MASTER NEGA TIVE NO. 91-80030 \ MICROFILMED 1991 COLUMBIA UNIVERSITY LIBRARIES/NEW YORK as part of the "Foundations of Western Civilization Preservation Project Funded by the NATIONAL ENDOWMENT FOR THE HUMANITIES Reproductions may not be made without permission from Columbia University Library ft' IV. COPYRIGHT STATEMENT The copyright law of the United States - Title 17, United States Code - concerns the making of photocopies or other reproductions of copyrighted material ... Columbia University Library reserves the right to refuse to accept a copy order if, in its judgement, fulfillment of the order would involve violation of the copyright law. if • AUTHOR: FOWLER, THOMAS TITLE: ELEMENTS OF DEDUCTIVE LOGIC PLACE: OXFORD DA TE : 1892 Master Negative # % COLUMBIA UNIVERSITY LIBRARIES PRESERVATION DEPARTMENT BIBLIOGRAPHIC MICROFORM TARGET Original Material as Filmed - Existing Bibliographic Record Restrictions on Use: i » y» W . II » n . ■ pp y 1 11 .p^ i j niy. ii . f mm 4 P i wmm TS! I ll i Jl W IfPJ >Fowler, Thomas, ilfb*: 1B3S- 1904, . ' r'^ •*>» The elements of deductive logic. .... Tenth edition, corrected and revised. xv,i92 p. S. (Clarendon Press series.) Oxford: Clarendon Press, i^9S.189f?« (His Logic deductive and inductive) . - . risqf^/: u TECHNICAL MICROFORM DATA REDUCTION RATIO: FILM SIZE: "^ ^ ^ ^ IMAGE PLACEMENT: lA DATE FILMED: HLMEDBY: RESEARCH PUBLICATIONS. INC WOODBRIDGE. CT //v INITIALS M^ c Association for information and Image Management 1100 Wayne Avenue. Suite 1100 Silver Spring, Maryland 20910 301/587-8202 Centimeter 1 2 3 mi iiiiliiiiliiiiliiiiliiiiliiiili I I 4 12 13 14 15 m I 5 6 7 8 9 10 11 iljiiil|iiil|iiil|iiiliiiiliiii m Inches 1.0 I.I 1.25 m m m ■ 6.3 2.8 1 I 3.2 3.6 4.0 1.4 2.5 2.2 2.0 1.8 1.6 MONUFflCTURED TO fillM STRNDRRDS BY fiPPLIED IMRGE, INC. Class I O h^ Book I 'C Columbia College Library Madison Av. and 49th St. New York. Beside ihe mam topic this book also treats of Suhject No. On page Suhject No. On page "''1 ill n i \A > €hmhn |wss Mm DEDUCTIVE LOGIC FOWLER a THE ELEMENTS OF HENRY FROWDE Oxford University Press Warehousk Amen Corner, £.C. DEDUCTIVE LOGIC DESIGNED MAINLY * FOR THE USE OF JUNIOR STUDENTS IN THE UNIVERSITIES BY THOMAS FOWLER, D.D, President of Corpus ChrisH College Wykeham Professor of I.o^ic in the University of Oxford And Honorary Doctor of Laws in the University of I-dinbur^rh NINTH EDITION CORRECTED AND REVISED AT THE CLARENDON PRESS MDCCCLXXXVII [ AH rights reserved'\ PREFACE. 1 THE precise object of the following pages is (without pre-supposing any technical acquaintance with logical terminology) to enable a student of average intelligence to acquire for himself an elementary knowledge of the main problems, principles, and rules of Deductive Logic. They are not designed to save him the trouble of after- wards consulting more advanced text-books, either in his own or other languages. The English student who wishes to gain an exact and detailed knowledge of the relations of Deduction to Induction, and consequently of the true place and value of the former process in any special science, must still have recourse to the works of Mr. Mill; or, if he wish to trace the history of logical terms and doctrines (one of the most important chapters in the history of both ancient and modern literature), he must still consult Sir W. Hamilton's Lectures, and the Appendices and Notes of Dr. Mansel to Aldrich's Logic. To these works, as well as to Archbishop Whately's luminous Chapter on Fallacies, and to the original and suggestive work of Mr. James Mill on the Analysts of the Phenomena of the Human Mind, the Author must, once for all, express his obligations. He has, however, endeavoured, on all disputed points, to reason out his 98188 VI PR EFA C E, own conclusions, feeling assured that no manual, how- ever elementary, can be of real service to the student, unless it express what may be called the 'reasoned opinions' of its author. The great difficulty to be encountered by any writer of an English Manual of Logic is the * unsetded state of our logical terminology. Many words have various significations, or are used in different senses by different writers, and often there are no recognised terms to express some distinction which it is still incumbent on the logician to notice. ' A fixed and sufficient terminology can, however, only be created by the habit of teaching Logic, and of carrying on our discussions on the science, in our own language. But though, in some respects, the Latin terminology may be superior to our own, there can be no question that the language in which men habitually think must be the fittest medium for analysing their thoughts. The Notes appended to the Chapters (as distinguished from the foot-notes) are designed to inform the student of any divergences from the ordinary mode of treatment, or to suggest to him further reading on topics which, if noticed at all, are only alluded to in the text. They may be omitted on the first reading. Besides the Notes appended to the various Chapters, it is perhaps desirable that the student, if he is entirely unacquainted with logical and psychological discussions, should omit, on the first reading, the Chapters on the Relation of Logic to Psychology, on the various Kinds PR EFA CE, Vll of Terms, on the Denotation and Connotation of Terms, on the Relation of the Predicate to the Subject of a Proposition, on Verbal and Real Propositions, on De- finitions, and on Divisions and Classifications. Unfor- tunately, the most difficult problems which the logician has to solve occur at the outset of his task. It is hoped that, independently of its bearing on University studies, a Short English Manual of Logic may be used with advantage in the Upper Forms of Schools, and that it may not be without interest to the general reader. The Manuals of Sanderson, Wallis, Aldrich, «fec., owing to the peculiar circumstances of the period in .which they were written (a period, which, being transitional, retained not only much of the scholastic terminology, but also much of the Realistic doctrine), have ceased to be adapted to modern instruction. The Author, with some misgivings, and a keen sense of the difficulties of the task, trusts that the present w^ork may be found use- fully to occupy their place. Its propositions cannot, however, be presented in the same curt and dogmatic shape, for we have learnt to regard many portions of Logic, like many portions of the sciences whose method it claims to analyse, as fairly open to differences of opinion. * . * In the Table of Contents it will be observed that the words Definitions, Divisions, Classifications, Infer- ences, Oppositions, Conversions, Permutations, occur in I VI 11 PREFA CE, place of the more ordinary forms Definition, Division, Classification, &c. The object of this change is to suggest to the student the importance of distinguishing the results from the processes by which they are gained. Many words employed in Logic and Psychology admit of both these meanings, and it is only by prefixing the indefinite article or using the plural number, as when we . speak of ' a definition ' or ' definitions,' that we can make it plain that we mean the result and not the process. It would be very difficult in all cases to mark the distinction, but I. have endeavoured to do so wherever it seemed to be of any importance ^ ' A similar confusion in many of the terms employed in Physics has been noticed by Mr. [now Sir W. R.] Grove {Correlation of Forces, fifth ed., p. 251). ' Another confusion of terms has arisen, and has, indeed, much embarrassed me in enunciating the propositions put forth in these pages on account of the imperfection of scientific language ; an imperfection in great measure unavoidable, it is true, but not the less embarrassing. Thus, the words light, heat, electricity, and magnetism, are constantly used in two senses— viz. that of the force producing, or the subjective idea of force or power, and of the effect produced, or the objective phenomenon. The word motion, indeed, is only applied to the effect, and not to the force, and the term chemical affinity is generally applied to the force, and not to the effect ; but the other four terms are, for want of a distinct termin- ology, applied indiscriminately to both.' PREFACE TO SUBSEQUENT EDITIONS. IN subsequent Editions several minor alterations and additions have been introduced into the Text of some of the Chapters. A few Notes, as well as an Index and a Collection of Examples, have also been added. In making these alterations and additions, the Author has gladly availed himself of suggestions kindly offered, from time to time, by various correspondents and friends, amongst whom he ought specially to mention Professor Park, of Belfast. He has, however, endeavoured to pre- vent the work from materially exceeding the limits of its original size. Several friends have suggested to the Author the intro- duction of two new chapters, one on the Categories, the other on the Formation of Terms. Influenced partly by a desire not to increase the bulk of the volume, and partly by still more important reasons, he has, after some hesi- tation, decided against the introduction of these chapters. The doctrine of Categories is important in the history of Logic ^ but it is not properly a branch of Logic, as that word is now generally understood and as it has been understood throughout this work. The Formation of Terms is a subject which properly belongs to Psychology and not to Logic, and moreover could not be adequately treated in a small compass. It is true that some ques- - tions properly belonging to Psychology or the History of ^ On the categories of Aristotle the reader is referred to Hamilton's Lectures on Logic, Lect. xi., and the Appendix to Mansel's Aldrich, Note B. Both Sir W. Hamilton and Dr. Mansel, especially the former, insist on their * wholly extra-logical ' character. PREFACE TO SUBSEQUENT EDITIONS, Logic have been noticed in various parts of the book, but they have only been casually alluded to, not treated in distinct chapters. Logic has always been over-weighted with extraneous matter, and, wherever it is possible, it is desirable to relieve it of its superfluities, though much discretion may be needed in the process, and though the requirements of examinations have a constant tendency to lead writers on Logic to consider not how little, but how much they can introduce into their works. In the Fifth and subsequent Editions, the Author has added an Appendix on the Heads of Predicables. There is no branch of elementary Logic so difficult to state in a form at once satisfactory to the teacher and intelligible to the learner. The attempt at a scientific treatment which he has made in Pt. IL ch. v. is, he believes, usually found too complicated for a beginner. As, however, he cannot state his own view of the doctrine in a simpler form, he has thought it the best course to add an Appendix, which, without any attempt at an exhaustive treatment, simply offers an explanation of the five w^ords. Genus, Species, Differentia, Property, Accident. This Appendix, on the first reading of the book, may be substituted by the student for Pt. IL ch. v., but should be combined with the formal definitions given on p. 45, of which, in fact, it furnishes an explanation. The principal changes in the eighth Edition are the addition of notes to Pt. III. chs. i. and vii., and some alterations in the text of the latter part of ch. vii. ; and, in the present Edition, certain additions to the latter part of ch. vii. and to the foot-note on p. 73. ) I CONTENTS. Introduction. CHAP. I. The Relation of Logic to Psychology II. Definition of Logic .... III. The Relation of Thought to Language IV. Division of the Products of Thought PAGE 1 5 7 9 PABT I. The Term. I. The Various Kinds of Terms . II. The Denotation and Connotation of Terms II 19 PABT II. The Proposition. I. The Subject and Predicate 23 II. The Copula 25 III. Division of Propositions according to their Quantity and Quality 28 IV. Distribution of Terms 33 V. Relation of the Predicate to the Subject of a Proposition (Heads of Predicables} . . . . .36 Xll CONTENTS, CHAP. VI. Verbal and Real Propositions VII. Definitions VIII. Divisions and Classifications PART III. Inferences. I. The Various Kinds of Inferences II. Immediate Inferences § I. Definition of an immediate Inference § 2. Oppositions § 3. Conversions § 4. Permutations .... III. Mediate Inference or Syllogism . § I. Structure of the Syllogism . § 2. Moods and Figures § 3. Determination of the Legitimate Moods of Syllogism, including the Syllogistic Rules, Reduction, and the Special Rules of the Figures , IV. Trains of Reasoning (the Sorites) . . . . V. Complex (Hypothetical) Propositions and Syllogisms . § I. Division of Complex Proposirions into Con- junctive and Disjunctive .... § 2. Conjunctive Syllogisms § 3. Disjunctive Syllogisms § 4. The Dilemma VI. On the words ' Most,' * Many,' &c., as expressing the Quantity of Propositions PAGE 48 49 58 68 77 77 77 So 82 84 84 88 90 109 112 112 114 116 119 124 { CONTENTS. Xlii CHAP. PAGE VIII. Fallacies 140 § I. Division of Fallacies 140 § 2. Fallacies due to the assumption of a False Premiss 141 ' § 3. Fallacies due to the neglect of the Laws of Deductive Inference 142 § 4. Fallacies due to Irrelevancy . . . 147 § 5. Fallacies due to Ambiguity of Language . 149 IX. On Method, as applied to the Arrangement of Syl- logisms in a Train of Reasoning . . . .156 APPENDIX 160 EXAMPLES 165 INDEX , . 185 ■''-^<^^j^^^^n^^i^dt^'' VIL Probable Reasoning, including Circumstantial Evidence 128 ELEMENTS OF DEDUCTIVE LOGIC 'I r H INTRODUCTION. -M- CHAPTER I. On the Relation of Logic to Psychology, % PSYCHOLOGY, or Mental Philosophy, may be defined as the science which classifies and analyses the pheno- mena of the human mind. It notes mental phenomena, considers them in their mutual relations, and investigates the mode of their generation. Of these mental pheno- mena some are called emotional, others intellectual. The intellectual phenomena may be regarded as the result, partly of Perception, partly of Imagination, partly of Comparison, Reflexion, or Thought. Perception is the act or operation of apprehending some present pheno- menon. Imagination is the act or operation of repre- senting to the mind some absent phenomenon. Com- parison, Reflexion, or Thought, is the act or operation of comparing phenomena, whether present or absent, as well as of comparing, either with these or with one another, the results themselves which we derive from such comparisons. Thus I perceive this particular rose before me, its colour, its smell, and the pleasure I derive W 1 B Ill II 2 RELATION OF LOGIC from seeing and smelling it. All these perceptions I can recall to-morrow, even though the rose be absent, by the act or operation of imagination. Lastly, I may compare this particular rose with another which lies on the table, or with one which I saw yesterday, and I may express their similarity by calling them both moss-roses, or their difference by calling one a moss-rose and the other a Tudor rose. Again, moss-rose and Tudor rose, which names are both results of the act or operation of com- parison, may themselves be compared, and their points of similarity expressed by the word ' rose.' So rose and dahlia may be compared, and their points of similarity expressed by the word ' flower.' Or I may compare the feeling with which I contemplate the rose with similar feelings which I have previously experienced, and call it 'pleasure'; or I may compare it with the feeling which I experience when I prick my finger with a thorn, and call one ^pleasure,' the other *pain'; or I may compare pleasure and pain themselves, and call them both 'feelings.' The act of making comparisons, and of apprehending similarities and differences, is usually called Thought or Thinking, and the results at which it arrives Thought or Thoughts. The act or operation itself, as distinguished from other mental acts or opera- tions, and the results which ensue from its correct or incorrect exercise, are alike legitimate subjects of inves-- ligation for the psychologist or mental philosopher. But the more detailed consideration of the latter, i.e. Thoughts or the results of Thinking, becomes the subject \ TO PSYCHOLOGY, 3 of a science with a distinct name, Logic, which is thus a subordinate branch of the wider science, Psychology. Note I. — The term Perception is here used in its or- dinary sense. The distinctions between External and Internal Perception, Perception Proper and Sensation Proper, are foreign to the present subject. They are discussed at great length in the works of Sir William Hamilton and Dr. Mansel, as well as in those of the Scottish^'school of metaphysicians generally, but, as in- volving some of the most abstruse and disputed questions in Psychology, it is not necessary or desirable that the student should acquaint himself with them till he com- mences the special study of that science. Note 2. — Imagination, as here defined, is what may be called Simple, or Reproductive, as distinguished from what may be called Complex, or Productive Imagination. The former simply represents to the mind absent objects of perception as they have already been perceived, the latter combines phenomena'/ or portions of phenomena, whether absent or present, into a new whole. Thus the notions of a particular man or a particular horse, if the man and horse be absent, are products of simple or reproductive imagination ; the notion of a centaur would be a product of complex or productive imagination. For further and more precise information on this distinc- tion, which is here necessarily stated somewhat roughly, B 2 RELATION OF LOGIC TO PSYCHOLOGY. see Sir W. Hamilton's Lectures on Metaphysics, Lect. xxxiii. The creations of poetry and art are results of complex imagination, or, in other words, of repeated processes of simple imagination and comparison. Inasmuch as com- plex imagination may be analysed into simple imagina- tion and comparison, and is thus not one of the ultimate acts or operations to which our mental phenomena are traceable, it would have been beside my purpose to have noticed it in the text. Note 3. — I have employed the expression 'act or operation/ avoiding the expression 'power or faculty/ as the latter implies a theory of mental phenomena which would be rejected by many psychologists. It has been necessary to speak of 'act or operation/ as the word ' act/ like many other terms of logic or psychology, may mean either the operation or the result. This is an in- stance at the very outset of the ambiguity noticed in the paragraph at the end of the preface. « ^"^Gr^^^S^Si-^J-^ i' CHAPTER II. Definition of Logic, IT is the province of Logic to distinguish correct from incorrect thoughts, i*e. to analyse those thoughts which are accepted by mankind as indubitably correct, and to point out wherein they differ from those which are re- garded as doubtful or incorrect ; and, as a consequence of this function, it is also its province to lay down rules for the attainment of correct thoughts and for the avoid- ance of incorrect thoughts. Thus Logic is both a Science and an Art. It is a Science, inasmuch as it furnishes us with a knowledge of what is, inasmuch as it is an analysis, and determines the conditions on which valid thoughts depend. It is an Art, inasmuch as it lays down rules for practice, and thus enables us to detect incorrect thoughts in the reasonings of others, and to avoid them in our own. Logic may therefore be defined as the science of the conditions on which correct thoughts depend, and the art of attaining to correct and avoiding incorrect thoughts. Note. — This definition is in substance that given by Mr. Mill in his Examination of Sir W. Hamilton's DEFINITION OF LOGIC, Philosophy, p. 391 (third ed., p. 448). * Logic/ he says, * is the art of thinking, which means of correct thinking, and the science of the conditions of correct thinking/ The word * thoughts' is substituted for * thinking/ in order to bring more prominently before the student, what Mr. Mill himself acknowledges, the fact that Logic is concerned with the products or results rather than with the process of thought, i. e. with thoughts rather than with thinking, though, in estimating the conditions on which correct thoughts depend, it is necessary, to some extent, to take account of the processes by which they are formed. It seems also desirable to introduce into the definition of Logic some reference to * incorrect thoughts,' as bringing out more distinctly the character of Logic as an art, and asserting for it the right of investigating fallacies. CHAPTER in. On the Relation of Thought to Language, WHETHER it is possible to think without the aid of language, is a question which has been a constant source of dispute amongst logicians and psychologists. It is not necessary, however, here to enter on this discussion. As all logicians are agreed that we cannot communicate our thoughts without the aid of language, or of equivalent signs, and that practically we do always think by means of language, by a sort of internal con- verse, it will be safer to adopt the terminology of those authors who regard our thoughts as expressed in lan- guage rather than that of those who consider or attempt to consider them in themselves as apart from their ex- pression in words. I shall therefore speak of Terms and ' Propositions, not of Concepts and Judgments. Note. — Sir W. Hamilton and his followers, regarding Logic as primarily and essentially concerned with thought, and only secondarily and accidentally with language, attempt to mark the products of thought by words which Ao not imply their expression in language. Thus, instead 8 RELATION OF THOUGHT TO LANGUAGE. of Terms and Propositions, they use respectively the words Concepts and Judgments. The word Syllo- gism, owing to the ambiguity of the Greek word X^oy, stands either for the internal thought or the external expression of it. (See Hamilton's Lectures on Logic, Lecture i.) «i-^^N»S)S>''3v^ w CHAPTER IV. Division of the Products of Thought, IT has been stated that thought is the act or operation of comparison. Its simplest result is that which is ex- pressed by the Term. Terms may be combined into Propositions, and Propositions, either singly or in con- junction with one or more other propositions, may lead to Inferences. I shall treat in order of the Term, of the Proposition, of Inferences. Before proceeding further, it may perhaps be useful to the student to give by anticipation instances of these products or results of thought. Man, good, manliness, goodness, the good- ness of man, the virtue of manliness, are all instances of terms. ' This man is good,' ' All citizens of a state are under an obligation to obey its laws,' are instances of Propositions, and, according to logical phraseology, * good ' is said to be predicated of ' this man,' and ' under an obligation to obey its laws ' is said to be predicated of ' all citizens of a state.* The term predicated is called the Predicate-^ and the term of which it is predicated is called the Subject^ the word * is ' or * are ' (or, in the case of negative propositions, 'is not' or *are not'), which connects the two, being called the Copula. Lastly, we may take as instances of Inferences the following: — } lO DIVISION OF THE PRODUCTS OF THOUGHT, ' No rectilineal figure is contained by less than three lines ; Therefore, no figure contained by less than three lines is a rectilineal figure. All Englishmen are of mixed descent, This is an Englishman ; Therefore, he is of mixed descent. The last proposition (which is called the conclusion) is said to be inferred from the proposition or pro- positions which precede it (called X\it premiss ox pre- misses). *^~--^^<5*>5fe^^>^i.'-» M PART I.— The Term. CHAPTER I. On the Various Kinds of Terms. A TERM (so called from terminus, a boundary, be- cause the terms are the two extremes or boundaries of the proposition) is a word or combination of words which may stand by itself as the subject or predicate of a Proposition ; it expresses either an individual, a group of individuals, an attribute, or a group of attributes. This definition obviously excludes all articles, adverbs, inter- jections, conjunctions, prepositions, and oblique cases of nouns. It also excludes verbs; for though a verb ex- presses attributes, and often serves at once for the copula and the predicate (or part of the predicate), as in the pro- positions ' John walks,' ' William fears Thomas,' it must, in a logical proposition stated according to strict form, always be analysed into the copula and participle : thus the above propositions, when stated logically, become *John is walking/ 'William is fearing Tliomas ^' A ^ A proposition of which the verb is not analysed into the copula and a participle is called by the older logicians ' sectmdl adjacentis^ in contradistinction to the form of proposition as ordinarily stated in Logic, which is called * tertii adjacejitis' Thus vir currit is a pro- position secimdi adjacentis, vir est currens is a proposition tertii adjacentis. 12 VARIOUS KINDS OF TERMS. VARIOUS KINDS OF TERMS. 13 pronoun is only significant as standing in the place of a substantive, and therefore we may limit ourselves to the consideration of substantives, adjectives, and participles. By the older logicians a terra was defined as a Cale- gorematic word (from KaT,,y6pr,iia ' something which can be predicated'), i.e. a word or combination of words which can stand by itself as the predicate of a proposition. All words or combinations of words which require to be joined with some other word or words in order to serve this purpose were called Syncategorematic words. It has been said that a term expresses an individual, a group of individuals, an attribute, or a group of attri- butes. If it expresses an individual, as Socrates, the present Queen of England, the sea, hie, ille, hoc, iUud &c., It IS called a (i) Singular Term. If it expresses a group of individuals, it may either be applicable to each mdividual of the group severally as well as to the group collectively, or to the group coUectively but not to each mdividual severally. Thus the terms, man, horse, flower, are not only applicable to each individual of the groups which they express, but also to the groups collectively; I can say, 'John is a man,' ' Thomas is a man,' ' this is a horse," that is a horse,' 'the rose is a flpwer,' 'the dahlia is a flower,' &c. But I cannot say ' Caius is the fourteenth legion,' 'Pompey is the Roman senate,' though I can predicate 'fourteenth legion' or 'the Roman senate ' of the groups, taken collectively, which these terms express. In the former case the term is called a (2) Common Term, in the latter a (3) Colleclive r Term, A collective term may by a slight change of phraseology be expressed as a common term; thus, * Roman senate' may become 'Roman Senators/ or * fourteenth legion * ' soldiers of the fourteenth legion.' But, as it stands, a collective term is, in predication, as will be noticed hereafter, equivalent to a singular term. A common term is equally applicable to each indi- vidual severally of the group which it expresses, and it is so in virtue of certain points of similarity which all the individuals possess in common. It is in fact because we have observed that they all possess certain attributes in common, that we are able to call them by a common name. Thus a common term, like man, horse, &c., at once suggests to me a certain group of individuals and a certain group of attributes which is predicable of each of these individuals severally. But there are other terms, called (4) Attributives, and (5) Abstract Terms, which ex- press attributes or groups of attributes only. Attribu- tives may be distinguished from Abstract Terms by the fact that they may form the predicate, but cannot, unless joined with a singular, collective, common, or abstract term, form the subject of a proposition. Grammatically, they are represented by adjectives and participles, when not used substantively. Of this class are the terms, human, red, heavy, brave, willing, thinking, running, &c., when not used substantively. Thus I may say, ' Socrates is human/ but, when I wish to employ the term ' human ' in the subject of a proposition, I must append to it some such word as * being,' and say * this human being is 14 VARIOUS KINDS OF TERMS, VARIOUS KINDS OF TERMS, 15 I i- SocratesV Such terms have, in fact, no signification, unless used substantively, or in conjunction with a sub- stantive, either as qualifying it or as predicated of it. Abstract Terms, on the other hand, not only express mere attributes or groups of attributes, but may be thought of without any reference to the individuals of which these attributes are predicable. Of this class are such terms as humanity, colour, figure, fortitude, &c. To it may be referred sentences employing the indicative mood and introduced by the word ' that,' infinitive moods, some instances of adjectives and participles used substan- tively, and generally every term, being neither singular, collective, nor common, which may be employed both as the subject and as the predicate of a proposition. All the terms discussed above may be employed as predicates of propositions. Attributives alone cannot be employed as subjects. We have thus divided terms into singular, collective, and common terms, attributives, and abstract terms. It must however be borne in mind that many of those attri- butives and abstract terms which are most frequently in use, or which have been for a long time in use, have come to be employed as common terms. Thus we some- times speak of red, good, virtue, figure, number, pleasure, ^ It is perhaps hardly necessary to remark that in such phrases as ' Great is Diana of the Ephesians,' ' Great is my rejoicmg,' the places of the subject and predicate are reversed, for the sake of laying greater emphasis on the predicate. In a logical analysis of such propositions, the subject and predicate must be restored to their normal positions. colour, &c. (both in the singular and plural number), exactly as if they were common terms, though they still retain in other connexions their use as abstract terms or attributives. This remark will be found of great importance in some of the subsequent sections. Note I. — Nothing has been said above of the com- mon distinction between abstract and concrete terms. An abstract term would be defined as a term expressive of an attribute or group of attributes considered apart from the individuals of which it is predicable; a con- crete term as a term expressive of an attribute or group of attributes considered in reference to the individuals of which it is predicable, as well as of an individual or group of individuals itself The terms John, the tenth legion, man, human, would all be called concrete; humanity would be called an abstract term^. It will be noticed that I have availed myself of the expression * abstract term,' but avoided, as too wide to be of prac- tical service, the contrasted expression 'concrete term.' Concrete terms include what I have called attributives, as well as singular, collective, and common terms. ^ Mr. Mill {Logicj bk. i. ch. ii. § 4) states that Locke and several later writers have applied the expression * abstract name' to all general names, that is, attributives and common terms as well as what I have called abstract terms. This statement, however, is not uniformly true of Locke. See, for instance, Essay, bk. iii. ch. viii. Mr. Mill himself, following the practice of the Schoolmen, takes the expression in the same limited sense as in the text. By the older logicians singular and collective terms were not regarded as concrete, 110 account being taken of them in this distinction. "I i6 VARIOUS KINDS OF TERMS. VARIOUS KINDS OF TERMS. 17 Note 2.— The term 'attributive' (which has already been employed by Harris and James Mill) is used in preference to the term 'adjective/ both because it in- cludes participles, and because it seems undesirable in a work on Logic to employ a technical term of Grammar. Harris (Hermes, bk. i. ch. vi.) includes amongst ' attribu- tives ' verbs; but a verb, as has already been stated, is, in Logic, always represented by the copula and a participle. I have placed attributives before abstract terms, be- cause they are more nearly allied to singular, collective, and common terms, being, for the most part, either pre- dicated of these terms or employed to qualify them. They seem also as a rule to precede abstract terms in their formation. Thus human, red, brave, good, willing, must have been employed before the corresponding terms humanity, redness, bravery, goodness, willingness. Note 3.— For the sake of completeness, I have spoken of a term as expressing an attribute or a group of attributes. There is however no distinct name for a term expressing a single attribute incapable of analysis, and the only peculiarity of such terms is, as will be seen below, that they are incapable of definition. Locke called attributes which were incapable of analysis ' simple ideas,' but the expression ' simple term ' would not be applicable in a corresponding sense. Note 4.— That common terms, attributives, and ab- stract terms are formed from a comparison of individual objects or groups of objects, and that consequently they are results of thought, is obvious. But it may not be so r easy to perceive that this is the case with singular and collective terms. These terms however are appropriated to individual objects or groups of objects in order to distinguish them from others, and the necessity for such distinction can only arise after a comparison of this or that individual or group with others, and a perception of certain points of resemblance and difference between them. Unless I had observed some difference between John and Thomas, this table and that, the thirteenth legion and the fourteenth, it would never have occurred to me to distinguish them by separate names ; but this very observation of a difference involves an act of com- parison, and consequently an act of thought. Note 5. — It is important to notice that in a series of terms, like man, human, humanity, all expressing the same attributes, the later and more abstract terms can hardly fail to suggest the earlier and more concrete, and it is so because the earlier terms of the series have been longer formed and are therefore, as a rule, more familiar to us. Thus 'humanity' can hardly fail to suggest to us the word ' human,' from which it is formed, and * human ' will suggest the word * man,' from the Latin equivalent of which it is also formed, and whose attributes it expresses. Nor can we use the word 'man' without thinking of this or that individual man with whom we are familiar. A common term, in fact, expresses simply an individual object divested of all its peculiar attributes, and regarded as possessing only those attributes which it has in common with all the other objects which are desig- c i8 VARIOUS KINDS OF TERMS. nated by the same name. But it is indifferent on which object of the group the mind concentrates its attention, and we are all along conscious that the particular object selected is simply representative of the group. And hence it is that a common name simultaneously suggests to the mind a group of individual objects and a bundle of attributes characteristic of that group. For a further discussion of this subject, see Hamilton's Lectures on Metaphysics, Lect. xxxv. and xxxvi. ; Hansel's Prolego- mena Logi'ca^ ch. i. ; and Mill's Examination of Hamilton, ch. xvii. Note 6. — Mr. Mill maintains that attributives, when employed as predicates, are really common terms. Thus the propositions ' All triangles are three-sided,' ' All wise men are just,' are regarded by him as only abbreviated modes of saying 'All triangles are three-sided figures,' 'All wise men are just men.' I should allow that the attributive in the predicate, when taken in conjunction with the subject, always suggests a common term which may be substituted for it, as in the syllogism *A11 wise men are virtuous, All virtuous men are happy ; .-. All wise men are happy/ But, though the attributive may always admit of being expressed as a common term, while it continues to be expressed as an attributive there seem to be present to the mind only attributes, whereas, when it becomes a common term, there seems also to be present a group of individuals possessing those attributes. CHAPTER H. On the Denotation and Connotation of Terms, A TERM may be said to denote or designate individuals or groups of individuals, to connote or mean attributes or groups of attributes ^ ^ It ought, perhaps, to have been stated in the earlier editions of this work that the term connotation is here employed in a some- what different sense from that which is attached to it either in the scholastic logic or in the system of Mr. Mill. In the scholastic logic, a cofttiotative term is ' one which primarily signifies an attribute, secondarily a subject,' as * white,' the contrasted term being called an * absolute term,' as ' man ' or * whiteness.' See Mansel's Aldrich, cap. i, § 3, note g. According to Mr. Mill's nomenclature, a connotative term is one which * denotes a subject and implies an attribute.' By Mr. Mill, not only singular and collective, but also abstract terms are regarded as non-connotative. In the scholastic logic, what I have called attributives are alone recognised as connotative terms. See Mill's Logic, Bk. I. ch. ii. § 5. As the term is already employed with so much uncertainty, it appears to me not inexcusable to claim still further licence, and to appropriate the expressions ' denotation ' and ' connotation ' of ' terms ' in a sense parallel to that which is expressed by the dis- tinction between the ' extension ' and the ' intension ' of a * notion ' or 'concept,' applying 'denotation' simply to the objects, and * connotation ' simply to the attributes which are signified by a term. This is a broad and exceedingly convenient distinction, and, notwith- standing the apparent paradox involved in it, namely, that ' abstract terms are not denotative,' I believe that the general employment of the expressions in this sense would considerably simplify the state- ment and explanation of many logical difficulties. C 2 20 DENOTATION AND CONNOTATION OF TERMS, 21 In the first place, a term may serve to denote or point out an individual object or group of individuals. Thus ' Socrates ' denotes or points out and distinguishes from all others the individual man Socrates. The ex- pression ' tenth legion * denotes or points out, and dis- tinguishes from all other collections of men, the particular group known as the tenth legion. Similarly, the word ' man ' denotes or points out, and distinguishes from all other groups, a certain group of individuals to each member of which and to each member of which only the word ' man ' may legitimately be applied. All terms of this kind, therefore, viz. singular, collective, and common terms, are denotative; but terms like human, white, humanity, whiteness, i.e. attributes and abstract terms, are not denotative, except mediately, that is, so far as they suggest the common terms 'human beings,' * white things.' In the second place, a term may serve to connote attributes or groups of attributes. Thus terms like humanity, human, man, viz. abstract, attributive, and comtnon terms, are all connotative, that is, they at once suggest or imply attributes. But singular and collective terms like * Socrates,* *the tenth legion,' are not connotative, except so far as they suggest common terms. This remark requires some explanation. A col- lective term like ^the tenth legion,' 'the House of Commons,' at once suggests the corresponding common term, 'soldiers of the tenth legion,' or 'members of the House of Commons ' ; and this common term may \ connote any number of attributes, but, as the ' attributes are suggested mediately through the common term and not directly by the collective term, the collective term is, strictly speaking, non-connotative. The same is the case with a singular term. A term like ' William ' may sug- gest to me ' man,' ' male,' * Englishman,' ' one of my friends,' &c., and so may become connotative, but it is in itself rightly regarded as non-connotative, inasmuch as it suggests to me these attributes only through the medium of the common terms to which it is referred. It appears therefore that common terms are both denotative and connotative; that singular and collective terms are denotative, but not connotative ; that abstract terms and attributives are connotative, but not denota- tive ; and finally, that mediately, as suggesting common terms, any non-connotative term may become connotative and any non-denotative term denotative. Note. — The distinction between the denotation and connotation of a term is often otherwise expressed, as that between the extension and intension (or comprehension)^ or the extensive and intensive (or comprehensive) capacity, or the breadth and depth, of a notion. Having adopted the phraseology which designates the simplest product of thought as a term, instead of a notion, I shall speak of the extensive and intensive (or comprehensive) ca- pacity of a term. The extensive capacity of a term is measured by the number of individuals which it 22 DENOTATION AND CONNOTATION OF TERMS, designates (denotes), the intensive or comprehensive capacity of a term by the number of attributes which it includes, suggests or implies (connotes). It is plain that in a series of common terms, standing to one another in a relation of subordination, the denotation and connota- tion, or the extensive and intensive capacities, of the term are so related, that as the one increases the other de- creases, and vice versa. Thus, if we arrange in order any series of common terms, as flower, rose, moss-rose, we see that 'flower,' which implies the smallest number of attributes, is applicable to the largest number of indi- viduals ; * moss-rose,' which is applicable to the smallest number of individuals, implies the largest number of attributes : and generally in any series of common terms arranged in subordination, the larger the denotation or extensive capacity, the smaller is the connotation or intensive capacity, and vice versa. In conformity with this principle, the singular term which stands for the individual, and is smallest in denotation, is, when we refer it to the various common terms which may be predicated of it, and so assign to it mediately an intensive capacity, the largest in connotation. Thus the term Socrates, when I regard it as expressing one who was a philosopher, a teacher, a martyr, a soldier, an Athenian citizen, &c., &c., suggests to me far more attributes than any one of these common terms singly. . PART II. — The Proposition. CHAPTER I. On the Subject and Predicate, A PROPOSITION asserts or denies, as the result of comparison, some word or combination of words of some other word or combination of words, as e. g. ' James is the man I saw yesterday'; 'No rectilineal figure is contained by less than three lines ' ; * Some stars are not planets.' As before stated, the words or combina- tions of words thus employed are called terms, the term affirmed or denied is called the predicate, the term of which it is affirmed or denied the subject, the con- necting verb, whether qualified or not by the negative particle, the copula, and the predicate is said to be predicated of the subject. In the above examples, ' the man I saw yesterday,' 'contained by less than three straight lines,' and ' planets ' are predicates and are /r^- ^//r^/^d/ respectively of 'James,' 'all rectilineal figures,' and ' some stars,' as subjects. In the first case the predicate is predicated affirmatively, a fact which is expressed by the copula ' is ' ; in the two last negatively, a fact which is 24 THE SUBJECT AND PREDICATE, expressed by the copula Ms not.' These remarks may appear inconsistent with the form of the second example, but ' no rectilineal figure is, &c/ is really an abbreviated and unambiguous mode of stating the longer and am- biguous proposition ' All rectilineal figures are not, &c.' The word * predicated,' as equivalent to 'asserted or denied,' is here used in a wider than its ordinary signifi- cation. In common language, we say such and such an attribute cannot be predicated of such and such a term, using 'predicated' as equivalent to 'asserted' and as opposed to ' denied.' All ambiguity may be avoided by speaking of the predicate as predicated affirmatively or predicated negatively of the subject. ^ t CHAPTER II. On the Copula. THE Logical Copula, it being its office simply to serve as a sign of predication, is limited to the present tense of the verb ' to be,' with or without the addition of the nega- tive particle, according as the proposition is negative or affirmative. This limitation follows from the fact that it is simply the office of the proposition to express my present judgment as to the compatibiUty or incompatibility of two terms. Hence all reference to time, past or future, and even to time present, as respects the terms themselves, and not my judgment as to their compatibility, must be expressed in the predicate and not in the copula. I may, for brevity's sake, say ' fire burns,' ' Alexander was the son of Philip,' ' The guns will be fired to-morrow,' and, in con- versation or discussion, it would undoubtedly be pedantic to express the propositions otherwise; but formally, for the purpose of being estimated logically, I must resolve them into their logical elements, and say ' Fire is burning,' ' Alexander is a person who was son of Philip,' 'The firing of the guns is an event which will take place to-morrow.' Not only does the logical copula convey no notion of time with reference to the terms themselves (or, to speak more accurately, the things signified by them), but it is also divested of the notion of existence. In other words, 26 THE COPULA, it is employed simply as a connecting particle, not as a substantive verb. Where the substantive verb is used in a logical proposition, it must be expressed in the predicate. Thus 'I am/ 'The king is not,' become 'I am existent,' ' The king is non-existent.' That the copula implies no notion of existence is evident from the fact that we can use such propositions as these : ' The labours of Hercules are a myth/ ' He is a nonentity.' Can we modify the copula so as to express certainty, probability, possibility, or other modes of connexion be- tween the subject and the predicate? This is the cele- brated question of Modality, a question which has been the source of much difference of opinion amongst logi- cians. Even though it were granted that the proposition simply expresses our present judgment on the compati- bility or incompatibility of two terms, it might be main- tained that it should express the nature of our judgment and the degree of our assent or dissent, whether it be certain, approximating to certainty, or faltering. Thus it might be maintained that the following should be accepted as instances of the ultimate analysis of a logical proposi- tion : * This is certainly the man I saw yesterday,' ' This is probably the man I saw yesterday,' * This is possibly the man I saw yesterday.' That we use these forms in conversation and discussion is unquestionable, but it is one main object of Logic to analyse our abbreviated infer- ences and statements into their full logical equivalents. In- stead therefore of admitting various descriptions of copulse (other than the affirmative and negative), in order to adapt ( THE COPULA. 27 Logic to ordinary language, it seems simpler, as well as more scientific, to insist on the uniform character of the copula, and to represent propositions like the foregoing as predicating our degree of assent to or dissent from the sentence in question. Thus, after asking myself the question ' Is this the man I saw yesterday ? ' I may either answer simply ' Thi^ is the man I saw yesterday,' or I may describe the degree of my assent by stating ' That this is the man I saw yesterday is certain, probable, possible,' &c. In fact, such propositions seem to be the result of an act of reflexion on the degree of our own conviction. I shall therefore regard the form A is or is not B as the ultimate and uniform logical analysis of all propositions, though I shall occasionally, for the sake of brevity, avail myself of the forms sanctioned by popular language. ]S!oie.—k.% regards the expression of time in the copula, the student will find the opposite opinion to that taken in the text adopted by Mr. Mill, Logic, vol. i. ch. iv. § 2. In support of my view, he may refer to Dr. Mansel's Prole- gomena Logica, pp. 63, 64. On the question of expressing in the copula the degrees of assurance with which a pro- position is entertained ('certainly/ 'probably/ &c.), see Sir W. Hamilton's Discussions, pp. i45-7» and, for a more qualified view than that taken either by Sir W. Hamilton or myself. Dr. Mansel's Prolegomena Logica, note G. f DIVISION OF PROPOSITIONS. 2Q CHAPTER III. Division of Propositions according to their Quantity and Quality, WE have already seen that propositions are either affirmative or negative, according as the copula used is of the form * is ' or 'is not/ This is called a division of propositions according to their Quatity, They are further divided, according to their Quantity^ into Universal and Particular. For, in affirming or denying a predicate of a subject, it is obvious that I may either affirm or deny the predicate of all the indi- viduals denoted by the subject, or of part only. Thus in affirming mortality of man, I may say * All men are mortal,' or ' Some men are mortal ' ; in denying wisdom of man, I may deny it of all men or only of some men, i. e. I may say ' No men are wise,' or ' Some men are not wise/ When the predicate is affirmed or denied of all the individuals denoted by the subject, the proposition is called an Universal Proposition; when of part only, a Particular Proposition. A Singular Pro- position, i.e. a proposition of which the subject is a singular term, ranks as an Universal, because the pre- dicate is affirmed or denied of everything (i. e. in this case, the one thing) denoted by the subject. The same remark holds good of a proposition in which the subject is a collective term. An attributive, as we have already seen, cannot, by itself, be used as the subject of a pro- position. Abstract terms which have come to be used as common terms, and admit of plurals, as figure, triangle, virtue, pleasure, &c., have a denotative power, and may, like common terms, form the subjects of either universal or particular propositions. But those abstract terms, like humanity, wisdom, &c., which retain their orio-inal characteristic of being connotative only, and admit of no plurals, simply express an attribute or group of attributes with which, as a whole, it is asserted or denied that the predicate is compatible; consequently, a proposition, of which such a term is the subject, ranks as an universal. Thus such propositions as ^Ambition is aggressive,' 'Wisdom is a virtue,' 'The fourteenth legion is dis- banded,' ' Socrates is an Athenian citizen,' are, on the very face of them, universals. But propositions in which the subject is a common term or an abstract term used as a common term, must be quantified ; that is, we must attach to the subject either an universal or a particular designation. It is not sufficient to say, 'triangles are figures/ 'horses are black'; we must state whether we mean that ' all triangles ' or ' some triangles ' are ' figures/ whether we mean that ' all horses ' or ' some horses ' are black. ' Indefinite ' or ' indesignate ' propositions, as they are called, i.e. propositions in which the subject, being '<\\ 30 DIVISION OF PROPOSITIONS, QUANTITY AND QUALITY. 31 a common term, is not quantified, are inadmissible . in Logic. By combining the division of propositions into uni- versal and particular with that into affirmative and nega- tive we obtain four forms, viz. — Universal Affirmative. All X is Y. (A) Universal Negative. No X is Y. (E) Particular Affirmative. Some X is Y. (I) Particular Negative. Some X is not Y. (O) I shall in future designate these forms of proposition respectively as A, E, I, 0\ • Note. — Sir W. Hamilton^, followed by several other logicians, maintains that in thought the predicate is always quantified as well as the subject. He proposes to reform the logical theory of the proposition accord- ^ It sometimes requires a little ingenuity to state a given proposi- tion in one of the above forms. Thus the propositions * None but the brave deserve the fair,' * The wise alone are good,' * Not every historian is worthy of credit,' ' All his acts are not defensible,' when stated in strictly logical form, become respectively. No not-brave (or None who are not brave) are deserving of the fair. No not-wise (or None who are not wise) are good, Some historians are not worthy of credit. Some of his acts are 7iot defensible. The simplest equi- valents of the two former propositions are. All who deserve the fair are brave. All good men are wise ; but these are arrived at by permutation and conversion, two forms of inference which have not yet been explained. Sir William Hamilton's theory was anticipated in a work now little read, but full of original suggestions on logical questions, Mr. George Bentham's Outline of a New System of Logic, published in 1827. ingly, and in lieu of the four ordinary forms of pro- position substitutes the following :— All X is all Y. All X is some Y. All X is not any Y. All X is not some Y. Some X is all Y. Some X is some Y. Some X is not any Y. Some X is not some Y. This scheme, if adopted, would, as Sir W. Hamilton shews, reduce all conversion to simple conversion, render nugatory any discussion as to the distribution of terms, and considerably simplify the forms of syllogism: see the Appendices to Sir W. Hamilton's Discussions, and to his Lectures on Logic. Amongst other criticisms may be seen Mr. Mill's in his Examination of Hamilton s Philosophy, ch. xxii. It wx)uld of course be undesirable to enter here into any discussion as to the merits of Sir W. Hamilton's theory, but, as reasons for not adopt- ing the quantification of the predicate in the present work, it may be sufficient to state (i) that, as to utility, the trouble entailed by quantifying the predicate in every proposition would probably far exceed that saved by simplifying the forms of Conversion and Syllogism ; (2) that the forms of expression given above are not merely unusual, but are such as we never do use ; whereas, though the analysis of our thoughts frequently leads to 32 DIVISION OF PROPOSITIONS. forms of expression which are unusual, this would, if admitted, be the only case in which it led to forms which are never used at all ; (3) that some of the above propositions really contain in a compressed form two ordinary propositions, as e. g. ' All A is all B,' contains the two ordinary propositions 'AH A is B' and 'All B is A,' the proposition 'Some A is all B' contains the two ordinary propositions ' Some A is B ' and ' All B is A,' whereas it is the object of Logic not to state our thoughts in a condensed form but to analyse them into their simplest elements. i»-«:iF^5ci^ii-s Element aire de Zoologie, edition septieme, § 367. expressing infimae species. Classes like 'black men,' 'Arabian horses,' &c. would not have been admitted to be species at all. We, on the contrary, conceive that there is no limit to our power of making classes ; how- ever specialised a group may be, we can almost always think of some attribute, the addition of which will make it more special still. : I fe»-3*A;'*^>=S» F 2 VARIOUS KINDS OF INFERENCES. 69 PART III.— Of Inferences. CHAPTER I. On the VarioiLs Kinds of Inferences. THE third and most important part of Logic treats of Inferences *. Wherever we assert a proposition in con- sequence of one or more other propositions, or, in other words, wherever we regard one or more propositions as justifying us in asserting a proposition distinct from any that has preceded, the combination of propositions may be regarded as an inference. Thus defined, inferences may be divided into inductive and deductive, and de- ductive inferences may be sub-divided into mediate and immediate. I shall attempt to make these distinctions clear by examples. ^ The word * inference ' is employed in no less than three different senses. It is sometimes used to express the conclusion in conjunction with the premiss or premisses from which it is derived, as when we speak of a syllogism or an induction as an inference ; sometimes it is used to express the conclusion alone ; sometimes the process by which the conclusion is derived from the premisses, as when we speak of Induction or Deduction as inferences or inferential processes. Except where the meaning is obvious from the context, I shall endeavour to confine the word to the first-named signification. The terms Induc- tion and Deduction will be appropriated to express processes which result, the former in inductiofis or inductive inferences, the latter in deductions or deductive infereiues, these last being sub-divided into syllogisms and immediate inferences. I mix tartaric acid and carbonate of soda* in certain proportions in water, and I observe that the mixture is followed by an effervescence; from this I infer that, whenever tartaric acid and carbonate of soda are mixed in water in these proportions, effervescence will follow. I put a poker into the fire, and I observe that after a time it becomes red-hot ; from this I infer that incandescence can always be produced in iron by a certain degree of heat. I observe five points in the orbit of a planet, and, from my knowledge of mathematics, perceive that they are situated in an ellipse : from this I infer that the entire orbit of the planet is elliptical, and that, in all future revolutions of the planet, a similar orbit will be described. Now what in these cases do I mean by the word ' infer ? ' That the mixture is followed by effervescence is a matter of observation ; but it is only an inferential process which justifies me in asserting that, inasmuch as it could have been produced by nothing else, the effervescence was produced by the mixture, and that, whenever in future I see a similar mixture, I may expect to see it followed by similar results. Two assumptions, it will be seen, under- lie this inference : ist, that every event has a cause, which leads me to assume that the effervescence must have been produced by some cause or other; 2nd, the belief in the uniformity of nature, which leads me to expect that, whenever similar circumstances are repeated, they will be followed by similar results. The reasoning therefore in these cases may be represented as follows : — The mixture of the tartaric acid with the carbonate 70 VARIOUS KINDS OF INFERENCES, VARIOUS KINDS OF INFERENCES. 71 of soda is followed by effervescence. (Original Proposition.) .-. (Owing to the special circumstances of the case and in accordance with the principle that every event must have a cause), the effervescence was produced by the mixture. .-. (In accordance with the principle of the uniformity of nature), a similar mixture will always be fol- lowed by an effervescence. We may represent the reasoning in the third example in the same manner : — We may assert (by virtue of our knowledge of mathematics) that five points which we have ob- served in the orbit of the planet Mars are situated in the arc of an ellipse. (Original Proposition.) .•.As there are, comparatively speaking, no causes de- termining the position of the planet at any given moment, except the attraction of the sun and the continued effects of the initial velocity, we may infer that the fact of the five points observed being situated in the arc of an ellipse is due to the com- bination of these two causes. .*. (In accordance with the principle of the uniformity of nature), it may be inferred that all other points in the orbit of the planet are situated in an ellipse, and that, in all future revolutions, a similar orbit will be described ; i. e. the orbit of the planet Mars may be regarded as elliptical. Now inferences of this kind are called IndMive. The instances I have selected are remarkably simple, but they are sufficient to shew that an induction may be defined as an inference in which we argue from parti- culars to adjacent particulars, or (if we speak of the adjacent particulars collectively) from particulars to uni- versals, in accordance with the laws of universal causation and of the uniformity of nature. As to the circumstances which justify us in asserting that one phenomenon or set of phenomena is the cause or the effect of, or is invariably conjoined with, another (for this is the problem of Induc- tion), the student is referred to works specially treating of Inductive Inference. It is sufficient here to distinguish inductive from the deductive inferences which it is our more special business to explain 2. 2 An Analogy is a form of imperfect induction, and, though justify- ing a conclusion more or less probable, never leads to certainty. If two objects resemble each other in several important respects, and we argue that any particular attribute which we know to be predicable of the one, and do not know to be either predicable or not predicable of the other, is, on account of the general resemblance of the two objects, also predicable of the other, the argument is called an argu- ment from analogy ; and, in the same way, if two objects are dis- similar, we may argue that an attribute which is predicable of the one, is, on account of their dissimilarity, not predicable of the other Thus, from the similaVity between the earth and the moon, we might argue that the latter is inhabited, or, from their dissimilarity, that it is not inhabited. The value of the inference always depends on the ratio of the ascertained resemblances to the ascertained differences (it being understood that the resemblances which we take into account are none of them derived, as properties, from each other, and so with the differences), providing that our knowledge of the objects is suf- ficiently large to justify us in drawing any inference at all. For a more 72 VARIOUS KINDS OF INFERENCES, VARIOUS KINDS OF INFERENCES, 73 Beginning where induction ended, we may state such a proposition as this: 'All iron when heated to a cer- tain degree becomes red-hot.' This, if combined with another proposition ' This is a piece of iron,' leads to the conclusion ' This piece of iron, if heated to a certain degree, will become red-hot/ Now it is plain that the con- detailed analysis of this mode of reasoning, and an estimate of the value to be attached to its conclusions, the student is referred to the author's Elements of Inductive Logic, ch. iv., and to Mr. Mill's Z^^V, Bk. III. ch. XX., one of the most instructive and important chapters in his work. It should be noticed that an analogy, as here described, is not identical with the Analogy of Aristotle, the Aristotelian Analogy being an equality of relations (tVoxT^s A(>7£yi/). Thus the expression. The intellect : the soul = the sight : the body, is an avako-^ia. From this analogy it is argued that anything which may be predicated of the one pair of terms may be predicated also of the other. Or, to take a non-Aristotelian instance, which will be more intelligible to beginners : A colony : the mother-country = a child : a parent. From this analogy it is argued that the reciprocal rights and duties of a colony and the mother-state are the same as those of a child and a parent. In this form of argument, if the relations between the two sets of terms were precisely the same in all respects, the conclusion would be invariably valid ; as it is usually found in practice, how- ever, the relations are the same in some respects, but not in others, and, consequently, the conclusion is valid, when based on those points in which the relations are the same, and invalid, when based on those points in which the relations are not the same. Thus, it might be maintained that, in many respects, the relation of the child to the parent is not the same as that of a colony to the mother- country, and, hence, that many of the rights and duties which exist in the one case do not exist in the other. A Metaphor is an analogy of this kind compressed into a single word or phrase. The fallacy of False Analogy will be noticed below. elusion we have just drawn w^as arrived at ill a different manner from those noticed above. Instead of being the conclusion of a process by which we argue from parti- culars to adjacent particulars or from particulars to universals (i. e. from cases which are within the range of our observation to others which are without), it is simply a combination of two propositions into one, being an obvious inference from what has been previously stated in the premisses. Induction has been not inaptly com- pared to the establishing of a formula, Deduction (for that is the appropriate name of the process which I am now discussing) to the reading it off. Induction leads to truths entirely new. Deduction combines, methodizes, and developes those which we have already gained. A Syllogism may be called a Mediate Inference, because the two terms of the conclusion are compared in the premisses by means of a third. It is thus opposed to an Immediate Inference, which consists of two propositions only, and in which the inferred propo- sition is derived from a single proposition without the aid of any other term or proposition, expressed or implied. Both mediate and immediate inferences may be styled deductive as opposed to inductive. This division may easily be shewn to be exhaustive. In any inference, we argue either to something already implied in the premisses^ or not; if the latter, the » If we state explicitly all the assumptions made in the inductive process, the conclusion is contained in the premisses, and the form of the reasoning becomes deductive ; but it is seldom that wc do state 74 VARIOUS KINDS OF INFERENCES. inference is inductive, if the former, deductive. If the deductive inference contain only a single premiss, it is immediate ; if it contain two premisses and the conclusion be drawn from these jointly, it is mediate and is called a syllogism. All deductive inferences which apparently contain more premisses than two admit of being analysed into a series of syllogisms. Nok I. — I am here departing from the ordinary scheme of division adopted by logicians. Inferences are generally divided into mediate and immediate, and mediate inferences are subdivided into inductive and deductive. As however I regard inductions as more strongly contrasted with both syllogisms and immediate inferences than either of these classes is with the other, it seems preferable to make inductions one of the main members, rather than our assumptions thus explicitly. The most essential distinction, however, between inductive and deductive reasoning consists not in the form of the inferences, but in the nature of the assumptions on which they rest. Deductive reasoning rests on certain assumptions with regard to language and co-existence (namely, the so-called Law of Identity, or some modification of that law, the Law of Con- tradiction, the Law of Excluded Middle, and the Canons of Syllogism), while inductive reasoning assumes over and besides these laws the truth of the Laws of Universal Causation of the Uniformity of Nature and, in certain cases, of the Conservation of Energy ; or, if it be of the unscientific description which is known as Inductio per Enumera- tionem Simplicem, it merely assumes, instead of them, the vague and wide principle that the unknown resembles, or will resemble, the known. It hardly needs to be added that all reasoning alike assumes the trustworthiness of present consciousness and of memory. VARIOUS KINDS OF INFERENCES, 75 one of the subordinate members of the division. Nor is there any reason why an immediate inference should not be regarded as deductive. It should also be noticed that Sir W. Hamilton would deny the title of inferences to inductions (as they have been here explained), whereas Mr. Mill would deny that either a syllogism or an immediate inference can properly be called an inference. Mr. Mill maintains that all In- ference is ' from the known to the unknown ' ; Sir W. Hamilton defines Inference as the 'carrying out into the last proposition what was virtually contained in the antecedent judgments.' j^oie 2.— The Aristotelian induction, in which the conclusion affirms or denies of a group what was in the premisses affirmed or denied of each member of the group severally, is, according to the above method of treatment, obviously regarded as a deductive inference. If I predicate some quality of each member of a group, and thence infer that all members of the group possess this quality, the conclusion is plainly contained in the premisses, and the inference is a syllogism. It may be represented in the form * — * By Aristode himself the inductive inference is analysed thus : — x, y, z are B, X, y, z are (i.e. constitute) A ; . • . A is B. The minor premiss, when stated in so peculiar a form, ot course admits of simple conversion, and thus assumes the form given in the text. 76 VARIOUS KINDS 01 INFERENCES, X, y, z are B, The individuals (or subordinate species) constituting the group A are x, y, z ; .*. The individuals (or subordinate species) constituting the group A are B. Such an inference is altogether different from what we now understand by an induction. On this subject the student may with advantage read Mr. Mill's chapter on ' Inductions improperly so called/ See Mill's Logic, Bk. III. ch. ii. An account of the Aristotelian induction will be found in Appendix G to Dr. Mansel's edition of Aldrich ; in SirW. Hamilton's -fiVj-^ on Logic, and in his Lectures on Logic, Lect. xvii. and Appendix vii. These authors, as already noticed in the case of Sir W. Hamil- ton, regard inductions, in the modern sense of the word, as extra-logical. The advanced student may also consult with advantage Mr. De Morgan's chapter on * Induction,' Formal Logic, ch. xi. CHAPTER II. On Immediate Infere^ices. § I. AN Immediate Inference may be formally defined as a combination of two propositions of which one is inferred from the other, the proposition inferred being virtually included in the proposition from which it is inferred. Of Immediate Inferences the most important forms are Oppositions, Conversions, Permutations ^ § 2. On Oppositions. Two propositions are said to be opposed when they have the same subject and predicate, but differ in quan- tity or quality or both. An Opposition may be defined as an immediate inference in which from the truth or falsity of one proposition we. infer either the truth or falsity of another, this proposition having the same subject and predicate as the former, but differing in quantity or quality or both. Thus from the propo- sition ' That all X is Y is true ' we may infer the pro- position ' That no X is Y is false,' or * That some X is Y is true,' or ' That some X is not Y is false/ ' It is the more common practice to speak of Opposition, Con- version, and Permutation, but I have adopted the plural number in order to draw attention to the fact that Logic is concerned with the results rather than with the processes by which they are arrived at. 78 OPPOSITIONS, The opposition between A and E is called a Contrary Opposition. I and O, a Subcontrary Opposition. A and I, or E and O, a Subaltern Opposition. A and O, or E and I, a Contradictory Opposithan. These forms of Opposition are exhibited in the annexed scheme : — A • . . Contrary . . £ • • CO n •1 S • • • V • • • C/5 e cr 9 • • I . . Subcontrary . . • o If A be true ; E is false, I true, O false. If A be false ; E is unknown, I unknown, O true. If E be true ; A is false, I false, O true. If E be false ; A is unknown, I true, O unknown. If I be true ; A is unknown, E false, O unknown. If I be false ; A is false, E true, O true. If O be true ; A is false, E unknown, I unknown. If O be false ; A is true, E false, I true. It will be observed that it is only in a Contradictory Opposition (where the opposed terms differ both in quantity and quality) that from the truth or falsity of one proposition we can invariably infer the truth or falsity of another, the conclusion which we draw in this case being from the truth or falsity of the one propo- OPPOSITIONS, 79 sition to the falsity or truth respectively of the other. Hence logicians have called contradictory the most per- fect form of opposition. It is a rule of practical Logic that a contradictory should always in disputations be used in preference to a contrary opposition ; for it serves equally well the purpose of contradicting an opponent, and the particular proposition which it asserts affords less ground for attack than an universal. Thus, if my opponent asserts A (as e.g. All philosophers are un- imaginative), I may meet his assertion by the contra- dictory O (Some philosophers, as e. g. Plato, Goethe, &c., are not unimaginative), and from this position I cannot well be dislodged. But suppose I assert in opposition to him an E proposition (No philosophers are unima- ginative), he will probably be able to adduce instances of some philosophers who, according to the ordinary mean- ing of the word * imaginative,' would be called unimagi- native, and so, by meeting my E with an I proposition, gain an apparent victory. As a fact, we should each have made assertions too wide, but he would have succeeded in dislodging me from my position, whereas (owing to my neglect of the laws of contradiction) I should not have succeeded in dislodging him from his. ]^ote.—\\. is plain that, according to the ordinary meaning of the word * opposition,' it is somewhat of an abuse of language to speak ^ of A and I, or E and O propositions as being opposed. It would be better 8o CONVERSIONS, if this form of inference were called Subalternation or Subordination. Nor, strictly speaking, can the relation between I and O be called one of opposition, for they may both be true together. Accordingly, Aristotle says that in reality (/car aKr]6eLav) there are three forms of opposition (those be- tween A and E, A and O, E and I), though in language [Kara rr)v \e^Lv) there are four (adding that between I and O). What is called Subaltern Opposition he does not recognise. § 3. On Conversions. A proposition is said to be converted when its terms are transposed, so that the subject becomes the predicate, and the predicate the subject. A Conversion may be defined as an immediate inference in which from one proposition we infer another having the same terms as the original proposition, but their order reversed. This inference in some cases necessitates a change of quantity in passing from one proposition to the other, and then it is called a Conversion per accidens ; when it necessitates no such change, it is called a Simple Conversion ^. I and E may both be converted simply. Thus, from * Some X is Y,' or * Some poets are philosophers,' I may infer * Some Y is X,' or ' Some philosophers are poets.' From * No X is Y,' or ' No savages are trustworthy,' I may infer ' No Y is X,' or ' No trustworthy persons are savages.* * It is proposed by Sir W? Hamilton to call the original proposi- tion the ' Convertend,' the inferred proposition the ' Converse.' CONVERSIONS, 81 A can only be converted per accidens. For though it may sometimes happen that the subject and predicate of an A proposition are co-extensive, and therefore conver- tible, this circumstance is not implied in the form of the proposition, and it is with what is implied in the form of the proposition that we are alone concerned. Thus, if I assert the proposition ' All plane triangles are three-sided rectilineal figures,' it happens in this pardcular case that I am justified, without any change of quantity, in stating the converse, ' All three-sided rectilineal figures are plane triangles.' But if I state that ' All plane triangles are rectilineal figures,' I am only justified in inferring that * Some rectilineal figures are plane triangles.' As, there- fore, the general form of an A proposition does not imply the simple convertibility of the subject and predi- cate, I am only justified in inferring from ' All X is Y,' that ' Some Y is X.' In those cases, however, in which the form of the pro- position implies that the subject .and predicate are co- extensive, the proposition, though an A proposition, may be converted simply. Thus, from the proposidons * The second legion is the only legion quartered in Britain,' 'Virtue is the condition of Happiness,' *A11 plane tri- angles may be defined as three-sided rectilineal figures,' it may be inferred by simple conversion that ' The only legion quartered in Britain is the second legion,* 'The condition of Happiness is Virtue,' ' All three-sided recti- lineal figures may be called plane triangles.' An O proposition cannot be converted at all. From G 82 PERMUTA TIONS, * Some X is not Y,' it does not follow that ' Some Y is not X,' for Y may stand to X in the relation of a species to a genus. Thus from the proposition ' Some Euro- peans are not Frenchmen/ I cannot infer that ' Some Frenchmen are not Europeans.' § 4. On Permutations ^ A Permutation may be defined as an immediate Infer- ence in which from one proposition we infer another differing in quality, and having, therefore, instead of the original predicate its contradictory. Thus : — From All X is Y, we may infer that No X is not-Y. From No X is Y, All X is not-Y. From Some X is Y, Some X is not not-Y. From Some X is not Y, . . . Some X is not-Y. The legitimacy of these inferences is apparent from the fact that contradictory terms (A and not-A) admit of no medium, so that, if I predicate the one affirmatively, I may always predicate the other negatively, and vice vers^. The O proposition, when permuted from * Some X is not Y,' into ' Some X is not-Y,' may of course be con- verted into 'Some not-Y is X.' This combination of permutation and conversion is improperly described by Whately and many previous logicians as a single inference, and styled ' Conversion by Contra-Position or Negation.' ' The terai Permutation is borrowed from Mr. Karslake's Aids to Logic, The same inference is sometimes called Infinitation, from the Nomen Infinitum, or, more properly, Nomen Indefinitum (not-Y, as the contradictory of Y), which is employed as the predicate. PERMUTA TIONS, 83 It may assist the student if I add some further in- stances of permutations : — All men are fallible, .*. No men are infallible. No men are infallible, .*. All men are fallible. Some poets are reflective, .*. Some poets are not unreflective. Some poets are not unreflective, .*. Some poets are reflective. All poets are men of genius, .'. (by permutation) No poets are not-men-of-genius ; .*. (by conversion) No not-men-of-genius (=None but men of genius) are poets. JSlote. — I have here employed an expression * Con- tradictory Terms,' which in most works on Logic is explained in the first part, as included under the doctrine of Opposition of Terms. It seemed, however, desirable to introduce there only those distinctions of terms which were likely to be frequently required in the sequel of the work. I may here state that ' Contradictory Terms,' such as white and not-white, lawful and un-lawful, are terms which admit of no medium, i.e. terms which are not both predicable of the same thing, while one or other of them must be predicable of it. ' Contrary Terms,' like good and bad, black and white, are terms which are the most opposed under the same genus; they cannot both be predicated of the same thing, but it is not necessary that one or other of them should be predicable of it. G 2 CHAPTER III. On Mediate Inference or Syllogism, § I. The Structure of the Syllogism. A SYLLOGISM may be defined as a combination of two propositions, necessitating a third in virtue of their mutual connexion; or as an inference in which one proposition is inferred from two others conjointly, the inferred proposition being virtually contained in the pro- positions from which it is inferred. This is obviously a definition of a legitimate syllogism. There may (as will appear below) be apparent syllogisms, which do not fulfil the conditions of this definition. We may take as instances of syllogisms : — (i) All B is A, AllCisB; .-.All C is A. (2) All sovereign powers are invested with supreme authority over their subjects, All republics are sovereign powers ; .•.All republics are invested with supreme authority over their subjects. (3) No rectilinear figure is bounded by one line, A circle is bounded by one line ; .'.A circle is not a rectilinear figure. STRUCTURE OF THE SYLLOGISM, 85 •» ") The proposition inferred is called the Conclusion, the pro- positions from which it is inferred the Premisses, either of them singly being called a Premiss, As the conclusion is virtually contained in the pre- misses conjointly, it is plain that the two terms of the conclusion must occur in the premisses, one in either. If both terms occurred in the same premiss, the other premiss would be entirely alien to the conclusion. The remaining term of each premiss must be the same ; else there would be nothing in common between the two premisses, and the conclusion could not be said to be inferred from the two conjointly. This third term, with which the two terms of the conclusion may ,be regarded as compared, is called the middle term. The predicate of the conclusion is called the major term^ and the subject the minor term ; the premiss, in which the major and middle terms are compared, is called the major Premiss^ and should always be stated first ; that in which the minor and middle terms are compared is called the minor premiss. Thus in a syllogism, formally stated, there are always three propositions including three terms, the premisses occurring first and the conclusion last. But practically, in reasoning, we frequently state the conclusion first, introducing one or both premisses with such a word as * for ' or ' because,' as stating our reason for the assertion. Thus I may say * I will not go out to-day, for it is raining,' or, * I will not go out to-day, for it is raining, and the rain may give me a cold.' When stated in this form, the conclusion is called by the older logicians the Problema 86 MEDIATE INFERENCE OR SYLLOGISM, or QucBsiio, being regarded as a question to which the reason or reasons assigned furnish the answer. It will also have occurred to the student that, as a fact, we usually state only one premiss, leaving the other (which may be either the major or minor) to be understood. Thus, instead of stating Syllogism (2) formally, as above, I should in an actual discussion say, * A republic is invested with supreme authority over its subjects, for every so- vereign power is invested with such authority,' or, *A republic is, &c., for it is a sovereign power,' or briefly, * A republic (being a sovereign power) is invested,' &c. ; or, the premiss coming first, ' Inasmuch as every so- vereign power is invested, &c., I maintain a republic to be invested with that authority,' or, * Inasmuch as a republic is a sovereign power, it is invested with,' &c.^ Instead of suppressing one of the premisses, I may, for brevity's sake, suppress the conclusion. Thus I may say ' Ever}^ sovereign power is invested with supreme authority over its subjects, and a republic is a sovereign power,' leaving it to the hearer or reader to draw the conclusion for him- self. The syllogism does not pretend to be the form, or even a form, in which our reasonings are usually stated, but simply one of the ultimate analyses of them. As every term in the syllogism occurs twice, it should be noticed that, on both occasions, it should be used in the same sense, or, to adopt technical language, every ' A syllogism with a suppressed premiss is by Aldrich wrongly identified with the Enthymeme of Aristotle. Such a syllogism was called by the Stoics a avWoyiafibs ixovoXrjiiiiaTos. STRUCTURE OF THE SYLLOGISM, 87 .^r, term in the syllogism should be used univocally. If we use a term equivocally, i.e. in two entirely different senses, or even analogously, i.e. in two different senses having some relation to each other, it is plain that, logically speaking, we are using two different terms, and con- sequently the syllogism will include four terms instead of three. This caution includes the rule usually given by logicians against an ambiguous middle. The neglect of it, palpable as it might be supposed to be, is often, espe- cially in a long course of reasoning, very difficult of detection, and is a fertile source of fallacy. I may adduce as very simple instances : — Humanity is a moral virtue, The study of polite letters is humanity ; .'.The study of polite letters is a moral virtue. The church is the aggregate of all Christian people. This particular congregation (or particular building) is the church (meaning at some particular place) ; .•.This particular congregation^ (or particular building) is the aggregate of all Christian people. In the former case, the term * humanity ' has come to be used in such widely different senses, that it may be re- garded as used equivocally; in the latter case, the senses of the word * church ' are perhaps sufficiently nearly allied to be regarded as analogous. All cases of what are termed ' Verbal Fallacies ' may be referred to this head. Note. — The words ' major,' * minor,' and ' middle,* as applied to the terms in a syllogism, have been inherited 88 MEDIATE INFERENCE OR SYLLOGISM, by all subsequent logicians from the nomenclature of Aristotle. He regarded what we shall presently call the First Figure (B is or is not A, C is B, .*. C is or is not A) as the perfect type of syllogism, and, amongst other modes, stated it in the form C is in B (to r eoriV eV oXo) TO) b), B is or is not in A, .*. C is or is not in A, Thus stated, C appears to be the smallest, B the inter- mediate, and A the largest term in extent. See Prior Analytics, Bk. I. ch. iv. In negative propositions, however, we have no means of determining the relative extent of the subject and predicate, and consequently Aristotle's nomenclature does not properly apply to negative syl- logisms. To affirmative syllogisms in the first figure, whether universal or particular, it applies only in a modified shape, for the propositions All X is Y, Some X is Y, though they imply that Y cannot be less in logical extent than All X in the one case or than Some X in the other, do not exclude the possibility of the subject and predicate being co-extensive. Hence, however con- venient it may be, Aristotle's nomenclature applies only, and that not with strict accuracy, to two forms of syllo- gism (Barbara and Darii) in the first figure. § 2. On Moods and Figures, I now proceed to consider the possible, not the legitimate, forms of syllogism. Here there are two cir- cumstances to be taken into consideration: ist, that syllogisms may vary according to the quantity and quality MOODS AND FIGURES. 89 of the propositions (A, E, I, O) of which they are com- posed ; 2nd, that they may vary according to the position of the terms in the premisses. The first consideration gives us the number of possible moods, the second the number of possible figures. It is by combining these two sources of variation that we shall obtain the number of possible syllogisms. There are, if we take into consideration the conclu- sion, sixty-four possible arrangements of the propositions A, E, I, O, i. e. in technical language, sixty-four possible moods, viz. AAA, AAE, AAI, AAO, &c. But if we con- sider the premisses only, the number of possible moods is limited to sixteen, viz. AA, AE, AI, AO, EA, EE, EI, EO, lA, IE, II, 10, OA, OE, 01, 00. In deter- mining what possible moods are legitimate, we may either ask ' Is this conclusion justified by these premisses ? * or * To what conclusion do these premisses lead ? ' If we ask the former question, we must examine the sixty-four possible moods in which the conclusion appears as well as the premisses ; if the latter, an examination of the six- teen possible arrangements of premisses is sufficient. With respect to the possible arrangements of the terms in the premisses (i. e. the figures, as they are technically called) there are also two methods of proceeding. Taking no account of the conclusion (and therefore not knowing which is the major term and which the minor), and asking simply ' In how many ways can the middle term be combined with the other terms in the premisses?' there are three possible figures ; viz. ist, that in which the 90 MEDIATE INFERENCE OR SYLLOGISM. middle term is subject in one premiss and predicate in the other ; 2nd, that in which it is predicate in both pre- misses ; 3rd, that in which it is subject in both. But if we take account of the conclusion, we are able to distinguish the major and minor terms, and consequently the major and minor premisses. In this case, there are four possible figures, viz. ist, that in which the middle term is subject in the major premiss and predicate in the minor; 2nd, that in which it is predicate in both premisses ; 3rd, that in which it is subject in both ; 4th, that in which it is predicate in the major premiss and subject in the minor. These four figures may be exhibited thus : — Fig. I. Fig. 2. Fig. 3. Fig. 4. BA AB BA AB CB CB BC BC .-.CA .-.CA .-.CA . * . vx A. If we take no account of the conclusion, either extreme in the premisses may become the major term, and the three figures may be represented thus : — Fig. I. Fig. 2. Fig. 3. BA AB BA CB CB BC /.CAorAC .-.CAorAC .-.CAorAC. § 3. Determination of the Legitimate Moods of Syllogism, Note, — Few difficulties in elementary Logic are more likely to embarrass the beginner than the variety of methods of constituting the Legitimate Moods of Syllo- gism. Sir W. Hamilton, as a consequence of quantifying LEGITIMATE MOODS OF SYLLOGISM, 91 the predicate, is able to represent all syllogisms as equa- tions, and thus to exhibit every afiirmative syllogism as a direct application of what is called the Law of Identity (Every A is A), and every Negative Syllogism as a direct appHcation of the Law of Contradiction (No A is not- A) 2. Besides Sir W. Hamilton, other logicians who do not, like him, quantify the predicate, have also attempted to enunciate general principles equally applicable to all syllogisms. See e.g. Port Royal Logic, Part III. ch. X. Others (as Abp. Thomson, Laws of Thought, § 96, and Lambert, as quoted by Dr. Mansel in his Notes on Aldrich^ ch. iii. § 6) enunciate a distinct principle for each figure. Others (and pre-eminently Aristotle) enunciate a canon for the first figure ^, and test "^ These two Laws (or Principles) together with the Law or Principle of Excluded Middle (A either is or is not B) and the Law or Principle of Sufficient Reason (Infer nothing without a ground or reason), are often called the Fundamental Laws of Thought. See Thomson's Laivs of Thought ^ § i I4j and Hamilton's Lectures on Logic ^ Lects. vi. and vii. ^ * lii'^oyiiv h\ TO KaTOi -navros Karriyopetadai, orav fir]d€v y Xa0€iv rS)v Tov vrroKfifxivov, KaO* oiv Oartpov ov Xex^^^^'''^^' ^^^ ^^ KaraL fjiTjSivos wffavTcos, An. Pr. I. I ; oaa Kara tov fcaTrjyopovixtvov \4y€TCU vdvTa Kol /caroL tov viroKuyLivov /irjO-qafrai, Cat. 3. This principle or canon in its Latin form was called by the Schoolmen the Dictum de Omni et Nullo. It is thus stated by Sanderson : * Dictum de Omni est hujusmodi : Quidquid affirmatur Universaliter de aliquo Subjecto, affirmari necesse est de iis quse sub eo continentur. Dictum de Nullo hujusmodi : Quidquid negatur Universaliter de aliquo Subjecto, negari necesse est de iis quae sub eo continentur.* By Aldrich it is stated in a more compressed form, as follows : ' Quod prgedicatur Universaliter de alio (i.e. de termino distributo), sive affirmative, sive negative, prsedicatur similiter de omnibus, sub eo contentis.* 93 MEDIATE INFERENCE OR SYLLOGISM. the validity of syllogisms in all other figures by reducing them to the first. Lastly, a favourite method amongst logicians is to enumerate the faults which are incident to a syllogism, and then reject those moods in which they are found. This method is often combined with one or more of the others. Aldrich, for instance, enun- ciates general canons of syllogism, then uses the method I have last explained, and finally reduces syllogisms in the other figures to their corresponding forms in the first. In accordance with the ordinary practice of elementary treatises, and as being perhaps at first more intelligible to the learner, I shall take into consideration the con- clusion, and consequently regard the number of figures as four, and that of possible moods as sixty-four, reserving for a note the shorter and more scientific procedure. The pro- blem therefore now before us is to determine which of the sixty-four moods are admissible in each of the four figures. In the first figure our task is easy. There we are able to establish a canon which will determine directly the legitimate moods. With a little attention the student will be able to per- ceive the truth of the following propositions : — (a) If one term be affirmed of another (provided the latter be taken in its entire extent), and this term of a third, the first may be affirmed of the third. (^) If one term be denied of another (provided the LEGITIMATE MOODS OF SYLLOGISM. 93 latter be taken in its entire extent), and this term affirmed of a third, the first may be denied of the third. (y) If one term be affirmed of another (even though the latter be taken in its entire extent), and this term denied of a third, we are not justified in drawing any conclusion as to the relation of the first to the third. For, if one term be denied of another, it does not follow that whatever may be predicated of this first term may also be denied of the other : thus I may deny * red ' of * blue,' but it does not follow that ' colour,' which I predicate of ' red,' may also be denied of ' blue.' (fi) If one term be denied of another (even though the latter be taken in its entire extent), and this term denied of a third, no conclusion can be drawn as to the relation of the first to the third. For, because I deny one term of another, it does not follow that whatever I can deny of the first can also be denied of the other, nor does it follow that it can be affirmed of it : thus, be- cause I can deny ' white ' of crows, and * black ' of * white ' (or rather of the corresponding com- mon term 'white things'), it does not follow that I can deny * black ' of crows ; nor, because I can deny * white ' of crows as well as * yellow ' of ' white ' (or * white things % does it follow that I can affirm * yellow ' of crows. Putting together these results, we obtain the following 94 MEDIATE INFERENCE OR SYLLOGISM. canon of reasoning in the first figure : If one term can be affirmed or denied of another (provided that the latter be taken in its entire extent), and this term affirmed of a third, the first can be affirmed or denied (respectively) of the third ; and, if these conditions are not fulfilled, no con- clusion can be drawn *. The onlj moods which fulfil the conditions of the canon are AAA, EAE, All, and EIO. The conclusions of two other moods, namely, AAI and EAO, might be inferred^ by Subaltern Opposition, from those' of AAA and EAE, and hence these are called Subaltern Moods, We have obtained, it will be observed, forms of syl- logism capable of proving any one of the four propo- sitions, A, E, I, or O, and into one or other of the types accepted as legitimate moods of the first figure all our mediate reasonings may be thrown. Here, then, our enquiry might terminate, if it were simply our object to obtain a sufficient number of legitimate types of reason- ing, but the problem before us is to state exhaustively all possible forms which can be accepted as legitimate. There being no canon which distinguishes with equal precision the legitimate and illegitimate moods of the other figures, we must, in discussing them, have recourse to some other method. I shall first enumerate and ex- plain certain syllogistic rules (derived from the definition * It will be plain from the statement of the Canon that the major premiss of a syllogism in the first figure must be universal. At the suggestion of Mr. St. G. Stock of Pembroke College, this fact has been made plainer in the statement of the Canon itself than was the case in the earlier editions. SYLLOGISTIC RULES, 95 of a syllogism) which will exclude illegitimate moods, and then, before accepting the remainder, I shall test them by reducing them to the first figure. Syllogistic Rules, I. The middle term must he distributed at least once. For, if in both premisses it were used in only a partial signifi- cation, it might denote entirely different objects in the one premiss from those which it denoted in the other, and so there might be no connexion between the two premisses. Thus, in the premisses ' All men are animals,' ' All horses are animals,' the part of the group ' animals ' which is coincident with 'men' may be, and here is, entirely distinct from that portion of the group which is coincident with ' horses,' and consequently we can draw no conclusion as to the relation betw^een men and horses. This fallacy is called Undistributed Middle, II. If a term be distributed in the conclusion, it must have been previously distributed in the premisses. The reason is obvious. If we use a term in a partial signification in the premisses, we cannot legitimately use it in its entire signification in the conclusion. To do so would be to argue from part to whole, or, in other words, to employ a term in a wider signification in the conclusion than that in which it is employed in the premisses. This fallacy is called illicit process of the major or minor, according as the term illegitimately distributed in the con- clusion is the major or minor term. In the syllogism. St I I ^1 g6 MEDIATE INFERENCE OR SYLLOGISM, Some A is not B, All B is C, .*. Some C is not A, we have illicit process of the major ; in the syllogism, All B is A, Some C is B, .-. All C A, illicit process of the minor. III. Two negative premisses prffve nothing. For they simply assert that there is no connexion between the middle term and the extremes; consequently we can draw no conclusion with respect to the relation of the extremes. IV. 1/ either of the premisses be negative, the conclusion must be negative. For the other premiss is affirmative, and, if in one premiss we affirm a connexion between the middle term and one of the extremes, and in the other premiss deny any connexion between the middle term and the other extreme, there can be no connexion between the two extremes. V. If the conclusion be negative, one of the premisses must he negative. For we cannot deny tha there is any con- nexion between the extremes, except we have previously denied that there is any connexion between one of the extremes and the middle term. VI. Two particular premisses prove nothing. For they ' cannot be both negative (O, O). Nor can they be both affirmative (I, I), for then the middle term would be undis- SYLLOGISTIC RULES. 97 tributed. The only remaining case is that of one affirma- tive and one negative premiss (I, O). But this combina- tion of premisses would leave no term to be distributed in the conclusion. Hence the conclusion would be an I pro- position, an affirmative conclusion inferred from a negative premiss, which (according to Rule IV) is illegitimate. VII. If one premiss be particular, the conclusion must be particular. « The premisses must be either both affirmative, or one affirmative and one negative (see Rule III). Now, if they are both affirmative, they will (by Rule VI) be A and I. This combination of premisses, as it contains only one distributed term, and the middle term must be distributed at least once (Rule I), leaves no term to be distributed in the conclusion : consequently the conclusion must be I. If the premisses are one affirmative and one negative, they must (by Rule VI) be either O and A or I and E. In either case the premisses distribute only two terms, and, as one of them must be the middle term (by Rule I), there remains only one term to be distributed in the conclusion. But the conclusion must be negative (by Rule IV). There- fore, it must be O ^ The converse of this Rule, viz. that a particular con- clusion necessitates a particular premiss, is not true. The only cases however in which we find a particular ^ This proof is substituted for the somewhat more elaborate proof given in the earlier editions. H 98 MEDIATE INFERENCE OR SYLLOGISM, conclusion without a particular premiss are those in which the premisses assume more than is required in order to prove the conclusion. This fact will be apparent to the student from an examination of the individual cases, and it might be laid down as a rule that, wherever there is a particular conclusion without a particular pre- miss, something superfluous is invariably assumed in the premisses ®. Of the above Rules, it is plain that Rules III, IV, V, VI, VII are applicable to the moods before they are re- ferred to the several figures, Rules I and II are applicable only when the moods are referred to some particular figure. By the application of the first set of Rules, the sixty- four possible moods are reduced to twelve, viz. AAA, AAI, AEE, AEO, All, AOO, EAE, EAO, EIO, lAI, IEO^ OAO. Thus EEE is rejected because it has two negative pre- misses, EAA because it has a negative premiss without a negative conclusion, AAE because it has a negative • The syllogistic rules are comprised in the mnemonic lines : — Distribuas medium ; nee quartus terminus adsit. Utraque nee prsemissa negans, nee particularis. SectetUr partem conclusio deteriorem. Et non distribuat, nisi cum prsemissa, negetve. ' It has been suggested to me by Professor Park that lEO might be rejected by the application of Rule II, even before it is referred to any particular figure. For the conclusion, O, distributes its predicate, that is the major term of the syllogism. But the major premiss, being I, evidently does not distribute this term. Consequently, there is invariably Illicit Process of the Major. SYLLOGISTIC RULES, \ 99 conclusion without a negative premiss, III because it has two particular premisses, lAA because it has a particular premiss without a particular conclusion. By the application of Rules I and II to these twelve moods, when referred to the several figures, there remain : — in fig. 2, EAE, AEE, EIO, AOO, EAO, AEO; in fig. 3, AAI, EAO, lAI, OAO, All, EIO; in fig. 4, AAI, AEE, lAI, EAO, EIO, AEO. I append a few examples of the methgd of testing the moods, when referred to the figures. Take AEE in figure 2. A ^ All A is B, E No C is B ; E . • . No C is A. No fault. Take lEO in figure 3. I Some B is A, E No B is C ; . • . Some C is not A. Illicit process of major. Take All in figure 4. A All A is B, 1 Some B is C ; I . • . Some C is A. Undistributed middle. It will be seen that of the sixty-four moods, when re- ferred to the four figures, there are only six in each which have not been rejected. It now remains further to test H 2 I lOO MEDIATE INFERENCE OR SYLLOGISM, these moods in the second, third, and fourth figures by reducing them to moods in the first. Reduction. As we have adopted no canon for the second, third, and fourth figures, we have as yet no positive proof that the six moods remaining in each of those figures are valid ; we merely know that they do not offend against any of the syllogistic rules. But, if we can reduce them, i. e. bring them back to the first figure, by shewing that they are only different statements of its moods, or, in other words, that precisely the same conclusions can be obtained from equivalent premisses in the first figure, their validity will be proved beyond question. There are two methods of performing this operation: ist, that called Osiensive Re- duction, which consists in employing one or more of the processes of conversion, permutation, and trans- position of premisses; 2nd, that called Reductio per im- possibile, which consists in shewing, by means of the first figure and the laws of opposition, that the contradictory of the conclusion is false, and therefore the conclusion itself true. Either of these methods is applicable to all the eighteen moods, and the result is that all are proved to be valid. I shall give instances of the application of each method. By ostensive reduction I shall test EAO in the fourth, lAI in the third, AEE and AOO in the second figures. REDUCTION ^ lOI Fig. 4. Fig. I. E No A is B. . • . No B is A. (Simple Conversion.) A All B is C. . • . Some C is B. (Conversion per ace.) 0. • . Some C is not A. Some C is not A. Fig. 3. Fig. I. I Some B is A. ->..^,^ ^^ All B is C. A All B is C. ^ *^^ . • . Some A is B. (Simple Conversion.) I.- . €ome C is A. Some A is C. . • . Some C is A. (Simple Conversion.) Fig. 2. Fig. I. A All A is B. -^.^^^ ^^ . • . No B is C. (Simple Conversion.) E No C is B. -^^ "^ All A is B. E. • . No C is A. No A is C. . • . No C is A. (Simple Conversion.) Fig. 2. Fig. I. A All A is B. . • . No A is not-B. (Permutation.) . • . No not-B is A. (Simple Conversion.) Some C is not B. . • . Some C is not-B. (Permutation.) 0. • . Some C is not A. Some C is not A. The mark x shews that the premisses are transposed ; the sign . * . on the right-hand side of the page is here appropriated to express the employment of conversion or permutation. The last example is interesting, because AOO in fig. 2, and OAO in fig. 3, inasmuch as they contain O premisses, cannot be reduced by the ordinary methods of transposition of premisses and conversion. Hence the older logicians (who, with few exceptions, did not recognise permutation) applied to them the tedious method of reductio per inipossibile (or, if we write it in full, reductio per deduciionem ad impossibtle). This method is equally applicable to all the imperfect moods, as the moods of the three last figures are often called. I now proceed , I0!2 MEDIATE INFERENCE OR SYLLOGISM. to give an example of it, and shall select AAI in the third figure. A All B is A, A All B is C ; I . • . Some C is A. This conclusion must be true : for, if not, suppose it to be fal^e ; Then its contradictory must be true, i. e. No C is A. \ But (from the premisses) All B is C. > Syll. II. . • . (By figure i ) No B is A. ) But (from the premisses) All B is A. Now these two (being contrary propositions) cannot both be true. But the proposition All B is A is assumed to be true. . • . The proposition No B is A must be false. Hence, either the reasoning of Syll. II. is faulty, or one of the premisses is untrue. But the reasoning (being in the first figure) must be valid. . • . One of the premisses is false. Now the premiss ' All B is C,' being one of the pre- misses of the original syllogism, is assumed to be true. . • . The other premiss (No C is A) must be false. . • . Its contradictory (Some C is A) is true. REDUCTION. 103 As the positive test of reduction confirms in every case the negative test of the syllogistic rules, it follows that six moods (though not the same six moods) are valid in each figure. These moods may be remembered by means of the mnemonic lines : — Barbara, Celarent, Darii, Ferioo^t, prioris : Cesar e, Camestres, Festtno, Baroko, secundse : Tertia, Darapti, Disamis, Daiisi, Felapton, Bokardo, Ferison, habet : Quarta insuper addit Bramantip, Camenes, Dimarts, Fesapo, Fresison : Quinque Subalierni totidem Generalibus orti, Nomen habent nullum, nee, si bene colligis, usum. In the above lines, the initial consonants, B, C, D, F, shew that the mood in the second, third, or fourth figure to which they are prefixed is to be reduced to the mood correspondingly marked in the first. Thus Disamis, when reduced, will become Darii. The vowels shew the moods ; thus Disamis represents lAI in the third figure. The letter j, when it occurs after a vowel, shews that the proposition for which that vowel stands is to be converted simply, the letter p that it is to be converted per accidens ^ * It should be noticed that when these letters follow the conclusion, as is the case in Camestres, Disamis, Bramantip^ Camenes, Dimaris, they apply not to the conclusion of the mood given for reduction but to the conclusion of the equivalent mood in the first figure, to which such mood is to be reduced. Thus, in reducing Bramantip, the student will find that the / affects not the I conclusion of Bramantip, but the A conclusion of the corresponding syllogism in Barbara. See, for other instances, p. loi. I04 MEDIATE INFERENCE OR SYLLOGISM, The letter m shews that the premisses are to be trans- posed, k that the mood is to be reduced per impossibile. It will be noticed that k occurs only in two moods, Baroko and Bokardo, but I have shewn that the per impossibile method is equally applicable to all imperfect moods, and that these two moods can be reduced osten- sively by means of permutation, so that any imperfect mood may be reduced either ostensively or per impos- sibile K The initial B in Baroko and Bokardo shews that the per impossibile method, in their case, assumes the validity of Barbara, but in other cases the operation may assume the validity of some one of the other moods in the first figure; thus, in the particular instance I have taken above, it is performed by means of Celarent. It is perhaps needless to add that all letters, not already explained in the mnemonic lines, are non- significant. The nature of the subaltern moods has already been explained. They are, AAI, EAO in fig. i, EAO, AEO in fig. 2, and AEO in fig. 4, included respectively in AAA, EAE, EAE, AEE, AEE. They cannot properly be re- garded as illegitimate, inasmuch as the conclusions are valid, but they are superfluous, inasmuch as they infer less than is justified by the premisses. * If we take k to signify the employment of permutation, Baroko and Bokardo may be replaced respectively by Fa/cso/f o and Do«samosit. These terms are adapted from Whately's Logic, Bk. II. ch. iii. § 5. ( ' SPECIAL RULES. The special Rules, 105 Besides the general syllogistic rules, already enunciated and proved, certain Special Rules have been enunciated for each figure. I give them below as generally stated. Those for the first figure have been proved in establishing its canon; those for the other figures the student may verify for himself by applying the rules, already laid down, on the distribution of terms. Fig. I. (a) The minor premiss must be affirmative. (^) The major premiss must be universal. Fig. 2. (a) One or other premiss must be negative. (p) The conclusion must be negative. (7) The major premiss must be universal. Fig- 3- («) The minor premiss must be affirmative. {0) The conclusion must be particular. Fig. 4. (a) When the major premiss is affirmative, the minor must be universal. {0) When the minor premiss is affirmative, the conclusion must be particular. (y) In negative moods the major premiss must be universal. (S) The conclusion cannot be an universal affir- mative, nor either of the premisses a par- ticular negative. / I06 MEDIATE INFERENCE OR SYLLOGISM, Note, — If, leaving out of consideration the conclusion, we regard the number of possible figures as three and that of possible moods as sixteen, we may proceed as follows. Having enunciated the canon of the first figure, we may constitute the moods Barbara^ Celarent, Darn, and Ferio. The subaltern moods, AAI and EAO, are not admissible, as the question here before us is not ' What conclusions are justified by such and such premisses,' but ' To what conclusions do such and such premisses lead ? ' Now, from this point of view, the conclusions of the subaltern moods are not directly inferred from the premisses, but are inferred by subalternation from the universal conclusions to which the premisses directly lead. The same observation will of course apply to the subaltern moods in the other figures. If we take no account of the conclusion, we have no means of determining which is the major and which is the minor term. Consequently, the premisses may lead to two kinds of conclusions : ist, those in which the predicate of the first premiss is predicated of the subject of the second ; 2nd, those in which the subject of the second premiss is predicated of the predicate of the first. Now the canon of the first figure applies only to the fir^st case; consequently we are bound to ask if any conclusions, falling under the second head, may be inferred from the premisses. These cannot be deter- mined directly by the canon, but must be determined in the same manner as conclusions in the second and third / INDIRECT MOODS, 107 figures. Here we proceed by a method similar to that employed in the text. The syllogistic rules exclude seven of the sixteen possible moods, viz. EE, EO, OE, 00, II, 10, 01. When the moods are referred to their several figures, we find that, where the extreme employed in the first premiss becomes the predicate, and the extreme employed in the second premiss the subject of the con- clusion, the results are, in the second figure, Cesare, Ca- mesires, Fesiino, Baroko ; in the third, Darapti, Disamis, Datm\ Felaplon, Bokardo, Feruon, the subaltern moods of the second figure EAO, AEO, being inadmissible. Where the extreme employed in the first premiss becomes the subject, and the extreme employed in the second premiss the predicate of the conclusion, the results are, in the first figure, AAI, AEO, All, EAE, lEO, which, when we transpose the premisses, become respectively Bramantip, Fesapo, Dimaris, Camenes, and Fresison in the fourth figure. Hence, according to this mode of treatment, the moods of the fourth figure are regarded as indirect moods of the first. Similarly, in the second figure we may constitute the indirect moods AEE, EAE,^ lEO, OAO. These, if we transpose the premisses, are merely a repetition of the ordinary moods of the second figure. This is also the case with the indirect moods of the third figure, viz. AAI, AEO, All, AOO, lAI, lEO. It will therefore be seen that, with the exception of rejecting the subaltern moods, which even there we regarded as superfluous, we arrive practically at the same results as in the text. The moods 108 MEDIATE INFERENCE OR SYLLOGISM, of the fourth figure are recognised, but, instead of being regarded as moods of a distinct figure, they are treated as indirect moods of the first. By the expression * indirect moods/ it will be seen, we mean moods in which the extreme employed in the first premiss becomes the sub- ject, and the extreme employed in the second premiss the predicate of the conclusion. /' — -s-^^^^J^H^^^""- CHAPTER IV. On Trains of Reasoning, {The Soi^ites) SYLLOGISMS may be combined in what is called a Train of Reasoning . Thus the major and minor pre- misses, or either, of our ultimate syllogisms may them- selves be proved by syllogisms; the major and minor premisses of these, or either, by other syllogisms, and so on, till at last we come to premisses not admitting of syllogistic proof. Such premisses are either assumed without any proof at all, or they are the result either of direct observation or of the testimony of others or of Induction. In a train of reasoning, any syllogism proving a pre- miss of a subsequent syllogism is called with reference to the subsequent syllogism a Pro-Syllogism, and the subsequent syllogism with reference to it an Epi-Syllo- gism. It is obvious that the very same syllogism in different relations may be called a Pro- Syllogism or an Epi- Syllogism. '«' The Sorites (or, as it is sometimes called, the chain- argument) is a common instance of a train of reasoning in a compressed form. It consists, in its proper and most usual form, of a series of propositions, in which the predicate of the preceding is always the subject of the succeeding premiss. The conclusion predicates the last predicate of the first subject. 110 TRAINS OF REASONING, THE SORITES. Ill Thus,— All A is B, All B is C, All C is D, All D is E ; . • . All A is E. When expanded, the Sorites contains as many syllo- gisms as there are propositions intermediate between the first proposition and the conclusion. These syllogisms are in the first figure, and the conclusion of each becomes the minor premiss of the next. Thus, the above Sorites contains three syllogisms, viz. — (i) All B is C, All A is B ; . • . All A is C. (2) All C is D, All A is C ; . • . All A is D. (3) All D is E, All A is D; . • . All A is E. In a Sorites, only one premiss can be particular, viz. the first ; and only one negative, viz. the last \ * The first premiss, if particular, may be stated in the form * Some B is A,' instead of in the usual form * Some A is B' ; the first syllogism of the expanded Sorites will then be in the third figure instead of the first. Similarly, the last premiss, if negative, may be stated in the form * No E is D,' instead of in the form ' No D is E,* which will /■ For, if there were a particular premiss in any place except the first, there would be a particular major, which, in the first figure, is inadmissible. Again, if any premiss except the last were negative, there would be a negative conclusion in one of the pre- vious syllogisms ; this conclusion would necessitate, in the following syllogisms, a negative minor, which, in the first figure, is inadmissible. IVofe. — Besides the above form of Sorites, there is another called the Regressive or Goclenian Sorites (so called from Goclenius, who first noticed it). Its pro- perties are exactly the reverse of those of the ordinary Sorites. The subject of each premiss becomes the pre- dicate of the next, the conclusion predicates the first predicate of the last subject, the conclusion of each of the expanded syllogisms becomes the major premiss of the next, and the rules by which it is governed are that the first premiss only can be negative and the last par- ticular. It may be stated thus : — All E js F, All D is E, All C is D, All B is C, All A is B ; .-.All A is F. make the last syllogism of the expanded Sorites a syllogism of the second figure. No advantage, however, is gained by this mode of statement, and it is not nearly so simple as the usual form. CHAPTER V. On Complex {Hypothetical) Propositions and Syllogisms. HITHERTO I have treated only of simple (or, as they have been inaccurately termed, Categorical) pro- positions and syllogisms. A Complex (or, as it is com- monly called, a Hypothetical) Syllogism is one in which one or more complex (or hypothetical) propositions occur. A complex proposition is a combination of two or more simple propositions in one sentence, the pro- positions being so related that the truth or falsity of one proposition or set of propositions is made to depend on the truth or falsity of the other proposition or set of propositions. If the two propositions or sets of propo- sitions be associated together, so that the truth of one depends on the truth of the other, the complex propo- sition may be called Conjunctive\ If they be dissociated, 1 The word categorical {Karri-^opiKos) properly means affirmative, and IS always used in that sense by Aristotle. '' I have, in accordance with more ancient usage, employed the word * conjunctive ' in place of the word * conditional,' which is employed by Aldrich and other logicians of his time. Instead of simple and complex propositions, they speak of categorical and hypothetical, subdividing hypothetical into conditional and disjunc- tive. Besides the improper use of the word categorical (noticed COMPLEX PROPOSITIONS AND SYLLOGISMS, II3 SO that the truth of one depends on the falsity of the other, and the falsity of one on the tryth of the other, the complex proposition may be called Disjunctive. Thus we may take as instances of Conjunctive Propositions : — If (or When, Where, Provided that, &c.) A is B, C is D ; If A is not B, C is D; If A is not B, C is not D ; If A is B and C is D, E is F; If A is B, either C is D or E is F or G is H ; If either A is B or C is D, E is F. It will be noticed in the second and third examples that negatives are introduced, but they are, notwith- standing, examples of conjunctive propositions, for C being D in the first case, and C not being D in the second, are made to depend on the truth of A not being B. As instances of Disjunctive Propositions we may take the following : — Either A is B, or C is D ; Either A is B, or C is D, or E is F ; A is either B or C or D ; Either A or B or C is D ; Either A is not B, or C is not D ; Either A is B, or C is not D. This form of proposition implies that the truth of one member involves the falsity of the other, and, vice versa, the falsity of one member the truth of the other. It should be noticed that both conjunctive and dis- junctive propositions admit of being reduced to the simple form. Thus : — above), it is extremely awkward to make hypothetical and conditional (which are synonyms) stand respectively for the genus and species. The words conjunctive and disjunctive serve also to point out that the division of complex propositions is exhaustive. I 114 COMPLEX PROPOSITIONS AND SYLLOGISMS, * If A is B, C is D ' becomes ' The case of A being B is a case of C being D ' or * A being B involves as a con- sequence C being D/ The disjunctive proposition, when analysed, contains four conjunctive propositions, each of which may be reduced to a simple proposition. Thus, * Either A is B, or C is D' is equivalent to the four conjunctive pro- positions : If A is B, C is not D ; If A is not B, C is D ; If C is D, A is not B ; If C is not D, A is B. Of these four propositions, however, the third is implied in the first, and the fourth in the second. I now proceed to consider Complex Syllogisms, i.e. syllogisms which contain Complex Propositions. J 2. I. Conjunctive Syllogisms, A Conjunctive Syllogism is a syllogism, one or both of whose premisses are conjunctive propositions; if only one premiss be conjunctive, the other must be simple. If both premisses be conjunctive, inasmuch as all con- junctive propositions rank as universal affirmatives, the syllogism, to be valid, must be conformed to Barbara in the first figure. Thus, IfAisB, CisD, • If C is D, E is F ; . • . If A is B, E is F, is a vallid syllogism ; but the following would not be valid : If A is B, C is D, If AisB, EisF; .-.IfCisD, EisF. CONJUNCTIVE SYLLOGISMS, 115 Far the most common form however of a conjunctive syllogism is that in which the major is a conjunctive, and the minor a simple proposition. Of this form there are four possible varieties, of which two are valid and two invalid. These may be represented thus : — If A is B, C is D. (Major premiss.) (i) AisB; . • . C is D. (a) A is not B ; No conclusion. 0) CisD; No conclusion. (2) CisnotD; . • . A is not B. Hence we obtain the rule that, if we affirm the antece- dent, we must affirm the consequent, or, if we deny the consequent, we must deny the antecedent ; but, if we deny the antecedent or affirm the consequent, no conclusion can be drawn. The reason of this rule will be obvious on a little reflexion. We assert that * A being B involves as a consequence C being D ' ; hence, if we grant that A is B, it must follow that C is D ; if we deny that C is D, it must follow that what involves it as a consequence must also be untrue ; but C might still be D, though A were not B, nor would it follow from C being D that A was also B. Syllogism (i) is called a Constructive conjunctive syllogism. Syllogism (2) is called a Destructive conjunctive syllogism. It may be useful to add a few examples of valid conjunctive syllogisms. x I 2 Il5 COMPLEX PROPOSITIONS AND SYLLOGISMS. (i) If A is B, C is not D. C is D ; . • . A is not B. (2) If A is not B, CisD. A is not B ; . • . C is D. (3) If A is not B, C is not D. A is not B ; . • . C is not D. C is not D ; •. A is B. CisD; . A is B. (4) If A is B, either C is D or F is G. AisB; .-.Either C is D or F is G. Neither C is D nor F is G ; . • . A is not B. (5) If either C is D or F is G, either X is Y or V is W. Either C is D or F is G ; NeitherXisYnorVisW; •.Either XisYor Vis W. .'. Neither C is D nor F is G. #3. II. Disjunctive Syllogisms. A Disjunctive Syllogism is a syllogism of which the major premiss is a disjunctive, and the minor a simple proposition, the latter affirming or denying one of the alternatives stated in the former. We may indeed combine two disjunctive propositions, and draw conclusions from them, but we can only do so after reducing the disjunctive propositions to the con- junctive form. Thus from the two propositions Either DISJUNCTIVE SYLLOGISMS, 117 A is B or C is D, Either A is B or E is F, we may draw four conclusions, viz. If C is D, E is F ; If C is not D, E is not F; If E is F, C is D ; If E is not F, C is not D. But, as these conclusions are really drawn from conjunctive propositions which are involved in the two disjunctive propositions, we are not justified in calling the syllogisms disjunctive. Hence, as will be noticed, my definition of disjunctive is not so wide as that of conjunctive syllo- gisms. The disjunctive syllogism admits of four conclusions, which may be exhibited thus : — Either A is B, or C is D. (I) (2) AisB; .'.C is not D. A is not B ; .-.CisD. (3) CisD; .', A is not B. - (4) Cisnot D; .'.AisB. I add a few examples : — Either A is B, or C is not D. (i) AisB; j 1(3) CisnotD; .-.CisD. (2) A is not B; (4) . • . C is not D. Either A is B, or C is D, or E is F. (i) A is B j .-. Neither C is D nor E is F. (2) A is not B ; .-. Either C is D or E is F. (3) Neither C is D nor E is F; .-.A is B. . * . A is not B. CisD. .-.A is B. Il8 COMPLEX PROPOSITIONS AND SYLLOGISMS, (4) Either C is D or E is F; .-.A is not B. (5) Either A is B or C is D ; .-.E is not F. &c. &c. He is either a fool or a knave. (i) He is a fool; . • . He is not a knave. (2) He is a knave; .'.He is not a fool. (3) He is not a fool ; . • . He is a knave. (4) He is not a knave ; .'.He is a fool. iVb/^.— Mr. Mill (in his Examination of Sir W. HamiU ion's Philosophy, ch. xxiii.) maintains that a disjunctive proposition merely implies that the two alternatives cannot both be false, but that it does not exclude the possibility of both of them being true. Thus, in the last example, he would maintain that there is nothing in the form of the assertion to exclude the supposition of the man being both a fool and a knave. In this opinion he is preceded by many other logicians, but it seems to me that in the expression 'either or ' we distinctly exclude the possibility of both alternatives being true, as well as of both being false. In fact, when we do not wish to ex- clude the possibility of both being true, we add the words * or both,' thus : ' He is either a fool or a knave, or both'; ' I shall come either to-day or to-morrow, or perhaps both days.' THE DILEMMA, 119 \ 4. The Dilemma, There remains the case in which one premiss of the complex syllogism is a conjunctive and the other a dis- junctive proposition, it being, of course, understood that the disjunctive proposition deals only with expressions which have already occurred in the conjunctive pro- position. This form is called a Dilemma, The order of the premisses is different, but it seems more natural that the conjunctive proposition should be the major. If we consider the case in which the major consists of one antecedent and several consequents, there is only one valid form of argument, and that is destructive. (i) If A is B, CisDandEisF; But either C is not D or E is not F ; .•.A is not B. If we asserted in the minor * C is D and E is F ' there would be no conclusion, and if we asserted * Neither C is D nor E is F,' the minor would not be disjunctive. The assertion * Either C is D or E is F ' is, according to my view of the significance of a disjunctive proposition, equivalent to the assertion ' Either C is not D or E is not F,' and leads to the same conclusion. If the major consist of several antecedents, and one consequent, there is only one valid form of argument, and that is constructive. (2) If AisBorifEisF, CisD; But either A is B or E is F ; .-.C is D. I20 COMPLEX PROPOSITIONS AND SYLLOGISMS, If we asserted in the minor * C is not D/ it would not satisfy the requirements of the definition by being a dis- junctive proposition. In the remaining case, where there are several ante- cedents and several consequents, there are two valid forms, one constructive and the other destructive. {3) If A is B, C is D ; and if E is F, G is H; But either A is B, or E is F ; .-.Either C is D, or G is H. (4) If A is B, C is D; and if E is F, G is H: But either C is not D, or G is not H ; .-. Either A is not B, or E is not F. It is evident that we may form a Trilemma, Tetra- lemma, &c., by increasing the number of antecedents or consequents or both, thus: — If A is B, or if E is F, or if G is H, C is D; But either A is B, or E is F, or G is H ; .•.C is D. If A is B, C is D; and if E is F, G is H; and if I is J, K is L ; But either A is B, or E is F, or I is J; .-. Either C is D, or G is H, or K is L. It is not uncommon to mistake for dilemma what is really only a conjunctive syllogism. Thus the two following syllogisms, when examined, will be found to be, the first a constructive, the second a destructive conjunctive. THE DILEMMA. lai (i) Whether geometry be regarded as a mental dis- cipline or as a practical science, it deserves to be studied; But geometry may be regarded as both a mental discipline and a practical science ; .'.It deserves to be studied. (2) If we go to war, we must either contract a debt, or increase the taxation, or indemnify ourselves at the enemy's expense ; We shall not be able to do any of these ; .-.We are not able to go to war. In disputation, the adversary who is refuted by a dilemma is said to be ' fixed on the horns of a dilemma '; he is said to rebut the dilemma, if he meet it by another with an opposite conclusion. Thus (to tell an old story) Protagoras the Sophist is said to have engaged with his pupil, Euathlus, that half the fee for instruction should be paid down at once, and the other half remain due till Euathlus should win his first cause. Euathlus deferred his appearance as ah advocate, till Protagoras became impatient and brought him into court. The Sophist then addressed his pupil as follows : ' Most foolish young man, whatever be the decision, you must pay your money; if the judges decide in my favour, I gain my fee by the decision of the court, if in yours by our bargain/ This dilemma Euathlus rebutted by the following : ' Most sapient master, whatever be the decision, you must lose your fee ; if the judges decide in my favour, you lose it 122 COMPLEX PROPOSITIONS AND SYLLOGISMS, by the decision of the court ; if in yours, by our bargain, for I shall not have gained my cause.* THE DILEMMA, 123 Note I. — Of the four cases of dilemma which I have given, the first would not be admitted by Abp. Whately and Dr. Mansel, who define dilemma as 'A syllogism having a conditional (i. e. conjunctive) major premiss with more than one antecedent, and a disjunctive minor.' Having however a disjunctive minor, it cannot properly be regarded as a conjunctive syllogism, and it seems less arbitrary and more systematic to define dilemma as 'a syllogism of which one premiss is a conjunctive and the other a disjunctive proposition * than to limit it as above. Note 2. — Few parts of Logic have occasioned more differences of opinion or nomenclature than the theory of complex (or hypothetical) propositions and syllogisms. Sir W. Hamilton (see his Lectures on Logic, Lecture xiii. and Appendix viii.) finally arrives at the opinion that * hypothetical and disjunctive judgments * are not more complex than ordinary propositions, and that ' hypo- thetical and disjunctive reasonings' are really forms of immediate inference. Thus he would represent the con- junctive syllogism in the form : — If A is B, C is D ; . • . A being B, C is D. The disjunctive syllogism he would represent in the form : — Either A is B, or C is D ; . • . A not being B, C is D. The other inferences from the premisses would, of course, be drawn similarly. % The dilemma would assume the form : — If A is B, C is D; and if E is F, G is H; . • . Either A being B, or E being F, it follows that C is D, or G is H ; or . • . Either C not being D, or G not being H, it follows that either A is not B, or E is not F. Without attempting to discuss the arguments that have been adduced on either side, I may express my own opinion that complex (or hypothetical) propositions and syllogisms are rightly so called, and that the latter are to be regarded as mediate, and not as immediate, forms of inference. Though no new term is introduced in the minor premiss, the major and minor premisses are entirely distinct propositions, and the conclusion is the result not of one proposition but of two propositions taken conjointly. »<'^bJ^Sr^«e«^3i=:*:>» CHAPTER VI. On the words 'Most', 'Many' &c., as express- ing the Quantity of Propositions. TO all particular propositions I have hitherto prefixed the word ' some/ Both in conversation and reasoning, however, it frequently happens that we use some other sign of particularity, such as ' many,' ' most,' &c. Nor does there seem any valid reason why these forms should not be recognised by Logic. From what has already been said of particular premisses, it will be seen that wherever one premiss is universal and the other modified by some sign of particularity, as ' some,' ' many,' ' most,' &c., the con- clusion must be particular, the degree of particularity in no case transcending (though, in some cases, it may fall below) that denoted by the particular premiss. Thus, if we state as our premisses ' All crimes are to be punished,' * Many offences against individual persons are crimes,' we must draw the conclusion that ' Many offences against individual persons are to be punished.' But if we state as our premisses ' All crimes are to be punished,' ' Most crimes are offences against individuals,' we are only justi- fied in drawing the conclusion that ' Some offences against 'most' 'MANYI £TC, 125 individuals are to be punished.' The student will, from the principles already laid down, be easily able to dis- tinguish between the two classes of cases. * Two particular premisses prove nothing.' This is a general rule, and is strictly true where the premisses are quantified as *some' ' some.' But there is one case in which two particular premisses necessitate a conclu- sion. I will begin with a simple instance of it. If two different predicates can both be predicated affirma- tively of the greater number of individuals denoted by the same common term, there must be some individuals of which they can both be predicated, i. e. in certain cases the predicates must be predicable of each other. Thus, from the premisses Most A are B, Most A are C, we must necessarily infer that Some A is both B and C, and consequently that Some B are C and Some C are B. But we may draw the same conclusion, even in those cases in which both premisses are not quantified by the word * most,' provided that the sum of the quantities by which the subjects are affected Exceeds unity. Thus from the premisses Three-fourths of A are B, One-third of A is C, it follows that at least one-twelfth of A is both B and C ; but if B and C be both predicable of the same objects, either must be, partially, predicable of the other. If, for instance, three men out of four exceed a certain height 125 'MOST' 'MANY' ETC, AS EXPRESSING THE QUANTITY OF PROPOSITIONS. 127 and one out of three a certain weight, at least one out of twelve must exceed both the given height and the given weight, and we may affirm both that Some men who exceed a certain height also exceed a certain weight, and that Some men who exceed a certain weight also exceed a certain height. Of course, when we use such an indefinite word as * most ' in either premiss, the other premiss must be quantified by an expression signifying at least one-half; else we cannot be sure that the quantities of the two premisses, when added together, exceed unity. A conclusion of this kind can only be drawn where the subject in both premisses is the same term, i.e. in the third figure ; for, in the mere form of a logical pro- position, we have no data to guide us with regard to the quantity of the predicate. Thus, from the premisses Nineteen-twentieths of A are B, Nine-tenths of B are C, we can draw no conclusion as to the relation of A and C ; for the tenth of B which is not C might be precisely that portion which was coincident with, or which contained, the nineteen-twentieths of A. Though, however, these syllogisms are confined to the third figure, they may be either affirmative or negative. Thus, from the premisses Three-fourths of A are not B, Two-thirds of A are C, we may infer that five-twelfths at least of A are C and not B, and consequently that some "C is not B, and some things which are not B are C. Note, — The propositions employed in the above chapter have usually been regarded as Particulars, though they have sometimes been classed with Universals. See Sir W. Hamilton's Lectures on Logic, vol. ii., first ed. p. 354 ; second ed. p. 361. j^ CHAPTER VII. On Probable Reasoning. IN discussing the copula, it was maintained that any modification of our assertions, such as the qualifications introduced by the words * probably,' * possibly,' &c., was, in the ultimate analysis of the proposition, to be referred to the predicate and not to the copula. Thus such a proposition as * A is probably B,' when stated in its strictly logical form, would become * That A is B is a probability/ It would however be tedious and prac- tically useless to reduce all our propositions to such a form. I may therefore proceed to lay down rules for reasoning from propositions whose copula is modified, remembering however that they are not stated in strictly logical language. The correctness of the following rule will be apparent. If a premiss whose copula is modified be combined with another premiss whose copula is unmodified, the copula of the conclusion must be modified also ; the modality, of course, never transcending that of the premiss. Thus from the premisses * All true poets are men of genius/ ' Sophocles is probably (certainly, possibly, &c.) a true poet,' I infer that Sophocles is probably (certainly, pos- sibly, &c.) a man of genius. From a certain and a PROBABLE REASONING, 129 probable premiss, therefore, arranged according to the ordinary laws of syllogism, we can never infqf more than a probable conclusion. To this head may most con- veniently be referred those syllogisms in which the major is a particular proposition introduced by the word ' most,' and the minor a singular proposition. Thus from the pre- misses, * Most philosophers are men of vivid imagination,' * A B is a philosopher,' I infer, as the conclusion, A B is probably a man of vivid imagination. If most phi- losophers possess certain characteristics, any particular philosopher will probably possess them, so that the major premiss is, in fact, equivalent to the proposition, * A philosopher is probably a man of vivid imagination.' Using the word * probable ' in the sense of ' more likely than not,' two probable premisses do not lead to a probable conclusion. This fact will be obvious from an easy example. Suppose there are in a bag four red, five blue, and six white balls : I may say with truth * Any ball drawn at random from the bag is probably a red or blue ball ' ; I may also say with truth ' Any ball drawn at random from the red and blue balls is probably a blue ball'; but I cannot infer that * Anyl)all drawn at random from the bag is probably a blue ball.' I shall only be justified in drawing the conclusion *Any ball drawn at random from the bag is possibly a blue ball.' But, where our information is so special as in the above instance, a conclusion of this kind is far too vague. Is there no method which will enable me to state in the conclusion the exact value of the expectation that any I30 PROBABLE REASONI]\rG, particular ball drawn at random may be blue, red, or white ? For such a method I must have recourse to mathematics. Though the word * probable ' is used in the sense of ' more likely than not,' the word prohahility is used as the equivalent of ' chance ' or ' expectation/ If it be three to two that a certain event will happen, 3 : 2 is called the odds for, 2:3 the odds against the event. Now the ' probability ' or * chance * of the event happening would be expressed by |, that of its not happening by f , the denominator in both cases being expressed by the sum of the terms of the odds, the numerator in the first case by the term of the odds for, in the latter case by the term of the odds against. If two events are independent of each other, the joint or compound probability that they will both happen must be much smaller than the pro- bability that either of them will happen alone, and it is discovered by multiplying together the fractions which express the probabilities of their happening separately ^ Thus, in the above instance, the chance of my drawing a red or blue ball = y\ ; the chance of my drawing out of the red and blue balls a blue ball = |; .-.the chance of my drawing a blue ball out of a bag which contains ^=iTX7 = i> a result at which, in this particular in- stance, I could of course have arrived more directly. Hence, when both premisses are affected by words like ' probably,' * possibly,' &c.,the probability of the conclusion ^ The truth of this proposition, with regard to two, three, or any nnmber of events, is proved at length in Peacock's Arithmetical Algebray § 469. * > PROBABLE REASONING, 131 may always be discovered by multiplying together the pro- babilities of the premisses, the conclusion being therefore less probable than either premiss. I append a few in- stances of conclusions drawn from probable premisses : — (i) This plant will probably sprout up during the winter months. (Let the probability = f .) Whatever plant sprouts up during the winter will probably be bitten by the frost. (Let the pro- bability = |.) .-.This plant may sprout up and be frost-bitten. (Here the probability = f x 4^ = ^f .) .'.The odds against the event are 23 to 12, and those m favour of it 12 to 23. (2) Two thirds of these men will be enlisted. Half the men enlisted are killed in battle. .-.The probability of any particular man being en- listed and killed in battle = f X J = -j. .-.It is two to one ci gainst any partipular man here being enlisted and killed in battle. (3) A warm day may possibly be a rainy day. (Let the probability = ^.) A rainy day is probably a calm day. (Let the probability = f .) .•.A warm day may possibly be both rainy and calm'^. (The probability = j-%.) * I am indebted to Professor Shaw, of Deny, for pointing out an inaccurac)' in the conclusion of this example, as stated in the earlier editions, and to Professor Park for the suggestion that it is a pecu- liarity of all arguments of this kind that, strictly speaking, both premisses should be expressed in the conclusion. K 2 w 132 PROBABLE REASONING, N. B. — It is most important for the student to bear in mind the ambiguous use of the words * probably/ * probable/ 'probability/ The adverb 'probably' we seem always to use in the sense of * more likely than not/ The adjective ' probable/ when employed as a predicate, seems also to be invariably used in the same sense ; thus we say ' It is probable that he will do so and so/ ' This event is probable/ But when used to qualify a substantive, as in such expressions 'probable premisses/ 'probable reasonings,' &c., it may be employed either in the above sense or simply as contrasted with certainty, and in the latter case the 'probability/ as we say, may vary from certainty to zero. Lastly, the word 'probability' may simply be equivalent to ' chance,' as explained above, or in some expressions it may have the meaning of * being more likely than not,' as when we say ' The probability is that he will do so and so/ By ' probable reasoning' at the head of this section I of course mean reasoning which falls short of certainty, and the value of which may vary to any extent so long as it does not rise to certainty or fall to zero. CORROBORATIVE EVIDENCE, ^2^?^ On the Combination of Probable Arguments, [Cumulative or Corroborative Evidence^ including Circumstantial Evidence?^ Probable arguments may be combined together in a chain ^ (or, as it has been more appropriately called, a coil) of reasoning, each argument leading to the same conclu- sion. Instead of weakening each other, as is the case with probable premisses, such arguments, being all in- dependent testimonies to the truth of the same conclusion, mutually strengthen each other. If the value of any single argument amounts to certainty, the conclusion must be true. In this case therefore we have to calculate the chances of failure in each separate argument ; these, when multiplied together, give the probability of all the arguments together failing to prove the conclusion ; and this fraction, when subtracted from unity (which repre- sents certainty), gives the probability, resulting from all the arguments joindy, in favour of the conclusion. Thus suppose the probabilities in favour of certain probable arguments to be represented respectively by \, |, f ; ^ Chain-reasoning would be the proper designation for a series of arguments (or links of evidence) which are all inter-dependent, so that, if one argument fails (or one link snaps), the reasoning breaks down altogether. Much legal evidence is of this kind, as when A testifies that he received a certain parcel from B, B that he received it from C, and so on. In this case, any one untrustworthy witness would invalidate the whole evidence. On the other hand, in corroborative evidence the various parts strengthen one another. ^34 PROBABLE REASONING. the chances of their failing to prove the conclusion will be represented Respectively by §, \, \ (or the differences between the favourable chances and unity); the chance therefore of their all failing to prove the conclusion =- f X T X T = ^V >* consequently the probability in favour of the conclusion, as based upon all the arguments jointly, is -j^j, i. e. the odds in favour of it are 29 to i. I may illustrate this case by an example, which will also serve to shew the characteristic uncertainty attaching to this method of reasoning. Suppose a man to be found lying dead on a road from the effects of a wound. On the same evening on which he died, another man was seen running away from the neighbourhood of the place. On this man's house being searched, his clothes are found to be stained with blood; his footsteps correspond with those leading to and from the place where the dead man was lying; and moreover he is known to have possessed a weapon, now not to be found, which was capable of inflicting the wounds. The presumption in favour of his guilt is very great; each argument, taken alone, possesses some cogency, and when all the argu- ments are taken together they appear to be irresistible. But suppose the suspected man, when arrested, to give this account of the affair : he was walking along the road, armed with a dagger; he was suddenly attacked by another man; a scuffle ensued, and in the scuffle he killed his assailant ; finding that he had killed him, he was seized with a sudden panic, threw away his weapon, and ran home. Such an account, in the case CORROBORATIVE EVIDENCE. 135 of a timid and secretive man, might possibly be true, and, in estimating the counter-probabilities, ;we should have to consider the characters of the accused and the dead man, and the nature of the motive, if any, which could have led to the supposed crime. Suppose the dead man's pockets were rifled, and the accused (who had been previously convicted of a felony) were in possession of his money, there could be little doubt that he had committed a murder; but suppose that the character of the accused was good, and no likely motive could be assigned for the commission of the crime, his own version of the affair might be accepted as probably true, or at least as throwing considerable doubt on the supposition of his guilt *. * As illustrating the danger of exaggerating the value to be attached to this kind of evidence, I could hardly adduce a more forcible example than the following passage from Lord Coke (quoted by Bentham" in his Rationale of Judicial Evidence, Bk. V. ch. XV. § 2) : — ' Violenta presumptio is many times plena probatio ; as if one be run thorow the bodie with a sword in a house, whereof he instantly dieth, and a man is seen to come out of that house with a bloody sword, and no other man was at that time in the house.' To this Bentham replies by two counter-suppositions : — * I. The deceased plunged the sword into his own body, as in the case of suicide ; the accused, not being in time to prevent him, drew out the sword, and so ran out, through confusion of mind, for chirurgical assistance. * 2. The deceased and the accused both wore swords. The deceased, in a fit of passion, attacked the accused. The accused, being close to the wall, had no retreat, and had just time enough to draw his sword, in the hope of keeping off the deceased ; the deceased, not seeing the sword in time, ran upon it, and so was killed. * Other suppositions might be started besides these ; nor do these 136 PROBABLE REASONING, The commonest example of evidence of this kind (which may be called Cumulative or Corroborative Evidence) is Circumstantial Evidence, so called from the fact that exculpative ones either of them seem in any considerable degree less probable than that criminative one : if so, the probability of de- linquency, instead of being conclusive, is but as 1 to 2.* Sometimes the individual arguments in evidence of this kind are of so little value that, even when several of them are accumulated, they have no practical force. ' Presumptio probabilis,' Lord Coke says rightly, ' moveth but little, but presumptio levis sen temeraria moveth not at all.' On the other hand, the force of this evidence may amount to that of moral certainty. ' If,' says Mr. Wills, ' it be proved that a party charged with crime has been placed in circumstances which com- monly operate as inducements to commit the act in question, that he has so far yielded to the operation of those inducements as to have manifested the disposition to commit the particular crime, — that he has possessed the requisite means and opportunities of effecting the object of his wishes,— that recently after the com- mission of the act he has become possessed of the fruits or other con- sequential advantages of the crime,— if he be identified with the corpus delicti by any conclusive mechanical circumstances, as by the im- pressions of his footsteps, or the discovery of any article of his apparel or property at or near the scene of the crime,— if there be relevant appearances of suspicion, connected with his conduct, person, or dress, and such as he might reasonably be presumed to be able, if innocent, to account for, but which, nevertheless, he cannot or will not explain,— if, being put upon his defence recently after the crime, under strong circumstances of adverse presumption, he cannot shew where he was at the time of its commission,— if he attempt to evade the force of those circumstances of presumption by false or incredible pretences, or by endeavours to evade or pervert the course of justice,— the concurrence of all or many of these cogent cir- cumstances, inconsistent with the supposition of his innocence and unopposed by facts leading to a counter-presumption, naturally, reasonably, and satisfactorily establishes the moral certainty of his CIRCUMSTANTIAL EVIDENCE. 137 several circumstances are adduced as all supporting the same conclusion. It will be readily seen that the utmost caution is required in estimating its value. We are bound to consider not only the circumstances which point to the conclusion, but also those which make against it, or in favour of any counter-supposition. It is only when we feel certain that we have exhausted all possible sup- positions, consistent with the circumstances of the case, and considered carefully the value of the arguments, or series of arguments, pointing to each of them, that we are entided to pronounce with confidence in favour of any particular conclusion. Any single syllogism in a coil of circumstantial evi- dence may be represented as an argument with one probable and one certain premiss, thus: A man who was seen running away, in order to escape observation, from the place where the dead man was lying, immediately after his death, was probably the murderer, This man was seen running away, &c. ; . • . He was probably the murderer. ^ The major premiss of each of these syllogisms is the result of an analogical argument; and its value must be estimated according to the rules of analogy, already explained. guilt, — if not with the same kind of assurance as if he had been seen to commit the deed, at least with all the assurance which the nature of the case and the vast majority of human actions admit.' Wills on Circumstantial Evidence, 4*^ ed., pp. 276, 7. 138 PROBABLE REASONING, CIRCUMSTANTIAL EVIDENCE. 139 Note I.— It is more usual to regard the separate syllo- gisms in a coil of circumstantial evidence as syllogisms in the second figure, involving an undistributed middle, and justifying a probable conclusion. Thus the above syllogism would be represented in the form : The murderer would, to escape observation, run away, &c., This man, to escape observation, ran away, &c. ; .-.This man is probably (or possibly) the murderer. It seems however extremely awkward to represent such reasonings as fallacies, and then, by way of compensation, to regard them as probable arguments. This mode of treatment, no doubt, originated in the desire to conform the arguments in circumstantial evidence to those Enthy- mematic syllogisms of Aristotle which employ the o-jy^elov in the second figure. For an account of the Enthymeme and its subdivisions, the student is referred to Trendelen- burg's Elements, § 37, and Dr. Mansel's Appendix to Aldrich, Note F. Note 2. — There are some cases, as, for instance, those in which the witnesses may be suspected of lying, or in which they are actuated by violent passions, where circum- stantial evidence may be of far greater value than direct evidence. See some remarks on this subject in an article contributed to the Fortnightly Review for January, 1873, by Sir H. Maine, and since reprinted at the end of Village Communities, On circumstantial evidence, gene- rally, the student would do well to consult Sir Fitzjames Stephen's Introduction to the Indian Evidence 4ct, Wills on Circumstantial Evidence '\ and Taylor's Treatise, on the Law 0/ Evidence, Pt. I. ch. 4. In Direct Evidence, the witness testifies to the precise fact alleged ; in Indirect or Circumstantial Evidence to some fact or facts indicative of the fact alleged. The rules for estimating the logical value of circumstantial evidence apply also to direct evi- dence in all those cases in which we have any reason for questioning either the veracity or the competence of the principal witness, and in which, therefore, his evidence requires corroboration from independent testimony. * The work of Mr. Wills is exceedingly interesting and instructive ; but the reader must be on his guard against an erroneous statement on the mathematical value of a coil of circumstantial evidence. — j^-s^a'jfe^^t^-^' CHAPTER VIII. On Fallacies, § 1. A FALLACY is, strictly speaking, a defective inference, but the word is, by common usage, extended to any error either in the premisses or in the conclusions of our arguments. In deductive Logic (for we are not here concerned with the fallacies incident to induction ^ 1 The fallacy of False Analogy (which consists either in over- estimating, in some particular case, the value of the argument from analogy, or in supposing an analogy where none exists) falls properly within the domain of Inductive Logic, and is discussed in the author's * Elements of Inductive Logic,' ch. vi. It is not to be confounded with the fallacy arising from the employment in a syllogism of a word used analogously, as if it were used univocally, which, as already noticed, is one case of the fallacy of ambiguous terms. Thus to argue, because there are certain points of re- semblance between the development of the individual and the development of the race, that, therefore, since the individual dies, the race will probably die also, or, because there are certain points of resemblance between the earth and the other planets, that, therefore, the other planets are certainly, or very probably, inhabited, would both be instances of false analogy, the former being the assumption of an analogy which appears to have no existence, the latter being an exaggeration of the value, in that particular case, of the probable argument. But to argue, because art (i. e. artistic skill) requires the highest intellectual gifts, and dissimulation is art (i.e. deceit), that, therefore, dissimulation requires the highest intellectual gifts, is obviously a mere play upon words, and owes its semblance of reason- ing simply to the ambiguity of language. FALLACIES. 141 or to the operation subsidiary to it, observation) these errors are traceable to one of four sources : the assump- tion of a false premiss, neglect of the laws of deductive inference, irrelevancy, and ambiguity of language. Any conclusions, therefore, or series of conclusions, which transgress no law of inference, which are derived from true premisses, which are relevant to the matter under discussion, and which, with their premisses, are expressed in unambiguous language, may be regarded as faultless. § 2. I. A false premiss, borrowed from some science which is not under investigation, can only be detected by a special knowledge of the science from which it is taken. Many fallacies, described in the old books on Logic, are really instances of the assumption of a false premiss, and therefore specially concern other sciences rather than logic. Thus the celebrated fallacy of Achilles and the tortoise (see p. 175) rests upon a false assumption, viz. that when the distances between the two become infini- tesimal they will be traversed by Achilles in finite and equal (and not, as will actually be the case, in infinitesimal and rapidly diminishing) times. The phrase ' False Analogy ' is also applied to a perversion of the Analogy of Aristotle (see p. 72) ; namely, to those cases in which the conclusion is based upon relations wherein the two sets of terms compared do not agree. Thus, though, in many respects, the relation of a king to his subjects- is the same as that of a father to his children, it is plain that invalid conclusions might be based on this general relation, by extending it to points in which it does not subsist. Such invalid conclusions would be called 'False Ana- logies.* 143 FALLACIES. § 3. II. The fallacies due to the neglect of the laws of deductive inference (which, strictly speaking, are the only fallacies to be detected by a mere knowledge of Deductive Logic) have already, to some extent, been discussed. The principal sources of fallacy in a single inference are illicit process and undistributed middle. Undistributed middle often occurs in the following form : All Conservatives (or Liberals, Roman Catholics, Protestants, Englishmen, Frenchmen, &c.) hold such and such opinions, do such and such things, or possess such and such characteristics; A B holds such and such opinions, does such and such things, or possesses such and such charac- teristics ; . • . A B is a Conservative (Liberal, Roman Catholic, &c.). We might, of course, argue quite as legitimately that, because both men and cats are animals, all men are cats. If, however, the argument assumed this shape, — None but Liberals (Roman Catholics, &c.) hold such and such opinions, or do such and such things ; This man holds such and such opinions, or does such and such things; .'.He is a Liberal (Roman Catholic, &c.), it would be perfectly legitimate. The major premiss in this case (see p. 83) is equivalent to the proposition *A11 men who hold such and such opinions, or do such and such things, are Liberals (Roman Catholics, &c.).' As was shewn in the Chapter on Proba- FALLACIES, 143 bilities, we might also advance a perfectly legitimate, argu- ment of the following kind : ' A man who holds such and such opinions, or does such and such things, is probably a Liberal (Roman Catholic, &c.) ; this man holds such and such opinions, or does such and such things; therefore he is probably a Liberal (Roman Catholic, &c.).' Of Illicit Process the following syllogisms may serve as examples : No form of democracy excludes the great mass of the people from political power. Any form of government which excludes the great mass of the people from political power is subject to violent revolutions ; .'.No form of democracy is subject to violent revo- lutions. The early history of some nations is full of incredible events, A history which is full of incredible events is not worthy of serious study ; .'.The early history of any nation is not worthy of serious study. The former syllogism (AEE in the first figure) involves illicit process of the major, the latter (EIE in the first figure) illicit process of the minor. It will be observed that the various propositions are not stated in strictiy logical form, though they easily admit of being so stated, and the premisses of both syllogisms require to be transposed. Where there is a long train of reasoning in which one 144 FALLACIES, syllogism is employed to prove the premiss of another, and so on, a fallacy frequently occurs, which often escapes detection. This is called the Argument in a Circle, A book is written, or a speech is made, with the object of proving some controverted opinion. The author or speaker, being full of one idea, after a litde preliminary matter, assumes the proposition to be proved, slightly dis- guised, probably, under some equivalent form ; from this proposition he deduces various conclusions, and these conclusions, when put together, of course triumphantly establish, from various sides, his view of the controversy. This is not an unfair analysis of many elaborate argu- ments, the fallacy being, in the great majority of in- stances, undesigned, and imposing on the author or speaker himself quite as much as on his readers or hearers. The fallacy, when expressed in its naked form, may be described as the assumption, in a train of reason- ing, of the conclusion of a subsequent syllogism as a premiss of a precedent syllogism. It may be represented thus: — Syll. (i) Bis A, Syll. (2) C is A, C is B ; B is C ; .'.C is A. .-.B is A. Thus, when asked why B is A, we reply, because C is A; and when asked why C is A, we reply, because B is A. Of course, in actual argument, hundreds of interme- diate syllogisms might occur between syll. (i) and syll. (2). The larger the number of intermediate steps, the more likely is the fallacy to escape detection, and, conversely, FALLACIES. 145 the true mode of exposing the fallacy, as Whately observes, is to narrow the circle by cutting out the intermediate steps, and exhibiting the assumption of the conclusion in its naked form. The fallacy is to be regarded as a breach of the laws of inference, because, when reduced to its simplest terms, it is a proving of the conclusion by means of itself, instead of by means of premisses which jointly necessitate it. But, unlike the fallacies of illicit process, &c., already treated, it cannot be detected from the inspection of a single syllogism, but requires the comparison of two syllogisms or more. The argument in a circle is the most important case of the fallacy called Petitio Principii (or, as it is more properly called, Petitio Qucesiti, begging the question). The other cases of petitio principii generally enumerated, though the enumeration is by no means exhaustive, are (2) when the conclusion simply re-asserts a premiss of the same syllogism-, (3) when it is exactly synonymous with** II * If a syllogism, in which one of the simpler forms of petitio principii occurs, were stated at length, one of the premisses wonld be otiose (i.e. would not contribute towards the conclusion), but, as already noticed, in our reasonings, as actually expressed, we generally suppress one of the premisses. ^ * The English language,' says Archbishop Whately {^Elements of Logic, Bk. III. § 13), ' is perhaps the more suitable for the fallacy ai petitio principii, from its being formed from two distinct lan- guages, and thus abounding in synonymous expressions which have no resemblance in sound and no connexion in etymology ; so that a sophist may bring forward a proposition expressed in words of Saxon origin, and give as a reason for it the very same proposition stated in words of Norman origin ; e.g. "to allow every man an unbounded L 146 FALLACIES, FALLACIES. 147 one of the premisses, (4) when one of the premisses is equally unknown with the conclusion, (5) when it is more unknown. Cases (4) and (5) are really instances of the assumption of a false premiss, or at least of a premiss which is not known to be true. Cases (2) and (3) arise from a neglect of the laws of syllogism, for, instead of proving the conclusion from the two premisses jointly^ we simply re-assert one of the premisses ■*. Connected with this fallacy is the rhetorical device of question-begging epithets. [ Th us, though the matter we are discussing is open to dispute, we may speak of a ne- farious project, a laudable ambition, an astute act, a far- sighted policy, and so on, attempting, by means of a carefully-selected epithet, to assume the point at issue, or at least to create an unfair prejudice in the mind of the hearer or reader whom we address. \ freedom of speech must always be, on the whole, advantageous to the State ; for it is highly conducive to the interests of the com- munity that each individual should enjoy a liberty, perfectly un- limited, of expressing his sentiments." * * It has sometimes been maintained that every syllogism is a petitio principii. Of course the controversy entirely turns upon the meaning of the terms, but, according to the account I have given of the two, a syllogism is so far from being a petitio principii, that every petitio principii is a distinct breach of the laws of syllogism. TTie conclusion of a syllogism is indeed implied by the two premisses taken in combination, and is, in fact, the compression into a single proposition of what, as premisses, were two distinct propositions, but, in a petitio principii, the conclusion is merely a re-assertion of one of the premisses ; in the simpler cases, of a premiss in the same syllogism ; in the argument in a circle, of a premiss in one of the preceding syllogisms of the series. § 4. III. The fallacy of Irrelevancy (or, as it is some- times called, shifting ground)/ is technically termed Igno- ratio Elenchi, i. e. ignorance of the syllogism required for the refutation of an adversary. Thus, in the strictest sense of the words, ignoratio elenchi is committed by a person who in a disputation does not confine himself to proving the contradictory or contrary of his adversary's assertion, or who proves a proposition other than the contradictory or contrary. But, like many other terms borrowed from the dialectical disputations of the ancients, this has now received a wider meaning. Whenever an argument is irrelevant to the object which a speaker or writer professes to have in view, it is called an ignoratio elenchi. vTlius, if I am endeavouring to convince a person that some particular measure is for his personal interest, and I adduce arguments to prove that it contributes to the general utility, or that it is the necessary consequence of other acts of legislation, I am guilty of an ignoratio elenchi, as I should also be if, when it was my object to establish either of the other two conclusions, I were to appeal to his personal interest. When the question at issue is the truth of an opinion, it is an ignoratio elenchi to attack it for its novelty, or for its coming from a foreign source, or for any supposed consequences which may result from it, or to try to throw discredit on its author by saying that it has often been started before, and so is no discovery of his. This fallacy is more common in spoken addresses than in books, as the feelings both of speaker and hearers L 2 A 148 FALLACIES, being more excited, and their judgment less critical, they are less likely to insist on relevancy of argument. On such occasions it most commonly takes the form of an argumentum ad hominem , rwher eby the speaker, in support of the truth of his assertions, or to throw discredit on an adversary, appeals, not to the unbiassed ju dgme nt of his auditors, but to their passions, interests, prejudices, sentiments and associations. The argumentum ad homi- nem, however, is not confined to set speeches; it some- times occurs in writings, and frequently in debates. In the latter, it often assumes the shape of an appeal to the previous acts, or the previously expressed convictions of the opponent; ' That measure, or that argument, or that proposal does not come well from you, who once pro- posed such a measure, or expressed such an opinion, or advanced such an argument, or did such and such acts.' There are occasions when the argumentum ad hominem may legitimately be used as a retort, but it must be advanced as such, and not as an argument. It is so called in opposition to the argumentum ad rem or ad judicium. Similar phrases are used to express other forms of the ignoratio elenchi, as e.g. the argumentum ad verecundiam, argumentum ad baculum^ &c. The argu- mentum ad populum I have treated as identical with the argumentum ad hominem; if called on to distinguish them, which seems unnecessary, I should refer the first to addresses made in the presence of a large auditory, the second to disputations with one or a few individuals ®. ^ The student will find some amusing examples of ignoratio % FALLACIES, 149 § 5. IV. The fallacy originating in ambiguity of language I noticed when warning the student against the employment of equivocal terms. This fallacy (whe- ther we call it that of equivocal terms, of ambiguous terms, or of ambiguity of language) of course includes fallacies arising from any ambiguity which may attach to the quantity of the subject, as e. g, the fallacy arising from the ambiguous use of the word ' all/ which will be noticed below. I now proceed to notice one or two common cases of this fallacy. The same term may often be used in one place distributively and in another collectively, and we may argue as if the term in both places had the same meaning. /This is called the fall acy of Composition^ox Division ; of composition, if we argue from a term taken distributively as if it were taken collectively; of division, if we argue from a term taken collectively as if it were taken distributively. Thus (to give common instances) 7 and 2 are (distributively) odd and even, nine is 7 and 2 (collectively) ; . ' . nine is odd and even. Here we argue from 7 and 2 taken distributively, as if they had been taken collectively, and the fallacy is one of composition. Five is one number, 3 and 2 (collectively) are five; .-.3 and 2 (distributively) are one number. Here the fallacy is one of division. Again, The people of England have a- prejudice against the French, he is one of the people of England ; . • . he has a prejudice elenchi, or irrelevant argument, in Sydney Smith's well-known jeu d'esprit, the Noodle's Oration. i I 150 FALLACIES. against the French. The major premiss might be quite true, and still the particular man spoken of might have a strong sympathy with the French, and be a warm admirer of their institutions. Here we argue from the term * people,' taken collectively, as if it had a distribu- tive signification and whatever were predicable of the English people might be predicated of every single in- dividual amongst them; hence the fallacy is one of division. The last instance is an example of a very common source of deception. A certain people, corpo- ration, or society, in its collective capacity, has certain characteristics, has performed certain acts, passed cer- tain resolutions, or is known to have expressed certain sentiments; hence it is unreflectingly supposed that any particular individual belonging to the class has the same characterisdcs, participates in the same sentiments, and has joined in the same acts. In many cases, of course, he may be a strong dissentient, and may have actively opposed the measures adopted. The ambiguous use of the word ' all ' furnishes a good instance of the fallacies of composition and division. We may argue from * all/ meaning all taken together, as if it meant all severally, and thus commit the fallacy of division ; or from * all,' meaning all severally, as if it meant all taken together, and thus commit the fallacy of composition. Thus, when I say, * All these boxes weigh so much/ or ' All these men can eat so much,' I leave it doubtful whether I mean all taken together or all taken severally. The ambiguity may be removed by substituting FALLACIES. 151 for the word * all/ when used in a distributive sense, ' every,' and, when used in a collective sense, * the whole of.' Another pair of fallacies which falls under the head of ' ambiguous terms ' is the pair known as the Fallacia \Accid entis (or the Fallacia a dido simpUciter ad dictum secundum quid) and the Fallacia a dido secundum quid ad dictum simpUciter, In the first we argue from what is true as a general rule (i. e. unless there be some modi- fying circumstances) as if it were true under all circum- stances; in the second from what is true under certain special circumstances as if it were true as a general rule. Thus a particular walk may be an agreeable one, but it does not follow that it would be so in wet or windy weather; plain-speaking, frugality, gene- rosity, may all be virtues, but it does not follow that it would be virtuous to practise them on all possible occasions. Or, to take instances of the second fallacy, a political revolution may, under particular circum- stances, be necessary to the welfare or existence of a country, but it does not follow that a ^tate of society, in which political revolutions are frequent, is either necessary or desirable; it may be necessary if I am suffering from a particular disease that I should take opium or abstain from labour, but it does not follow that these practices would be good for me when I am restored to health. The fallacies are due to our not sufficiently qualifying the terms which we use, and, by insisting on precision of language, they may always be avoided. i I 's 152 FALLACIES, Though the definitions I have given of this pair of fallacies are conformable to the usage of most modern logicians ^ and are stated in a form which is most likely to be of practical service to the student, they do not exactly correspond with the original meaning of the expressions. The ' Fallacia Accidentis ' and the * Fallacia a dicto secundum quid ad dictum simpliciter/ according to their original usage, applied to those cases in which a term, when not implying accidents, was confounded with the same term, when implying accidents. Thus, to take the common instance (which is sufficiently absurd) : ' What you buy in the market you eat ; raw meat is what you buy in the market ; . • . raw meat is what you eat/ Here it may be replied that what we buy in the market we do indeed eat, but not necessarily in the same state in which we buy it at market"^. This particular instance is an * As, for instance, Mill {Logic, Bk. V. ch. vi. § 4), Port Royal Logic (Part III. ch. xix. § 5, 7). The latter virtually treats both fallacies as if they were a dicto secundum quid ad dictum simpliciter. '' Mr. de Morgan adduces one of Boccaccio's stories as affording an amusing instance of the fallacia accidentis. It is the old example of the * raw meat ' in another form : — * A servant who was roasting a stork for his master was prevailed upon by his sweetheart to cut off a leg for her to eat. When the bird came upon table, the master desired to know what was become of the other leg. The man answered that storks had never more than one leg. The master, very angry, but determined to strike his servant dumb before he punished him, took him next day into the fields where they saw storks, standing each on one leg, as storks do. The servant turned triumphantly to his master, on which the latter shouted, and the birds put down their other legs and flew away. " Ah, sir," said the servant, "you did not shout to the stork at dinner FALLACIES, ^5i example of the Fallacia Accidentis. From their technical meaning, these fallacies would easily pass into their pre- sent signification, which is both more intelligible and of greater practical service. I may notice one more example of the errors due to ambiguous language, viz. the fallacy of what may be called Paronymous Terms. The same word may often assume different forms, as substantive, adjective, adverb, or verb, but it does not follow, when it has assumed these different forms, that they all retain corresponding meanings. It has been already noticed that the words probably, probable, probability, though the two last are themselves ambiguous, vary in meaning according as we use the adverb, the adjective, or the substantive. Thus, if I hear some one ask the question ' What is the pro- bability of my throwing an ace with a die at a single throw?' I cannot infer that in any single throw I shall probably throw an ace. Again, because a man has done something unjust (i.e. has committed an act which in its results is unjust), I cannot infer that he has acted un- justly (i. e. with intentional injustice), nor, even if he has acted unjustly (i. e. in one or more instances), can I infer that he is an unjust man (i. e. a man of unjust habits or character). To take an old instance, because projectors are unfit to be trusted, and this man has formed a pro- ject, it does not follow that he is unfit to be trusted. Nor from the meaning attached to the expressions, kingly, yesterday ; if you had done so, he would have shewn his other leg too.'" !! 154 FALLACIES. nobly, gentlemanly, can we argue to the usual qualities of a king, a nobleman, or a gentleman ; nor, on the meanings of the words ' to trow,' * to represent,' can we base any sound argument as to the nature of truth, or the duties of a representative. All instances of this fal- lacy, when stated syllogistically, involve four terms, and so offend against the rules for the construction of a syllogism, but, as we do not ordinarily state our argu- ments in a syllogistic shape, and these fallacies undoubt- edly impose on us through the ambiguity of language, it is better to consider them here rather than under the second head. Many other forms of fallacy may be regarded as due to ambiguities of language, but it has perhaps been the tendency of modern logicians, and especially of Whately, to overload this division of fallacies, and to treat as merely differences of language what are in reality radical differ- ences of opinion. At the same time it cannot be denied that terms expressive of fundamental conceptions in their several sciences, such as faith, church, election, law, loyalty, federation, justice, value, capital force, nature, natural, &c., are frequently used, in the same discussion, in the most widely divergent senses, and are consequently the source of endless confusion in our reasonings. Thus the term * faith ' may mean either a belief m certain propositions, or confidence J trust, and repose in a certain person ; the word * church ' may mean the whole body of Christians (and, of course, in this sense its signification will vary accord- ing to the meaning attached to the term Christian), a FALLACIES, 155 particular section of Christians, a congregation meeting in a certain place, the place of meeting, and, lastly, by a strange perversion of the term, the clergy as dis- tinguished from the laity ; the term ' loyalty ' may mean either attachment to the laws of a country in general, special attachment to some particular portion of the laws, or, in its most restricted sense, personal attachment to the supreme ruler ; ' capital ' may mean either the amount of money possessed by a trader, or his whole stock of commodities available for future production ; ' natural ' may express either the original condition of a thing, or the state into which it is ultimately developed, besides having countless other meanings. On account of the various significations which may be attached to the same term, it is necessary, in entering on any investigation, carefully to define the terms to be employed, and never, without express notice, to deviate from the sense thus imposed upon them ^ ' The so-called ' Fallacia plurium interrogationum ' has not been noticed in the text, because it is a rhetorical artifice, rather than a logical fallacy. It consists in covertly putting as a single question what is in reality two, as for instance, * Are gall and honey sweet ? * * Have you cast your horns ? ' (known as * comutus '). * What did you take, when you broke into my house last night ? ' * Have you given up beating your father ? ' The object is to entrap the re- spondent into an admission which he would otherwise not be likely to make. !|^ CHAPTER IX. ■u 'm9 On Method as applied to the arrangemefit of Syllogisms in a Train of Reasoning. I DO not propose to treat of Method in general (for this would involve a discussion of induction and the various relations in which it stands to deductive inference), but it may be useful to the student if I offer a few remarks on Method under the limitation stated in the heading of this chapter. When syllogisms are combined in a train of reasoning, we may either commence with the conclusion, and ask what reasons we have for believing it, and then go on to ask the reason for believing the premisses, and so on, till at last we arrive at some propositions of which there is no doubt, or in which we at least can acquiesce ; or else we may follow the reverse process, and commencing with propositions which are the result of some previous investigation, or which we at all events accept as true, may go on combining them with each other, till at last we arrive at some conclusion which we regard as suffi- ciently important to terminate our enquiries. The former method will be familiar to my readers as that by which we solve what are called 'geometrical deductions,' and in fact as the method which we generally though not ANALYTICAL AND SYNTHETICAL METHODS, 157 universally employ when we are attempting to resolve difficulties for ourselves; the latter as the method by which the propositions in Euclid are proved, and in fact as the method which we generally though not uni- versally employ, when it is our object to teach others, either orally or by book. Now the former method is called Analytical (from the Greek word ai/aXvo-t?), because it may be regarded as the breaking up of a whole into its parts, the resolution of the final conclusion of a series of syllogisms into the various premisses on which it de- pends, and of which it is, as it were, the total expression. The latter method is called Synthetical (from the Greek word (rvv6€ais), because it may be regarded as the putting together of the parts into a whole, the combination of the various premisses into a conclusion which is, as it were, their total result. The Synthetical Method is also sometimes called Progressive, and the Analytical Method Regressive, for reasons which will be apparent from what has already been said. The words a priori and a posteriori may also be used to express the same distinction. In inductive inference (to which these terms are more properly applied) we are said to proceed a posteriori, when, a certain event having taken place, we attempt to trace the steps by which it came about, or, a certain phenomenon being presented to us for examination, we attempt to infer the mode of its production; and, vice versa, we are said to proceed a priori, when, from our knowledge of certain circum- stances, we attempt to predict an event, or, by putting in 158 METHOD AS APPLIED TO THE ARRANGEMENT operation certain causes, we attempt to discover their effect. Similarly, in deductive inference, if, a conclusion being assumed as provisionally true, we attempt to discover reasons for it, we may be said to proceed a posteriori ; if, starting with the premisses, we go on combining them to see whither they will lead us, we may be said to proceed a priori. In the former method of reasoning, we are pecu- liarly liable to impose on ourselves or others by availing ourselves of premisses which are fanciful, obscure, in- capable of proof, questionable, or untrue, especially if the conclusion express some cherished conviction or some position which it is the interest of ourselves, our class, or our party to accept and to disseminate. Whenever, therefore, we argue from our conclusions backwards, especial caution is required, if it be our sincere desire to test our convictions impartially. There is also a more general employment of these latter expressions, according to which the term a posteriori is appropriated to designate inductive reasoning, which ought always to be based on individual facts of observa- tion ; a priori to designate deductive reasoning, which proceeds from general principles. The expression a priori is often applied by way of censure to deductive reasoning, when the general principles from which it proceeds are supposed to rest on no ultimate inductions from fact, but to be mere assumptions arbitrarily taken for granted by the author who employs them. OF SYLLOGISMS IN A TRAIN OF REASONING. 159 Note, — For an account of the various senses in which the words * analysis' and * synthesis' are or have been employed, the student is referred to Sir W. Hamilton's Lectures on Logic, Lect. xxiv, and Dr. Hansel's edition of Aldrichy Appendix G. fi 11'' I APPENDIX. On the Five Words ' Genus ^ 'Species^ 'Differential 'Property I 'Accident! WHEN two classes are so related to each other that one is contained under the other, the larger or containing class is called the Genus^ and the smaller or contained class is called the Species. Thus, if we compare animal and man, man is a species, and animal the genus ; if a man and Englishman, man is the genus, and Englishman a species : if we compare moss-rose and rose, moss-rose is a species, and rose the genus ; if rose and flower, flower is the genus, and rose a species. Genera and Species are denoted by Common Terms, which themselves also, as well as the groups which they denote, are called Genera and Species. In the A proposition, a genus can be predicated of a species, but not a species of a genus. Thus we can say * All men are animals,' or * Man is an animal,' but not * All animals are men,' or * An animal is a man.* In the I pro- position, on the other hand, the genus may be the subject, and in the O proposition, if the terms compared are genus and species, must be. The distinction between Differentia, Property, and Acci- dent is more difficult, but may be explained by reference to what has been said (Pt. I. ch. ii.) on the Connotation of Terms. As the former distinction was applicable to classes and the common terms which denote them, so this is applicable to APPENDIX. i6i attributes and the attributives by which attributes are ex- pressed. It may be noticed also that a Differentia, Property, or Accident, being expressed by an Attributive, must be a predicate and cannot be a subject. Now, taking any common term, or any abstract term used as a common term, like triangle, an attributive predicated of it may ex- press either part of its connotation or not. Thus, if we assert that ^ A triangle is a three-sided rectilineal figure,* the term * three-sided,' like the genus ^ figure,' is connoted by the very term * triangle.' Moreover, the term * three-sided ' serves to differentiate or distinguish * triangle ' from all other rectilineal figures, such as quadrilateral, pentagon, &c. Hence we may define differentia as on p. 45. Here however a difficulty occurs. Two or more species falling under the same genus may be distinguished by more than one dif- ferencing attribute. In this case we should speak of the differentice, or we might speak of the differentia as the sum of the differeniice. But, in case the attributive does not express any part of the connotation of the term, it may, nevertheless, express some attribute which foUows from the connotation. Thus, if we say * A triangle is a rectilineal figure having the sum of its angles equal to two right angles,' we are predicating by the expression * having ' &c., not part of the connotation of the word triangle, but an attribute which nvay be directly inferred from part of the connotation. Or, again, if we were dealing with some moral or physical phenomenon, the attri- bute might follow, not as a conclusion from a premiss, but as an effect from a cause. Thus we might predicate of man that he is * capable of progress,' this capability being regarded as an effect of his rationality ; or of animal and vegetable tissues that they are liable to decay, this liability being regarded as an effect of the material of which they are composed and the influences to which they are subject. Hence the definition oi Property on p. 45. M y tfi i \\ i62 APPENDIX, Lastly, the attributive may neither express any part of the connotation of the term, nor any attribute which follows from part of the connotation. In this case it is called an Accident. But accidents are of two kinds. If an accident may be predicated of all the individuals denoted by a common term, as, to take the conventional instance, blackness of crows, it is called an Inseparable Accident. In the most common case, where it is predicable of some of the individuals and not of others, as, for instance, blackness of men, it is called a Separable Accident. The student is recommended to compare, throughout, this explanation with the definitions given on p. 45. EXAMPLES. ^1 # c-cNS»<5S$3fic9''e>*S)o-» ^'i M 2 EXAMPLES. EXAMINE the following Definitions or Descriptions (pointing out their faults, if any, and, where they are sufficient, stating under what head of Defini- tion or Description they fall). ( 1 ) A square is a four-sided rectilinear figure, having all its sides equal. (2) Monarchy is a form of political government in which one man is sovereign. (3) An University is a corporation which grants learned degrees. (4) Logic is the Art of Reasoning. (5) A plane triangle is a figure generated by the section of a cone passing through the vertex and perpendicular to the base. (6) Wealtli is the sum of things useful, necessary, and agreeable. (7) Man is a mammal having hands and cooking his own food. (8) An animal is a sentient organised being. (9) A liquid is that which can be poured out. (10) A Federation is a political union the members of which are bound together for purposes of offence and de- fence. (11) Man is a mammal possessing the power of articulate speech. (12) Political Philosophy is the science of the laws which govern the equilibrium and development of human society. . ll 1 66 EXAMPLES, Examine the following Divisions, substituting, where they are incorrect, one or more correct ones. (i) Men into Aryans, Mongolians, Africans, and Americans. (2) Quadrilateral Figures into Squares, Rectangles, Paral- lelograms, and Rhomboids. ^ (3) The Fine Arts into Painting, Drawing, Sculpture, Architecture, Poetry, and Photography. (4) Governments into Monarchies, Tyrannies, Oligarchies, and Democracies. (5) Books into entertaining and unentertaining. (6) Men into those who lend and those who borrow. (7) The Sciences into Physical, Social, Ethical, Logical, and Metaphysical. (8) Plants into Flowering Plants, Mosses, Ferns, and Pines. (9) The origin of Colonies is to be traced either to the necessity for frontier garrisons (as amongst the Romans) or to the poverty or discontent of the inhabitants of the mother- country (as amongst the Greeks and ourselves). (10) *The general stock of any country or society is the same with that of all its inhabitants or members, and therefore naturally divides itself into the same three portions, each of which has a distinct function or office : 1st, that portion which is reserved for immediate con- sumption, and of which the characteristic is that it affords no revenue or profit ; 2nd, the fixed capital, of which the characteristic is that it affords a revenue or profit without circulating or changing masters ; 3rd, the circulating capital, of which the characteristic is that it affords a revenue only by circulating or changing masters.'— Adam Smith's IVea/^k of Nations, Vol. ii. ch. i. EXAMPLES, 167 Convert the following propositions (previously permuting them, where necessary). (I (2 (3 (4 (5 (6 (7 (8 (9 (10: (II (12 (13 (14 (15 (16 (17 All plane triangles are rectilinear figures. All plane triangles are three-sided rectilinear figures. ^^ All plane triangles may be defined as three-sided rec- tilinear figures. Some branches of Mathematics admit of a direct prac- tical application. Men of fair promises are often not to be trusted. Some members of the Government are not prepared to accept the measure. Virtue is a condition of Happiness. Virtue is the condition of Happiness. A syllogism is a form of inference. y<^ With man many things are impossible. ^ A \\ ^ Some men of great powers of imagination are not poets. Cvwx,^ U« None but persons of great powers of imagination are poets. What I have written, I have written. Propositions are either simple or complex. The proper study of mankind is man. Only the ignorant affect to despise knowledge. He can't be wrong whose life is in the right. ft State in logical form (where necessary) and examine the following arguments. (i) Every book is liable to error, Every book is a human production ; .'. All human productions are liable to error. (2) All tulips are beautiful flowers, No roses are tulips ; .*. No roses are beautiful flowers. i68 EXAMPLES, (3) Some men are wise, Some men are good ; .*. Some wise men are good. (4) All wise men are good, Socrates was a wise man ; .'. He was good. (5) Some mathematicians are logicians, No logicians are unacquainted with the works of Aristotle ; /.Some mathematicians are not unacquainted with the works of Aristotle. (6) No persons destitute of imagination are true poets, Some persons destitute of imagination are good logicians ; /.^Some good logicians are not true poets. (7) No persons destitute of imagination are true poets. Some persons destitute of imagination are good logicians ; « .*. Some true poets are not good logicians. (8) If Caesar was a tyrant, he deserved to die, Caesar was not a tyrant ; .'. He did not deserve to die. (9) If virtue is involuntary, vice is also involuntary. Vice is voluntary ; .*. Virtue is also voluntary. (10) All valid syllogisms have three terms. This syllogism has three terms ; .*. It is a valid syllogism. (ii) Some learned men have become mad. He is not a learned man ; .*. He will not become mad. (12) The Reformers were bitter enemies of the Papal Supremacy, EXAMPLES, 169 A B was a Reformer (for he was prominent in sup- porting the Reform Bill of 1832) ; .*. He was a bitter enemy of the Papal Supremacy. (13) Logic is either a science or an art. It is a science ; .'. It is not an art. (14) Six and seven are even and uneven. Thirteen is six and seven ; .*. Thirteen is even and uneven. (15) He must be a Mahommedan, for only Mahommedans hold these opinions. (16) He must be a Mahommedan, for all Mahommedans hold these opinions. (17) To reject this proposal would be unreasonable, and consequently to accept it is reasonable. (18) This event happened either at Rome, Naples, or Florence; it did not happen at Rome or Naples, and consequently it must have happened at Florence. (19) Logic is indeed worthy of being cultivated, if Aristotle is to be regarded as infallible ; but he is not : Logic therefore is not worthy of being cultivated. (20) An indissoluble association of ideas conHnands belief, and consequently every belief is the consequence of an indissoluble association of ideas. (21) This measure would be destructive of the national prosperity, and I cannot adduce a more cogent argument than that, five years ago, you were your- self of the same opinion. (22) If a man cannot make progress towards perfection, he must either be a brute or a divinity ; but no man is either: therefore every man is capable of such progress. (23) Lias lies above New Red Sandstone, New Red Sand- III 170 EXAMPLES, stone lies above coal ; therefore Lias lies above Coal. (24) My hand touches the pen ; the pen touches the paper : therefore my hand touches the paper. (25) A, B, C, D, and E, are the only German students I know : they are all men of considerable intellectual attainments, and consequently I may infer that all German students are men of considerable intellec- tual attainments. (26) All equilateral triangles are equiangular, and therefore all equiangular triangles must be equilateral. (27) These two figures are equal to the same figure, and therefore to one another. (28) For those who are bent on cultivating their minds by diligent study, the incitement of academical honours is unnecessary ; and it is ineffectual for the idle and such as are indifferent to mental improvement : therefore the incitement of academical honours is either unnecessary or ineffectual. (29) He has no appreciation of beauty, for he has no taste for Pictures. (30) Warm countries alone produce wines, Spain is a warm country ; therefore Spain produces wines. (31) The Germans are a literary nation ; therefore A B, who is a German, is a literary man. (32) We must increase the income-tax, for war has become a necessity, and we cannot go to war without money, and money can only be raised by taxation, and the only tax which the resources of the country can bear is the income-tax, which will fall on the richer part of the population. (33) Governors of dependencies should be vested with ab- solute power, for otherwise it is impossible to crush rebellion. EXAMPLES. 171 (34) A is larger than B, and B is larger than C ; therefore, a fortiori, A is larger than C. (35) Alexander is the son of Philip, and therefore Philip is the father of Alexander. (36) If * to improve is to change, and to be perfect is to have changed often,' what hope can we entertain of those who oppose change ? (37) Luxury is at once beneficial and injurious to society; for luxury is the using the gifts of Providence, either to the injury of the user, or to the injury of others towards whom the user stands in any relation which obliges him to aid and assistance ; but luxury causes expenditure of money, and therefore is beneficial to society. (38) Old age is wiser than youth ; therefore it is only reasonable that we should be guided by the de- cisions of our ancestors. (39) A man cannot always be right in his opinions, and therefore we ought continually to distrust our judgments. (40) Had an armistice been beneficial to France and Ger- many, it would have been agreed upon by those powers ; but such has not been the case : it is plain, therefore, that an armistice would not have been advantageous to either of the bellfgerents. (41) If education is popular, compulsion is unnecessary ; if unpopular, compulsion will not be tolerated. (42) I will not do this act, because it is unjust ; I know that it is unjust, because my conscience tells me so, and my conscience tells me so, because the act is wrong. (43) This proposition is too good to be practicable. (44) Such and such a system of education has produced several distinguished men ; therefore it does not admit of any improvement. iT% EXAMPLES. EXAMPLES, 173 (45) Slavery is a natural institution ; but what is natural is just, and what is just it is unjust to abrogate ; and consequently it would be unjust to abrogate slavery. (46) * Mercy but murders, pardoning those that kill.* Romeo and Juliet^ Act iii. Sc. i. (47) A, B, and C have distinguished themselves both in athletic sports and intellectual pursuits; therefore those who are most famous for their excellence in athletic sports are generally most famous for their intellectual attainments as well. (48) What is called a * constitutional monarchy' is impos- sible, for the sovereign authority of a state cannot be limited by law ; now the king is sovereign, and, as such, cannot be subject to any superior authority. (49) Parallel lines are equi-distant ; for, if from two points in one of them perpendiculars be drawn to the other, these perpendiculars are parallel (Euc. I. 28), and the two lines intercepted between them are parallel ; therefore a parallelogram is formed, of which the perpendiculars are opposite sides and therefore equal. (50) If Bacon's opinion is right, it is improper to stock a new colony with the refuse of jails : but this course we must allow not to be improper, if our method of colonising New South Wales be a wise one : if this be wise, therefore. Bacon's opinion is not right. (51) * Profit' is interpreted in. the dictionary * advantage'; to take profit, then, is to take advantage : it is wrong to take advantage of one's neighbour : there- fore it is wrong to take profit. (52) Romulus must be a historical person, because it is not at all likely that the Romans, whose memory was only burdened with seven kings, should have for- gotten the most famous of them, namely, the first. (53) You maintain that an action can only be called virtuous, if it contribute to the welfare of mankind or of some section of mankind : hence you are botftid to regard every convenient object, for instance a horse, a tree, or a chair, as virtuous. (54) The knowledge of things is more useful than the know- ledge of words ; and, therefore, the study of nature is more improving to the mind than the study of language. (55) In a lottery it is improbable that any particular ticket- holder will draw the prize. But some one must draw the prize. Therefore something improbable must happen. (56) My informant A heard his story from B, who would certainly tell it as originally told to him ; B heard it from C, who would probably tell it accurately ; C from D, who would also probably tell it accurately ; D from E, who, I have no reason to suppose, would tell it inaccurately : I may consequently receive A's story as probably accurate. (57) Large colonies are as detrimental to the power of a state as overgrown limbs to the vigour of the human • body. (58) All law is an abridgment of liberty, and consequently of happiness. (59) I am under an obligation to do it, but he Who is obliged has no power of resistance ; consequently I have no choice about the matter. (60) You never give an opinion without believing yourself to be right, and therefore you must suppose yourself to be infallible. (61) If man be not a necessary agent, determined by plea- sure and pain, there is no foundation for rewards and punishments. These would be useless, unless men were necessary agents, and were determined by pleasure and pain: because, if men were free and li^' I 174 EXAMPLES. EXAMPLES. 175 indifferent to pleasure and pain, pain could be no motive to cause men to observe the law. (62) Night invariably precedes day, and therefore night must be the cause of day. (63) The planet Mars resembles the Earth in the possession of an atmosphere, clouds, and water, and has a tempera- ture in which terrestrial life might exist, and, therefore, it is probably inhabited as the earth is. (64) * If it be fated that you recover from your present dis- ease, you will recover, whether you call in a doctor or not ; again, if it be fated that you do not re- cover from your present disease, you will not recover, whether you call in a doctor or not : but one or other of these contradictories is fated, and therefore it can be of no service to call in a doctor.*— {/^nava Ratio.) (65) The story of the formation of the human race by Prome- theus must be true, for the clay from which he formed it was shown in Greece within historical times. (66) The Latin word * virtus ' originally meant * manliness * ; hence the virtue of manliness or courage is the highest virtue and the type of all other virtues. (67) This person may reasonably be supposed to have com- mitted the theft, for he can give no satisfactory ac- count of himself on the night of the alleged offence ; moreover he is a person of bad character, and, being poor, is naturally liable to a temptation to steal. (68) Opium produces sleep, for it possesses a soporific virtue. (69) The student of History is compelled to admit the truth of the Law of Progress, for he finds that Society has never stood still. (70) You are inconsistent with yourself, for you told me yesterday that there was a presumption of this man's guilt, and now, when I say that I may presume his guilt, you contradict me. 71) 'Suppose one man should by fraud or violence take from another the fruit of his labour with intent to give it to a third, who, he thought, would have as much pleasure from it as would balance the pleasures which the first possessor would have had in the enjoyment and his vexation in the loss of it ; suppose also that no bad con- sequences would follow; yet such an action would surely be vicious.' Butler, On the Nature of Virtue. (72) There exist many differences of opinion and much uncertainty with regard to many questions connected with Geology ; consequently Geology is not a science, and any arguments which assume the truth of geo- logical theories must invariably be regarded with considerable suspicion. (73) * Suppose Achilles to move ten times as fast as the Tortoise, but the Tortoise to have the start of Achilles, say, by one-tenth of the distance to be traversed: when Achilles has arrived at the point from which the Tortoise started, the Tortoise will still be one-hundredth part of the whole distance in advance of him ; when Achilles has reached this point, the Tortoise will still be one-thousandth part of the whole distance in advance of him ; and so on. Thus Achilles will never be able to pass the Tortoise.' {Fallacy of Achilles and the Tortoise) (74) * Epimenides the Cretan says that " all the Cretans are liars," but Epimenides is himself a Cretan ; therefore he is himself a liar. But if he be a liar, what he says is untrue, and consequently the Cretans are veracious ; but Epimenides is a Cretan, and there- fore what he says is true ; hence the Cretans are liars, Epimenides is himself a liar, and what he says is untrue. Thus we may go on alternately proving that Epimenides and the Cretans are truthful and untruthful.' — {Fallacy of Mentiens.) (75) The idea of the obligation to virtue is innate, for it is 176 EXAMPLES, EXAMPLES, 177 found in all men, and it could not be universal if it were acquired by experience. (76) Berkeley's Theory of the Non-existence of Matter is palpably absurd, for it is impossible even to place one's foot on the ground without experiencing the resistance of matter. {']^') I have no hesitation in saying that the proposition, however good in theory, is in practice utterly absurd. (78) If any objection that can be urged would justify a change of established laws, no laws could reasonably be main- tained ; but some laws can reasonably be maintained : therefore no objection that can be urged will justify a change of established laws. (79) I cannot accept your opinion as true, for it seems to me that its general recognition would be attended with the most injurious consequences to society. (80) Why should any but professional moralists trouble themselves with the solution of moral difficulties ? For, as we resort to a physician in case of any physical disease, so, in the case of any moral doubt or any moral disorganisation, it seems natural that we should rely on the judgment of some man spe- cially skilled in the treatment of such subjects. (81) *Wood, stones, fire, water, flesh, iron, and the like things, which I name and discourse of, are things that I know. And I should not have known them but that I per- ceived them bymy senses ; and things perceived by the senses are immediately perceived ; and things imme- diately perceived are ideas ; and ideas cannot exist without the mind ; their existence therefore consists in being perceived ; when, therefore, they are actually perceived there can be no doubt of their existence.' Berkeley, Third Dialogue between Hy las andPhilonous, (82) *And because the greatest part of men are such as prefer their own private good before all things, even that good which is sensual before whatsoever is most divine ; and for that the labour of doing good, toge- ther with the pleasure arising from the contrary, doth make men for the most part slower to the one and proner to the other, than that duty prescribed them by law can prevail sufficiently with them : there- fore unto laws that men do make for the benefit of men it hath seemed always needful to add rewards, which may more allure unto good than any hardness deter- reth from it, and punishments, which may more deter from evil than any sweetness thereto allureth.' Hooker, Reel, Pol. Bk. I. x. (6.) (83) ' The scarcity of a dear year, by diminishing the de- mand for labour, tends to lower its price, as the high price of provisions tends to raise it. The plenty of a cheap year, on the contrary, by increasing the demand, tends to raise the price of labour, as the cheapness of provisions tends to lower it. In the ordinary variations of the price of provisions, those two opposite causes seem to counterbalance one another; which is probably in part the reason why the wages of labour are everywhere so much more steady and permanent than the price of provisions.' Adam Smith, Wealth of Nations^ Bk. I. ch. viii. (84) * I am a Jew. Hath not a Jew eyes? hath" not a Jew hands, organs, dimensions, senses, affections, passions? fed with the same food, hurt with the same weapons, subject to the same diseases, healed by the same means, warmed and cooled by the same winter and summer, as a Christian is ? If you prick us, do we not bleed ? If you tickle us, do we not laugh ? If you poison us, do we not die ? and if you wrong us, shall we not revenge ? If we are like you in the rest, we will resemble you in that.' Merchant of Venice, Act iii. Sc. I. N ;? t k ^' W 178 EXAMPLES, EXAMPLES. J79 (85) * The most striking and important of the effects of heat consist, however, in the liquefaction of soHd substances, and the conversion of the liquids so produced into vapour. There is no solid substance known which, by a sufficiently intense heat, may not be melted, and finally dissipated in vapour ; and this analogy is so extensive and cogent, that we cannot but suppose that all those bodies which are liquid under ordinary cir- cumstances, owe their liquidity to heat, and would freeze or become solid if their heat could be suffi- ciently reduced. In many we see this to be the case in ordinary winters ; for some, severe frosts are requi- site ; others freeze only with the most intense arti- ficial colds ; and some have hitherto resisted all our endeavours ; yet the number of these last is few, and they will probably cease to be exceptions as our means of producing cold become enlarged. A similar analogy leads us to conclude that all aeriform fluids are merely liquids kept in the state of vapour by heat. Many of them have been actually condensed into the liquid state by cold accompanied with violent pressure ; and as our means of applying these causes of condensation have improved, more and more refractory ones have successively yielded. Hence we are fairly entitled to extend our conclusion to those which we have not yet been able to succeed with ; and thus we are led to regard it as a general fact, that the liquid and aeriform or vaporous states are entirely dependent on heat, that were it not for this cause, there would be nothing but solids in nature ; and that, on the other hand, no- thing but a sufficient intensity of heat is requisite to destroy the cohesion of every substance, and reduce all bodies, first to liquids, and then into vapour.* Herschel, On the Study of Natural Philosophy. (86) * We are not inclined to ascribe much practical value to that analysis of the inductive method which Bacon has given in the second book of the Novum Organum. It is indeed an elaborate and correct analysis. But it is an analysis of that which we are all doing from morn- ing to night, and which we continue to do even in our dreams.' Macaulay, Essay on Bacon, (87) * Promises are not binding where the performance is unlawful. There are two cases of this : one, where the unlawfulness is known to the parties, at the time of making the promise ; as where an assassin promises his employer to despatch his rival or his enemy ; or a servant to betray his master. The parties in these cases are not obliged to perform what the promise requires, because they were under a prior obligation to the contrary. From which prior obligation what is there to discharge them ? Their promise, their own act and deed. But an obligation, from which a man can discharge himself by his own act, is no obligation at all. The guilt therefore of such promises lies in the making, not in the breaking of them ; and if, in the interval betwixt the promise and the performance, a man so far recover his reflection, as to repent of his engagements, he ought certainly to break through them.' Paley, Moral and Political Philosophy, Bk. III. Part I. ch. V. (88) ^This, I think, any one may observe in himself, and others, that the greater visible Good does not alwaj^s raise men's desires in proportion to the greatness, it appears, and is acknowledged to have : Though every little Trouble moves us, and sets us on work to get rid of it. The Reason whereof is evident from the Nature of our Happiness and Misery itself. All pre- seht Pain, whatever it be, makes a part of our present Misery : But all absent Good does not at any time • N 2 n i8o EXAMPLES. \ make a necessary part of our present Happiness^ nor the absence of it make a part of our Misery. If it did, we should be constantly and infinitely miserable ; there being infinite degrees of Happiness, which are not in cur possession. All Uneasiness therefore being re- moved, a moderate.portion of Good serves at present to content men ; and some few degrees of pleasure in a succession of ordinary Enjoyments make up a Happi- ness, wherein they can be satisfied. If this were not so, there could be no room for those indifferent, and visibly trifling Actions, to which our Wills are so often determined ; and wherein we voluntarily waste so much of our Lives ; which remissness could by no means consist with a constant determination of Will or Desire to the greatest apparent Good.* Locke, Essay concerning Human Understandings N Bk. II. ch. xxi. § 44. (89) ' There is only one part of the Protectionist scheme which requires any further notice : its policy towards colonies, and foreign dependencies ; that of compelling them to trade exclusively with the dominant country. A country which thus secures to itself an extra foreign demand for its commodities, undoubtedly gives itself some advantage in the distribution of the general gains of the commercial world. Since, however, it causes the industry and capital of the colony to be diverted from channels, which are proved to be the most pro- ductive, inasmuch as they are those into which industry and capital spontaneously tend to flow ; there is a loss, on the whole, to the productive powers of the world, and the mother country does not gain so much as she makes the colony lose. If, therefore, the mother country refuses to acknowledge any reciprocity of obligation, she imposes a tribute on the'colony in an indirect mode, greatly more oppressive and in- EXAMPLES, 181 jurious than the direct. But if, with a more equitable spirit, she submits herself to corresponding restrictions for the benefit of the colony, the result of the whole transaction is the ridiculous one, that each party loses much, in order that the other may gain a little.' Mill's Political Economy, Bk. V. ch. x. § I. (90) * The money to replace what has been burned will not be sent abroad to enrich foreign manufactures ; but, thanks to the wise policy of protection which has built up American industries, it will stimulate our own manufactures, set our mills running fasteit, and give employment to thousands of idle workmen. Thus in a short time our abundant natural resources will restore what has been lost, and in converting the raw material our manufacturing interests will take on a new activity.* New York Tribune of Oct. 24, 1871, quoted by Professor Cairnes in * Some Leading Prin- ciples of Political Economy.' a- % usually appears to contain four, or perhaps more, terms. We must not, however, reject it on that account, till we have previously attempted to translate the propositions of which it is composed into equivalent propositions, con- taining among them three terms only. The cases in which this cannot be done are either those in which the terms of the conclusion, or at least one of them, are distinct, not in form only, but in meaning, from any of the terms employed in the premisses, or those in which there is no term common, or capable of being repre- sented as common, to the two premisses. Thus the syllogism ' Lias lies above New Red Sandstone, New Red A syllogism, when not stated in logical form, i«** 11 i82 EXAMPLES, Sandstone lies above Coal; therefore Lias lies above Coal/ obviously admits of being stated in the form, 'Whatever lies above New Red Sandstone lies above Coal, Lias lies above New Red Sandstone ; therefore Lias lies above Coal,' and consequently ought not to be rejected as having four terms. But the premisses * Lias lies above New Red Sandstone, the Cretaceous System lies above the Oolitic,' contain no term common, nor any term capable (from a mere inspection of the language) of being represented as common, to the pre- misses, and hence they might fairly be rejected as con- taining four terms, and consequently leading to no con- clusion. Even here, however, by any one who possessed sufficient special knowledge of Geology to be aware that the Oolitic System lies above Lias, the premisses might be represented as containing three terms only, and as necessitating the conclusion ' The Cretaceous System lies above New Red Sandstone.' But from such premisses as *Lias lies above New Red Sandstone,' 'A Painting should represent beauty of colour as well as beauty of form,' no conclusion whatever could be drawn by means of either logical or special knowledge. The premisses are, in fact, utterly alien to each other, or, in other words, they are not in pari materid. In examining an argument, the student may always avail himself of any special know- ledge which he may possess, provided that, in his answer, he carefully distinguish between what is due to such special knowledge and what to a knowledge of the ordinary usages of language and of the rules of Logic. EXAMPLES, 183 In the first example given above, a mere knowledge of the rules of Logic, without any reference whatever to the usages of language, would not justify us in drawing any conclusion from the premisses ; the slightest acquaintance with the usages of language would however enable us to represent the terms, which are apparently four, as three, and to infer the conclusion 'Lias lies above Coal' But in the second example no acquaintance either with the rules of Logic or with the ordinary usages of language would enable us to draw the conclusion, ' The Cretaceous System lies above New Red Sandstone;' such a conclusion, though valid, is only justified by a special knowledge of Geology: no person, unacquainted with the facts of Geology, would be justified in admitting the conclusion as an inference from the premisses. Where it is obvious that an argument is intended to be syllogistic, and only one premiss is stated, it is of course expected that the student will supply the other premiss. In some of the examples, it may be an useful exercise to discuss the truth of the premisses as well as the legitimacy of the conclusion. INDEX. Abstract terms, p. 1 3. — different senses in which the expression has been employ- ed, 15. — sometimes used as common terms, 14, 15. Accident, 44. — defined, 45. — inseparable, 41, 42. — separable, 42, 43. Accidental propositions, 48. Accident is fallacia, 1 51-15 3. Act, ambiguity of the word, 4. — or operation employed in preference to power or faculty, 4. ' All,* ambiguous use of the word, 150-151. Ambiguous middle, 87. — terms, fallacy of, 87, 141, 149-155. Ampliative judgments, 48. Analogously, terms used, 87. Analogy, 71, 72. — fallacy of false, 140-141. — of Aristotle, 72, 141. Analytical judgments, 48. — method, 157. A Posteriori method, 157-158. A Priori method, 157-158. Argument in a circle, 143-145. Argumentum ad hominem, 148. — ad populum, ad verecundiam, &c., 148. Attributives, 13, 14-16. — sometimes used as common terms, 14, 15. — when employed as predicates, regarded by Mr. Mill as com- mon terms, 18. Begging the question, fallacy of, 145- Canon of reasoning in the first figure, 92-94. Categorematic words, 12. Categorical propositions and syl- logisms, these expressions not here employed, 1 1 2-1 13, Categories of Aristotle, ix. 66. Chain-argument, 109. Chain-reasoning, 133. Circle, argument in a, 143-145. Circumstantial evidence, 133- 139- Classifications, 65-67. — rules for, 65-66. — difference between Natural and Artificial, 62. Collective terms, 12, 13. — results of thought, 1 7, i86 INDEX. INDEX. 187 Common terms, 12, 13, 17-18. Comparison (reflexion or thought), definition of, i. Composition, fallacy of, 149-150. Concepts employed by Sir W. Hamilton in preference to terms, 8. Conclusion, 10, 84. — negative, 96. — particular, 97, 98. Concrete terms, 1 5. Conditional propositions and syl- logisms, these expressions not here employed, 1 1 2-1 1 3. Connotation of terms, 19-22. Contradiction, law of, 74, 91. Contradictory terms, 83. Contrary terms, 83. Conversion by contraposition or negation, 82. Conversions, 80-82. Copula, 9, 23-27, 128. Corroborative Evidence, 133- 139- Cross-division, 60. Cumulative Evidence, 133-139. Definitions, 37, 44, 49-57- — final, 51, 52. . — incomplete, 52, 53. — provisional, 52, 53. — real and nominal, 56, 57. — what terms are incapable of, 49- — why discussed under the se- cond part of logic, 46. — rules for legitimate, 55. Denotation of terms, 19-22. Description of a singular or col- lective term, 49. — of a common term, 53, 54. Designations, 39, 44. Dichotomy, division by, 61. Differentia, 38. — defined, 45. — generic, 64. — specific, 64. Differentiae, 50, 51. — difficulty of distinguishing from properties, 55, 56. Dilemma, 119-123. Distinction distinguished from division, 59. Distribution of terms, 33-35- Divided term, 58. Dividing members, 58. Division, principle of, 59-60. — by dichotomy, 61. — fallacy of, 149-150. — rules for a legitimate, 62. — what terms are capable of being divided, 58. Divisions, 58-67. — why discussed under the se- cond part of logic, 46. Enthymeme of Aristotle, 86, 138- Enumeration distinguished from division, 59. Epi-syllogism, 109. Equivocal terms, fallacy of, 149- 155- Equivocally, terms used, 87. Essential propositions, 48. Evidence, cumulative or corro- borative (including circum- stantial), 133-139- Excluded middle, law of, 74, 91. Explicative judgments, 48. Extensive capacity of a term, 21. Fallacia accidentis, 151-153- — a dicto simpliciter ad dictum secundum quid, 1 51-153. • — plurium interrogationum, 155. Fallacies, 140-155. Fallacy of Achilles and the Tor- toise, 141, 175. — of ambiguous or equivocal terms, or of ambiguity of language, 141, 1 49-155- — of composition, 149-150. — of division, 149-150. — of false analogy, 140- 141. — of illicit process, 95, 143. — of irrelevancy or ignoratio elenchi, 147-149. — of paronymous terms, 153- 154- — of petitio principii, 144-146. — of undistributed middle, 95, 142-143. Figures, 89-90. — are there three or four? 89-90. — moods of the fourth may be re- presented as indirect moods of the first, io6-io8. — special rules of the, 105. Four terms, 85, 87, 181-183. Fundamentum divisionis, 59-60. Genus, 37, 38. — cognate, 64. Genus, defined, 45. — subaltern, 64. — summum, 63, 66. Heads of predicables, 36-47, 160-162. — why discussed under the se- cond part of logic, 45, 46. Identity, law of, 74, 91. rSioK of Aristotle, 37-41, 44. Ignoratio elenchi, 147-149. Illicit process of major or minor term, 95, 143. Imagination, definition of, i. — simple or reproductive as distinguished from complex or productive, 3. Import of propositions, 46, 47. Indefinite or indesignate propo- sitions, 29, 30. Induction, sense in which the word is employed, 68. Inductions, Aristotelian, 75-76. — distinguished from deductions, 69-74. — instances of, 69-70. — sense in which the word is employed, 68. Inference, different senses of the word, 68. — limitations of the word by Sir W. Hamilton and Mr. Mill, 74-75- Inferences, deductive, 68, 72-75. — defined, 68. — immediate, 68, 73-75, 77-83. — inductive, 68-71, 74-76- — instances of, 10. ; I f i88 INDEX, INDEX. 189 Inferences, mediate, 68, 73~7^- — various kinds of, 68-76. Infinitation, 82. Inseparable accident, 41, 42. Intensive capacity of a term, 21. Irrelevancy, fallacy of, 147-149. Judgments, employed by Sir W. Hamilton in preference to propositions, 8. Language, its relation to thought, 7- Laws of inductive and deductive reasoning, 74, 91. Logic, definition of, 5, 6. — its relation to psychology, 3. — Mr. Mill's definition of, 6. Major and minor premisses, 85. Major and minor terms, 85. — illicit process of, 95-96, 143. - — sense in which they were em- ployed by Aristotle, 88. Mental philosophy (or psycho- logy), definition of, i. Metaphor, 72. Method as applied to the arrange- ment of syllogisms in a train of reasoning, 156-159. Middle term, 85. — ambiguous, 87. — sense in which it was employ- ed by Aristotle, 88. — undistributed, 95, 142-143. Mnemonic lines for the syllogistic rules, 98. — for the valid moods, 103. Modality, question of, 26, 27, 1 28. Moods, 89. — determination of the legiti- mate, 90-92. — indirect, 107-108. — subaltern, 94, 104, 106-107. * Most,* ' many,* &c., as express- ing the quantities of propo- sitions, 124-127, 129. • Nomen infinitum or indefinitum, 82. Odds, meaning of the expression, 130. Oppositions, 77-80. •— Aristotelian, 80. Paronymous terms, fallacy of, 153-154- Partition, 59. Perception, definition of, i. Permutations, 82, 83. Predicables, heads of, 36-47, 160-162. Predicate of a proposition, 9, 23. — its relation to the subject of a proposition, 39-47. Predicated, meaning of, 23, 24. Predication, theory of, 46, 47. Premisses, 10, 85. — major and minor, 85. — negative, 96. — particular, 96-98. Principle of division, 59, 60. Probable, signification of the word, 139, 132. — reasoning, 128-139. Probability, signification of the word, 130, 132. Probably, signification of the word, 132. Problema, 85. Progressive method, 157, Property, 40, 41. — defined, 45. — difficulty of distinguishing from differentia, 55, 60. — generic, 64-65. — specific, 64-65. Propositions, complex (hypothe- tical), 1 1 2-1 14, 122-123. — conjunctive and disjunctive, 1 1 2-1 14. — definition of, 23. — disjunctive, dispute as to their meaning, 118. — division of, according to their quantity and quality, 28- 32. — employed in preference to judgments, 7. — import of, 46, 47. — instances of, 9. — quality of, as expressed by 'most,' *many,' &c., 124- 127. — secundi adjacentis and tertii adjacentis, 11. — verbal and real, 48. — whose copula is modified, 128-129. Pro-syllogism, 109. Psychology, definition of, i. — its relation to logic, 3. Qusestio, 86. Quality of propositions, a 8. Quantification of the subject, 29- 30. — predicate, 30-32. Quantity of propositions, 28-30. — as expressed by * most,' ' many,* &c., 124-127. Question — beggmg epithets, 146. Real propositions, 48. Reasoning, probable, 128-139. Reduction, 100-104. — ostensive, 1 00-101. — per impossibile, 100-104. Reflexion (comparison or thought), definition of, i. Regressive or Goclenian sorites, III. — method, 157. ^Tjnuov of Aristotle, 138. Separable accident, 42, 43. Simple ideas, 16. Singular terms, 12. — propositions rank as univer- sal, 28, 29. Sorites, 109-1 11. — Goclenian or regressive, in. Special rules for the syllogistic figures, 105. Species, 38-39. — cognate or co-ordinate, 64. — co-ordinate but exclusive, 44. — defined, 45. — infima, 63-64, 66-67. — overlapping, 43, 44. — subaltern, 64. Subaltern opposition, 78, 80. — moods, 94, 104, 106-107. 190 INDEX. Subaltemation, 80. Sub-division, 63-64. Subject of a proposition, 9, 23. Subordination, 80. Sufficient reason, law of, 91. Syllogism, ambiguity of the word, 8. — is the a petitio principii, 146. — legitimate forms of, 90-108. — possible forms of, 88-90. — structure of the, 84-88. 2t/AXo7tff;x^s fiovoK-fjfifMTOSj 86. Syllogisms, complex (hypothe- tical), 112-123. — conjunctive, 114-116, 122- 123. — disjunctive, 116-118, 1 22-1 23. Syllogistic rules, 95-100. 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