one special field. Metaphysics, on the. other hand, inquires into the most fundamental and general relations of all things, and tries to find out what the inmost nature of anv thing must be in order that all of these relations should belong to it together. Logic, like metaphysics,

he will find consciousness itself flickering and disappearing. — If simple apprehensions exist, they form no part of our knowledge, of our coherent thought, or of our reasoning. All these involve judgments.

TRUTH IMPERSONAL. 7

has a very general aim; it too inquires into the most fundamental relations of things and the way in which one involves another. But its inquiry is not so profound as that of metaphysics; it does not ask what the inmost nature of things must be in order that these relations should exist together in them; and the knowledge that it does try to gain about relations and their mutual implications it regards as a means, not as an end. We cannot reason at all in science or anything else unless we have some idea of them, and we cannot reason correctly unless our idea of them is essentially correct; but it may be correct enough to enable us to reason well about most subjects without being nearly so profound as advanced metaphysical inquiries try to make it. Thus in so far as logic tries to make us reason correctly by giving us correct conceptions of things and the way in w^hich their relations involve each other, it is a kind of simple metaphysics studied for a practical end.

There is a sense, however, in which i^ is perfectly true that logic deals with ' laws of thought \ A kw of t"bGught tells how people actually do think, just as a law of astronomy tells how heavenly bodies actually move, and the real science of the laws of thought is therefore psychology; but inasmuch as there are certain natural ways of thinking that lead to various kinds of logical blunders, it is necessary to understand them in order to understand why we make the blunders. Thus in so far as logic deals with various kinds of fallacies which we naturally commit and tries to explain their origin, it is touching on the field of psychology and dealing with ' laws of thought '.

Every judgment, true or false, asserts something about some supposed reality beyond itself, and the difference between the true and the false is that the state of Truth imaffairs asserted by the former really exists and P^^^®^^^* that asserted by the latter does not. Whether it does or does not depends altogether upon the nature of things and the presence or absence of the conditions that might naturally produce it. It does not depend at all upon the judgment about it. Whether the sun is shining or not at a certain place depends altogether upon the time of day and the presence or absence of clouds, fog, and an eclipse. If such conditions as these are all favorable, the sun is certainly shining whether I happen to think so or not; and if I say it is not shining when it really is, my statement is false no matter who I am or how sincere I may be in making it; so likewise if I have nervous prostration or a broken leg, I have it no matter who says that I have or that I have not. Hence it is arrant nonsense to say that something may be * true for one person and false for another'. One person may believe that a statement is true and another may believe that it is not, but the facts are what they are wholly regardless of these conflicting beliefs, and one of the persons must be wrong. If every possible statement could really be true for one person and false for another, then every one would always be right in what he thought no matter what it was, and there would be no such thing as an error or mistake, and therefore no distinction between correct thinking and incorrect. All logic is based on the assumption that there is such a distinction, and therefore from the standpoint of logic no blunder could be more fundamental or destructive than that which is involved in the serious belief that something may be ' true for one person and false for another \

The notion that some fact might exist 'for' one person and not ' for ' another doubtless arises from the existence of individual differences in matters of taste and a certain confusion about their meaning. If a picture pleases me, I say it is beautiful ; and if it displeases you, you say it is ugly; and all that either of us has any right to mean by the statement is that the picture does please him or displease him. Each statement thus tells about the relation between two things, the picture and the beholder; and because of the difference between the two beholders both statements understood in this way may be perfectly true; the picture really is beautiful

COROLLARY. 9

for me and really is ugly for you, and there is nothing more to be said — de gustibus nil disputandum. But these words * beautiful ' and ' ugly ' and others like them have the same grammatical form as words like ' square ' and ' round ' which really tell about the thing itself, quite regardless of its relations to the beholder; and this helps to make us ignore the difference betw^een them and assume that somehow or other we can describe the thing itself (as we do with such words as ' round ' and ' square '), wdiile at the same time the truth of the description depends (as it does w^ith such words as ' beautiful ' and ' ugly ') upon who it is that gives it.

Somewhat like the statement that something may be true for one person and false for another is the statement that it may be true in one science and false in another. Towards the end of the middle ages the monkish philosophers found themselves reaching conclusions that w^ere quite contrary to the doctrine of the Church wdiich they were bound to accept. So they said that tliere was a difference between theology and philosophy, that a doctrine might be true in one though false in the other, and that they accepted all the teachings of the Church as true in theology, though they might reject a part of them as false in philosophy. By this subterfuge they tried to give an excuse for continuing their thinking as freely as possible and yet save their heads by remaining in nominal subservience to the Church. Of course this doctrine of a ' double truth ' was nothing but a subterfuge, and it disappeared when men gained the righc to exercise their own individual judgment in matters of belief.