QIartteU Httiaeraita Eibrarg 3tl)aca, £?eni ^nrh BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 Cornell University Library arV11987 An elementary handbook of logic / 3 1924 031 474 764 olin.anx Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/cletails/cu31924031474764 AN ELEMENTARY HANDBOOK OF LOGIC AN ELEMENTARY HANDBOOK OF LOGIC BY JOHN J. TOOHEY, S. J. PROFESSOR OF LOGIC AND METAPHYSICS IN GEORGETOWN UNIVERSITY, WASHINGTON, u. «J. NEW YORK SCHWARTZ, KIRWIN & FAUSS 42 BARCLAY STREET 3rnt{inmi l^ottet ANTHONY J. MAAS, S. J. Prcepositus Prov. Marylandia Neo-Ehoracensis. jRibil ©iJitat ARTHUR J. SCANLAN, S.T.D. Censor Lihrorum. STinprimattir -I- JOHN CARDINAL FARLEY, Archbishop of New York. V^ Copyright, ipiS, By SCHWARTZ, KIRWIN & FAUSS Entered at Stationers' Hall, London PREFACE The present volume makes no pretensions to being anything more than is implied in its title. It is ele- mentary, and it is a handbook. Being elementary, it omits all the subtler questions which frequently find a place in extended treatises on Logic. Being a hand- book, it is not designed for private study, but for use in the classroom. It does not attempt to provide a detailed explanation of the various topics as they come up for study. This has been left to the teacher, whose exposition of the doctrine would probably be embarrassed and rendered less effective if his pupils were confronted with long and unnecessary comments in the text. A special effort has been made to combine clear- ness with brevity, so that once the student has re- ceived the teacher's explanation, he may have a lucid epitome which will enable him to recall with facility all the essential principles of Logic. The volume departs in two particulars from the common method of treatriient. First, a distinction has been drawn between the Act of Inference and the Process of Inference, and a separate chapter has been devoted to each. This will probably be found to simplify the explanation of Reasoning and to bring out the essential identity of form which underlies the various types of argurftent. Secondly, the h3'^pothesis of the Distribution of the Predicate has been abandoned. This course was vi PREFACE deemed necessary, owing to the inconsistencies which seem to beset that hypothesis. A discussion of these inconsistencies will be found in the Appendix. A sec- ond reason for this departure lay in the fact that a far more direct and simple explanation of the Cate- gorical Syllogism is achieved by the use of the Dicta of the three first Figures of the Syllogism than by re- course to the hypothesis of the Distribution of the Predicate. If it be permitted to make a suggestion to those who decide to adopt this volume as a text-book for their classes, it is that the Dicta be printed on charts which may be hung up in the classroom. This will be found to facilitate yery materially the pupil's mas- tery of the subject. The Dicta need not be committed to memory, but the pupil should be required to de- duce from them all the Rules and Moods of the Cate- gorical Syllogism. Georgetown University, January, 1918 TABLE OF CONTENTS INTRODUCTION SECTION* 1. The Definition of Logic PAGE . 1 CHAPTER I APPREHENSION AND THE IDEA 2. Apprehension: Its Material and Formal Object: Idea 3. Comprehension and Extension of an Idea 4. Prescission : Abstraction : Reflection 5. Direct and Reflex Idea .... 6. Clear, Distinct, and Comprehensive Idea 7. Concrete and Abstract Idea . 8. Singular, Universal, and Collective Idea 9. Contradictory, Contrary, Relative, and Disparate Ideas CHAPTER n THE TERM 10. Sign: Natural and Arbitrary 11 11. Word: Categorematic and Syncategorematic. . . 11 12. Term: Simple and Complex ...... 12 13. Concrete and Abstract Term. 12 14. Absolute and Relative Term. 12 15. Singular, Collective, and Common Term . . .13 16. Univocal, Equivocal, Distributed, and Undistributed Term .......... 13 17. Connotation and Denotation of a Term. . . .14 18. Supposition of Terms : Its Various Kinds . . .14 vii TABLE OF CONTENTS CHAPTER III JUDGMENT SECTION PAGE 19. Judgment: Its Matter and Form: Affirmative -and Negative 18 20. A priori, A posteriori, Immediate, and Mediate Judg- ment 20 CHAPTER IV THE CATEGORICAL PROPOSITION 21. Categorical Proposition : Its Matter and Form . . 22 22. Quality of Categorical Propositions: Affirmative and Negative . 23 23. Quantity of Categorical Propositions : Universal and Particular 24 24. Indesignate Proposition: Singular Proposition in Argu- ment .......... 26 25. Signs of Quantity: Force of the Sign "Few" . . 27 26. Relation of the Comprehension of the Predicate to that of the Subject ........ 28 27. Natural and Unnatural Proposition . . . .29 28. Symbolical Representation of Propositions . . .30 29. Distribution of the Predicate. . . . . .30 30. Conversion of Affirmative Propositions. . . .31 31. Conversion of Negative Propositions . . . .32 32. Simple and Compound Categorical Propositions . . 33 33. Formal Compound Categorical Propositions. . . 33 34. Elliptical Compound Categorical Propositions . . 34 35. Assertoric and Modal Categorical Propositions . . 36 36. Opposite Categorical Propositions: Contradictory, Con- trary, Subcontrary, and Subaltern Propositions , , 38 TABLE OF CONTENTS CHAPTER V EDUCTION SECTION PAGE 37. Import and Implication of Categorical Proposition : Eduction: Positive and Negative Terms . . .42 38. Conversion 44 39. Obversion 47 40. Contraposition . 48 41. Inversion 51 42. Eduction by an Added Determinant . . . .53 43. Eduction by an Omitted Determinant . . . .54 44. Eduction by Complex Conception 54 45. Eduction by Converse Relation 55 CHAPTER VI THE ACT OF INFERENCE 46. Act of Inference : Its Matter and For4n : Logical De- pendence or Sequence: Act of Inference not a Judg- ment .......... 57 47. A priori, A posteriori, Immediate, and Mediate Inference 61 CHAPTER VII THE HYPOTHETICAL PROPOSITION 48. Hypothetical Proposition : Its Matter and Form . . 63 49. Opposite Hypothetical Propositions .... 64 50. Import and Implication of Hypothetical Proposition . 65 51. Disjunctive Proposition : Its Matter and Form . . 67 52. Interpretation of the Proposition "No man can be noble and base" ......... 69 53. Fundamental Laws of Thought 70 TABLE OF CONTENTS CHAPTER VIII THE PROCESS OF INFERENCE AND THE SYLLOGISM SECTION PAGE 54. Process of Inference: Its Premises and Conclusion . 73 55. Argument: Syllogism . 74 CHAPTER IX THE MIXED HYPOTHETICAL SYLLOGISM 56. Mixed Hypothetical Syllogism 75 57. Rules of the Mixed Hypothetical Syllogism . . .76 58. Fallacies of the Mixed Hypothetical Syllogism . . 79 59. Disjunctive Syllogism . . . . ' . . .79 60. Conjunctive Syllogism ....... 82 CHAPTER X THE SIMPLE CATEGORICAL SYLLOGISM 61. Simple Categorical Syllogism. . . . . .83 62. Full Expression of the Simple Categorical Syllogism . 84 63. Axioms of Identity and Diversity : Essence of the Simple Categorical Syllogism . . . . .85 64. Matter and Form of the Simple Categorical Syllogism . 86 65. Laws of the Truth and Falsity of Valid Conclusions . 87 CHAPTER XI FIGURES AND MOODS OF THE CATEGORICAL SYLLOGISM 66. Figure of a Categorical Syllogism. .... 90 67. Mood of a Categorical Syllogism . . . . .92 68. Advantages of the Method of the Dicta. . . .92 69. Dicta of the First, Second, and Third Figures . . 93 TABLE OF CONTENTS xi SECTION PAGE 70. Interpretation of the Dicta of the Second and Third Figures ......... 96 71. Rules and Moods of the First Figure . . . .98 72. Rules and Moods of the Second Figure . . .99 73. Rules and Moods of the Third Figure: Rules of the Three First Figures Proved Independently of the Dicta 100 74. Rules arid Moods of the Fourth Figure. . . . 105 75. Subaltern Moods: The Mnemonic Lines . . . 106 CHAPTER XII GENERAL RULES OF THE CATEGORICAL SYLLOGISM ' 76. Rules of' the Three First Figures Compared with the General Rules of the Categorical Syllogism . . 108 77. Statement of the General Rules of the Categorical Syllogism 109 78. Comment on Three of the General Rules . . . 109 79. Explanation of the First General Rule . . . .110 80. Apparent Exceptions to the First General Rule . . 112 81. Explanation of the. Second General Rule . . . 114 82. Explanation of the Fourth General Rule . . . 115 83. Apparent Exceptions to the. Fourth General Rule . .116 84. Explanation of the Fifth and Sixth General Rules. . 117 85. Apparent Exceptions to the Fifth General Rule . .118 86. Apparent Exceptions to the Sixth General Rule . .119 87. Explanation of the Eighth General Rule . . .119 88. Apparent Exceptions to the Eighth General Rule . . 120 CHAPTER XIII REDUCTION OF CATEGORICAL SYLLOGISMS 89. Reduction : Original Purpose of Reduction : Direct and Indirect Reduction ....... 122 90. Direct and Indirect Reduction Illustrated . . . 123 91. Explanation of the Mnemonic Lines .... 124 TABLE OF CONTENTS CHAPTER XIV THE PURE HYPOTHETICAL SYLLOGISM AND OTHER TYPES OF ARGUMENT SECTION PAGE 92. General Remark ........ 128 93. Pure Hypothetical Syllogism: Its Figures and Rules . 128 94. Dilemma 131 95. Rules of the Dilemma 134 %. Some Famous Dilemmas and Sophisms. . . . 137 97. Enthymeme ....;.... 139 98. Polysyllogism 140 99. Sorites : Aristotelian, Godenian, and Pure Hypothetical 141 100. Rules of the Aristotelian and the Gocleniau Sorites . 14S 101. Expository Syllogism 146 CHAPTER XV THE PREDICABLES AND THE CATEGORIES 102. The Predicables : Genus : Specific Difference : Species : Property: Accident 148 103. Remarks on the Genus and Specific Difference . . 151 104. Proximate, Supreme, and Subaltern Genus . . .153 105. The Categories 154 106. Tree of Porphyry 155 107. Predicables Represented by Direct Universal Ideas . 155 CHAPTER XVI LOGICAL DIVISION r 108. Logical Division : Basis of Division : Dichotomy . . 157 109. Physical, Metaphysical, and Verbal Division . . . 158 110. Rules of Logical Division 159 TABLE OF CONTENTS CHAPTER XVII DEFINITION SECTION PAGE 111. Explanation of Definition 162 112. Nominal and Real Definition. 163 113. Essential, Genetic, Distinctive, Descriptive, Physical, and Causal Definition ....... 164 114. Limits of Real Definition 166 lis. Rules of Real Definition 168 CHAPTER XVIII FALLACIES 116. General Remark on Fallacies. 171 117. Fallacies in dictione and extra dictionem . . . 172 118. Equivocation 173 119. Amphiboly 174 120. Composition ......... 175 121. Division 175 122. Composition and Division 176 123. Accent : Special Pleading : Quibbling .... 178 124. Verbal Form or Figura dictionis ..... 179 125. Accident or Moral Universal 180 126. Secundum quid or Special Case : False Analogy . . 182 127. Ignoratio elenchi or Evading the Question . . . 182 128. Petitio principii or Begging the Question : Vicious Circle : Question-begging Epithet .... 187 129. Non causa pro causa or Fabricated Absurdity: False Cause or Post hoc, ergo propter hoc .... 190 130. Consequent or Non sequitur 191 131. Complex or Insinuating Question: Insinuation or Innuendo 197 xiv TABLE OF CONTENTS APPENDIX PAGE Note on Section 21 : The Copula 199 Note on Section 29 : I. The Distribution of the Predicate . 200 II. Class Mode of Reading Proposi- tions : Quantification of the Pred- icate ...... 214 Note on Section 45: Immediate Inferences. . . . 217 Note on Section 55 : The Definition of the Syllogism . . 218 Note on Section 74: The Fourth Figure . . . .221 Note on Section 80: Relative Terms in the Categorical Syl- logism .......... 223 Note on Section 93 : The Pure Hypothetical Syllogism . 225 References 227 INTRODUCTION THE DEFINITION OF LOGIC 1. Logic is the science of valid reasoning. Logic is a science, because it is a system of demon- strated truths which relate to a particular object. A speculative science is a science of which the direct aim is the ascertainment of truth. A practical science is a science of which the direct aim is the application to practice of the truths it ascertains. Like Ethics and JEsthetics, Logic is a practical science. It aims at determining the laws of valid reasoning, not for the sake of the knowledge thus acquired, but in order to aid the mind in reasoning correctly and in detecting fallacies. The material object of a science is the thing or things with which the science Is occupied, as they exist independently of the science. Thus, the earth is the material object of Geography and Geology. The formal object of a science is that aspect of the material object which is explicitly contemplated by the science. For example, the surface of the earth is the formal object of Geography, and the constitution of the earth is the formal object of Geology. The material object of Logic is reasoning, the 1 Z AN ELEMENTARY HANDBOOK OF LOGIC elements of which it is composed, and its expression in language. The formal object of Logic is the validity of the reasoning. The validity of reasoning is the logical dependence of one element in the reasoning upon the remaining element or elements (cf. 46, 54). CHAPTER I APPREHENSION AND THE IDEA 2. The mind has three cognitive acts, viz. Appre- hension, Judgment, and Reasoning. A cognitive act is an act of the mind by which some- thing is known, that is, represented or asserted. The formal object of a cognitive act is that which is explicitly represented or attained by that act. Apprehension is a cognitive act which merely repre- sents an object and does not involve in itself a mental assertion; thus, the act of the mind which represents "tree" or "gold" or "Cicero" is an apprehension. Absolute apprehension is an apprehension which has for its formal object something absolute, that is, an object apart from its relations; e.g. the apprehension of "man," "animal," "America." Comparative apprehension is an apprehension which has for its formal object a relation or an object as related to something; e.g. the apprehension of "father," "master," "similar," "thing as white as snow." Simple apprehension is an absolute or a comparative apprehension considered apart from mental assertion. The material object of apprehension is the thing or things which are apprehended, as they are in them- selves, with all their attributes or aspects, independ- ently of the mind's contemplation of them. The formal object of apprehension is that aspect of 3 4 AN ELEMENTARY HANDBOOK OF LOGIC the material object which is explicitly represented by the apprehension. It may also be defined as that aspect of the material object under which the ma- terial object is explicitly represented by the apprehen- sion. Or again, it is the material object under that aspect under which it is explicitly represented by the apprehension. Of course, the formal object of an ap- prehension need not be only one aspect ; in many cases it is two or more. Since almost every object outside the representation of the mind has hundreds of aspects, it is plain that hundreds of apprehensions (or ideas) may have one and the same material object, while their formal ob- jects are all different. Other words for aspect are Attribute, Note, Form, and the like. These words signify that which deter- mines or marks a thing so that it can be known or recognized. Apprehension may be viewed as an act of the mind or as a representation. Viewed as a representation, that is, as representing an object, apprehension is called an Idea, Notion, or Concept. 3. The comprehension of an idea is the sum-total of notes or attributes which the idea explicitly repre- sents in the object. The sum-total of notes which are not explicitly represented by the idea, but which may be determined by an analysis of the formal object of the idea, may be called the implicit comprehension of the idea. The extension of an idea is the sum-total of indi- viduals or objects which are severally represented by the idea; that is, it is the sum-total of individuals of APPREHENSION AND THE IDEA 5 which the idea can be predicated when they are taken one by one. These objects or individuals are called the Inferiors of the idea. Usually comprehension and extension vary in- versely; that is, the wider the comprehension, the narrower the extension, and vice versa; thus, the idea of "red man" has a wider comprehension, but a nar- rower extension, than the idea of "man." However, if the note which is added to the comprehension is al- ready contained in the implicit comprehension of the idea, or necessarily characterizes the formal object which is represented by the idea, the addition of the note does not narrow the extension of the idea ; thus, the idea of "mortal man" has the same extension as the idea of "man." 4. Attention is the application of the mind to some- thing. Prescission is an act of the mind by which it attends to one out of several aspects of an object without at- tending to the others. It would be still more accurate to say that prescission is an act of the mind by which it attends to an object under one of its aspects with- out attending to it under its other aspects. For ex- ample, the mind, contemplating Peter Jones, attends to the aspect "soldier" in him and does not attend to the aspects "American," "young," "handsome," or "tall." Here the mind is said to prescind from "American," "young," etc. Prescission requires that that which is prescinded shall not be really distinct from the material object. Two objects of thought are really distinct from each other, when they are physically separated or can be physically separated; b AN ELEMENTARY HANDBOOK OF LOGIC thus, the head of a man is really distinct from his shoulders. Hence, if we were to attend to the head of Peter Jones without attending to his shoulders, we should not be prescinding from his shoulders. On the other hand, a semicircle is convex when viewed from one direction, and concave when viewed from the opposite direction; yet it is not true that one part of it is concave and the other part con- vex: the whole semicircle is convex and the whole of it is concave. Hence, when the mind attends tb the aspect "concave" without attending to the aspect "con- vex," it prescinds from "convex." It would also be an act of prescission if the mind attended to two or more aspects of an object without attending to the others. In that case the mind would be said to attend to a complex aspect. The thing outside the representation of the mind which is characterized by the aspect or attribute is called the subject; the aspect is usually called a form. Abstraction is an act of the mind by which it at- tends to the aspect or form obtained by prescission and positively excludes the subject in which the form resides; e.g. "tallness," "courage." The name "abstraction" is also applied by many authors to the act of prescission. Reflection is an act of the mind by which it turns to contemplate its own acts. PsycHblogical reflection is an act of the mind by which it turns to contemplate its own acts so far as they are acts or modifications of the soul. Ontological reflection is an act of the mind by which it turns to contemplate its own acts so far as they are APPREHENSION AND THE IDEA 7 representations, that is, so far as they represent an object. It is also called ontological reflection, when the mind contemplates the formal object of a previous cognitive act for the purpose of analysis or comparison. Analysis is the act of resolving the formal object of an idea or other cognitive act into its notes or ele- ments. Synthesis is the act of combining two or more notes into the formal object of one idea. 5. A direct idea is an idea which represents some- thing outside the mind ; e.g. the idea of a horse. A reflex idea is an idea which represents something inside the mind; e.g. the idea of an abstraction. 6. A clear idea is an idea which distinguishes an object from other objects ; e.g. the idea of "tree." The opposite of a clear idea is an obscure or vague idea. A distinct idea is an idea which not only distin- guishes an object from other objects, but also distin- guishes two or more notes or aspects of the object; e.g. the idea of "tall pine tree." The opposite of a distinct idea is a confused idea. Every idea is a clear idea as far as it goes. An idea is called obscure only by comparison with another idea which represents the same object more clearly. A comprehensive or adequate idea is an idea which represents explicitly all there is to be known about an object. Only a being of infinite intelligence can have a comprehensive idea of anything. 7. A concrete idea is an idea which represents. the form along with the subject; e.g. the idea of "man" or "white (horse)." 8 AN ELEMENTARY HANDBOOK OF LOGIC An abstract idea is an idea which represents the form without the subject; that is, it represents the form as standing by itself ; e.g. the idea of "humanity" or "whiteness" or "rashness." The abstract idea is the result of abstraction. 8. A singular or individual idea is an idea which represents one determinate object or certain determi- nate objects; e.g. the idea of "Plato" or "this man" or "these horses." A universal idea is an idea which represents sev- erally many individual objects, and hence it can be predicated of each of them; e.g. the idea of "man" or "king" or "dog." The objects represented by a universal idea are called the Inferiors of the idea. . It is to be observed that the universal idea rep- resents severally many objects, whether the mind adverts to those objects or not. The mind may at- tend to the one or more notes or attributes which the idea represents without attending to the various indi- viduals which possess these notes or attributes. When a universal idea is used in a judgment, the mind some- times adverts to the individuals which are represented by it, that is, to the extension of the idea, and some- times it does not. When the mind attends separately to the objects in the extension of the universal idea, it uses the idea distributively. When it does not at- tend to the extension of the idea, it uses the idea absolutely. The distributive use of an idea is com- mop in th.e subject of a judgment ; the absolute use, in the predicate ; e.g. "Every man is rational." In the distributive use of a universal idea, the mind APPREHENSION AND THE IDEA \J sometimes adverts to all the individuals in the exten- sion of the idea, as in the subject of the foregoing example; sometimes it adverts to an indeterminate individual or to an indeterminate number of the indi- viduals in the extension, as in the example following : "Some men are wise." "Some men" in this proposi- tion is by certain authors called the expression of a particular idea. A transcendental idea is an idea which represents sev- erally all objects whatever, and hence it can be predi- cated of each of them ; e.g. the idea of "being" or "one." A collective idea is an idea which represents a num- ber of similar individuals as constituting one whole; e.g. the idea of "army" or "cavalry" or "senate." A collective idea ihay be either singular or universal; e.g. the idea of "this army," the idea of "army." 9. Incompatible ideas are ideas whose formal objects cannot co-exist in the same respect or in the same part in one individual; e.g. the ideas of "hot" and "cold," of "first" and "second." The idea which represents a form or an object as having that form is called a positive idea; e.g. the idea of "combatant." The idea which represents the absence of a form or an object as lacking that form is called a negative idea; e.g. the idea of "non-combatant." Privation is the absence or negation of a form which is found in a thing when the thing is in its normal condition or which a thing is fitted to possess; e.g. "blindness." Every privation is a negation, but not every negation is a privation. Lack of sight is a pri- vation In a man, but a mere negation in a tree. Contradictory ideas are a pair of ideas one of which 10 AN ELEMENTARY HANDBOOK OF LOGIC represents a form, and the other, the simple absence of that form ; or they are a pair of ideas one of which rep- resents an object as having a certain form, and the other, an object as simply lacking that form; e.g. the ideas of "combatant" and "non-combatant," of "met- allic" and "non-metallic." Contradictory ideas are also called complementary ideas, because between them they comprise all things whatsoever. Contrary ideas are a pair of ideas representing forms which in a given respect are at the extremes of opposition to each other; e.g. the ideas of "hot" and "cold" in respect to temperature, of "first" and "last" in respect to order. Relative ideas are ideas representing objects so far as they are related to each other; e.g. the ideas of "father" and "son," of "Creator" and "creature." Here we have spoken of relative ideas, that is, of at least a pair of ideas. There is also a relative or comparative idea, which is the same as a comparative apprehension (cf. 2) ; such an idea represents a relation or an object as related to something; e.g. the idea of "brother," "equal," "thing larger than a man." Contradictory, contrary, and relative ideas are all incompatible ideas; but there are some incompatible ideas which do not fall under any of these three heads ; e.g. the ideas of "first" and "second," of "cold" and "lukewarm." It is, however, not unusual to classify such ideas under contrary ideas. Disparate ideas are ideas representing forms which are not opposed to each other and are not necessarily related to each other; e.g. the ideas of "holy" and "learned," of "hot" and "yellow." CHAPTER II THE TERM 10. A sign is anything from which or by which something beyond itself is known; e.g." smoke is a sign of fire; breathing, of life; a footprint, of an animal. Signification is the connection between the sign and the thing signified. A natural sign is one whose signification comes from nature; e.g. smoke, a groan. An arbitrary sign is one whose signification depends on convention, that is, on agreement between men ; thus, the palm is a sign of victory. Language, in general, is a natural sign; but any given word is an arbitrary sign. 11. A word is a vocal sound uttered by a man and having a signification from the free convention of men. "Word" is also applied to the letter or letters which are used to represent a word. A categorematic word is a word which by itself has a determinate signification; e.g. "tiger," "red," "humanity." A syncategorematic word is a word which has a determinate signification only when used along with another word ; e.g. "every," "as," "from," "by." Categorematic words are substantives and adjectives 11 ' 12 AN ELEMENTARY HANDBOOK OF LOGIC and such words as may be used as substantives or ad- jectives, and finally, pronouns in the nominative case. In Logic all categorematic words are called names or terms. 12. A term is the verbal expression of an idea. A simple term is a term consisting of one word ; e.g. "animal." A complex term is a term consisting of several words; e.g. "rational animal." Of the words composing the complex term one is called the principal term, and the other word or words, the incident term. The principal term is the term which denotes the subject of the form; thus, "animal" in the foregoing example. The incident term is the term which denotes the form that is in the subject; e.g. "rational" in the same example. The incident term is either explicative or restrictive. An explicative or explanatory term is a term which denotes something that is found in the whole exten- sion of the idea expressed by the principal term; e.g. "mortal man." A restrictive term is a term which denotes some- thing that is found in only part of the extension of the idea expressed by the principal term, and hence it restricts that term ; e.g. "learned man." 13. A term is concrete or abstract according as it expresses a concrete or an abstract idea (cf. 7). 14. An absolute term is a term which expresses an THE TERM 13 absolute idea; that is, it is a term which denotes an object without referring it to another object; e.g. "tree," "table." A relative term is a term which expresses a relative idea; that is, it is a term which denotes a relation or an object as related to another object; e.g. "father," "king," "thing stronger than iron." The other object is called the correlative (cf. 2, 9). 15. A singular term is a term which denotes one determinate object "or certain determinate objects; e.g. "Plato," "this house," "these men." A collective term is a term which denotes a number of similar objects taken together as constituting one whole; e.g. "group," "army." A common term is a term which denotes a number of objects taken separately; e.g. "lion," "pillar." 16. A univocal term is a common term which ex- presses only one idea, and hence is applied severally to many objects in the same sense; e.g. "giraffe." An equivocal term is a common term which ex- presses two or more ideas, and hence is applied to different objects in a different sense; e.g. "bow" ap- plied to a nod of the head and to the forward part of a ship. For the logician a word employed in two senses in the same argument is equivalently two terms. A univocal term is also called a general term. A distributed term is a general term which refers explicitly to each of the objects in the extension of a universal idea; e.g. "all men," "every tree." An undistributed term is a general term which refers explicitly to an indeterminate object or to each of 14 AN ELEMENTARY HANDBOOK OF LOGIC an indeterminate number of the objects in the exten- sion of a universal idea; e.g. "some tree," "some men." 17. The connotation of a term is the sum-total of notes or attributes in an object which are conventionally sig- nified by the term. Other names for connotation are Comprehension, Intension, and Implication (cf. 3). The denotation of a term is the extension of the idea which it expresses; that is, it is the sum-total of objects to which the term can be severally applied in the same sense. Denotation is also called Extension and Application. SUPPOSITION OF TERMS 18. The supposition of a term is the use of a term in a proposition. Material supposition is the use of a term without regard to what it denotes; e.g. "Man is a word of three letters." Formal supposition is the use of a term to denote something ; e.g. "Man is mortal." Formal supposition is logical or real. Logical supposition is the use of a term to denote an object as it is in the representation of the mind; e.g. "Man is a universal idea;" "Man is a species." Real supposition is the use of a term to denote an object as it is outside the representation of the mind; e.g. "Man is rational." Real supposition is singular, absolute, or general. Singular supposition is the use of a term to denote a definite individual or certain definite individuals; e.g. "This man is learned ;" "Those men are upright." SUPPOSITION OF TERMS 15 Absolute supposition is the use of a general term to denote a form or attribute considered in itself; e.g. "Man is mortal." This supposition regards only the comprehension of the idea expressed by the term and leaves out of account the extension. It is the kind of supposition which is usual in the predicate of a propo- sition when the predicate is a general term (cf. 8). General supposition is the use of a general term to denote a form or attribute considered as existing in an indeterminate object or in a number of objects; e.g. "All men are mortal." This supposition regards both comprehension and extension. General supposition is collective or distributive. Collective supposition is the use of a general term for the objects severally denoted by it taken together; e.g.. "All the angles of .a triangle are equal to two right angles"=:"Angles A and B and C together are equal to two right angles." Definite collective supposition is the use of a gen- eral term for all the objects severally denoted by it taken together ; e.g. "All the Apostles are twelve." Indefinite collective supposition is the use of a gen- eral term for an indeterminate number of the objects severally denoted by it taken together; e.g. "Some soldiers built the hut ;" "Many m.osquitoes (together) weigh a pound." N. B. — Collective supposition must be carefully dis- tinguished from the collective term (cf. 15, 8). Distributive supposition is the use of a general term for the objects denoted by it taken separately; e.g. "Every man is mortar'="This man is mortal, and that 16 AN ELEMENTARY HANDBOOK OF LOGIC man is mortal, and that other man is mortal," and so on (cf. 8). Distributive supposition is universal or particular. Universal supposition is the use of a general term for each and every object denoted by it; e.g. "Every man is mortal." There is a supposition, called incomplete universal supposition, which is sometimes mentioned in works on Logic. This supposition is the use of a general term, not for every object denoted by it, but for a specimen of every kind of object denoted by it; e.g. "Every animal was in Noah's Ark." Particular supposition is the distributive use of a general term indeterminately for one or a number of the objects denoted by it; e.g. "Some American was chosen ;" "Some men are wise." . Particular supposition is disjunctive or confused (vague). Disjunctive supposition is the distributive use of a general term indeterminately for one or a number of the objects denoted by it in such a way that what is asserted can be verified in at least one individual ob- ject taken by itself; e.g. "Some Apostle was a traitor." This supposition is called disjunctive, because the proposition in which it occurs is resolvable into a disjunctive proposition (cf. 51). Thus, the example we have just used can be resolved as follows : "Either Peter was a traitor, or John was a traitor, or Judas was a traitor," etc. In disjunctive supposition what is asserted must be verified in at least one individual object which in itself is fixed and determined, though it is not determined by the term. SUPPOSITION OF TERMS 17 Confused or vague supposition is the distributive use of a general term indeterminately for one or a number of the objects denoted by it in such a way that what is asserted cannot be verified in any individual object taken by itself ; e.g. "Some eye is requisite for seeing;" "Some hoat is necessary for sailing.'' The proposition in which confused supposition occurs can- not be resolved into a disjunctive proposition. Thus, we cannot say, "Either the right eye is requisite for seeing, or the left eye is requisite for seeing;" but we may say, "Either the right or the left eye is requisite for seeing;" or we may say, "Some eye or other is requisite for seeing." Similarly, we may say, "Either the affirmative or the negative side is sure to win;" but we cannot resolve this into a disjunctive proposi- tion. When confused supposition occurs, not only is the individual object not determined by the. term, but it is not determined in itself. In confused supposition what is asserted can be verified determinately in one object, only in case all the other objects are lacking. Divided supposition is the use of a term for some- thing which is in existence at a different time from that indicated by the verb; e.g. "The blind see;" "The deaf hear,"="Those who were blind now see," etc. Such propositions are said to be false in sensu com- posito (that is, when the form of the subject and that of the predicate are considered as combined), and true in sensu diviso (that is, when the forms are considered as divided and as existing at different times). CHAPTER III JUDGMENT 19. Judgment is an act of the mind asserting that in the world of reality the formal objects of two ideas are one and the same thing or that they are different (that is, distinct) things (cf.4) ; or more briefly, but less accurately, it is the mental assertion of the objective identity or diversity of two ideas. Judgment may be more loosely defined as an act of the mind asserting that the object represented by one idea possesses or lacks the attribute represented by another idea. N. B.- — The meaning of the first part of the defini- tion of judgment is not that the mind asserts that the formal object of one idea is the formal object of an- other idea, but that the thing represented under one aspect by one idea is one and the same with the thing represented under another aspect by another idea (cf. 2). In section 2 we saw that the formal object of a cognitive act is that which is explicitly represented or attained by that act ; hence— The formal object or form of a judgment is the ob- jective identity or diversity of two ideas. More accu- rately, it is the identity or diversity (in the world of reality) of the formal objects of two ideas. ' The material object or matter of a judgment are two ideas, or rather, the formal objects of two ideas. 18 JUDGMENT 19 The world of reality is the sum-total of actual and possible objects of thought which do not depend upon the mind's thought about them for being what they are, whether in themselves or in their relations. The world of reality embraces all objects which are outside the representation of the mind and all objects which have a foundation outside the representation of the mind. A reality is anything that is independent of the mind's thought about it for being what it is. An unreality is anything that is dependent upon the mind's thought about it for being what it is. An affirmative judgment is a judgment which asserts the objective identity of two ideas; e.g. "Gold is yel- low." An affirmative judgment is also called an affirmation, and is said to affirm something. A negative judgment is a judgment which asserts the objective diversity of two ideas ; e.g. "The horse is not rational." A negative judgment is also called a negation, and is said to deny something. The subject of a judgment is the idea, or rather, the object of which something is affirmed or denied; for instance, "gold" and "horse" in the two foregoing examples. The predicate of a judgment is that which is affirmed or denied of the subject; for instance, "yellow" and "rational" in the same examples. The subject or predicate of a judgment may be either an idea representing an object absolutely or an idea representing an object as related to something else ; that is, it may be either an absolute or a relative idea. For example, it may be "sentient thing" or it 20 AN ELEMENTARY HANDBOOK OF LOGIC may be "thing smaller than a bird." Thus, we may say, "The ant is a sentient thing," and "The ant is a thing smaller than a bird," or more briefly, "The ant is smaller than a bird." In the second judgment we assert an objective identity between "ant" and "thing smaller than a bird." "Smaller than a bird" is a rela- tive attribute which is predicated of the ant. Every judgment is preceded not only by two simple apprehensions by which the mind acquires the two ideas which constitute the matter of the judgment; it is also preceded by a comparative apprehension by which the mind apprehends (perceives) the objective identity or diversity -of those two ideas. This com- parative apprehension is called Complex Apprehen- sion. If the mind, upon comparing together the formal objects of these ideas, does not apprehend their iden- tity or diversity, then there is comparison, but no com- plex apprehension ; and consequently, unless the mind is influenced by the will, there can be no judgment in regard to the objects as compared, but only a state of doubt. 20. An a priori judgment is a judgment which as- serts an objective identity or diversity perceived by means of a mere comparison of the formal objects of two ideas either with each other or with the formal object of another idea; e.g. "The whole is greater than any of its parts;" "The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the two other sides;" "The triangle is not round." An h. posteriori judgment is a judgment which as- serts an objective identity or diversity perceived by JUDGMENT 21 means of experience in addition to comparison; e.g. "This iron is hot;" "The earth moves round the sun;" "The moon is not self-luminous." An a priori judgment is also called analytical, be- cause by analysis of the subject we find the predicate contained in it or excluded from it; pure, because we find the identity or diversity of the formal objects of the two ideas without recurring to experience ; neces- sary, because the identity or diversity is necessary and cannot under any circumstances be lacking. An o posteriori judgment, for opposite reasons, is called synthetic, experimental (or empirical), and con- tingent. The matter of an a priori judgment is called neces- sary matter. The matter of an a posteriori judgment is called contingent matter. An immediate judgment is a judgment which as- serts an objective identity or diversity perceived with- out recourse to proof; e.g. "The whole is greater than any of its parts;" "The triangle is not round;" "This iron is hot." A mediate judgment is a judgment which asserts an objective identity or diversity perceived by means of proof; e.g. "The square of the hypotenuse of a right- angled triangle is equal to the sum of the squares of the two other sides;" "The earth moves round the sun." CHAPTER IV THE CATEGORICAL PROPOSITION 21. The categorical proposition is the verbal expres- sion of the matter and form of a judgment. It consists of two terms and the copula. The matter of a judgment, that is, the formal objects of the two ideas which enter into it, is expressed by the two terms which stand in the positions of subject and predicate. Accordingly, the two terms which stand as subject and predicate are called the matter of the categorical proposition. The form of a judgment, that is, the objective iden- tity or diversity of the two ideas, is expressed by the copula. Hence, the copula is called the form of the categorical proposition. The copula is always "is" or "is not" or some other part of the present tense, indica- tive mood, of the verb "to be." Cf. Appendix: Note on Section 21. Care must be taken not to confuse the logical predi- cate of a proposition with the grammatical predicate. The grammatical predicate is a finite verb; e.g. "The bird sings.'' Here the grammatical predicate is "sings." In order to throw this proposition into logical shape as the formal expression of the judgment, we must write it as follows: "The bird is singing" (cf. p. 46). A categorical proposition is said to be in logical form, when it expresses formally and sepa- 22 QUALITY OF CATEGORICAL PROPOSITIONS 23 rately the subject, the copula, and the predicate. It will be well for the beginner in Logic to set down in logical form 'all the propositions with which he has to deal ; repeated practice of this exercise will enable him to detect at once the matter and form of the judg- ments which are expressed by the propositions. Once this facility is acquired, there will be no need to insist upon the full logical form of the proposition, and he may say indifferently, even in argument, "The bird sings" or "The bird is singing." For purposes of rhetorical effect, the predicate is sometimes written first ; for example, "Great is Diana of the Ephesians." Again, we sometimes find a judg- ment expressed by a single word ; for example, "Dinner!" which means "Dinner is prepared." It is to be observed that a term, whether it be simple or complex (cf. 12), is nevertheless a single term. No matter what be the number of words in a complex term, the term itself is one and expresses only one idea. Hence, the subject, copula, and predicate of a proposition may consist of only three words, as in the proposition, "Man is rational," or they may consist of many more, as in the proposition, "The learned econo- mist who was here last Saturday has met with a seri- ous accident." QUALITY OF CATEGORICAL PROPOSITIONS 22. The quality of a categorical proposition is the character of the copula. An affirmative proposition is a categorical proposi- tion in which the copula consists either formally or 24 AN ELEMENTARY HANDBOOK OF LOGIC elliptically of the verb "is;" e.g. "Gold is yellow;" "The King rules England." The affirmative proposition is the verbal expression of the matter and form of an affirmative judgment. A negative proposition is a categorical proposition in which the copula consists either formally or elliptic- ally of the verb "is" modified by a negative particle ; e.g. "A feather is not heavy ;" "The horse does not fly." The negative proposition is the verbal expression of the matter and form of a negative judgment. If the negative particle, instead of modifying the copula, immediately modifies the subject or the predi- cate, the proposition is affirmative; e.g. "These men are non-combatants;" "Non-members oppose the or- ganization;" "Those who are not of age are exempt." In an affirmative proposition the subject and predi- cate stand for the same thing, but in a different way; that is, the subject stands for the thing which has the form signified by the predicate, and the predicate stands for the thing as having this form. In a negative proposition the subject and predicate stand for two different or distinct things. QUANTITY OF CATEGORICAL PROPOSITIONS 23. The quantity of a term is the completeness with which the term refers severally to the individuals comprised in the extension of the universal idea ex- pressed by it. According as the term refers severally, on the one hand, to all the objects, or, on the other, to an indeterminate object or an indeterminate num- ber of the objects in the extension of a universal idea^ QUANTITY OF CATEGORICAL PROPOSITIONS 25 it is said to be distributed or undistributed (cf. 16, 8). The quantity of a categorical proposition is the quantity of the subject-term. A universal proposition is a categorical proposition which has a distributed term for its subject; that is, it is a categorical proposition in which the predicate is affirmed or denied of each and every individual com- prised under the extension of the subject-idea; e.g. "Every just man is deserving of praise;" "No horse is rational." N. B. — A proposition is called universal, not only when the subject is a distributed term, but often also when the subject is a general term which is used ab- solutely and the predicate, too, is a general term (cf. 18) ; e.g. "Man is mort^al;" "The circle is round." It is plain that the subject of each of these propositions can be changed into a distributed term on demand. The reason is, that "man" and "circle" are formal objects of universal ideas (cf. 8), and hence, what is true of "man" and "circle" without qualification or restriction is true of "every man" and "every circle." A particular proposition is a categorical proposition which has an undistributed term for its subject; that is, it is a categorical proposition in which the predi- cate is affirmed or denied severally of an indetermi- nate individual or ah indeterminate number of the in- dividuals comprised under the extension of the subject- idea ; e.g. "Some men are truthful ;" " "Some men are not strong." The particular proposition is also called an Indefinite Proposition. A singular proposition is a categorical proposition which has a singular term for its subject; that is, it is 26 AN ELEMENTARY HANDBOOK OF LOGIC a categorical proposition in which the predicate is affirmed or denied of one determinate individual or of certain determinate individuals; e.g. "Cicero is famous;" "This dog is mad;" "These stones are not valuable." An indesignate proposition is a categorical proposi- tion virhich has for its subject a general term with no sign of quantity accompanying it; e.g. "Man is rational;" "Frenchmen are polite;" "Crows are not white." An absolutely universal proposition is a proposition which states something that holds good of the full extension of the subject-idea; e.g. "Every circle is round ;" "No triangle is four-sided." A morally universal proposition is a proposition which states something that holds good of only the greater part of the extension of the subject-idea; e.g. "All old people praise past times;" "No mother hates her child." For the logician a morally universal proposition is a particular proposition. When he speaks of universal propositions, he means propositions which are abso- lutely universal. 24. Whether an indesignate proposition is absolutely or morally universal is to be determined by the judg- ment which it is intended to express. When an in- designate proposition, which is only morally universal, is put forth in argument as absolutely universal, it is to be denied. When the subject-term is used collectively, that is, when we have collective supposition, the proposition is frequently, though by no means always, equivalent to a QUANTITY OF CATEGORICAL PROPOSITIONS 27 singular proposition; e.g. "All the Apostles are twelve." When the subject of a proposition is a collective term, the proposition may be either universal, particu- lar, or singular. So far as its use in argument goes, the singular proposition is equivalent to a universal proposition; because in it the predicate is affirmed or denied of the full extension of the subject-idea, though this exten- sion is only one individual or certain definite indi- viduals. 25. Signs of Quantity. In an affirmative proposition the sign "every" or "each," prefixed to the subject, makes the proposition universal. The sign "all" generally does so, but not always ; for it is some- times used in collective supposition ; e.g. "All the angles of a triangle are equal to two right angles" (cf. 18). The sign "every" should be used instead of "all" when the latter leaves an opening for ambiguity. In Logic we shall, for the sake of convenience, employ the formula "All S is P" for the universal affirmative proposition. In a negative proposition "every," "each," and "all," prefixed to the subject, are equivalent to the sign "some," and they make the proposition particular; e.g. "All men are not wealthy" or "Not all men are wealthy" = "Some men are not wealthy." The usual sign of universality in a negative prop- osition is "no" prefixed to the subject; e.g. "No horse is rational." This does not conflict with the definition of negative proposition in section 22 ; for this proposi- tion is the same as the following: "Horses are not rational." 28 AN ELEMENTARY HANDBOOK OF LOGIC "Some" is the sign usually prefixed to the subject to make the proposition particular. In Logic "some" has the force of "one at least" or "a certain (or uncertain) number of." Hence, "some" in Logic is not inconsist- ent with "all ;" for example, the following propositions afe not inconsistent with each other: "All men are mortal," "Some men are mortal." A proposition beginning with "whoever," "what- ever," "he who," "those who," "that which," or an equivalent expression, is universal. The sign "a few" makes the proposition particular, but does not of itself affect the quality of the proposi- tion. The sign "few"' commonly has the same force as "only a few;" and the proposition in which it occurs can usually be resolved into two propositions, one affirmative, and the other negative ; e.g. "Few lawyers achieve world-wide fame" = "A few (or some) law- yers achieve world-wide fame" and "Most (or more) lawyers do not achieve world-wide fame." What has been said of "few" is also true of the signs "hardly any" and "scarcely any." N. B. — The sign of quantity is not part of the term. 26. Relation of the Comprehension of the Predicate to that of the Subject. In an affirmative proposition we assert that all the notes in the explicit and implicit com- prehension of the predicate, whether taken separately or collectively, 3^rtio\xaA in the object denoted by the subject. The reason is, that in an affirmative proposition we assert that the object denoted by the subject is (that is, is identical with) the object denoted by the predicate, and COMPREHENSION OF THE PREDICATE 29 hence, the object denoted by the subject has each and all the notes in the explicit and implicit comprehension of the predicate. For example, "Man is an animal :" "man" has each and all the notes that make up the explicit and implicit comprehension of "animal," e.g. sentient, organic, corporeal, substance. In a negative proposition we assert that the notes in the comprehension of the predicate, taken collectively, are not found in the object denoted by the subject; but we do not assert that no portion of those notes is found in that object. The reason is, that in a nega- tive proposition we assert that the objects denoted by the subj.ect and predicate are different or distinct things; and the possession by the predicate-object of one note which is not found in the object denoted by the subject constitutes the predicate-object a different thing from the object denoted by the subject. When we say "A man is not a horse," we mean the collection of notes which make up the comprehension of "horse" (e.g. equine, sentient, organic, corporeal, substance) is not found in "man;" we do not mean that he pos- sesses none of these notes. 27. A natural proposition is either an affirmative proposition in which the idea expressed by the predi- cate has an extension at least as wide as that of the idea expressed by the subject, or it is a negative propo- sition ; e.g. "All men are mortal ;" "All men are rational animals;" "Some men are white;" "Some men are not lawyers." An unnatural proposition is an affirmative proposi- tion in which the idea expressed by the predicate has a narrower extension than that of the idea expressed 30 AN ELEMENTARY HANDBOOK OF LOGIC by the subject; e.g. "Some mortal beings are men;" "Some animals are horses." 28. In Logic the two first vowels of the La.tin word "affirmo" (■= "I affirm") are used to denote respect- ively the universal affirmative and the particular af- firmative proposition; the two vowels of the word "nego"( = "I deny") denote respectively the univer- sal negative and the particular negative . proposition. Thus, we have the four typical propositions symbol- ized as, follows: A = Universal affirmative E = Universal negative I = Particular affirmative O = Particular negative DISTRIBUTION OF THE PREDICATE 29. In works on Logic the predicate of an affirma- tive proposition is usually set down as undistributed, and the predicate of a negative proposition, as distrib- uted. Thus, according to this doctrine we should have the following table : In A, the subject is distributed, the predicate, undistributed In E, " " " distributed, " " distributed In I, " " " undistributed, " " undistributed In O, " " " undistributed, " " distributed The hypothesis of the distribution of the predicate was devised in order to simplify the explanation of Conversion and of the Categorical Syllogism. But when we come to the Categorical Syllogism, we shall find that the explanation is much easier and simpler if this hypothesis is discarded. Our main reason for CONVERSION OF AFFIRMATIVE PROPOSITIONS 31 not adopting it will be explained in the Appendix (cf. Note on Section 29). For the present, we shall at- tempt to show, without recourse to this hypothesis, where the process of Conversion is possible and what its result is "in the case of each proposition. 30. Conversion of Affirmative Propositions. In an affirmative proposition we assert that the object de- noted by the subject and the object denoted by the predicate are one and the same thing, that the subject- object has the attribute signified by the predicate; but we do not advert to the question whether there are any other objects which have this attribute. Whether any object besides the subject-object, possesses this attribute, we cannot learn from the proposition; to find this out, we must appeal to some other source of information. "Every man is an animal," "Every man is a rational animal :" — prior or sub- sequent to the utterance of these propositions we may know whether or not "animal" and "rational ani- mal" are co-extensive in application with "man ;" but the proposition itself does not decide the question for us. Hence, if we wish to make the subject and predi- cate change places, it is not the proposition itself, but information secured elsewhere, which will determine what sign of quantity we are objectively warranted in prefixing to the proposition. Since "Every man is an animal," we know by reflection that there are at least as many objects with the attribute signified by "ani- mal" as there are men, and that each of these objects has the attribute signified by "man ;" hence, we are warranted in predicating "man" of each of these ob- jects. But since we do not know from the proposition 32 AN FXEMENTARY HANDBOOK OF LOGIC whether or not there are any objects besides men which possess the attribute signified by "animal," we cannot say "Every animal is a man," but only "Some animals are men." What has been said of the proposition, "Every man is an animal," can be applied mutatis mutandis to the particular affirmative proposition, "Some horses are black." Our rule, then, for the conversion of affirma- tive propositions is that the subject of the converse must have the particular sign of c[uantity. Therefore, the converse of "All S is P" and "Some S is P" is "Some P is S." 31. Conversion of Negative Propositions. In Si nega- tive proposition we assert that the object denoted by the subject and the object denoted by the predicate are dififerent (distinct) things, that the subject- object lacks the attribute signified by the predicate. In the universal negative proposition, "No horse is rational," we assert that the horse lacks the attribute "rational." Hence, by reflection we know that any object with the attribute "rational" lacks the attribute signified by "horse." Consequently, if we wish to make the subject and predicate change places, we are warranted by information furnished by the proposition itself in prefixing the universal sign of quantity to the converse; thus, "No rational being is a horse." For this reason, the proposition, "No S is P," may always be changed into "No P is S." The particular negative proposition, "Some swans are not white," cannot be changed into "No white things are swans;" for this could again be changed COMPOUND CATEGORICAL PROPOSITIONS 33 into "No swans are white," which asserts more than the original proposition, and hence is not one of its implications. It cannot be changed into "Some white things are not swans;" for this supposes information as to the number of white things which cannot be ob- tained from the original proposition : so far as we know from the original proposition, all the white things may be swans. This would be still more evi- dent, if we attempted to convert the proposition, "Some animals are not horses." Hence, the particular negative proposition, "Some S is not P," cannot be converted. COMPOUND CATEGORICAL PROPOSITIONS 32. A simple categorical proposition is a proposi- tion which expresses the matter and form of only one judgment, and hence has only one subject and one predicate. A compound categorical proposition is a proposi- tion which expresses the matter and form of two or more judgments, and hence has several subjects or several predicates. It is either formal or elliptical. 33. A formal compound categorical proposition is a proposition in which the several judgments expressed are evident from the structure of the proposition. A conjunctive categorical proposition is a proposi- tion in which there are several subjects or several predicates connected by the particle "and" or an equivr alent, expressed or understood; e.g. "All lawyers and physicians are professional men." This proposition is equivaleijt to the two following simple propositions : 34 AN ELEMENTARY HANDBOOK OF LOGIC "All lawyers are professional men," "All physicians are professional men." The foregoing compound prop- osition may also be written in the form, "Any one who is either a lawyer or a physician is a professional man." A remotive proposition is a negative proposition in which there are several subjects or several predicates connected by "neither . . . nor" or equivalent particles ; e.g. "The horse is neither horned nor cloven- footed" = "The horse is not horned" and "The horse is not cloven-footed;" "Neither aliens nor minors nor criminals are voters" = "Aliens are not voters," "Minors are not voters," and "Criminals are not voters." These two examples may also be written as follows : "No horse is either horned or cloven-footed," "No one who is either an alien, or a minor, or a criminal is a voter." An adversative categorical proposition is a proposi- tion in which there are several subjects or several predicates connected by the particle "but" or an equiv- alent ; e.g. "Cicero was not a great general, but a great orator ; "Not Paul, but Peter, was Bishop of Rome." 34. An elliptical compound categorical proposition is a proposition in which the several judgments ex- pressed are not evident from the structure of the proposition, but become evident when the proposition is resolved. This kind of proposition is also called an Exponible Proposition. An exclusive proposition is a proposition which, by means of a word such as "only," "alone," "none but," excludes what is asserted from everything except the subject ; e.g. "Only graduates are eligible" (or "Gradu- ates are the only eligible persons") ; "None but Sdhiors COMPOUND CATEGORICAL PROPOSITIONS 35 "were present." The principal judgment expressed by these propositions is generally universal negative, viz. "No non-graduates are eligible;" "No non-Seniors were present." The other judgment is affirmative, usually particular, viz. "Some graduates are eligible;" "Some Seniors were present." An exceptive proposition is a proposition which, by means of a word like "except", excludes what is as- serted from one or more of the inferiors of the subject- idea (cf. 3, 8) ; e.g. "All animals except men are ir- rational" = "Men are not irrational," "All other ani- mals are irrational." A comparative proposition is a proposition which affirms or denies that an attribute belongs to the sub- ject in the same degree as it does to something else or in a greater or a less degree; e.g. "Philosophy is more important than eloquence." This proposition, in addition to the judgment formally expressed, involves the two following judgments: "Philosophy is impor- tant," "Eloquence is important." An inceptive proposition is a proposition which makes an assertion as to the commencement of some- thing; e.g. "Wilson became President on March 4, 1913." This proposition is equivalent to the following : "Wilson was not President before March 4, 1913" and "Wilson was President on and after March 4, 1913." A desitive proposition is a proposition which makes an assertion as to the ending of something : e.g. "Taft ceased to be President on March 4, 1913." This prop- osition is equivalent to the following: "Taft was President on and before March 4, 1913" and "Taft was not President after March 4, 1913." 36 AN ELEMENTARY HANDBOOK OF LOGIC Reduplicative and specificative propositions are propositions which contain an iterative particle such as "inasmuch as," "in so far as," "as such," or "as," the force of which is to indicate in what way or in what sense the predicate belongs to the subject. If the particle indicates the form according to which the predicate must belong to the subject, the proposition is reduplicative; e.g. "Man, inasmuch as he is intelli- gent, is free" = "Man, because he is intelligent, is free." If the particle merely indicates the form ac- cording to which the predicate does or can belong to the subject, the proposition is specificative; e.g. "The physician, so far as he is a man, reads and speaks" = "The physician reads and speaks, and the form ac- cording to which he does or can read and speak is his humanity." MODAL CATEGORICAL PROPOSITIONS 35. An assertoric categorical proposition is a propo- sition which asserts the fact of the objective identity or diversity of two ideas. The propositions we have been dealing with thus far are mostly assertoric. A modal categorical proposition is a proposition which asserts the necessity or the possibility of the objective identity or diversity of two given ideas. Modal propositions are divided into apodeictic and problematic. An apodeictic proposition is a modal proposition which asserts the necessity of the objective identity or diversity of two given ideas; e.g. "That a circle should be round is necessary;" "That a square should MODAL CATEGORICAL PROPOSITIONS 37 not be round is necessary." To say that the objective diversity of two ideas is necessary is equivalent to say- ing that the objective identity of those two ideas is impossible. For this reason, the second of the two examples is usually written, "That a square should be round is impossible." The matter of an apodeictic proposition is called necessary matter. A problematic proposition is a modal proposition which asserts the possibility of the objective identity or diversity of two given ideas ; e.g. "That a king should be wise is possible;" "That the weather should not be fair is possible." The word "contingent" is used in Logic in the sense of "not necessary." To say that the objective diversity of two ideas is possible is equivalent to saying that the objective identity of those two ideas is contingent. P'or this reason, it is cus- tomary to write the second of the two examples as follows: "That the weather should be fair is con- tingent." The matter of a problematic proposition is called contingent matter. The objective identity of the two given ideas, whether it be necessary, impossible, possible, or con- tingent, is called the dictum, when expressed as the subject of the modal proposition; the predicate is called the mode. Since the predicate is one or other of the four terms, "necessary," "impossible," "pos- sible," "contingent," there are four modes. Since the modal proposition makes an assertion about the objective identity or diversity of two given ideas, and since the objective identity or diversity is 38 AN ELEMENTARY HANDBOOK OF LOGIC expressed by the copula, the mode may be expressed by a verb which immediately modifies the copula. Thus, the four propositions we have employed in illustration may be worded as follows: "A circle must be round" (necessary) ; "A square cannot be round" (impos- sible) ; "A king may be wise" (possible) ; "The weather need not be fair" (contingent). OPPOSITE CATEGORICAL PROPOSITIONS 36. Opposite categorical propositions are categorical propositions which are identical in matter but differ- ent in form. They are illustrated in the following Square of Opposition : Contraries A E All soldiers No soldiers" are' brave are brave 1 ^^"^%^ C^^^ "I 3 on i,^^-;""^ ^~~~~---.o m Some soldiers Some soldiers are brave are not brave Subcontraries Contradictory propositions are two opposite prop- ositions one of which is universal and the other par- ticular; e.g. "All soldiers are brave" — "Some soldiers are not brave;" "No soldiers are brave" — "Some sol- diers are brave." In each of these pairs one proposi- OPPOSITE CATEGORICAL PROPOSITIONS 39 tion affirms or denies jus-t enough to make the other false. Contradictory propositions cannot both be true nor both be false at the same time; but one is true and the other false. Hence, from the truth of one we can infer the falsity of the other; and from the falsity of one we can infer the truth of the other. In order to disprove a given proposition, it is neces- sary and sufficient to prove its contradictory. Contrary propositions are two opposite propositions both of which are universal ; e.g. "All soldiers are brave'' — "No soldiers are brave." One of the proposi- tions affirms or denies more than is necessary to make the other false. Contrary propositions cannot both be true, but both may be false, at the same time. Hence, from the truth of one we may infer the falsity of the other; but from the falsity of one we cannot infer the truth of the other. To refute a universal proposition, it is not necessary to prove its contrary; but it is frequently of great ad- vantage, if it can be done ; for then the refutation is overwhelming and manifest to everyone. Subcontrary propositions are two opposite proposi- tions both of which are particular; e.g. "Some soldiers are brave" — "Some soldiers are not brave." Sub- contrary propositions cannot both be false, but both may be true, at the same time. If they were in any case false together, then their respective contradict- ories would have to be true together, and such prop- ositions as the following would both be true : "All soldiers are brave," "No soldiers are brave ;" but this is impossible. 40 AN ELEMENTARY HANDBOOK OF LOGIC When it is said that subcontraries may be true to- gether, the meaning is that, unless we have evidence that the contradictory of one of them is true, we have no right to assume that the subcontraries are not both true. Subaltern propositions are two propositions iden- tical in matter and form, but different in quantity ; e.g. "All soldiers are brave" — "Some soldiers are brave;" "No soldiers are brave" — "Some soldiers are not brave." Subaltern propositions may both be true and may both be false at the same time. From the truth of the universal we may infer the truth of the par- ticular, but not vice versa. From the falsity of the particular we may infer the falsity of the universal, but not vice versa. The following table indicates the inferences which may be drawn from the truth or falsity of the proposi- tions A, E, I, O, in terms of each other: If A is true, E is false, I is true, O is false If A is false, E is unknown, I is unknown, O is true If E is true, A is false, I is false, O is true If E is false, A is unknown, I is true, O is unknown If I is true, A is unknown, E is false, O is unknown If I is false, A is false, E is true, O is true If O is true, A is false, E is unknown, I is unknown If O is false, A is true, E is false, I is true The definitions and remarks in the preceding para- graphs of this section are based on the supposition that we are dealing exclusively with the four typical forms of the categorical proposition. In order to ex- tend the doctrine of Opposition so that it shall apply to all kinds of propositions, we shall have to employ the definitions which are given below in section 48. OPPOSITE CATEGORICAL PROPOSITIONS 41 The opposition between modal propositions brought out in the following diagram: Contraries IS A E ^Iron must Iron cannot be solid be solid ^"~-^^^ ""^^^.. jjisee-^ 4-J 3 ^^0^^>^ -%;5g 3 m Iron may be solid Iron need not be solid LO Subcontraries A singular proposition, like "Plato is clear," is vague, and it would be difficult to say offhand whether "Plato is not clear" is its contradictory or its contrary. When a proposition of this kind is qualified by some word or expression, such as "always," "everywhere," "partly," "in some places," or "in some respects," it is possible to employ it as a basis for a square of oppo- sites. Thus : 1. Plato is always clear. 2. Plato is never clear. 3. Plato is sometimes clear. 4. Plato is sometimes not clear. 1 and 4 and also 2 and 3 are contradictories; 1 and 2 are contraries; 3 and 4 are subcontraries; 1 and 3 and also 2 and 4 are subalterns. CHAPTER V EDUCTION 37. The import or meaning of a categorical proposi- tion is that which is explicitly asserted by the judg- ment which it expresses; it is that which is adverted to in the act of judging. In other words, it is the formal object of the judgment. The implication of a categorical proposition is a judgment or several judgments involved in the import of the proposition. Thus, the judgment expressed by the proposition, "All men are rational," involves the judgment, "No irrational beings are men." The im- plications of a given judgment are all the additional judgments to which a man necessarily commits him- self in pronouncing that judgment, though he may not have actually formulated these other judgments in his mind. Eduction is the process of drawing out the implica- tion of a single proposition. There are four kinds of eduction which have a prominent place in works on Logic, viz. Conversion, Obversion, Contraposition, and Inversion. As there will be frequent mention of positive and negative terms in the next few sections, a word should be said in explanation of them. A positive term is a term which expresses a positive idea (cf. 9) ; that is, it is a term which denotes a form 42 EDUCTIOK 43 or an object as possessing that form; e.g. "rational," "rational being." A negative term is a term which expresses a nega- tive idea; that is, it is a term which denotes the ab- sence of a form or an object as lacking that form; e.g. "irrational," "irrational being," "non-metallic." Positive terms we shall symbolize by the letters S and P (S standing for subject, and P for predicate), and negative terms, by non-S and non-P. The positive and negative terms of which we speak in eduction are contradictory of each other; that is, they are such that any pair of them comprises all objects whatever (cf. 9). Care must, therefore, be taken that the negative term which is employed shall be the contradictory, and not the contrary, of the posi- tive term. Almost always the contradictory of a simple term may be obtained by prefixing "non" to it. Other negative prefixes and suffixes frequently have this force, but more often not. "Invisible," "irrational," "un- tainted," "useless," have the same meaning respectively as "non-visible," "non-rational," "non-tainted," "non- useful ;" but "unpleasant," "unholy," "immoral," "dis- courteous," are not the same as "non-pleasant," "non- holy," "non-moral," "non-courteous." We may speak, for instance, of a lump of coal as non-holy, but we cannot speak of it as unholy. S means "thing (or being) that is S," and non-S, "thing (or being) that is not S ;" P means "thing (or being) that is P," and non-P, "thing (or being) that is not P." Thus, the Partial Inverse of "Every truth- ful man is mortal" is "Some beings that are not truth- ful men are not mortal." "Non-truthful man" is not 44 AK ELEMENTARY HANDBOOK OF LOGIC the contradictory of "truthful man ;" for the two terms, "truthful man" and "non-truthful man," do not comprise between them all objects whatever. We have seen that a pair of contradictory terms, such as S — non-S, P — non-P, comprises all objects whatever. If in any case the one term S (or P) ex- tends by itself to all objects, then there is no non-S (or non-P) ; that is, there is no "thing that is not S" (or "thing that is not P"). "Thing," for instance, extends by itself to all objects whatever, and there- fore there is no object which can be denoted by the negative of "thing;" that is, there is no "thing that is not a thing." When non-S is lacking, any eduction which involves the existence of non-S is impossible; the same is true mutatis mutandis when non-P is lacking, and again when S is lacking, and still again when P is lacking. CONVERSION 38. Conversion is an' eduction by which from a given proposition another is derived having for its subject the original predicate and for its predicate the original subject. The original proposition is called the Convertend, and the derived proposition, the Converse. RULE for conversion: The quality of the converse must be the same as that of the convertend. The reason for the rule is that an identity or a diversity remains an identity or a diversity (cf. 19), whether we view it from the standpoint of the subject or from the standpoint of the predicate. CONVERSION 45 A second rule is usually laid down in works on Logic as follows : In the converse no term may he distributed which was not distributed in the conver- tend. But this rule assumes that the predicate of a categorical proposition is distributed or undistributed. This assumption we have seen to be unnecessary, and hence there is no need for the rule (cf. 29, 30, 31). Simple conversion is a conversion in which the con- verse has the same quantity as the convertend. Conversion per accidens is a conversion in which a particular converse is derived from a universal con- vertend. E and I can be converted simply: E No trees are sentient — No sentient things are trees. I Some flowers are fragrant — Some fragrant things are flowers. A can be converted only per accidens; E may be converted per accidens: A All men are mortal — Some mortal beings are men. E No trees are sentient — Some sentient things are not trees. O cannot be converted. The justification of these statements will be found in sections 30 and 31. The results we have reached in this section are ex- hibited in the following table : Convertend Converse A All S is P Some P is S E No S is P No P is S I Some S is P Some P is S O Some S is not P 46 AN ELEMENTARY HANDBOOK OF LOGIC Before a proposition is converted, it should be thrown into logical form; that is, the subject, the copula, and the predicate should be set down defi- nitely and distinctly. "A stitch in time saves nine :" — in logical form this proposition will read "A stitch in time is a thing that saves nine stitches;" and its converse is "Something that saves nine stitches is a stitch in time." Moreover, unless the matter of the proposition makes it imperative, we must be careful not to change the predicate into a singular term by the use of the definite article "the ;" for example, the logical form of "Peter struck James" is not "Peter is the person who struck James," but "Peter is a person who struck James," and the converse is "Some one who struck James is Peter." Sometimes, however, the matter of the proposition makes it necessary to use the definite article before the predicate, because the predicate is seen from the mat- ter to be a singular term. Thus, the proposition, "Dickens wrote David Copperfield," in logical form is "Dickens is the person who wrote David Copperfield," from which we derive the converse, "The person who wrote David Copperfield is Dickens." Again, when casting a proposition into logical form, we shovild not place at the beginning of the predicate such words as "he who," "that which," "those who," "those which," "the persons who," "the things which," ■ — unless, indeed, the matter evidently calls for them. The insertion of such words in the predicate will gen- erally make the new form of the proposition say more than the original form. OBVERSION 47 OBVERSION 39. Obversion is an eduction by which from a given proposition another is derived having for its subject the original subject and for its predicate the contra- dictory of the original predicate. The original proposition is called the Obvertend, and the derived proposition, the Obverse. RULE : To obtain the obverse, negative the predi- cate and change the quality, but not the quantity, of the obvertend. Examples of obversion : Obvertend Obverse A All metals are material — No metals are non-material (immaterial). E No horses are rational — All horses are non-rational (irrational) . I Some men are tactful — Some men are not non- tactful (tactless). O Some substances are not visible — Some substances are non-visible (invisible). A obverts to E; E to A; I to O; O to I. We have then the following table : Obvertend Obverse A All S is P No S is non-P E No S is P All S is non-P I Some S is P Some S is not non-P O Some S is not P Some S is non-P The process of obversion is justified as follows: 48 AN ELEMENTARY HANDBOOK OF LOGIC When the subject-object is identical with the predi- cate-object (All S is P, Some S is P), it is really dis- tinct from anything that is really distinct from the predicate-object {No S is non-P, Some S is not non-P) (cf. 19, 4). When the subject-object is really distinct from the predicate-object (No S is P, Some S is not P), it is identical with something that is really distinct from the predicate-object (All S is non-P, Some S is non-P) ; that is, it is identical with itself, and this is all the information the proposition gives us beyond the fact that the subject-object is really distinct from the predicate-object. No S is non-P and All S is non-P may be expressed respectively as follows : S is really distinct from any object which is really distinct from P ; S is iden- tical with some object which is really distinct from P. We justify the process of obversion, as expressed in language, by saying that two negatives are equivalent to an affirmative. CONTRAPOSITION 40. Contraposition is an eduction by which from a given proposition another is derived having for its subject the contradictory of the original predicate and for its predicate the contradictory of the original subject. The original proposition we shall call the Contra- ponend; the derived proposition is called the Contra- positive. When the derived proposition has for its predicate the subject of the original proposition, and not its con- tradictory, it is called the Partial Contrapositive. CONTRAPOSITION 49 RULE : To obtain the partial contrapositive, obvert the original proposition and then convert the obverse. To obtain the contrapositive, obvert the partial contra- positive. A, E, and O may be contraposited. I cannot be contraposited ; because, after being ob- verted, it becomes O, and O cannot be converted. Examples of contraposition : Contraponend Obverse Partial Contrapositive Contrapositive All residents are combat- ants. No residents are non-com- batants. No non-combatants are res- idents. All non-combatants are non- residents. The partial contrapositive of A may also be written in the form, "Only combatants are residents" or "None but combatants are residents." Contraponend Obverse Partial Contrapositive Contrapositive No professionals are mem- bers. All professionals are non- members. Some non-members are pro- fessionals. Some non-members are not non-professionals. so AN ELEMENTARY HANDBOOK OF LOGIC o Some Americans are not voters. Some Americans are non- voters. Some non-voters are Amer- icans. Some non-voters are not non-Americans. The following table sums up these results in sym- bolical form : Contraponend Obverse Partial Contrapositive Contrapositive Contraponend A All S is P E No S is P I Some S is P O Some S is not P Partial Contrapositive No non-P is S Some non-P is S Contrapositive All non-P is non-S Some non-P is not non-S Some non-P is S Some non-P is not non-S It is to be observed that the contrapositive is of the same quality as the contraponend, whereas the partial contrapositive is of opposite quality. Again, the par- tial contrapositive and the contrapositive of All S is P, being both universal, allow us to pass back from them to the original proposition. All S is P. But this reverse process is not possible in the case of the proposition No S is P, because its partial contrapositive and its con- trapositive are both particular. When "No non-P is S" is derived from "All S is P," it is called the partial contrapositive of "All S is P." When "All S is P" is derived from "No non-P is S," it is called the obverted converse of "No non-P is S." Thus, the obverted converse of "No beings that are not mortal are men" is "All men are mortal." INVERSION 51 INVERSION 41. Inversion is an eduction by which from a given proposition another is derived having for its subject the contradictory of the original subject and for its predicate the contradictory of the original predicate. The original proposition is called the Invertend, and the derived proposition, the Inverse. When the derived proposition has for its predicate the predicate of the original proposition, and not its contradictory, it is called the Partial Inverse. RULE : To obtain the inverse of A, ohvert and con- vert alternately through four steps. To obtain the inverse of E, convert and obvert alter- nately through four steps. To obtain the partial inverse of A and E, ohvert the inverse of A and E respectively. A and E may be inverted. I and O cannot be inverted ; for if we attempt to in- vert either of them, we shall be confronted in the proc- ess by an O proposition to be converted, and O does not admit of conversion. Examples of inversion : A Invertend All residents are combatants. By obversion No residents are non-combatants. By conversion No non-combatants are residents. By obversion All non-combatants are non-resi- dents. By conversion Some non-residents are non-com- batants. 52 AN ELEMENTARY HANDBOOK OF LOGIC The proposition "Some non-residents are non-com- batants" is the inverse of "All residents are combat- ants." By obverting the inverse we obtain the partial in- verse, viz. "Some non-residents are not combatants." E Invertend No professionals are members. By conversion No members are professionals. By obversion All members are non-professionals. By conversion Some non-professionals are mem- bers. By obversion Some non-professionals are not non-members. "Some non-professionals are not .non-members" is the inverse of "No professionals are members." By obverting the inverse we obtain the partial inverse, viz. "Some non-professionals are members." The inverse is of the same quality as the original proposition, but the partial inverse is of opposite qual- ity. The quantity, whether of the inverse or of the partial inverse, is always particular. Since only universal propositions can be inverted, we may lay down the following simple rule for in- version : RULE FOR THE INVERSE: Replace the subject and the predicate of the invertend by their respective con- tradictories and change the quantity from universal to particular. The following table shows the results we have reached : Invertend Partial Inverse Inverse A All S is P Some non-S is not P Some non-S is non-P E No S is P Some non-S is P Some non-S is not non-P OTHER FORMS OF EDUCTION S3 Since the letters A, E, I, O represent the four typ- ical forms of the categorical proposition, and since S and P stand for subject and predicate respectively, we may represent the four propositions concisely by the symbols SaP, SeP, SiP, SoP. Letting S' and P' denote non-S and non-P respectively, we may bring together in the following table the results of the four processes of eduction we have thus far considered : 1. Original proposition 2. Obverse 3. Converse 4. Obverted converse 5. Partial contrapositive 6. Contrapositive 7. Partial inverse 8. Inverse OTHER FORMS OF EDUCTION 42. Eduction by an added determinant is an eduction by which from a given proposition another is derived in which the original subject and predicate are quali- fied by the same incident term (cf. 12). For example, "AH heroes are benefactors" — "All American heroes are American benefactors." It is to be observed that the same incident term must qualify the subject and the predicate of the derived proposition. If the term has a different meaning or force in the subject from what it has in the predi- cate, the eduction is not valid. For example, "A ball- player is a man" — "A poor ball-player is a poor man." A E I o SaP SeP SiP SoP SeP' SaP' SoP' SiP' PiS PeS PiS PoS' PaS' PoS' P'eS P'iS P'iS P'aS' P'oS' P'oS' S'oP S'iP S'iP' S'oP' 54 AN ELEMENTARY HANDBOOK OF LOGIC Quantitative terms, like "small" aiid "large," are espe- cially apt to lead to fallacy ; e.g. "An ostrich is a bird" — "A small ostrich is a small bird." 43. Eduction by an omitted determinant is an educ- tion by which from an affirmative proposition in which the predicate contains an incident term is de- rived another proposition in which the predicate lacks the incident term (cf. 12). For example, "These men are famous lawyers" — "These men are lawyers." If the incident term is such as to change the mean- ing of the principal term, this eduction is impossible ; e.g. "What this book relates are imaginary facts" — "What this book relates are facts." 44. Eduction by complex conception is an eduction by which from a given proposition another is derived having the same relation added to the original subject and predicate. For example, "A lark is a bird" — "The feathers of a lark are the feathers of a bird ;" "Gold is a valuable metal" — "A ring made of gold is a ring made of a valuable metal," or "Anything made of gold is (a thing) made of a valuable metal." It is to be observed, however, that there is danger of committing a fallacy, if quantitative relations are employed; e.g. "Physicians are professional men"^ "A majority of physicians are a majority of profes- sional men ;" "A carpenter is a mechanic" — "The most skilful body of carpenters is the most skilful body of mechanics." This kind of eduction from negative propositions is apt to be invalid, if the relation em- ployed is not a relation of part to whole; e.g. "A dog is not a horse" — "The color of a dog is not the color OTHER FORMS OF EDUCTION 55 of a horse," or "The owner of a dog is not the owner of a horse." 45. Eduction by converse relation is an eduction by which from a given proposition, in which a relation of the subject to another object is affirmed or denied, is derived another proposition, in which the reverse relation of the other object to the original subject is affirmed or denied respectively. For example, "The ele- phant is larger than a horse"^ — "The horse is smaller than an elephant;" "John is the brother of Mary" — "Mary is the sister of John ;" "Washington is north of New Orleans" — ^"New Orleans is south of Washington ;" "A is equal to B" — ^"B is equal to A ;" "Robert is not the father of James" — "James is not the son of Robert." Akin to eduction by converse relation are such educ- tions as the following: "The horse is larger than a dog" — "Whatever is larger than a horse is larger than a dog," "Whatever is as large as a horse is larger than a dog," "Whatever is smaller than a dog is smaller than a horse," "Whatever is as small as a dog is smaller than a horse," "Whatever is as small as a dog is not as large as a horse," etc. Note. — The various processes we have considered under the general head of Eduction are usually called Immediate Inferences. But this name is misleading; for it confuses the act with the process of inference (cf . 47) . Cf . Appendix : Note on Section 45. We said above that the name "Eduction" is applied to the process of drawing out the implication of a single proposition. This name might also be applied to processes like the following : "All men are mortal ; 56 AN ELEMENTARY HANDBOOK OF LOGIC All men are rational; Therefore all men are mortal and rational;" "Therefore no being that is either im- mortal or irrational is a man" (cf. 78). But it will be observed that in both cases the conclusion is a com- pound categorical proposition. CHAPTER VI THE ACT OF INFERENCE 46. Inference is either an act or a process. The act of inference, primarily, is an act of the mind asserting that in the world of reality the formal object of one potential judgment is logically depend- ent upon the formal object of one or more potential judgments (cf. 19). More briefly, but less accurately, the act of inference, primarily, is the mental assertion of the logical dependence of one potential judgment upon one or more potential judgments. A potential judgment is a judgment which is not pronounced, but which can be pronounced. The formal object of a potential judgment is a formal object which is not asserted* but which can be asserted, by a judgment. For the sake of brevity, we shall call the act of infer- ence by the simple name of Inference. The word "Reasoning" is a synonym for Inference. The formal object or form of an inference, primarily, is the' logical dependence of one potential judgment upon one or more potential judgments; or rather, it is the logical dependence of the formal object of one potential judgment upon the formal object of one or more potential judgments (cf. 19, 2). Logical dependence is also called Sequence. The material object or matter of an inference, 57 58 AN ELEMENTARY HANDBOOK OF LOGIC primaril}^ are two or more potential judgments, or rather, the formal objects of two or more potential judgm.ents. The act of inference, secondarily, is an act of the mind asserting that in the world of reality the formal object of one potential inference is logically dependent upon the formal object of one or more potential in- ferences ; or more briefly, but less accurately, it is the mental assertion of the logical dependence of one potential inference upon one or more potential in- ferences. A potential inference is an inference which is not pronounced, but which can be pronounced. The formal object of a potential inference is a formal object which is not asserted, but which can be asserted, by an act of inference. Examples of inference in its primary manifesta- tion : "If John committed this robbery, then he is deserving of imprisonment ;" "If all anarchists are un- patriotic, and Peter is an anarchist, then Peter is un- patriotic." Examples of inference in its secondary manifesta- tion : "If it is true that, if it has rained, the grass is wet, then it is true that, if the grass is not wet, it has not rained;" "If it is true that, if John committed this robbery, he is deserving of imprisonment, and that, if he was in the house at midnight, he committed this robbery, then it is true that, if John was in the house at midnight, he is deserving of imprisonment." From the inference, "If all anarchists are unpa- triotic, and Peter is an anarchist, then Peter is unpa- triotic," we can derive by eduction the following: "If THE ACT OF INFERENCE 59 it is true that Peter is not unpatriotic, then it is true that, if all anarchists are unpatriotic, Peter is not an anarchist." In this second inference we assert the logical dependence of a potential inference upon a po- tential judgment. Hereafter, for the sake of convenience, we shall refer chiefly to the act of inference in its primary manifesta- tion ; for what we shall say of it under this form will apply to it mutatis mutandis in its secondary manifesta- tion. The consequent is the formal object of a potential , judgment which is asserted to be logically dependent upon the formal object of one or more potential judg- ments. More briefly, it is a potential judgment which is asserted to be logically dependent upon one or more potential judgments. Thus, in the inference, "If John committed this robbery, then he is deserving of im- prisonment," the consequent is "he is deserving of imprisonment." The antecedent is the formal object of the one or more potential judgments upon which the consequent is asserted to be logically dependent. Less accurately, it is the potential judgment or potential judgments upon which the consequent is asserted to be logically dependent. Thus, the antecedent of the foregoing in- ference is the potential judgment, "John committed this robbery." The formal object of a potential judgment is logic- ally dependent upon the formal object of another potential judgment, when the reality of the latter involves the reality of the former, and consequently, when the as- sertion of the latter involves the assertion of the former. 60 AN ELEMENTARY HANDBOOK OF LOGIC Instead of the cumbersome expression, "the reality of the formal object of the one or more potential judg- ments upon which the consequent is asserted to be logically dependent," we shall adopt as its equivalent the shorter phrase, "the truth of the antecedent." We may theji construct a concise definition of logical de- pendence as follows : Logical dependence or sequence means that the truth of the antecedent involves the truth of the consequent. Note. — The act of inference is not a judgment; for it is not the mental assertion of the objective identity or diversity of two ideas (cf. 19). What is implied in the inference we have been employing in illustra- tion is that the evidence which shall warrant our pro- nouncing John guilty of this robbery will also warrant our pronouncing him deserving of imprisonment. But in the inference itself we do not make either of these pronouncements, and hence the inference does not contain a judgment. In inference the mind prescinds from the presence or absence of any evidence for pronouncing any of the potential judgments contained in the matter of the inference. One or more of the judg- ments whose formal objects are contained in the matter of the inference may have been made prior to the inference; but in the act of inference itself none of these judg- ments is made. Frequently the inference is called a conditional judgment. It would be more correct to say that it contains the formal object of a judgment which is held in abej'-ance till the antecedent is proved ; and a judgment in abeyance is not a judgment at all. There is no such thing as a conditional judgment, any more than there is a conditional act of jumping. THE ACT OF INFERENCE 61 When it is contended that the example we have been using is a conditional judgment, it is not meant that the hypothetical proposition can be resolved into a cate- gorical ; for, when it is resolved into a categorical proposi- tion, no one would call it the expression of a conditional judgment. What is meant by the contention we refer to is that in our example we assert conditionally that "John is deserving of imprisonment." As a matter of fact, in that example we do not assert at all, whether condition- ally or otherwise, that "John is deserving of imprison- ment ;" just as in the second example we do not assert that "Peter is unpatriotic." 47. An a priori inference is an inference which as- serts a logical dependence perceived by means of a mere comparison of the formal objects of the potential judgments which enter into it either with themselves or with the formal object of another potential judg- ment ; e.g. "If he walked along a straight line from one point to another, then he travelled over the shortest distance between those two points;" "If a line is drawn through a circle perpendicular to a tangent from the point of junction between the tangent and the circle, then the line will pass through the centre of the circle." An a posteriori inference is an inference which as- serts a logical dependence perceived by means of ex- perience in addition to comparison; e.g. "If the iron is hot, then it will melt the wax;" "If a feather and a coin are allowed to fall together in a vacuum, then they will descend at the same rate of speed." An immediate inference is an inference which asserts a logical dependence perceived without recourse to 62 AN ELEMENTARY HANDBOOK OF LOGIC proof ; e.g. "If he walked along a straight line from one point to another, then he travelled over the shortest distance between those two points." A mediate inference is an inference which asserts a logical dependence perceived by means of proof; e.g. "If a line is drawn through a circle perpendicular to a tangent from the point of junction between the tan- gent and the circle, then the line will pass through the centre of the circle." CHAPTER VII THE HYPOTHETICAL PROPOSITION 48. The hypothetical or conditional proposition is the normal verbal expression of the matter and form of an act of inference ; e.g. "If John committed this robbery, then he is deserving of imprisonment." The matter of an inference, that is, the two or more potential judgments which enter into it, are expressed by the two parts of the proposition called antecedent and consequent. The antecedent and consequent of the hypothetical proposition correspond to the ante- cedent and consequent of the act of inference. In the foregoing example the antecedent is "John committed this robbery," and the consequent is "he is deserving of imprisonment." Since the antecedent and conse- quent express the matter of the inference, they are called the matter of the hypothetical proposition. The form of an inference, that is, the sequence or logical dependence, is expressed by "If . . . then," though it is not unusual to omit one or both of these particles. For this reason, "If . . . then" is called the form of the hypothetical proposition." The hypothetical proposition is neither affirmative nor negative ; for it asserts neither the objective iden- tity nor the objective diversity of two ideas (cf. 19). The presence or absence of a negative particle in the antecedent or the consequent or both does not affect 63 64 AN ELEMENTARY HANDBOOK OF LOGIC the form of the hypothetical proposition. "If this ani- mal is not rational, then it is not a man :" — this prop- osition is neither affirmative nor negative. We do, however, speak of denying a hypothetical proposition. But denying such a proposition does not mean making it negative; it means pronouncing it false. The act of the mind which pronounces a hypo- thetical proposition false is sometimes a judgment and sometimes an inference. In general, we may say that any pair of propositions are contradictories, which are such that they cannot both be true jior both be false ; e.g. "If a pupil is studious, he is successful" — "Some- times if a pupil is studious, he is not successful;" "If a pupil is studious, he is successful"— "Some studious pupils are not successful." If the propositions are such that they cannot both be true, but both may be false, at the same time, they are contraries; e.g. "If a pupil is studious, he is successful"- — "If a pupil is studious, he is not successful ;" "If a pupil is studious, he is successful" — "No studious pupil is successful." A hypothetical proposition is true, when the sequence or logical dependence which it asserts is real; is false, when the sequence is not real (cf. 19). A hypothetical proposition may be true, though the formal objects of the potential judgments entering into it are not only unreal, but impossible; e.g. "If this circle is square, it has four right angles." 49. If the same term stands as subject in the ante- cedent and the consequent, and is not a singular term, the hypothetical proposition can easily be resolyed into a categorical ; e.g. "If a man is just, he is brave" — "All just men are brave." In like manner, universal THE HYPOTHETICAL PROPOSITION 65 and particular categorical propositions can frequently be resolved into hypotheticals. Thus, by resolving the propositions which were used in the square of cate- gorical opposites in section 36, we may construct a square of hypothetical opposites : 1. If a man is a soldier, he is brave. 2. If a man is a soldier, he is not brave. 3. Sometimes if a man is a soldier, he is brave. 4. Sometimes if a man is a soldier, he is not brave. 1 and 4 and also 2 and 3 are contradictories ; 1 and 2 are contraries ; 3 and 4 are subcontraries ; 1 and 3 and also 2 and 4 are subalterns. In this connection some remark should be made con- cerning propositions like the one which in the next section is symbolized thus: "If X is true, then Y is true." The opposition between propositions of this kind may be illustrated as follows: 1. If X is true, Y is true. 2. If X is true, Y is not true. 3. If X is true, Y may be true. 4. If X is true, Y need not be true. If 3 and 4 be contemplated, not by themselves, but simply as the contradictories of 2 and 1 respectively, they may also be written as follows : "Even though X is true, still Y may be true," "Even though X is true, still Y need not be true." 50. The import or meaning of a hypothetical prop- osition is that which is explicitly asserted by the in- ference which it expresses ; it is that which is adverted to in the act of inferring. In other words, it is the formal object of the inference. DO AN ELEMENTARY HANDBOOK OF LOGIC The implication of a hypothetical proposition is an inference or several inferences involved in the import of the proposition. If we allow A, B, C, D to stand for terms, the hypo- thetical proposition will usually have the form, "If A is B, C is D ;" e.g. "If this child is disobedient, his parents suffer." When the antecedent and the conse- quent have the same subject, the form will be "If A is B, it is D ;" e.g. "If this child is disobedient, he is selfish." If we allow X and Y to stand for ante- cedent and consequent respectively, the proposition will have the form, "If X is true, then Y is true ;" e.g. "If the earth is immovable, the sun moves round the earth." This last form may be written still more briefly, thus : "If X, then Y." We saw in section 46 that what is asserted by an act of inference is that the truth of the antecedent involves the truth of the consequent. But in this act we neither assert nor imply that the truth of the consequent involves the truth of the antecedent. Thus, in the proposition, "If this child is disobedient, his parents suffer," we neither assert nor imply that "If the parents of this child suffer, he is disobedient." The parents might suffer for many reasons without the child being disobedient. If, however, the truth of the antecedent is a neces- sary condition of the truth of the consequent, then the truth of the consequent involves the truth of the ante- cedent. For example, "If this triangle is equilateral, all its angles are equal;" from this we may. conclude, "If all the angles of this triangle are equal, it is equi- lateral." However, the presence of a necessary con- THE DISJUNCTIVE PROPOSITION 67 dition cannot be learned from the form of the proposi- tion, but only from an inspection of its matter. N. B. — The matter of propositions of a given kind varies indefinitely, but the form is the same in all of them. Hence, whatever 'we. can learn from the form of any proposition we can learn from the form of every proposition of the same kind. From the fact that if the antecedent is true, the consequent is true, it follows that if the consequent is false, the antecedent is false ; for, if the antecedent were not false, it would be true, and that would make the consequent true, which is against the supposition. For example, from the proposition, "If this child is disobedient, his parents suffer," we may pass to the proposition, "If the parents of this child do not suffer, he is not disobedient." The falsity of the antecedent does not involve the falsity of the consequent, except in the case of the necessary condition. For example, "If this child is disobedient, his parents suffer :'' from this proposition we cannot argue, "If this child is not disobedient, his , parents do not suffer." We may sum up these results as follows : The prop- osition, "If A is B, C is D," yields "If C is not D, A is not B," but neither of the following propositions : "If C is D, A is B," "If A is not B, C is not D." THE DISJUNCTIVE PROPOSITION 51. The disjunctive or alternative proposition is the verbal expression of the matter and form of an act of inference, the form being expressed by the particles 68 AN ELEMENTARY HANDBOOK OF LOGIC "either ... or ;" consequently, it can be resolved into a hypothetical proposition. For example, "Either George or Peter is a lawyer" = "If George is not a lawyer, Peter is," "If Peter is not a lawyer, George is ;" "George is either a lawyer or a physician" = "If George is not a lawyer, he is a physician," "If George is not a physician, he is a lawyer;" "Either the earth moves round the sun, or astronomy is an illusion'' = "If the earth does not move round the sun, astronomy is an illusion," "If astronomy is not an illusion, the earth moves round the sun." These three examples may be represented symbolically as follows : "Either A or B is G," "A is either B or C," "Either X is true or Y is true." The elements of the disjunctive proposition con- nected by "or" are called Alternants. The alternants are the matter, and "either . . . or" is the form, of the disjunctive proposition. So far as we are able to judge from the form of the disjunctive proposition, the alternants are not mutually exclusive. Thus, in the proposition, "The pupil is either diligent or talented," we do not mean to exclude the possibility of the pupil being both diligent and talented (cf. 50, N. B.). Sometimes, however, we see from the alternants themselves — that is, from the matter of the proposition — that they are mutually exclusive ; e.g. "He either passed or failed in the examination."- A proposition of this kind yields four hypotheticals, namely : "If he did not pass in the examination, he failed," "If he did not fail in the examination, he passed," "If he passed in the examination, he did not fail," "If he failed in the examination, he did not pass." THE DISJUNCTIVE PROPOSITION 69 A disjunctive proposition with mutually exclusive alternants corresponds to a hypothetical with a neces- sary condition. When a disjunctive proposition contains only two al- ternants, the rule for resolving it into a hypothetical is as follows : Negative one of the alternants, make it the ante- cedent, and leave the other as it stands for the consequent. When a hypothetical proposition has a single antece- dent and a single consequent, the rule for resolving it into a disjunctive is as follows: Negative the antece- dent, leave the consequent as it stands, and connect the two by "or." 52. Propositions like "No man can be noble and base," "No man is both noble and base," are really disjunctive propositions with negative alternants, and may be expressed as follows : "Either a man is not noble or he is not base." This proposition is resolv- able into the hypothetical, "If a man is noble, he is not base," "If a man is base, he is not noble." The proposition, "John is noble," means "John pos- sesses the attribute noble" (cf. 19). "John is non- noble" is equivalent to "John lacks the attribute noble." "No man can be noble and non-noble" : — this proposition, in the ordinary disjunctive form, is writ- ten thus : "Either a man is not noble or he is not non- noble." The full interpretation of this proposition is as follows : "Either a man does not possess the attribute noble or he does not lack the attribute noble." This gives us the hypothetical, "If a man possesses the at- tribute noble, he does not lack the attribute noble," "If a man lacks the attribute noble, he does not possess the attribute noble." 70 AN ELEMENTARY HANDBOOK OF LOGIC THE FUNDAMENTAL LAWS OF THOUGHT OR FIRST PRINCIPLES 53. In general, a Law is a norm which must be followed in order to reach a certain end. A Law of Thought is a norm or principle with which our judgments and inferences must be in accord or at least not at variance, if we are to think correctly. The laws of thought are countless in number, and the vast majority of them have never been formu- lated. When they are formulated, they are frequently self-evident. Even the laws that have been formu- lated we follow for the most part without explicit advertence to them, Nevertheless, if in a given process of thought we violate any one of them, the process will be chaotic and fruitless. Three laws of thought are usually set down by logicians as being the most fundamental of all, viz. the Law of Identity, the Law of Contradiction, and the Law of Excluded Middle. The first is a judg- ment; the second and third, as we shall see presently, are acts of inference. LAW OF IDENTITY: Everything is what it is, or Everything is itself. This law emphasizes the unchangeable character of objective truth and its independence of thought and will (cf. 19). Hence, once a truth is ascertained, we are not at liberty to disregard it in our judgments and inferences. LAW OF CONTRADICTION: Nothing can at the same time and in the same respect be and not he. LAWS OF THOUGHT OR FIRST PRINCIPLES 71 This proposition expresses an act of inference; for we saw in the preceding section that its ordinary dis- junctive form is as follows : "Either a thing does not possess a certain attribute or it does not lack that at- tribute;" and this is resolvable into the hypothetical, "If a thing possesses a certain attribute, it does not lack that attribute," "If a thing lacks a certain attribute, it does not possess that attribute." In the usual formulation of the law special attention should be called to the words, "at the same time and in the same respect," which are inserted in order to eliminate the ambiguity which would otherwise lurk in this formula. LAW OF EXCLUDED MIDDLE: Everything either is or is not. This again is the expression of an act of inference. The meaning of the proposition, more fully expressed, is, "Either a thing possesses a certain attribute or it lacks that attribute." Casting this into strictly logical form, we have the hypothetical proposition, "If a thing does not possess a certain attribute, it lacks that attribute," "If a thing does not lack a certain attribute, it pos- sesses that attribute." We saw in section 49 that when a hypothetical proposition has the same general term as subject in the antecedent and the consequent, it can be resolved into a categorical. Hence, the Law of Contradiction and the Law of Excluded Middle may be stated as follows : Law of Contradiction : "A thing which possesses a certain attribute does not lack that attribute," "A thing which lacks a certain attribute, does not possess 72 AN ELEMENTARY HANDBOOK OF LOGIC that attribute." More briefly: "A thing with a cer- tain attribute is not a thing without that attribute," "A thing without a certain attribute is not a thing with that attribute." Law of Excluded Middle : "A thing which does not possess a certain attribute lacks that attribute," "A thing which does not lack a certain attribute possesses that attribute." CHAPTER VIII THE PROCESS OF INFERENCE AND THE SYLLOGISM 54. The process of inference is the process of estab- lishing the reality of the formal object of a judgment or of an act of inference (cf. 19). More briefly, it is the process of establishing the truth of a judgment or of an act of inference. The process of inference involves three acts, viz. an act of inference, the assertion of the antecedent of this act of inference, and the assertion of its conse- quent. For example — If the defendant is innocent, the court should acquit him, But the defendant is innocent, There;fore the court should acquit him. In this example the act of inference is contained in the first line, and is called the Major Premise. The assertion of the antecedent is in the second line, and is called the Minor Premise. The assertion of the consequent is in the third line, and is called the Con- clusion. Since the consequent is known to be logically de- pendent upon the antecedent, the function of proof is to furnish evidence of the antecedent; for, once the antecedent is proved, the consequent also is proved (cf. 46). In many cases the proof is at hand before 73 74 AN ELEMENTARY HANDBOOK OF LOGIC the act of inference is made. Speaking figuratively, all proof consists in the elimination of the antecedent, thus leaving us to assert the consequent. The antece- dent is eliminated in the minor premise. The consequent or conclusion is true, when its formal object is a reality (cf. 19). It is valid, when it is log- ically dependent upon the antecedent. 55. An argument is the verbal expression of one or more processes of inference. The syllogism is an argument which expresses only one process of inference, and in this order: the act of inference, the assertion of the antecedent, and the assertion of the consequent. Because of its concise- ness and accuracy in exhibiting the reasoning process, the syllogism is called the perfect expression of the .process of inference. N. B. — The name "syllogism" is also applied to one or two forms of argument which, as they stand, are an incomplete expression of a process of inference. We shall refer to this again in sections 56 and 62 and also in the Appendix. Cf. Appendix: Note on Sec- tion 55. The Syllogism is composed of three propositions which are called Major Premise, Minor Premise, and Conclusion, corresponding to the major premise, minor premise, and conclusion of the reasoning process. The example which was given in the preceding section to illustrate the process of inference will also serve to illustrate the syllogism. If the major or the minor premise or the conclusion is omitted from the verbal expression of the reasoning process, the argument is called an Enthymeme. CHAPTER IX THE MIXED HYPOTHETICAL SYLLOGISM 56. We begin the explanation of the syllogism with the mixed hypothetical syllogism, because this is the least complex of all forms of argument. The mixed hypothetical or mixed conditional syl- logism is a syllogism in which the major premise is a hypothetical proposition with a single antecedent and a single consequent, and the minor premise, a simple categorical proposition either positing the antecedent or sublating the consequent of the major premise. To posit is to set down as true. To sublate is to set down as false. The character of this syllogism is shown in the fol- lowing formulas : If A is B, C is D, AisB, Therefore C is D. If A is B, C is D, C is not D, Therefore A is not B. There are two "moods" of the mixed hypothetical syllogism, viz. the Modus Ponens or Constructive Syllogism and the Modus ToUens or Destructive Syl- logism. In the modus ponens the minor premise posits the antecedent of the major premise. In the 75 76 AN ELEMENTARY HANDBOOK OF LOGIC modus tollens the minor premise sublates the consequent of the major premise. The modus ponens (or modus ponendo ponens) pre- sents no difficulty. The justification of the modus tollens (or modus tollendo tollens) lies in a suppressed major premise which, if expressed, would show that the modus tollens is the modus ponens. We saw in section 50 that the proposition "If this child is dis- obedient, his parents suffer" yields by eduction the proposition "If his parents do not suffer, this child is not disobedient." If, then, in the modus tollens we express the suppressed major premise, the argument will run as follows: If this child is disobedient, his parents suffer. Therefore If his parents do not suffer, this child is not disobedient. But his parents do not suffer. Therefore This child is not disobedient. This argument is now in the modus ponens; for the minor premise — "His parents do not suffer" — posits the antecedent of the premise immediately preceding it. The question we have just been discussing will be considered at greater length in the Appendix. (Cf. pp. 218-221). 57. Rules of the Mixed Hypothetical Syllogism: RULE I: Positing the antecedent in the minor premise necessitates positing the consequent in the con- clusion. RULE II : Sublating the antecedent in the minor premise does not warrant sublating the consequent in the conclusion. THE MIXED HYPOTHETICAL SYLLOGISM 77 The reason for Rule I is that in a hypothetical proposition we assert that the consequent is logically dependent upon the antecedent, that the truth of the antecedent involves the truth of the consequent (cf. 46). The reason for Rule II is that we do not assert that the truth of the antecedent is a necessary condition of the truth of the consequent. For all we know from the forw, of the proposition, the consequent may be true, though the antecedent is false (cf. SO). Hence, we can- not sa);- that the consequent is false because the antece- dent is false. RULE III : Sublating the consequent in the minor premise necessitates sublating the antecedent in the con- clusion. RULE IV: Positing the consequent in the minor premise does not warrant positing the antecedent in the conclusion. The reason for Rule III is that the modus tollens becomes the modus ponens when the suppressed major premise is expressed; and sublating the original con- sequent is the same as positing the antecedent of the suppressed major premise. The reason for Rule IV is that in a hypothetical proposition we do not assert that the truth of the antecedent is a necessary condition of the truth of the consequent, and hence, so far as we know from the form of the proposition, the consequent may be true without the antecedent being true. If in a given hypothetical proposition the matter reveals that the truth of the antecedent is a necessary condition of the truth of the consequent, then we are justified in positing the consequent in the minor 78 AN ELEMENTARY HANDBOOK OF LOGIC premise and the antecedent in the conclusion; we may also in this case sublate the antecedent in the minor premise and the consequent in the conclusion. The first and third rules are illustrated in the follow- ing examples : Modus ponens : If it has rained, the grass is wet, But it has rained. Therefore The grass is wet. or Modus tollens: But the grass is not wet. Therefore It has not rained. Modus ponens: If the boy has not studied, he will fail. But the boy has not studied. Therefore He will fail. or Modus tollens: But the boy will not fail. Therefore He has studied. Modus ponens: If the general is skilful, he will hot lose the battle. But the general is skilful. Therefore He will not lose the battle. or Modus tollens: But the general will lose the battle, Therefore He is not skilful. Caution: When the consequent is sublated in the minor premise, the conclusion will be the contra- dictory, not the contrary, of the antecedent. Sublating the consequent in the minor premise warrants us only in asserting that the antecedent is not true, that is, in THE DISJUNCTIVE SYLLOGISM 79 asserting that its contradictory is true. If all men were upright, crimes would not occur. But crimes do occur. Therefore Some men are not upright. It would be a fallacy to conclude, "Therefore no men are upright." 58. Fallacies of the Mixed Hypothetical Syllogism. There are two fallacies which may easily be committed by the unwary in using the mixed hypothetical syl- logism : 1. The fallacy of sublating the antecedent in the minor premise, that is, of deducing the falsity of the consequent from the falsity of the antecedent. For example — If it has rained, the grass is wet. But it has not rained. Therefore The grass is not wet. 2. The fallacy of positing the consequent in the minor premise, that is, of deducing the truth of the antecedent from the truth of the consequent. For example — If it has rained, the grass is wet, But the grass is wet, Therefore It has rained. THE DISJUNCTIVE SYLLOGISM 59. The- disjunctive or alternative syllogism is a syllogism in which the major premise is a disjunctive 80 AN ELEMENTARY HANDBOOK OF LOGIC proposition, and the minor premise, a proposition sub- lating part of the alternants of the major premise. For example — A is either B or C, A is not B, Therefore A is C. A is either B or C or D, A is not B, Therefore A is either C or D. A is either B or C or D, A is neither B nor C, Therefore A is D. Either X is true or Y is true, Y is not true, Therefore X is true. The minor premise in each of the foregoing syl- logisms suhlates part of the alternants of the dis- junctive premise, and the conclusion posits what remains. A syllogism of this type is called the Modus ToUendo Ponens. As was pointed out in section 51, the form of a dis- junctive proposition does not warrant us in interpreting the alternants as mutually exclusive. In the proposition, "He was either first or second in the race," the alternants are mutually exclusive; but this is revealed by the matter of the proposition, not by its form. In the Modus Ponendo ToUens the minor premise posits part of the disjunctive premise, and the con- clusion suhlates the remainder. The modus tollendo ponens is always valid. THE DISJUNCTIVE SYLLOGISM 81 The modus ponendo tollens is valid only when the alternants are known to be mutually exclusive. Modus tollendo ponens — Valid: He is either diligent or talented, He is not talented, Therefore He is diligent. Modus ponendo tollens — Invalid: He is either diligent or talented, He is talented, Therefore He is not diligent. In the first paragraph of section 51 it was ob- served that the disjunctive proposition is resolvable •into the hypothetical. Hence, the disjunctive syl- logism with but two alternants is only the mixed hypo- thetical syllogism worded differently. Expressing the foregoing examples in hypothetical form, we shall find that the modus tollendo ponens is either the modus ponens or the modus tollens of the mixed hypothetical syllogism, and that the modus ponendo tollens commits one of the fallacies indicated in the preceding section. The modus tollendo ponens becomes : If he is not talented, he is diligent, He is not talented. Therefore He is diligent. The modiis ponendo tollens becomes : If he is not talented, he is diligent. He is talented. Therefore He is not diligent. N. B. — Contradictory alternants are not only mutu- ally exclusive, but collectively exhaustive (cf . 9, 36, 37) ; 82 AN ELEMENTARY HANDBOOK OF LOGIC e.g. "Either it rains or it does not rain;" "He is either learned or not learned." 60. The Conjunctive Syllogism. The so-called con- junctive syllogism is a disjunctive syllogism in which the alternants of the disjunctive premise are all nega- tive. For example — Every gem is either nnt a diamond or not a ruby, This gem is a diamond, Therefore It is not a ruby. The major premise is usually stated as in the fol- lowing syllogism, and it is when this premise is so stated that the syllogism is called conjunctive: No gem can be both a diamond and a ruby, This gem is a diamond. Therefore It is not a ruby. CHAPTER X THE SIMPLE CATEGORICAL SYLLOGISM 6L The simple categorical syllogism is the verbal expression of a process of inference that contains an act of inference in which is asserted the real logical dependence of the objective identity or diversity of two ideas upon two potential judgments in which the formal objects of the two ideas are compared separately with the formal object of a third idea. The simple- categorical syllogism, as commonly set forth in text-books on Logic, expresses merely three judgments, the sequence or logical dependence being indicated by the particle "therefore." In the two first judgments the formal objects of two ideas are com- pared separately with the formal object of a third idea and their identity with or diversity from the formal object of the third idea is separately asserted (cf. 19, N. B.). In the third judgment the objective identity or diversity of the two first ideas is asserted. The proposi- tions expressing the two first judgments are called the antecedent or premises of the syllogism. The proposi- tion expressing the third judgment is called the con- sequent or conclusion. This type of syllogism is called categorical, because the three propositions which usually compose it are 83 84 AN ELEMENTARY HANDBOOK OF LOGIC categorical. For example — All men are mortal, But all kings are men. Therefore AH kings are mortal. 62. It is important to observe that, if we express in full the process of inference which issues in the conclusion, "All kings are mortal," then the two prop- ositions, "All men are mortal" and "All kings are men," will form one compound categorical proposition in the minor premise, and the major premise will be a hypothetical proposition expressing an act of in- ference in which the antecedent consists of two poten- tial judgments comparing the formal objects of two ideas separately with the formal object of a third idea, and the consequent consists of a potential judgment whose formal object is the objective identity or diver- sity of those two ideas (cf. Appendix: Note on Sec- tion 55), Thus: If all men are mortal and all kings are men, then all kings are mortal, But all men are mortal and all kings are men. Therefore All kings are mortal. In text-books on Logic it is customary to omit the major premise of the foregoing syllogism and to treat the two simple propositions constituting the corn- pound proposition of the minor premise as two sep- arate premises. It will be convenient to follow this practice, and we shall write the simple categorical syllogism as it was written in the preceding section. THE SIMPLE CATEGORICAL SYLLOGISM 85 63. The following are the two axioms on which the simple categorical syllogism is based : AXIOM OF IDENTITY: When the formal objects of tivo ideas are identical with the formal object of a third idea, they are identical with each other. AXIOM OF DIVERSITY: When the formal ob- jects of two ideas are one of them identical zvith and the other different {distinct) from the formal object of a third idea, they are different from each other. It is of the essence of the simple categorical syl- logism that the formal objects of two ideas be com- pared with the formal object of a third idea. The formal object of the third idea forms the common basis for comparing the formal objects of the two other ideas with each other in such a way as to determine their identity or diversity. The whole purpose of the treatment of the simple categorical syllogism is to find out in what way the formal objects of two ideas may be related as regards identity or diversity with the formal object of a third idea so that their own identity or diversity shall be a necessary sequence from that relation (cf. 19, N. B.). If, instead of comparing the formal objects of two ideas with the formal object of a third idea, we com- pared one of them with the formal object of a third, and the other with the formal object of a fourth idea, the comparison would be useless, and we should be unable to determine anything as regards the objective identity or diversity of the two first ideas; for in that case we should not be employing a basis of comparison which was common to the formal objects of those two ideas. 86 AN ELEMENTARY HANDBOOK OF LOGIC MATTER AND FORM OF THE SIMPLE CATE- GORICAL SYLLOGISM 64. The form of the categorical syllogism is the sequence, that is, the logical dependence of the con- clusion or consequent upon the premises or antece- dent. If the sequence is lacking, we have only an apparent syllogism, not a real one. The proximate matter of the simple categorical syl- logism are the three propositions which enter into it. The remote matter are the terms which constitute the matter of the three propositions. The terms are three, two of which are compared separately with the third term. The third term is called the Middle Term. The two other terms are called the Extremes. When we speak of the matter of the syllogism with- out qualification, we mean the proximate matter. The syllogism we have already employed may be represented symbolically as follows : All men are mortal. All M is P But all kings are men, All S is M Therefore All kings are mortal. Therefore All S is P There are three propositions. These propositions contain three terms, each appearing tivice: the middle term, twice in the two first propositions, but not in the third proposition; the extremes, each once in the third proposition and once in one of the other propositions. The premises or antecedent are the propositions in which the extremes are compared with the middle term. The conclusion or consequent is the proposition MATTER AND FORM OF THE SYLLOGISM ' 87 which asserts the objective identity or diversity of the ideas expressed by the extremes. The major term or major extreme is the predicate of the conclusion. It is denoted by the letter P. The minor term or minor extreme is the subject of the conclusion. It is denoted by the letter S. The predicate of the conclusion is called the major term, because the predicate of an affirmative proposi- tion generally has a wider extension than the subject (cf. 27). The major premise is the premise which contains the major term (P). The minor premise is the premise which contains the minor term (S). The order in which the premises are arranged does not affect the validity of the syllogism. Sometimes the minor premise is placed first; but usually it is found after the major premise. LAWS OF THE TRUTH AND FALSITY OF VALID CONCLUSIONS 65. The conclusion is valid, when it is logically de- pendent upon the premises. The conclusion is true, when what it asserts is a reality (cf. 19). Special attention should be directed to the two fol- lowing laws, which relate to the truth and falsity of valid conclusions : I. A valid conclusion from true premises is always true. This law follows from the very nature of a sequence 88 AN ELEMENTARY HANDBOOK OF LOGIC or logical dependence (cf. 46, 50), which means that the truth of the antecedent (or premises) involves the truth of the consequent (or conclusion). Hence, if a valid conclusion is false, at least one of the premises is false. II. A valid conclusion from premises one or both of which are false may be false and may he true. This law follows from what was said in section 50, namely, that unless the truth of the antecedent is a necessary condition of the truth of the consequent, the falsity of the antecedent does not involve the. falsity of the consequent (or conclusion), nor does the truth of the consequent involve the truth of the antecedent. The following examples illustrate the law: "Every science is useless; But physics is a science; Therefore physics is useless." Here a valid hut false conclusion is derived from premises one of which is false. "No horses are able to fly ; But all dogs are horses ; Therefore no dogs are able to fly." Here we have a conclusion which is both valid and true derived from premises one of which is false. "Every stone is rational ; But every man is a stone ; Therefore every man is rational." In this syllogism both premises are false, but the conclusion is both valid and true. Two important practical corollaries flow from the second law : (1) It does not follow that a doctrine is false be- cause the argum,ents adduced in support of it are false. (2) It does not follow that the arguments adduced TRUTH AND FALSITY OF VALID CONCLUSIONS 89 in support of a doctrine are true because the doctrine itself is true. A conclusion is true per se, when it is not only valid and true, but follows from true premises. A conclusion is true per accidens, when it is true, but not valid, and also when it is both true and valid, but follows from premises one or both of which are false. CHAPTER XI FIGURES AND MOODS OF THE CATEGORICAL SYLLOGISM 66. The figure of a categorical syllogism is the rela- tion between the position of the middle term in the major premise and its position in the minor. If we state the major premise first, and denote the major, middle, and minor terms by the letters P, M, S respectively, we may arrange the terms of the premises in four different ways so as to obtain four figures, as follows : Fig. 1 Fig. 2 Fig. 3 Fig. 4 M P P M M P P M S- M S M M S M S In the first figure the middle term is subject in the major premise and predicate in the minor. In the second figure the middle term is predicate in both premises. In the third figure the middle term is subject in both premises. In the fourth figure the middle term is predicate in the major premise and subject in the minor. The diagram below shows the relative positions of the middle term in the four figures, the letters X and Y 90 FIGURES AND MOODS OF THE SYLLOGISM 91 denoting respectively the subject and the predicate of the major premise, and X' and Y', the subject and the predicate of the minor. Fig. 2 If the lines are extended through X' and Y' and their extremities numbered backwards, the lines num- bered 1, 2, 3, 4 indicate the relative positions of the middle term in the first, second, third, and fourth fig- ures respectively. The four figures are illustrated concretely in the following examples: Figure 1 : All men are mortal, All kings are men. Therefore All kings are mortal. Figure 2 : All animals are sentient. No plants are sentient. Therefore No plants are animals. Figure 3: All men are mortal. All men are rational beings. Therefore Some rational beings are mortal. Figure 4: No soldiers are women. Some women are brave, Therefore Some brave persons are not soldiers. 92 AN ELEMENTARY HANDBOOK OF LOGIC 67. The. mood of a categorical syllogism is the logical dependence of the quality and quantity of the con- clusion upon the quality and quantity of the premises; For example, All is a mood in which the major prem- ise is universal afifirmative, the minor premise, particu- lar affirmative, and the conclusion, particular affirmative. It now remains for us to determine the rules of the different figures and to find out what combinations of quality and quantity in the premises of each figure will yield a valid conclusion. In undertaking to do this, we suppose that the following conditions are ful- filled: (1) that the predicate of the major premise is not a singular term; (2) that "some"' in the sense of "one at least" is the only sign of a particular proposi- tion employed in the syllogism; (3) that we have no information about the matter of a proposition, but only about its quality and quantity. It is necessary to pre- suppose these conditions in order to simplify the ex- planation of the categorical syllogism. DICTA OF THE THREE FIRST FIGURES 68. The usual method of procedure in works on Logic is first to prove the general rules of the syl- logism and then to take up the figures and moods. But the whole treatment of the categorical syllogism will be much simpler and more intelligible if this method is reversed. In the method which is com- monly adopted the explanation of the syllogism is needlessly complicated. Moreover, in this method the proof of the general rules proceeds on the hypothe- sis that the predicate is undistributed in affirmative, DICTA OF THE THREE FIRST FIGURES 93 and distributed in negative, propositions (cf. Appen- dix: Note on Section 29). Again, the general rules, as usually derived, do not of themselves make it evident that a syllogism which conforms to them is valid : all they do is to w^arn us that a syllogism cannot violate them without being invalid. If, instead of following the usual order, we start with the dicta of the three first figures, we shall find that our treatment of the categorical syllogism is clear of these disadvantages. In the first place, by means of the dicta we make evident from the outset the validity of arguing in any of the three first figures. Secondly, we get rid of any need of referring to the predicate as distributed or undistributed. Thirdly, the rules of the different figures will be evident from their respective dicta. Fourthly, the dicta will supply us immediately with all the moods of the three first figures. Fifthly, the general rules of the syllogism will be seen to follow at once as corollaries from the dicta. 69. Aristotle laid down the Dictum de omni et nuUo as the axiom upon which rests the validity of the categorical syllogism. But it applies only to the first figure. This dictum has been worded in various ways by different authors. Our own wording of it will be found in the dictum of the first figure. For the sake of clearness, the letters P, M, S will be inserted in brackets in the statement of the dicta. It must be borne in mind that P denotes the predicate of the conclusion, and that the premise in which it is compared with M is the major premise. Again, S denotes the subject of the conclusion and is compared with M in the minor premise. 94 AN ELEMENTARY HANDBOOK OF LOGIC In section 19 we saw that judgment may be defined as an act of the mind asserting that the object repre- sented by one idea possesses or lacks the attribute represented by another idea. For example, "Man is rational" = "Man possesses the attribute rational ;" "The horse is not rational" = "The horse lacks the at- tribute rational." When the words "possess" and "lack" occur in the dicta, they signify affirmative and negative propositions respectively. Dictum of the First Figure : "Any attribute [P] which is affirmed or denied of the formal object [M] of a uni- versal idea may be affirmed or denied respectively of any- thing [6*] which possesses that formal object [M]." Cf. 23, N. B. (More' exactly stated, the dictum would run as fol- lows : "Any object with a given attribute [P] which is asserted to be identical with or different from the formal object [M] of a universal idea may be asserted respectively to be identical with or different from anything [S] which is identical with that formal ob- ject [Ml"). "Man" is the formal object of a universal idea. "This man" is the formal object of a singular idea. The formal object of an idea is that which is explicitly rep- resented by the idea (cf. 2). A universal idea is an idea which represents severally many individuals, and hence it can be predicated of each of them (cf. 8). We may illustrate the dictum of the first figure by a concrete example : Every man is mortal. Every king is a man. Therefore Every king is mortal. DICTA OF THE THREE FIRST FIGURES 95 If we insert in the dictum the terms of this syllogism, the dictum will run as follows: "Any attribute ['mortal'] which is affirmed of the formal object ['man'] of a universal idea may be affirmed of anything ['eyery king'] which possesses that formal object ['man']." Dicta of the Second Figure: 1. "The formal object [P] of a universal idea which possesses a certain attri- bute [M] may be denied of anything {S\ which lacks that attribute [M]." 2. "The formal object [P] of a universal idea which lacks a certain attribute [M] may be denied of any- thing \S\ which possesses that attribute {M\." (The two dicta of the second figure may be com- bined into one as follows: "The formal object [P] of a universal idea which possesses or lacks a certain at- tribute [M] may be denied of anything [S] which respectively lacks or possesses that attribute [M]"). The second dictum of the second figure is illustrated in the following syllogism : No plant is sentient, Every animal is sentient. Therefore No animal is a plant. "The formal object ['plant'] of a universal idea which lacks a certain attribute ['sentient'] may be denied of anything ['every animal'] which possesses that attribute ['sentient']." Dicta of the Third Figure: 1. "Any attribute [P] which is affirmed or denied of the formal object [M] of a universal idea which [Af] in any case possesses a second attribute \_S'\ may be affirmed or denied re- spectively of something {S\ having the second attribute." 96 AN ELEMENTARY HANDBOOK OF LOGIC 2. "Any attribute [P] which in some cases is affirmed or denied of the formal object [M] of a universal idea which [M] always possesses a second attribute [6"] may be affirmed or denied respectively of something [5"] hav- ing the second attribute." The following syllogism will serve to illustrate the first dictum oi the third figure: Every man is mortal, Every man is rational, Therefore Some rational beings are mortal. "Any attribute ['mortal'] which is affirmed of the formal object ['man'] of a universal idea which ['man'] in any case possesses a second attribute ['rational'] may be affirmed of something ['some rational beings'] having the second attribute." 70. The dicta of the first and second figures are self- evident. The dicta of the third figure are at least as evident as the converse of an affirmative proposition. The dicta of the second figure amount to this: "An object without a certain attribute is not an object with that attribute," "An object with a certain attribute is not an object without that attribute." Thus, it will be observed that the dicta of the second figure are nothing more or less than the Law of Contradiction stated categorically (cf. 53 ad fin.). The dicta of the third figure may be interpreted as follows: 1. If "All M is S" or "Some M is S" (minor), then (by conversion) some objects with the attribute S have the attribute M, and consequently, some ob- jects with the attribute S have any given attribute [P] which always accompanies M (major = "A\\ M DICTA OF THE THREE FIRST FIGURES 97 is P") or 'lack any given attribute [P] which never accompanies M (major="'No M is P"). 2. If "All M is S" (minor), that is, if all objects with the at- tribute M have the attribute S, then (by conversion) some objects with the attribute S have any given at- tribute [P] which is sometimes combined with M (majors "Some M is P") or lack any given at- tribute [P] which is sometimes not combined with M (major ^= "Some M is not P"). It is to be observed that in the third and fourth figures we must convert the minor premise in order that S, its predicate, may become subject of the con- clusion ; biit when we do convert it, we still retain in mind all the information that was provided by the minor premise. Hence, a particular negative proposi- tion cannot appear as minor premise in the third or the fourth figure (cf. 31). It will be noticed that in the minor premise of the last syllogism in the preceding section the term "rational" is in the predicate position and is used abso- lutely; in the conclusion it is in the subject position and is used distributively (cf. 18, 8). When the sub- ject-term of a proposition is used distributively and is universal, e.g. "All men are mortal," we can easily change it into the absolute use without converting the proposition, thus, "Man is mortal ;" and we can just as easily change it back again to the distributive use. But we cannot change the absolute use of a predicate-term to the distributive use, unless we con- vert the proposition. The first and second figures involve ea:ch only one process of reasoning. The third and fourth iigiires 98 AN ELEMENTARY HANDBOOK OF LOGIC involve two processes ; for they each involve a process of conversion. Hence, the third and fourth figures are much inferior to the first and second in point of simplicity. The first figure was regarded by Aristotle as the most perfect; and it is so for three reasons. First, it is the only figure in which the conclusion can be universal affirmative. Secondly, the subject and the predicate of the conclusion occupy respectively a subject and a predicate position in the premises. Thirdly, no term is used distributively or absolutely in the conclusion which was not used in the same way in the premises (cf. 18, 8). RULES AND MOODS OF THE FIRST FIGURE 71. In the dictum of the first figure we read, "Any attribute [ P] which is affirmed or denied of the form- al object [M] of a universal idea." Since the words "the formal object [M] of a universal idea" are un- qualified (i. e. unrestricted or unlimited), this dictum provides that the major premise shall be universal (cf. 23, N. B.). It further provides that the minor premise shall be affirmative; for it says "anything [S] which possesses that formal object [M]." Hence, the two rules of the first figure are — RULE I : The major premise must he universal. RULE II : The minor premise must be affirmative. The dictum of the first figure warrants four and only four moods. The words "Any attribute [P] which is RULES AND MOODS OF THE SECOND FIGURE 99 affirmed or denied" allow the major premise to be either affirmative or negative; and the words "any- thing [S] which possesses" allow the minor premise to be either universal or particular affirmative. The words "affirmed or denied respectively" indicate that the conclusion is to be of the same quality as the major premise, while the word "anything" shows that the conclusion is to be of the same quantity as the minor. Thus, in the first figure we have the four following moods : . AAA,. EAE, All, EIO. These moods are contained in the following mnemonic line: Barbara, Celarent, Darii, Ferioque prioris (cf. 91). RULES AND MOODS OF THE SECOND FIGURE 72. In the first dictum of the second figure we read, "The formal object [P] of a universal idea which pos- sesses a certain attribute [M] ;" and in the second dictum, "The formal object [P] of a universal idea which lacks a certain attribute [M]." Since the words "The formal object [P] of a universal idea" are un- restricted in both dicta, these dicta provide that the major premise shall be universal. They further pro- vide that one premise shall be negative; for in the first dictum we read "anything [S] which lacks that attribute [M]," and in the second dictum, "The formal object [P] . . . which lacks a certain attribute [M]." Again, they provide that one premise shall be affirmative; for in the first dictum are the words "The formal object [P] . . . which possesses a certain attribute [M]," and in the second dictum, "anything 100 AN ELEMENTARY HANDBOOK OF LOGIC [S] which possesses that attribute [M]." Hence, the three rules of the second figure are — RULE I : The major premise must be universal. RULE II : One premise must he negative. RULE III : One premise must he affirmative. The first dictum of the second figure warrants two and only two moods. It allows a universal affirmative major premise ("The formal object [P] of a universal idea which possesses") with a universal or particular negative minor ("anything [S] which lacks"). This dictum, then, gives us the moods, AEE, AOO. In like manner, the second dictum warrants two and only two moods. It allows a universal negative major premise ("The formal object [P] of a universal idea which lacks") with a universal or particular affirma- tive minor {"anything [S] which possesses"). The two moods, then, warranted by the second dictum are EAE, EIO. These two moods together with the two allowed by the first dictum are contained in the fol- lowing mnemonic line : Cesare, Camestres, Festino, Baroco, secunda. RULES AND MOODS OF THE THIRD FIGURE 73. In the first dictum of the third figure we read, "Any attribute [P] which is affirmed or denied of the formal object [M] of a universal idea which [M] in any case possesses a second attribute [S] ;" and in the second dictum, "Any attribute [P] which in some cases is affirmed or denied of the formal object [M] RULES AND MOODS OF THE THIRD FIGURE 101 of a universal idea which [M] always possesses a second attribute [S]." Both dicta provide that one premise shall be universal; for in the first dictum the words "The formal object [M] of a universal idea" are unrestricted ; and in the second dictum we read "which [M.\ always possesses a second attribute [S]." Again, both dicta provide that the minor premise shall be affirmative; for in both we read, "which [M] . . . possesses a second attribute [S]." Moreover, they both provide that the conclusion shall be particular; for both contain the words, "may be affirmed or denied of something [S]." Hence, the three rules of the third figure are — RULE I : One premise must be universal. RULE II : The minor premise must he affirmative. RULE III : The conclusion must be particular. The first dictum of the third figure warrants four and only four moods. It allows the major premise to be either universal affirmative or universal negative: universal, because the words "the formal object [M] of a universal idea" are unrestricted ; either affirmative or negative, because the dictum reads "Any attribute [P] which is affirmed or denied." Whether the major premise be affirmative or negative, this dictum allows the minor premise to be either universal or particular affirmative ("which [M] in any case possesses a sec- ond attribute [S]"). Thus, this dictum warrants the moods, AAI, All, EAO, EIO. The second dictum warrants two and only two moods. The major premise may be either particular affirmative or particular negative ("Any attribute [P] which in some cases is affirmed or denied"); but the 102 AN ELEMENTARY HANDBOOK OF LOGIC minor premise must be universal affirmative ("which [M] always possesses a second attribute [S]"). This gives us the moods lAI, OAO. The six moods which are warranted by the two dicta of the third figure are usually combined as follows : Tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet. Note. — If it be desired to prove the rules of the three first figures independently of the dicta, it may be done as follows: First, we must show that two negative premises cannot yield a conclusion. When both premises are negative, the subject-object in both premises is really distinct from the predicate-object (cf. 19, 4, 22 ad fin.). Suppose the subject-object to be the same in both premises (Figure 3). Then we have no means of comparing the predicate-object of the major premise with the predicate-object of the minor; for both premises may be true, whether the predicate-objects are one and the same thing or different things. For example, the two premises "No M is P" and "No M is S" may be true, whether S is or is not P. And the same applies mutatis mutandis to the case in which the predicate-object is the same in both premises (Figure 2), and also to the case in which the subject- object in one premise is the same as the predicate- object in the other (Figures 1 and 4). Therefore, from two negative premises no conclusion can be drawn. Figure 1 : If M has the attribute P, it does not follow that M alone has this attribute ; and hence, there RULES OF THE THREE FIRST FIGURES 103 may be objects which have the attribute P and lack the attribute M. Consequently, if S lacks the attri- bute M, we cannot tell whether S lacks the attribute P; for S may be among the objects which have the attribute P and lack the attribute M. Therefore, the minor premise of the first figure cannot be negative. When some M has the attribute P, some other M may lack the attribute P. Hence, if S has the attri- bute M, we cannot conclude that it has the attribute P; for S may be among the objects which have the attribute M and lack the attribute P, When some M lacks the attribute P, some other M may have the at- tribute P. Hence, if S has the attribute M, we cannot conclude it lacks the attribute P ; for S may be among the objects which have both the attributes M and P. Therefore, the major premise of the first figure cannot be particular. Figure 2 : If both premises are affirmative, then both P and S have the attribute M. But there may be some objects which have the attribute M and lack the attribute P. Hence, from the fact that S has the attribute M we cannot conclude that it has the attri- bute P ; for S may be among the objects which have the attribute M and lack the attribute P. Therefore, no conclusion can be drawn in the second figure from two affirmative premises. When some P is not M, that is, when some P lacks the attribute M, there may be some objects with the attribute P which have the attribute M. Hence, if S has the attribute M, it does not follow that it lacks the attribute P ; for.S may be among the objects which have both the attributes M and P. When some P is M, 104 AN ELEMENTARY HANDBOOK OF LOGIC that is, when some P has the attribute M, there may be some objects with the attribute P which lack the attribute M. Consequently, if S lacks the attri- bute M, it does not follow that it lacks the attribute P; for S may be among the objects which have the attribute P and lack the attribute M. Therefore, no conclusion can be drawn in the second figure when the major premise is particular. Figure 3: If M has the attribute P, it does not follow that M alone has this attribute ; and hence, there may be objects which have the attribute P and lack the attribute M. Consequently, if no M is S, that is, if every M lacks the attribute S, which is equivalent to saying, if every S lacks the attribute M, we cannot tell whether S lacks also the attribute P; for S may be among the objects which have the attribute P and lack the attribute M. If some Mis not S, there can be no conclusion, because a particular negative cannot be converted (cf. 70). Therefore, the minor premise of the third figure cannot be negative. If some M possesses or lacks the attribute P, and some M has the attribute S, we have no means of comparing S with P ; for the M's which possess or lack the attribute P may be entirely different objects from the M's which have the attribute S. Therefore, one of the premises in the third figure must be universal. Since the minor premise of the third figure must be affirmative, and since this premise must be converted in order to obtain S, its predicate, for the subject of the conclusion, the subject of the conclusion must receive the particular sign of quantity (cf. 38), There- RULES AND MOODS OF THE FOURTH FIGURE 105 fore, the conclusion of the third figure must be par- ticular. RULES AND MOODS OF THE FOURTH FIGURE 74. With the possible exception of the moods EAO and EIO, the fourth figure is worthless. The position of the subject and the predicate in the conclusion is the reverse of what it was in the premises. Hence, whether a term be used distribiitively or absolutely in the premises, it is used in the opposite way in the conclusion (cf. 18, 8). In order, then, to frame an argument on the plan of the fourth figure, we have to twist and torture it into an unnatural shape. It may safely be said, that the arguments of everyday life, whether scientific or concerned with practical ques- tions, never ^ssume the form of the fourth figure. We should, therefore, have contented ourselves with a bare mention of this figure, were it not the tradition to give it a place in works on Logic. As it is, it would be a waste of time to construct dicta for this figure; for its moods are easily derived from the moods of the first and third figures. Cf. Appendix : Note on Section 74. The rules of the fourth figure are as vague as the general rules of the syllogism, and afford us no direct aid for the construction of syllogisms or the detection of fallacies. The following are the rules of this figure : RULE I : // the major premise is affirmative, the minor must be universal. 106 AN ELEMENTARY HANDBOOK OF LOGIC RULE II: // either premise is negative, the major must be universal. RULE III: // the minor premise is affirmative, the conclusion must be particular. If we let the major and minor premises of the three first moods in the first figure become minor and major premises respectively, we shall obtain three moods in the fourth figure. Thus, by transposing the premises of Barbara, Celarent, and Darii, we obtain the follow- ing moods in figure four: AAI, AEE, lAI. Again, if we convert simply the major premise of the moods Felapton and Ferison in the third figure, we obtain two moods in the fourth figure, viz. EAO and EIO. These five moods are contained in the line — Bramantip, Camenes, Dimaris, Fesapo, Fresison. 75. The moods we have mentioned are the only ones in the four figures, unless we wish to draw a par- ticular conclusion when we may draw a conclusion which is universal. Thus, in the first figure, instead of the moods AAA and EAE, we may have AAI and EAO ; in the second figure, instead of EAE and AEE, we may have EAO and AEO ; in the fourth figure, instead of AEE, we may have AEO. When we draw a particular conclusion from prem- ises which warrant a universal conclusion, the par- ticular conclusion is called a "weakened" conclusion, and the mood is called a Subaltern Mood. As we saw in the preceding paragraph, there are five subaltern moods in the four figures. Apart from the subaltern moods, the four figures have nineteen moods, which are enumerated in the following mnemonic lines: THE MNEMONIC LINES 107 Barbara, Celarent, Darii, Ferioque prions; Cesare, Camestres, Festino, Baroco, secundce; Tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, hahet; Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison. CHAPTER XII GENERAL RULES OF THE CATEGORICAL SYLLOGISM 76. The rules of the three first figures are direct and easy of application. Given the figure, two rules, or at the most three, are a sufficient test of the validity of a syllogism. The general rules of the syllogism, on the contrary, applying as they do to all the figures, are more indefinite in character; and, by reason of the numerous details which they require to be attended to, their employment for the detection of fallacy is much more difficult. It is plain, then, that the validity of a categorical syllogism should ordinarily be tested by the rules of the figures rather than by the general rules. It has been observed that all the rules and moods of the three first figures are contained in their dicta. It will now be shown that in these dicta are contained also the general rules of the syllogism, so far as they are true and useful. Consequently, to know the dicta and their more obvious implications is to have mas- tered the categorical syllogism. Since the fourth figure is unnatural, and since its moods are easily derivable from the moods of the first and third figures, its special requirements will be dis- regarded in the comments we are about to make upon the general rules. 108 GENERAL RULES OF THE CATEGORICAL SYLLOGISM 109 77. The Latin wording of the general rules is as follows : 1. Terminus esto triplex: major, mediusque, minorque. 2. Latins hunc, quam prjemissse, conclusio non vult. (Nequaquam medium capiat conclusio oportet). 3. Aut semel aut iterum medius generaliter esto. 4. Utraque si prasmissa neget, nil inde sequetur. 5. x\mbae afifirmantes nequeunt generare negantem. 6. 7. Pejorem sequitur semper conclusio partem. 8. Nil sequitur geminis ex particularibus umquam. In English the rules run as follows : 1. The simple categorical syllogism contains three and only three terms, and is composed of three and only three propositions. 2. No term may be distributed in the conclusion which was not distributed in the premises. (The conclusion must not contain the middle term). 3. The middle term must be distributed at least once in the premises. 4. From two negative premises no conclusion can be drawn. , 5. Two affirmative premises cannot yield a negative conclusion. 6. A negative premise requires a negative conclusion. 7. A particular premise requires a particular conclu- sion. 8. From two particular premises no conclusion can be drawn. 78. Few occasions occur for appealing to the rule in- closed in the parenthesis. In a categorical syllogism the conclusion merely expresses the objective identity or diversity of the two ideas which in the premises were 110 AN ELEMENtARY HANDBOOK OF LOGIC compared with the idea expressed by thp middle term. Consequently, an argument which should express the middle term in the conclusion would not be a categor- ical syllogism even in appearance (cf. 45, Note). It should, however, be remarked in passing that the intro- duction of the middle term into the conclusion may be attended by a fallacy not unlike the one that was noticed in connection with Eduction by an Added Determinant (cf. 42). For example, "These men are tennis-players ; These men are poor ; Therefore these men are poor tennis-players." Rule 3 is necessary only for the first figure. It is cov- ered by Rule 8 for the third figure. It is meaningless when applied to the second figure. In the second figure the middle term is neither distributed nor undistributed ; for, being predicate in both premises, it is not used dis- tributively at all. Cf. Appendix: Note on Section 29. The rule which is numbered 7 in the English word- ing of the Rules is superfluous. It is covered by Rule 2 for all three figures. Since the only term which can be distributed or undistributed in the conclusion is the minor term, Rule 2 means that the minor term must not be distributed in the conclusion, if it was not dis- tributed in the premises ; it can be neither distributed nor undistributed in the premises of the third figure; for in this figure it is predicate in its premise. EXPLANATION OF THE GENERAL RULES 79. RULE I : The simple categorical syllogism con- tains three and only three terr,is, and is composed of three and only three propositions. EXPLANATION OF THE GENERAL RULES 111 Rule 1 is provided for in all five dicta; for everyone of them requires three and only three terms, viz. P, M, and S, and prescribes that P and S shall be com- bined separately with M — thus giving us the two . premises, — and then combined with each other — thus giving us the conclusion. This rule follows from the very nature of the simple categorical syllogism. Since this type of syllogism deals with the formal objects of three and only three ideas, it is evident that three and only three ideas can be expressed by the terms of any argument which is a simple categorical syllogism. We saw in section 12 that a term is the verbal ex- pression of an idea. Hence, if two different words or two different phrases express one and the same idea, these words or phrases, though differing in spelling, are one and the same term, and consequently, may be substituted for each other in a syllogism without alter- ing it in any way. On the other hand, in order to have three and only three terms in an argument, it is essen- tial that three and only three ideas be expressed by the words of the argument. If only three words are used, but one of them expresses one idea in one part of the argument and another idea in another part, the three words are four terms. Rule 1, then, warns us particularly against the use of an ambiguous word in the syllogism. The follow- ing examples violate this rule : "The bow is the fore- most part of a ship; But a nod of the head is a bow; Therefore a nod of the head is the foremost part of a ship." "Man is a species ; But Cicero is a man ; There- fore Cicero is a species" (cf. 18, 107). 112 AN ELEMENTARY HANDBOOK OF LOGIC The violation of Rule 1 is called Quarternio Termi- norvim or the Fallacy of Four Terms. 80. Apparent Exceptions to Rule 1. It is a point of special importance to remark that the syllogism is an argument which expresses only one process of infer- ence (cf. 55). When, therefore, we say that the simple categorical syllogism can have only three terms, we are contemplating the verbal expression of a single process of inference. Doubtless, if an argument ex- presses more than one process, it may contain more than three terms. Consequently, no argument of four or more terms, even though conveyed in three cate- gorical propositions, can be fairly adduced as an ex- ception to Rule 1, if it can be shown to be an elliptical expression of more than one process. The horse is larger than a dog, The elephant is larger than a horse, Therefore The elephant is larger than a dog. It is not unusual to find examples of this kind cited by logicians as a proof that the categorical syllogism may contain more than three terms. This argument, they say, is valid, though it has four terms, namely, "horse," "larger than a dog," "elephant," and "larger than a horse." But, as a matter of fact, this argument is not a complete expression of a process of inference. More than that, it is an elliptical expression of two processes. "The horse is larger than a dog:" — this is not a premise of the process of inference by which we reach the conclusion, "Therefore the elephant is larger than a dog;" it is a datum from which we derive by eduction a major premise which is not expressed, viz. EXPLANATION OF THE GENERAL RULES 113 "Every thing larger than a horse is larger than a dog" (cf. 45). This eduction is one process that is not com- pletely stated. Again, we could not arrive at the con- clusion, "Therefore the elephant is larger than a dog,'' unless we had some common term or basis with which to compare the subject and predicate of the conclusion, viz. "elephant" and "(thing) larger than a dog" (cf. Appendix: Note on Section 21). In the original argu- ment this common or middle term is not expressed in two premises. Consequently, the two propositions, "The elephant is (a thing) larger than a horse," "Therefore the elephant is (a thing) larger than a dog," are in reality an enthymeme (cf. 55). The argu- ment, fully stated, would run as follows : The horse is (a thing) larger than a dog, Therefore Every thing larger than a horse is (a thing) larger than a dog. The elephant is (a thing) larger than a horse. Therefore The elephant is (a thing) larger than a dog. The middle term is "thing larger than a horse." It is obvious that we could not have drawn the conclu- sion in the argument as originally stated, had we not recognized that everything larger than a horse is larger than a dog. Cf. Appendix: Note on Section 80. Gold is a precious metal. This cup is plated with gold. Therefore This cup is plated with a precious metal. There arc five terms in this argument, viz. "gold," "precious metal," "this cup," "plated with gold," and "plated with a precious metal." But here also there are 114 AN ELEMENTARY HANDBOOK OF LOGIC two processes of inference underlying the argument. From the proposition, "Gold is a precious metal," an unexpressed major premise is derived by means of Eduction by Complex Conception (cf. 44). Expressed in full, the argument would run — Gold is a precious metal. Therefore Every thing plated with gold is plated with a precious metal, This cup is plated with gold, Therefore This cup is plated with a precious metal. 81. RULE II: No term may be distributed in the conclusion which was not distributed in the premises. This rule is needlessly vague. It should read, "The minor term should not be distributed in the conclu- sion, if it was not distributed in its premise" (cf. 78). Rule 2 is provided for in the dicta of the third figure, which allow only a particular conclusion. It is pro- vided for in the dicta of the first and second figures which warrant a conclusion only with reference to the things ("anything") mentioned in the minor premise. The reason for Rule 2 is that, if the minor term were distributed in the conclusion without having been dis- tributed in its premise, the conclusion would be as- serting more than was implied in the premises, and hence it would not be logically dependent upon them. This rule is violated in the following examples: "All criminals should be imprisoned ; But some Ameri- cans are criminals; Therefore all Americans should be imprisoned." "No birds are rational ; But all birds are bipeds; Therefore no bipeds are rational." EXPLANATION OF THE GENERAL RULES 115 The violation of Rule 2 is called Illicit Process 6f the Minor or Illicit Minor. RULE III: The middle term must be distributed at least once in the premises. We commented on this rule in section 78. 82. RULE IV: From two negative premises no con- clusion can be drawn. Rule 4 is provided for in the dicta of the first and third figures, all of which require the minor premise to be affirmative. It is provided for in the first dictum of the second figure, v^^hich requires an affirmative major premise ("The formal object [P] .... which possesses a certain attribute"), and in the second dictum of the second figure, which requires an affirma- tive minor premise ("anything . [S] which possesses that attribute"). This rule may be proved independently of the dicta as follows : If both premises are negative, the subject- object in both premises is asserted to be really distinct from the predicate-object (cf. 19, 4, 22 ad fin.). Sup- pose the subject-object is the same in both premises (Figure 3) : — Then we have no means of comparing the predicate-object of the major premise with the predicate-object of the minor so as to determine their identity or diversity; for both premises may be true, whether the predicate-objects are identical or different (distinct). For example, the two premises "No M is P" and "No M is S" may be true, whether S is or is not P. This is shown concretely in the following examples: "No horses are rational; But no horses are men ; Therefore all men are rational." "No horses 116 AN FXEMENTARY HANDBOOK OF LOGIC are rational ; But no horses are dogs ; Therefore no dogs are rational." What has just been said of the case in which the subject-object is the same in both premises holds true mutatis mutandis in the case where the predicate- object is the same in both premises (Figure 2), and also in the case where the subject-object of oije premise is the same as the predicate-object of the other (Figures 1 and 4). The violation of Rule 4 is called the Fallacy of Two Negatives. 83. Apparent Exceptions to Rule 4. The remark which was made in the first paragraph of section 80 should be recalled in connection with the present rule. No man who is not secure is happy, No tyrant is secure, Therefore No tyrant is happy. This argument is not a simple categorical syllogism ; for it is an elliptical expression of two processes of inference, viz. an eduction and a process whose verbal expression is the simple categorical syllogism. The common term or basis which served for the compari- son of the subject and predicate of the conclusion is not expressed in two premises in the argument, and therefore the argument is not a simple categorical syl- logism. It is rather a double enthymeme. The con- clusion was reached by first inferring either the ob- verted converse of the major premise (cf. 40) or the obverse of the minor. Which of these two processes is performed by a given individual depends on his mental habits or the particular way in which the argumeiit EXPLANATION OF THE GENERAL RULES 117 Strikes him at the moment of presentation. The aver- age mind would probably obvert the minor. In any case, the conclusion was reached in one of the fol- lowing ways : No man who is not secure is happy, Therefore Everyone who is happy is secure. No tyrant is secure. Therefore No tyrant is happy. The first line being omitted, the syllogism is in the second figure with "secure" as middle term. No man who is not secure is happy. No tyrant is secure. Therefore Every tyrant is a man who is not secure. Therefore No tyrant is happy. Omitting the second line, we have a syllogism in the first figure in which the middle term is "man who is not secure." 84. RULE V: Two affirmative premises cannot yield a negative conclusion. RULE VI: A negative premise requires a negative conclusion. Rules 5 and 6 are provided for in the dicta of the second figure, both of which prescribe a negative premise and a negative conclusion. They are also provided for in the dicta of the first and third figures; for these dicta warrant a negative conclusion only when one of the premises is negative, and they pre- scribe a negative conclusion when one of the premises is negative, as is evident from the following words which appear in all of them: "Any attribute [P] 118 AN ELEMENTARY HANDBOOK OF LOGIC which is affirmed or denied . . . may be affirmed or denied respectively." Without reference to the dicta these rules may be proved as follows: Proof of Rule V: If both premises are affirm- ative, they assert that the objects denoted by the major and minor terms are both identical with the object denoted by the middle term (cf. 19). Hence, by the Axiom of Identity (cf. 63) they are identical with each other, and the conclusion must be affirma- tive. The following example violates Rule 5 : "All negroes are black; But some Americans are negroes; Therefore some Americans are not black." Proof of Rule VI: If one premise is negative, then, by Rule 4, the other premise must be affirmative. The negative premise asserts that the object denoted by one of the extremes (cf. 64) is different from the object denoted by the middle term; the affirmative premise asserts that the object denoted by the other extreme is identical with the object denoted by the middle term. Hence, by the Axiom of Diversity, the objects denoted by the extremes, that is, by the major and minor terms, are different from each other, and the conclusion must be negative. Rule 6 is violated in the following example: "No noble-minded men are selfish; But some statesmen are noble-minded men; Therefore some statesmen are selfish." 85. Apparent Exceptions to Rule 5, The apparent exceptions to Rule 5 generally involve a process of obversion in addition to the process of the categorical syllogism. "All men are rational; This animal is EXPLANATION OF THE GENERAL RULES 119 irrational; Therefore this animal is not a man." The two processes of inference of which this example is an elliptical expression are stated in full as follows : All men are rational, This animal is irrational, Therefore This animal is not rational. Therefore This animal is not a man. Omitting the second line, we have a syllogism in the second figure. 86. Apparent Exceptions to Rule 6. The apparent exceptions to Rule 6 involve at least two processes of inference. "Every material that is not compound is an element ; Gold is not compound ; Therefore gold is an element." The complete expression of the infer- ences underlying this argument is as follows : Every material that is not compound is an element. Gold is not compound, Therefore Gold is a material that is not compound, Therefore Gold is an element. With the second line omitted, the argument is a syl- logism in the first figure having for its middle term "material that is not compound." RULE VII: A particular premise requires a particu- lar conclusion. As we saw in section 78, this rule is superfluous: it is covered by Rule 2. 87. RULE VIII: From two particular premises no conclusion can he drawn. Rule 8 is provided for in all the dicta, everyone of 120 AN ELEMENTARY HANDBOOK OF LOGIC which prescribes a universal premise. This may be seen from the rules of the three first figures. For a theoretical proof of Rule 8 which is inde- pendent of the dicta, consult the Note at the end of section 73. Here it will be sufficient to prove the rule concrefely. Take, first, a case in which both premises are affirm- ative. The conclusion will then be affirmative (Rule 5). Let the letters P, M, S stand for "palace," "mag- nificent," and "statue," respectively. Arrange these terms in particular affirmative propositions in any of the figures, and it will be found that the premises are true. Nevertheless, we know that the conclusion, "Some statues are palaces," is false (cf. 65). Take, secondly, a case in which one premise is negative. This should give us a negative conclusion (Rule 6). Let P, M, S denote "pointed," "massive," and "spear," respectively. These terms may be ar- iranged in particular propositions according to any of the figures. Moreover, in any figure either the major or the minor premise may be negative. In every case we know that the premises are true. And yet we can- not draw the conclusion, "Some spears are not pointed;" for we know that this is not true. 88. Apparent Exceptions to Rule 8. In section 67 we said that in attempting to determine the rules of the categorical syllogism we supposed that "some," in the sense of "one at least," was the only sign of a particu- lar proposition employed in the syllogism. If signs like "most" and "two-thirds" are used, Rule 8 does not apply universally ; for then the proof of the rule would EXPLANATION OF THE GENERAL RULES 121 not hold. Consequently, the argument given below is not an exception to Rule 8; for it disregards the con- dition on which the rule was laid down. Most Americans are free, Most Americans are white. Therefore Some white persons are free. "Most" has the same force as the expression "at least one more than half," and hence the conclusion is valid. By reason of the sign "most" we know that the Americans who are free, referred to in the major premise, overlap the Americans who are white, re- ferred to in the minor. The following dictum applies to syllogisms like the one we have just been considering: "Any attribute [P] which in most cases is affirmed or denied of the formal object [M] of a universal idea which [M] in the majority of cases possesses a second attribute [S] may be affirmed or denied respectively of something [S] having the second attribute." CHAPTER XIII REDUCTION OF CATEGORICAL SYLLOGISMS 89. The reduction of a categorical syllogism is the process of reconstructing a syllogism of the second, third, or fourth figure upon the plan of the first so as to obtain the same conclusion. Since the general rules of the categorical syllogism, when proved apart from the dicta, are negative in character, not making manifest by themselves the validity of a conclusion, but putting us on our guard against certain causes of invalidity, the original pur- pose of reduction was to submit every mood outside the first figure to the test of the Dictum de omni et nulla (cf. 69). But this could not be done except by refashioning the mood according to the first figure, because the Dictum applies to this figure alone. How- ever, recourse to this test is no longer necessary, for we have dicta which apply to all the moods of the three first figures. Nevertheless, though the reduction of syllogisms is unnecessary as a means of verification, it is a good logical exercise, because it increases our knowledge of the relations between the various terms and the various propositions which enter into the syllogism. The employment of concrete examples will show that many arguments will not fit naturally into th§ first figure, 123 REDUCTION OF CATEGORICAL SYLLOGISMS 123 Direct reduction is a reduction by means of conver- sion alone or by means of conversion combined with transposition of the premises. Indirect reduction is a reduction in which the con- tradictory of the conclusion is shown to be inconsist- ent with the premises. We, prove a proposition in- directly, when we show that its contradictory is in- compatible with what is already held to be true (cf. 36). 90. We may illustrate direct reduction by the fol- lowing example, in which the syllogism to be reduced is in the third figure: All metals are easily combined with oxygen. Some metals are lighter than water, Therefore .Some things lighter than Water are easily combined with oxygen. If the minor premise is converted simply, the syl- logism will be in the first figure. Indirect reduction may be illustrated by means of the following syllogism in the second figure : All horses are quadrupeds, Baroco Some animals are not quadrupeds, Therefore Sonie animals are not horses. If this conclusion is not true, its contradictory ("All animals are horses") is true (cf. 36). Combining the contradictory of the conclusion with the premises, which are granted to be true, the three following propositions must be true together : All horses are quadrupeds, .Some animals are not quadrupeds, AU animalg are horses. 124 AN ELEMENTARY HANDBOOK OF LOGIC Omitting the second proposition, we have a syl- logism in the first figure : All horses are quadrupeds, Barbara All animals are horses, Therefore All animals are quadrupeds. The person who admitted the premises of the orig- inal syllogism, but denied the conclusion, would have to hold both the following propositions to be true: "Some animals are not quadrupeds," "All animals are quadrupeds;" but these propositions are incon- sistent with each other, since they are contradictories. Consequently, anyone who grants the original prem- ises must also grant the conclusion which was de- rived from them. Indirect reduction is also called Reductio ad impos- sibile or Reductio ad absurdum. It is only by this method that we can reduce Baroco of the second figure and Bocardo of the third to the first figfure so long as we retain the terms of the original syllogism. THE MNEMONIC LINES 91. The mnemonic lines are here repeated for con- venience of reference : Barbara, Celarent, Darii, Feriogwe prioris; Cesare, Camestres, Festino, Baroco, secundce; Tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet; Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison. De Morgan speaks of these lines as "the magic words by which the diflferent moods have been de- noted for many centuries, words which I take to be THE MNEMONIC LINES 125 more full ol meaning than any that ever were made." The mnemonic lines inform us in detail how each mood of figures 2, 3, and 4 is to be reduced to figure 1. The consonants, b (not initial), d (not initial), 1, n, r, t, are the only letters that have no meaning. The vowels, a, e, i, o, indicate the quality and quan- tity of the propositions in the mood. Thus, the fol- lowing syllogism in the fourth figure is represented by Camenes: A All animals are sentient things, E No sentient things are plants, E Therefore No plants are animals. The initial letters, B, C, D, F, appearing in figures 2, 3, and 4, signify that any given mood in those figures is to be reduced to that mood of the first figure which has the same initial letter. Camestres, for example, is reduced to Celarent: All P is M No M is S No S is M All P is M Therefore No S is P Therefore No P is S Therefore No S is P All unselfish men are lovable. No cruel men are lovable, Therefore No cruel men are unselfish. Reduced to Celarent, this syllogism reads: No lovable men are cruel. All unselfish men are lovable, Therefore No unselfish men are cruel. Therefore No cruel men are unselfish. 126 AN ELEMENTARY HANDBOOK OF LOGIC The use of arrows may help to make the reduction clearer, as follows: All P is M -_^ > No M is S No S is M > All P is M No S is P < ■ No P is S S (in the middle of a word) signifies that the premise immediately preceding it is to be converted simply. Thus, in the reduction of Camestres we converted the minor premise simply. s (at the end of a word) means that the conclusion of the new syllogism is to be converted simply, so that we' may obtain the original conclusion. This is also illustrated in the foregoing reduction of Camestres. p (in the middle of a word) indicates that the premise immediately preceding it is to be converted per accidens. This is done, for example, in reducing Felapton to Ferio: No M is P No M is P All M is S Some S is M Therefore Some S is not P Therefore Some S is not P No metals are organic. All metals combine with oxygen. Therefore Some things that combine With oxygen are not organic. Reduced to Ferio, the syllogism reads : No metals are organic. Some things that combine with oxygen are metals. Therefore Some things that combine with oxygen are not organic. THE MNEMONIC LINES 127 p (at the end o£ a word) shows that the new con- clusion is to be converted per accidens in order to give us the original conclusion. This, for example, is what happens in the reduction of Bramantip to Barbara: All P is M All M is S All M is S All P is M Therefore Some S is P Therefore All P is S Therefore Some S is P All emperors are men, All men are mortal. Therefore Some mortal beings are emperors. Reduce this syllogism to Barbara, and it reads : All men are mortal. All emperors are men, Therefore All emperors are mortal. Therefore Some mortal beings are emperors. m signifies that the premises are to be transposed (metathesis pramissarum) , that is, that the major and minor premises are to become minor and major prem- ises respectively. This is illustrated in the foregoing reduction of Bramantip, as well as in that of Camestres. c indicates that the mood is to be reduced indirectly, and that in the process the premise immediately pre- ceding this letter is to be replaced by the contradictory of the conclusion. CHAPTER XIV THE PURE HYPOTHETICAL SYLLOGISM AND OTHER TYPES OF ARGUMENT 92. The pure hypothetical syllogism is the only type of argument in the present chapter which has not been explained in the chapters preceding, at least in its essential features. If we regard the disjunctive proposition as a variant of the h3'-potheticaI, we shall find that the dilemma is a modification of the mixed or of the pure hypothetical syllogism. The remain- ing types of argument either abridge or modify those already discussed or else they combine several of them into one. THE PURE HYPOTHETICAL SYLLOGISM 93. The pure hypothetical or pure conditional syl- logism is a syllogism which proves a hypothetical conclusion by means of two hypothetical premises in which the antecedent and the consequent of the con- clusion are compared separately with a third potential judgment (cf. 46). For example — If A is B, C is D, If E is F, A is B, Therefore If E is F, C is D. The process of inference expressed by the cate- gorical syllogism has for its remote matter the formal 128 THE PURE HYPOTHETICAL SYLLOGISM 129 objects of three ideas, one of which aiifords a basis of comparison for determining the objective identity or diversity of the two others. The remote matter of the process of inference expressed by the pure hypothetical syllogism are the formal objects of three potential judgments, one of which affords a basis of comparison for determining the logical de- pendence of one of the others upon the remaining one. In the categorical syllogism the subject and predicate of the conclusion (that is, the minor and major terms), besides appearing in the conclusion, appear each once in the premises, and the middle term appears twice in the premises. In the pure hypothetical syllogism the antecedent and consequent of the conclusion, besides appearing in the conclu- sion, appear each once in the premises; and there is a third potential judgment ("A is B" in the foregoing example) which appears twice in the premises and is the common basis of comparison for the antece- dent and consequent of the conclusion. Hence, the antecedent and consequent of the conclusion corre- spond in general to the minor and major terms re- spectively, and the potential judgment which appears only in the premises, to the middle term. Cf. Appen- dix : Note on Section pj. We may, therefore, determine the "figure" of a hy- pothetical syllogism as we did that of a categorical, our criterion being the relative positions of the com- mon potential judgment in the premises. The dicta of the three figures of the pure hypo- thetical syllogism will be found at the end of the Appendix. 130 AN ELEMENTARY HANDBOOK OF LOGIC In Figure 1 the common potential judgment is ante- cedent in the major premise and consequent in the minor. In Figure 2 it is consequent in both premises. In Figure 3 it is antecedent in both premises. If the minor premise be denominated affirmative or negative, according as the potential judgment in its consequent is the same as it was in the major premise or the contradictory of what it was, the following rules may be laid down for these figures: In Figure 1 the minor premise must he affirmative. In Figure 2 the minor premise must be negative. In Figure 3 the conclusion must be particular. The figures are illustrated concretely in the follow- ing examples : Fig. 1 : If the man is not guilty, he should be ac- quitted. If he was away from home, he is not guilty. Therefore If he was away from home, he should be acquitted. Fig. 2 : If the animal is rational, it is a man, If the animal is a quadruped, it is not a man, Therefore If the animal is a quadruped, it is not ra- tional. Fig. 3 : If a man is a genei^al, he is brave. If a man is a general, he is intelligent. Therefore Sometimes if a man is intelligent, he is brave. THE DILEMMA 131 Note. — There are syllogisms in which a hypothet- ical or a disjunctive conclusion is derived from prem- ises, one of which is categorical. For example — If A is B, it is C, DisA, Therefore If D is B, it is C. A is either B or C, DisA, Therefore D is either B or C. No special names have been assigned to these types of syllogism. The name "hypothetico-categorical" has sometimes been employed to designate the mixed hy- pothetical syllogism. If we interpret this name as signifying a syllogism with a hypothetical premise and a categorical conclusion, there will be a certain appro- priateness in designating the two foregoing types of syllogism respectively by the names "categorico- hypothetical" and "categorico-disjunctive." THE DILEMMA 94. The dilemma is an argument in which the ma- jor premise is a compound hypothetical proposition, and the minor premise, a disjunctive proposition alter- natively positing the antecedents or sublating the con- sequents of the major. The force of the constructive dilemma is more stri- king when the disjunctive premise is stated first. Strictly speaking, the word "dilemma" implies only two alternants ; but it is commonly used even when there are three or more. The constructive dilemma is one in which the minor 132 AN ELEMENTARY HANDBOOK OF LOGIC premise alternatively posits the antecedents of the major. The destructive dilemma is one in which the minor premise alternatively suhlates the consequents of the major. In the constructive dilemma the major premise must have at least two different antecedents ; other- wise, the minor premise could not posit alternatively. The consequents may be either the same or different. When the consequents are the same, the dilemma is simple constructive ; when they are different, it is com- plex constructive. In the simple constructive dilemma the conclusion is a simple categorical proposition pos- iting the consequent ; in the complex constructive the conclusion is a disjunctive proposition positing the consequents alternatively. (1) Simple constructive dilemma — If A is B, Eis F; and if C is D, E is F; But either A is B or C is D ; Therefore E is F. (2) Complex constructive dilemma — If A is B, E is F ; and if C is D, G is H ; But either A is B or C is D ; Therefore Either E is F or G is H. In the destructive dilemma the major premise must have at least two different consequents. The ante- cedents may be either the same or different ; and the dilemma will be simple destructive or complex destruc- tive accordingly. (1) Simple destructive dilemma — If A is B, C is D ; and if A is B, E is F ; But either C is not D or E is not F; Therefore A is not B. THE DILEMMA 133 (2) Complex destructive dilemma — li A is B, E is F ; and if C is D, G is H ; But either E is not F or G is not H ; Therefore Either A is not B or C is not D. Concrete examples of the dilemma — If Logic furnishes useful principles, it is worthy of study; and if it trains the mind, it is worthy of study ; But it. either furnishes useful principles or trains the mind; Therefore Logic is worthy of study. If iEschines joined in the public rejoicings, he is inconsistent ; and if he did not, he is unpatriotic ; But he either joined in them or he did not; Therefore He is either inconsistent or unpatriotic. If a man is a leader, he is attentive to de- tails ; and if he is a leader, he has a strong influence upon others ; But either George will not be attentive to details or he will not have a strong in- fluence upon others ; Therefore George will not be a leader. If the man were intelligent, he would per- ceive the mistake ; and if he were honest, he would acknowledge it; But either he does not perceive the mistake or he will not acknowledge it ; Therefore He is not intelligent or he is not honest. The simple constructive and the simple destructive dilemmas may sometimes be stated as follows with a 134 AN ELEMENTARY HANDBOOK OF LOGIC gain in conciseness and force : (1) Simple constructive — If Logic either furnishes useful principles or trains the mind, it is worthy of study; But it either furnishes useful principles or trains the mind; Therefore It is worthy of study. (2) Simple destructive — If a man is a leader, he both attends to de- tails and influences other men; But George either will not attend to details or he will not influence other men ; Therefore George will not be a leader. Note. — Arguments like the following are also fre- quently regarded as dilemmas : If A is B, C is either D or E, But C is neither D nor E, Therefore A is not B. RULES OF THE DILEMMA 95. RULE I: All the pertinent alternants must he stated in the disjunctive premise. If a boy, upon quitting college, has mastered the branches of the college course, he has a well equipped mind ; and if he has not mastered them, he has not profited by the course. But every boy, upon quitting college, has mastered the branches of the college course or he has not mastered them. Therefore every boy, upon quitting college, has a well equipped mind or he has not profited by the course. RULES OF THE DILEMMA 135 In this argument the disjunctive minor premise does not exhaust all the pertinent alternants. The disjunct- ive premise should read : "But every boy, upon quit- ting college, has mastered all the branches of the col- lege course or he has mastered none of them or he has mastered some of them." When we add an alternant to a dilemma that has been directed against us, we are said to escape be- tween the horns of the dilemma. It is to be observed that the number of alternants in the disjunctive premise determines the number of elements in the compound hypothetical premise. RULE II : The logical dependence of each conse- quent upon its antecedent in the major premise must be real and manifest. If I devote myself to my worldly interests, I shall lose my soul; and if I devote myself to the inter- ests of my soul, I shall ruin the position of my family. But I must either devote myself to my worldly interests or to those of my soul. Therefore I shall either lose my soul or ruin the position of my family. In reply to this argument we might say that it does not follow that a man will lose his soul because he devotes himself to his worldly interests; nor does it follow that he will ruin the position of his family because he devotes himself to the interests of his soul. Answering a dilemma in this way is called taking the dilemma by the horns. 136 AN ELEMENTARY HANDBOOK OF LOGIC It may be observed in addition that the disjunctive premise in the last example omits the third alternant of attending to the interests of both. RULE III : The hypothetical premise must state all the pertinent consequents warranted by each antecedent. An Athenian mother, attempting to dissuade her son from entering public life, used the following argu- ment: If you say what is just, men will hate you ; and if you say what is unjust, the gods will hate you. But you must say what is just or what is unjust. Therefore You will be hated. The son answered as follows: If I say what is just, the gods will love me; and if I say what is unjust, men will love me. But I must say what is just or what is un- just. Therefore I shall be loved. * The original argument failed to state all the perti- nent consequents that followed from each of the ante- cedents in the hypothetical premise. The omitted consequents were supplied in the rejoinder and the others left out. When we answer a dilemma in this way we are said to retort the dilemma. FAMOUS DILEMMAS AND SOPHISMS 137 96. Some Famous Dilemmas and Sophisms. The following sophistical arguments have come down to us from ancient Greece : The "Litigiosus." Protagoras, the Sophist, is said to have agreed to teach Euathlus the art of plead- ing, the stipulation being that Euathlus should pay one-half of the fee when fully instructed, and the other half when he won his first case in court. Euathlus put off practising his profession, and finally he was sued by Protagoras for the rest of the fee. The fol- lowing argument was advanced by Protagoras : If Euathlus loses this case, he must pay me, by the judgment of the court; and if he wins it, he must pay me, by the terms of the contract. But he must either win it or lose it. Therefore he must pay me in any case. Euathlus retorted as follows: If I win this case, I ought not to pay, by the judg- ment of the court; and if I lose it, I ought not to pay, by the terms of the contract. But I must 'either win it or lose it. Therefore I ought not to pay in any case. Comment: Protagoras and Euathlus were both guilty of sophistical reasoning. Both started by pronouncing the decision of the court to be binding ; and then they immediately and unconcernedly implied that it was not binding by appealing to the contract. Now, either the litigants regarded the decision of the court as binding, or they did not. If they regarded it as binding, they had no right to appeal from it to the contract. If they did not regard it as binding, they were dishonest in appealing to it at all in their argument. 138 AN ELEMENTARY HANDBOOK OF LOGIC The "Liar." Epimenides, the Cretan, says that all Cretans are liars. If Epimenides' statement is not true, he is a liar; and if it is true, he is a liar, for he is a Cretan. But his statement is either true or not true. Therefore he is a liar. But since he is a liar, his statement is not true that all Cretans are liars. Therefore some Cretans are not liars. But since some Cretans are not liars, Epimenides is not necessarily a liar because he is a Cretan. Therefore we may accept his statement that all Cretans are liars. And so on. Comment: There is no difificulty, if the statement of Epimenides means that Cretans generally utter falsehoods. But if we suppose it to be a fact that Cretans never speak the truth, we cannot suppose a Cretan to say so without involving ourselves in a con- tradiction. The two propositions — "Cretans never speak the truth" and "Epimenides, the Cretan, said so" — cannot be true together {Keynes). Argument of Zeno against Motion: If an arrow moves, it must move either in the place where it is or in the place where it is not. But it cannot move in the place where it is, else it would not be there; nor can it move in the place where it is not, for it is not there to move. Therefore an arrow cannot move. Comment: A body can be in a place in two ways: first, it may be where it was before, and this is rest; secondly, it may be in a place in which it was not THE ENTHYMEME 139 before and from which it immediately recedes, that is, it does not remain where it is, and this is motion. Hence, the arrow not only moves from the place where it was to a place where it is not ; it also moves in the place where it is, but where it is transiently ; that is, it does not remain in the place where it is, but acquires another place, in which again it is, and from which again it recedes without remaining there ; and so on, till it remains in the place it occupies, and this is to rest. If in the major premise of Zeno's argument we sub- stitute for "move" words which express its meaning clearly and unambiguously, we shall see immediately that the premise is not true or, at least, that it is open to two interpretations. Thus : "If an arrow occu- pies in successive instants of time successive positions in space, it must occupy those successive positions either where it is or where 't is not." In order to get rid of the ambiguity in the major premise, we should employ three alternants instead of two, viz. "Either where it is permanently (i.e. where it remains) or where it is transiently (i.e. where it is momentarily but does not remain) or where it is not." THE ENTHYMEME 97. The enthymeme is an abridged syllogism, one of the premises or the conclusion being omitted. The arguments employed in speaking and writing are usually in the form of an enthymeme. This is also the form frequently assumed by the fallacious argu- ment, which often gains in plausibility by failing in explicitness. 140 AN ELEMENTARY HANDBOOK OF LOGIC According as the major premise, the minor premise, or the conclusion is omitted, the enthymeme is said to be of the first, the second, or the third order. The syllogism : All bullies are cowards, But Verres is a bully. Therefore Verres is a coward. The first order: Verres is a bully, Therefore He is a coward. The second order: All bullies are cowards. Therefore Verres is a coward. The third order: All bullies are cowards. And Verres is a bully. For purposes of rhetorical effect, the enthymeme of the third order is much preferable to the complete syllogism. Like the categorical syllogism, the hypothetical syl- logism often appears in abbreviated form ; e.g. "There is a just God ; Therefore the heroic virtues of the Martyrs have been rewarded." THE POLYSYLLOGISM 98. The polysyllogism is a series of syllogisms so connected that the conclusion of one is a premise of another. A prosyllogism is a syllogism the conclusion of which is a premise of another syllogism. An episyllogism is a syllogism a premise of which is the conclusion of another syllogism. Any intermediate syllogism in a polysyllogism is an episyllogism with reference to the syllogism immedi- ately preceding, and a prosyllogism with reference to the one immediately following. THE SORITES 141 The following example illustrates the polysyllogism : A man who desires more than he has is dis- contented, An avaricious man desires more than he has, Therefore An avaricious man is discontented. A miser is an avaricious man, Therefore A miser is discontented. Balbus is a miser, Therefore Balbus is discontented. THE SOEITES 99. The sorites is a polysyllogism in which all the conclusions except the last are omitted, the premises being so arranged that any two successive premises have a common term or a common potential judgment. The two common forms of the sorites are the Aris- totelian and the Goclenian. The Aristotelian sorites was the only form mentioned in works on Logic be- fore the sixteenth century. It is called Aristotelian to distinguish it from the Goclenian sorites. Rudolf Goclenius of Marburg (1547 to 1628) was the first to call attention to the Goclenian form; hence its name. The Aristotelian sorites is one in which the first premise contains the subject of the conclusion and every term common to two successive premises ap- pears first as predicate and then as subject. The Goclenian sorites is one in which the first premise contains the predicate of the conclusion and every term common to two successive premises ap- pears first as subject and then as predicate. 142 AN ELEMENTARY HANDBOOK OF LOGIC Aristotelian Sorites All A is B All B is C All C is D All D is E Therefore All A is fe Goclenian Sorites All D is E All C is D All B is C All A is B Therefore All A is E In the Aristotelian sorites the first premise is a minor premise; for it contains the subject of the conclusion. The remaining premises are major premises. In the Goclenian sorites the first premise is a major premise; for it contains the predicate of the conclusion. The remaining premises are minor premises. When the argument of the Aristotelian sorites is stated in full, all the suppressed conclusions appear as minor premises in successive syllogisms. Thus, the foregoing Aristotelian sorites is an abridged expres- sion of the three following syllogisms : (1) All B All A Therefore All A All C All A Therefore All A All D All A Therefore All A When the argument of the Goclenian sorites is stated in full, all the suppressed conclusions appear as major premises in successive syllogisms. Thus, the foregoing Goclenian sorites is an abridged expres- sion of the three following syllogisms : (2) (3) s C, s B, s C. s D, s C, s D. s E, s D, s E. THE SORITES 143 E, D, E. E, C, E. E, B, s E. (1) All D All C Therefore All C (2) All C All B Therefore All B (3) All B All A Therefore All A The following is a concrete example of the Aris- totelian sorites: Balbus is a miser, A miser is an avaricious man, An avaricious man desires more than he has, A man vvrho desires more than he has is discontented, Therefore Balbus is discontented. The constituent syllogisms of the foregoing sorites are as follows: (1) A miser is an avaricious man, Balbus is a miser, Therefore Balbus is an avaricious man. (2) An avaricious man desires more than he has, Balbus is an avaricious man, Therefore Balbus desires more than he has. (3) A man who desires more than he has is discontented, Balbus desires more than he has, Therefore Balbus is discontented. 144 AN ELEMENTARY HANDBOOK OF LOGIC The following example illustrates the Goclenian sorites : A man who desires more than he has is dis- contented, An avaricious man desires more than he has, A miser is an avaricious man, Balbus is a miser. Therefore Balbus is discontented. This sorites is resolvable into the following syl- logisms : (1) A man who desires niore than he has is discontented, An avaricious man desires more than he has, Therefore An avaricious man is discontented. (2) An avaricious man is discontented, A miser is an avaricious man, Therefore A miser is discontented. (3) A miser is discontented, Balbus is a miser. Therefore Balbus is discontented. The following is an example of the pure hypothet- ical sorites : If Balbus hoards his gold, he is a miser, If he is a miser, he is avaricious. If he is avaricious, he desires more than he has, If he desires more than he has, he is discon- tented. Therefore If Balbus hoards his gold, he is discon- tented. RULES OF THE SORITES 145 100. It will have been observed that the syllogisms into which both the Aristotelian and the Goclenian sorites may be resolved are all in the FirstFigure. Con- sequently, the rules of both forms of sorites will be deducible from the rules of the first figure. Rules of the Aristotelian Sorites. It is to be noticed that in the Aristotelian sorites all the premises except the first are major premises ; the first premise and all the suppressed conclusions are minor premises. RULE I : Every premise, except the first, must he universal. RULE II : Every premise, except the last, must he affirmative. Proof of Rule 1 : Every premise in the Aristotelian sorites, except the first, is a major premise: hence, every premise, except the first, must be universal ; for the major premise in the first figure must be universal (cf. 71). The first premise and all the suppressed conclu- sions are minor premises: therefore, the first premise may be particular. Proof of Rule 2: Every premise, except the last, must be affirmative ; for if any other than the last were negative, it would yield a negative conclusion (cf . 84) . This negative conclusion would become a minor prem- ise, and the minor premise in the first figure cannot be negative. The last premise is the major premise of the last syllogism, and therefore it may be negative. Rules of the Goclenian Sorites. In the Goclenian sorites all the premises, except the first, are minor 146 AN ELEMENTARY HANDBOOK OF LOGIC premises ; the first premise and all the suppressed con- clusions are major premises. RULE I : Every premise, except the last, must he universal. RULE II : Every premise, except the first, must he affirmative. Proof of Rule 1 : Every premise, except the last, must be universal ; for if any other than the last were particular, it would yield a particular conclusion. This particular conclusion would become a major premise, and the major premise in the first figure cannot be particular. The last premise is the minor premise of the last syllogism, and therefore it may be particular. Proof of Rule 2: Every premise in the Goclenian sorites, except the first, is a minor premise ; hence, every premise, except the first, must be affirmative; for the minor premise in the first figure must be affirmative. The first premise and all the suppressed conclusions are major premises: therefore, the first premise may be negative. THE EXPO.SITORY SYLLOGISM 101. The expository syllogism is a syllogism in which the middle term is singular. The major and minor terms may or may not be singular. The ex- pository syllogism is illustrated in the following ex- ample : Cicero was a Roman, Cicero was an orator. Therefore Some orator was a Roman. THE EXPOSITORY SYLLOGISM 147 This type of syllogism is called expository, because it exposes the matter, as it were, before our eyes. The expository syllogism may be constructed ac- cording to any figure ; but it is commonly found in the third, where the middle term is subject in both prem- ises. It is unusual to meet a proposition with a singular term as predicate, unless the subject too is singular. Our warrant for employing a singular term as mid- dle term in the first and third figures is that in argu- ment a proposition with a singular term as subject is equivalent to a universal proposition (cf. 24). If a singular term is middle term in the second figure, the validity of the syllogism will sometinies be de- termined only by a knowledge of its matter. CHAPTER XV THE PREDICABLES AND THE CATEGORIES 102. THE PREDICABLES. One of the busiest functions of the human mind is the detection of points of resemblance and dissimilarity between the multitu- dinous objects which fall under its observation. At first, these objects are apparently a disordered "and chaotic mass. When confronted with this medley of disorganized materials, the mind grows restive, and strives to discover some principle of order. By scruti- nizing, comparing, and analyzing the maze of ma- terials before it, the mind comes to recognize that the objects are not absolutely diverse from each other, that many of them have certain features in common which are not shared by others, while large numbers of the latter also have their own points of agreement. The mind makes a mental note of the characteristics or, at least, of some important characteristic pos- sessed in common by a number of the objects, and these objects it arranges in a group. It then proceeds in a similar way with the remaining materials. But the mind is not content to leave the groups in isola- tion with no point of contact between them. It en- deavors to discover some connecting link, proximate or remote, between group and group. It is for this reason that, in forming the groups, the mind usually 148 THE PREDICABLES 149 fixes upon an attribute or aspect which the objects it is arranging in a group have in common with other objects. In this way the original chaos is gradually- converted into a system. This process of systematic grouping is called Classification. When a number of objects ofiFer themselves for classification, they confront us with a vast and diversi- fied array of attributes. Thus, in the race of men are found such various attributes as sentient, rational, capable of laughing, white, learned, strong, warm- blooded, vertebrate, mortal, and so on. The logician arranges all the attributes in the objects under five heads, called the Predicables, the principle of arrange- ment being the relation which the attributes severally bear to the purpose of the classification. The five heads are Genus, Specific Difference, Species, Prop- erty, and Accident. The predicables are defined as follows : The predicables are the attributes or aspects of an object arranged according to their fitness or unfitness to serve the purpose of a given classification. The fitness of an attribute to serve the purpose of a given classification is generally measured by the amount or depth of information which, consistently with the framing of the class, it conveys concerning the objects to be classified. At times the fitness of an attribute to serve this purpose is determined by con- siderations of convenience or symmetry. The predi- cables may also be defined as a division of the attri- butes of an object which must be kept in view and of those which should be disregarded in the process of a given classification. ISO AN ELEMENTARY HANDBOOK OF LOGIC The attributes which must be kept in view are called essential: they are the Genus and the Specific Difference. The .attributes which should be disregarded are called hon-essential : they are the Property and the Accident. For the purpose of illustrating the different predi- cables, we will suppose that we are classifying men, and that we wish to connect them with objects in the visible world. The genus is an attribute which the objects to be classified have in common with other objects and which best serves the end of the classification. Thus, "animal" is the genus of man. The specific difference is an attribute which is found in all the objects to be classified, but not in the other objects having the generic attribute, and which again best serves the end of the classification. For example, "rational" is the specific difference of man. The genus and specific difference together constitute the Species or Essence of an object with reference to the particular classification; hence — The species or essence of an object in a particular classification is its genus and specific difference in that classification. The species of man, for instance, is "rational animal." N. B. — There is a special sense assigned to "species" or "essence" which has the sanction of immemorial usage, namely, it is the sum of the most fundamental at- tributes or aspects of an object. When "species" is taken in this sense, then any aspect over and above the most fundamental attributes is at the utmost a property. THE PREDICABLES 151 A property is an attribute which is found in all the objects to be classified and not elsewhere, but is not best suited to the end of the classification. Thus, "capable of laughing" is a property of man. An accident is an attribute which is not perma- nently-present in the objects to be classified or is found in only some of them. For example, "sleeping" and "white" are accidents of man. The accident which has just been defined is com- monly called the separable accident. It is called sep- arable, because it is not necessarily found in all the objects to be classified. There is another accident, called inseparable, which is defined as follows : An inseparable accident is an attribute which is found in all the objects to be classified and in other objects as well, but is not best suited to the end of the classification. For example, "warm-blooded" and "vertebrate" are inseparable accidents of man. 103. The genus and specific difference must together suffice to mark off the objects to be classified from all other objects. If any one of several aspects is capable of combining with a given genus for the accomplish- ment of this purpose, that aspect is usually best suited to the end of the classification which either implies the others or is more fundamental than they. Thus, "rational," in the case of man, is more fundamental than "risible." What has just been said of the aspect which is to be employed for the specific difference jnust also be applied to the aspect which is to be selected for the genus. For example, "rational corporeal substance" and "rational organism" both suffice to mark off man 152 AN ELEMENTARY HANDBOOK OF LOGIC from all other individual objects; but "corporeal sub- stance" and "organism" are both implied in "animal ;" hence, "animal" is best suited as a genus to the end of the classification. In classification our selection of an attribute to serve as genus will be primarily determined by the world of objects with which we wish to connect the individuals to be classified. Suppose, for example, that men are the subject-matter of our classification. If we wish to connect them with the material creation, we shall select "animal" for our genus, and "rational" for our specific difference. If we wish to connect them with the spiritual world, our genus will be "^rational being" or "rational substance," and our specific difference will be "sentient." Thus, either "rational animal" or "sentient rational being" is the species or essence of a man considered as a man. "Man gifted with the power of persuasion in public speech" is the species or essence of a man considered as an orator. The attributes off an individual which have been selected as genus and specific difference are not com- monly called the species or essence of the individual when it is considered in itself, but only when it' is viewed as a member of a particular class. Take, for example, an individual who is an orator. The power of persuasion in public speech is not essential to him considered in himself; but it is essential to him when viewed as belonging to the class of orators. "Risible" and "having the power of speech" are properties of an individual, Vvhen viewed as a member of the class of men; but they are accidents, when the individual is viewed as a member of the class of ani- THE PREDICABLES 153 mals. "Sleeping" is an accident of an individual, when regarded as belonging to the class of men; but it is essential, when the individual is regarded as belonging to the class of beings that are asleep. "Rational animal" is the species of man, considered as man and as connected with the material creation. If we wish to widen the class,, so as to include in it beasts, birds, fishes, etc., we shall have to drop the specific difference "rational," select for our genus an aspect which characterizes men and beasts, etc., and also the objects with which we wish to connect them, and then fix upon an aspect as specific difference which marks them off from the other objects. "Organism" will serve our purpose as genus, and "sentient" as specific difference. The species, then, is "sentient organism." 104. The proximate genus of an individual is the genus which in a given order implies all the other genera in the same order. For example, "animal," in the same order with "substance," and as connecting man with the material creation, is the proximate genus of man. The supreme genus of an individual is the genus which in a given order implies none of the other genera in the same order. For example, "substance," in the same order with "animal," is the supreme genus of man. A subaltern genus is a genus intermediate between the proximate and the supreme genus. For example, in the same order with "animal" and "substance," "organism" is a subaltern genus of man. The supreme genus and all subaltern genera are called Remote Genera. 154 AN ELEMENTAKY HANDBOOK OF LOGIC Of the ideas representing the genera of the same order the idea which represents the proximate genus has the narrowest extension, and the idea which rep- resents the supreme genus has the widest. 105. THE CATEGORIES. Aristotle maintained that all genera whatever and all predicates are in- cluded in one or other of ten great orders. The supreme genus in each of these orders he called a Category. In Latin the categories are called Praedi- camenta. The categories, then, according to Aristotle, are ten supreme genera under which may be grouped all genera and all predicates whatever. The following is the list of the categories ; Sub- stance, Quantity, Quality, Relation, Activity, Passiv- ity, Place, Time, Posture, Apparel. The answer to any question that can be put con- cerning an individual will, according to Aristotle, fall under one of these categories. This may be illustrated as follows: SUBSTANCE: K^/saf is Tom Brown? A man. QUANTITY: How tall is he? Six feet. QUALITY: What kind of a man is he? White. RELATION: How is he related to Peter Brown? His son. ACTIVITY: What is he doing? Playing tennis. PASSIVITY: What is he undergoing? Defeat and ridicule. PLACE: Where is he? Behind the college. TIME: What is the time at which he is playing? Three o'clock. THE TREE OF PORPHYRY 155 POSTURE: What is his posture f Upright, with legs apart. APPAREL: What is he wearing f A tennis shirt and flannel trousers. 106. The Tree of Porphyry is an outline of a descend- ing series of genera and species, beginning with Sub- stance and ending with Man. The outline is as fol- lows: ^^..^-'^Substance Incorporeal Corporeal <;^ J];;>-Body Organic Organism Non-sentient 7^^^;> Animal Rational