LOGIC

DEDUCTIVE AND INDUCTIVE

First Edition, June 1898. (Grant Richards. )Second Edition, November 1901. (Grant Richards. )Third Edition, January 1906. (A. Moring Ltd. )Reprinted, January 1908. (A. Moring Ltd. )Reprinted, May 1909. (A. Moring Ltd. )Reprinted, July 1910. (A. Moring Ltd. )Reprinted, September 1911. (A. Moring Ltd. )Reprinted, November 1912. (A. Moring Ltd. )Reprinted, April 1913. (A. Moring Ltd. )Reprinted, May 1920. (Simpkin. )

LOGIC

DEDUCTIVE AND INDUCTIVE

BY

CARVETH READ, M. A. 

AUTHOR OF

"THE METAPHYSICS OF NATURE"

"NATURAL AND SOCIAL MORALS"

ETC. 

FOURTH EDITION

ENLARGED, AND PARTLY REWRITTEN

SIMPKIN, MARSHALL, HAMILTON, KENT & CO. LTD. , 4 STATIONERS' HALL COURT. LONDON, E. C. 4

PREFACE

In this edition of my _Logic_, the text has been revised throughout, several passages have been rewritten, and some sections added. The chiefalterations and additions occur in cc. I. , v. , ix. , xiii. , xvi. , xvii. , xx. 

The work may be considered, on the whole, as attached to the school ofMill; to whose _System of Logic_, and to Bain's _Logic_, it is deeplyindebted. Amongst the works of living writers, the _Empirical Logic_ ofDr. Venn and the _Formal Logic_ of Dr. Keynes have given me mostassistance. To some others acknowledgments have been made as occasionarose. 

For the further study of contemporary opinion, accessible in English, one may turn to such works as Mr. Bradley's _Principles of Logic_, Dr. Bosanquet's _Logic; or the Morphology of Knowledge_, Prof. Hobhouse's_Theory of Knowledge_, Jevon's _Principles of Science_, and Sigwart's_Logic_. Ueberweg's _Logic, and History of Logical Doctrine_ isinvaluable for the history of our subject. The attitude toward Logic ofthe Pragmatists or Humanists may best be studied in Dr. Schiller's_Formal Logic_, and in Mr. Alfred Sidgwick's _Process of Argument_ andrecent _Elementary Logic_. The second part of this last work, on the"Risks of Reasoning, " gives an admirably succinct account of theirposition. I agree with the Humanists that, in all argument, theimportant thing to attend to is the meaning, and that the most seriousdifficulties of reasoning occur in dealing with the matter reasonedabout; but I find that a pure science of relation has a necessary placein the system of knowledge, and that the formulæ known as laws ofcontradiction, syllogism and causation are useful guides in the framingand testing of arguments and experiments concerning matters of fact. Incisive criticism of traditionary doctrines, with some remarkablereconstructions, may be read in Dr. Mercier's _New Logic_. 

In preparing successive editions of this book, I have profited by thecomments of my friends: Mr. Thomas Whittaker, Prof. Claude Thompson, Dr. Armitage Smith, Mr. Alfred Sidgwick, Dr. Schiller, Prof. Spearman, andProf. Sully, have made important suggestions; and I might have profitedmore by them, if the frame of my book, or my principles, had been moreelastic. 

As to the present edition, useful criticisms have been received from Mr. S. C. Dutt, of Cotton College, Assam, and from Prof. M. A. Roy, ofMidnapore; and, especially, I must heartily thank my colleague, Dr. Wolf, for communications that have left their impress upon nearly everychapter. 

CARVETH READ. 

LONDON, _August_, 1914

CONTENTS

 PAGE

PREFACE v

CHAPTER I

INTRODUCTORY

§1. Definition of Logic 1§2. General character of proof 2§3. Division of the subject 5§4. Uses of Logic 6§5. Relation of Logic to other sciences 8 to Mathematics (p. 8); to concrete Sciences (p. 10); to Metaphysics (p. 10); to regulative sciences (p. 11)§6. Schools of Logicians 11 Relation to Psychology (p. 13)

CHAPTER II

GENERAL ANALYSIS OF PROPOSITIONS

§1. Propositions and Sentences 16§2. Subject, Predicate and Copula 17§3. Compound Propositions 17§4. Import of Propositions 19§5. Form and Matter 22§6. Formal and Material Logic 23§7. Symbols used in Logic 24

CHAPTER III

OF TERMS AND THEIR DENOTATION

§1. Some Account of Language necessary 27§2. Logic, Grammar and Rhetoric 28§3. Words are Categorematic or Syncategorematic 29§4. Terms Concrete or Abstract 30§5. Concrete Terms, Singular, General or Collective 33

CHAPTER IV

THE CONNOTATION OF TERMS

§1. Connotation of General Names 37§2. Question of Proper Names 38 other Singular Names (p. 40)§3. Question of Abstract Terms 40§4. Univocal and Equivocal Terms 41 Connotation determined by the _suppositio_ (p. 43)§5. Absolute and Relative Terms 43§6. Relation of Denotation to Connotation 46§7. Contradictory Terms 47§8. Positive and Negative Terms 50 Infinites; Privitives; Contraries (pp. 50-51)

CHAPTER V

CLASSIFICATION OF PROPOSITIONS

§1. As to Quantity 53 Quantity of the Predicate (p. 56)§2. As to Quality 57 Infinite Propositions (p. 57)§3. A. I. E. O. 58§4. As to Relation 59 Change of Relation (p. 60); Interpretation of 'either, or' (p. 63); Function of the hypothetical form (p. 64)§5. As to Modality 66§6. Verbal and Real Propositions 67

CHAPTER VI

CONDITIONS OF IMMEDIATE INFERENCE

§1. Meaning of Inference 69§2. Immediate and Mediate Inference 70§3. The Laws of Thought 72§4. Identity 73§5. Contradiction and Excluded Middle 74§6. The Scope of Formal Inference 76

CHAPTER VII

IMMEDIATE INFERENCES

§1. Plan of the Chapter 79§2. Subalternation 79§3. Connotative Subalternation 80§4. Conversion 82 Reciprocality (p. 84)§5. Obversion 85§6. Contrary Opposition 87§7. Contradictory Opposition 87§8. Sub-contrary Opposition 88§9. The Square of Opposition 89§10. Secondary modes of Immediate Inference 90§11. Immediate Inferences from Conditionals 93

CHAPTER VIII

ORDER OF TERMS, EULER'S DIAGRAMS, LOGICAL EQUATIONS, EXISTENTIAL IMPORT OF PROPOSITIONS

§1. Order of Terms in a proposition 95§2. Euler's Diagrams 97§3. Propositions considered as Equations 101§4. Existential Import of Propositions 104

CHAPTER IX

FORMAL CONDITIONS OF MEDIATE INFERENCE

§1. Nature of Mediate Inference and Syllogism 107§2. General Canons of the Syllogism 108 Definitions of Categorical Syllogism; Middle Term; Minor Term; Major Term; Minor and Major Premise (p. 109) Illicit Process (p. 110); Distribution of the Middle (p. 110); Negative Premises (p. 112); Particular Premises (p. 113)§3. _Dictum de omni et nullo_ 115§4. Syllogism in relation to the Laws of Thought 116§5. Other Kinds of Mediate Inference 118

CHAPTER X

CATEGORICAL SYLLOGISMS

§1. Illustrations of the Syllogism 121§2. Of Figures 122§3. Of Moods 123§4. How valid Moods are determined 124§5. Special Canons of the Four Figures 126§6. Ostensive Reduction and the Mnemonic Verses 127§7. Another version of the Mnemonic Verses 132§8. Indirect Reduction 132§9. Uses of the several Figures 134§10. Scientific Value of Reduction 135§11. Euler's Diagrams for the Syllogism 136

CHAPTER XI

ABBREVIATED AND COMPOUND ARGUMENTS

§1. Popular Arguments Informal 138§2. The Enthymeme 139§3. Monosyllogism, Polysyllogism, Prosyllogism, Episyllogism 141§4. The Epicheirema 142§5. The Sorites 142§6. The Antinomy 145

CHAPTER XII

CONDITIONAL SYLLOGISMS

§1. The Hypothetical Syllogism 147§2. The Disjunctive Syllogism 152§3. The Dilemma 154

CHAPTER XIII

TRANSITION TO INDUCTION

§1. Formal Consistency and Material Truth 159§2. Real General Propositions assert more than has been directly observed 160§3. Hence, formally, a Syllogism's Premises seem to beg the Conclusion 162§4. Materially, a Syllogism turns upon the resemblance of the Minor to the Middle Term; and thus extends the Major Premise to new cases 163§5. Restatement of the _Dictum_ for material reasoning 165§6. Uses of the Syllogism 167§7. Analysis of the Uniformity of Nature, considered as the formal ground of all reasoning 169§8. Grounds of our belief in Uniformity 173

CHAPTER XIV

CAUSATION

§1. The most important aspect of Uniformity in relation to Induction is Causation 174§2. Definition of "Cause" explained: five marks of Causation 175§3. How strictly the conception of Cause can be applied depends upon the subject under investigation 183§4. Scientific conception of Effect. Plurality of Causes 185§5. Some condition, but not the whole cause, may long precede the Effect; and some co-effect, but not the whole effect, may long survive the Cause 187§6. Mechanical Causes and the homogeneous Intermixture of Effects; Chemical Causes and the heteropathic Intermixture of Effects 188§7. Tendency, Resultant, Counteraction, Elimination, Resolution, Analysis, Reciprocity 189

CHAPTER XV

INDUCTIVE METHOD

§1. Outline of Inductive investigation 192§2. Induction defined 196§3. "Perfect Induction" 196§4. Imperfect Induction methodical or immethodical 197§5. Observation and Experiment, the material ground of Induction, compared 198§6. The principle of Causation is the formal ground of Induction 201§7. The Inductive Canons are derived from the principle of Causation, the more readily to detect it in facts observed 202

CHAPTER XVI

THE CANONS OF DIRECT INDUCTION

§1. The Canon of Agreement 206 Negative Instances (p. 208); Plurality of Causes (p. 208) Agreement may show connection without direct Causation (p. 209)§2. The Canon of Agreement in Presence and in Absence 212 It tends to disprove a Plurality of Causes (p. 213)§3. The Canon of Difference 216 May be applied to observations (p. 221)§4. The Canon of Variations 222 How related to Agreement and Difference (p. 222); The Graphic Method (p. 227); Critical points (p. 230); Progressive effects (p. 231); Gradations (p. 231)§5. The Canon of Residues 232

CHAPTER XVII

COMBINATION OF INDUCTION WITH DEDUCTION

§1. Deductive character of Formal Induction 236§2. Further complication of Deduction with Induction 238§3. The Direct Deductive (or Physical) Method 240§4. Opportunities of Error in the Physical Method 243§5. The Inverse Deductive (or Historical) Method 246§6. Precautions in using the Historical Method 251§7. The Comparative Method 255§8. Historical Evidence 261

CHAPTER XVIII

HYPOTHESES

§1. Hypothesis defined and distinguished from Theory 266§2. An Hypothesis must be verifiable 268§3. Proof of Hypotheses 270 (1) Must an hypothetical agent be directly observable? (p. 270); _Vera causa_ (p. 271) (2) An Hypothesis must be adequate to its pretensions (p. 272); _Exceptio probat regulam_ (p. 274) (3) Every competing Hypothesis must be excluded (p. 275); Crucial instance (p. 277) (4) Hypotheses must agree with the laws of Nature (p. 279)§4. Hypotheses necessary in scientific investigation 280§5. The Method of Abstractions 283 Method of Limits (p. 284); In what sense all knowledge is hypothetical (p. 286)

CHAPTER XIX

LAWS CLASSIFIED; EXPLANATION; CO-EXISTENCE; ANALOGY

§1. Axioms; Primary Laws; Secondary Laws, Derivative or Empirical; Facts 288§2. Secondary Laws either Invariable or Approximate Generalisations 292§3. Secondary Laws trustworthy only in 'Adjacent Cases' 293§4. Secondary Laws of Succession or of Co-existence 295 Natural Kinds (p. 296); Co-existence of concrete things to be deduced from Causation (p. 297)§5. Explanation consists in tracing resemblance, especially of Causation 299§6. Three modes of Explanation 302 Analysis (p. 302); Concatenation (p. 302); Subsumption (p. 303)§7. Limits of Explanation 305§8. Analogy 307

CHAPTER XX

PROBABILITY

§1. Meaning of Chance and Probability 310§2. Probability as a fraction or proportion 312§3. Probability depends upon experience and statistics 313§4. It is a kind of Induction, and pre-supposes Causation 315§5. Of Averages and the Law of Error 318§6. Interpretation of probabilities 324 Personal Equation (p. 325); meaning of 'Expectation' (p. 325)§7. Rules of the combination of Probabilities 325 Detection of a hidden Cause (p. 326); oral tradition (p. 327); circumstantial and analogical evidence (p. 328)

CHAPTER XXI

DIVISION AND CLASSIFICATION

§1. Classification, scientific, special and popular 330§2. Uses of classification 332§3. Classification, Deductive and Inductive 334§4. Division, or Deductive Classification: its Rules 335§5. Rules for testing a Division 337§6. Inductive Classification 339§7. Difficulty of Natural Classification 341§8. Darwin's influence on the theory of Classification 342§9. Classification of Inorganic Bodies also dependent on Causation 346

CHAPTER XXII

NOMENCLATURE, DEFINITION, PREDICABLES

§1. Precise thinking needs precise language 348§2. Nomenclature and Terminology 349§3. Definition 352§4. Rules for testing a Definition 352§5. Every Definition is relative to a Classification 353§6. Difficulties of Definition 356 Proposals to substitute the Type (p. 356)§7. The Limits of Definition 357§8. The five Predicables 358 Porphyry's Tree (p. 361)§9. Realism and Nominalism 364§10. The Predicaments 366

CHAPTER XXIII

DEFINITION OF COMMON TERMS

§1. The rigour of scientific method must be qualified 369§2. Still, Language comprises the Nomenclature of an imperfect Classification, to which every Definition is relative; 370§3. And an imperfect Terminology 374§4. Maxims and precautions of Definition 375§5. Words of common language in scientific use 378§6. How Definitions affect the cogency of arguments 380

CHAPTER XXIV

FALLACIES

§1. Fallacy defined and divided 385§2. Formal Fallacies of Deduction 385§3. Formal Fallacies of Induction 388§4. Material Fallacies classified 394§5. Fallacies of Observation 394§6. Begging the Question 396§7. Surreptitious Conclusion 398§8. Ambiguity 400§9. Fallacies, a natural rank growth of the Human mind, not easy to classify, or exterminate 403

QUESTIONS 405

LOGIC

CHAPTER I

INTRODUCTORY

§ 1. Logic is the science that explains what conditions must befulfilled in order that a proposition may be proved, if it admits ofproof. Not, indeed, every such proposition; for as to those that declarethe equality or inequality of numbers or other magnitudes, to explainthe conditions of their proof belongs to Mathematics: they are said tobe _quantitative_. But as to all other propositions, called_qualitative_, like most of those that we meet with in conversation, inliterature, in politics, and even in sciences so far as they are nottreated mathematically (say, Botany and Psychology); propositions thatmerely tell us that something happens (as that _salt dissolves inwater_), or that something has a certain property (as that _ice iscold_): as to these, it belongs to Logic to show how we may judgewhether they are true, or false, or doubtful. When propositions areexpressed with the universality and definiteness that belong toscientific statements, they are called laws; and laws, so far as theyare not laws of quantity, are tested by the principles of Logic, if theyat all admit of proof. 

But it is plain that the process of proving cannot go on for ever;something must be taken for granted; and this is usually considered tobe the case (1) with particular facts that can only be perceived andobserved, and (2) with those highest laws that are called 'axioms' or'first principles, ' of which we can only say that we know of noexceptions to them, that we cannot help believing them, and that theyare indispensable to science and to consistent thought. Logic, then, maybe briefly defined as the science of proof with respect to _qualitative_laws and propositions, except those that are axiomatic. 

§ 2. Proof may be of different degrees or stages of completeness. Absolute proof would require that a proposition should be shown to agreewith all experience and with the systematic explanation of experience, to be a necessary part of an all-embracing and self-consistentphilosophy or theory of the universe; but as no one hitherto has beenable to frame such a philosophy, we must at present put up withsomething less than absolute proof. Logic, assuming certain principlesto be true of experience, or at least to be conditions of consistentdiscourse, distinguishes the kinds of propositions that can be shown toagree with these principles, and explains by what means the agreementcan best be exhibited. Such principles are those of Contradiction (chap. Vi. ), the Syllogism (chap. Ix. ), Causation (chap. Xiv. ), andProbabilities (chap. Xx. ). To bring a proposition or an argument underthem, or to show that it agrees with them, is logical proof. 

The extent to which proof is requisite, again, depends upon the presentpurpose: if our aim be general truth for its own sake, a systematicinvestigation is necessary; but if our object be merely to remove someoccasional doubt that has occurred to ourselves or to others, it may beenough to appeal to any evidence that is admitted or not questioned. Thus, if a man doubts that _some acids are compounds of oxygen_, butgrants that _some compounds of oxygen are acids_, he may agree to theformer proposition when you point out that it has the same meaning asthe latter, differing from it only in the order of the words. This iscalled proof by immediate inference. 

Again, suppose that a man holds in his hand a piece of yellow metal, which he asserts to be copper, and that we doubt this, perhapssuggesting that it is really gold. Then he may propose to dip it invinegar; whilst we agree that, if it then turns green, it is copper andnot gold. On trying this experiment the metal does turn green; so thatwe may put his argument in this way:--

 _Whatever yellow metal turns green in vinegar is copper; This yellow metal turns green in vinegar; Therefore, this yellow metal is copper. _

Such an argument is called proof by mediate inference; because onecannot see directly that the yellow metal is copper; but it is admittedthat any yellow metal is copper that turns green in vinegar, and we areshown that this yellow metal has that property. 

Now, however, it may occur to us, that the liquid in which the metal wasdipped was not vinegar, or not pure vinegar, and that the greenness wasdue to the impurity. Our friend must thereupon show by some means thatthe vinegar was pure; and then his argument will be that, since nothingbut the vinegar came in contact with the metal, the greenness was due tothe vinegar; or, in other words, that contact with that vinegar was thecause of the metal turning green. 

Still, on second thoughts, we may suspect that we had formerly concededtoo much; we may reflect that, although it had often been shown thatcopper turned green in vinegar, whilst gold did not, yet the same mightnot always happen. May it not be, we might ask, that just at thismoment, and perhaps always for the future gold turns, and will turngreen in vinegar, whilst copper does not and never will again? He willprobably reply that this is to doubt the uniformity of causation: he mayhope that we are not serious: he may point out to us that in everyaction of our life we take such uniformity for granted. But he will beobliged to admit that, whatever he may say to induce us to assent to theprinciple of Nature's uniformity, his arguments will not amount tological proof, because every argument in some way assumes thatprinciple. He has come, in fact, to the limits of Logic. Just as Eucliddoes not try to prove that 'two magnitudes equal to the same third areequal to one another, ' so the Logician (as such) does not attempt toprove the uniformity of causation and the other principles of hisscience. 

Even when our purpose is to ascertain some general truth, the results ofsystematic inquiry may have various degrees of certainty. If Logic wereconfined to strict demonstration, it would cover a narrow field. Thegreater part of our conclusions can only be more or less probable. Itmay, indeed, be maintained, not unreasonably, that no judgmentsconcerning matters of fact can be more than probable. Some say that allscientific results should be considered as giving the average of cases, from which deviations are to be expected. Many matters can only betreated statistically and by the methods of Probability. Our ordinarybeliefs are adopted without any methodical examination. But it is theaim, and it is characteristic, of a rational mind to distinguish degreesof certainty, and to hold each judgment with the degree of confidencethat it deserves, considering the evidence for and against it. It takesa long time, and much self-discipline, to make some progress towardrationality; for there are many causes of belief that are not goodgrounds for it--have no value as evidence. Evidence consists of (1)observation; (2) reasoning checked by observation and by logicalprinciples; (3) memory--often inaccurate; (4) testimony--oftenuntrustworthy, but indispensable, since all we learn from books or fromother men is taken on testimony; (5) the agreement of all our results. On the other hand, belief is caused by many influences that are notevidence at all: such are (1) desire, which makes us believe in whateverserves our purpose; fear and suspicion, which (paradoxically) make usbelieve in whatever seems dangerous; (2) habit, which resists whateverdisturbs our prejudices; (3) vanity, which delights to think oneselfalways right and consistent and disowns fallibility; (4) imitativeness, suggestibility, fashion, which carry us along with the crowd. All these, and nobler things, such as love and fidelity, fix our attention uponwhatever seems to support our prejudices, and prevent our attending toany facts or arguments that threaten to overthrow them. 

§ 3. Two departments of Logic are usually recognised, Deduction andInduction; that is, to describe them briefly, proof from principles, andproof from facts. Classification is sometimes made a third department;sometimes its topics are distributed amongst those of the former two. Inthe present work the order adopted is, Deduction in chaps. Ii. To xiii. ;Induction in chaps. Xiii. To xx. ; and, lastly, Classification. But suchdivisions do not represent fundamentally distinct and opposed aspects ofthe science. For although, in discussing any question with an opponentwho makes admissions, it may be possible to combat his views with merelydeductive arguments based upon his admissions; yet in any question ofgeneral truth, Induction and Deduction are mutually dependent and implyone another. 

This may be seen in one of the above examples. It was argued that acertain metal must be copper, because every metal is copper that turnsgreen when dipped in vinegar. So far the proof appealed to a generalproposition, and was deductive. But when we ask how the generalproposition is known to be true, experiments or facts must be alleged;and this is inductive evidence. Deduction then depends on Induction. Butif we ask, again, how any number of past experiments can prove a generalproposition, which must be good for the future as well as for the past, the uniformity of causation is invoked; that is, appeal is made to aprinciple, and that again is deductive proof. Induction then dependsupon Deduction. 

We may put it in this way: Deduction depends on Induction, if generalpropositions are only known to us through the facts: Induction dependson Deduction, because one fact can never prove another, except so far aswhat is true of the one is true of the other and of any other of thesame kind; and because, to exhibit this resemblance of the facts, itmust be stated in a general proposition. 

§ 4. The use of Logic is often disputed: those who have not studied it, often feel confident of their ability to do without it; those who havestudied it, are sometimes disgusted with what they consider to be itssuperficial analysis of the grounds of evidence, or needlesstechnicality in the discussion of details. As to those who, not havingstudied Logic, yet despise it, there will be time enough to discuss itsutility with them, when they know something about it; and as for thosewho, having studied it, turn away in disgust, whether they are justifiedevery man must judge for himself, when he has attained to equalproficiency in the subject. Meanwhile, the following considerations maybe offered in its favour:

Logic states, and partly explains and applies, certain abstractprinciples which all other sciences take for granted; namely, the axiomsabove mentioned--the principles of Contradiction, of the Syllogism andof Causation. By exercising the student in the apprehension of thesetruths, and in the application of them to particular propositions, iteducates the power of abstract thought. Every science is a model ofmethod, a discipline in close and consecutive thinking; and this meritLogic ought to possess in a high degree. 

For ages Logic has served as an introduction to Philosophy that is, toMetaphysics and speculative Ethics. It is of old and honourabledescent: a man studies Logic in very good company. It is the warp uponwhich nearly the whole web of ancient, mediæval and modern Philosophy iswoven. The history of thought is hardly intelligible without it. 

As the science of proof, Logic gives an account of the _general_ natureof evidence deductive and inductive, as applied in the physical andsocial sciences and in the affairs of life. The _general_ nature of suchevidence: it would be absurd of the logician to pretend to instruct thechemist, economist and merchant, as to the _special_ character of theevidence requisite in their several spheres of judgment. Still, byinvestigating the general conditions of proof, he sets every man uponhis guard against the insufficiency of evidence. 

One application of the science of proof deserves special mention:namely, to that department of Rhetoric which has been the mostdeveloped, relating to persuasion by means of oratory, leader-writing, or pamphleteering. It is usually said that Logic is useful to convincethe judgment, not to persuade the will: but one way of persuading thewill is to convince the judgment that a certain course is advantageous;and although this is not always the readiest way, it is the mosthonourable, and leads to the most enduring results. Logic is thebackbone of Rhetoric. 

It has been disputed whether Logic is a science or an art; and, in fact, it may be considered in both ways. As a statement of general truths, oftheir relations to one another, and especially to the first principles, it is a science; but it is an art when, regarding truth as an enddesired, it points out some of the means of attaining it--namely, toproceed by a regular method, to test every judgment by the principles ofLogic, and to distrust whatever cannot be made consistent with them. Logic does not, in the first place, teach us to reason. We learn toreason as we learn to walk and talk, by the natural growth of our powerswith some assistance from friends and neighbours. The way to developone's power of reasoning is, first, to set oneself problems and try tosolve them. Secondly, since the solving of a problem depends upon one'sability to call to mind parallel cases, one must learn as many facts aspossible, and keep on learning all one's life; for nobody ever knewenough. Thirdly one must check all results by the principles of Logic. It is because of this checking, verifying, corrective function of Logicthat it is sometimes called a Regulative or Normative Science. It cannotgive any one originality or fertility of invention; but it enables us tocheck our inferences, revise our conclusions, and chasten the vagariesof ambitious speculation. It quickens our sense of bad reasoning both inothers and in ourselves. A man who reasons deliberately, manages itbetter after studying Logic than he could before, if he is sincere aboutit and has common sense. 

§ 5. The relation of Logic to other sciences:

(a) Logic is regarded by Spencer as co-ordinate with Mathematics, bothbeing Abstract Sciences--that is, sciences of the _relations_ in whichthings stand to one another, whatever the particular things may be thatare so related; and this view seems to be, on the whole, just--subject, however, to qualifications that will appear presently. 

Mathematics treats of the relations of all sorts of things considered asquantities, namely, as equal to, or greater or less than, one another. Things may be quantitatively equal or unequal in _degree_, as incomparing the temperature of bodies; or in _duration_; or in _spatialmagnitude_, as with lines, superficies, solids; or in _number_. And itis assumed that the equality or inequality of things that cannot bedirectly compared, may be proved indirectly on the assumption that'things equal to the same thing are equal, ' etc. 

Logic also treats of the relations of all sorts of things, but not as totheir quantity. It considers (i) that one thing may be like or unlikeanother in certain attributes, as that iron is in many ways like tin orlead, and in many ways unlike carbon or sulphur: (ii) that attributesco-exist or coinhere (or do not) in the same subject, as metalliclustre, hardness, a certain atomic weight and a certain specific gravitycoinhere in iron: and (iii) that one event follows another (or is theeffect of it), as that the placing of iron in water causes it to rust. The relations of likeness and of coinherence are the ground ofClassification; for it is by resemblance of coinhering attributes thatthings form classes: coinherence is the ground of judgments concerningSubstance and Attribute, as that iron is metallic; and the relation ofsuccession, in the mode of Causation, is the chief subject of thedepartment of Induction. It is usual to group together these relationsof attributes and of order in time, and call them qualitative, in orderto contrast them with the quantitative relations which belong toMathematics. And it is assumed that qualitative relations of things, when they cannot be directly perceived, may be proved indirectly byassuming the axiom of the Syllogism (chap. Ix. ) and the law of Causation(chap. Xiv. ). 

So far, then, Logic and Mathematics appear to be co-ordinate anddistinct sciences. But we shall see hereafter that the satisfactorytreatment of that special order of events in time which constitutesCausation, requires a combination of Logic with Mathematics; and so doesthe treatment of Probability. And, again, Logic may be said to be, in acertain sense, 'prior to' or 'above' Mathematics as usually treated. Forthe Mathematics assume that one magnitude must be either equal orunequal to another, and that it cannot be both equal and unequal to it, and thus take for granted the principles of Contradiction and ExcludedMiddle; but the statement and elucidation of these Principles are leftto Logic (chap. Vi. ). The Mathematics also classify and definemagnitudes, as (in Geometry) triangles, squares, cubes, spheres; but theprinciples of classification and definition remain for Logic todiscuss. 

(b) As to the concrete Sciences, such as Astronomy, Chemistry, Zoology, Sociology--Logic (as well as Mathematics) is implied in them all; forall the propositions of which they consist involve causation, co-existence, and class-likeness. Logic is therefore said to be prior tothem or above them: meaning by 'prior' not that it should be studiedearlier, for that is not a good plan; meaning by 'above' not in dignity, for distinctions of dignity amongst liberal studies are absurd. But itis a philosophical idiom to call the abstract 'prior to, ' or 'higherthan, ' the concrete (see Porphyry's Tree, chap. Xxii. § 8); and Logic ismore abstract than Astronomy or Sociology. Philosophy may thank thatidiom for many a foolish notion. 

(c) But, as we have seen, Logic does not investigate the truth, trustworthiness, or validity of its own principles; nor doesMathematics: this task belongs to Metaphysics, or Epistemology, thecriticism of knowledge and beliefs. 

Logic assumes, for example, that things are what to a careful scrutinythey seem to be; that animals, trees, mountains, planets, are bodieswith various attributes, existing in space and changing in time; andthat certain principles, such as Contradiction and Causation, are trueof things and events. But Metaphysicians have raised many plausibleobjections to these assumptions. It has been urged that natural objectsdo not really exist on their own account, but only in dependence on somemind that contemplates them, and that even space and time are only ourway of perceiving things; or, again, that although things do reallyexist on their own account, it is in an entirely different way from thatin which we know them. As to the principle of Contradiction--that if anobject has an attribute, it cannot at the same time and in the same waybe without it (e. G. , if an animal is conscious, it is false that it isnot conscious)--it has been contended that the speciousness of thisprinciple is only due to the obtuseness of our minds, or even to thepoverty of language, which cannot make the fine distinctions that existin Nature. And as to Causation, it is sometimes doubted whether eventsalways have physical causes; and it is often suggested that, grantingthey have physical causes, yet these are such as we can neither perceivenor conceive; belonging not to the order of Nature as we know it, but tothe secret inwardness and reality of Nature, to the wells and reservoirsof power, not to the spray of the fountain that glitters in oureyes--'occult causes, ' in short. Now these doubts and surmises aremetaphysical spectres which it remains for Metaphysics to lay. Logic hasno direct concern with them (although, of course, metaphysicaldiscussion is expected to be logical), but keeps the plain path of plainbeliefs, level with the comprehension of plain men. Metaphysics, asexamining the grounds of Logic itself, is sometimes regarded as 'thehigher Logic'; and, certainly, the study of Metaphysics is necessary toevery one who would comprehend the nature and functions of Logic, or theplace of his own mind and of Reason in the world. 

(d) The relation of Logic to Psychology will be discussed in the nextsection. 

(e) As a Regulative Science, pointing out the conditions of trueinference (within its own sphere), Logic is co-ordinate with (i) Ethics, considered as assigning the conditions of right conduct, and with (ii)Æsthetics, considered as determining the principles of criticism andgood taste. 

§ 6. Three principal schools of Logicians are commonly recognised:Nominalist, Conceptualist, and Materialist, who differ as to what it isthat Logic really treats of: the Nominalists say, 'of language'; theConceptualists, 'of thought'; the Materialists, 'of relations of fact. 'To illustrate these positions let us take authors who, if some of themare now neglected, have the merit of stating their contrasted views witha distinctness that later refinements tend to obscure. 

(a) Whately, a well-known Nominalist, regarded Logic as the Science andArt of Reasoning, but at the same time as "entirely conversant aboutlanguage"; that is to say, it is the business of Logic to discover thosemodes of statement which shall ensure the cogency of an argument, nomatter what may be the subject under discussion. Thus, _All fish arecold-blooded_, ∴ _some cold-blooded things are fish:_ this is a soundinference by the mere manner of expression; and equally sound is theinference, _All fish are warm-blooded_, ∴ _some warm-blooded things arefish_. The latter proposition may be false, but it follows; and(according to this doctrine) Logic is only concerned with the consistentuse of words: the truth or falsity of the proposition itself is aquestion for Zoology. The short-coming of extreme Nominalism lies inspeaking of language as if its meaning were unimportant. But Whately didnot intend this: he was a man of great penetration and common-sense. 

(b) Hamilton, our best-known Conceptualist, defined Logic as the scienceof the "formal laws of thought, " and "of thought as thought, " that is, without regard to the matter thought about. Just as Whately regardedLogic as concerned merely with cogent forms of statement, so Hamiltontreated it as concerned merely with the necessary relations of thought. This doctrine is called Conceptualism, because the simplest element ofthought is the Concept; that is, an abstract idea, such as is signifiedby the word _man, planet, colour, virtue_; not a representative orgeneric image, but the thought of all attributes common to any class ofthings. Men, planets, colours, virtuous actions or characters, have, severally, something in common on account of which they bear thesegeneral names; and the thought of what they have in common, as theground of these names, is a Concept. To affirm or deny one concept ofanother, as _Some men are virtuous_, or _No man is perfectly virtuous_, is to form a Judgment, corresponding to the Proposition of which theother schools of Logic discourse. Conceptualism, then, investigates theconditions of consistent judgment. 

To distinguish Logic from Psychology is most important in connectionwith Conceptualism. Concepts and Judgments being mental acts, orproducts of mental activity, it is often thought that Logic must be adepartment of Psychology. It is recognised of course, that Psychologydeals with much more than Logic does, with sensation, pleasure and pain, emotion, volition; but in the region of the intellect, especially in itsmost deliberate and elaborate processes, namely, conception, judgment, and reasoning, Logic and Psychology seem to occupy common ground. Infact, however, the two sciences have little in common except a fewgeneral terms, and even these they employ in different senses. It isusual to point out that Psychology tries to explain the subjective_processes_ of conception, judgment and reasoning, and to give theirnatural history; but that Logic is wholly concerned with the _results_of such processes, with concepts, judgments and reasonings, and merelywith the validity of the results, that is, with their truth orconsistency; whilst Psychology has nothing to do with their validity, but only with their causes. Besides, the logical judgment (in FormalLogic at least) is quite a different thing from the psychological: thelatter involves feeling and belief, whereas the former is merely a givenrelation of concepts. _S is P_: that is a model logical judgment; therecan be no question of believing it; but it is logically valid if _M isP_ and _S is M_. When, again, in Logic, one deals with belief, itdepends upon evidence; whereas, in Psychology belief is shown to dependupon causes which may have evidentiary value or may not; for Psychologyexplains quite impartially the growth of scientific insight and thegrowth of prejudice. 

(c) Mill, Bain, and Venn are the chief Materialist logicians; and toguard against the error of confounding Materialism in Logic with theontological doctrine that nothing exists but Matter, it may suffice toremember that in Metaphysics all these philosophers are Idealists. Materialism in Logic consists in regarding propositions as affirming ordenying relations (_cf. _ § 5) between matters-of-fact in the widestsense; not only physical facts, but ideas, social and moral relations;it consists, in short, in attending to the meaning of propositions. Ittreats the first principles of Contradiction and Causation as true ofthings so far as they are known to us, and not merely as conditions ortendencies of thought; and it takes these principles as conditions ofright thinking, because they seem to hold good of Nature and human life. 

To these differences of opinion it will be necessary to recur in thenext chapter (§ 4); but here I may observe that it is easy to exaggeratetheir importance in Logic. There is really little at issue betweenschools of logicians as such, and as far as their doctrines runparallel; it is on the metaphysical grounds of their study, or as to itsscope and comprehension, that they find a battle-field. The present workgenerally proceeds upon the third, or Materialist doctrine. If Deductionand Induction are regarded as mutually dependent parts of one science, uniting the discipline of consistent discourse with the method ofinvestigating laws of physical phenomena, the Materialist doctrine, thatthe principles of Logic are founded on fact, seems to be the mostnatural way of thinking. But if the unity of Deduction and Induction isnot disputed by the other schools, the Materialist may regard them asallies exhibiting in their own way the same body of truths. TheNominalist may certainly claim that his doctrine is indispensable:consistently cogent forms of statement are necessary both to theConceptualist and to the Materialist; neither the relations of thoughtnor those of fact can be arrested or presented without the aid oflanguage or some equivalent system of signs. The Conceptualist may urgethat the Nominalist's forms of statement and argument exist for the sakeof their meaning, namely, judgments and reasonings; and that theMaterialist's laws of Nature are only judgments founded upon ourconceptions of Nature; that the truth of observations and experimentsdepends upon our powers of perception; that perception is inseparablefrom understanding, and that a system of Induction may be constructedupon the axiom of Causation, regarded as a principle of Reason, just aswell as by considering it as a law of Nature, and upon much the samelines. The Materialist, admitting all this, may say that a judgment isonly the proximate meaning of a proposition, and that the ultimatemeaning, the meaning of the judgment itself, is always somematter-of-fact; that the other schools have not hitherto been eager torecognise the unity of Deduction and Induction or to investigate theconditions of trustworthy experiments and observations within the limitsof human understanding; that thought is itself a sort of fact, ascomplex in its structure, as profound in its relations, as subtle in itschanges as any other fact, and therefore at least as hard to know; thatto turn away from the full reality of thought in perception, and toconfine Logic to artificially limited concepts, is to abandon the effortto push method to the utmost and to get as near truth as possible; andthat as to Causation being a principle of Reason rather than of Nature, the distinction escapes his apprehension, since Nature seems to be thatto which our private minds turn upon questions of Causation forcorrection and instruction; so that if he does not call Nature theUniversal Reason, it is because he loves severity of style. 

CHAPTER II

GENERAL ANALYSIS OF PROPOSITIONS

§ 1. Since Logic discusses the proof or disproof, or (briefly) thetesting of propositions, we must begin by explaining their nature. Aproposition, then, may first be described in the language of grammar as_a sentence indicative_; and it is usually expressed in the presenttense. 

It is true that other kinds of sentences, optative, imperative, interrogative, exclamatory, if they express or imply an assertion, arenot beyond the view of Logic; but before treating such sentences, Logic, for greater precision, reduces them to their equivalent sentencesindicative. Thus, _I wish it were summer_ may be understood to mean, _The coming of summer is an object of my desire_. _Thou shalt not kill_may be interpreted as _Murderers are in danger of the judgment_. Interrogatories, when used in argument, if their form is affirmative, have negative force, and affirmative force if their form is negative. Thus, _Do hypocrites love virtue?_ anticipates the answer, _No_. _Arenot traitors the vilest of mankind?_ anticipates the answer, _Yes_. Sothat the logical form of these sentences is, _Hypocrites are not loversof virtue_; _Traitors are the vilest of mankind_. Impersonalpropositions, such as _It rains_, are easily rendered into logical formsof equivalent meaning, thus: _Rain is falling_; or (if that betautology), _The clouds are raining_. Exclamations may seem capricious, but are often part of the argument. _Shade of Chatham!_ usually means_Chatham, being aware of our present foreign policy, is muchdisgusted_. It is in fact, an appeal to authority, without theinconvenience of stating what exactly it is that the authority declares. 

§ 2. But even sentences indicative may not be expressed in the way mostconvenient to logicians. _Salt dissolves in water_ is a plain enoughstatement; but the logician prefers to have it thus: _Salt is soluble inwater_. For he says that a proposition is analysable into threeelements: (1) a Subject (as _Salt_) about which something is asserted ordenied; (2) a Predicate (as _soluble in water_) which is asserted ordenied of the Subject, and (3) the Copula (_is_ or _are_, or _is not_ or_are not_), the sign of relation between the Subject and Predicate. TheSubject and Predicate are called the Terms of the proposition: and theCopula may be called the sign of predication, using the verb 'topredicate' indefinitely for either 'to affirm' or 'to deny. ' Thus _S isP_ means that the term _P_ is given as related in some way to the term_S_. We may, therefore, further define a Proposition as 'a sentence inwhich one term is predicated of another. '

In such a proposition as _Salt dissolves_, the copula (_is_) iscontained in the predicate, and, besides the subject, only one elementis exhibited: it is therefore said to be _secundi adjacentis_. When allthree parts are exhibited, as in _Salt is soluble_, the proposition issaid to be _tertii adjacentis_. 

For the ordinary purposes of Logic, in predicating attributes of a thingor class of things, the copula _is_, or _is not_, sufficientlyrepresents the relation of subject and predicate; but when it isdesirable to realise fully the nature of the relation involved, it maybe better to use a more explicit form. Instead of saying_Salt--is--soluble_, we may say _Solubility--coinheres with--the natureof salt_, or _The putting of salt in water--is a cause of--itsdissolving_: thus expanding the copula into a full expression of therelation we have in view, whether coinherence or causation. 

§ 3. The sentences of ordinary discourse are, indeed, for the mostpart, longer and more complicated than the logical form of propositions;it is in order to prove them, or to use them in the proof of otherpropositions, that they are in Logic reduced as nearly as possible tosuch simple but explicit expressions as the above (_tertii adjacentis_). A Compound Proposition, reducible to two or more simple ones, is said tobe exponible. 

The modes of compounding sentences are explained in every grammar-book. One of the commonest forms is the copulative, such as _Salt is bothsavoury and wholesome_, equivalent to two simple propositions: _Salt issavoury; Salt is wholesome. Pure water is neither sapid nor odorous_, equivalent to _Water is not sapid; Water is not odorous_. Or, again, _Tobacco is injurious, but not when used in moderation_, equivalent to_Much tobacco is injurious; a little is not_. 

Another form of Exponible is the Exceptive, as _Kladderadatsch ispublished daily, except on week-days_, equivalent to _Kladderadatsch ispublished on Sunday; it is not published any other day_. Still anotherExponible is the Exclusive, as _Only men use fire_, equivalent to _Menare users of fire; No other animals are_. Exceptive and exclusivesentences are, however, equivalent forms; for we may say, _Kladderadatsch is published only on Sunday_; and _No animals use fire, except men_. 

There are other compound sentences that are not exponible, since, thoughthey contain two or more verbal clauses, the construction shows thatthese are inseparable. Thus, _If cats are scarce, mice are plentiful_, contains two verbal clauses; but _if cats are scarce_ is conditional, not indicative; and _mice are plentiful_ is subject to the conditionthat _cats are scarce_. Hence the whole sentence is called a ConditionalProposition. For the various forms of Conditional Propositions see chap. V. § 4. 

But, in fact, to find the logical force of recognised grammatical formsis the least of a logician's difficulties in bringing the discourses ofmen to a plain issue. Metaphors, epigrams, innuendoes and other figuresof speech present far greater obstacles to a lucid reduction whether forapproval or refutation. No rules can be given for finding everybody'smeaning. The poets have their own way of expressing themselves;sophists, too, have their own way. And the point often lies in what isunexpressed. Thus, "barbarous nations make, the civilised writehistory, " means that civilised nations do not make history, which noneis so brazen as openly to assert. Or, again, "Alcibiades is dead, but Xis still with us"; the whole meaning of this 'exponible' is that X wouldbe the lesser loss to society. Even an epithet or a suffix may imply aproposition: _This personage_ may mean _X is a pretentious nobody_. 

How shall we interpret such illusive predications except by cultivatingour literary perceptions, by reading the most significant authors untilwe are at home with them? But, no doubt, to disentangle the compoundpropositions, and to expand the abbreviations of literature andconversation, is a useful logical exercise. And if it seem a laborioustask thus to reduce to its logical elements a long argument in a speechor treatise, it should be observed that, as a rule, in a long discourseonly a few sentences are of principal importance to the reasoning, therest being explanatory or illustrative digression, and that a closescrutiny of these cardinal sentences will frequently dispense us fromgiving much attention to the rest. 

§ 4. But now, returning to the definition of a Proposition given in § 2, that it is 'a sentence in which one term is predicated of another, ' wemust consider what is the import of such predication. For thedefinition, as it stands, seems to be purely Nominalist. Is aproposition nothing more than a certain synthesis of words; or, is itmeant to correspond with something further, a synthesis of ideas, or arelation of facts?

Conceptualist logicians, who speak of judgments instead ofpropositions, of course define the judgment in their own language. According to Hamilton, it is "a recognition of the relation ofcongruence or confliction in which two concepts stand to each other. " Tolighten the sentence, I have omitted one or two qualifications(Hamilton's _Lectures on Logic_, xiii. ). "Thus, " he goes on "if wecompare the thoughts _water_, _iron_, and _rusting_, we find themcongruent, and connect them into a single thought, thus: _water rustsiron_--in that case we form a judgment. " When a judgment is expressed inwords, he says, it is called a proposition. 

But has a proposition no meaning beyond the judgment it expresses? Mill, who defines it as "a portion of discourse in which a predicate isaffirmed or denied of a subject" (_Logic_, Book 1. , chap. Iv. § 1. ), proceeds to inquire into the import of propositions (Book 1. , chap. V. ), and finds three classes of them: (a) those in which one proper name ispredicated of another; and of these Hobbes's Nominalist definition isadequate, namely, that a proposition asserts or denies that thepredicate is a name for the same thing as the subject, as _Tully isCicero_. 

(b) Propositions in which the predicate means a part (or the whole) ofwhat the subject means, as _Horses are animals_, _Man is a rationalanimal_. These are Verbal Propositions (see below: chap. V. § 6), andtheir import consists in affirming or denying a coincidence between themeanings of names, as _The meaning of 'animal' is part of the meaning of'horse. '_ They are partial or complete definitions. 

But (c) there are also Real Propositions, whose predicates do not meanthe same as their subjects, and whose import consists in affirming ordenying one of five different kinds of matter of fact: (1) That thesubject exists, or does not; as if we say _The bison exists_, _The greatauk is extinct_. (2) Co-existence, as _Man is mortal_; that is, _thebeing subject to death coinheres with the qualities on account of whichwe call certain objects men_. (3) Succession, as _Night follows day_. (4) Causation (a particular kind of Succession), as _Water rusts iron_. (5) Resemblance, as _The colour of this geranium is like that of asoldier's coat_, or _A = B_. 

On comparing this list of real predications with the list of logicalrelations given above (chap. I. § 5 (a)), it will be seen that the twodiffer only in this, that I have there omitted simple Existence. Nothingsimply exists, unrelated either in Nature or in knowledge. Such aproposition as _The bison exists_ may, no doubt, be used in Logic(subject to interpretation) for the sake of custom or for the sake ofbrevity; but it means that some specimens are still to be found in N. America, or in Zoological gardens. 

Controversy as to the Import of Propositions really turns upon adifference of opinion as to the scope of Logic and the foundations ofknowledge. Mill was dissatisfied with the "congruity" of concepts as thebasis of a judgment. Clearly, mere congruity does not justify belief. Inthe proposition _Water rusts iron_, the concepts _water_, _rust_ and_iron_ may be congruous, but does any one assert their connection onthat ground? In the proposition _Murderers are haunted by the ghosts oftheir victims_, the concepts _victim_, _murderer_, _ghost_ have a highdegree of congruity; yet, unfortunately, I cannot believe it: thereseems to be no such cheap defence of innocence. Now, Mill held thatLogic is concerned with the grounds of belief, and that the scope ofLogic includes Induction as well as Deduction; whereas, according toHamilton, Induction is only Modified Logic, a mere appendix to thetheory of the "forms of thought as thought. " Indeed, Mill endeavoured inhis _Logic_ to probe the grounds of belief deeper than usual, andintroduced a good deal of Metaphysics--either too much or notenough--concerning the ground of axioms. But, at any rate, his greatpoint was that belief, and therefore (for the most part) the RealProposition, is concerned not merely with the relations of words, oreven of ideas, but with matters of fact; that is, both propositions andjudgments point to something further, to the relations of things whichwe can examine, not merely by thinking about them (comparing them inthought), but by observing them with the united powers of thought andperception. This is what convinces us that _water rusts iron_: and thedifficulty of doing this is what prevents our feeling sure that_murderers are haunted by the ghosts of their victims_. Hence, althoughMill's definition of a proposition, given above, is adequate forpropositions in general; yet that kind of proposition (the Real) withregard to which Logic (in Mill's view) investigates the conditions ofproof, may be more explicitly and pertinently defined as 'a predicationconcerning the relation of matters of fact. '

§ 5. This leads to a very important distinction to which we shall oftenhave to refer in subsequent pages--namely, the distinction between theForm and the Matter of a proposition or of an argument. The distinctionbetween Form and Matter, as it is ordinarily employed, is easilyunderstood. An apple growing in the orchard and a waxen apple on thetable may have the same shape or form, but they consist of differentmaterials; two real apples may have the same shape, but contain distinctounces of apple-stuff, so that after one is eaten the other remains tobe eaten. Similarly, tables may have the same shape, though one be madeof marble, another of oak, another of iron. The form is common toseveral things, the matter is peculiar to each. Metaphysicians havecarried the distinction further: apples, they say, may have not only thesame outward shape, but the same inward constitution, which, therefore, may be called the Form of apple-stuff itself--namely, a certainpulpiness, juiciness, sweetness, etc. ; qualities common to all dessertapples: yet their Matter is different, one being here, anotherthere--differing in place or time, if in nothing else. The definition ofa species is the form of every specimen of it. 

To apply this distinction to the things of Logic: it is easy to see howtwo propositions may have the same Form but different Matter: not using'Form' in the sense of 'shape, ' but for that which is common to manythings, in contrast with that which is peculiar to each. Thus, _All malelions are tawny_ and _All water is liquid at 50° Fahrenheit_, are twopropositions that have the same form, though their matter is entirelydifferent. They both predicate something of the whole of their subjects, though their subjects are different, and so are the things predicated ofthem. Again, _All male lions have tufted tails_ and _All male lions havemanes_, are two propositions having the same form and, in theirsubjects, the same matter, but different matter in their predicates. If, however, we take two such propositions as these: _All male lions havemanes_ and _Some male lions have manes_, here the matter is the same inboth, but the form is different--in the first, predication is madeconcerning _every_ male lion; in the second of only _some_ male lions;the first is _universal_, the second is _particular_. Or, again, if wetake _Some tigers are man-eaters_ and _Some tigers are not man-eaters_, here too the matter is the same, but the form is different; for thefirst proposition is _affirmative_, whilst the second is _negative_. 

§ 6. Now, according to Hamilton and Whately, pure Logic has to do onlywith the Form of propositions and arguments. As to their Matter, whetherthey are really true in fact, that is a question, they said, not forLogic, but for experience, or for the special sciences. But Mill desiredso to extend logical method as to test the material truth ofpropositions: he thought that he could expound a method by whichexperience itself and the conclusions of the special sciences may beexamined. 

To this method it may be objected, that the claim to determine MaterialTruth takes for granted that the order of Nature will remain unchanged, that (for example) water not only at present is a liquid at 50°Fahrenheit, but will always be so; whereas (although we have no reasonto expect such a thing) the order of Nature may alter--it is at leastsupposable--and in that event water may freeze at such a temperature. Any matter of fact, again, must depend on observation, either directly, or by inference--as when something is asserted about atoms or ether. Butobservation and material inference are subject to the limitations of ourfaculties; and however we may aid observation by microscopes andmicrometers, it is still observation; and however we may correct ourobservations by repetition, comparison and refined mathematical methodsof making allowances, the correction of error is only an approximationto accuracy. Outside of Formal Reasoning, suspense of judgment is youronly attitude. 

But such objections imply that nothing short of absolute truth has anyvalue; that all our discussions and investigations in science or socialaffairs are without logical criteria; that Logic must be confined tosymbols, and considered entirely as mental gymnastics. In this bookprominence will be given to the character of Logic as a formal science, and it will also be shown that Induction itself may be treated formally;but it will be assumed that logical forms are valuable as representingthe actual relations of natural and social phenomena. 

§ 7. Symbols are often used in Logic instead of concrete terms, not onlyin Symbolic Logic where the science is treated algebraically (as by Dr. Venn in his _Symbolic Logic_), but in ordinary manuals; so that it maybe well to explain the use of them before going further. 

It is a common and convenient practice to illustrate logical doctrinesby examples: to show what is meant by a Proposition we may give _salt issoluble_, or _water rusts iron:_ the copulative exponible is exemplifiedby _salt is savoury and wholesome_; and so on. But this procedure hassome disadvantages: it is often cumbrous; and it may distract thereader's attention from the point to be explained by exciting hisinterest in the special fact of the illustration. Clearly, too, so faras Logic is formal, no particular matter of fact can adequatelyillustrate any of its doctrines. Accordingly, writers on Logic employletters of the alphabet instead of concrete terms, (say) _X_ instead of_salt_ or instead of _iron_, and (say) _Y_ instead of _soluble_ orinstead of _rusted by water_; and then a proposition may be representedby _X is Y_. It is still more usual to represent a proposition by _S is(or is not) P, S_ being the initial of Subject and _P_ of Predicate;though this has the drawback that if we argue--_S is P_, therefore _P isS_, the symbols in the latter proposition no longer have the samesignificance, since the former subject is now the predicate. 

Again, negative terms frequently occur in Logic, such as _not-water_, or_not-iron_, and then if _water_ or _iron_ be expressed by _X_, thecorresponding negative may be expressed by _x_; or, generally, if acapital letter stand for a positive term, the corresponding small letterrepresents the negative. The same device may be adopted to expresscontradictory terms: either of them being _X_, the other is _x_ (seechap. Iv. , §§ 7-8); or the contradictory terms may be expressed by _x_and _x̄_, _y_ and _ȳ_. 

And as terms are often compounded, it may be convenient to express themby a combination of letters: instead of illustrating such a case by_boiling water_ or _water that is boiling_, we may write _XY_; or sincepositive and negative terms may be compounded, instead of illustratingthis by _water that is not boiling_, we may write _Xy_. 

The convenience of this is obvious; but it is more than convenient; for, if one of the uses of Logic be to discipline the power of abstractthought, this can be done far more effectually by symbolic than byconcrete examples; and if such discipline were the only use of Logic itmight be best to discard concrete illustrations altogether, at least inadvanced text-books, though no doubt the practice would be too severefor elementary manuals. On the other hand, to show the practicalapplicability of Logic to the arguments and proofs of actual life, oreven of the concrete sciences, merely symbolic illustration may be notonly useless but even misleading. When we speak of politics, or poetry, or species, or the weather, the terms that must be used can rarely havethe distinctness and isolation of X and Y; so that the perfunctory useof symbolic illustration makes argument and proof appear to be muchsimpler and easier matters than they really are. Our belief in anyproposition never rests on the proposition itself, nor merely upon oneor two others, but upon the immense background of our general knowledgeand beliefs, full of circumstances and analogies, in relation to whichalone any given proposition is intelligible. Indeed, for this reason, itis impossible to illustrate Logic sufficiently: the reader who is inearnest about the cogency of arguments and the limitation of proofs, andis scrupulous as to the degrees of assent that they require, mustconstantly look for illustrations in his own knowledge and experienceand rely at last upon his own sagacity. 

CHAPTER III

OF TERMS AND THEIR DENOTATION

§ 1. In treating of Deductive Logic it is usual to recognise threedivisions of the subject: first, the doctrine of Terms, words, or othersigns used as subjects or predicates; secondly, the doctrine ofPropositions, analysed into terms related; and, thirdly, the doctrine ofthe Syllogism in which propositions appear as the grounds of aconclusion. 

The terms employed are either letters of the alphabet, or the words ofcommon language, or the technicalities of science; and since the wordsof common language are most in use, it is necessary to give some accountof common language as subserving the purposes of Logic. It has beenurged that we cannot think or reason at all without words, or somesubstitute for them, such as the signs of algebra; but this is anexaggeration. Minds greatly differ, and some think by the aid ofdefinite and comprehensive picturings, especially in dealing withproblems concerning objects in space, as in playing chess blindfold, inventing a machine, planning a tour on an imagined map. Most peopledraw many simple inferences by means of perceptions, or of mentalimagery. On the other hand, some men think a good deal without anycontinuum of words and without any imagery, or with none that seemsrelevant to the purpose. Still the more elaborate sort of thinking, thegrouping and concatenation of inferences, which we call reasoning, cannot be carried far without language or some equivalent system ofsigns. It is not merely that we need language to express our reasoningsand communicate them to others: in solitary thought we often depend onwords--'talk to ourselves, ' in fact; though the words or sentences thatthen pass through our minds are not always fully formed or articulated. In Logic, moreover, we have carefully to examine the grounds (at leastthe proximate grounds) of our conclusions; and plainly this cannot bedone unless the conclusions in question are explicitly stated andrecorded. 

Conceptualists say that Logic deals not with the process of thinking(which belongs to Psychology) but with its results; not with conceivingbut with concepts; not with judging but with judgments. Is the conceptself-consistent or adequate? Logic asks; is the judgment capable ofproof? Now, it is only by recording our thoughts in language that itbecomes possible to distinguish between the process and the result ofthought. Without language, the act and the product of thinking would beidentical and equally evanescent. But by carrying on the process inlanguage and remembering or otherwise recording it, we obtain a resultwhich may be examined according to the principles of Logic. 

§ 2. As Logic, then, must give some account of language, it seemsdesirable to explain how its treatment of language differs from that ofGrammar and from that of Rhetoric. 

Grammar is the study of the words of some language, their classificationand derivation, and of the rules of combining them, according to theusage at any time recognised and followed by those who are consideredcorrect writers or speakers. Composition may be faultless in itsgrammar, though dull and absurd. 

Rhetoric is the study of language with a view to obtaining some specialeffect in the communication of ideas or feelings, such aspicturesqueness in description, vivacity in narration, lucidity inexposition, vehemence in persuasion, or literary charm. Some of theseends are often gained in spite of faulty syntax or faulty logic; butsince the few whom bad grammar saddens or incoherent arguments divertare not carried away, as they else might be, by an unsophisticatedorator, Grammar and Logic are necessary to the perfection of Rhetoric. Not that Rhetoric is in bondage to those other sciences; for foreignidioms and such figures as the ellipsis, the anacoluthon, the oxymoron, the hyperbole, and violent inversions have their place in themagnificent style; but authors unacquainted with Grammar and Logic arenot likely to place such figures well and wisely. Indeed, common idioms, though both grammatically and rhetorically justifiable, both correct andeffective, often seem illogical. 'To fall asleep, ' for example, is aperfect English phrase; yet if we examine severally the words itconsists of, it may seem strange that their combination should meananything at all. 

But Logic only studies language so far as necessary in order to state, understand, and check the evidence and reasonings that are usuallyembodied in language. And as long as meanings are clear, good Logic iscompatible with false concords and inelegance of style. 

§ 3. Terms are either Simple or Composite: that is to say, they mayconsist either of a single word, as 'Chaucer, ' 'civilisation'; or ofmore than one, as 'the father of English poetry, ' or 'modern civilisednations. ' Logicians classify words according to their uses in formingpropositions; or, rather, they classify the uses of words as terms, notthe words themselves; for the same word may fall into different classesof terms according to the way in which it is used. (Cf. Mr. AlfredSidgwick's _Distinction and the Criticism of Beliefs_, chap. Xiv. )

Thus words are classified as Categorematic or Syncategorematic. A wordis Categorematic if used singly as a term without the support of otherwords: it is Syncategorematic when joined with other words in order toconstitute the subject or predicate of a proposition. If we say _Venusis a planet whose orbit is inside the Earth's_, the subject, 'Venus, 'is a word used categorematically as a simple term; the predicate is acomposite term whose constituent words (whether substantive, relative, verb, or preposition) are used syncategorematically. 

Prepositions, conjunctions, articles, adverbs, relative pronouns, intheir ordinary use, can only enter into terms along with other wordshaving a substantive, adjectival or participial force; but when they arethemselves the things spoken of and are used substantively (_suppositiomaterialis_), they are categorematic. In the proposition, _'Of' was usedmore indefinitely three hundred years ago than it is now_, 'of' iscategorematic. On the other hand, all substantives may be usedcategorematically; and the same self-sufficiency is usually recognisedin adjectives and participles. Some, however, hold that thecategorematic use of adjectives and participles is due to an ellipsiswhich the logician should fill up; that instead of _Gold is heavy_, heshould say _Gold is a heavy metal_; instead of _The sun is shining_, _The sun is a body shining_. But in these cases the words 'metal' and'body' are unmistakable tautology, since 'metal' is implied in gold and'body' in sun. But, as we have seen, any of these kinds of word, substantive, adjective, or participle, may occur syncategorematically inconnection with others to form a composite term. 

§ 4. Most terms (the exceptions and doubtful cases will be discussedhereafter) have two functions, a denotative and a connotative. A term'sdenotative function is, to be the name or sign of something or somemultitude of things, which are said to be called or denoted by the term. Its connotative function is, to suggest certain qualities andcharacteristics of the things denoted, so that it cannot be usedliterally as the name of any other things; which qualities andcharacteristics are said to be implied or connoted by the term. Thus'sheep' is the name of certain animals, and its connotation prevents itsbeing used of any others. That which a term directly indicates, then, is its _Denotation_; that sense or customary use of it which limits theDenotation is its _Connotation_ (ch. Iv. ). Hamilton and others use'Extension' in the sense of Denotation, and 'Intension' or'Comprehension' in the sense of Connotation. Now, terms may beclassified, first according to what they stand for or denote; that is, according to their _Denotation_. In this respect, the use of a term issaid to be either Concrete or Abstract. 

A term is Concrete when it denotes a 'thing'; that is, any person, object, fact, event, feeling or imagination, considered as capable ofhaving (or consisting of) qualities and a determinate existence. Thus'cricket ball' denotes any object having a certain size, weight, shape, colour, etc. (which are its qualities), and being at any given time insome place and related to other objects--in the bowler's hands, on thegrass, in a shop window. Any 'feeling of heat' has a certain intensity, is pleasurable or painful, occurs at a certain time, and affects somepart or the whole of some animal. An imagination, indeed (say, of afairy), cannot be said in the same sense to have locality; but itdepends on the thinking of some man who has locality, and is definitelyrelated to his other thoughts and feelings. 

A term is Abstract, on the other hand, when it denotes a quality (orqualities), considered by itself and without determinate existence intime, place, or relation to other things. 'Size, ' 'shape, ' 'weight, ''colour, ' 'intensity, ' 'pleasurableness, ' are terms used to denote suchqualities, and are then abstract in their denotation. 'Weight' is notsomething with a determinate existence at a given time; it exists notmerely in some particular place, but wherever there is a heavy thing;and, as to relation, at the same moment it combines in iron withsolidity and in mercury with liquidity. In fact, a quality is a point ofagreement in a multitude of different things; all heavy things agree inweight, all round things in roundness, all red things in redness; and anabstract term denotes such a point (or points) of agreement among thethings denoted by concrete terms. Abstract terms result from theanalysis of concrete things into their qualities; and conversely aconcrete term may be viewed as denoting the synthesis of qualities intoan individual thing. When several things agree in more than one quality, there may be an abstract term denoting the union of qualities in whichthey agree, and omitting their peculiarities; as 'human nature' denotesthe common qualities of men, 'civilisation' the common conditions ofcivilised peoples. 

Every general name, if used as a concrete term, has, or may have, acorresponding abstract term. Sometimes the concrete term is modified toform the abstract, as 'greedy--greediness'; sometimes a word is adaptedfrom another language, as 'man--humanity'; sometimes a composite term isused, as 'mercury--the nature of mercury, ' etc. The same concrete mayhave several abstract correlatives, as 'man--manhood, humanity, humannature'; 'heavy--weight, gravity, ponderosity'; but in such cases theabstract terms are not used quite synonymously; that is, they implydifferent ways of considering the concrete. 

Whether a word is used as a concrete or abstract term is in mostinstances plain from the word itself, the use of most words being prettyregular one way or the other; but sometimes we must judge by thecontext. 'Weight' may be used in the abstract for 'gravity, ' or in theconcrete for a measure; but in the latter sense it is syncategorematic(in the singular), needing at least the article 'a (or the) weight. ''Government' may mean 'supreme political authority, ' and is thenabstract; or, the men who happen to be ministers, and is then concrete;but in this case, too, the article is usually prefixed. 'The life' ofany man may mean his vitality (abstract), as in "Thus following life increatures we dissect"; or, the series of events through which he passes(concrete), as in 'the life of Nelson as narrated by Southey. '

It has been made a question whether the denotation of an abstract termmay itself be the subject of qualities. Apparently 'weight' may begreater or less, 'government' good or bad, 'vitality' intense or dull. But if every subject is modified by a quality, a quality is alsomodified by making it the subject of another; and, if so, it seems thento become a new quality. The compound terms 'great weight, ' 'badgovernment, ' 'dull vitality, ' have not the same denotation as the simpleterms 'weight, 'government, ' 'vitality': they imply, and may be said toconnote, more special concrete experience, such as the effort felt inlifting a trunk, disgust at the conduct of officials, sluggish movementsof an animal when irritated. It is to such concrete experiences that wehave always to refer in order fully to realise the meaning of abstractterms, and therefore, of course, to understand any qualification ofthem. 

§ 5. Concrete terms may be subdivided according to the number of thingsthey denote and the way in which they denote them. A term may denote onething or many: if one, it is called Singular; if many, it may do sodistributively, and then it is General; or, as taken all together, andthen it is Collective: one, then; any one of many; many in one. 

Among Singular Terms, each denoting a single thing, the most obvious areProper Names, such as Gibraltar or George Washington, which are merelymarks of individual things or persons, and may form no part of thecommon language of a country. They are thus distinguished from otherSingular Terms, which consist of common words so combined as to restricttheir denotation to some individual, such as, 'the strongest man onearth. '

Proper Terms are often said to be arbitrary signs, because their usedoes not depend upon any reason that may be given for them. Gibraltarhad a meaning among the Moors when originally conferred; but no one nowknows what it was, unless he happens to have learned it; yet the nameserves its purpose as well as if it were "Rooke's Nest. " Every Newtonor Newport year by year grows old, but to alter the name would causeonly confusion. If such names were given by mere caprice it would makeno difference; and they could not be more cumbrous, ugly, or absurd thanmany of those that are given 'for reasons. '

The remaining kinds of Singular Terms are drawn from the commonresources of the language. Thus the pronouns 'he, ' 'she, ' 'it, ' aresingular terms, whose present denotation is determined by the occasionand context of discourse: so with demonstrative phrases--'the man, ''that horse. ' Descriptive names may be more complex, as 'the wisest manof Gotham, ' which is limited to some individual by the superlativesuffix; or 'the German Emperor, ' which is limited by the definitearticle--the general term 'German Emperor' being thereby restrictedeither to the reigning monarch or to the one we happen to be discussing. Instead of the definite, the indefinite article may be used to makegeneral terms singular, as 'a German Emperor was crowned at Versailles'(_individua vaga_). 

Abstract Terms are ostensively singular: 'whiteness' (e. G. ) is onequality. But their full meaning is general: 'whiteness' stands for allwhite things, so far as white. Abstract terms, in fact, are onlyformally singular. 

General Terms are words, or combinations of words, used to denote anyone of many things that resemble one another in certain respects. 'George III. ' is a Singular Term denoting one man; but 'King' is aGeneral Term denoting him and all other men of the same rank; whilst thecompound 'crowned head' is still more general, denoting kings and alsoemperors. It is the nature of a general term, then, to be used in thesame sense of whatever it denotes; and its most characteristic form isthe Class-name, whether of objects, such as 'king, ' 'sheep, ' 'ghost'; orof events, such as 'accession, ' 'purchase, ' 'manifestation. ' Things andevents are known by their qualities and relations; and every suchaspect, being a point of resemblance to some other things, becomes aground of generalisation, and therefore a ground for the need and use ofgeneral terms. Hence general terms are far the most important sort ofterms in Logic, since in them general propositions are expressed and, moreover (with rare exceptions), all predicates are general. For, besides these typical class-names, attributive words are general terms, such as 'royal, ' 'ruling, ' 'woolly, ' 'bleating, ' 'impalpable, ''vanishing. '

Infinitives may also be used as general terms, as '_To err is human_';but for logical purposes they may have to be translated into equivalentsubstantive forms, as _Foolish actions are characteristic of mankind_. Abstract terms, too, are (as I observed) equivalent to general terms;'folly' is abstract for 'foolish actions. ' '_Honesty is the bestpolicy_' means _people who are honest may hope to find their account inbeing so_; that is, in the effects of their honest actions, providedthey are wise in other ways, and no misfortunes attend them. Theabstract form is often much the more succinct and forcible, but forlogical treatment it needs to be interpreted in the general form. 

By antonomasia proper names may become general terms, as if we say _'AJohnson' would not have written such a book_--i. E. , any man of hisgenius for elaborate eloquence. 

A Collective Term denotes a multitude of similar things considered asforming one whole, as 'regiment, ' 'flock, ' 'nation': not distributively, that is, not the similar things severally; to denote them we must say'soldiers of the regiment, ' 'sheep of the flock, ' and so on. If in amultitude of things there is no resemblance, except the fact of beingconsidered as parts of one whole, as 'the world, ' or 'the town ofNottingham' (meaning its streets and houses, open spaces, people, andcivic organisation), the term denoting them as a whole is Singular; but'the world' or 'town of Nottingham, ' meaning the inhabitants only, isCollective. 

In their strictly collective use, all such expressions are equivalent tosingular terms; but many of them may also be used as general terms, aswhen we speak of 'so many regiments of the line, ' or discuss the'plurality of worlds'; and in this general use they denote any of amultitude of things of the same kind--regiments, or habitable worlds. 

Names of substances, such as 'gold, ' 'air, ' 'water, ' may be employed assingular, collective, or general terms; though, perhaps, as singularterms only figuratively, as when we say _Gold is king_. If we say withThales, '_Water is the source of all things_, ' 'water' seems to be usedcollectively. But substantive names are frequently used as generalterms. For example, _Gold is heavy_ means 'in comparison with otherthings, ' such as water. And, plainly, it does not mean that theaggregate of gold is heavier than the aggregate of water, but only thatits specific gravity is greater; that is, bulk for bulk, any piece ofgold is heavier than water. 

Finally, any class-name may be used collectively if we wish to assertsomething of the things denoted by it, not distributively butaltogether, as that _Sheep are more numerous than wolves_. 

CHAPTER IV

THE CONNOTATION OF TERMS

§ 1. Terms are next to be classified according to theirConnotation--that is, according to what they imply as characteristic ofthe things denoted. We have seen that general names are used to denotemany things in the same sense, because the things denoted resemble oneanother in certain ways: it is this resemblance in certain points thatleads us to class the things together and call them by the same name;and therefore the points of resemblance constitute the sense or meaningof the name, or its Connotation, and limit its applicability to suchthings as have these characteristic qualities. 'Sheep' for example, isused in the same sense, to denote any of a multitude of animals thatresemble one another: their size, shape, woolly coats, cloven hoofs, innocent ways and edibility are well known. When we apply to anythingthe term 'sheep, ' we imply that it has these qualities: 'sheep, 'denoting the animal, connotes its possessing these characteristics; and, of course, it cannot, without a figure of speech or a blunder, be usedto denote anything that does not possess all these qualities. It is by afigure of speech that the term 'sheep' is applied to some men; and toapply it to goats would be a blunder. 

Most people are very imperfectly aware of the connotation of the wordsthey use, and are guided in using them merely by the custom of thelanguage. A man who employs a word quite correctly may be sadly posed bya request to explain or define it. Moreover, so far as we are aware ofthe connotation of terms, the number and the kind of attributes wethink of, in any given case, vary with the depth of our interest, andwith the nature of our interest in the things denoted. 'Sheep' has onemeaning to a touring townsman, a much fuller one to a farmer, and yet adifferent one to a zoologist. But this does not prevent them agreeing inthe use of the word, as long as the qualities they severally include inits meaning are not incompatible. 

All general names, and therefore not only class-names, like 'sheep, ' butall attributives, have some connotation. 'Woolly' denotes anything thatbears wool, and connotes the fact of bearing wool; 'innocent' denotesanything that habitually and by its disposition does no harm (or has notbeen guilty of a particular offence), and connotes a harmless character(or freedom from particular guilt); 'edible' denotes whatever can beeaten with good results, and connotes its suitability for mastication, deglutition, digestion, and assimilation. 

§ 2. But whether all terms must connote as well as denote something, hasbeen much debated. Proper names, according to what seems the betteropinion, are, in their ordinary use, not connotative. To say that theyhave no meaning may seem violent: if any one is called John Doe, thisname, no doubt, means a great deal to his friends and neighbours, reminding them of his stature and physiognomy, his air and gait, his witand wisdom, some queer stories, and an indefinite number of otherthings. But all this significance is local or accidental; it only existsfor those who know the individual or have heard him described: whereas ageneral name gives information about any thing or person it denotes toeverybody who understands the language, without any particular knowledgeof the individual. 

We must distinguish, in fact, between the peculiar associations of theproper name and the commonly recognised meaning of the general name. This is why proper names are not in the dictionary. Such a name asLondon, to be sure, or Napoleon Buonaparte, has a significance notmerely local; still, it is accidental. These names are borne by otherplaces and persons than those that have rendered them famous. There areLondons in various latitudes, and, no doubt, many Napoleon Buonapartesin Louisiana; and each name has in its several denotations an altogetherdifferent suggestiveness. For its suggestiveness is in each applicationdetermined by the peculiarities of the place or person denoted; it isnot given to the different places (or to the different persons) becausethey have certain characteristics in common. 

However, the scientific grounds of the doctrine that proper names arenon-connotative, are these: The peculiarities that distinguish anindividual person or thing are admitted to be infinite, and anythingless than a complete enumeration of these peculiarities may fail todistinguish and identify the individual. For, short of a completeenumeration of them, the description may be satisfied by two or moreindividuals; and in that case the term denoting them, if limited by sucha description, is not a proper but a general name, since it isapplicable to two or more in the same sense. The existence of otherindividuals to whom it applies may be highly improbable; but, if it belogically possible, that is enough. On the other hand, the enumerationof infinite peculiarities is certainly impossible. Therefore propernames have no assignable connotation. The only escape from thisreasoning lies in falling back upon time and place, the principles ofindividuation, as constituting the connotation of proper names. Twothings cannot be at the same time in the same place: hence 'the man whowas at a certain spot on the bridge of Lodi at a certain instant in acertain year' suffices to identify Napoleon Buonaparte for that instant. Supposing no one else to have borne the name, then, is this itsconnotation? No one has ever thought so. And, at any rate, time andplace are only extrinsic determinations (suitable indeed to events likethe battle of Lodi, or to places themselves like London); whereas theconnotation of a general term, such as 'sheep, ' consists of intrinsicqualities. Hence, then, the scholastic doctrine 'that individuals haveno essence' (see chap. Xxii. § 9), and Hamilton's dictum 'that everyconcept is inadequate [sic] to the individual, ' are justified. 

General names, when used as proper names, lose their connotation, asEuxine or Newfoundland. 

Singular terms, other than Proper, have connotation; either inthemselves, like the singular pronouns 'he, ' 'she, ' 'it, ' which aregeneral in their applicability, though singular in application; or, derivatively, from the general names that combine to form them, as in'the first Emperor of the French' or the 'Capital of the BritishEmpire. '

§ 3. Whether Abstract Terms have any connotation is another disputedquestion. We have seen that they denote a quality or qualities ofsomething, and that is precisely what general terms connote: 'honesty'denotes a quality of some men; 'honest' connotes the same quality, whilst denoting the men who have it. 

The denotation of abstract terms thus seems to exhaust their force ormeaning. It has been proposed, however, to regard them as connoting thequalities they directly stand for, and not denoting anything; but surelythis is too violent. To denote something is the same as to be the nameof something (whether real or unreal), which every term must be. It is abetter proposal to regard their denotation and connotation ascoinciding; though open to the objection that 'connote' means 'to markalong with' something else, and this plan leaves nothing else. Millthought that abstract terms are connotative when, besides denoting aquality, they suggest a quality of that quality (as 'fault' implies'hurtfulness'); but against this it may be urged that one quality cannotbear another, since every qualification of a quality constitutes adistinct quality in the total ('milk-whiteness' is distinct from'whiteness, ' _cf. _ chap. Iii. § 4). After all, if it is the mostconsistent plan, why not say that abstract, like proper, terms have noconnotation?

But if abstract terms must be made to connote something, should it notbe those things, indefinitely suggested, to which the qualities belong?Thus 'whiteness' may be considered to connote either snow or vapour, orany white thing, apart from one or other of which the quality has noexistence; whose existence therefore it implies. By this course thedenotation and connotation of abstract and of general names would beexactly reversed. Whilst the denotation of a general name is limited bythe qualities connoted, the connotation of an abstract name includes allthe things in which its denotation is realised. But the whole difficultymay be avoided by making it a rule to translate, for logical purposes, all abstract into the corresponding general terms. 

§ 4. If we ask how the connotation of a term is to be known, the answerdepends upon how it is used. If used scientifically, its connotation isdetermined by, and is the same as, its definition; and the definition isdetermined by examining the things to be denoted, as we shall see inchap. Xxii. If the same word is used as a term in different sciences, as'property' in Law and in Logic, it will be differently defined by them, and will have, in each use, a correspondingly different connotation. Butterms used in popular discourse should, as far as possible, have theirconnotations determined by classical usage, i. E. , by the sense inwhich they are used by writers and speakers who are acknowledged mastersof the language, such as Dryden and Burke. In this case the classicalconnotation determines the definition; so that to define terms thus usedis nothing else than to analyse their accepted meanings. 

It must not, however, be supposed that in popular use the connotationof any word is invariable. Logicians have attempted to classifyterms into Univocal (having only one meaning) and Æquivocal (orambiguous); and no doubt some words (like 'civil, ' 'natural, ' 'proud, ''liberal, ' 'humorous') are more manifestly liable to ambiguous use thansome others. But in truth all general terms are popularly andclassically used in somewhat different senses. 

Figurative or tropical language chiefly consists in the transfer ofwords to new senses, as by metaphor or metonymy. In the course of years, too, words change their meanings; and before the time of Dryden ourwhole vocabulary was much more fluid and adaptable than it has sincebecome. Such authors as Bacon, Milton, and Sir Thomas Browne often usedwords derived from the Latin in some sense they originally had in Latin, though in English they had acquired another meaning. Spenser andShakespeare, besides this practice, sometimes use words in a way thatcan only be justified by their choosing to have it so; whilst theircontemporaries, Beaumont and Fletcher, write the perfect modernlanguage, as Dryden observed. Lapse of time, however, is not the chiefcause of variation in the sense of words. The matters which terms areused to denote are often so complicated or so refined in the assemblage, interfusion, or gradation of their qualities, that terms do not exist insufficient abundance and discriminativeness to denote the things and, atthe same time, to convey by connotation a determinate sense of theiragreements and differences. In discussing politics, religion, ethics, æsthetics, this imperfection of language is continually felt; and theonly escape from it, short of coining new words, is to use such words aswe have, now in one sense, now in another somewhat different, and totrust to the context, or to the resources of the literary art, in orderto convey the true meaning. Against this evil the having been born sinceDryden is no protection. It behoves us, then, to remember that terms arenot classifiable into Univocal and Æquivocal, but that all terms aresusceptible of being used æquivocally, and that honesty and lucidityrequire us to try, as well as we can, to use each term univocally in thesame context. 

The context of any proposition always proceeds upon some assumption orunderstanding as to the scope of the discussion, which controls theinterpretation of every statement and of every word. This was called byDe Morgan the "universe of discourse": an older name for it, revived byDr. Venn, and surely a better one, is _suppositio_. If we are talking ofchildren, and 'play' is mentioned, the _suppositio_ limits thesuggestiveness of the word in one way; whilst if Monaco is the subjectof conversation, the same word 'play, ' under the influence of adifferent _suppositio_, excites altogether different ideas. Hence toignore the _suppositio_ is a great source of fallacies of equivocation. 'Man' is generally defined as a kind of animal; but 'animal' is oftenused as opposed to and excluding man. 'Liberal' has one meaning underthe _suppositio_ of politics, another with regard to culture, and stillanother as to the disposal of one's private means. Clearly, therefore, the connotation of general terms is relative to the _suppositio_, or"universe of discourse. "

§ 5. Relative and Absolute Terms. --Some words go in couples or groups:like 'up-down, ' 'former-latter, ' 'father-mother-children, ''hunter-prey, ' 'cause-effect, ' etc. These are called Relative Terms, and their nature, as explained by Mill, is that the connotations of themembers of such a pair or group are derived from the same set of facts(the _fundamentum relationis_). There cannot be an 'up' without a'down, ' a 'father' without a 'mother' and 'child'; there cannot be a'hunter' without something hunted, nor 'prey' without a pursuer. Whatmakes a man a 'hunter' is his activities in pursuit; and what turns achamois into 'prey' is its interest in these activities. The meaning ofboth terms, therefore, is derived from the same set of facts; neitherterm can be explained without explaining the other, because the relationbetween them is connoted by both; and neither can with propriety be usedwithout reference to the other, or to some equivalent, as 'game' for'prey. '

In contrast with such Relative Terms, others have been called Absoluteor Non-relative. Whilst 'hunter' and 'prey' are relative, 'man' and'chamois' have been considered absolute, as we may use them withoutthinking of any special connection between their meanings. However, ifwe believe in the unity of Nature and in the relativity of knowledge(that is, that all knowledge depends upon comparison, or a perception ofthe resemblances and differences of things), it follows that nothing canbe completely understood except through its agreements or contrasts witheverything else, and that all terms derive their connotation from thesame set of facts, namely, from general experience. Thus both man andchamois are animals; this fact is an important part of the meaning ofboth terms, and to that extent they are relative terms. 'Five yards' and'five minutes' are very different notions, yet they are profoundlyrelated; for their very difference helps to make both notions distinct;and their intimate connection is shown in this, that five yards aretraversed in a certain time, and that five minutes are measured by themotion of an index over some fraction of a yard upon the dial. 

The distinction, then, between relative and non-relative terms mustrest, not upon a fundamental difference between them (since, in fact, all words are relative), but upon the way in which words are used. Wehave seen that some words, such as 'up-down, ' 'cause-effect, ' can onlybe used relatively; and these may, for distinction, be calledCorrelatives. But other words, whose meanings are only partiallyinterdependent, may often be used without attending to their relativity, and may then be considered as Absolute. We cannot say 'the hunterreturned empty handed, ' without implying that 'the prey escaped'; but wemay say 'the man went supperless to bed, ' without implying that 'thechamois rejoiced upon the mountain. ' Such words as 'man' and 'chamois'may, then, in their use, be, as to one another, non-relative. 

To illustrate further the relativity of terms, we may mention some ofthe chief classes of them. 

Numerical order: 1st, 2nd, 3rd, etc. ; 1st implies 2nd, and 2nd 1st;and 3rd implies 1st and 2nd, but these do not imply 3rd; and so on. 

Order in Time or Place: before-after; early-punctual-late;right-middle-left; North-South, etc. 

As to Extent, Volume, and Degree: greater-equal-less;large-medium-small; whole and part. 

Genus and Species are a peculiar case of whole and part (_cf. _ chaps. Xxi. -ii. -iii. ). Sometimes a term connotes all the attributes thatanother does, and more besides, which, as distinguishing it, are calleddifferential. Thus 'man' connotes all that 'animal' does, and also (as_differentiæ_) the erect gait, articulate speech, and other attributes. In such a case as this, where there are well-marked classes, the termwhose connotation is included in the others' is called a Genus of thatSpecies. We have a Genus, triangle; and a Species, isosceles, marked offfrom all other triangles by the differential quality of having two equalsides: again--Genus, book; Species, quarto; Difference, having eachsheet folded into four leaves. 

There are other cases where these expressions 'genus' and 'species'cannot be so applied without a departure from usage, as, e. G. , if wecall snow a species of the genus 'white, ' for 'white' is not arecognised class. The connotation of white (i. E. , whiteness) is, however, part of the connotation of snow, just as the qualities of'animal' are amongst those of 'man'; and for logical purposes it isdesirable to use 'genus and species' to express that relativity ofterms which consists in the connotation of one being part of theconnotation of the other. 

Two or more terms whose connotations severally include that of anotherterm, whilst at the same time exceeding it, are (in relation to thatother term) called Co-ordinate. Thus in relation to 'white, ' snow andsilver are co-ordinate; in relation to colour, yellow and red and blueare co-ordinate. And when all the terms thus related stand forrecognised natural classes, the co-ordinate terms are called co-ordinatespecies; thus man and chamois are (in Logic) co-ordinate species of thegenus animal. 

§ 6. From such examples of terms whose connotations are related as wholeand part, it is easy to see the general truth of the doctrine that asconnotation decreases, denotation increases: for 'animal, ' with lessconnotation than man or chamois, denotes many more objects; 'white, 'with less connotation than snow or silver, denotes many more things, Itis not, however, certain that this doctrine is always true in theconcrete: since there may be a term connoting two or more qualities, allof which qualities are peculiar to all the things it denotes; and, ifso, by subtracting one of the qualities from its connotation, we shouldnot increase its denotation. If 'man, ' for example, has among mammalsthe two peculiar attributes of erect gait and articulate speech, then, by omitting 'articulate speech' from the connotation of man, we couldnot apply the name to any more of the existing mammalia than we can atpresent. Still we might have been able to do so; there might have beenan erect inarticulate ape, and perhaps there once was one; and, if so, to omit 'articulate' from the connotation of man would make the term'man' denote that animal (supposing that there was no other differenceto exclude it). Hence, potentially, an increase of the connotation ofany term implies a decrease of its denotation. And, on the other hand, we can only increase the denotation of a term, or apply it to moreobjects, by decreasing its connotation; for, if the new things denotedby the term had already possessed its whole connotation, they mustalready have been denoted by it. However, we may increase the _known_denotation without decreasing the connotation, if we can discover thefull connotation in things not formerly supposed to have it, as whendolphins were discovered to be mammals; or if we can impose therequisite qualities upon new individuals, as when by annexing somemillions of Africans we extend the denotation of 'British subject'without altering its connotation. 

Many of the things noticed in this chapter, especially in this sectionand the preceding, will be discussed at greater length in the chapterson Classification and Definition. 

§ 7. Contradictory Relative Terms. --Every term has, or may have, anothercorresponding with it in such a way that, whatever differentialqualities (§ 5) it connotes, this other connotes merely their absence;so that one or the other is always formally predicable of any Subject, but both these terms are never predicable of the same Subject in thesame relation: such pairs of terms are called Contradictories. WhateverSubject we take, it is either visible or invisible, but not both; eitherhuman or non-human, but not both. 

This at least is true formally, though in practice we should thinkourselves trifled with if any one told us that 'A mountain is eitherhuman or non-human, but not both. ' It is symbolic terms, such as X andx, that are properly said to be contradictories in relation to anysubject whatever, S or M. For, as we have seen, the ordinary use ofterms is limited by some _suppositio_, and this is true ofContradictories. 'Human' and 'non-human' may refer to zoologicalclassification, or to the scope of physical, mental, or moral powers--asif we ask whether to flourish a dumbbell of a ton weight, or to know thefuture by intuition, or impeccability, be human or non-human. Similarly, 'visible' and 'invisible' refer either to the power of emitting orreflecting light, so that the words have no hold upon a sound or ascent, or else to power of vision and such qualifications as 'with thenaked eye' or 'with a microscope. '

Again, the above definition of Contradictories tells us that they cannotbe predicated of the same Subject "in the same relation"; that is, atthe same time or place, or under the same conditions. The lamp isvisible to me now, but will be invisible if I turn it out; one side ofit is now visible, but the other is not: therefore without thisrestriction, "in the same relation, " few or no terms would becontradictory. 

If a man is called wise, it may mean 'on the whole' or 'in a certainaction'; and clearly a man may for once be wise (or act wisely) who, onthe whole, is not-wise. So that here again, by this ambiguity, termsthat seem contradictory are predicable of the same subject, but not "inthe same relation. " In order to avoid the ambiguity, however, we haveonly to construct the term so as to express the relation, as 'wise onthe whole'; and this immediately generates the contradictory 'not-wiseon the whole. ' Similarly, at one age a man may have black hair, atanother not-black hair; but the difficulty is practically removable bystating the age referred to. 

Still, this case easily leads us to a real difficulty in the use ofcontradictory terms, a difficulty arising from the continuous change or'flux' of natural phenomena. If things are continually changing, it maybe urged that contradictory terms are always applicable to the samesubject, at least as fast as we can utter them: for if we have just saidthat a man's hair is black, since (like everything else) his hair ischanging, it must now be not-black, though (to be sure) it may stillseem black. The difficulty, such as it is, lies in this, that the humanmind and its instrument language are not equal to the subtlety ofNature. All things flow, but the terms of human discourse assume acertain fixity of things; everything at every moment changes, but forthe most part we can neither perceive this change nor express it inordinary language. 

This paradox, however, may, I suppose, be easily over-stated. The changethat continually agitates Nature consists in the movements of masses ormolecules, and such movements of things are compatible with aconsiderable persistence of their qualities. Not only are the molecularchanges always going on in a piece of gold compatible with its remainingyellow, but its persistent yellowness depends on the continuance of someof those changes. Similarly, a man's hair may remain black for someyears; though, no doubt, at a certain age its colour may begin to beproblematical, and the applicability to it of 'black' or 'not-black' maybecome a matter of genuine anxiety. Whilst being on our guard, then, against fallacies of contradiction arising from the imperfectcorrespondence of fact with thought and language, we shall often have toput up with it. Candour and humility having been satisfied by the aboveacknowledgment of the subtlety of Nature, we may henceforward proceedupon the postulate--that it is possible to use contradictory terms suchas cannot both be predicated of the same subject in the same relation, though one of them may be; that, for example, it may be truly said of aman for some years that his hair is black; and, if so, that during thoseyears to call it not-black is false or extremely misleading. 

The most opposed terms of the literary vocabulary, however, such as'wise-foolish, ' 'old-young, ' 'sweet-bitter, ' are rarely truecontradictories: wise and foolish, indeed, cannot be predicated of thesame man in the same relation; but there are many middling men, of whomneither can be predicated on the whole. For the comparison ofquantities, again, we have three correlative terms, 'greater--equal--less, ' and none of these is the contradictory of eitherof the others. In fact, the contradictory of any term is one thatdenotes the sum of its co-ordinates (§ 6); and to obtain acontradictory, the surest way is to coin one by prefixing to the giventerm the particle 'not' or (sometimes) 'non': as 'wise, not-wise, ''human, non-human, ' 'greater, not-greater. '

The separate word 'not' is surer to constitute a contradictory than theusual prefixes of negation, 'un-' or 'in-, ' or even 'non'; sincecompounds of these are generally warped by common use from a purelynegative meaning. Thus, 'Nonconformist' does not denote everybody whofails to conform. 'Unwise' is not equivalent to 'not-wise, ' but means'rather foolish'; a very foolish action is not-wise, but can only becalled unwise by meiosis or irony. Still, negatives formed by 'in' or'un' or 'non' are sometimes really contradictory of their positives; as'visible, invisible, ' 'equal, unequal. '

§ 8. The distinction between Positive and Negative terms is not of muchvalue in Logic, what importance would else attach to it being absorbedby the more definite distinction of contradictories. For contradictoriesare positive and negative in essence and, when least ambiguously stated, also in form. And, on the other hand, as we have seen, when positive andnegative terms are not contradictory, they are misleading. As with'wise-unwise, ' so with many others, such as 'happy-unhappy'; which arenot contradictories; since a man may be neither happy nor unhappy, butindifferent, or (again) so miserable that he can only be called unhappyby a figure of speech. In fact, in the common vocabulary a formalnegative often has a limited positive sense; and this is the case withunhappy, signifying the state of feeling in the milder shades ofPurgatory. 

When a Negative term is fully contradictory of its Positive it is saidto be Infinite; because it denotes an unascertained multitude of things, a multitude only limited by the positive term and the _suppositio_; thus'not-wise' denotes all except the wise, within the _suppositio_ of'intelligent beings. ' Formally (disregarding any _suppositio_), such anegative term stands for all possible terms except its positive: xdenotes everything but X; and 'not-wise' may be taken to include stones, triangles and hippogriffs. And even in this sense, a negative term hassome positive meaning, though a very indefinite one, not a specificpositive force like 'unwise' or 'unhappy': it denotes any and everythingthat has not the attributes connoted by the corresponding positive term. 

Privative Terms connote the absence of a quality that normally belongsto the kind of thing denoted, as 'blind' or 'deaf. ' We may predicate'blind' or 'deaf' of a man, dog or cow that happens not to be able tosee or hear, because the powers of seeing and hearing generally belongto those species; but of a stone or idol these terms can only be usedfiguratively. Indeed, since the contradictory of a privative carrieswith it the privative limitation, a stone is strictly 'not-blind': thatis, it is 'not-something-that-normally-having-sight-wants-it. '

Contrary Terms are those that (within a certain genus or _suppositio_)severally connote differential qualities that are, in fact, mutuallyincompatible in the same relation to the same thing, and thereforecannot be predicated of the same subject in the same relation; and, sofar, they resemble Contradictory Terms: but they differ fromcontradictory terms in this, that the differential quality connoted byeach of them is definitely positive; no Contrary Term is infinite, butis limited to part of the _suppositio_ excluded by the others; so that, possibly, neither of two Contraries is truly predicable of a givensubject. Thus 'blue' and 'red' are Contraries, for they cannot both bepredicated of the same thing in the same relation; but are notContradictories, since, in a given case, neither may be predicable: if aflower is blue in a certain part, it cannot in the same part be red; butit may be neither blue nor red, but yellow; though it is certainlyeither blue or not-blue. All co-ordinate terms are formal Contraries;but if, in fact, a series of co-ordinates comprises only two (asmale-female), they are empirical Contradictories; since each includesall that area of the _suppositio_ which the other excludes. 

The extremes of a series of co-ordinate terms are Opposites; as, in alist of colours, white and black, the most strongly contrasted, are saidto be opposites, or as among moods of feeling, rapture and misery areopposites. But this distinction is of slight logical importance. Imperfect Positive and Negative couples, like 'happy and unhappy, ' which(as we have seen) are not contradictories, are often called Opposites. 

The members of any series of Contraries are all included by any one ofthem and its contradictory, as all colours come under 'red' and'not-red, ' all moods of feeling under 'happy' and 'not-happy. '

CHAPTER V

THE CLASSIFICATION OF PROPOSITIONS

§ 1. Logicians classify Propositions according to Quantity, Quality, Relation and Modality. 

As to Quantity, propositions are either Universal or Particular; that isto say, the predicate is affirmed or denied either of the whole subjector of a part of it--of _All_ or of _Some S_. 

 _All S is P_ (that is, _P_ is predicated of _all S_). _Some S is P_ (that is, _P_ is predicated of _some S_). 

An Universal Proposition may have for its subject a singular term, acollective, a general term distributed, or an abstract term. 

(1) A proposition having a singular term for its subject, as _The Queenhas gone to France_, is called a Singular Proposition; and someLogicians regard this as a third species of proposition with respect toquantity, distinct from the Universal and Particular; but that isneedless. 

(2) A collective term may be the subject, as _The Black Watch is orderedto India_. In this case, as well as in singular propositions, apredication is made concerning the whole subject as a whole. 

(3) The subject may be a general term taken in its full denotation, as_All apes are sagacious_; and in this case a Predication is madeconcerning the whole subject distributively; that is, of each andeverything the subject stands for. 

(4) Propositions whose subjects are abstract terms, though they mayseem to be formally Singular, are really as to their meaningdistributive Universals; since whatever is true of a quality is true ofwhatever thing has that quality so far as that quality is concerned. _Truth will prevail_ means that _All true propositions are accepted atlast_ (by sheer force of being true, in spite of interests, prejudices, ignorance and indifference). To bear this in mind may make one cautiousin the use of abstract terms. 

In the above paragraphs a distinction is implied between Singular andDistributive Universals; but, technically, every term, whether subjector predicate, when taken in its full denotation (or universally), issaid to be 'distributed, ' although this word, in its ordinary sense, would be directly applicable only to general terms. In the aboveexamples, then, 'Queen, ' 'Black Watch, ' 'apes, ' and 'truth' are alldistributed terms. Indeed, a simple definition of the UniversalProposition is 'one whose subject is distributed. '

A Particular Proposition is one that has a general term for its subject, whilst its predicate is not affirmed or denied of everything the subjectdenotes; in other words, it is one whose subject is not distributed: as_Some lions inhabit Africa_. 

In ordinary discourse it is not always explicitly stated whetherpredication is universal or particular; it would be very natural to say_Lions inhabit Africa_, leaving it, as far as the words go, uncertainwhether we mean _all_ or _some_ lions. Propositions whose quantity isthus left indefinite are technically called 'preindesignate, ' theirquantity not being stated or designated by any introductory expression;whilst propositions whose quantity is expressed, as _Allfoundling-hospitals have a high death-rate_, or _Some wine is made fromgrapes_, are said to be 'predesignate. ' Now, the rule is thatpreindesignate propositions are, for logical purposes, to be treated asparticular; since it is an obvious precaution of the science of proof, in any practical application, _not to go beyond the evidence_. Still, the rule may be relaxed if the universal quantity of a preindesignateproposition is well known or admitted, as in _Planets shine withreflected light_--understood of the planets of our solar system at thepresent time. Again, such a proposition as _Man is the paragon ofanimals_ is not a preindesignate, but an abstract proposition; thesubject being elliptical for _Man according to his proper nature_; andthe translation of it into a predesignate proposition is not _All menare paragons_; nor can _Some men_ be sufficient, since an abstract canonly be adequately rendered by a distributed term; but we must say, _Allmen who approach the ideal_. Universal real propositions, true withoutqualification, are very scarce; and we often substitute for them_general_ propositions, saying perhaps--_generally, though notuniversally, S is P_. Such general propositions are, in strictness, particular; and the logical rules concerning universals cannot beapplied to them without careful scrutiny of the facts. 

The marks or predesignations of Quantity commonly used in Logic are: forUniversals, _All_, _Any_, _Every_, _Whatever_ (in the negative _No_ or_No one_, see next §); for Particulars, _Some_. 

Now _Some_, technically used, does not mean _Some only, _ but _Some atleast_ (it may be one, or more, or all). If it meant '_Some only_, 'every particular proposition would be an exclusive exponible (chap. Ii. § 3); since _Only some men are wise_ implies that _Some men are notwise_. Besides, it may often happen in an investigation that all theinstances we have observed come under a certain rule, though we do notyet feel justified in regarding the rule as universal; and thissituation is exactly met by the expression _Some_ (_it may be all_). 

The words _Many_, _Most_, _Few_ are generally interpreted to mean_Some_; but as _Most_ signifies that exceptions are known, and _Few_that the exceptions are the more numerous, propositions thuspredesignate are in fact exponibles, mounting to _Some are_ and _Someare not_. If to work with both forms be too cumbrous, so that we mustchoose one, apparently _Few are_ should be treated as _Some are not_. The scientific course to adopt with propositions predesignate by _Most_or _Few_, is to collect statistics and determine the percentage; thus, _Few men are wise_--say 2 per cent. 

The Quantity of a proposition, then, is usually determined entirely bythe quantity of the subject, whether _all_ or _some_. Still, thequantity of the predicate is often an important consideration; andthough in ordinary usage the predicate is seldom predesignate, Logiciansagree that in every Negative Proposition (see § 2) the predicate is'distributed, ' that is to say, is denied altogether of the subject, andthat this is involved in the form of denial. To say _Some men are notbrave_, is to declare that the quality for which men may be called braveis not found in any of the _Some men_ referred to: and to say _No menare proof against flattery_, cuts off the being 'proof against flattery'entirely from the list of human attributes. On the other hand, everyAffirmative Proposition is regarded as having an undistributedpredicate; that is to say, its predicate is not affirmed exclusively ofthe subject. _Some men are wise_ does not mean that 'wise' cannot bepredicated of any other beings; it is equivalent to _Some men are wise_(_whoever else may be_). And _All elephants are sagacious_ does notlimit sagacity to elephants: regarding 'sagacious' as possibly denotingmany animals of many species that exhibit the quality, this propositionis equivalent to '_All elephants are_ some _sagacious animals_. ' Theaffirmative predication of a quality does not imply exclusive possessionof it as denial implies its complete absence; and, therefore, to regardthe predicate of an affirmative proposition as distributed would be togo beyond the evidence and to take for granted what had never beenalleged. 

Some Logicians, seeing that the quantity of predicates, though notdistinctly expressed, is recognised, and holding that it is the part ofLogic "to make explicit in language whatever is implicit in thought, "have proposed to exhibit the quantity of predicates by predesignation, thus: 'Some men are _some_ wise (beings)'; 'some men are not _any_ brave(beings)'; etc. This is called the Quantification of the Predicate, and leads to some modifications of Deductive Logic which will bereferred to hereafter. (See § 3; chap. Vii. § 4, and chap. Viii. § 3. )

§ 2. As to Quality, Propositions are either Affirmative or Negative. AnAffirmative Proposition is, formally, one whose copula is affirmative(or, has no negative sign), as _S--is--P, All men--are--partial tothemselves_. A Negative Proposition is one whose copula is negative (or, has a negative sign), as _S--is not--P, Some men--are not--proof againstflattery_. When, indeed, a Negative Proposition is of UniversalQuantity, it is stated thus: _No S is P, No men are proof againstflattery_; but, in this case, the detachment of the negative sign fromthe copula and its association with the subject is merely an accident ofour idiom; the proposition is the same as _All men--are not--proofagainst flattery_. It must be distinguished, therefore, from such anexpression as _Not every man is proof against flattery_; for here thenegative sign really restricts the subject; so that the meaningis--_Some men at most_ (it may be _none) are proof against flattery_;and thus the proposition is Particular, and is rendered--_Some men--arenot--proof against flattery_. 

When the negative sign is associated with the predicate, so as to makethis an Infinite Term (chap. Iv. § 8), the proposition is called anInfinite Proposition, as _S is not-P_ (or _p), All men are--incapable ofresisting flattery_, or _are--not-proof against flattery_. 

Infinite propositions, when the copula is affirmative, are formally, themselves affirmative, although their force is chiefly negative; for, as the last example shows, the difference between an infinite and anegative proposition may depend upon a hyphen. It has been proposed, indeed, with a view to superficial simplification, to turn allNegatives into Infinites, and thus render all propositions Affirmativein Quality. But although every proposition both affirms and deniessomething according to the aspect in which you regard it (as _Snow iswhite_ denies that it is any other colour, and _Snow is not blue_affirms that it is some other colour), yet there is a great differencebetween the definite affirmation of a genuine affirmative and the vagueaffirmation of a negative or infinite; so that materially an affirmativeinfinite is the same as a negative. 

Generally Mill's remark is true, that affirmation and denial stand fordistinctions of fact that cannot be got rid of by manipulation of words. Whether granite sinks in water, or not; whether the rook lives a hundredyears, or not; whether a man has a hundred dollars in his pocket, ornot; whether human bones have ever been found in Pliocene strata, ornot; such alternatives require distinct forms of expression. At the sametime, it may be granted that many facts admit of being stated withnearly equal propriety in either Quality, as _No man is proof againstflattery_, or _All men are open to flattery_. 

But whatever advantage there is in occasionally changing the Quality ofa proposition may be gained by the process of Obversion (chap. Vii. §5); whilst to use only one Quality would impair the elasticity oflogical expression. It is a postulate of Logic that the negative signmay be transferred from the copula to the predicate, or from thepredicate to the copula, without altering the sense of a proposition;and this is justified by the experience that not to have an attributeand to be without it are the same thing. 

§ 3. A. I. E. O. --Combining the two kinds of Quantity, Universal andParticular, with the two kinds of Quality, Affirmative and Negative, weget four simple types of proposition, which it is usual to symbolise bythe letters A. I. E. O. , thus:

A. Universal Affirmative -- All S is P. I. Particular Affirmative -- Some S is P. E. Universal Negative -- No S is P. O. Particular Negative -- Some S is not P. 

As an aid to the remembering of these symbols we may observe that A. AndI. Are the first two vowels in _affirmo_ and that E. And O. Are thevowels in _nego_. 

It must be acknowledged that these four kinds of proposition recognisedby Formal Logic constitute a very meagre selection from the list ofpropositions actually used in judgment and reasoning. 

Those Logicians who explicitly quantify the predicate obtain, in all, eight forms of proposition according to Quantity and Quality:

U. Toto-total Affirmative -- All X is all Y. A. Toto-partial Affirmative -- All X is some Y. Y. Parti-total Affirmative -- Some X is all Y. I. Parti-partial Affirmative -- Some X is some Y. E. Toto-total Negative -- No X is any Y. η. Toto-partial Negative -- No X is some Y. O. Parti-total Negative -- Some X is not any Y. ω. Parti-partial Negative -- Some X is not some Y. 

Here A. I. E. O. Correspond with those similarly symbolised in the usuallist, merely designating in the predicates the quantity which wasformerly treated as implicit. 

§ 4. As to Relation, propositions are either Categorical or Conditional. A Categorical Proposition is one in which the predicate is directlyaffirmed or denied of the subject without any limitation of time, place, or circumstance, extraneous to the subject, as _All men in England aresecure of justice_; in which proposition, though there is a limitationof place ('in England'), it is included in the subject. Of this kind arenearly all the examples that have yet been given, according to the form_S is P_. 

A Conditional Proposition is so called because the predication is madeunder some limitation or condition not included in the subject, as _If aman live in England, he is secure of justice_. Here the limitation'living in England' is put into a conditional sentence extraneous to thesubject, 'he, ' representing any man. 

Conditional propositions, again, are of two kinds--Hypothetical andDisjunctive. Hypothetical propositions are those that are limited by anexplicit conditional sentence, as above, or thus: _If Joe Smith was aprophet, his followers have been unjustly persecuted_. Or in symbolsthus:

 If A is, B is; If A is B, A is C; If A is B, C is D. 

Disjunctive propositions are those in which the condition under whichpredication is made is not explicit but only implied under the disguiseof an alternative proposition, as _Joe Smith was either a prophet or animpostor_. Here there is no direct predication concerning Joe Smith, butonly a predication of one of the alternatives conditionally on the otherbeing denied, as, _If Joe Smith was not a prophet he was an impostor_;or, _If he was not an impostor, he was a prophet_. Symbolically, Disjunctives may be represented thus:

 A is either B or C, Either A is B or C is D. 

Formally, every Conditional may be expressed as a Categorical. For ourlast example shows how a Disjunctive may be reduced to two Hypotheticals(of which one is redundant, being the contrapositive of the other; seechap. Vii. § 10). And a Hypothetical is reducible to a Categorical thus:_If the sky is clear, the night is cold_ may be read--_The case of thesky being clear is a case of the night being cold_; and this, though aclumsy plan, is sometimes convenient. It would be better to say _The skybeing clear is a sign of the night being cold_, or a condition of it. For, as Mill says, the essence of a Hypothetical is to state that oneclause of it (the indicative) may be inferred from the other (theconditional). Similarly, we might write: _Proof of Joe Smith's not beinga prophet is a proof of his being an impostor_. 

This turning of Conditionals into Categoricals is called a Change ofRelation; and the process may be reversed: _All the wise are virtuous_may be written, _If any man is wise he is virtuous_; or, again, _Eithera man is not-wise or he is virtuous_. But the categorical form isusually the simplest. 

If, then, as substitutes for the corresponding conditionals, categoricals are formally adequate, though sometimes inelegant, it maybe urged that Logic has nothing to do with elegance; or that, at anyrate, the chief elegance of science is economy, and that therefore, forscientific purposes, whatever we may write further about conditionalsmust be an ugly excrescence. The scientific purpose of Logic is toassign the conditions of proof. Can we, then, in the conditional formprove anything that cannot be proved in the categorical? Or does aconditional require to be itself proved by any method not applicable tothe Categorical? If not, why go on with the discussion of Conditionals?For all laws of Nature, however stated, are essentially categorical. 'Ifa straight line falls on another straight line, the adjacent angles aretogether equal to two right angles'; 'If a body is unsupported, itfalls'; 'If population increases, rents tend to rise': here 'if' means'whenever' or 'all cases in which'; for to raise a doubt whether astraight line is ever conceived to fall upon another, whether bodies areever unsupported, or population ever increases, is a superfluity ofscepticism; and plainly the hypothetical form has nothing to do with theproof of such propositions, nor with inference from them. 

Still, the disjunctive form is necessary in setting out the relation ofcontradictory terms, and in stating a Division (chap. Xxi. ), whetherformal (_as A is B or not-B_) or material (as _Cats are white, or black, or tortoiseshell, or tabby_). And in some cases the hypothetical form isuseful. One of these occurs where it is important to draw attention tothe condition, as something doubtful or especially requiringexamination. _If there is a resisting medium in space, the earth willfall into the sun; If the Corn Laws are to be re-enacted, we had bettersell railways and buy land_: here the hypothetical form draws attentionto the questions whether there is a resisting medium in space, whetherthe Corn Laws are likely to be re-enacted; but as to methods ofinference and proof, the hypothetical form has nothing to do with them. The propositions predicate causation: _A resisting medium in space is acondition of the earth's falling into the sun; A Corn Law is a conditionof the rise of rents, and of the fall of railway profits_. 

A second case in which the hypothetical is a specially appropriate formof statement occurs where a proposition relates to a particular matterand to future time, as _If there be a storm to-morrow, we shall miss ourpicnic_. Such cases are of very slight logical interest. It is asexercises in formal thinking that hypotheticals are of most value;inasmuch as many people find them more difficult than categoricals tomanipulate. 

In discussing Conditional Propositions, the conditional sentence of aHypothetical, or the first alternative of a Disjunctive, is called theAntecedent; the indicative sentence of a Hypothetical, or the secondalternative of a Disjunctive, is called the Consequent. 

Hypotheticals, like Categoricals, have been classed according toQuantity and Quality. Premising that the quantity of a Hypotheticaldepends on the quantity of its Antecedent (which determines itslimitation), whilst its quality depends on the quality of its consequent(which makes the predication), we may exhibit four forms:

 A. _If A is B, C is D;_ I. _Sometimes when A is B, C is D;_ E. _If A is B, C is not D;_ O. _Sometimes when A is B, C is not D. _

But I. And O. Are rarely used. 

As for Disjunctives, it is easy to distinguish the two quantities thus:

 A. _Either A is B, or C is D;_ I. _Sometimes either A is B or C is D. _

But I. Is rarely used. The distinction of quality, however, cannot bemade: there are no true negative forms; for if we write--

 _Neither is A B, nor C D, _

there is here no alternative predication, but only an Exponibleequivalent to _No A is B, and No C is D_. And if we write--

 _Either A is not B, or C is not D, _

this is affirmative as to the alternation, and is for all methods oftreatment equivalent to A. 

Logicians are divided in opinion as to the interpretation of theconjunction 'either, or'; some holding that it means 'not both, ' othersthat it means 'it may be both. ' Grammatical usage, upon which thequestion is sometimes argued, does not seem to be established in favourof either view. If we say _A man so precise in his walk and conversationis either a saint or a consummate hypocrite_; or, again, _One who ishappy in a solitary life is either more or less than man_; we cannot insuch cases mean that the subject may be both. On the other hand, if itbe said that _the author of 'A Tale of a Tub' is either a misanthrope ora dyspeptic_, the alternatives are not incompatible. Or, again, giventhat _X. Is a lunatic, or a lover, or a poet_, the three predicates havemuch congruity. 

It has been urged that in Logic, language should be made as exact anddefinite as possible, and that this requires the exclusiveinterpretation 'not both. ' But it seems a better argument, that Logic(1) should be able to express all meanings, and (2), as the science ofevidence, must not assume more than is given; to be on the safe side, itmust in doubtful cases assume the least, just as it generally assumes apreindesignate term to be of particular quantity; and, therefore'either, or' means 'one, or the other, or both. '

However, when both the alternative propositions have the same subject, as _Either A is B, or A is C_, if the two predicates are contrary orcontradictory terms (as 'saint' and 'hypocrite, ' or 'saint' and'not-saint'), they cannot in their nature be predicable in the same wayof the same subject; and, therefore, in such a case 'either, or' meansone or the other, but not both in the same relation. Hence it seemsnecessary to admit that the conjunction 'either, or' may sometimesrequire one interpretation, sometimes the other; and the rule is that itimplies the further possibility 'or both, ' except when both alternativeshave the same subject whilst the predicates are contrary orcontradictory terms. 

If, then, the disjunctive _A is either B or C_ (_B_ and _C_ beingcontraries) implies that both alternatives cannot be true, it can onlybe adequately rendered in hypotheticals by the two forms--(1) _If A isB, it is not C_, and (2)_If A is not B, it is C_. But if the disjunctive_A is either B or C_ (_B_ and _C_ not being contraries) implies thatboth may be true, it will be adequately translated into a hypotheticalby the single form, _If A is not B, it is C_. We cannot translate itinto--_If A is B, it is not C_, for, by our supposition, if '_A is B_'is true, it does not follow that '_A is C_' must be false. 

Logicians are also divided in opinion as to the function of thehypothetical form. Some think it expresses doubt; for the consequentdepends on the antecedent, and the antecedent, introduced by 'if, ' mayor may not be realised, as in _If the sky is clear, the night is cold_:whether the sky is, or is not, clear being supposed to be uncertain. Andwe have seen that some hypothetical propositions seem designed to drawattention to such uncertainty, as--_If there is a resisting medium inspace_, etc. But other Logicians lay stress upon the connection of theclauses as the important matter: the statement is, they say, that theconsequent may be inferred from the antecedent. Some even declare thatit is given as a necessary inference; and on this ground Sigwart rejectsparticular hypotheticals, such as _Sometimes when A is B, C is D_; forif it happens only sometimes the connexion cannot be necessary. Indeed, it cannot even be probably inferred without further grounds. But this isalso true whenever the antecedent and consequent are concerned withdifferent matter. For example, _If the soul is simple, it isindestructible_. How do you know that? Because _Every simple substanceis indestructible_. Without this further ground there can be noinference. The fact is that conditional forms often cover assertionsthat are not true complex propositions but a sort of euthymemes (chap. Xi. § 2), arguments abbreviated and rhetorically disguised. Thus: _Ifpatience is a virtue there are painful virtues_--an example from Dr. Keynes. Expanding this we have--

 Patience is painful; Patience is a virtue: ∴ Some virtue is painful. 

And then we see the equivocation of the inference; for though patiencebe painful _to learn_, it is not painful _as a virtue_ to the patientman. 

The hypothetical, '_If Plato was not mistaken poets are dangerouscitizens_, ' may be considered as an argument against the laureateship, and may be expanded (informally) thus: 'All Plato's opinions deserverespect; one of them was that poets are bad citizens; therefore itbehoves us to be chary of encouraging poetry. ' Or take thisdisjunctive, '_Either Bacon wrote the works ascribed to Shakespeare, orthere were two men of the highest genius in the same age and country_. 'This means that it is not likely there should be two such men, that weare sure of Bacon, and therefore ought to give him all the glory. Now, if it is the part of Logic 'to make explicit in language all that isimplicit in thought, ' or to put arguments into the form in which theycan best be examined, such propositions as the above ought to beanalysed in the way suggested, and confirmed or refuted according totheir real intention. 

We may conclude that no single function can be assigned to allhypothetical propositions: each must be treated according to its ownmeaning in its own context. 

§ 5. As to Modality, propositions are divided into Pure and Modal. AModal proposition is one in which the predicate is affirmed or denied, not simply but _cum modo_, with a qualification. And some Logicians haveconsidered any adverb occurring in the predicate, or any sign of past orfuture tense, enough to constitute a modal: as 'Petroleum is_dangerously_ inflammable'; 'English _will be_ the universal language. 'But far the most important kind of modality, and the only one we needconsider, is that which is signified by some qualification of thepredicate as to the degree of certainty with which it is affirmed ordenied. Thus, 'The bite of the cobra is _probably_ mortal, ' is called aContingent or Problematic Modal: 'Water is _certainly_ composed ofoxygen and hydrogen' is an Assertory or Certain Modal: 'Two straightlines _cannot_ enclose a space' is a Necessary or Apodeictic Modal (theopposite being inconceivable). Propositions not thus qualified arecalled Pure. 

Modal propositions have had a long and eventful history, but they havenot been found tractable by the resources of ordinary Logic, and are nowgenerally neglected by the authors of text-books. No doubt suchpropositions are the commonest in ordinary discourse, and in some roughway we combine them and draw inferences from them. It is understoodthat a combination of assertory or of apodeictic premises may warrant anassertory or an apodeictic conclusion; but that if we combine either ofthese with a problematic premise our conclusion becomes problematic;whilst the combination of two problematic premises gives a conclusionless certain than either. But if we ask 'How much less certain?' thereis no answer. That the modality of a conclusion follows the less certainof the premises combined, is inadequate for scientific guidance; sothat, as Deductive Logic can get no farther than this, it has abandonedthe discussion of Modals. To endeavour to determine the degree ofcertainty attaching to a problematic judgment is not, however, beyondthe reach of Induction, by analysing circumstantial evidence, or bycollecting statistics with regard to it. Thus, instead of 'The cobra'sbite is _probably_ fatal, ' we might find that it is fatal 80 times in100. Then, if we know that of those who go to India 3 in 1000 arebitten, we can calculate what the chances are that any one going toIndia will die of a cobra's bite (chap. Xx. ). 

§ 6. Verbal and Real Propositions. --Another important division ofpropositions turns upon the relation of the predicate to the subject inrespect of their connotations. We saw, when discussing Relative Terms, that the connotation of one term often implies that of another;sometimes reciprocally, like 'master' and 'slave'; or by inclusion, likespecies and genus; or by exclusion, like contraries and contradictories. When terms so related appear as subject and predicate of the sameproposition, the result is often tautology--e. G. , _The master hasauthority over his slave; A horse is an animal; Red is not blue; Britishis not foreign_. Whoever knows the meaning of 'master, ' 'horse, ' 'red, ''British, ' learns nothing from these propositions. Hence they are calledVerbal propositions, as only expounding the sense of words, or as ifthey were propositions only by satisfying the forms of language, not byfulfilling the function of propositions in conveying a knowledge offacts. They are also called 'Analytic' and 'Explicative, ' when theyseparate and disengage the elements of the connotation of the subject. Doubtless, such propositions may be useful to one who does not know thelanguage; and Definitions, which are verbal propositions whosepredicates analyse the whole connotations of their subjects, areindispensable instruments of science (see chap. Xxii. ). 

Of course, hypothetical propositions may also be verbal, as _If the soulbe material it is extended_; for 'extension' is connoted by 'matter';and, therefore, the corresponding disjunctive is verbal--_Either thesoul is not material, or it is extended_. But a true divisionaldisjunctive can never be verbal (chap. Xxi. § 4, rule 1). 

On the other hand, when there is no such direct relation between subjectand predicate that their connotations imply one another, but thepredicate connotes something that cannot be learnt from the connotationof the subject, there is no longer tautology, but an enlargement ofmeaning--e. G. , _Masters are degraded by their slaves; The horse is thenoblest animal; Red is the favourite colour of the British army; If thesoul is simple, it is indestructible_. Such propositions are calledReal, Synthetic, or Ampliative, because they are propositions for whicha mere understanding of their subjects would be no substitute, since thepredicate adds a meaning of its own concerning matter of fact. 

To any one who understands the language, a verbal proposition can neverbe an inference or conclusion from evidence; nor can a verbalproposition ever furnish grounds for an inference, except as to themeaning of words. The subject of real and verbal propositions willinevitably recur in the chapters on Definition; but tautologies are suchcommon blemishes in composition, and such frequent pitfalls in argument, that attention cannot be drawn to them too early or too often. 

CHAPTER VI

CONDITIONS OF IMMEDIATE INFERENCE

§ 1. The word Inference is used in two different senses, which are oftenconfused but should be carefully distinguished. In the first sense, itmeans a process of thought or reasoning by which the mind passes fromfacts or statements presented, to some opinion or expectation. The datamay be very vague and slight, prompting no more than a guess or surmise;as when we look up at the sky and form some expectation about theweather, or from the trick of a man's face entertain some prejudice asto his character. Or the data may be important and strongly significant, like the footprint that frightened Crusoe into thinking of cannibals, oras when news of war makes the city expect that Consols will fall. Theseare examples of the act of inferring, or of inference as a process; andwith inference in this sense Logic has nothing to do; it belongs toPsychology to explain how it is that our minds pass from one perceptionor thought to another thought, and how we come to conjecture, concludeand believe (_cf. _ chap. I. § 6). 

In the second sense, 'inference' means not this process of guessing oropining, but the result of it; the surmise, opinion, or belief whenformed; in a word, the conclusion: and it is in this sense thatInference is treated of in Logic. The subject-matter of Logic is aninference, judgment or conclusion concerning facts, embodied in aproposition, which is to be examined in relation to the evidence thatmay be adduced for it, in order to determine whether, or how far, theevidence amounts to proof. Logic is the science of Reasoning in thesense in which 'reasoning' means giving reasons, for it shows what sortof reasons are good. Whilst Psychology explains how the mind goesforward from data to conclusions, Logic takes a conclusion and goes backto the data, inquiring whether those data, together with any otherevidence (facts or principles) that can be collected, are of a nature towarrant the conclusion. If we think that the night will be stormy, thatJohn Doe is of an amiable disposition, that water expands in freezing, or that one means to national prosperity is popular education, and wishto know whether we have evidence sufficient to justify us in holdingthese opinions, Logic can tell us what form the evidence should assumein order to be conclusive. What _form_ the evidence should assume: Logiccannot tell us what kinds of fact are proper evidence in any of thesecases; that is a question for the man of special experience in life, orin science, or in business. But whatever facts constitute the evidence, they must, in order to prove the point, admit of being stated inconformity with certain principles or conditions; and of theseprinciples or conditions Logic is the science. It deals, then, not withthe subjective process of inferring, but with the objective grounds thatjustify or discredit the inference. 

§ 2. Inferences, in the Logical sense, are divided into two greatclasses, the Immediate and the Mediate, according to the character ofthe evidence offered in proof of them. Strictly, to speak of inferences, in the sense of conclusions, as immediate or mediate, is an abuse oflanguage, derived from times before the distinction between inference asprocess and inference as result was generally felt. No doubt we oughtrather to speak of Immediate and Mediate Evidence; but it is of littleuse to attempt to alter the traditional expressions of the science. 

An Immediate Inference, then, is one that depends for its proof upononly one other proposition, which has the same, or more extensive, terms (or matter). Thus that _one means to national prosperity ispopular education_ is an immediate inference, if the evidence for it isno more than the admission that _popular education is a means tonational prosperity:_ Similarly, it is an immediate inference that _Someauthors are vain_, if it be granted that _All authors are vain_. 

An Immediate Inference may seem to be little else than a verbaltransformation; some Logicians dispute its claims to be called aninference at all, on the ground that it is identical with the pretendedevidence. If we attend to the meaning, say they, an immediate inferencedoes not really express any new judgment; the fact expressed by it iseither the same as its evidence, or is even less significant. If from_No men are gods_ we prove that _No gods are men_, this is nugatory; ifwe prove from it that _Some men are not gods_, this is to emasculate thesense, to waste valuable information, to lose the commanding sweep ofour universal proposition. 

Still, in Logic, it is often found that an immediate inference expressesour knowledge in a more convenient form than that of the evidentiaryproposition, as will appear in the chapter on Syllogisms and elsewhere. And by transforming an universal into a particular proposition, as _Nomen are gods_, therefore, _Some men are not gods_, --we get a statementwhich, though weaker, is far more easily proved; since a single instancesuffices. Moreover, by drawing all possible immediate inferences from agiven proposition, we see it in all its aspects, and learn all that isimplied in it. 

A Mediate Inference, on the other hand, depends for its evidence upon aplurality of other propositions (two or more) which are connectedtogether on logical principles. If we argue--

 No men are gods; Alexander the Great is a man; ∴ Alexander the Great is not a god:

this is a Mediate Inference. The evidence consists of two propositionsconnected by the term 'man, ' which is common to both (a Middle Term), mediating between 'gods' and 'Alexander. ' Mediate Inferences compriseSyllogisms with their developments, and Inductions; and to discuss themfurther at present would be to anticipate future chapters. We must nowdeal with the principles or conditions on which Immediate Inferences arevalid: commonly called the "Laws of Thought. "

§ 3. The Laws of Thought are conditions of the logical statement andcriticism of all sorts of evidence; but as to Immediate Inference, theymay be regarded as the only conditions it need satisfy. They are oftenexpressed thus: (1) The principle of Identity--'_Whatever is, is_'; (2)The principle of Contradiction--'_It is impossible for the same thing tobe and not be_'; (3) The principle of Excluded Middle--'_Anything musteither be or not be_. ' These principles are manifestly not 'laws' ofthought in the sense in which 'law' is used in Psychology; they do notprofess to describe the actual mental processes that take place injudgment or reasoning, as the 'laws of association of ideas' account formemory and recollection. They are not natural laws of thought; but, inrelation to thought, can only be regarded as laws when stated asprecepts, the observance of which (consciously or not) is necessary toclear and consistent thinking: e. G. , Never assume that the same thingcan both be and not be. 

However, treating Logic as the science of thought only as embodied inpropositions, in respect of which evidence is to be adduced, or whichare to be used as evidence of other propositions, the above laws orprinciples must be restated as the conditions of consistent argument insuch terms as to be directly applicable to propositions. It was shown inthe chapter on the connotation of terms, that terms are assumed byLogicians to be capable of definite meaning, and of being usedunivocally in the same context; if, or in so far as, this is not thecase, we cannot understand one another's reasons nor even pursue insolitary meditation any coherent train of argument. We saw, too, thatthe meanings of terms were related to one another: some being fullcorrelatives; others partially inclusive one of another, as species ofgenus; others mutually incompatible, as contraries; or alternativelypredicable, as contradictories. We now assume that propositions arecapable of definite meaning according to the meaning of their componentterms and of the relation between them; that the meaning, the factasserted or denied, is what we are really concerned to prove ordisprove; that a mere change in the words that constitute our terms, orof construction, does not affect the truth of a proposition as long asthe meaning is not altered, or (rather) as long as no fresh meaning isintroduced; and that if the meaning of any proposition is true, anyother proposition that denies it is false. This postulate is plainlynecessary to consistency of statement and discourse; and consistency isnecessary, if our thought or speech is to correspond with the unity andcoherence of Nature and experience; and the Laws of Thought orConditions of Immediate Inference are an analysis of this postulate. 

§ 4. The principle of Identity is usually written symbolically thus: _Ais A; not-A is not-A_. It assumes that there is something that may berepresented by a term; and it requires that, in any discussion, _everyrelevant term, once used in a definite sense, shall keep that meaningthroughout_. Socrates in his father's workshop, at the battle of Delium, and in prison, is assumed to be the same man denotable by the same name;and similarly, 'elephant, ' or 'justice, ' or 'fairy, ' in the samecontext, is to be understood of the same thing under the same_suppositio_. 

But, further, it is assumed that of a given term another term may bepredicated again and again in the same sense under the same conditions;that is, we may speak of the identity of meaning in a proposition aswell as in a term. To symbolise this we ought to alter the usualformula for Identity and write it thus: _If B is A, B is A; if B isnot-A, B is not-A_. If Socrates is wise, he is wise; if fairies frequentthe moonlight, they do; if Justice is not of this world, it is not. _Whatever affirmation or denial we make concerning any subject, we arebound to adhere to it for the purposes of the current argument orinvestigation. _ Of course, if our assertion turns out to be false, wemust not adhere to it; but then we must repudiate all that we formerlydeduced from it. 

Again, _whatever is true or false in one form of words is true or falsein any other_: this is undeniable, for the important thing is identityof meaning; but in Formal Logic it is not very convenient. If Socratesis wise, is it an identity to say 'Therefore the master of Plato iswise'; or, further that he 'takes enlightened views of life'? If _Everyman is fallible_, is it an identical proposition that _Every man isliable to error_? It seems pedantic to demand a separate propositionthat _Fallible is liable to error_. But, on the other hand, theinsidious substitution of one term for another speciously identical, isa chief occasion of fallacy. How if we go on to argue: therefore, _Everyman is apt to blunder, prone to confusion of thought, inured toself-contradiction_? Practically, the substitution of identities must beleft to candour and good-sense; and may they increase among us. FormalLogic is, no doubt, safest with symbols; should, perhaps, content itselfwith A and B; or, at least, hardly venture beyond Y and Z. 

§ 5. The principle of Contradiction is usually written symbolically, thus: _A is not not-A_. But, since this formula seems to be adapted to asingle term, whereas we want one that is applicable to propositions, itmay be better to write it thus: _B is not both A and not-A_. That is tosay: _if any term may be affirmed of a subject, the contradictory termmay, in the same relation, be denied of it_. A leaf that is green on oneside of it may be not-green on the other; but it is not both green andnot-green on the same surface, at the same time, and in the same light. If a stick is straight, it is false that it is at the same timenot-straight: having granted that two angles are equal, we must denythat they are unequal. 

But is it necessarily false that the stick is 'crooked'; must we denythat either angle is 'greater or less' than the other? How far is itpermissible to substitute any other term for the formal contradictory?Clearly, the principle of Contradiction takes for granted the principleof Identity, and is subject to the same difficulties in its practicalapplication. As a matter of fact and common sense, if we affirm any termof a Subject, we are bound to deny of that Subject, in the samerelation, not only the contradictory but all synonyms for this, and alsoall contraries and opposites; which, of course, are included in thecontradictory. But who shall determine what these are? Without anauthoritative Logical Dictionary to refer to, where all contradictories, synonyms, and contraries may be found on record, Formal Logic willhardly sanction the free play of common sense. 

The principle of Excluded Middle may be written: _B is either A ornot-A_; that is, _if any term be denied of a subject, the contradictoryterm may, in the same relation, be affirmed_. Of course, we may denythat a leaf is green on one side without being bound to affirm that itis not-green on the other. But in the same relation a leaf is eithergreen or not-green; at the same time, a stick is either bent ornot-bent. If we deny that A is greater than B, we must affirm that it isnot-greater than B. 

Whilst, then, the principle of Contradiction (that 'of contradictorypredicates, one being affirmed, the other is denied ') might seem toleave open a third or middle course, the denying of bothcontradictories, the principle of Excluded Middle derives its name fromthe excluding of this middle course, by declaring that the one or theother must be affirmed. Hence the principle of Excluded Middle does nothold good of mere contrary terms. If we deny that a leaf is green, weare not bound to affirm it to be yellow; for it may be red; and then wemay deny both contraries, yellow and green. In fact, two contraries donot between them cover the whole predicable area, but contradictoriesdo: the form of their expression is such that (within the _suppositio_)each includes all that the other excludes; so that the subject (ifbrought within the _suppositio_) must fall under the one or the other. It may seem absurd to say that Mont Blanc is either wise or not-wise;but how comes any mind so ill-organised as to introduce Mont Blanc intothis strange company? Being there, however, the principle is inexorable:Mont Blanc is not-wise. 

In fact, the principles of Contradiction and Excluded Middle areinseparable; they are implicit in all distinct experience, and may beregarded as indicating the two aspects of Negation. The principle ofContradiction says: _B is not both A and not-A_, as if _not-A_ might benothing at all; this is abstract negation. But the principle of ExcludedMiddle says: _Granting that B is not A, it is still something_--namely, _not-A_; thus bringing us back to the concrete experience of a continuumin which the absence of one thing implies the presence of somethingelse. Symbolically: to deny that B is A is to affirm that B is not A, and this only differs by a hyphen from B is not-A. 

These principles, which were necessarily to some extent anticipated inchap. Iv. § 7, the next chapter will further illustrate. 

§ 6. But first we must draw attention to a maxim (also alreadymentioned), which is strictly applicable to Immediate Inferences, though(as we shall see) in other kinds of proof it may be only a formalcondition: this is the general caution _not to go beyond the evidence_. An immediate inference ought to contain nothing that is not contained(or formally implied) in the proposition by which it is proved. Withrespect to quantity in denotation, this caution is embodied in the rule'not to distribute any term that is not given distributed. ' Thus, ifthere is a predication concerning 'Some S, ' or 'Some men, ' as in theforms I. And O. , we cannot infer anything concerning 'All S. ' or 'Allmen'; and, as we have seen, if a term is given us preindesignate, we aregenerally to take it as of particular quantity. Similarly, in the caseof affirmative propositions, we saw that this rule requires us to assumethat their predicates are undistributed. 

As to the grounds of this maxim, not to go beyond the evidence, not todistribute a term that is given as undistributed, it is one of thethings so plain that to try to justify is only to obscure them. Still, we must here state explicitly what Formal Logic assumes to be containedor implied in the evidence afforded by any proposition, such as 'All Sis P. ' If we remember that in chap. Iv. § 7, it was assumed that everyterm may have a contradictory; and if we bear in mind the principles ofContradiction and Excluded Middle, it will appear that such aproposition as 'All S is P' tells us something not only about therelations of 'S' and 'P, ' but also of their relations to 'not-S' and'not-P'; as, for example, that 'S is not not-P, ' and that 'not-P isnot-S. ' It will be shown in the next chapter how Logicians havedeveloped these implications in series of Immediate Inferences. 

If it be asked whether it is true that every term, itself significant, has a significant contradictory, and not merely a formal contradictory, generated by force of the word 'not, ' it is difficult to give any betteranswer than was indicated in §§ 3-5, without venturing further intoMetaphysics. I shall merely say, therefore, that, granting that somesuch term as 'Universe' or 'Being' may have no significantcontradictory, if it stand for 'whatever can be perceived or thoughtof'; yet every term that stands for less than 'Universe' or 'Being' has, of course, a contradictory which denotes the rest of the universe. Andsince every argument or train of thought is carried on within a special'universe of discourse, ' or under a certain _suppositio_, we may saythat _within the given suppositio every term has a contradictory_, andthat every predication concerning a term implies some predicationconcerning its contradictory. But the name of the _suppositio_ itselfhas no contradictory, except with reference to a wider and inclusive_suppositio_. 

The difficulty of actual reasoning, not with symbols, but about mattersof fact, does not arise from the principles of Logic, but sometimes fromthe obscurity or complexity of the facts, sometimes from the ambiguityor clumsiness of language, sometimes from the deficiency of our ownminds in penetration, tenacity and lucidity. One must do one's best tostudy the facts, and not be too easily discouraged. 

CHAPTER VII

IMMEDIATE INFERENCES

§ 1. Under the general title of Immediate Inference Logicians discussthree subjects, namely, Opposition, Conversion, and Obversion; to whichsome writers add other forms, such as Whole and Part in Connotation, Contraposition, Inversion, etc. Of Opposition, again, all recognisefour modes: Subalternation, Contradiction, Contrariety andSub-contrariety. The only peculiarities of the exposition upon which weare now entering are, that it follows the lead of the three Laws ofThought, taking first those modes of Immediate Inference in whichIdentity is most important, then those which plainly involveContradiction and Excluded Middle; and that this method results inseparating the modes of Opposition, connecting Subalternation withConversion, and the other modes with Obversion. To make up for thisdeparture from usage, the four modes of Opposition will be broughttogether again in § 9. 

§ 2. Subalternation. --Opposition being the relation of propositions thathave the same matter and differ only in form (as A. , E. , I. , O. ), propositions of the forms A. And I. Are said to be Subalterns inrelation to one another, and so are E. And O. ; the universal of eachquality being distinguished as 'subalternans, ' and the particular as'subalternate. '

It follows from the principle of Identity that, the matter of thepropositions being the same, if A. Is true I. Is true, and that if E. Istrue O. Is true; for A. And E. Predicate something of _All S_ or _Allmen_; and since I. And O. Make the same predication of _Some S_ or_Some men_, the sense of these particular propositions has already beenpredicated in A. Or E. If _All S is P, Some S is P_; if _No S is P, SomeS is not P_; or, if _All men are fond of laughing, Some men are_; if _Nomen are exempt from ridicule, Some men are not_. 

Similarly, if I. Is false A. Is false; if O. Is false E. Is false. If wedeny any predication about _Some S_, we must deny it of _All S_; sincein denying it of _Some_, we have denied it of at least part of _All_;and whatever is false in one form of words is false in any other. 

On the other hand, if I. Is true, we do not know that A. Is; nor if O. Is true, that E. Is; for to infer from _Some_ to _All_ would be goingbeyond the evidence. We shall see in discussing Induction that the greatproblem of that part of Logic is, to determine the conditions underwhich we may in reality transcend this rule and infer from _Some_ to_All_; though even there it will appear that, formally, the rule isobserved. For the present it is enough that I. Is an immediate inferencefrom A. , and O. From E. ; but that A. Is not an immediate inference fromI. , nor E. From O. 

§ 3. Connotative Subalternation. --We have seen (chap. Iv. § 6) that ifthe connotation of one term is only part of another's its denotation isgreater and includes that other's. Hence genus and species stand insubaltern relation, and whatever is true of the genus is true of thespecies: If _All animal life is dependent on vegetation, All human lifeis dependent on vegetation_. On the other hand, whatever is not true ofthe species or narrower term, cannot be true of the whole genus: If itis false that '_All human life is happy_, ' it is false that '_All animallife is happy_. '

Similar inferences may be drawn from the subaltern relation ofpredicates; affirming the species we affirm the genus. To take Mill'sexample, if _Socrates is a man, Socrates is a living creature_. On theother hand, denying the genus we deny the species: if _Socrates is notvicious, Socrates is not drunken_. 

Such cases as these are recognised by Mill and Bain as immediateinferences under the principle of Identity. But some Logicians mighttreat them as imperfect syllogisms, requiring another premise tolegitimate the conclusion, thus:

 _All animal life is dependent on vegetation; All human life is animal life; ∴ All human life is dependent on vegetation. _

Or again:

 _All men are living creatures; Socrates is a man; ∴ Socrates is a living creature. _

The decision of this issue turns upon the question (_cf. _ chap. Vi. § 3)how far a Logician is entitled to assume that the terms he uses areunderstood, and that the identities involved in their meanings will berecognised. And to this question, for the sake of consistency, one oftwo answers is required; failing which, there remains the rule of thumb. First, it may be held that no terms are understood except those that aredefined in expounding the science, such as 'genus' and 'species, ''connotation' and 'denotation. ' But very few Logicians observe thislimitation; few would hesitate to substitute 'not wise' for 'foolish. 'Yet by what right? Malvolio being foolish, to prove that he is not-wise, we may construct the following syllogism:

 _Foolish is not-wise; Malvolio is foolish; ∴ Malvolio is not-wise. _

Is this necessary? Why not?

Secondly, it may be held that all terms may be assumed as understoodunless a definition is challenged. This principle will justify thesubstitution of 'not-wise' for 'foolish'; but it will also legitimatethe above cases (concerning 'human life' and 'Socrates') as immediateinferences, with innumerable others that might be based upon thedoctrine of relative terms: for example, _The hunter missed his aim_:therefore, _The prey escaped_. And from this principle it will furtherfollow that all apparent syllogisms, having one premise a verbalproposition, are immediate inferences (_cf. _ chap. Ix. § 4). 

Closely connected with such cases as the above are those mentioned byArchbishop Thomson as "Immediate Inferences by added Determinants"(_Laws of Thought_, § 87). He takes the case: '_A negro is afellow-creature_: therefore, _A negro in suffering is a fellow-creaturein suffering_. ' This rests upon the principle that to increase theconnotations of two terms by the same attribute or determinant does notaffect the relationship of their denotations, since it must equallydiminish (if at all) the denotations of both classes, by excluding thesame individuals, if any want the given attribute. But this principle istrue only when the added attribute is not merely the same verbally, buthas the same significance in qualifying both terms. We cannot argue _Amouse is an animal_; therefore, _A large mouse is a large animal_; for'large' is an attribute relative to the normal magnitude of the thingdescribed. 

§ 4. Conversion is Immediate Inference by transposing the terms of agiven proposition without altering its quality. If the quantity is alsounaltered, the inference is called 'Simple Conversion'; but if thequantity is changed from universal to particular, it is called'Conversion by limitation' or '_per accidens. _' The given proposition iscalled the 'convertend'; that which is derived from it, the 'converse. '

Departing from the usual order of exposition, I have taken up Conversionnext to Subalternation, because it is generally thought to rest upon theprinciple of Identity, and because it seems to be a good method toexhaust the forms that come only under Identity before going on to thosethat involve Contradiction and Excluded Middle. Some, indeed, disputethe claims of Conversion to illustrate the principle of Identity; andif the sufficient statement of that principle be 'A is A, ' it may be aquestion how Conversion or any other mode of inference can be referredto it. But if we state it as above (chap. Vi. § 3), that whatever istrue in one form of words is true in any other, there is no difficultyin applying it to Conversion. 

Thus, to take the simple conversion of I. , _Some S is P; ∴ Some P is S. _ _Some poets are business-like; ∴ Some business-like men are poets. _

Here the convertend and the converse say the same thing, and this istrue if that is. 

We have, then, two cases of simple conversion: of I. (as above) and ofE. For E. :

 _No S is P; ∴ No P is S. _ _No ruminants are carnivores; ∴ No carnivores are ruminants. _

In converting I. , the predicate (P) when taken as the new subject, beingpreindesignate, is treated as particular; and in converting E. , thepredicate (P), when taken as the new subject, is treated as universal, according to the rule in chap. V. § 1. 

A. Is the one case of conversion by limitation:

 All S is P; ∴ Some P is S. 

 All cats are grey in the dark; ∴ Some things grey in the dark are cats. 

The predicate is treated as particular, when taking it for the newsubject, according to the rule not to go beyond the evidence. To inferthat _All things grey in the dark are cats_ would be palpably absurd;yet no error of reasoning is commoner than the simple conversion of A. The validity of conversion by limitation may be shown thus: if, _All Sis P_, then, by subalternation, _Some S is P_, and therefore, by simpleconversion, _Some P is S_. 

O. Cannot be truly converted. If we take the proposition: _Some S isnot P_, to convert this into _No P is S_, or _Some P is not S_, wouldbreak the rule in chap. Vi. § 6; since _S, _ undistributed in theconvertend, would be distributed in the converse. If we are told that_Some men are not cooks_, we cannot infer that _Some cooks are not men_. This would be to assume that '_Some men_' are identical with '_Allmen_. '

By quantifying the predicate, indeed, we may convert O. Simply, thus:

 _Some men are not cooks_ ∴ _No cooks are some men. _

And the same plan has some advantage in converting A. ; for by the usualmethod _per accidens_, the converse of A. Being I. , if we convert thisagain it is still I. , and therefore means less than our originalconvertend. Thus:

 _All S is P ∴ Some P is S ∴ Some S is P. _

Such knowledge, as that _All S_ (the whole of it) _is P_, is tooprecious a thing to be squandered in pure Logic; and it may be preservedby quantifying the predicate; for if we convert A. To Y. , thus--

 _All S is P ∴ Some P is all S--_

we may reconvert Y. To A. Without any loss of meaning. It is the chiefuse of quantifying the predicate that, thereby, every proposition iscapable of simple conversion. 

The conversion of propositions in which the relation of terms isinadequately expressed (see chap. Ii. , § 2) by the ordinary copula (_is_or _is not_) needs a special rule. To argue thus--

 _A is followed by B_ ∴ _Something followed by B is A_--

would be clumsy formalism. We usually say, and we ought to say--

 _A is followed by B_ ∴ _B follows A_ (or _is preceded by A_). 

Now, any relation between two terms may be viewed from either side--_A:B_ or _B: A_. It is in both cases the same fact; but, with the alteredpoint of view, it may present a different character. For example, in theImmediate Inference--_A > B_ ∴ _B < A_--a diminishing turns into anincreasing ratio, whilst the fact predicated remains the same. Given, then, a relation between two terms as viewed from one to the other, thesame relation viewed from the other to the one may be called theReciprocal. In the cases of Equality, Co-existence and Simultaneity, thegiven relation and its reciprocal are not only the same fact, but theyalso have the same character: in the cases of Greater and Less andSequence, the character alters. 

We may, then, state the following rule for the conversion ofpropositions in which the whole relation explicitly stated is taken asthe copula: Transpose the terms, and for the given relation substituteits reciprocal. Thus--

 _A is the cause of B ∴ B is the effect of A. _

The rule assumes that the reciprocal of a given relation is definitelyknown; and so far as this is true it may be extended to more concreterelations--

 _A is a genus of B ∴ B is a species of A A is the father of B ∴ B is a child of A. _

But not every relational expression has only one definite reciprocal. Ifwe are told that _A is the brother of B_, we can only infer that _B iseither the brother or the sister of A_. A list of all reciprocalrelations is a desideratum of Logic. 

§ 5. Obversion (otherwise called Permutation or Æquipollence) isImmediate Inference by changing the quality of the given proposition andsubstituting for its predicate the contradictory term. The givenproposition is called the 'obvertend, ' and the inference from it the'obverse. ' Thus the obvertend being--_Some philosophers are consistentreasoners_, the obverse will be--_Some philosophers are not inconsistentreasoners_. 

The legitimacy of this mode of reasoning follows, in the case ofaffirmative propositions, from the principle of Contradiction, that ifany term be affirmed of a subject, the contradictory term may be denied(chap. Vi. § 3). To obvert affirmative propositions, then, the ruleis--Insert the negative sign, and for the predicate substitute itscontradictory term. 

 A. _All S is P ∴ No S is not-P All men are fallible ∴ No men are infallible. _

 I. _Some S is P ∴ some S is not-P Some philosophers are consistent ∴ Some philosophers are not inconsistent. _

In agreement with this mode of inference, we have the rule of modernEnglish grammar, that 'two negatives make an affirmative. '

Again, by the principle of Excluded Middle, if any term be denied of asubject, its contradictory may be affirmed: to obvert negativepropositions, then, the rule is--Remove the negative sign, and for thepredicate substitute its contradictory term. 

 E. _No S is P ∴ All S is not-P No matter is destructible ∴ All matter is indestructible. _

 O. _Some S is not P ∴ Some S is not-P Some ideals are not attainable ∴ Some ideals are unattainable. _

Thus, by obversion, each of the four propositions retains its quantitybut changes its quality: A. To E. , I. To O. , E. To A. , O. To I. And allthe obverses are infinite propositions, the affirmative infinites havingthe sense of negatives, and the negative infinites having the sense ofaffirmatives. 

Again, having obtained the obverse of a given proposition, it may bedesirable to recover the obvertend; or it may at any time be requisiteto change a given infinite proposition into the corresponding directaffirmative or negative; and in such cases the process is stillobversion. Thus, if _No S is not-P_ be given us to recover the obvertendor to find the corresponding affirmative; the proposition being formallynegative, we apply the rule for obverting negatives: 'Remove thenegative sign, and for the predicate substitute its contradictory. ' Thisyields the affirmative _All S is P_. Similarly, to obtain the obvertendof _All S is not-P_, apply the rule for obverting Affirmatives; and thisyields _No S is P_. 

§ 6. Contrariety. --We have seen in chap. Iv. § 8, that contrary termsare such that no two of them are predicable in the same way of the samesubject, whilst perhaps neither may be predicable of it. Similarly, Contrary Propositions may be defined as those of which no two are everboth true together, whilst perhaps neither may be true; or, in otherwords, both may be false. This is the relation between A. And E. Whenconcerned with the same matter: as A. --_All men are wise_; E. --_No menare wise_. Such propositions cannot both be true; but they may both befalse, for some men may be wise and some not. They cannot both be true;for, by the principle of Contradiction, if _wise_ may be affirmed of_All men, not-wise_ must be denied; but _All men are not-wise_ is theobverse of _No men are wise_, which therefore may also be denied. 

At the same time we cannot apply to A. And E. The principle of ExcludedMiddle, so as to show that one of them must be true of the same matter. For if we deny that _All men are wise_, we do not necessarily deny theattribute 'wise' of each and every man: to say that _Not all are wise_may mean no more than that _Some are not_. This gives a proposition inthe form of O. ; which, as we have seen, does not imply its subalternans, E. 

If, however, two Singular Propositions, having the same matter, butdiffering in quality, are to be treated as universals, and therefore asA. And E. , they are, nevertheless, contradictory and not merelycontrary; for one of them must be false and the other true. 

§ 7. Contradiction is a relation between two propositions analogous tothat between contradictory terms (one of which being affirmed of asubject the other is denied)--such, namely, that one of them is falseand the other true. This is the case with the forms A. And O. , and E. And I. , in the same matter. If it be true that _All men are wise_, it isfalse that _Some men are not wise_ (equivalent by obversion to _Somemen are not-wise_); or else, since the 'Some men' are included in the'All men, ' we should be predicating of the same men that they are both'wise' and 'not-wise'; which would violate the principle ofContradiction. Similarly, _No men are wise_, being by obversionequivalent to _All men are not-wise_, is incompatible with _Some men arewise_, by the same principle of Contradiction. 

But, again, if it be false that _All men are wise_, it is always truethat _Some are not wise_; for though in denying that 'wise' is apredicate of 'All men' we do not deny it of each and every man, yet wedeny it of 'Some men. ' Of 'Some men, ' therefore, by the principle ofExcluded Middle, 'not-wise' is to be affirmed; and _Some men arenot-wise_, is by obversion equivalent to _Some men are not wise_. Similarly, if it be false that _No men are wise_, which by obversion isequivalent to _All men are not-wise_, then it is true at least that_Some men are wise_. 

By extending and enforcing the doctrine of relative terms, certain otherinferences are implied in the contrary and contradictory relations ofpropositions. We have seen in chap. Iv. That the contradictory of agiven term includes all its contraries: 'not-blue, ' for example, includes red and yellow. Hence, since _The sky is blue_ becomes byobversion, _The sky is not not-blue_, we may also infer _The sky is notred_, etc. From the truth, then, of any proposition predicating a giventerm, we may infer the falsity of all propositions predicating thecontrary terms in the same relation. But, on the other hand, from thefalsity of a proposition predicating a given term, we cannot infer thetruth of the predication of any particular contrary term. If it be falsethat _The sky is red_, we cannot formally infer, that _The sky is blue_(_cf. _ chap. Iv. § 8). 

§ 8. Sub-contrariety is the relation of two propositions, concerning thesame matter that may both be true but are never both false. This is thecase with I. And O. If it be true that _Some men are wise_, it may alsobe true that _Some (other) men are not wise_. This follows from themaxim in chap. Vi. § 6, not to go beyond the evidence. 

For if it be true that _Some men are wise_, it may indeed be true that_All are_ (this being the subalternans): and if _All are_, it is (bycontradiction) false that _Some are not_; but as we are only told that_Some men are_, it is illicit to infer the falsity of _Some are not_, which could only be justified by evidence concerning _All men_. 

But if it be false that _Some men are wise_, it is true that _Some menare not wise_; for, by contradiction, if _Some men are wise_ is false, _No men are wise_ is true; and, therefore, by subalternation, _Some menare not wise_ is true. 

§ 9. The Square of Opposition. --By their relations of Subalternation, Contrariety, Contradiction, and Sub-contrariety, the forms A. I. E. O. (having the same matter) are said to stand in Opposition: and Logiciansrepresent these relations by a square having A. I. E. O. At its corners:

A. Contraries E. 

S Co s Su nt e ub ra i ba di r al ct o lt ct o te di r er ra i rn nt e ns Co s s

I. Sub-contraries O. 

As an aid to the memory, this diagram is useful; but as an attempt torepresent the logical relations of propositions, it is misleading. For, standing at corners of the same square, A. And E. , A. And I. , E. And O. , and I. And O. , seem to be couples bearing the same relation to oneanother; whereas we have seen that their relations are entirelydifferent. The following traditional summary of their relations inrespect of truth and falsity is much more to the purpose:

 (1) If A. Is true, I. Is true, E. Is false, O. Is false. (2) If A. Is false, I. Is unknown, E. Is unknown, O. Is true. (3) If I. Is true, A. Is unknown, E. Is false, O. Is unknown. (4) If I. Is false, A. Is false, E. Is true, O. Is true. (5) If E. Is true, A. Is false, I. Is false, O. Is true. (6) If E. Is false, A. Is unknown, I. Is true, O. Is unknown. (7) If O. Is true, A. Is false, I. Is unknown, E. Is unknown. (8) If O. Is false, A. Is true, I. Is true, E. Is false. 

Where, however, as in cases 2, 3, 6, 7, alleging either thefalsity of universals or the truth of particulars, it follows that twoof the three Opposites are unknown, we may conclude further that one ofthem must be true and the other false, because the two unknown arealways Contradictories. 

§ 10. Secondary modes of Immediate Inference are obtained by applyingthe process of Conversion or Obversion to the results already obtainedby the other process. The best known secondary form of ImmediateInference is the Contrapositive, and this is the converse of the obverseof a given proposition. Thus:

 DATUM. OBVERSE. CONTRAPOSITIVE. 

 A. _All S is P_ ∴ _No S is not-P_ ∴ _No not-P is S_ I. _Some S is P_ ∴ _Some S is not not-P_ ∴ (none) E. _No S is P_ ∴ _All S is not-P_ ∴ _Some not-P is S_ O. _Some S is not P_ ∴ _Some S is not-P_ ∴ _Some not-P is S_

There is no contrapositive of I. , because the obverse of I. Is in theform of O. , and we have seen that O. Cannot be converted. O. , however, has a contrapositive (_Some not-P is S_); and this is sometimes giveninstead of the converse, and called the 'converse by negation. '

Contraposition needs no justification by the Laws of Thought, as it isnothing but a compounding of conversion with obversion, both of whichprocesses have already been justified. I give a table opposite of theother ways of compounding these primary modes of Immediate Inference. 

 A I E O--------------------------------------------------------------------------------

 1 All A is B Some A is B No A is B Some A is not B

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Obverse 2 No A is b Some A is not b All A is b Some A is b

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Converse 3 Some B is A Some B is A No B is A

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Obverseof 4 Some B is not a Some B is not a All B is aConverse

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Contra-positive 5 No b is A Some b is A Some b is A

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Obverseof 6 All b is a Some b is not a Some b is not aContrapos

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ConverseofObverse 7 Some a is BofConverse

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ObverseofConverseof 8 Some a is not bObverseofConverse

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ConverseofObverse 9 Some a is bofContrapos

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ObverseofConverseof 10 Some a is not BObverseofContrapos

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In this table _a_ and _b_ stand for _not-A_ and _not-B_ and had betterbe read thus: for _No A is b, No A is not-B_; for _All b is a_ (col. 6), _All not-B is not-A_; and so on. 

It may not, at first, be obvious why the process of alternatelyobverting and converting any proposition should ever come to an end;though it will, no doubt, be considered a very fortunate circumstancethat it always does end. On examining the results, it will be found thatthe cause of its ending is the inconvertibility of O. For E. , whenobverted, becomes A. ; every A, when converted, degenerates into I. ;every I. , when obverted, becomes O. ; O cannot be converted, and toobvert it again is merely to restore the former proposition: so that thewhole process moves on to inevitable dissolution. I. And O. Areexhausted by three transformations, whilst A. And E. Will each endureseven. 

Except Obversion, Conversion and Contraposition, it has not been usualto bestow special names on these processes or their results. But theform in columns 7 and 10 (_Some a is B--Some a is not B_), where theoriginal predicate is affirmed or denied of the contradictory of theoriginal subject, has been thought by Dr. Keynes to deserve adistinctive title, and he has called it the 'Inverse. ' Whilst theInverse is one form, however, Inversion is not one process, but isobtained by different processes from E. And A. Respectively. In this itdiffers from Obversion, Conversion, and Contraposition, each of whichstands for one process. 

The Inverse form has been objected to on the ground that the inference_All A is B ∴ Some not-A is not B_, distributes _B_ (as predicate of anegative proposition), though it was given as undistributed (aspredicate of an affirmative proposition). But Dr. Keynes defends it onthe ground that (1) it is obtained by obversions and conversions whichare all legitimate and (2) that although _All A is B_ does notdistribute _B_ in relation to _A_, it does distribute _B_ in relation tosome _not-A_ (namely, in relation to whatever _not-A_ is _not-B_). Thisis one reason why, in stating the rule in chap. Vi. § 6, I havewritten: "an immediate inference ought to contain nothing that is notcontained, _or formally implied_, in the proposition from which it isinferred"; and have maintained that every term formally implies itscontradictory within the _suppositio_. 

§ 11. Immediate Inferences from Conditionals are those whichconsist--(1) in changing a Disjunctive into a Hypothetical, or aHypothetical into a Disjunctive, or either into a Categorical; and (2)in the relations of Opposition and the equivalences of Obversion, Conversion, and secondary or compound processes, which we have alreadyexamined in respect of Categoricals. As no new principles are involved, it may suffice to exhibit some of the results. 

We have already seen (chap. V. § 4) how Disjunctives may be read asHypotheticals and Hypotheticals as Categoricals. And, as to Opposition, if we recognise four forms of Hypothetical A. I. E. O. , these plainlystand to one another in a Square of Opposition, just as Categoricals do. Thus A. And E. (_If A is B, C is D_, and _If A is B, C is not D_) arecontraries, but not contradictories; since both may be false (_C_ maysometimes be _D_, and sometimes not), though they cannot both be true. And if they are both false, their subalternates are both true, beingrespectively the contradictories of the universals of opposite quality, namely, I. Of E. , and O. Of A. But in the case of Disjunctives, wecannot set out a satisfactory Square of Opposition; because, as we saw(chap. V. § 4), the forms required for E. And O. Are not trueDisjunctives, but Exponibles. 

The Obverse, Converse, and Contrapositive, of Hypotheticals (admittingthe distinction of quality) may be exhibited thus:

 DATUM. OBVERSE. 

A. _If A is B, C is D_ _If A is B, C is not d_I. Sometimes _when A is B, C is D_ Sometimes _when A is B, C is not d_E. _If A is B, C is not D_ _If A is B, C is d_O. Sometimes _when A is B, C is not D_ Sometimes _when A is B, C is d_

 CONVERSE. CONTRAPOSITIVE. 

Sometimes _when C is D, A is B_ _If C is d, A is not B_Sometimes _when C is D, A is B_ (none)_If C is D, A is not B_ Sometimes _when C is d, A is B_ (none) Sometimes _when C is d, A is B_

As to Disjunctives, the attempt to put them through these differentforms immediately destroys their disjunctive character. Still, given anyproposition in the form _A is either B or C_, we can state thepropositions that give the sense of obversion, conversion, etc. , thus:

 DATUM. --_A is either B or C;_ OBVERSE. --_A is not both b and c;_ CONVERSE. --_Something, either B or C, is A;_ CONTRAPOSITIVE. --_Nothing that is both b and c is A_. 

For a Disjunctive in I. , of course, there is no Contrapositive. Given aDisjunctive in the form _Either A is B or C is D_, we may write for itsObverse--_In no case is A b, and C at the same time d_. But no Converseor Contrapositive of such a Disjunctive can be obtained, except by firstcasting it into the hypothetical or categorical form. 

The reader who wishes to pursue this subject further, will find itelaborately treated in Dr. Keynes' _Formal Logic_, Part II. ; to whichwork the above chapter is indebted. 

CHAPTER VIII

ORDER OF TERMS, EULER'S DIAGRAMS, LOGICAL EQUATIONS, EXISTENTIAL IMPORTOF PROPOSITIONS

§ 1. Of the terms of a proposition which is the Subject and which thePredicate? In most of the exemplary propositions cited by Logicians itwill be found that the subject is a substantive and the predicate anadjective, as in _Men are mortal_. This is the relation of Substance andAttribute which we saw (chap. I. § 5) to be the central type ofrelations of coinherence; and on this model other predications may beformed in which the subject is not a substance, but is treated as if itwere, and could therefore be the ground of attributes; as _Fame istreacherous, The weather is changeable_. But, in literature, sentencesin which the adjective comes first are not uncommon, as _Loud was theapplause, Dark is the fate of man, Blessed are the peacemakers_, and soon. Here, then, 'loud, ' 'dark' and 'blessed' occupy the place of thelogical subject. Are they really the subject, or must we alter the orderof such sentences into _The applause was loud_, etc. ? If we do, and thenproceed to convert, we get _Loud was the applause_, or (morescrupulously) _Some loud noise was the applause_. The last form, it istrue, gives the subject a substantive word, but 'applause' has becomethe predicate; and if the substantive 'noise' was not implied in thefirst form, _Loud is the applause_, by what right is it now inserted?The recognition of Conversion, in fact, requires us to admit that, formally, in a logical proposition, the term preceding the copula issubject and the one following is predicate. And, of course, materiallyconsidered, the mere order of terms in a proposition can make nodifference in the method of proving it, nor in the inferences that canbe drawn from it. 

Still, if the question is, how we may best cast a literary sentence intological form, good grounds for a definite answer may perhaps be found. We must not try to stand upon the naturalness of expression, for _Darkis the fate of man_ is quite as natural as _Man is mortal_. When thepurpose is not merely to state a fact, but also to express our feelingsabout it, to place the grammatical predicate first may be perfectlynatural and most effective. But the grounds of a logical order ofstatement must be found in its adaptation to the purposes of proof andinference. Now general propositions are those from which most inferencescan be drawn, which, therefore, it is most important to establish, iftrue; and they are also the easiest to disprove, if false; since asingle negative instance suffices to establish the contradictory. Itfollows that, in re-casting a literary or colloquial sentence forlogical purposes, we should try to obtain a form in which the subject isdistributed--is either a singular term or a general term predesignate as'All' or 'No. ' Seeing, then, that most adjectives connote a singleattribute, whilst most substantives connote more than one attribute; andthat therefore the denotation of adjectives is usually wider than thatof substantives; in any proposition, one term of which is an adjectiveand the other a substantive, if either can be distributed in relation tothe other, it is nearly sure to be the substantive; so that to take thesubstantive term for subject is our best chance of obtaining anuniversal proposition. These considerations seem to justify the practiceof Logicians in selecting their examples. 

For similar reasons, if both terms of a proposition are substantive, theone with the lesser denotation is (at least in affirmativepropositions) the more suitable subject, as _Cats are carnivores_. Andif one term is abstract, that is the more suitable subject; for, as wehave seen, an abstract term may be interpreted by a correspondingconcrete one distributed, as _Kindness is infectious_; that is, _Allkind actions suggest imitation_. 

If, however, a controvertist has no other object in view than to refutesome general proposition laid down by an opponent, a particularproposition is all that he need disentangle from any statement thatserves his purpose. 

§ 2. Toward understanding clearly the relations of the terms of aproposition, it is often found useful to employ diagrams; and thediagrams most in use are the circles of Euler. 

These circles represent the denotation of the terms. Suppose theproposition to be _All hollow-horned animals ruminate_: then, if wecould collect all ruminants upon a prairie, and enclose them with acircular palisade; and segregate from amongst them all the hollow-hornedbeasts, and enclose them with another ring-fence inside the other; oneway of interpreting the proposition (namely, in denotation) would befigured to us thus:

[Illustration: FIG. 1. ]

An Universal Affirmative may also state a relation between two termswhose denotation is co-extensive. A definition always does this, as _Manis a rational animal_; and this, of course, we cannot represent by twodistinct circles, but at best by one with a thick circumference, tosuggest that two coincide, thus:

[Illustration: FIG. 2. ]

The Particular Affirmative Proposition may be represented in severalways. In the first place, bearing in mind that 'Some' means 'some atleast, it may be all, ' an I. Proposition may be represented by Figs. 1and 2; for it is true that _Some horned animals ruminate_, and that_Some men are rational_. Secondly, there is the case in which the 'Somethings' of which a predication is made are, in fact, not all; whilst thepredicate, though not given as distributed, yet might be so given if wewished to state the whole truth; as if we say _Some men are Chinese_. This case is also represented by Fig. 1, the outside circle representing'Men, ' and the inside one 'Chinese. ' Thirdly, the predicate mayappertain to some only of the subject, but to a great many other things, as in _Some horned beasts are domestic_; for it is true that some arenot, and that certain other kinds of animals are, domestic. This case, therefore, must be illustrated by overlapping circles, thus:

[Illustration: FIG. 3. ]

The Universal Negative is sufficiently represented by a single Fig. (4):two circles mutually exclusive, thus:

[Illustration: FIG. 4. ]

That is, _No horned beasts are carnivorous_. 

Lastly, the Particular Negative may be represented by any of the Figs. 1, 3, and 4; for it is true that _Some ruminants are not hollow-horned_, that _Some horned animals are not domestic_, and that _Some hornedbeasts are not carnivorous_. 

Besides their use in illustrating the denotative force of propositions, these circles may be employed to verify the results of Obversion, Conversion, and the secondary modes of Immediate Inference. Thus theObverse of A. Is clear enough on glancing at Figs. 1 and 2; for if weagree that whatever term's denotation is represented by a given circle, the denotation of the contradictory term shall be represented by thespace outside that circle; then if it is true that _All hollow hornedanimals are ruminants_, it is at the same time true that _Nohollow-horned animals are not-ruminants_; since none of thehollow-horned are found outside the palisade that encloses theruminants. The Obverse of I. , E. Or O. May be verified in a similarmanner. 

As to the Converse, a Definition is of course susceptible of SimpleConversion, and this is shown by Fig. 2: 'Men are rational animals' and'Rational animals are men. ' But any other A. Proposition is presumablyconvertible only by limitation, and this is shown by Fig. 1; where _Allhollow-horned animals are ruminants_, but we can only say that _Someruminants are hollow-horned_. 

That I. May be simply converted may be seen in Fig. 3, which representsthe least that an I. Proposition can mean; and that E. May be simplyconverted is manifest in Fig. 4. 

As for O. , we know that it cannot be converted, and this is made plainenough by glancing at Fig. 1; for that represents the O. , _Someruminants are not hollow-horned_, but also shows this to be compatiblewith _All hollow-horned animals are ruminants_ (A. ). Now in conversionthere is (by definition) no change of quality. The Converse, then, of_Some ruminants are not hollow-horned_ must be a negative proposition, having 'hollow-horned' for its subject, either in E. Or O. ; but thesewould be respectively the contrary and contradictory of _Allhollow-horned animals are ruminants_; and, therefore, if this be true, they must both be false. 

But (referring still to Fig. 1) the legitimacy of contrapositing O. Isequally clear; for if _Some ruminants are not hollow-horned_, _Someanimals that are not hollow-horned are ruminants_, namely, all theanimals between the two ring-fences. Similar inferences may beillustrated from Figs. 3 and 4. And the Contraposition of A. May beverified by Figs. 1 and 2, and the Contraposition of E. By Fig. 4. 

Lastly, the Inverse of A. Is plain from Fig. 1--_Some things that arenot hollow-horned are not ruminants_, namely, things that lie outsidethe outer circle and are neither 'ruminants' nor 'hollow-horned. ' Andthe Inverse of E may be studied in Fig. 4--_Some things that arenot-horned beasts are carnivorous_. 

Notwithstanding the facility and clearness of the demonstrations thusobtained, it may be said that a diagrammatic method, representingdenotations, is not properly logical. Fundamentally, the relationasserted (or denied) to exist between the terms of a proposition, is arelation between the terms as determined by their attributes orconnotation; whether we take Mill's view, that a proposition assertsthat the connotation of the subject is a mark of the connotation of thepredicate; or Dr. Venn's view, that things denoted by the subject (ashaving its connotation) have (or have not) the attribute connoted by thepredicate; or, the Conceptualist view, that a judgment is a relation ofconcepts (that is, of connotations). With a few exceptions artificiallyframed (such as 'kings now reigning in Europe'), the denotation of aterm is never directly and exhaustively known, but consists merely in'all things that have the connotation. ' If the value of logical trainingdepends very much upon our habituating ourselves to construepropositions, and to realise the force of inferences from them, according to the connotation of their terms, we shall do well not toturn too hastily to the circles, but rather to regard them as means ofverifying in denotation the conclusions that we have already learnt torecognise as necessary in connotation. 

§ 3. The equational treatment of propositions is closely connected withthe diagrammatic. Hamilton thought it a great merit of his plan ofquantifying the predicate, that thereby every proposition is reduced toits true form--an equation. According to this doctrine, the proposition_All X is all Y_ (U. ) equates X and Y; the proposition _All X is some Y_(A. ) equates X with some part of Y; and similarly with the otheraffirmatives (Y. And I. ). And so far it is easy to follow his meaning:the Xs are identical with some or all the Ys. But, coming to thenegatives, the equational interpretation is certainly less obvious. Theproposition _No X is Y_ (E. ) cannot be said in any sense to equate X andY; though, if we obvert it into _All X is some not-Y_, we have (in thesame sense, of course, as in the above affirmative forms) X equated withpart at least of 'not-Y. '

But what is that sense? Clearly not the same as that in whichmathematical terms are equated, namely, in respect of some mode ofquantity. For if we may say _Some X is some Y_, these Xs that are alsoYs are not merely the same in number, or mass, or figure; they are thesame in every respect, both quantitative and qualitative, have the samepositions in time and place, are in fact identical. The proposition2+2=4 means that any two things added to any other two are, _in respectof number_, equal to any three things added to one other thing; and thisis true of all things that can be counted, however much they may differin other ways. But _All X is all Y_ means that Xs and Ys are the samethings, although they have different names when viewed in differentaspects or relations. Thus all equilateral triangles are equiangulartriangles; but in one case they are named from the equality of theirangles, and in the other from the equality of their sides. Similarly, 'British subjects' and 'subjects of King George V' are the same people, named in one case from the person of the Crown, and in the other fromthe Imperial Government. These logical equations, then, are in truthidentities of denotation; and they are fully illustrated by therelations of circles described in the previous section. 

When we are told that logical propositions are to be considered asequations, we naturally expect to be shown some interesting developmentsof method in analogy with the equations of Mathematics; but fromHamilton's innovations no such thing results. This cannot be said, however, of the equations of Symbolic Logic; which are thestarting-point of very remarkable processes of ratiocination. As thesubject of Symbolic Logic, as a whole, lies beyond the compass of thiswork, it will be enough to give Dr. Venn's equations corresponding withthe four propositional forms of common Logic. 

According to this system, universal propositions are to be regarded asnot necessarily implying the existence of their terms; and therefore, instead of giving them a positive form, they are translated into symbolsthat express what they deny. For example, the proposition _All devilsare ugly_ need not imply that any such things as 'devils' really exist;but it certainly does imply that _Devils that are not ugly do notexist_. Similarly, the proposition _No angels are ugly_ implies that_Angels that are ugly do not exist_. Therefore, writing _x_ for'devils, ' _y_ for 'ugly, ' and _ȳ_ for 'not-ugly, ' we may express A. , the universal affirmative, thus:

 A. _xȳ_ = 0. 

That is, _x that is not y is nothing_; or, _Devils that are not-ugly donot exist_. And, similarly, writing _x_ for 'angels' and _y_ for 'ugly, 'we may express E. , the universal negative, thus:

 E. _xy_ = 0. 

That is, _x that is y is nothing_; or, _Angels that are ugly do notexist_. 

On the other hand, particular propositions are regarded as implying theexistence of their terms, and the corresponding equations are so framedas to express existence. With this end in view, the symbol v is adoptedto represent 'something, ' or indeterminate reality, or more thannothing. Then, taking any particular affirmative, such as _Somemetaphysicians are obscure_, and writing _x_ for 'metaphysicians, ' and_y_ for 'obscure, ' we may express it thus:

 I. _xy_ = v. 

That is, _x that is y is something_; or, _Metaphysicians that areobscure do occur in experience_ (however few they may be, or whetherthey all be obscure). And, similarly, taking any particular negative, such as _Some giants are not cruel_, and writing _x_ for 'giants' and_y_ for 'not-cruel, ' we may express it thus:

 O. _xȳ_ = v. 

That is, _x that is not y is something_; or, _giants that are not-crueldo occur_--in romances, if nowhere else. 

Clearly, these equations are, like Hamilton's, concerned withdenotation. A. And E. Affirm that the compound terms xȳ and xy have nodenotation; and I. And O. Declare that xȳ and xy have denotation, orstand for something. Here, however, the resemblance to Hamilton's systemceases; for the Symbolic Logic, by operating upon more than two termssimultaneously, by adopting the algebraic signs of operations, +, -, ×, ÷(with a special signification), and manipulating the symbols byquasi-algebraic processes, obtains results which the common Logicreaches (if at all) with much greater difficulty. If, indeed, the valueof logical systems were to be judged of by the results obtainable, formal deductive Logic would probably be superseded. And, as a mentaldiscipline, there is much to be said in favour of the symbolic method. But, as an introduction to philosophy, the common Logic must hold itsground. (Venn: _Symbolic Logic_, c. 7. )

§ 4. Does Formal Logic involve any general assumption as to the realexistence of the terms of propositions?

In the first place, Logic treats primarily of the _relations_ implied inpropositions. This follows from its being the science of proof for allsorts of (qualitative) propositions; since all sorts of propositionshave nothing in common except the relations they express. 

But, secondly, relations without terms of some sort are not to bethought of; and, hence, even the most formal illustrations of logicaldoctrines comprise such terms as S and P, X and Y, or x and y, in asymbolic or representative character. Terms, therefore, of some sort areassumed to exist (together with their negatives or contradictories) _forthe purposes of logical manipulation_. 

Thirdly, however, that Formal Logic cannot as such directly involve theexistence of any particular concrete terms, such as 'man' or 'mountain, 'used by way of illustration, is implied in the word 'formal, ' that is, 'confined to what is common or abstract'; since the only thing common toall terms is to be related in some way to other terms. The actualexistence of any concrete thing can only be known by experience, as with'man' or 'mountain'; or by methodically justifiable inference fromexperience, as with 'atom' or 'ether. ' If 'man' or 'mountain, ' or'Cuzco' be used to illustrate logical forms, they bring with them anexistential import derived from experience; but this is the import oflanguage, not of the logical forms. 'Centaur' and 'El Dorado' signify tous the non-existent; but they serve as well as 'man' and 'London' toillustrate Formal Logic. 

Nevertheless, fourthly, the existence or non-existence of particularterms may come to be implied: namely, wherever the very fact ofexistence, or of some condition of existence, is an hypothesis or datum. Thus, given the proposition _All S is P_, to be P is made a condition ofthe existence of S: whence it follows that an S that is not P does notexist (_xȳ_ = 0). On the further hypothesis that S exists, it followsthat P exists. On the hypothesis that S does not exist, the existence ofP is problematic; but, then, if P does exist we cannot convert theproposition; since _Some P is S_ (P existing) would involve theexistence of S; which is contrary to the hypothesis. 

Assuming that Universals _do not_, whilst Particulars _do_, imply theexistence of their subjects, we cannot infer the subalternate (I. Or O. )from the subalternans (A. Or E. ), for that is to ground the actual onthe problematic; and for the same reason we cannot convert A. _peraccidens_. 

Assuming, again, a certain _suppositio_ or universe, to which in a givendiscussion every argument shall refer, then, any propositions whoseterms lie outside that _suppositio_ are irrelevant, and for the purposesof that discussion are sometimes called "false"; though it seems betterto call them irrelevant or meaningless, seeing that to call them falseimplies that they might in the same case be true. Thus propositionswhich, according to the doctrine of Opposition, appear to beContradictories, may then cease to be so; for of Contradictories one istrue and the other false; but, in the case supposed, both aremeaningless. If the subject of discussion be Zoology, all propositionsabout centaurs or unicorns are absurd; and such speciousContradictories as _No centaurs play the lyre--Some centaurs do play thelyre_; or _All unicorns fight with lions--Some unicorns do not fightwith lions_, are both meaningless, because in Zoology there are nocentaurs nor unicorns; and, therefore, in this reference, thepropositions are not really contradictory. But if the subject ofdiscussion or _suppositio_ be Mythology or Heraldry, such propositionsas the above are to the purpose, and form legitimate pairs ofContradictories. 

In Formal Logic, in short, we may make at discretion any assumptionwhatever as to the existence, or as to any condition of the existence ofany particular term or terms; and then certain implications andconclusions follow in consistency with that hypothesis or datum. Still, our conclusions will themselves be only hypothetical, depending on thetruth of the datum; and, of course, until this is empiricallyascertained, we are as far as ever from empirical reality. (Venn:_Symbolic Logic_, c. 6; Keynes: _Formal Logic_, Part II. C. 7: _cf. _Wolf: _Studies in Logic_. )

CHAPTER IX

FORMAL CONDITIONS OF MEDIATE INFERENCE

§ 1. A Mediate Inference is a proposition that depends for proof upontwo or more other propositions, so connected together by one or moreterms (which the evidentiary propositions, or each pair of them, have incommon) as to justify a certain conclusion, namely, the proposition inquestion. The type or (more properly) the unit of all such modes ofproof, when of a strictly logical kind, is the Syllogism, to which weshall see that all other modes are reducible. It may be exhibitedsymbolically thus:

 M is P; S is M: ∴ S is P. 

Syllogisms may be classified, as to quantity, into Universal orParticular, according to the quantity of the conclusion; as to quality, into Affirmative or Negative, according to the quality of theconclusion; and, as to relation, into Categorical, Hypothetical andDisjunctive, according as all their propositions are categorical, or one(at least) of their evidentiary propositions is a hypothetical or adisjunctive. 

To begin with Categorical Syllogisms, of which the following is anexample:

 All authors are vain; Cicero is an author: ∴ Cicero is vain. 

Here we may suppose that there are no direct means of knowing thatCicero is vain; but we happen to know that all authors are vain andthat he is an author; and these two propositions, put together, unmistakably imply that he is vain. In other words, we do not at firstknow any relation between 'Cicero' and 'vanity'; but we know that thesetwo terms are severally related to a third term, 'author, ' hence calleda Middle Term; and thus we perceive, by mediate evidence, that they arerelated to one another. This sort of proof bears an obvious resemblance(though the relations involved are not the same) to the mathematicalproof of equality between two quantities, that cannot be directlycompared, by showing the equality of each of them to some thirdquantity: A = B = C ∴ A = C. Here B is a middle term. 

We have to inquire, then, what conditions must be satisfied in orderthat a Syllogism may be formally conclusive or valid. A speciousSyllogism that is not really valid is called a Parasyllogism. 

§ 2. General Canons of the Syllogism. 

(1) A Syllogism contains three, and no more, distinct propositions. 

(2) A Syllogism contains three, and no more, distinct univocal terms. 

These two Canons imply one another. Three propositions with less thanthree terms can only be connected in some of the modes of ImmediateInference. Three propositions with more than three terms do not showthat connection of two terms by means of a third, which is requisite forproving a Mediate Inference. If we write--

 All authors are vain; Cicero is a statesman--

there are four terms and no middle term, and therefore there is noproof. Or if we write--

 All authors are vain; Cicero is an author: ∴ Cicero is a statesman--

here the term 'statesman' occurs without any voucher; it appears in theinference but not in the evidence, and therefore violates the maxim ofall formal proof, 'not to go beyond the evidence. ' It is true that ifany one argued--

 All authors are vain; Cicero wrote on philosophy: ∴ Cicero is vain--

this could not be called a bad argument or a material fallacy; but itwould be a needless departure from the form of expression in which theconnection between the evidence and the inference is most easily seen. 

Still, a mere adherence to the same form of words in the expression ofterms is not enough: we must also attend to their meaning. For if thesame word be used ambiguously (as 'author' now for 'father' and anon for'man of letters'), it becomes as to its meaning two terms; so that wehave four in all. Then, if the ambiguous term be the Middle, noconnection is shown between the other two; if either of the others beambiguous, something seems to be inferred which has never been reallygiven in evidence. 

The above two Canons are, indeed, involved in the definition of acategorical syllogism, which may be thus stated: A Categorical Syllogismis a form of proof or reasoning (way of giving reasons) in which onecategorical proposition is established by comparing two others thatcontain together only three terms, or that have one and only one term incommon. 

The proposition established, derived, or inferred, is called theConclusion: the evidentiary propositions by which it is proved arecalled the Premises. 

The term common to the premises, by means of which the other terms arecompared, is called the Middle Term; the subject of the conclusion iscalled the Minor Term; the predicate of the conclusion, the Major Term. 

The premise in which the minor term occurs is called the Minor Premise;that in which the major term occurs is called the Major Premise. And aSyllogism is usually written thus:

Major Premise--All authors (Middle) are vain (Major);

Minor Premise--Cicero (Minor) is an author (Middle):

Conclusion--∴ Cicero (Minor) is vain (Major). 

Here we have three propositions with three terms, each term occurringtwice. The minor and major terms are so called, because, when theconclusion is an universal affirmative (which only occurs in Barbara;see chap. X. § 6), its subject and predicate are respectively the lessand the greater in extent or denotation; and the premises are calledafter the peculiar terms they contain: the expressions 'major premise'and 'minor premise' have nothing to do with the order in which thepremises are presented; though it is usual to place the major premisefirst. 

(3) No term must be distributed in the conclusion unless it isdistributed in the premises. 

It is usual to give this as one of the General Canons of the Syllogism;but we have seen (chap. Vi. § 6) that it is of wider application. Indeed, 'not to go beyond the evidence' belongs to the definition offormal proof. A breech of this rule in a syllogism is the fallacy ofIllicit Process of the Minor, or of the Major, according to which termhas been unwarrantably distributed. The following parasyllogismillicitly distributes both terms of the conclusion:

 All poets are pathetic; Some orators are not poets: ∴ No orators are pathetic. 

(4) The Middle Term must be distributed at least once in the premises(in order to prove a conclusion in the given terms). 

For the use of mediate evidence is to show the relation of terms thatcannot be directly compared; this is only possible if the middle termfurnishes the ground of comparison; and this (in Logic) requires thatthe whole denotation of the middle should be either included or excludedby one of the other terms; since if we only know that the other termsare related to _some_ of the middle, their respective relations may notbe with the same part of it. 

It is true that in what has been called the "numerically definitesyllogism, " an inference may be drawn, though our canon seems to beviolated. Thus:

 60 sheep in 100 are horned; 60 sheep in 100 are blackfaced: ∴ at least 20 blackfaced sheep in 100 are horned. 

But such an argument, though it may be correct Arithmetic, is not Logicat all; and when such numerical evidence is obtainable the comparativelyindefinite arguments of Logic are needless. Another apparent exceptionis the following:

 Most men are 5 feet high; Most men are semi-rational: ∴ Some semi-rational things are 5 feet high. 

Here the Middle Term (men) is distributed in neither premise, yet theindisputable conclusion is a logical proposition. The premises, however, are really arithmetical; for 'most' means 'more than half, ' or more than50 per cent. 

Still, another apparent exception is entirely logical. Suppose we aregiven, the premises--_All P is M_, and _All S is M_--the middle term isundistributed. But take the obverse of the contrapositive of bothpremises:

 All m is p; All m is s: ∴ Some s is p. 

Here we have a conclusion legitimately obtained; but it is not in theterms originally given. 

For Mediate Inference depending on truly logical premises, then, it isnecessary that one premise should distribute the middle term; and thereason of this may be illustrated even by the above supposed numericalexceptions. For in them the premises are such that, though neither ofthe two premises by itself distributes the Middle, yet they alwaysoverlap upon it. If each premise dealt with exactly half the Middle, thus barely distributing it between them, there would be no logicalproposition inferrible. We require that the middle term, as used in onepremise, should necessarily overlap the same term as used in the other, so as to furnish common ground for comparing the other terms. Hence Ihave defined the middle term as 'that term common to both premises bymeans of which the other terms are compared. '

(5) One at least of the premises must be affirmative; or, from twonegative premises nothing can be inferred (in the given terms). 

The fourth Canon required that the middle term should be givendistributed, or in its whole extent, at least once, in order to affordsure ground of comparison for the others. But that such comparison maybe effected, something more is requisite; the relation of the otherterms to the Middle must be of a certain character. One at least of themmust be, as to its extent or denotation, partially or wholly identifiedwith the Middle; so that to that extent it may be known to bear to theother term, whatever relation we are told that so much of the Middlebears to that other term. Now, identity of denotation can only bepredicated in an affirmative proposition: one premise, then, must beaffirmative. 

If both premises are negative, we only know that both the other termsare partly or wholly excluded from the Middle, or are not identical withit in denotation: where they lie, then, in relation to one another wehave no means of knowing. Similarly, in the mediate comparison ofquantities, if we are told that A and C are both of them unequal to B, we can infer nothing as to the relation of C to A. Hence the premises--

 No electors are sober; No electors are independent--

however suggestive, do not formally justify us in inferring anyconnection between sobriety and independence. Formally to draw aconclusion, we must have affirmative grounds, such as in this case wemay obtain by obverting both premises:

 All electors are not-sober; All electors are not-independent: ∴ Some who are not-independent are not-sober. 

But this conclusion is not in the given terms. 

(6) (a) If one premise be negative, the conclusion must be negative: and(b) to prove a negative conclusion, one premise must be negative. 

(a) For we have seen that one premise must be affirmative, and that thusone term must be partly (at least) identified with the Middle. If, then, the other premise, being negative, predicates the exclusion of theremaining term from the Middle, this remaining term must be excludedfrom the first term, so far as we know the first to be identical withthe Middle: and this exclusion will be expressed by a negativeconclusion. The analogy of the mediate comparison of quantities may hereagain be noticed: if A is equal to B, and B is unequal to C, A isunequal to C. 

(b) If both premises be affirmative, the relations to the Middle of boththe other terms are more or less inclusive, and therefore furnish noground for an exclusive inference. This also follows from the functionof the middle term. 

For the more convenient application of these canons to the testing ofsyllogisms, it is usual to derive from them three Corollaries:

(i) Two particular premises yield no conclusion. 

For if both premises be affirmative, _all_ their terms areundistributed, the subjects by predesignation, the predicates byposition; and therefore the middle term must be undistributed, and therecan be no conclusion. 

If one premise be negative, its predicate is distributed by position:the other terms remaining undistributed. But, by Canon 6, the conclusion(if any be possible) must be negative; and therefore its predicate, themajor term, will be distributed. In the premises, therefore, both themiddle and the major terms should be distributed, which is impossible:e. G. , Some M is not P; Some S is M: ∴ Some S is not P. 

Here, indeed, the major term is legitimately distributed (though thenegative premise might have been the minor); but M, the middle term, isdistributed in neither premise, and therefore there can be noconclusion. 

Still, an exception may be made by admitting a bi-designate conclusion:

 Some P is M; Some S is not M: ∴ Some S is not some P. 

(ii) If one premise be particular, so is the conclusion. 

For, again, if both premises be affirmative, they only distribute oneterm, the subject of the universal premise, and this must be the middleterm. The minor term, therefore, is undistributed, and the conclusionmust be particular. 

If one premise be negative, the two premises together can distributeonly two terms, the subject of the universal and the predicate of thenegative (which may be the same premise). One of these terms must be themiddle; the other (since the conclusion is negative) must be the major. The minor term, therefore, is undistributed, and the conclusion must beparticular. 

(iii) From a particular major and a negative minor premise nothing canbe inferred. 

For the minor premise being negative, the major premise must beaffirmative (5th Canon); and therefore, being particular, distributesthe major term neither in its subject nor in its predicate. But sincethe conclusion must be negative (6th Canon), a distributed major term isdemanded, e. G. , Some M is P; No S is M: ∴ ------

Here the minor and the middle terms are both distributed, but not themajor (P); and, therefore, a negative conclusion is impossible. 

§ 3. First Principle or Axiom of the Syllogism. --Hitherto in thischapter we have been analysing the conditions of valid mediateinference. We have seen that a single step of such inference, aSyllogism, contains, when fully expressed in language, threepropositions and three terms, and that these terms must stand to oneanother in the relations required by the fourth, fifth, and sixthCanons. We now come to a principle which conveniently sums up theseconditions; it is called the _Dictum de omni et nullo_, and may bestated thus:

 Whatever is predicated (affirmatively or negatively) of a term distributed, With which term another term can be (partly or wholly) identified, May be predicated in like manner (affirmatively or negatively) of the latter term (or part of it). 

Thus stated (nearly as by Whately in the introduction to his _Logic_)the _Dictum_ follows line by line the course of a Syllogism in the FirstFigure (see chap. X. § 2). To return to our former example: _All authorsare vain_ is the same as--Vanity is predicated of all authors; _Cicerois an author_ is the same as--Cicero is identified as an author;therefore _Cicero is vain_, or--Vanity may be predicated of Cicero. The_Dictum_ then requires: (1) three propositions; (2) three terms; (3)that the middle term be distributed; (4) that one premise beaffirmative, since only by an affirmative proposition can one term beidentified with another; (5) that if one premise be negative theconclusion shall be so too, since whatever is predicated of the middleterm is predicated _in like manner_ of the minor. 

Thus far, then, the _Dictum_ is wholly analytic or verbal, expressing nomore than is implied in the definitions of 'Syllogism' and 'MiddleTerm'; since (as we have seen) all the General Canons (except the third, which is a still more general condition of formal proof) are derivablefrom those definitions. However, the _Dictum_ makes a further statementof a synthetic or real character, namely, that _when these conditionsare fulfilled an inference is justified_; that then the major and minorterms are brought into comparison through the middle, and that the majorterm may be predicated affirmatively or negatively of all or part of theminor. It is this real assertion that justifies us in calling the_Dictum_ an Axiom. 

§ 4. Whether the Laws of Thought may not fully explain the Syllogismwithout the need of any synthetic principle has, however, been made aquestion. Take such a syllogism as the following:

 All domestic animals are useful; All pugs are domestic animals: ∴ All pugs are useful. 

Here (an ingenious man might urge), having once identified pugs withdomestic animals, that they are useful follows from the Law of Identity. If we attend to the meaning, and remember that what is true in one formof words is true in any other form, then, all domestic animals beinguseful, of course pugs are. It is merely a case of subalternation: wemay put it in this way:

 All domestic animals are useful: ∴ Some domestic animals (e. G. , pugs) are useful. 

The derivation of negative syllogisms from the Law of Contradiction (hemight add) may be shown in a similar manner. 

But the force of this ingenious argument depends on the participialclause--'having once identified pugs with domestic animals. ' If this isa distinct step of the reasoning, the above syllogism cannot be reducedto one step, cannot be exhibited as mere subalternation, nor be broughtdirectly under the law of Identity. If 'pug, ' 'domestic, ' and 'useful'are distinct terms; and if 'pug' and 'useful' are only known to beconnected because of their relations to 'domestic': this is somethingmore than the Laws of Thought provide for: it is not ImmediateInference, but Mediate; and to justify it, scientific method requiresthat its conditions be generalised. The _Dictum_, then, as we have seen, does generalise these conditions, and declares that when such conditionsare satisfied a Mediate Inference is valid. 

But, after all (to go back a little), consider again that proposition_All pugs are domestic animals_: is it a distinct step of the reasoning;that is to say, is it a Real Proposition? If, indeed, 'domestic' is nopart of the definition of 'pug, ' the proposition is real, and is adistinct part of the argument. But take such a case as this:

 All dogs are useful; All pugs are dogs. 

Here we clearly have, in the minor premise, only a verbal proposition;to be a dog is certainly part of the definition of 'pug. ' But, if so, the inference 'All pugs are useful' involves no real mediation, and theargument is no more than this:

 All dogs are useful; ∴ Some dogs (e. G. , pugs) are useful. 

Similarly, if the major premise be verbal, thus:

 All men are rational; Socrates is a man--

to conclude that 'Socrates is rational' is no Mediate Inference; for somuch was implied in the minor premise, 'Socrates is a man, ' and themajor premise adds nothing to this. 

Hence we may conclude (as anticipated in chap. Vii. § 3) that 'anyapparent syllogism, having one premise a verbal proposition, is reallyan Immediate Inference'; but that, if both premises are realpropositions, the Inference is Mediate, and demands for its explanationsomething more than the Laws of Thought. 

The fact is that to prove the minor to be a case of the middle term maybe an exceedingly difficult operation (chap. Xiii. § 7). The difficultyis disguised by ordinary examples, used for the sake of convenience. 

§ 5. Other kinds of Mediate Inference exist, yielding valid conclusions, without being truly syllogistic. Such are mathematical inferences ofEquality, as--

 A = B = C ∴ A = C. 

Here, according to the usual logical analysis, there are strictly fourterms--(1) A, (2) equal to B, (3) B, (4) equal to C. 

Similarly with the argument _a fortiori_, A > B > C ∴ (much more) A > C. 

This also is said to contain four terms: (1) A, (2) greater than B, (3)B, (4) greater than C. Such inferences are nevertheless intuitivelysound, may be verified by trial (within the limits of sense-perception), and are generalised in appropriate axioms of their own, corresponding tothe _Dictum_ of the syllogism; as 'Things equal to the same thing areequal to one another, ' etc. 

Now, surely, this is an erroneous application of the usual logicalanalysis of propositions. Both Logic and Mathematics treat of the_relations_ of terms; but whilst Mathematics employs the sign = for onlyone kind of relation, and for that relation exclusive of the terms;Logic employs the same signs (_is_ or _is not_) for all relations, recognising only a difference of quality in predication, and treatingevery other difference of relation as belonging to one of the termsrelated. Thus Logicians read _A--is--equal to B_: as if _equal to B_could possibly be a term co-relative with A. Whence it follows that theargument _A = B = C ∴ A = C_ contains four terms; though everybody seesthat there are only three. 

In fact (as observed in chap. Ii. § 2) the sign of logical relation(_is_ or _is not_), whilst usually adequate for class-reasoning(coinherence) and sometimes extensible to causation (because a causeimplies a class of events), should never be stretched to include otherrelations in such a way as to sacrifice intelligence to formalism. And, besides mathematical or quantitative relations, there are others(usually considered qualitative because indefinite) which cannot bejustly expressed by the logical copula. We ought to read propositionsexpressing time-relations (and inferences drawn accordingly) thus:

 B--is before--C; A--is before--B: ∴ A--is before--C. 

And in like manner _A--is simultaneous with--B; etc. _ Such arguments (aswell as the mathematical) are intuitively sound and verifiable, andmight be generalised in axioms if it were worth while: but it is not, because no method could be founded on such axioms. 

The customary use of relative terms justifies some Mediate Inferences, as, _The father of a father is a grand-father_. 

Some cases, however, that at first seem obvious, are really delusiveunless further data be supplied. Thus _A co-exists with B, B with C; ∴ Awith C_--is not sound unless _B_ is an instantaneous event; for where Bis perdurable, _A_ may co-exist with it at one time and _C_ at another. 

Again: _A is to the left of B, B of C; ∴ A of C_. This may pass; but itis not a parallel argument that if _A is north of B and B west of C_, then _A is north-west of C_: for suppose that A is a mile to the northof B, and B a yard to the west of C, then A is practically north of C;at least, its westward position cannot be expressed in terms of themariner's compass. In such a case we require to know not only thedirections but the distances of A and C from B; and then the exactdirection of A from C is an affair of mathematical calculation. 

Qualitative reasoning concerning position is only applicable to thingsin one dimension of space, or in time considered as having onedimension. Under these conditions we may frame the followinggeneralisation concerning all Mediate Inferences: Two terms definitelyrelated to a third, and one of them positively, are related to oneanother as the other term is related to the third (that is, positivelyor negatively); provided that the relations given are of the same kind(that is, of Time, or Coinherence, or Likeness, or Equality). 

Thus, to illustrate by relations of Time--

 B is simultaneous with C; A is not simultaneous with B: ∴ A is not simultaneous with C. 

Here the relations are of the same kind but of different logicalquality, and (as in the syllogism) a negative copula in the premisesleads to a negative conclusion. 

An examination in detail of particular cases would show that the abovegeneralisation concerning all Mediate Inferences is subject to too manyqualifications to be called an Axiom; it stands to the real Axioms (the_Dictum_, etc. ) as the notion of the Uniformity of Nature does to thedefinite principles of natural order (_cf. _ chap. Xiii. § 9). 

CHAPTER X

CATEGORICAL SYLLOGISMS

§ 1. The type of logical, deductive, mediate, categorical Inference is aSyllogism directly conformable with the _Dictum_: as--

 All carnivores (M) are excitable (P); Cats (S) are carnivores (M): ∴ Cats (S) are excitable (P). 

In this example P is predicated of M, a term distributed; in which term, M, S is given as included; so that P may be predicated of S. 

Many arguments, however, are of a type superficially different from theabove: as--

 No wise man (P) fears death (M); Balbus (S) fears death (M): ∴ Balbus (S) is not a wise man (P). 

In this example, instead of P being predicated of M, M is predicated ofP, and yet S is given as included not in P, but in M. The divergence ofsuch a syllogism from the _Dictum_ may, however, be easily shown to besuperficial by writing, instead of _No wise man fears death_, thesimple, converse, _No man who fears death is wise_. 

Again:

 Some dogs (M) are friendly to man (P); All dogs (M) are carnivores (S): ∴ Some carnivores (S) are friendly to man (P). 

Here P is predicated of M undistributed; and instead of S being includedin M, M is included in S: so that the divergence from the type ofsyllogism to which the _Dictum_ directly applies is still greater thanin the former case. But if we transpose the premises, taking first

 All dogs (M) are carnivores (P), then P is predicated of M distributed; and, simply converting the otherpremise, we get--

 Some things friendly to man (S) are dogs (M):

whence it follows that--

 Some things friendly to man (S) are carnivores (P);

and this is the simple converse of the original conclusion. 

Once more:

 No pigs (P) are philosophers (M); Some philosophers (M) are hedonists (S): ∴ Some hedonists (S) are not pigs (P). 

In this case, instead of P being predicated of M distributed, M ispredicated of P distributed; and instead of S (or part of it) beingincluded in M, we are told that some M is included in S. Still there isno real difficulty. Simply convert both the premises, and we have:

 No philosophers (M) are pigs (P); Some hedonists (S) are philosophers (M). 

Whence the same conclusion follows; and the whole syllogism plainlyconforms directly to the _Dictum_. 

Such departures as these from the normal syllogistic form are said toconstitute differences of Figure (see § 2); and the processes by whichthey are shown to be unessential differences are called Reduction (see §6). 

§ 2. Figure is determined by the position of the Middle Term in thepremises; of which position there are four possible variations. Themiddle term may be subject of the major premise, and predicate of theminor, as in the first example above; and this position, being directlyconformable to the requirements of the _Dictum_, is called the FirstFigure. Or the middle term may be predicate of both premises, as in thesecond of the above examples; and this is called the Second Figure. Orthe middle term may be subject of both premises, as in the third of theabove examples; and this is called the Third Figure. Or, finally, themiddle term may be predicate of the major premise, and subject of theminor, as in the fourth example given above; and this is the FourthFigure. 

It may facilitate the recollection of this most important point if weschematise the figures thus:

 I. II. III. IV. 

M---P P---M M---P P---M \ | | / \ | | / \ | | /S---M S---M M---S M---S

The horizontal lines represent the premises, and at the angles formedwith them by the slanting or by the perpendicular lines the middle termoccurs. The schema of Figure IV. Resembles Z, the last letter of thealphabet: this helps one to remember it in contrast with Figure I. , which is thereby also remembered. Figures II. And III. Seem to standback to back. 

§ 3. The Moods of each Figure are the modifications of it which arisefrom different combinations of propositions according to quantity andquality. In Figure I. , for example, four Moods are recognised: A. A. A. , E. A. E. , A. I. I. , E. I. O. 

 A. All M is P; A. All S is M: A. ∴ All S is P. 

 E. No M is P; A. All S is M: E. ∴ No S is P. 

 A. All M is P; I. Some S is M: I. ∴ Some S is P. 

 E. No M is P; I. Some S is M: O. ∴ Some S is not P. 

Now, remembering that there are four Figures, and four kinds ofpropositions (A. I. E. O. ), each of which propositions may be majorpremise, minor premise, or conclusion of a syllogism, it appears that ineach Figure there may be 64 Moods, and therefore 256 in all. Onexamining these 256 Moods, however, we find that only 24 of them arevalid (i. E. , of such a character that the conclusion strictly followsfrom the premises), whilst 5 of these 24 are needless, because theirconclusions are 'weaker' or less extensive than the premises warrant;that is to say, they are particular when they might be universal. Thus, in Figure I. , besides the above 4 Moods, A. A. I. And E. A. O. Are valid inthe sense of being conclusive; but they are superfluous, becauseincluded in A. A. A. And E. A. E. Omitting, then, these 5 needless Moods, which are called 'Subalterns' because their conclusions are subaltern(chap. Vii. § 2) to those of other Moods, there remain 19 Moods that arevalid and generally recognised. 

§ 4. How these 19 Moods are determined must be our next inquiry. Thereare several ways more or less ingenious and interesting; but all dependon the application, directly or indirectly, of the Six Canons, whichwere shown in the last chapter to be the conditions of MediateInference. 

(1) One way is to begin by finding what Moods of Figure I. Conform tothe _Dictum_. Now, the _Dictum_ requires that, in the major premise, Pbe predicated of a term distributed, from which it follows that no Moodcan be valid whose major premise is particular, as in I. A. I. Or O. A. O. Again, the _Dictum_ requires that the minor premise be affirmative("with which term another is identified"); so that no Mood can be validwhose minor premise is negative, as in A. E. E. Or A. O. O. By suchconsiderations we find that in Figure I. , out of 64 Moods possible, onlysix are valid, namely, those above-mentioned in § 3, including the twosubalterns. The second step of this method is to test the Moods of theSecond, Third, and Fourth Figures, by trying whether they can be reducedto one or other of the four Moods of the First (as briefly illustratedin § 1, and to be further explained in § 6). 

(2) Another way is to take the above six General or Common Canons, andto deduce from them Special Canons for testing each Figure: aninteresting method, which, on account of its length, will be treated ofseparately in the next section. 

(3) Direct application of the Common Canons is, perhaps, the simplestplan. First write out the 64 Moods that are possible without regard toFigure, and then cross out those which violate any of the Canons orCorollaries, thus:

AAA, [\A\A\E] (6th Can. B). AAI. [\A\A\O] (6th Can. B). [\A\E\A] (6th Can. A) AEE, [\A\E\I] (6th Can. A) AEO, [\A\I\A] (Cor. Ii. ) [\A\I\E] (6th Can. B) AII, [\A\I\O] (6th Can. B)[\A\O\A] (6th Can. A) [\A\O\E] (Cor. Ii. ) [\A\O\I] (6th Can. A) AOO. 

Whoever has the patience to go through the remaining 48 Moods willdiscover that of the whole 64 only 11 are valid, namely:

A. A. A. , A. A. I. , A. E. E. , A. E. O. , A. I. I. , A. O. O. , E. A. E. , E. A. O. , E. I. O. , I. A. I. , O. A. O. 

These 11 Moods have next to be examined in each Figure, and if valid inevery Figure there will still be 44 moods in all. We find, however, thatin the First Figure, A. E. E. , A. E. O. , A. O. O. Involve illicit process ofthe major term (3rd Can. ); I. A. I. , O. A. O. Involve undistributed Middle(4th Can. ); and A. A. I. , E. A. O. Are subalterns. In the Second Figure allthe affirmative Moods, A. A. A. , A. A. I. , A. I. I. , I. A. I. , involveundistributed Middle; O. A. O. Gives illicit process of the major term;and A. E. O. , E. A. O. Are subalterns. In the Third Figure, A. A. A. , E. A. E. , involve illicit process of the minor term (3rd Can. ); A. E. E. , A. E. O. , A. O. O. , illicit process of the major term. In the Fourth Figure, A. A. A. And E. A. E. Involve illicit process of the minor term; A. I. I. , A. O. O. , undistributed Middle; O. A. O. Involves illicit process of the major term;and A. E. O. Is subaltern. 

Those moods of each Figure which, when tried by these tests, are notrejected, are valid, namely:

Fig. I. --A. A. A. , E. A. E. , A. I. I. , E. I. O. (A. A. I. , E. A. O. , Subaltern);

Fig. II. --E. A. E. , A. E. E. , E. I. O. , A. O. O. (E. A. O. , A. E. O. , Subaltern);

Fig. III. --A. A. I. , I. A. I. , A. I. I. , E. A. O. , O. A. O. , E. I. O. ;

Fig. IV. --A. A. I. , A. E. E. , I. A. I. , E. A. O. , E. I. O. (A. E. O. , Subaltern). 

Thus, including subaltern Moods, there are six valid in each Figure. InFig. III. Alone there is no subaltern Mood, because in that Figure therecan be no universal conclusion. 

§ 5. Special Canons of the several Figures, deduced from the CommonCanons, enable us to arrive at the same result by a somewhat differentcourse. They are not, perhaps, necessary to the Science, but afford avery useful means of enabling one to thoroughly appreciate the characterof formal syllogistic reasoning. Accordingly, the proof of each rulewill be indicated, and its elaboration left to the reader. There is nodifficulty, if one bears in mind that Figure is determined by theposition of the middle term. 

Fig. I. , Rule (a): _The minor premise must be affirmative_. 

For, if not, in negative Moods there will be illicit process of themajor term. Applying this rule to the eleven possible Moods given in §4, as remaining after application of the Common Canons, it eliminatesA. E. E. , A. E. O. , A. O. O. 

(b) _The major premise must be universal_. 

For, if not, the minor premise being affirmative, the middle term willbe undistributed. This rule eliminates I. A. I. , O. A. O. ; leaving sixMoods, including two subalterns. 

Fig. II. (a) _One premise must be negative. _

For else neither premise will distribute the middle term. This ruleeliminates A. A. A. , A. A. I. , A. I. I. , I. A. I. 

(b) _The major premise must be universal. _

For else, the conclusion being negative, there will be illicit processof the major term. This eliminates I. A. I. , O. A. O. ; leaving six Moods, including two subalterns. 

Fig. III. (a) _The minor premise must be affirmative. _

For else, in negative moods there will be illicit process of the majorterm. This rule eliminates A. E. E. , A. E. O. , A. O. O. 

(b) _The conclusion must be particular. _

For, if not, the minor premise being affirmative, there will be illicitprocess of the minor term. This eliminates A. A. A. , A. E. E. , E. A. E. ;leaving six Moods. 

Fig. IV. (a) _When the major premise is affirmative, the minor must beuniversal. _

For else the middle term is undistributed. This eliminates A. I. I. , A. O. O. 

(b) _When the minor premise is affirmative the conclusion must beparticular. _

Otherwise there will be illicit process of the minor term. Thiseliminates A. A. A. , E. A. E. 

(c) _When either premise is negative, the major must be universal. _

For else, the conclusion being negative, there will be illicit processof the major term. This eliminates O. A. O. ; leaving six Moods, includingone subaltern. 

§ 6. Reduction is either--(1) Ostensive or (2) Indirect. OstensiveReduction consists in showing that an argument given in one Mood canalso be stated in another; the process is especially used to show thatthe Moods of the second, third, and fourth Figures are equivalent to oneor another Mood of the first Figure. It thus proves the validity of theformer Moods by showing that they also essentially conform to the_Dictum_, and that all Categorical Syllogisms are only superficialvarieties of one type of proof. 

To facilitate Reduction, the recognised Moods have all had names giventhem; which names, again, have been strung together into mnemonic versesof great force and pregnancy:

 Barbara, Celarent, Darii, Ferioque prioris: Cesare, Camestres, Festino, Baroco, secundæ: Tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet: Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison. 

In the above verses the names of the Moods of Fig. I. Begin with thefirst four consonants B, C, D, F, in alphabetical order; and the namesof all other Moods likewise begin with these letters, thus signifying(except in Baroco and Bocardo) the mood of Fig. I. , to which each isequivalent, and to which it is to be reduced: as Bramantip to Barbara, Camestres to Celarent, and so forth. 

The vowels A, E, I, O, occurring in the several names, give the quantityand quality of major premise, minor premise, and conclusion in the usualorder. 

The consonants s and p, occurring after a vowel, show that theproposition which the vowel stands for is to be converted either (s)simply or (p) _per accidens_; except where s or p occurs after the thirdvowel of a name, the conclusion: then it refers not to the conclusion ofthe given Mood (say Disamis), but to the conclusion of that Mood of thefirst Figure to which the given Mood is reduced (Darii). 

M (_mutare_, metathesis) means 'transpose the premises' (as ofCa_m_estres). 

C means 'substitute the contradictory of the conclusion for theforegoing premise, ' a process of the Indirect Reduction to be presentlyexplained (see Baroco, § 8). 

The other consonants, r, n, t (with b and d, when not initial), occurring here and there, have no mnemonic significance. 

What now is the problem of Reduction? The difference of Figures dependsupon the position of the Middle Term. To reduce a Mood of any otherFigure to the form of the First, then, we must so manipulate itspremises that the Middle Term shall be subject of the major premise andpredicate of the minor premise. 

Now in Fig. II. The Middle Term is predicate of both premises; so thatthe minor premise may need no alteration, and to convert the majorpremise may suffice. This is the case with Cesare, which reduces toCelarent by simply converting the major premise; and with Festino, whichby the same process becomes Ferio. In Camestres, however, the minorpremise is negative; and, as this is impossible in Fig. I. , the premisesmust be transposed, and the new major premise must be simply converted:then, since the transposition of the premises will have transposed theterms of the conclusion (according to the usual reading of syllogisms), the new conclusion must be simply converted in order to prove thevalidity of the original conclusion. The process may be thus represented(_s. C. _ meaning 'simply convert')

 Camestres. Celarent. 

 All P is M; ----\ /---> No M is S; \ c/ \/ s/\ / \ No S is M: ----/ \---> All P is M:

 s. C. ∴ No S is P. No m (not-M) is P;

 _obv_ Some S is not M: -----------------------> Some S is m (not-M): ∴ Some S is not P. ∴ Some S is not P. 

In Fig. III. The middle term is subject of both premises; so that, toreduce its Moods to the First Figure, it may be enough to convert theminor premise. This is the case with Darapti, Datisi, Felapton, andFerison. But, with Disamis, since the major premise must in the FirstFigure be universal, we must transpose the premises, and then simplyconvert the new minor premise; and, lastly, since the major and minorterms have now changed places, we must simply convert the new conclusionin order to verify the old one. Thus:

 Disamis. Darii. 

 Some M is P; ----\ /---> All M is S; \s. / \/ /\c. / \ All M is S: ----/ \---> Some P is M:

 s. C. ∴ Some S is P. All M is S; \ / \/ /\ _contrap_ / \ All M is S: ----------/ \---------> Some p (not-P) is M:

 _convert & obvert_∴ Some S is not P. All M is S; \/ /\All M is S: ----------/ \------> All P is M:

 convert per acc. ∴ Some S is P.