rX 
 
 LIBRARY 
 
 UNIVERSITY Of \ 
 C All FORNIA/ 
 
 I 
 
 IHJE EARL OF 
 
 Witstmmhrib. 
 
* 
 
 s?J( 
 
SYLLABUS OF LOGIC, 
 
 IN WHICH 
 
 THE VIEWS OF KANT ARE GENERALLY ADOPTED, AND 
 
 THE LAWS OF SYLLOGISM SYMBOLICALLY 
 
 EXPRESSED. 
 
 BY 
 
 THOMAS SOLLY, ESQ. 
 
 LATE OP CAIUS COLLEGE, CAMBRIDGE. 
 
 J. & J. J. DEIGHTON; 
 J. W. PARKER, LONDON; & J. H. PARKER, OXFORD. 
 
 1839. 
 
CAMBRIDGE: 
 
 PRINTED BY METCALFE AND PALMER, TRINITY STREET. 
 
BC7/ 
 
 S7 
 
 PREFACE. 
 
 THE object which I have proposed to myself 
 in writing this Treatise on Logic, is the com- 
 bination of a brief but complete account of 
 the Aristotelian system, with some of Kant's 
 philosophical views of the nature and divi- 
 sions of the science. With respect to the 
 first-mentioned part of my task, I have en- 
 deavoured to give a strictly h priori charac- 
 ter to the derivation of the fundamental laws 
 of syllogism, and the results of their combi- 
 nation in the various forms of reasoning. 
 This is attempted, partly by employing a 
 method in the derivation of these laws of a 
 rather more exhaustive character than that 
 which has usually been adopted by logicians, 
 
11 PREFACE. 
 
 and partly by introducing mathematical ana- 
 lysis, for the exhibition of the symmetry in 
 their forms. Symbolical representation is also 
 employed in the second section of the Intro- 
 duction, for the purpose of explaining the 
 nature of abstract conceptions, and the method 
 of thinking some of them purely, or inde- 
 pendently of sense. . 
 
 I have derived from the deductio ad absurdum 
 the grounds for a division of the twenty-four 
 categorical syllogisms into eight systems of 
 three each. By means of this arrangement, 
 which to the best of my knowledge is en- 
 tirely new, the equality of the number of sound 
 moods in the first three figures, and several 
 other properties of categorical syllogism, may 
 be demonstrated a priori without the assist- 
 ance of symbolical reasoning. And although 
 these results admit of no immediate application 
 to practice, yet are they useful in giving the 
 student such firm hold upon the fundamental 
 principles of the science, that they will never 
 afterwards desert him. Besides, it should be 
 considered a sufficient merit that they add a 
 theoretical completeness to the science, which 
 it can never obtain when the symmetry of 
 
PREFACE. Ill 
 
 its results is arrived at by empirical methods 
 alone. 
 
 There is another subject which has been 
 very much neglected in all the works on Logic 
 with which I am acquainted. Aristotle's Ana- 
 lytics is the only book in which I have dis- 
 covered any attempt at a theory of the 
 modality of syllogism, and on this particular 
 point he appears to me to have failed. I have 
 accordingly devoted an entire section to the 
 consideration of this subject, at the end of 
 the second book, in which I have endeavoured 
 to expose the fallacious nature of his reason- 
 ing. It must, moreover, be remembered, that 
 this subject is one of the most important 
 connected with Logic, for any misconception 
 respecting it may give birth to fallacies of a 
 very complicated nature, and extremely diffi- 
 cult to detect. Perhaps not a few of the 
 errors on the nature of the will might be 
 ultimately traced to this source. 
 
 I will now mention as briefly as may be 
 the extent to which I am indebted to Kant, 
 or rather his translators, for it is by their 
 means alone that I have had any access to 
 his works. 
 
IV PREFACE. 
 
 The division of the science into Transcen- 
 dental and Universal, (for the latter of which 
 terms I have substituted ' Formal ' as being 
 more generally intelligible) is adopted from 
 the Criticism, and cannot perhaps be entirely 
 comprehended without a reference to that 
 work. The first part of the second section 
 in the Introduction, and a great part of the 
 third, are little more than an abridged para- 
 phrase of some portion of his Logic. There 
 are also several other places throughout the 
 Introduction which have at least originated 
 in some idea derived from his works, though 
 it is impossible individually to specify them 
 here. 
 
 In the first book, which contains the merely 
 formal analysis, I have adopted the opinion 
 of Kant respecting the distinct nature of 
 Categorical, Hypothetical, and Disjunctive 
 propositions. But as the latter part of the 
 first book is entirely opposed to his opinion 
 on the philosophical correctness of the dis- 
 tinction of figures, it will be necessary to 
 account for my having thought proper to 
 retain the scholastic theory on that subject. 
 Many other reasons might probably be urged 
 
PREFACE. V 
 
 in favour of this view, but one alone is amply 
 convincing to myself. The deductio ad absur- 
 dum can only be applied to some syllogisms in 
 the first figure, by the introduction of those 
 moods of other figures which cannot be re- 
 duced to the first by conversion ; and the 
 indirect proof of these syllogisms would only 
 restore the original syllogisms in the first 
 figure. There are one or two other points 
 in which I have not followed Kant, and which 
 are immediately consequent upon the adoption 
 of the system;' of figures. These it is not 
 necessary to specify here ; for if the reader is 
 acquainted with the works of that philosopher, 
 he will readily detect them for himself; and if 
 he is not, he would not understand my expla- 
 nation. 
 
 Material fallacies, ambiguous terms, and 
 many other similar subjects which are usually 
 considered in works on this science, have found 
 no place in the following pages. All these 
 subjects have been already discussed by Dr. 
 Whately in such a very able and lucid manner, 
 that nothing more remains to be said about 
 them. And even had this not been the case, 
 my silence respecting them could not have 
 
VI PREFACE. 
 
 been considered an omission, as they never 
 entered into the plan of my work. 
 
 I have attempted little more than an ana- 
 lysis of the formal laws of reasoning, and how 
 far this attempt has been attended with suc- 
 cess, the reader can now determine for himself. 
 
 Cambridge, May 13, 1839. 
 
TABLE OF CONTENTS. 
 
 INTRODUCTION. 
 
 SECTION I NATURE AND DIVISIONS OF THE SCIENCE. 
 
 Art. 
 
 Legality of the Understanding .. ... ... 1 
 
 Legality of the Reason ... ... ... 2 
 
 A priori Character of their Laws ... ... 3 
 
 Division of Logic into Transcendental and Uni- 
 versal or Formal ... ... ... ... 4 
 
 Proper Province of Logic ... 5 
 
 Logic not an Organum ... ... ... 6 
 
 Notice on Whately's Logic ... ... 79 
 
 SECTION II. COGNITION. 
 
 Definition of Cognition ... ... ... 10 
 
 Distinct and Indistinct Cognition ... .. 11 
 
 Conceptions, how obtained ... ... ... 12 
 
 Their Matter and Sphere 13 
 
 Their Rank ... .14 
 
 Symbolical expression of the Nature of Abstract 
 Conceptions ... ... ... ... 15 17 
 
 Of certain pure Conceptions ... 18 20 
 
Vlll CONTENTS. 
 
 SECTION III. JUDGMENTS. 
 
 Art. 
 
 Judgments distinguished as to their ... ... 21 
 
 Quantity, into Universal, Particular, Singular 22 
 
 Quality, into Affirmative, Negative, Indefinite ... 23 
 Relation, into Categorical, Hypothetical, Disjunctive 24 
 
 Modality, into Problematical, Assertive, Necessary 25 
 
 Propositions ... ... ... ... .. 26 
 
 Synthetical and Analytical Judgments ... 27 
 
 How confounded ... 28 
 
 Definition 29 
 
 Several applicable to the same thing 30 
 
 BOOK I. 
 
 SECTION I. CATEGORICAL PROPOSITIONS. 
 
 Their Form ... ... 31 
 
 Their Number 32 
 
 Law for Quantity of Predicates ... ... 33 
 
 SECTION II. MUTUAL RELATION OF CATEGORICAL 
 
 PROPOSITIONS. 
 
 Opposition of three kinds Contradictory, Con- 
 trary, Sub-contrary .,,., ... ... 34 36 
 
 Subalternation .. ... ... ... 37 
 
 Conversion ... .. ... ... ... 38 
 
 SECTION III. HYPOTHETICAL AND DISJUNCTIVE PRO- 
 POSITIONS. 
 
 Form of Hypothetical Propositions 39 
 
 Admits no Variations ... . ;,, _ ... .-. 40 
 
 Their Contradictory Categorical - - k t ' t { -; 44 
 Their Nature distinct from that of Categorical 
 
 Propositions ... ... .. . v .'.; .', 5 ... f - 4-2 
 
 Disjunctive Propositions ... ... .... 4*3 
 
 Have no Contradictory ... ... .. 44 
 
CONTENTS. IX 
 
 SECTION IV. SYLLOGISM. Art. 
 
 Syllogism defined 45 
 
 Its Divisions ... .. ^< ... ... .. 46 
 
 Categorical Syllogism .. ... ... 47 
 
 Method to be adopted in determining its Laws .. 48 
 
 Its Elements ... 49 
 
 Division of the Enquiry ... .. -'^0 50 
 
 Law for the Middle Term ... l'c. j&i 51 
 
 Quality of the Premises ... ... 52 
 
 Forms of Premises from which a Conclusion for 
 
 the Reason is possible ... .. ... 53 
 
 Conclusions inexpressible in the legitimate Cate- 
 gorical forms ... ... ... .. 54 
 
 Illicit Processes ... .. ... ... 55 
 
 Quality of Conclusion... ... 56 
 
 Recapitulation of the Laws of Categorical Syllo- 
 gism 57 
 
 Secondary Laws ... ... ... ... 58 
 
 Table of Sound Moods 59 
 
 Division into Figures and complete Table of Cate- 
 gorical Syllogism ... ... ... ... 60 
 
 Laws peculiar to different Figures ... 61 
 
 Transformation of Figures and Table 62 
 
 Rejection of certain forms of Syllogism considered. 
 
 Table of Moods for each Figure ... ... 63 
 
 Hypothetical Syllogism .. 64 
 
 Disjunctive Syllogism ... ... ... 65 
 
 Dilemma 66 
 
 Other forms of Syllogism ... .. ... 67 
 
 Enthymeme ... ... ... ... ... 68 
 
 Sorites ^U^I 69 
 
 SECTION V. THE DEDUCTIO AD ABSURDUM, ITS 
 
 NATURE 70 
 
 Affords the grounds for a Symmetrical Division 
 
 of the 24 Syllogisms into eight Systems ... 71 
 
X CONTENTS. 
 
 SECTION VI. SYMBOLICAL EXPRESSION FOR THE 
 
 SYLLOGISTIC LAWS. Art> 
 
 Fundamental Equations ... ... ... 72 
 
 Symmetry of the Equations to the first three 
 
 Figures ... 73 
 
 Number of possible Solutions 74 
 
 Derivation of Secondary Laws 75 
 
 Truth of Premises 5 >W, 76 
 
 BOOK II. 
 
 SECTION I. LIMITATIONS OF THE FORM OF JUDG- 
 MENTS. 
 
 Conception of Substance never a Predicate ... 77 
 
 Law between Predicate and Copula ... ... 78 
 
 No formal proposition of Identity ... ... 79 
 
 Relative merits of the four Figures ... ... 80 
 
 True Conclusions from false Premises ... 81 
 
 Conclusions of the Reason ... ... ... 82 
 
 SECTION II. MODALITY OF SYLLOGISM. 
 
 Does not affect the form of Conclusion ... 83 
 
 Proper and Consequential Modality distinguished 84 
 
 Proper and Derived Modality distinguished ... 85 
 
 No conclusion from Problematical Premises ... 86 
 
 The law of Derived Modality 87 
 
 Fallacies in Aristotle's Analytics arising from the 
 breach of this law 88 90 
 
 APPENDIX. 
 
 Examples for practice 155 
 
 Mathematical Note 160 
 
 Index to the principal Technical Terms ... 161 
 
ERRATA. 
 
 Page 4, lines 5 and 7, from the bottom, for "represented," read 
 
 " imaged." 
 
 Page 40, line 1, for " cognitions," read " representation." 
 
 Page 89, lines 5, 6, in the table, dele "Subject Premiss." 
 
 Page 108, line 10 from bottom, insert "not" between "B" and 
 
 " being." 
 
SYLLABUS OF LOGIC 
 
 INTRODUCTION. 
 SECTION I. 
 
 ON THE NATURE AND DIVISIONS OF THE SCIENCE. 
 
 (1.) THE subjection of the Understanding 
 to certain invariable laws is the first indis- 
 pensable condition to all knowledge. Kant 
 has shewn, in his Criticism on the pure Reason, 
 that the mind legislates for matter, or in 
 other words, that the laws we discover in 
 the external world derive their very possi- 
 bility from the laws of Mind. But let us for 
 an instant imagine the possibility of nature 
 following fixed laws as an object of our senses, 
 quite independently of any laws in our under- 
 standing, which for argument's sake we will 
 suppose to be without them. It is evident 
 that upon this hypothesis the Understand- 
 ing could never take cognizance of these 
 
 u 
 
2 A SYLLABUS OF LOGIC. 
 
 laws of nature, nor even of the existence 
 of the objects they concerned. For these 
 objects can only become known to us by 
 means of certain laws, according to which 
 we can severally distinguish them from each 
 other : and how could we even distinguish 
 between ourselves and nature, or between the 
 'me* and the ' not me/ without some law in 
 the faculty by which objects are known? 
 Every change in our representations might 
 either arise from a change in nature, or from 
 a change in our own indeterminate state. 
 But even could we separate ourselves from 
 nature, there would still remain the question, 
 ' How can we conjoin any phenomena in a 
 synthesis for the purposes of knowledge, unless 
 we are conscious of something fixed and 
 determinate to which we can refer all the 
 phenomena to be conjoined, and also of some 
 law for the mode of their conjunction?' With- 
 out these requisites the cessation of the phe- 
 nomena and of their conjunction in the mind 
 must be simultaneous. And, moreover, even 
 while the phenomena lasted, no act of con- 
 joining them in any one instant of time could 
 ever be considered the same as, or be united 
 with a similar act of conjoining them in any 
 succcessive instant, unless we were to grant 
 something fixed in our consciousness, and 
 
INTRODUCTION. O 
 
 quite independent of time, as a common 
 ground for the unity of these successive 
 syntheses. Hence, upon every hypothesis, the 
 legality of the knowing faculty is a necessary 
 condition to its use; and it now only remains to 
 extend this remark to the other logical faculty 
 the Constructive Reason. 
 
 (2.) The Understanding has been defined 
 by Kant, as ' the faculty of rules ;' by Coleridge, 
 as ' the faculty of judging according to sense/ 
 Its operation may be considered as two-fold, 
 accordingly as it dissects a representation by 
 analysis, _or conjoins several representations 
 in a synthesis. But there can be no act of 
 analysis without the consciousness of a prior 
 synthesis ; and hence it follows, that the latter 
 is an indispensable condition to every act of 
 the faculty. Now the Reason, considered as 
 to its logical use, (its transcendental use has 
 nothing to do with our present purpose, and 
 need not be considered here,) differs from 
 the Understanding in this : Whereas the 
 Understanding merely conjoins the diversity 
 of representations in synthesis, or dissects 
 them by analysis, in either case referring this 
 diversity to the unity of consciousness; the 
 Constructive Reason on the other hand con- 
 joins the very unities of these syntheses, 
 which unities are acts in the consciousness 
 
 B2 
 
A SYLLABUS OF LOGIC. 
 
 or logical functions of the Understanding, and 
 entirely distinct from the diverse representa- 
 tions contained in them.* But as this kind 
 of reasoning is rather difficult to comprehend 
 in the abstract, we shall endeavour to ex- 
 plain our meaning by an example: If any 
 person were to state the two propositions, 
 ' all tyrants are unhappy/ and ' Nero was a 
 tyrant/ my Understanding alone could never 
 have enabled me to discover that ' Nero was 
 unhappy/ For this faculty unaided by the 
 Reason could only conjoin certain concep- 
 tions according to a rule. For instance, 1 
 
 * If we might be permitted to offer an illustration of so abstract 
 a subject, we would represent the synthesis of representations in 
 the Understanding by the arc of a circle connecting two points, 
 whose centre or unity of the syntheses may be of course at a 
 finite distance. But if another point be taken in the same right 
 line with the first two, we can only join all these points, and make 
 the centres of the two arcs correspond, by placing the common 
 centre at an infinite distance, where it will represent the Reason 
 conjoining the unities of the syntheses of the Understanding. It 
 must also be observed, that as on the one hand the centre of this 
 circle is infinitely distant, but its arc, which is a straight line joining 
 the three points, is finite and determinate ; so on the other hand 
 the rational act of conjoining the unities of the syntheses of the 
 Understanding transcends all possible experience, and cannot be 
 represented, but can only be thought, whereas the synthesis, 
 which results from the act of reason, is quite as easily repre- 
 sented as those from which it was originally derived. We are 
 perfectly aware of the manifold objections which might be urged 
 against this illustration, but we are inclined to think it may, in 
 some measure, assist the reader in comprehending the idea we 
 wish to convey. 
 
INTRODUCTION. 
 
 may severally conjoin the conceptions of un- 
 happiness and Nero with the conception of 
 tyrant, as in each case belonging to the same 
 subject : but in order to conjoin the two 
 extreme parts of these judgments, namely, 
 Nero and unhappiness, I must simultaneously 
 reflect on my previous conjunction of each 
 of them with the conception of tyrants, and 
 therefore on the two acts of the Understand- 
 ing by which these judgments take place. 
 Before then I can arrive at any conclusion 
 from the two given propositions, I must pos- 
 sess some faculty by which I can conjoin the 
 very acts of the Understanding that are con- 
 tained in them, and this is what is meant by 
 the Constructive Reason. 
 
 The ease with which the Reason arrives at 
 a conclusion from judgments that have been 
 reduced to their most simple logical form, 
 and the perfect similarity of such a conclu- 
 sion when obtained to any immediate act of 
 the Understanding, are circumstances which 
 have rendered the peculiar function of the 
 Reason, in its logical use, extremely liable to 
 be overlooked. But a little attention to the 
 subject will make it evident, that the con- 
 junction of the acts of the Understanding 
 requires quite another faculty than that of the 
 Understanding itself, which merely conjoins 
 
6 A SYLLABUS OF LOGIC. 
 
 the parts of possible experience according to 
 rules.* 
 
 Having given this short account of the 
 logical use of the Reason, we shall now pro- 
 ceed to shew that its functions are entirely 
 determined by laws. For as the conclusions 
 of the Reason are constructed with materials 
 derived from the use of the Understanding, 
 they can only concern the objects of the latter 
 faculty, and must in their intrinsic nature be 
 possible as its immediate acts. But this har- 
 mony of the two faculties can only be secured 
 by the subjection of the Reason to laws. Here 
 then at length we are justified in assuming, 
 that the whole use of our intellectual and ra- 
 tional faculties is based upon certain universal 
 laws, and that the Science which treats of them 
 is not a mere chimera of the imagination, but 
 founded on a reality which is the primary con- 
 dition to all other knowledge whatever. 
 
 * After the author had written his account of the logical use of 
 the Reason, he met with the following passage from Kant. It 
 refers, however, to the whole use of the faculty : 
 
 " The Understanding may be a faculty of the unity of pheno- 
 mena.by means of rules ; Reason is thus the faculty of the unity 
 of the rules of the Understanding under principles. Reason, 
 therefore, never refers directly to experience, or to an object, but 
 to the Understanding, in order to give to the diverse cognitions 
 of this, unity a priori by means of conceptions, which may be 
 termed unity of Reason, and which is of quite another kind to 
 that which can be derived from the Understanding." Anonymous 
 Translation of Kant's Criticism. 
 
INTRODUCTION. 7 
 
 (3.) The laws of the use of the Understand- 
 ing must be a priori, and cannot be derived by 
 the method of induction from any number of 
 mental phenomena. For it has been shewn 
 that they are necessary to all use of the Un- 
 derstanding, (as it cannot even determine an 
 object without them,) and consequently to 
 experience itself; and we should therefore be 
 guilty of a glaring circle in our reasoning, if 
 we endeavoured to derive from experience 
 those laws which have previously been made 
 the very grounds of its possibility.* 
 
 But if the laws of the Understanding are 
 
 * The commonest example of this most fallacious attempt to 
 deduce the pure conceptions of the Understanding from expe- 
 rience, is to be found in the grounds that are sometimes given for 
 the causal relation of phenomena. For instance : Q. Why do I 
 expect a spark from the concussion of flint and steel? A. Because 
 in all past time such a concussion has been immediately succeeded 
 by a spark Q. But why does this succession having taken 
 place in all past time, give me any right to expect that it will take 
 place again? A. Because any succession that has taken place 
 very often, has been afterwards observed to take place again. And 
 if we were to ask for a reason, why the uniformity of nature up 
 to the present moment should be considered as any guide for the 
 future, we should get precisely the same answer, or in other 
 words, the fact would be stated as the ground for itself. And 
 we might thus continue the question and answer to infinity, with- 
 out ever getting any nearer the point. The difficulties of this 
 subject were first brought to light by the subtilty of Hume. And 
 although he never succeeded in offering any satisfactory solution 
 of them, yet his " sceptical doubts" aroused the attention of Kant, 
 and became the occasion of the Criticism on Pure Reason, in which 
 the difficulty is satisfactorily explained. 
 
8 A SYLLABUS OF LOGIC. 
 
 independent of all empirical matter, a fortiori 
 must this be the case with the formal laws of 
 the Reason, which simply regard the conjunc- 
 tion of the acts of the former faculty, and 
 are therefore removed one step farther from 
 their empirical contents. Hence it follows 
 that both Reason and Understanding are en- 
 tirely self-regulated, or subject to a priori 
 laws. And the determination of these metho- 
 dically, and their arrangement in a system, 
 is the business of Logic in the widest accep- 
 tation of the word. 
 
 Division of Logic. 
 
 (4.) Logic is divided by Kant into Trans- 
 cendental, and Universal or Formal.* 
 
 Transcendental Logic agrees with Formal 
 in excluding all consideration of particular 
 objects, but differs from it in admitting 
 that of the pure conception of an object in 
 general. 
 
 Formal Logic entirely excludes all consider- 
 ation of the objects thought, and merely re- 
 
 * The reader who has no acquaintance with the Kantian 
 system, must not be surprised if he does not clearly under- 
 stand the distinction between these branches of the science. 
 As, however, Formal Logic is exclusively the subject of the first 
 book, and a few considerations from Transcendental are intro- 
 duced in the second the two books together may throw some 
 light upon the nature of the division in question. 
 
INTRODUCTION. 9 
 
 gards the form of our judgments, and their 
 relations to each other. 
 
 It will be seen from the above definitions 
 that Transcendental Logic embraces a very 
 wide field, and includes within its limits much 
 that is metaphysical, and entirely foreign to 
 the other branch of the science.* 
 
 As Formal Logic alone is the proposed sub- 
 ject of this treatise, we should have been jus- 
 tified in taking leave of Transcendental Logic 
 here: but certain limitations are imposed upon 
 the very forms of the judgments of the Un- 
 derstanding by conceptions which are peculiar 
 to the last-mentioned science. And although 
 they may be justly assumed in every treatise 
 on Formal Logic, yet, as they are the origin of 
 several very peculiar results,f we thought the 
 notice of them in our second book might be 
 deemed not wholly superfluous. 
 
 (5.) This apology has been offered for an 
 intended reference to what properly lies out 
 of the sphere of the subject, on account of 
 the great injury which the sciences must in- 
 
 For instance, the conceptions of substance, causality, &c. 
 
 f Some of the limitations are alluded to in the commencement 
 of the first book. The absence of a formal proposition of 
 identity is one of them. For if we say A is B, we do not know 
 but that other things may be B also. Qne of the peculiar results is 
 the " possibility of a true conclusion from false premises in every 
 form of reasoning." 
 
10 A SYLLABUS OF LOGIC. 
 
 variably incur whenever their boundaries are 
 not strictly recognised. Formal Logic has, 
 perhaps, suffered more on this score than 
 any other science : for, to say nothing of many 
 encroachments on the side of Transcendental 
 Logic, it has been a common custom with 
 logicians to introduce into their treatises a 
 great deal of matter derived entirely from 
 empirical psychology, which is a distinct 
 branch of knowledge. Thus, the choice and 
 arrangement of arguments, the best applica- 
 tion of particular syllogistic forms, and other 
 similar considerations, have frequently been 
 made the subject of rules which can only 
 be derived from practice, and should never 
 be mixed up with Logic, which is an a priori 
 science. Observations of this kind can never 
 form part of a system, but are merely an 
 aggregate of information, which is very useful 
 no doubt, but belongs more particularly to the 
 Art of Rhetoric. Knowledge of this kind may 
 properly be termed an art, and so far the old 
 logicians were at least consistent. But this 
 term can never be applied to Logic without an 
 absurdity as manifest as if we were to speak of 
 the Art of the Differential Calculus, or Conic 
 Sections.* 
 
 * We have thought it better to put a few observations on Dr. 
 Whately's Definitions of Logic in a notice by themselves at the 
 end of this section. 
 
INTRODUCTION. 11 
 
 (6.) Logic is not an organum of the sciences; 
 for it does not contain a single reference to 
 any one branch of knowledge, and therefore 
 has no resting-place or fulcrum from which 
 to commence its investigations : it is how- 
 ever of the greatest use in testing the work 
 of another organum, by exposing all its results. 
 When a new law has been arrived at by Induc- 
 tion, Logic will determine all its consequences; 
 and should one of these prove at variance with 
 truth, it is certain that the induction is errone- 
 ous, and that new observations are necessary. 
 
 Notice on Whately's Logic. 
 
 (7.) As the distinction between the theory 
 of Logic and its application to practice has 
 been well explained in the very valuable trea- 
 tise by the Archbishop of Dublin, we cannot 
 help feeling in some measure surprised that 
 he has united these very different subjects in 
 the same definition. In the introduction to 
 his work, Dr. Whately defines Logic as the 
 "Science, and also the Art of Reasoning;" 
 and afterwards explains in a note, " that as 
 a science is conversant about knowledge only, 
 an art is the application of knowledge to 
 practice." Now it follows from the latter 
 unobjectionable definition, that the rules for 
 the art must be merely practical, and derived 
 
12 A SYLLABUS OF LOGIC. 
 
 from the experience of " how we reason best/' 
 or in other words from empirical psychology ; 
 for if they were a priori, they would ipso 
 facto become laws of the science. Hence 
 we see that the rules of the art, which are 
 all empirical and depend on observation, dif- 
 fer from the laws of the science, which are 
 a priori and derived from the reason, not 
 only in their use and nature, but also in their 
 origin. The question however is not one of 
 fact, or even of theory, as Dr. Whately has 
 himself introduced the very distinction for 
 which we are contending : but the question 
 is, whether that arrangement can be consi- 
 dered conducive to the interests of science, 
 which combines under the same name two 
 branches of knowledge whose nature and origin 
 are equally distinct from each other. 
 
 (8.) There are two other passages in Dr. 
 Whately's work, which appear to the author 
 to contradict each other, and to convey equally 
 false notions of the nature of the science. 
 Soon after the definition upon which we have 
 just commented, Dr. Whately adds, " Its most 
 appropriate office however is, that of instituting 
 an analysis of the process of the mind in reason- 
 ing." Now it is not the process of the mind in 
 reasoning, but the principles with which that 
 process must accord, that is the proper object of 
 
INTRODUCTION. 13 
 
 Logic.* We hardly ever reason to ourselves in 
 syllogism, but only in a manner which agrees 
 with its laws. For instance, if I think of the 
 future death of a hale and hearty man, in whom 
 there do not appear the slightest symptoms of 
 decay, I do not consider it a merely probable 
 circumstance, but morally certain. Now this 
 conclusion could only be logically deduced by 
 my virtually assuming the proposition that ' All 
 men must die/ and therefore that John being 
 a man must die also. But although the actual 
 law which connects humanity and mortality 
 must have been thought in my mind, when 
 I first considered it as morally certain that 
 John must die, yet the proposition that ' all 
 men must die' (which is, as it were, the ex- 
 ponent of the law) had, in all probability, never 
 occurred to me : perhaps, indeed, so far from 
 thinking of all mankind, I had not thought 
 of any other person than John. We see then 
 that the object of Logic is not the actual pro- 
 cess , of our reasoning, but rather the princi- 
 ples to which that process can always be re- 
 ferred.f And it must be remembered that 
 
 * Dr. Whately admits that Logic " investigates the principles 
 on which augmentation is conducted ; " and had his definition 
 terminated here, it would not have been liable to any objections. 
 
 f The simple state of the case is this : In the actual process 
 of our minds we generally connect the mere conceptions in a law, 
 without thinking of all that is contained in their sphere. But 
 
14 A SYLLABUS OF LOGIC. 
 
 this is no unimportant point in the defini- 
 tion of Logic, as an erroneous conception 
 on this head is extremely calculated to bring 
 the science into disrepute. For, if any one 
 discovers that the process of reasoning in his 
 own mind is rarely, if ever, in the form of 
 a complete syllogism, he immediately leaves 
 the science with the conviction that it cer- 
 tainly does not contain the principles of all 
 reasoning, and probably not of any. 
 
 (8.) The other passage to which we would 
 allude, is to be found in a note near the com- 
 mencement of the second book, in which Dr. 
 Whately states, that " Logic is entirely conver- 
 sant about language." 
 
 This appears to us to be at variance with 
 the passage we have just quoted, in which the 
 most appropriate office of Logic is said to be 
 the analysis of the process of the mind in 
 reasoning. Now, that language follows cer- 
 tain laws, is unquestionably true. But whence 
 do these laws arise, unless from the necessity 
 
 in the logical development we consider the aggregate of indi- 
 viduals that come under the conception contained in the subject 
 of an universal proposition. As, for instance, in the above ex- 
 ample, if I look at John and think of his future death, I merely 
 connect the conception of death with that of humanity, to the 
 conditions of which I .see that h'e corresponds. But if I am 
 asked to express logically my reasons for expecting his death, I 
 immediately commence with ' All men die,' &c. 
 
INTRODUCTION. 15 
 
 of language conforming itself to the mental 
 laws of which it is the exponent ? If language 
 did not receive its stamp from mind (the laws 
 of which we assume to be universally the same 
 in all countries and ages), we could never be 
 certain that we might not at some future 
 time meet with a people whose language re- 
 quired quite a new logic, and in that case 
 the science would lose its a priori character, 
 and rest on probability alone. 
 
16 
 
 A SYLLABUS OF LOGIC. 
 
 SECTION IT. 
 
 ON COGNITION. 
 
 (10.) COGNITION is the generic name for all 
 representations that are sufficiently completed 
 for the logical use, and must therefore contain 
 a reference to our consciousness on the one 
 hand, and to an object on the other. 
 
 If a cognition contain sensation, it is em- 
 pirical ; if it does not, it is pure. 
 
 Cognition may be divided into the two spe- 
 cies Intuition and Conception, each of which 
 may be either pure or empirical. 
 
 Intuition is a merely sensual* cognition of 
 an object. 
 
 Conception is an intellectual cognition, and 
 refers to many objects. 
 
 Perhaps we can exhibit these co-divisions 
 of cognition in a more intelligible manner by 
 means of the following scheme : 
 
 COGNITION TABLE. 
 
 . 
 
 INTUITION. 
 
 CONCEPTION. 
 
 Pure 
 
 A straight line. 
 
 Substance. 
 
 Empirical.. 
 
 A horse that is seen. 
 
 f Abstract conception 
 \ of a horse. 
 
 * Pure Intuitions concern the mere forms of the Sensitivity, 
 Space, and Time, and are therefore sensual without containing 
 sensation. 
 
INTRODUCTION. 17 
 
 To investigate the ultimate original of cog- 
 nition, and shew by what process a merely 
 sensual representation obtains objective vali- 
 dity,* and thus becomes an intuition, is entirely 
 foreign to our present purpose, as it belongs to 
 the science of Metaphysics. We shall there- 
 fore only explain that formal property of our 
 cognition upon which its logical perfection 
 mainly depends, and conclude the section with 
 investigating those logical acts of the under- 
 standing by which it arrives at conceptions. 
 
 (11.) Cognition may be divided into distinct 
 and indistinct. 
 
 Indistinct cognition is that in which a diver- 
 sity is thought, that is not exhibited objectively 
 to the consciousness. 
 
 Distinct cognition, on the contrary, is that 
 in which no diversity is thought, which is not 
 exhibited objectively to the consciousness. 
 
 Let us take an example of an indistinct 
 intuition. 
 
 While walking on the sea-shore, I perceive 
 a large object in the offing, which, from its 
 general appearance, I immediately know to be 
 a ship. Now, from my previous knowledge of 
 
 * We give objective validity to a representation, when we 
 conceive it to arise from something independent of the peculiar 
 state or nature of our own mind, and therefore believe that other 
 people would view it as we do. 
 
 C 
 
18 A SYLLABUS OF LOGIC. 
 
 the construction of a ship, I know that it must 
 have a rudder, shrouds, and a great variety of 
 tackle, which are not to be found in the image 
 as it appears to me, although I must really see 
 them : I therefore say that my intuition of the 
 ship is indistinct. If, on the other hand, I had 
 never seen a ship before, and had no conception 
 of the nature of the object presented to me, 
 I should say that my intuition of the object 
 was distinct; and this would arise from my 
 ignorance of what there was in it to distin- 
 guish. But if any person then asked me ' if 
 I could see what the object was ?' the question 
 ' what ?' would superadd in my mind a con- 
 ception of this object, having some nature 
 peculiar to itself, or having been made for 
 some particular purpose, which was not ex- 
 hibited to me in my intuition, and I should 
 therefore answer that I could not see distinctly 
 what it was. 
 
 Hence we see that an intuition is indistinct, 
 when it suggests a conception of the diverse, 
 which it does not exhibit objectively to the 
 consciousness. And it is the relation of our 
 conception to the representation, and not the 
 latter alone, upon which distinctness must 
 depend. For if distinctness were defined as 
 the consciousness of the diversity of the repre- 
 sentation, without introducing any limitation 
 
INTRODUCTION. 19 
 
 to this diversity, distinctness would be impos- 
 sible, as the diversity in any object is infinite, 
 and can never be exhausted.* 
 
 In the same manner we. say that a conception 
 is indistinct, when we attach to it the concep- 
 tion that an analysis of it is possible, or in 
 other words that it contains a diversity, and 
 yet cannot dissect, and severally represent to 
 our consciousnessf the parts of this diversity. 
 For instance, we may be said to have an in- 
 distinct conception of the nature of the human 
 mind, when we attach to it the conception 
 that some analysis of its different powers is 
 possible, and yet cannot exhibit these powers 
 separately to our consciousness. 
 
 Conceptions. 
 
 (12.) The logical acts by which the un- 
 derstanding arrives at conceptions are three, 
 Comparison, Reflection, and Abstraction. 
 
 Comparison is that act of the understanding 
 by which several representations are referred 
 to the consciousness simultaneously. 
 
 * It is evident from this that distinctness is relative, not posi- 
 tive. And we have dwelt the more upon this, as it does not seem 
 to be very clearly laid down in Kant's Logic. 
 
 f If the remarks in the text are founded on truth, it will follow 
 that an increase of knowledge may sometimes render a previously 
 distinct conception indistinct, by destroying the equality be- 
 tween that which we know to be in it, and that of which we are 
 conscious. 
 
 C2 
 
20 A SYLLABUS OF LOGIC. 
 
 Reflection is that act of the understanding 
 by which we determine what is common to 
 several representations, and consequently how 
 we may embrace them in one conception. 
 
 Abstraction is that act of the understanding 
 by which we separate all that is not common 
 to several representations. 
 
 For example, if I compare several men, 
 I reflect on their resemblance as bipeds that 
 walk erect, and abstract their differences as to 
 height, complexion, &c., and thus arrive at 
 a conception that answers to all of them. 
 
 Matter and Sphere of Conceptions. 
 
 Conceptions may be considered either as to 
 their matter, or as to their sphere. 
 
 (13.) The matter of a conception consists of 
 the various representations contained in it. 
 
 The sphere of a conception consists of the 
 things that come under it, or answer its condi- 
 tions. 
 
 As any addition to the matter of a con- 
 ception is a new condition to be answered, 
 and as a part of the original sphere of the 
 conception probably does not answer this new 
 condition ; it follows that the conception thus 
 altered by an addition to its matter will have 
 a less sphere than that of the original concep- 
 tion. Hence it is evident that the sphere and 
 
INTRODUCTION. 21 
 
 matter of a conception vary inversely, and that 
 the more matter there is in a conception, the 
 less is its sphere. Let us take an example. 
 The conception of a horse has more matter 
 than the conception of an animal : for the 
 former conception must contain all that is 
 contained in the latter conception, and some- 
 thing more besides. But the sphere of the 
 conception of a horse is less than the sphere of 
 the conception of an animal ; as there are 
 many more things that answer the conditions 
 of the latter than the conditions of the former 
 conception. 
 
 Rank of Conceptions. 
 
 (l^,) Superior and inferior conceptions are 
 merely relative terms. The inferior conception 
 contains all the matter of the superior concep- 
 tion, and something more besides, but is itself 
 contained under the superior conception. For 
 example, the conception ' horse' is inferior to 
 that of ' animal/ 
 
 A superior conception is also termed a genus, 
 and an inferior conception a species. There 
 can be a highest genus, but there cannot be 
 a lowest species ; for we may continue to 
 abstract matter from a conception till we have 
 left so little in it, that the next step would 
 take the conception away entirely. This con- 
 
22 A SYLLABUS OF LOGIC. 
 
 ception must then be the highest, for it is 
 impossible to think of another superior to it, 
 under which it might rank. But we can never 
 arrive at a lowest species ; for though we con- 
 tinue to increase the matter of the conception, 
 and thereby lessen its sphere, we can never be 
 certain that we have exhausted all the possible 
 partial representations, which may make dis- 
 tinct species under this conception, and thus 
 render it a genus : for instance, I may gradually 
 abstract from my conception of* horse* through 
 the steps of ' quadruped/ 'animal/ and 'organic 
 being/ till at last I arrive at mere being, 
 from which I cannot abstract any thing more. 
 This conception must therefore be considered 
 as a highest genus. But if I take the con- 
 ception of 'horse/ and continue gradually to 
 increase its contents, I can never be certain 
 that there may not be other unknown dis- 
 tinctions in horses, which might constitute 
 the grounds of a division into still lower 
 species. 
 
 But although we can never arrive at a lowest 
 species (which must of course be a conception), 
 we can very easily complete the determination 
 of a cognition by fixing its individuality in 
 Time and Space. 
 
INTRODUCTION. 23 
 
 Symbolical expression of the nature of Abstract 
 Conceptions. 
 
 (15.) A conception may be symbolically 
 represented as the common measure of the 
 representations from which it is derived. For 
 if we express the various parts of a repre- 
 sentation by the letters a, b, c, &c., and the 
 whole representation as their product,* we 
 may consider the following quantities as the 
 expressions for three different representations, 
 
 abode f t b c df, abode, 
 
 where it is evident that the conception ob- 
 tained in the manner already explained, will 
 be symbolically represented by the common 
 measure bed. 
 
 It frequently happens that we cannot image 
 to our minds the conceptions we obtain by 
 reflection and abstraction; for sometimes the 
 representations from which we wish to obtain 
 a conception, have not a single sensible partial 
 representation in common, but only a law con- 
 necting their parts. And although this law 
 may always be thought, yet it cannot be 
 imaged to the mind, as the image could only 
 
 * In the following symbolical exposition of the nature of ab- 
 stract conceptions, we have thought it better to retain the names 
 for the operations that are suggested by arithmetical algebra. The 
 terms, ' product,' multiplication, '&c. must therefore be understood 
 as merely referring to the corresponding symbolical operations. 
 
24 A SYLLABUS OF LOGIC. 
 
 represent a particular case. If, for instance, 
 I witness a great many cases of the rebound 
 of elastic bodies from a smooth surface, per- 
 haps the only thing these cases have in 
 common is the equality of the angles of in- 
 cidence and reflection, which cannot be imaged 
 to the mind, but can only be thought. For 
 an image could only give us the represen- 
 tation of the equality of two particular angles, 
 but the conception of the equality generally 
 could only be thought in the understanding. 
 This may be symbolically expressed as follows. 
 Let the representations be 
 
 b . (ft), e . (e), h . (A). 
 
 Now, in arithmetical algebra these quantities 
 would have no common measure, but in sym- 
 bolical algebra we may separate the symbol of 
 affection 0, and consider that as the symbolical 
 representation of the conception of the law. 
 And it must be observed, that as a general law 
 may be thought and employed in reasoning, 
 but cannot be imaged (which is only possible 
 for a particular case), in like manner the func- 
 tion 0, which is a symbol of affection, may be 
 employed in analysis as a medium of reasoning, 
 but can never be itself interpreted in all its 
 generality in any subordinate science, as its 
 interpretation is only possible by its union 
 with some particular symbol of quantity. 
 
INTRODUCTION. 25 
 
 (16.) It is impossible to obtain a correct 
 abstract conception of the organic productions 
 of nature. For let us take as an example the 
 conception of man. If we were to proceed 
 in accordance with the simple method already 
 proposed, and abstract every thing that is not 
 common to all men, we should obtain a con- 
 ception. in which no one could ever recognise 
 the least semblance of humanity : it must 
 possess neither arms, legs, eyes, ears, nose, 
 teeth, hair, or the power of speech, for men 
 exist who are separately deprived of each of 
 these things. Our abstract conception of the 
 human form divine would therefore contain 
 a trunk without a single limb, and a scull 
 without a single feature. 
 
 Again, if we determine in our own minds 
 the greater part of a horse (all but the tail 
 for instance), in conformity with our general 
 conception of that animal, we may allow 
 variations in that one part between very 
 wide limits, and yet consider the whole 
 result as coming under our conception of 
 a horse. In the same manner we might 
 allow any other part of the horse to vary 
 between very wide limits, provided the rest 
 resembled the corresponding parts of horses 
 generally, or of any well made horse in par- 
 ticular : but were we to introduce all these 
 
26 A SYLLABUS OF LOGIC. 
 
 variations simultaneously, the result would 
 be a monster, to which our conception of 
 ' horse ' would be no longer applicable. Hence 
 we see that the several parts of our conception 
 must be considered as functions of other parts, 
 and all mutually dependent on one another. 
 And retaining the symbolical language we 
 have already employed, where u, v, w, &c. 
 express the partial variable representations, 
 we might write 
 
 abstract conception = F(n,v, w, x, y, *) 
 
 where (u, v f w, x 9 y, &) = ; 
 and every set of values that answers the con- 
 ditions of the equation will correspond to an 
 individual of the species. 
 
 (17.) A friend once suggested to the author 
 that there was a great analogy between an 
 abstract conception and an enveloping curve; 
 and after a little consideration he discovered 
 that the symbolical expression for the envelope 
 is not a mere illustration, but a strictly correct 
 representation of the nature of an abstract 
 conception. 
 
 Instead of representing each individual of a 
 species by a function of a particular set of values 
 for u, v, w, &c., let us consider the whole locus* 
 
 * As the original (i.e. geometrical) signification of the word 
 'locus' is necessarily confined to functions containing not more 
 than three variables, it may be as well to remark, that we have 
 adopted it here in the strictly analogous sense of ' interpretation 
 of an equation.' - . 
 
INTRODUCTION. 27 
 
 of an equation as but one individual. If then 
 this equation contains a constant (a)* to which 
 we give continuously successive values, we 
 shall obtain a family of these equations to 
 individuals whose loci will intersect each 
 other (in a locus of one dimension less than 
 themselves), and the envelope or locus of 
 these intersections will be a correct symboli- 
 cal representation of the conception abstracted 
 from them all. For we may consider an ab- 
 stract conception as one which contains all 
 the points of resemblance between any in- 
 dividual, and those immediately preceding and 
 succeeding it. 
 
 We therefore discover a perfect analogy 
 between abstract conceptions and enveloping 
 surfaces or curves : and the only difference 
 in their respective symbolical representations 
 consists in the number of variables which we 
 are justified in introducing in either case. 
 For the equations whose interpretations relate 
 to space cannot of course have more variables 
 
 * It may be as well to remind the reader, that though the con- 
 stant (a) is a symbol of quantity in the function considered simply 
 symbolically, yet it becomes a symbol of affection (in fact apart of 
 the function) in the interpretation : the symbol (a) is therefore 
 to be considered as a part of the law between the members of the 
 individual ; just in the same manner as a constant in the equation 
 to a surface or curve is only interpreted through its affecting the 
 variables. 
 
23 A SYLLABUS OF LOGIC. 
 
 than space has dimensions, and are therefore 
 confined to three : whereas the variables intro- 
 duced in our abstract conceptions concern 
 many other things besides space, and are 
 therefore unlimited in number. We subjoin 
 the symbolical expression for the enveloping 
 or abstract conception. 
 
 Let F(w, v, w, x, y, *,- a) represent the locus 
 of an individual : then the locus of the en- 
 veloping conception will be expressed by the 
 equations 
 
 F(u, v,w, x, y, z,a) = ... (1) 
 
 da 
 
 And if a is eliminated from (1) and (2) we 
 shall have an equation of the form 
 
 $ (u, v, w, x, y, z) = 0, 
 
 which represents the locus of the abstract 
 conception. 
 
 (18.) Before we conclude this section, it may 
 be as well to add a few remarks on the method 
 of thinking certain conceptions purely or inde- 
 pendently of sense. 
 
 The idea of a limit, which is the basis of the 
 differential calculus, supplies the only possible 
 means of thinking the conceptions Substance 
 and Causality purely, i. e. independently of all 
 empirical matter. For in order to think the 
 
INTRODUCTION. 29 
 
 conception of Substance at all, I must imagine 
 certain phenomena, and then conceive the ob- 
 jective existence of a substratum for their 
 support. But this alone is insufficient, for 
 although the conception of Substance is ob- 
 tained, yet it is not pure as long as it con- 
 tains any sensation or phenomenon. I must, 
 therefore, diminish my sensation till it approx- 
 imate to zero. This, however, must not be 
 effected by lessening its extension in space, 
 for in that case the object itself would vanish, 
 as its identity and objective existence can 
 only be determined in that form of sensation. 
 Neither, on the other hand, may the sensation 
 be diminished as to its intensity : for although 
 this operation would only affect the conception 
 of its being an object for me, and not of its 
 being an object at all, yet this mode of dimi- 
 nution would introduce the conception of suc- 
 cessive states, and therefore other matter 
 besides the bare conception of Substance. 
 
 The only way that remains is that of dimi- 
 nishing the time in which the phenomenon is 
 viewed, till we arrive at the limit. If -then 
 we consider time as the independent variable, 
 and the phenomenon as a function of it, the 
 pure conception of Substance will be properly 
 represented as the limiting ratio between the 
 phenomenon and the time in which it is viewed. 
 
30 A SYLLABUS OF LOGIC. 
 
 Let S = the pure conception of Substance, 
 P = phenomenon, and T = time in which the 
 phenomenon is viewed, we shall then have 
 s dP 
 '- dT 
 
 And as the conception of Substance contains 
 permanence, it follows that S is constant. We 
 may therefore consider 
 
 P = S T. 
 
 as the equation to a phenomenon to which 
 the pure conception of Substance alone is 
 applicable. 
 
 (19.) Very nearly the same reasoning will 
 apply to the conception of Causality. This 
 can only be imaged to the mind in a suc- 
 cession of phenomena occupying finite por- 
 tions of time. But while our conception con- 
 tains sensible phenomena, it is not pure : we 
 therefore diminish the conceived duration of 
 the times in which they exist, till they approx- 
 imate to zero. When they arrive at this limit 
 the conception becomes pure, for the pheno- 
 mena will only occupy successive instants of 
 time; and must therefore vanish. Hence we 
 see that the two cases resemble each other, in- 
 asmuch as they are each represented by the 
 limiting ratio of the phenomenon to the variable 
 time. They differ however in this : whereas 
 the phenomenon from which the conception 
 
INTRODUCTION. 31 
 
 of Substance alone was derived did not vary 
 in time, but bore a constant relation to it ; 
 in the latter case the phenomenon will vary, 
 and consequently be some unknown function 
 of time. If then the pure conception of 
 Causality = C, and the other symbols retain 
 their former signification, we shall have 
 P = $ (T, S), 
 
 (20.) If the reader will examine his con- 
 
 ception of -^ and also of one of these cate- 
 dx 
 
 gories, the analogy between them will become 
 
 immediately evident. The ratio -^ cannot 
 
 dx 
 
 be imaged to our minds (because we cannot 
 image the ratio between two nothings), but it 
 can be thought. For we can very well under- 
 stand that there may be such a relation between 
 two quantities, as that the nearer they approx- 
 imate to zero, the nearer does their ratio ap- 
 proximate to equality with some given ratio, 
 and that the two approximations are com- 
 pleted simultaneously. Exactly in the same 
 manner the pure conception of Substance can- 
 not be imaged;* for we cannot image that, 
 
 * " The invisible was assumed as the supporter of the apparent, 
 rtav (j>aivo/j.V(av as their substance, a term which in any other 
 interpretation expresses only the striving of the imaginative power 
 under conditions that involve the necessity of its frustration." 
 Coleridge's Friend. 
 
32 A SYLLABUS OF LOGIC. 
 
 which, considered as to space and time, and 
 the sensation contained in them, is equal to 
 zero. But it may be thought : for we can 
 very well understand that there should be in 
 our minds a law or a priori relation between 
 phenomena and the time in which they are 
 viewed, and that this law should retain its 
 signification even when the phenomena and 
 their time both = 0. It is, however, quite 
 beyond the powers of symbolical representa- 
 tion to give the nature of these conceptions 
 themselves. Nothing more is here intended, 
 than to express, by an analogy to the mathe- 
 matical idea of a limit, the only method by 
 which these conceptions can be thought pure, 
 and independent of the possible experience to 
 which they are necessary. 
 
33 
 
 SECTION III. 
 
 JUDGMENTS. 
 
 (21.) A judgment is that act of the under- 
 standing by which it determines how certain 
 representations may be conjoined in the con- 
 sciousness according to some rule of its own. 
 
 Kant has divided the consideration of judg- 
 ments as to their form into the four points 
 of Quantity, Quality, Relation, and Modality; 
 and each of these again into three subdivisions 
 or moments,* from which he derives the twelve 
 categories, or pure conceptions of the under- 
 standing. 
 
 Quantity. 
 
 (22.) The three moments of Quantity are 
 the Universal, the Particular, and the Singular. 
 These determine the quantity of the subject of 
 a judgment, and refer respectively to the whole 
 sphere of a conception, to a part of the sphere 
 
 * Our choice of the word ' moment,' was entirely determined by 
 its having been employed in this signification by the anonymous 
 Translator of the Criticism. 
 
 D 
 
34 A SYLLABUS OF LOGIC. 
 
 of a conception, or to an individual ; e. g. All 
 men are mortal ; some men are mortal ; John is 
 mortal. 
 
 In Transcendental Logic we distinguish be- 
 tween the Universal and Singular, because the 
 subject of the former is in respect of its formal 
 quantity unlimited, whereas the subject of the 
 latter is completely determined. If, for in- 
 stance, I speak of ' all men/ there is nothing 
 in my bare conception of man which can put 
 any limit to the number of individuals that 
 may come under it, or answer its conditions; 
 and the quantum of the sphere of the con- 
 ception is therefore formally indeterminate : 
 whereas, if I speak of an individual, John, my 
 cognition contains all that is found in the 
 conception of man, and also the possibility of 
 his complete determination in space and time, 
 for on that his identity depends.* But For- 
 
 * That our conception of identity requires us to think the pos- 
 sibility of determining the object in a particular space at a parti- 
 cular time, is evident from the consideration that there is no limit 
 to the number of individuals who may exactly resemble each other 
 in all other respects. Now the particularity of space and time is 
 determined by a reference to ourselves ; for we can only think of 
 a particular time as being at such a period before or after the pre- 
 sent time, or that in which we are thinking, and of a particular 
 space as being at a certain distance from the present space, or that 
 in which we are now sensible : hence, the identity of all external 
 objects ultimately depends on a reference to the identity of the 
 thinking subject. But if I endeavour to represent my own iden- 
 tity, I must have recourse to the identity of external objects; 
 
INTRODUCTION. 35 
 
 mal Logic takes no cognizance of the object 
 thought, and consequently does not recognise 
 the distinction : for in speaking of an indi- 
 vidual man, I speak of all that answers to my 
 cognition, and that is all that Formal Logic 
 can require for an Universal. This science 
 therefore acknowledges only the two moments, 
 Universal and Particular. 
 
 Quality. 
 
 (23.) The three moments of Quality are the 
 Affirmative, the Negative, and the Indefinite. 
 
 The first, as its name implies, affirms the 
 predicate of the subject ; the second denies 
 it ; but the third affects the matter of a con- 
 ception which it limits by entirely excluding it 
 from some particular conception. Thus, ' not 
 A' is indefinite, as it applies to any thing that 
 lies out of the sphere of A. As every thing 
 
 e. g. I may represent myself to myself as the person who was 
 in a particular room at a certain time : hence it follows, that I 
 think of the identity of other objects by means of a reference to 
 myself, and of myself by means of a reference to other objects. 
 There is, then, between the 'me* and the 'not me' a kind of 
 polarity, which existing in space and time according to some un- 
 discoverable law of the consciousness, gives rise to the conception 
 of identity. This may be illustrated by representing space and time 
 as the axes of co-ordinates, and our own mind as the origin from 
 whence any values of x and y may be measured for the determi- 
 nation of an object. Or if we think of our own identity by means 
 of that of another object, we must place the origin at the object, 
 and consider our own mind as the point to be determined. 
 
 D 2 
 
36 A SYLLABUS OF LOGIC. 
 
 must be either A or ' not A,' it is the same 
 thing whether ' not A* be affirmed or A be 
 denied of any subject; and for this reason 
 the Indefinite is not considered in Formal 
 Logic, as the Negative moment can always 
 take its place. 
 
 Relation. 
 
 (24.) The three moments of Relation are 
 the Categorical, the Hypothetical, and the 
 Disjunctive. 
 
 The first of these considers the relation of 
 cognitions to the same substratum or sub- 
 stance. The second considers the dependence 
 of one cognition as consequent upon another 
 as antecedent. The third considers a recipro- 
 city of relation between cognitions in such a 
 manner that any one of them can be known 
 by the determination of all the rest. The 
 most general form of a Categorical judgment 
 would be ' A is B,' or ' No A is B,' in which 
 the conception B is asserted to belong, or not 
 to belong, to the subject A. An Hypotheti- 
 cal is of the form t If A is B, C is I)/ in which 
 the judgment that f C is D ' is made to depend 
 as consequence upon the judgment that ' A is 
 B' as antecedent. Either ' A is B, or ' C is D,' 
 is a Disjunctive judgment, and (the determina- 
 tion of either part would determine the other. 
 
INTRODUCTION. 37 
 
 On these three points of a judgment we 
 touch but lightly here, because they affect 
 the laws of Formal Logic, and are therefore 
 more fully considered in another place. 
 
 Modality. 
 
 (25). Modality concerns the manner in 
 which we think a judgment with regard to its 
 truth, and is divided into the three moments 
 of Problematical, Assertive, and Necessary. 
 The first degree of holding true is that of 
 problematical, which merely signifies the pos- 
 sibility of the judgment, inasmuch as it does 
 not violate any of the universal laws of think- 
 ing. 
 
 The second degree accords to a judgment 
 the agreement with the matter of the senses, 
 as well as with their necessary forms. The third 
 degree makes some a priori law of thinking the 
 matter of the judgment. 
 
 A problematical judgment merely implies a 
 formal, but not a material possibility. For 
 instance, the conception of an elderly lady 
 riding through the air on a broomstick, is 
 formally possible, as it involves no contradic- 
 tion, but is not generally considered materially 
 possible. 
 
 A judgment may be assertive without its 
 contents having come immediately under the 
 
38 A SYLLABUS OF LOGIC. 
 
 cognizance of the person who forms it. No- 
 thing more is meant by the above definition, 
 than that assertive judgments must have 
 sufficient and empirical grounds for their 
 truth. 
 
 The third moment, Necessity, is applicable to 
 any a priori law, whether rational, intellectual, 
 or intuitive. For instance, that A is C, if A is 
 B, and B is C, is a necessary truth of reason. 
 That no change in phenomena can be self- 
 originated, or that every effect must have its 
 cause, is a necessary truth of the understand- 
 ing. That the straight line is the shortest 
 between two given points, is a necessary truth 
 of intuition. 
 
 (26.) Propositions are judgments whose mo- 
 dality is either assertive or necessary: for a 
 problematical judgment, so far as its modality 
 alone is concerned,* merely implies that a cer- 
 tain judgment is not necessarily false, but bears 
 no positive testimony to its truth. And we 
 could not therefore say with propriety that 
 any thing is proposed in a problematical judg- 
 
 * It was necessary to introduce this restrictive clause ; for 
 problematical judgments, considered independently of their moda- 
 lity, are generally intended to imply some anticipation of a truth, 
 and their main use is as a step to the assertive. For instance, 
 there may be such a person as Mr. Stiggins in New York, but no 
 man would ever make such a judgment unless he had some 
 grounds for believing it true. 
 
INTRODUCTION. 39 
 
 ment, as that term has evidently a positive 
 signification. 
 
 Propositions alone can enter into the consi- 
 deration of Formal Logic ; for the laws of 
 Reason require something determinate for the 
 matter of the judgments whose combinations 
 they regulate, and are not conversant about 
 mere possibilities which have only a negative 
 value in relation to truth. This science more- 
 over regards all propositions as assertive : for it 
 cannot admit the distinction between assertive 
 and necessary modality, as this is determined 
 by the matter of judgments, and is quite inde- 
 pendent of their form. 
 
 Synthetical and Analytical Judgments. 
 
 (27.) Judgments may be divided, as to their 
 matter, into Synthetical and Analytical. 
 
 Synthetical judgments are those whose 
 predicates are not contained in the conception 
 of their subjects. 
 
 Analytical judgments, on the other hand, 
 predicate of their subjects something that is 
 already contained in them, or, in other words, 
 their predicate is a superior conception to 
 their subject. 
 
 Hence we see that synthesis, or the principle 
 of synthetical judgments is the conjunction of 
 
40 A SYLLABUS OF LOGIC. 
 
 two different cognitions, neither of which is 
 contained in the other. 
 
 Analysis, or the principle of analytical judg- 
 ments, is the dissection of a conception into its 
 partial representations. 
 
 As the matter contained in the conceptions 
 of the subject and predicate determine the 
 nature of the judgment in respect of the above 
 division, it follows that the same proposition 
 may express an analytical judgment to one 
 person, and a synthetical judgment to another. 
 The proposition ' all members of the Univer- 
 sity are members of some particular college/ is 
 an analytical judgment to a person who is well 
 acquainted with the constitution of the Univer- 
 sity, and whose conception of a member of the 
 University already contains the conception of 
 his belonging to some particular college. But 
 the same proposition is synthetical to a person 
 who is entirely ignorant of that constitution, 
 and is therefore unacquainted with the neces- 
 sary conditions to being a member of the 
 University. 
 
 (28.) But the most remarkable instance of 
 confusion between synthesis and analysis arises 
 from the circumstance, that what is analysis 
 considered objectively, is very frequently a 
 synthesis if considered subjectively. Thus 
 the process by which a chemist examines any 
 
INTRODUCTION. 41 
 
 substance for the purpose of discovering its 
 qualities, is rightly named analysis when re- 
 ferred to the object examined, but is synthesis 
 when considered in relation to himself. For 
 instance, if a person, who is perfectly ignorant 
 of the constituent elements of water, separates 
 the two gases by means of some chemical 
 process, the investigation is correctly styled 
 analysis, if considered in reference to the ob- 
 ject ( water/ as he has disjoined its component 
 parts : but the result in the operator's own 
 mind is a synthesis ; for his previous concep- 
 tion of water contained nothing more than 
 fluidity, the absence of all colour and taste, 
 and perhaps the property of dissolving a great 
 many salts. But by the analysis of the object 
 he has increased this conception, by adding 
 to it, that water is composed of oxygen and 
 hydrogen ; and consequently his mental act is 
 a synthesis. 
 
 Definition. 
 
 (29.) Definition is a judgment which deter- 
 mines all the partial representations contained 
 in any conception, and is therefore equivalent 
 to a completed analysis. 
 
 There is a specious resemblance of definition 
 which is grounded on a synthesis, and has 
 really no right to the name. 
 
42 A SYLLABUS OF LOGIC. 
 
 Definition, in the strict acceptation of the 
 word, regards only the matter of a conception, 
 and not its sphere. But sometimes the term is 
 improperly applied to the exposition of any 
 conception whose sphere is the same with 
 that of the conception to be defined. Let us 
 suppose that a conception a b, or one which 
 contains the partial representations a and b y 
 is connected by a certain law with the con- 
 ception c d, and in such a manner that the 
 sphere of a b is identical with the sphere of 
 c d. In this case it would not be a correct 
 definition of the conception a b, to state that it 
 contained the partial representations c and d ; 
 though such a judgment would lead us to 
 the same individuals as if the correct defi- 
 nition had been given. Such a judgment 
 would be a true synthetical judgment, but 
 not a definition. 
 
 (30.) It follows from this, that two persons 
 may give different definitions of the same 
 thing, and yet both of them be equally correct : 
 for though the things be the same in each 
 case, yet the conceptions by which each indivi- 
 dual may recognise these things may be very 
 different, and perhaps have hardly a single 
 point in common. Let us take as an example 
 the conception of water. The common con- 
 ception contains little more than that water 
 
INTRODUCTION. 43 
 
 is a perfectly tasteless and colourless liquid. 
 Now these properties are connected by a law 
 of nature with a combination of oxygen and 
 hydrogen, in certain proportions and under 
 certain circumstances; and a definition which 
 contained either account of water would equally 
 refer to the same thing, but to very different 
 conceptions. The rustic would probably not 
 have much faith in the chemist's proof that 
 a certain liquid was water by his exhibiting 
 the gases in a separate state ; neither would the 
 chemist have much faith in the rustic's defini- 
 tion : and yet the definition would be correct 
 in each case, as it would express that concep- 
 tion by which each was in the habit of recog- 
 nising the thing itself. The example in Mr. 
 Newman's Logic is a very good one. He 
 observes that it would be extremely incorrect 
 to define man as a cooking animal, although it 
 is highly probable that man, and man only, 
 answers to that description. But as we do not 
 recognise men by this peculiarity in their 
 nature in short, as it does not form part of 
 our conception of humanity, such a definition 
 would only mark out the sphere, and by no 
 means determine the matter of our conception 
 of man. When two conceptions are united by 
 some law which is immediately and univer- 
 sally recognised, it becomes a matter of indif- 
 
44 A SYLLABUS OF LOGIC. 
 
 ference from which of these conceptions we 
 derive our definition. For instance, the com- 
 mon definition of a triangle is ' a figure of 
 three sides/ whereas in strictness it should be 
 'a figure of three angles/ The usual defini- 
 tion is in reality a pure synthetical judgment 
 of intuition. 
 
FORMAL LOGIC. 
 
 BOOK I. 
 SECTION I. 
 
 PROPOSITION S. 
 
 (30.) Propositions are of three kinds 
 Categorical, Hypothetical, and Disjunctive. 
 
 Form of Categorical Propositions. 
 
 Categorical propositions consist of two terms 
 and a copula, of which the terms designate the 
 matter, and the copula their relation to each 
 other. Their most general form may be re- 
 presented by the proposition ' A is B,' in which 
 the terms A and B are connected by the copula 
 * is/ Particular examples are, ' horses are 
 animals/ 'men are not monkies/ 
 
 Many categorical propositions do not come 
 immediately under this simple form, but are 
 easily reduced to it by a periphrasis ; e. g. 
 the proposition ' I took a ride this morning* 
 
46 A SYLLABUS OF LOGIC. 
 
 maj be put under the form, ' I am a person 
 who took a ride this morning/ 
 
 The terms are susceptible of no other* for- 
 mal variations than the moments of quantity, 
 which determine to how much of the sphere 
 of a conception the proposition refers. 
 
 There can be only two such formal varia- 
 tions the whole, and less than the whole, or 
 part. For we evidently cannot speak of more 
 than the whole, as the idea involves an absur- 
 dity : neither can we make any distinction 
 between different quantities that are less than 
 the whole, without introducing the considera- 
 tion of the matter of a conception as well as 
 of its form. It follows, therefore, that ' the 
 whole' and 'less than the whole' are the only 
 variations of quantity which Formal Logic can 
 recognise : e. g. we may speak of ' all albinos/ 
 i. e. the whole sphere of the conception ' albino,' 
 
 * It is sometimes convenient to consider the matter of a pro- 
 position with respect to its quality as well as quantity, and to 
 speak of ' not A/ ' not B,' &c. : in this case, the * not A' is 
 called the external sphere of A. The attribute of quality is, 
 however, but rarely accorded to the terms : for if the predicate 
 is an external sphere, the form of the proposition is precisely 
 the same as if the negation had been applied to the copula. 
 Thus, A is a t not B ' is exactly equivalent to A 'is not ' B. 
 And an external sphere is but rarely introduced in the subject 
 for other reasons, as well as on account of its extreme awkward- 
 ness. For instance, we do not say, ' not men' are ' not English- 
 men,' but 'all Englishmen are men,' though the meaning of the 
 two propositions is precisely the same. 
 
PROPOSITIONS. 47 
 
 or of ' some albinos/ i. e. a part of the sphere 
 of that conception. 
 
 When the whole of either term is compared 
 with the other, it is said to be distributed ; 
 when a part only is so compared, it is said to 
 be undistributed. For instance, in the pro- 
 position ( All A is B,' the term A is distri- 
 buted ; but in the proposition ( Some A is B,' 
 it is undistributed. 
 
 The only formal variations of which the 
 copula is susceptible, are two moments of 
 quality, affirmation, and negation. For if we 
 compare any two terms A and B in a cate- 
 gorical proposition, we can only affirm or deny 
 the one of the other : e. g. A is B, A is not 
 B. In the first example the quality is said 
 to be affirmative, in the latter negative. 
 
 Hence, the two terms and the copula, which 
 constitute the three elements of a categorical 
 proposition, severally admit of two gradations 
 or variations in form : and if there were no 
 law for their limitation, the number of possible 
 combinations of these elements would = 2 3 = 8. 
 These eight combinations would be, 
 
 * All A is Some B. All A is All B. 
 
 * Some A is Some B. Some A is All B. 
 
 * All A is not All B. All A is not Some B. 
 
 * Some A is not All B. Some A is not Some B. 
 
48 A SYLLABUS OF LOGIC. 
 
 (32.) But we find by experience, that of 
 these eight forms of categorical propositions, 
 only four are ever introduced in practice : 
 for the quantity of the term that is placed 
 last in the general categorical form (Le. A is B) 
 is entirely determined by the quality of the 
 copula,* The variations of the copula and of 
 this latter term must therefore be taken toge- 
 ther, and the whole number of combinations 
 for categorical propositions will 2 2 = 4. 
 
 As the first and lastf terms of a categorical 
 proposition do not bear precisely the same 
 relation to the copula, independently of their 
 mere position, they are distinguished respec- 
 tively by the names of subject and predicate: 
 
 * The law for the dependence of the quantity of th 
 upon the quality of the copula, can only be assumed in Formal 
 Logic : for the a priori grounds upon which it rests are to be 
 found in the consideration of one of the categories or conceptions 
 of the understanding, and therefore belong to the Transcendental 
 branch of the science. This subject will be considered in a 
 future section, but a cursory view of it may not be wholly out of 
 place here. The category of substance can never be introduced 
 in the predicate of any proposition, and this term must therefore 
 be a conception. But the quantum of the sphere of a conception 
 is entirely indeterminate, as there is no formal limit to the number 
 of individuals that may answer its conditions. It is therefore 
 impossible to compare the limits of the sphere of the predicate 
 with those of the subject, as those of the former are unknown. 
 The subject then can only be placed wholly out of, or wholly in, 
 the sphere of the predicate, and the latter will be distributed or 
 undistributed accordingly. 
 
 f i. e. the first and last in the simplest form of a categorical 
 proposition, e.g. ' All A is B.' 
 
PROPOSITIONS. 49 
 
 and the propriety of introducing such a dis- 
 tinction into Formal Logic depends entirely 
 on this formal difference of relation, and not 
 at all on the real nature of the distinction, 
 which is a subject of Transcendental Logic, 
 and cannot therefore be considered here. 
 Neither does it at all depend on the order in 
 which the terms are arranged ; * for this is 
 frequently reversed in some languages, and 
 occasionally even in English. 
 
 Of the eight combinations given in the last 
 page, those marked by an asterisk are the four 
 legitimate Categorical propositions. As, how- 
 ever, the quantity of the predicate is a known 
 function of the quality of the copula, it is 
 never expressed, but always understood ; the 
 four propositions will therefore assume the 
 following form : 
 
 All A is B. No A is B. 
 
 Some A is B. Some A is not B ; 
 
 where it must be observed that the proposition 
 ' No A is B ' is equivalent to the proposition 
 
 * This order is seldom introduced in English, except in poetry, 
 emphatic diction, or where the subject is a sentence. The fol- 
 lowing are examples of these three cases : 
 
 " Oh, many are the Poets that are sown 
 
 By Nature \-Wordsworth. 
 
 i. e. the poets are many that, &c. "Brave indeed is that man 
 who," &c. 2. e. that man who, &c. is brave indeed. " It is very 
 easy to say a great deal in a letter that cannot be hinted at in a 
 personal interview," i. e. to say a great deal in a letter, &c. is very 
 easy. E 
 
oO 
 
 A SYLLABUS OF LOGIC. 
 
 ' All A is not B/ or the entire exclusion of 
 A from B. 
 
 As the difference in form of these four pro- 
 positions arises from variations in the subject 
 and copula, they will admit of two correspond- 
 ing co-divisions. One of these is into Univer- 
 sal and Particular, and respects the quantity of 
 the subject ; the other is into Affirmative and 
 Negative, and respects the quantity of the 
 copula. They are thus divided into Universal 
 Affirmative represented by A ; Universal Nega- 
 tive by E ; Particular Affirmative by I ; and 
 Particular Negative by O : 
 
 
 AFFIRMATIVE. 
 
 NEGATIVE. 
 
 Universal. 
 
 A. 
 All A is B. 
 
 E. 
 
 No A is B. 
 
 Particular. 
 
 I. 
 
 Some A is B. 
 
 0. 
 
 Some A is not B. 
 
 (33.) The law by which the number of legi- 
 timate combinations is reduced to four, may 
 be stated as follows : 
 
 Affirmative copulas have undistributed pre- 
 dicates ; Negative copulas have distributed 
 predicates. For if B is predicated of A, A is 
 subsumed under the conception B as being a 
 part of its sphere ; for instance, in the pro- 
 
PROPOSITIONS. 51 
 
 position ' all horses are animals : ' the subject 
 ' horses ' is subsumed under the conception of 
 the predicate animals/ as being a part of 
 the sphere of that conception. But it is evi- 
 dent that the proposition does not refer to 
 the whole sphere of the conception e animals/ 
 for in that case there could be no other ani- 
 mals than horses. 
 
 The annexed diagram is a general illustra- 
 tion of the law : 
 
 All A is B. Some A is B. 
 In each of these diagrams we see that A is 
 compared with only a part of B ; that is to 
 say, B is undistributed. 
 
 It frequently happens that the sphere of the 
 predicate is not any larger than the sphere of 
 the subject : this however is incidental, and 
 can only arise from the peculiar nature of the 
 matter of the proposition, and never from its 
 logical form : for in the example s all carnivor- 
 ous animals have teeth of a certain form/ it 
 may be equally true that ' all animals with 
 teeth of this form are carnivorous/ But the 
 latter proposition requires additional knowledge 
 of the subject matter, and cannot be deduced 
 from the mere form of the original statement. 
 
 E2 
 
52 A SYLLABUS OF LOGIC. . 
 
 If, on the other hand, the predicate B is 
 denied of the subject A, the sphere of A is 
 entirely excluded from the whole sphere of B : 
 for instance, in the proposition ' no durable 
 friendship can be based on a participation in 
 crime,' friendship is entirely excluded from all 
 those things which can be based on a partici- 
 pation in crime : or in the particular propo- 
 sition ' some of the most talented men do not 
 possess the best private characters; these ' some 
 men 'are entirely excluded from all who pos- 
 sess the best private characters. In each case 
 then the predicate is distributed, as the pro- 
 position refers to the whole of its sphere. The 
 diagrams will take the following forms : 
 
 No A is B Some A is not B 
 
 in which it is evident that the whole of B lies 
 out of as much of A as is introduced in the 
 subject. 
 
 The accompanying table contains the results 
 of these remarks : 
 
 SUBJECTS. PREDICATES. 
 
 Universal Affirmative A ... Distributed Undistributed. 
 Particular Affirmative I ... Undistributed Undistributed. 
 Universal Negative E ... Distributed Distributed. 
 Particular Negative O ...Undistributed Distributed. 
 
PROPOSITIONS. 53 
 
 SECTION II. 
 
 MUTUAL RELATIONS OF CATEGORICAL 
 PROPOSITIONS. 
 
 (34.) The mutual relations of the four propo- 
 sitions A, E, I, O, have been usually classed by 
 logicians under the generic term of Opposition. 
 But as there is no opposition whatever in some 
 of these relations, it may be more correct to 
 restrict the acceptation of the word to those 
 cases which contain its meaning, and apply 
 that of Subalternation to the rest. 
 
 Opposition. 
 
 Opposition is of three kinds Contradictory, 
 Contrary, and Sub-contrary. 
 
 Contradictory opposition is the relation which 
 exists between two propositions that simply 
 contradict each other. One of them therefore 
 must be false, and the other true. For each 
 of them must be either true or false, and if 
 one of them is true, the other which asserts 
 that it is false, must be itself false ; and if 
 one of them is false, the other which asserts 
 that it is false, must be itself true. 
 
 Contradictories differ from each other in 
 the quality of their copulas ; for as one denies 
 
54? A SYLLABUS OF LOGIC. 
 
 what the other asserts, the former must be 
 affirmative, and the latter negative. 
 
 They differ also in the quantity of their sub- 
 jects. For a contradictory contains nothing 
 more than the falsity of that proposition to which 
 it is opposed, and must therefore express the 
 change that takes place in the relation of the 
 terms, when its opposite first ceases to be true. 
 But the first formal change affecting the truth 
 of an universal proposition, affects only a part 
 of its subject ; as that cannot be true of the 
 whole, which is false of a part. And the first 
 formal change affecting the truth of a parti- 
 cular proposition, must necessarily affect the 
 whole of the subject; as that cannot be false 
 of every part, which is not at the same time 
 false of the whole. 
 
 These remarks will be better understood by 
 the assistance of the accompanying diagrams : 
 
 B 
 
 m 
 
 Fig. 1. a. Fig. 1. b. Fig. 2. a. Fig. 2. b. 
 
 Fig. 1. 0, represents the proposition, All A 
 is B ; and Fig. 2. , represents the proposition, 
 No A is B. Now it is very evident that the 
 first changes which will render these proposi- 
 tions no longer true, must take place when 
 the sphere of A first begins to emerge from 
 
PROPOSITIONS. 55 
 
 that of B, as in Fig. 1. b, or when it first begins 
 to impinge on it, as in Fig. 2. b. And the pro- 
 positions which express these changes must 
 necessarily be the respective contradictories of 
 the original propositions. c Some A is not B ' 
 will therefore contradict the proposition ' All 
 A is B/ and ' Some A is B/ the proposition 
 ' No A is B.' Contradictories are therefore op- 
 posed to each other both in quality and quan- 
 tity : the universal affirmative A is opposed to 
 the particular negative O; and the universal 
 negative E, to the particular affirmative I. 
 Example : ' Some Englishmen are as light- 
 hearted as Frenchmen ;' e No Englishmen are 
 as light-hearted as Frenchmen.' 
 
 (35.) Contrary opposition exists between 
 two propositions which contain each other's 
 contradictory, and something more besides. 
 Hence it follows that they cannot be both true, 
 but may be both false : for if one is false, that 
 part of its contrary which merely contradicts 
 it, must be itself true ; but the other part, or 
 surplus statement, may be either true or false. 
 
 As contraries state more than each other's 
 contradictories, the quantity of their subjects 
 must be greater, and therefore distributed; 
 whence it follows that A and E are the only 
 propositions between which this species of 
 opposition exists. Examples of this opposi- 
 tion are 'All smuggling is dishonest'; 'No 
 
56 A SYLLABUS OF LOGIC. 
 
 smuggling is dishonest.' 'All Englishmen are 
 haughty'; 'No Englishmen are haughty/ &c. 
 (36.) Subcontrary opposition, as its name 
 in some measure indicates, is the opposition 
 of propositions contained under contraries. 
 The subjects of the subcontraries must of 
 course be less than the subjects of the con- 
 traries which contain them, and are there- 
 fore undistributed. As I and O the sub-con- 
 traries are contradictories of the two contra- 
 ries E and A, and as the contraries cannot 
 be both true, but may be both false, it fol- 
 lows that the subcontraries may be both true^ 
 but cannot be both false. Examples are, ( Some 
 men are liars'; ' Some men are not liars': both 
 of which are true. ( Some men are perfect' ; 
 ' Some men are not perfect': of which propo- 
 sitions one is false. 
 
 Subalternation . 
 
 (37.) Subalternation is the relation which 
 exists between an universal proposition, and 
 the particular that is contained in it. The 
 former is usually called the Subalternant, the 
 latter the Subalternate. 
 
 The truth of the subalternant necessarily 
 involves the truth of its subalternate, as what 
 is true of the whole, must be also true of a 
 part ; and the falsity of the subalternate in- 
 volves the falsity of the subalternant, as what 
 
PROPOSITIONS. 57 
 
 is false of a part cannot be true of the whole. 
 But the converse of these propositions does not 
 hold ; for what is true of a part, is not neces- 
 sarily true of the whole, and what is false of 
 the whole, is not necessarily false of a part. 
 Hence, we cannot infer the truth of the uni- 
 versal from that of the particular subalternate 
 to it, nor the falsity of the particular from 
 that of its subalternant universal. Examples 
 are ' All dogs are animals/ whence it follows, 
 ' Some dogs are animals.' ' Some politicians are 
 no better than they should be/ whence it cannot 
 be logically inferred that ' All politicians are 
 no better than they should be/ The annexed 
 scheme is usually employed to elucidate the 
 relations of categorical propositions : 
 
 A f Contraries f E 
 
 f f 
 
 c c 
 
 s > * 
 
 ? X **'o 2 
 
 5 0^ X I 
 
 t * * 
 
 / / 
 
 / c Subcontraries c O 
 
 f means false ; t means true ; and c contingent. 
 
 The letters f, f, c, that are placed against 
 the Universal Affirmative A, are intended to 
 
58 
 
 A SYLLABUS OF LOGIC. 
 
 represent that A is false when E or O are true, 
 and contingent when I is true, and similarly of 
 the rest of the table. 
 
 Perhaps the following table is more imme- 
 diately intelligible 
 
 If one 
 
 The other 
 
 
 ' s 
 
 
 O 
 
 
 '43 
 
 
 ^ 
 
 
 r 
 
 
 c 
 
 
 a 
 
 fc 
 
 
 O 
 
 i. 
 
 H 
 
 
 
 * i 
 
 - 
 
 a I 
 
 
 ^' 
 
 
 O 
 
 G < 
 
 t- 
 
 O 
 
 ' 
 
 2 
 
 
 3 
 
 
 . OT 
 
 HH 
 
 i 
 
 1" 
 
 1 
 
 i 
 
 pq 
 
 PH x 
 
 s 
 
 d 
 
 sj 
 
 
 
 eq 
 
 
 ts 
 Ol 
 
 w 
 
 G 
 
 
 <u 
 
 
 i 
 
 
 i 
 
 
 . M 
 
 Between A and 
 
 a Between E and I.. 
 
 Between A and E. 
 
 Between I and O. 
 
 (True False. 
 
 \ False. 
 
 -True. 
 
 'f True False. 
 
 \ False Contingent. 
 
 f False.. True. 
 
 \ True Contingent. 
 
PROPOSITIONS. 59 
 
 Conversion. 
 
 (38.) Conversion is the transposition of the 
 terms of a proposition, in such a manner that 
 the subject becomes the predicate, and the 
 predicate the subject. Conversion is of two 
 kinds, simple and limited. The former is the 
 simple transposition of the terms, retaining 
 the previous quantities ; the latter requires 
 the limitation of the subject. 
 
 In all conversion, no term must be distri- 
 buted in the converted proposition, that is not 
 distributed in the original proposition : hence 
 E and I are simply convertible, as in E 
 both terms are distributed, and in I, neither. 
 No term then is distributed in the converted 
 proposition, that was not previously distri- 
 buted in the original proposition. But A is 
 not simply convertible, as its predicate is 
 undistributed, and would become distributed 
 by simple conversion. It admits however of 
 limited conversion, and then becomes I ; for 
 in I, no terms are distributed, and therefore 
 none can be distributed in it, which are un- 
 distributed in the original proposition. E may 
 also be converted by limitation, as well as 
 simply, and then becomes O. O can never be 
 converted, as its subject is undistributed, and 
 therefore can never become the predicate of 
 
60 A SYLLABUS OF LOGIC. 
 
 a negative proposition, which is always dis- 
 tributed. 
 
 CONVERSION. 
 
 Simple. Limited. 
 
 E to E A to I 
 
 I to I E to O 
 
 Examples : 
 
 No cats are cows. All dogs are animals. 
 
 No cows are cats. Some animals are dogs. 
 
PROPOSITIONS. 61 
 
 SECTION III. 
 
 HYPOTHETICAL AND DISJUNCTIVE PROPOSITIONS. 
 
 (39.) An hypothetical proposition consists 
 of two judgments and a copula, r.-y 
 
 The judgments constitute the matter of the 
 proposition, and the copula expresses their 
 relation, or that function of the understanding 
 by which they are conjoined in one con- 
 sciousness. 
 
 The whole proposition represents that one 
 of these judgments which is called the ante- 
 cedent, contains all the necessary grounds for 
 the truth of the other, which is accordingly 
 called the consequent. 
 
 The most general form of these propo- 
 sitions is the following. f If A is B, C is D/ in 
 which the judgment ' A is B' is the antecedent, 
 ( C is D' the consequent, and the word e if the 
 copula. 
 
 (40.) As an hypothetical proposition merely 
 asserts such a connexion between two judg- 
 ments, that the truth of the one may always 
 
62 A SYLLABUS OF LOGIC. 
 
 be inferred from the truth of the other,* it 
 follows that both the form and the matter of 
 these judgments constitute the matter alone of 
 the hypothetical, and cannot be the means 
 of introducing any variations into its form. 
 Neither can such variations arise .from the 
 form of the copula; for the conception of 
 a law connecting a consequent with its grounds, 
 may be thought, or not thought, with regard 
 to any judgments, but will neither admit of 
 a negative quality, or of a quantum or degree. 
 Hence there is but one form for hypothetical 
 propositions, and all the variations that take 
 place in them must be considered as affecting 
 their matter alone. For instance, the propo- 
 sition ' if some A is B, some C is D/ does not 
 differ in form from the proposition ' if all 
 A is B, all C is D.' It is true that the cate- 
 gorical judgments in these examples have 
 formal differences, as in the former they are 
 both particular, and in the latter both univer- 
 sal. But in either case, such a relation is 
 asserted to exist between them, that the truth 
 of the one must be the invariable consequence 
 
 * The understanding merely asserts a relation of such a nature 
 between the judgments that the truth of the one may be inferred 
 from that of the other. But in order that this result should be 
 actually inferred, another judgment is necessary which shall 
 assert the truth of the antecedent, and then the reason will 
 deduce the truth of the consequent. 
 
PROPOSITIONS. 63 
 
 of the truth of the other ; and it is this 
 universality alone, and not that of the judg- 
 ments themselves, with which the form of 
 an hypothetical proposition is concerned. 
 
 (41.) From these observations it will readily 
 be seen that an hypothetical proposition can- 
 not have an hypothetical contradictory : for 
 a contradictory states nothing more than the 
 falsity of that proposition to which it is op- 
 posed, and, if this latter is an hypothetical, 
 merely denies that there is any law by which 
 the truth of the consequent can be rightly 
 inferred from that of the antecedent. But 
 this does not establish any similar relation 
 between any other two judgments, and is not 
 therefore hypothetical in its nature. The con- 
 tradictory must accordingly be a categorical 
 proposition, in which I predicate of my consci- 
 ousness, that it does not give objective validity 
 to such a relation between two judgments, that 
 one contains the grounds for the other. 
 
 (42.) Many logicians have considered the 
 hypothetical proposition as merely another 
 form of a categorical ; and this error has been 
 rather favoured by the seeming ease with 
 which an hypothetical may be put under such 
 a categorical form as shall answer all the 
 logical ends of the original proposition. But 
 the two propositions have been shewn by 
 
64 A SYLLABUS OF LOGIC. 
 
 Kant to be based on fundamentally different 
 acts of the understanding, which ought never 
 to be confounded with each other. For in- 
 stance, the hypothetical proposition < if A is, 
 B is/ appears to be fully expressed in the 
 categorical form, ' all the cases of A being, are 
 cases of B being ;' and so far as any logical 
 deductions from either one or the other are 
 concerned, the propositions are equivalent. 
 But there is this fundamental difference be- 
 tween them : the former represents the depen- 
 dence of one judgment upon grounds contained 
 in the other, and therefore enunciates a law 
 to which the understanding has accorded its 
 assent as being universally valid ; whereas the 
 latter form either merely asserts the fact ' that 
 all the cases of A being are cases of B being,' 
 without superadding the conception that they 
 are so universally by a law ; or else, if it means 
 that 'all possible cases of A being are ne- 
 cessarily cases of B being/ it has only placed 
 what constituted the form in the hypothetical 
 proposition in the matter of the categorical, 
 but has by no means transfused the virtue of 
 the form of the one into the form of the other. 
 In fact, this method is no more a formal reduc- 
 tion of an hypothetical to a categorical proposi- 
 tion, than if we merely said, " the hypothetical 
 proposition ' if A is, B is/ is true." 
 
PROPOSITIONS. 65 
 
 Disjunctive Propositions. 
 (43.) Disjunctive propositions consist of any 
 number of judgments, which they disjoin in the 
 relation of reciprocal dependence. Hence the 
 whole conception of their possibility is divided 
 between them, and the truth of either may be 
 inferred from the falsity of the others, or in- 
 versely, the falsity of all the others from the 
 truth of one. Their general form is ' either A 
 is B, or C is D.' All the observations that have 
 been made on hypothetical propositions, in 
 order to prove the impossibility of any vari- 
 ations in their form, apply with equal force 
 to the disjunctive : and as the disjunctive 
 expresses a distinct function of thinking, 
 any attempt to bring it into a categorical 
 form will meet with no better success than in 
 the case of hypotheticals. It is true that we 
 can express the meaning of a disjunctive in 
 a categorical form ; but then, what constitutes 
 the form in the one proposition must be in- 
 troduced in the matter of the other. For in- 
 stance, the disjunctive proposition ' A is either 
 B or C ' may be expressed thus : ' the cases 
 of A being B are identical with the cases of 
 A being not C/ But here we see that the com- 
 pleteness of the division, which is the real 
 principle of disjunctive propositions, is ex- 
 pressed in the conception of the identity of 
 
66 A SYLLABUS OF LOGIC. 
 
 one judgment with the negation of another, 
 which constitutes the matter of the catego- 
 rical proposition. And hence, as in the case of 
 the hypothetical, we have no more succeeded 
 in the transformation of the disjunctive to 
 the categorical form than, if we had said, " the 
 proposition ' A is either B or C ' is true," 
 
 (44.) Disjunctives resemble hypothetical 
 propositions in not admitting a contradictory 
 of the same form with themselves : for the 
 contradictory of a disjunctive must merely 
 contain the falsity of that proposition, and 
 therefore denies that a certain cognition is 
 exactly divided out among the members of 
 the disjunction, but by no means gives a new 
 and correct division. Hence the contradicto- 
 ries to both these forms of propositions are 
 simply categorical in their nature. 
 
SYLLOGISM. 67 
 
 SECTION IV. 
 
 SYLLOGISM. 
 
 (45.) Kant has defined syllogism as that 
 function of thinking by which we derive one 
 judgment from another, and has in this 
 manner included all those judgments which 
 may be formally derived from another single 
 judgment by the methods of opposition, sub- 
 alternation, &c. 
 
 These syllogisms, styled immediate from 
 their wanting a middle term, he arranges 
 under the title of syllogisms of the under- 
 standing, and the propriety of such a clas- 
 sification, as far as the meaning alone is 
 concerned, is evident from the nature of the 
 faculty employed. But as it is rather ques- 
 tionable whether the derivation of the word 
 will bear out this general acceptation, we 
 shall follow the usual practice of logicians, 
 and restrict its signification to those syllo- 
 gisms which Kant defines as the syllogisms of 
 reason. 
 
 Syllogism, .then, is that function of the 
 reason by which a third judgment is derived 
 
 F 2 
 
68 A SYLLABUS OF LOGIC. 
 
 from the union of two others, neither of which 
 contain it when taken separately. 
 
 (46.) The first division of which syllogism 
 is susceptible, is determined by the nature of 
 the judgments in respect of their moments of 
 relation. These have already been shewn to 
 be three the categorical, hypothetical, and 
 disjunctive ; and as there are a great many 
 combinations of judgments of these several 
 moments, from all of which other judgments 
 may be deduced, it follows that there must 
 also be a great many different syllogistic forms 
 of ratiocination. But as the principles in- 
 volved in all of them are the same as those in 
 the three simplest forms (which are usually 
 named after the three moments), it will be 
 sufficient to investigate the laws which regu- 
 late these alone, and, with one or two excep- 
 tions, leave the consideration of the rest to 
 the ingenuity of the reader. In these three 
 species of syllogism, the categorical, the hypo- 
 thetical, and the disjunctive, a categorical 
 conclusion is deduced from one judgment of 
 categorical form, and another of that form 
 from which the syllogism takes its name. 
 We shall now proceed to the separate consi- 
 deration of each of these three forms of rea- 
 soning. 
 
SYLLOGISM. 69 
 
 Categorical Syllogism. 
 
 (47.) In this syllogism the two given judg- 
 ments, which are also called premises, and 
 the derived judgment or conclusion, are all 
 categorical propositions. The nature of this 
 species of argument may be popularly stated 
 as follows : ' If two cognitions are severally 
 compared with a third cognition, (that is, ob- 
 jectively the same in each case,*) they may 
 afterwards be compared f with one another. J 
 
 * Perhaps an objection may be brought against this exposition, 
 on the grounds of its admitting negative premises. But it must 
 be observed that when both premises are of this quality, the real 
 middle is an external sphere, and consequently undistributed, or 
 virtually two middles. Arid this fallacy is quite excluded by the 
 clause in parentheses. 
 
 f The affirming or denying one cognition of another is what is 
 here intended by comparison. For if two cognitions are referred 
 to the consciousness simultaneously, they must either be thought 
 as belonging to the same subject or substratum, in which case 
 one may be affirmed of the other ; or as not belonging to the 
 same, in which case one may be denied of the other. 
 
 J As it is our present object to discover the universal laws of 
 all categorical syllogism, we have preferred this simple though not 
 very elegant definition to the celebrated ' dictum de omni out 
 nullo' of Aristotle, which is merely a particular case of it, and 
 may be stated thus : ' What is affirmed or denied of all, is affirmed 
 or denied of each ;' but this would only have given a particular 
 class of categorical syllogisms, and is therefore insufficient for 
 our present purpose. Were we however to admit either the em- 
 pirical considerations of the use of the reason, or the metaphy- 
 sical considerations of the conceptions of the understanding, we 
 should then be justified in confining our attention to the dictum 
 alone. For in the actual use of the reason we discover the fact, 
 that this form of syllogistic ratiocination is at once more natural 
 
70 A SYLLABUS OF LOGIC. 
 
 This is the principle of all categorical syllo- 
 gism ; and whenever the conditions are really 
 answered, and the comparison made according 
 to certain laws which we are about to deter- 
 mine, the conclusion is necessary, and the 
 reasoning incontrovertible. Let us take as an 
 example, ' Animals of the same species are sup- 
 posed to have originated from the same pair ; 
 all dogs are animals of the same species, 
 therefore all dogs are supposed to have ori- 
 ginated from the same pair/ Here, dogs and 
 animals supposed to have originated from the 
 same pair, are severally compared with animals 
 of the same species, and are afterwards com- 
 pared with one another. 
 
 (48.) We must now determine those laws, 
 in conformity with which the various forms 
 of categorical propositions may be so com- 
 bined as to answer the required conditions of 
 syllogism. 
 
 For this purpose we must not give a mere 
 aggregate of rules, even though those rules 
 should be in themselves sufficient. This me- 
 thod, which has been adopted in most (if not 
 
 and intelligible than the rest, and in the nature of one of the con- 
 ceptions of the understanding, we can discover the reason of the 
 fact. But we must not regard what lies out of the field of Formal 
 Logic, and have therefore given a definition which will include all 
 the forms of categorical syllogism that are possible in theory, 
 however awkward some of them may be in practice. 
 
SYLLOGISM. 71 
 
 all) works on Logic, is exceedingly unscien- 
 tific, as it only shews that certain rules are 
 requisite and sufficient to provide against cer- 
 tain forms of paralogism, but by no means 
 proves that these are the only forms to which 
 a syllogism is exposed. Now the method of 
 exhaustion will secure our reasoning against 
 all fallacy, so far at least as mere Formal Logic 
 is concerned.* We shall therefore dissect cate- 
 gorical syllogism into its ultimate elements, 
 and discover what rules are necessary and 
 sufficient for the prevention of fallacy in each 
 of them. 
 
 (49.) Every categorical syllogism contains 
 three terms the major term, the minor term, 
 and the middle term ; and three propositions 
 the major premiss, the minor premiss, and the 
 conclusion. In the major premiss, the major 
 term is compared with the middle term. In 
 
 * It may be as well to notice here, the very prevalent custom 
 of introducing into treatises on Logic, the consideration of 
 material as well as formal fallacies. In all the works on this 
 science that have come under our observation, (with the excep- 
 tion at least of Kant's,) nearly the first rule for syllogism is to the 
 effect that the middle term must not have different meanings in 
 the two premises. Now Logic merely considers the formal laws 
 of reasoning, but has nothing whatever to do with its matter ; and 
 the introduction of such a rule as this in a work on that science, 
 is something like beginning a treatise on Geometry with an 
 injunction to the student to draw his circles correctly: as if 
 mathematical reasoning could be at all affected by the perfection 
 of the diagram. 
 
72 A SYLLABUS OF LOGIC. 
 
 the minor premiss, the minor term is com- 
 pared with the middle term. And in the con- 
 clusion, the major and minor are compared 
 together. The major term is always the pre- 
 dicate, and the minor the subject of the con- 
 clusion. Hence may be deduced the following 
 rules for distinguishing the different terms and 
 premises in any given syllogism : 
 
 1st. The term that is common to the two 
 
 premises is the middle term. 
 2nd. The term that is the predicate of the 
 conclusion is the major term ; and the 
 premiss that contains it, the major pre- 
 miss. 
 
 3rd. The term that is the subject of the 
 conclusion is the minor term ; and the 
 premiss that contains it, the minor pre- 
 miss. 
 
 The three propositions are usually placed in 
 the following order : the major premiss, the 
 minor premiss, and the conclusion. And the 
 particular form of a syllogism, as far as it 
 depends on the particular categorical forms 
 of its three propositions, is termed its mood : 
 though this name is also given to any ternary 
 combination of the symbols A, E, I, O, with- 
 out reference to its conformity to the syllo- 
 gistic laws. 
 
 (50.) For the simplification of the subject 
 
SYLfeOGlSM. 73 
 
 we shall divide it into the three following 
 
 questions : 
 
 1st. What forms of premiss are sufficient for 
 a conclusion of the reason,* without con- 
 sidering its capability of being expressed 
 in any of the four categorical forms, (i.e. 
 
 2nd. When can the conclusion be expressed 
 in any of the legitimate forms, and when 
 can it not ? 
 
 3rd. What laws are sufficient and necessary 
 to secure the legitimacy of the con- 
 clusion ? 
 
 As the first of these questions excludes all 
 reference to the categorical form of the con- 
 clusion, and only seeks for premises that give 
 a conclusion valid for the reason, it does away 
 with the distinction of subject and predicate, 
 so far as the conclusion is concerned ; and 
 therefore with the distinction of major and 
 minor premiss. Hence, in this part of our 
 investigation, both premises are on exactly the 
 same footing. 
 
 * As it is impossible that the reader should understand the 
 following pages, unless he has a clear conception of what is meant 
 by a conclusion ' possible for the reason, but not expressible in 
 any of the four legitimate categorical forms,' we subjoin the fol- 
 lowing example. If the premises are 'some B is A, some C is not 
 B,' the reason may logically deduce that some C is not some A. 
 But this conclusion is not in one of the four legitimate forms, and 
 is therefore styled a conclusion only for the reason. 
 
74- A SYLLABUS OF LOGIC. 
 
 We may also discard all considerations of 
 the extremes. For any peculiarities in the 
 form of these terms, can merely require cor- 
 responding peculiarities when they recur in 
 the conclusion, as they constitute the matter 
 only of that judgment. They may therefore 
 affect the nature of the conclusion, but cannot 
 affect the possibility of its existence. 
 
 All that remains then for our consideration, 
 is the middle term, arid the copula in both the 
 premises. 
 
 (51.) Now the middle term admits of no 
 variations but those of quantity. Every law 
 therefore regarding it must respect this, and 
 this alone. And we find accordingly that the 
 middle term must be distributed in at least one 
 of the premises. 
 
 For if the middle term is undistributed, 
 a part of it only is compared with each 
 premiss. And as it cannot be formally known 
 that the major and minor are compared with 
 the same parts, the middle term becomes 
 virtually two terms, and the major and minor 
 terms cannot be considered as formally com- 
 pared with the same. But it is not necessary 
 that the middle term should be distributed in 
 more than one premiss. For if the whole 
 middle term is compared with one of the 
 extremes in one premiss, and only a part of it 
 
SYLLOGISM. 75 
 
 with the other extreme in the other premiss, 
 that part must be compared with both ex- 
 tremes, and in this case therefore the middle 
 term cannot be considered as two terms, or 
 ambiguous. 
 
 (52.) The first law for the quality of the 
 premises may be derived from the law for the 
 distribution of the middle term. This could 
 not be the case if every possible comparison 
 of the spheres of two terms, both internal and 
 external, positive and negative, found a cor- 
 responding expression among the categorical 
 forms : for in that case, a law regarding 
 quantity could not affect quality. This how- 
 ever has been shewn not to be the case, as 
 external spheres are never made subjects ; and 
 hence arises the possibility of deducing from 
 the law for the distribution of the middle 
 term the following law for the quality of the 
 premises. 
 
 No conclusion is possible from two negative 
 premises. For in premises of this form, the 
 extremes are each placed in the external 
 sphere of as much of the middle term as is 
 compared with them, and consequently the 
 external sphere becomes the real term with 
 which both extremes are compared. But the 
 external sphere is not distributed in any 
 categorical form, and consequently the virtual 
 
76 A SYLLABUS OF LOGIC. 
 
 middle term would be undistributed, and no 
 conclusion possible. 
 
 No. B is A. No. C is B. 
 
 Fig.l. ^\ C* J Fig. 2. 
 
 In the accompanying diagram we see that 
 A and C are each placed in the external sphere 
 of B ; but as the external sphere is undistri- 
 buted in both propositions, we do not know 
 that A and C are compared with the same 
 parts of it, and cannot therefore compare them 
 with one another. In Fig. 1. we find that ' no 
 C is A/ and in Fig. 2. the contradictory ' some 
 
 c is A; 
 
 As the external spheres of conceptions do 
 not admit the variations of quantity, or bear 
 the same mutual relations in any categorical 
 propositions as the internal spheres, they are 
 never understood unless they are expressly 
 mentioned. It is for this reason that the law 
 for the distribution of the middle term is 
 generally understood to relate to the internal 
 spheres only, and that the law against two 
 negative premises cannot be subsumed under 
 it as a particular case. It is therefore given 
 
SYLLOGISM. 77 
 
 as a separate law, that at least one premiss 
 must be affirmative. 
 
 The extreme term contained in the affirma- 
 tive premiss will of course agree and coincide 
 with the middle term, and will therefore 
 agree or disagree with all with which the 
 middle term agrees or disagrees in the other 
 premiss. Hence the latter premise may be 
 either affirmative or negative, and there is no 
 other law for the quality of the premises, than 
 that given above. 
 
 (53.) We have now exhausted the elements 
 of the premises, (for we have shewn that our 
 present inquiry does not involve the distinction 
 of the premises, or the consideration of the 
 major and minor terms,) and have arrived at 
 the following laws for the middle term, and 
 the quality of the two copulas. 
 
 In order that a conclusion for the reason, 
 though not necessarily categorically expressible, 
 be possible from any particular forms of 
 premises, it is only necessary 
 
 1st. That these categorical forms contain at 
 least one distributed term. 
 
 2nd. That one of these forms be affirmative. 
 
 As at present there is no distinction between 
 the premises, we are not considering permuta- 
 tions but combinations ; and as each proposition 
 may be combined with itself, the number will 
 
78 
 
 A SYLLABUS OF LOGIC. 
 
 be 
 
 5.4 
 1 .2 
 
 = 10, of which we shall find by the 
 
 following table, that 6 are unobjectionable, 
 that 3 are excluded by their negative premises, 
 and 1 by the want of a distributed term. 
 
 
 - 
 
 ?orms of premises from 
 which no conclusion 
 
 is possible. 
 .^ 
 
 Forms of premi- 
 ses from which 
 a conclusion at 
 
 
 to 
 
 3 
 
 No 
 
 No 
 
 least .for the 
 
 
 'cj 
 
 O 
 
 distributed 
 
 affirmative 
 
 reason, is pos- 
 sible 
 
 
 O 
 
 term. 
 
 term. 
 
 
 r 
 
 A 
 
 _ 
 
 
 
 A A 
 
 A combined with / 
 
 E 
 
 T 
 
 
 
 
 A E 
 A T 
 
 
 J 
 
 
 
 
 A O 
 
 f 
 
 E 
 
 _ 
 
 EE 
 
 
 
 E combined with -/ 
 
 T 
 
 . 
 
 . 
 
 E I 
 
 I combined with 4 
 
 O 
 I 
 O 
 
 I I 
 
 E O 
 
 I 
 
 
 
 O combined with... 
 
 
 
 
 00 
 
 
 
 
 The above table is only intended to state, 
 that some conclusion, though perhaps not 
 categorically expressible, is sometimes possible 
 in each of the six forms in the last column, 
 but always impossible in the four forms in 
 the other two columns. And it must be ob- 
 served that we can only say sometimes pos- 
 sible ; for though all these six forms contain 
 one or more distributed terms, yet if the 
 middle is not one of those terms, it will be 
 undistributed, and no conclusion possible. For 
 
SYLLOGISM. 79 
 
 instance, let B be the middle term, and A I the 
 form of the premises, if we say ' All B is A/ 
 ' Some C is B,' the conclusion that ' Some C 
 is A/ is strictly deducible. But if we say ' All 
 A is B; ' Some C is B,' therefore ' Some C is A/ 
 our conclusion would not be logically correct, 
 as the middle term B would be undistributed 
 in each premiss. We repeat, therefore, that 
 the last table merely indicates the possibility 
 of deducing a conclusion for the reason from 
 some premises of certain forms, and the im- 
 possibility of deducing any conclusion what- 
 ever from premises of certain other forms. 
 
 (54.) We may now dismiss the first part of 
 our present investigation, and consider the 
 second question. When can the conclusion 
 be expressed in any of the legitimate forms, 
 and when can it not ? 
 
 As the variations of quality have not been 
 allowed to affect the terms * in the categorical 
 forms of the admissible premises, the conclu- 
 sion from them that is possible for the reason, 
 but not categorically expressible, cannot be so 
 restricted, on account of its being valid for 
 
 * Perhaps the following explanation will be more easily under- 
 stood than the text. As external spheres have not been admitted 
 as terms in the forms of premises given in the above table, it is 
 impossible that they should appear in the conclusion ; the imprac- 
 ticability therefore of these conclusions cannot arise from the 
 appearance of an external sphere in either of their terms. 
 
80 A SYLLABUS OF LOGIC. 
 
 only the external sphere of a conception. It 
 must therefore belong to one of the four 
 rejected forms (mentioned in Art. 32,) whose 
 predicates do not as functions of the copula fol- 
 low the same law with the accepted forms. In 
 other words, the conclusion must belong to 
 one of these four forms, in which the predicate 
 of an affirmative is distributed, and of a ne- 
 gative undistributed. But it cannot belong 
 to a proposition of the former class, as in that 
 case one of the legitimate categorical proposi- 
 tions, in which the predicate is undistributed, 
 would state less than this sound conclusion, 
 and would therefore be contained in it, and be 
 itself true ; but this is contrary to the hypothe- 
 sis. Hence the only case that remains in 
 which a conclusion can be valid for the reason, 
 but not categorically expressible, is when a 
 conclusion whose quality is negative has 
 a predicate whose quantity ought to be un- 
 distributed. And we shall accordingly find 
 that the rule for the quantity of the predicate 
 of the conclusion will exclude all those pre- 
 mises which give conclusions only possible for 
 the reason. 
 
 (55.) Our next object is to determine what 
 laws are sufficient and necessary to secure the 
 legitimacy of the conclusion. 
 
 For this, let us examine the three elements 
 
SYLLOGISM. 81 
 
 of the conclusion the subject, the predicate, 
 and the copula. 
 
 As the only variations of which the terms 
 admit are those of quantity, the laws concern- 
 ing them must respect that, and that only. 
 
 As, moreover, logic can only consider the 
 formal quantity of a term, either extreme may 
 contain in the conclusion the same quantity as 
 in its premiss, but not more. Hence the only 
 rule for the terms is this : 
 
 If the major or minor terms are undistri- 
 buted in the premises, the predicate and 
 subject must be respectively undistributed 
 in the conclusion. 
 
 When this rule is violated in a syllogism 
 whose major term is undistributed in the 
 major premiss, but whose predicate is distri- 
 buted in the conclusion, the resulting fallacy 
 is called an illicit process of the major. 
 
 When the minor term is undistributed in 
 the minor premiss, and the subject is distri- 
 buted in the conclusion, the resulting fallacy 
 is called an illicit process of the minor. 
 
 (56.) With regard to the quality of the 
 copula, the only law for the conclusion is this : 
 
 If either premiss be negative, the conclusion 
 must also be negative ; but if both premises are 
 affirmative, the conclusion must also be affir- 
 mative. 
 
 G 
 
82 A SYLLABUS OF LOGIC. 
 
 For as one premiss must be affirmative, the 
 extreme term which it contains must agree 
 with the middle term in that premiss, and 
 therefore agrees or disagrees in the conclu- 
 sion, with whatever the middle term agrees or 
 disagrees with in the other premiss. What- 
 ever therefore is the quality of this latter pre- 
 miss must also be the quality of the conclusion. 
 
 (57.) The elements of a categorical syllo- 
 gism have now been completely exhausted, 
 and it is absolutely certain that if all the given 
 rules are preserved inviolate, a formally incor- 
 rect conclusion can never be obtained. 
 
 All then that is necessary to ensure the 
 legitimacy of a syllogism, is comprised in the 
 five following rules : 
 
 1. There cannot be more than one negative 
 premiss, 
 
 2. If there is one negative premiss, the con- 
 clusion is negative ; if there is no nega- 
 tive premiss, the conclusion is affirmative. 
 
 3. The middle term must be distributed in 
 at least one of the premises. 
 
 4. If the predicate of the conclusion is 
 distributed, the major term must be dis- 
 tributed in the major premise. 
 
 5. If the subject of the conclusion is dis- 
 tributed, the minor term must be distri- 
 buted in the minor premiss. 
 
SYLLOGISM. 83 
 
 The rules that have been given for the 
 premises will only ensure a conclusion pos- 
 sible for the reason, and hence it will follow 
 that the rules for the conclusion will in certain 
 cases affect the premises also. For if the 
 conclusion is negative, its predicate must be 
 distributed ; and therefore, by Rule 4, the major 
 term must also be distributed in the major 
 premiss. 
 
 (58.) Although the five given rules are quite 
 sufficient in themselves, yet two others are 
 derived from them, with which the student 
 should be well acquainted, as they are of very 
 easy application. They are 
 
 1. No conclusion can be drawn from par- 
 ticular premises. 
 
 2. Only a particular conclusion can be drawn 
 where one of the premises is particular. 
 
 As the middle term must always be distri- 
 buted in one premiss at least, and as no term 
 can be distributed in the conclusion that is 
 not distributed in the premises, it follows 
 that there must be at least one more distri- 
 bution in the premises than in the conclusion. 
 But there can be only one distribution in the 
 predicates of the premises, as there can be 
 only one negative premiss, and in that case 
 there must be a distribution in the predicate 
 of the conclusion, as that must also be nega^- 
 
 02 
 
84 A SYLLABUS OF LOGIC. 
 
 tive. Hence it follows that the number of 
 distributions in the subjects of the premises 
 must exceed by at least unity, the number 
 of distributions in the subject of the con- 
 clusion. If then both premises are particular, 
 there will be no distributions in the subjects of 
 the premises ; and as the above condition will 
 not be answered, there can be no conclusion. 
 And if one premiss is particular, there will 
 be only one distribution in the subjects of the 
 premises, and therefore there can be none 
 in the subject of the conclusion, i. e. the con- 
 clusion must be particular. 
 
 (59.) We shall now be able to extend our 
 table for the premises only, and determine the 
 moods in which a syllogism, perfect in all its 
 parts, is possible. It must however be remem- 
 bered, that as the table of sound premises paid 
 no respect to the distinction of major and 
 minor premiss, the order of the premises was 
 there indifferent. But as the conclusion is 
 here considered in its legitimate categorical 
 forms, the distinction holds, and we must 
 therefore examine the two permutations of 
 each of those forms of premises. 
 
SYLLOGISM. 
 
 85 
 
 Premises. 
 
 Conclu- 
 sion. 
 
 Moods in which Paralogism is 
 unavoidable. 
 
 Moods in which 
 Syllogism is 
 possible. 
 
 A AA 
 A A I 
 
 Illicit Process 
 of Minor. 
 
 Undistributed 
 Middle. 
 
 N 
 
 Illicit Process 
 of Major. 
 
 A A with 1 
 
 A 
 I 
 
 _. 
 
 
 
 _. 
 
 A I with | 
 
 A 
 I 
 
 A I A 
 
 . 
 
 
 
 A I I 
 
 I A with | 
 
 A 
 I 
 
 I A A 
 
 or I A A 
 
 -_ 
 
 I A I 
 
 A E with | 
 
 E 
 O 
 
 
 
 
 
 
 
 AE E 
 AEO 
 
 E A with | 
 
 E 
 O 
 
 _ 
 
 
 
 
 
 E A E 
 E A O 
 
 A O with | 
 
 E 
 
 
 AOE 
 
 or A O E 
 
 or AOE 
 
 AOO 
 
 O A with | 
 
 E 
 O 
 
 O A E 
 
 or AE 
 
 or O A E 
 
 O AO 
 
 E I with/ 
 
 E 
 O 
 
 EIE 
 
 
 
 
 
 E I O 
 
 I E with / 
 
 E 
 O 
 
 
 
 
 
 IEE 
 IE O 
 
 
 
 I O with / 
 
 E 
 
 
 IO E 
 
 or I O E 
 
 I OE 
 100 
 
 
 
 I with | 
 
 E 
 O 
 
 I E 
 
 O I E 
 I O 
 
 or OIE 
 or O I O 
 
 
 
 SUBJECT. PREDICATE. 
 
 A Distributed Undistributed. 
 
 E Distributed Distributed. 
 
 I Undistributed Undistributed. 
 
 O Undistributed Distributed. 
 
86 A SYLLABUS OF LOGIC. 
 
 The following example will shew how the 
 above table should be studied. 
 
 Let us take the mood O A E. In this mood 
 the predicate is distributed in the major pre- 
 miss, and the subject in the minor premiss, 
 and both subject and predicate in the con- 
 clusion. Hence it follows that either some 
 term must be distributed in the conclusion 
 which was not distributed in the premises, or 
 else the middle term cannot be distributed in 
 either premiss. We cannot therefore determine 
 at once which form the fallacy will take, but 
 may be quite certain that there must be either 
 an illicit process of major or minor, or else an 
 undistributed middle. Again, in the mood 
 O I E, both subject and predicate are distri- 
 buted in the conclusion, whereas no term is 
 distributed in the minor premiss, and it there- 
 fore follows that there must be an illicit process 
 of the minor. It is also evident that the 
 middle^ term cannot be distributed in the 
 minor premiss, and that if it is distributed in 
 the major premiss, the major term must be 
 undistributed, and consequently there must be 
 a fallacy either of undistributed middle or 
 illicit major. 
 
 (60.) It will be seen from the last column in 
 the table, that there are eleven moods in which 
 syllogism is possible, or in which the syllogistic 
 
SYLLOGISM. 87 
 
 rules may be observed. But it does not follow 
 that the rules of syllogism are necessarily 
 observed in these eleven moods, as paralogism 
 is possible in all but two of them ;* for in the 
 premises of some of the moods we have but 
 one distributed term, and in others two or 
 more ; and unless the position of the terms is 
 such as to obey the laws for the distribution 
 of the middle term, and prevention of illicit 
 process, fallacy will still be the inevitable 
 result. 
 
 The position of the terms will accord- 
 ingly give rise to a further classification of 
 categorical syllogisms. And as each premise 
 admits of two permutations, in one of which 
 the middle term is predicate, and in the other 
 subject, there will be four combinations deter- 
 mined by its position, which are usually called 
 the four syllogistic figures. 
 
 In the first figure, the middle term is the 
 subject of the major premiss, and predicate of 
 the minor. 
 
 In the second figure, the middle term is the 
 predicate of both premises. 
 
 In the third figure, the middle term is the 
 subject of both premises. 
 
 In the fourth figure, the middle term is the 
 
 * These two moods are E A O, and E I 0, which are true in 
 all the four figures. 
 
88 
 
 A SYLLABUS OF LOGIC. 
 
 predicate of the major, and subject of the 
 minor premiss. 
 
 The following scheme will enable the reader 
 to understand the distinction at a glance. 
 
 Let S represent the minor term (or subject 
 of conclusion), M the middle term, and P the 
 major term, (or predicate of conclusion). 
 
 
 1st Fig. 
 
 2nd Fig. 
 
 3rd Fig. 
 
 4th Fig. 
 
 Major Premiss 
 
 M P 
 
 P M 
 
 M P 
 
 P M 
 
 Minor Premiss 
 
 S M 
 
 S M 
 
 M S 
 
 M S 
 
 Conclusion 
 
 S P 
 
 S P 
 
 S P 
 
 S P 
 
 In each of these figures several moods are 
 sound, and others unsound. For instance, the 
 mood A A A, which is sound in the first 
 figure, in the second figure would have its 
 middle undistributed in each premiss, while 
 the mood A E E, which is sound in the second 
 figure, would contain an illicit process of the 
 major in the first figure. The accompanying 
 table will shew what moods are sound, and 
 what unsound, in each figure, and in the latter 
 case, the peculiar form of the fallacy. 
 
SYLLOGISM. 
 
 TABLE OF CATEGORICAL SYLLOGISM. 
 
 Same of 
 Mood. 
 
 ?orm of 
 Mood. 
 
 1st 
 Figure. 
 
 2nd 
 Figure. 
 
 3rd 
 
 Figure. 
 
 4th 
 Figure. 
 
 A A A< 
 A A U 
 
 Subject 
 Premiss 
 
 o-in 
 
 D-ul 
 D-UJ 
 
 D-U^j 
 D-ul 
 
 u-u/ 
 
 M P 
 
 S M 
 S P 
 
 P M 
 
 S M 
 S P 
 
 M P 
 M S 
 S P 
 
 P M 
 
 M S 
 S P 
 
 Sound 
 
 Undistributed 
 Middle 
 
 Illicit 
 Minor 
 
 Illicit 
 Minor 
 
 Sound 
 
 undistributed 
 Middle 
 
 Sound 
 
 Sound 
 
 ,{ 
 
 D-U>| 
 
 u-ul 
 u-uj 
 
 Sound 
 
 Undistributed 
 Middle 
 
 Sound 
 
 Undistributed 
 Middle. 
 
 I A I I 
 
 U-U) 
 D-UV 
 
 u-uj 
 
 Undistributed 
 Middle 
 
 Jndistributed 
 Middle 
 
 Sound 
 
 Sound 
 
 AEE^ 
 
 D-in 
 
 D-DV 
 D-D/ 
 
 Illicit 
 Major 
 
 Sound 
 
 Illicit 
 Major 
 
 Sound 
 
 AEO<j 
 
 D-U>| 
 D-Dl 
 U-Dj 
 
 Illicit 
 Major 
 
 Sound 
 
 Illicit 
 
 Major 
 
 Sound 
 
 AOOJ 
 
 D U^ 
 U-D) 
 U-Dj 
 
 Illicit 
 Major 
 
 Sound 
 
 Illicit 
 Major 
 
 Undistributec 
 Middle 
 
 EAE<[ 
 
 D D^j 
 D U V 
 D Dj 
 
 Sound 
 
 Sound 
 
 Illicit 
 Minor 
 
 Illicit 
 Minor 
 
 BACK 
 
 D D^k 
 D-UV 
 U Dj 
 
 Sound 
 
 Sound 
 
 Sound 
 
 Sound 
 
 E I O<j 
 
 U D^ 
 D U 
 U Dj 
 
 Sound 
 
 Sound 
 
 Sound 
 
 Sound 
 
 OACK 
 
 U D^ 
 D U 1 
 U Dj 
 
 Undistributec 
 Middle. 
 
 Illicit 
 Major 
 
 Sound 
 
 Illicit 
 Major 
 
90 A SYLLABUS OF LOGIC. 
 
 In the first column of this table, the 
 name of the mood is given. In the second, its 
 form with regard to the distribution of its 
 terms, where D signifies distributed, and U 
 signifies undistributed. In the other four 
 columns, which are headed by the forms of the 
 four figures, the nature of each mood in each 
 figure is expressed by the word e sound,' if the 
 reasoning is unobjectionable, and if not so, by 
 the name of its particular fallacy. 
 
 By comparing the form of one of the 
 moods with the general form for any one of 
 the figures given in the upper line of the table, 
 we shall see if any D in the one corresponds in 
 position with an M in the other, in which case 
 the middle will be distributed. We must also 
 observe if either S and P in the conclusion 
 correspond with a D in the form of the mood, 
 and if so, they must correspond respectively 
 with a D in the premises, or else there will be 
 an illicit process. With this explanation there 
 will be no difficulty in understanding the 
 manner in which the table is formed. For 
 instance, let us take the mood A A A. We 
 find in the form of the mood that the subject 
 in each premiss is distributed, and upon turning 
 to the form of the first figure, we find that the 
 subject of the major premiss is the middle 
 term, and are therefore justified in concluding 
 
SYLLOGISM. 91 
 
 that the mood A A A in the first figure has 
 its middle distributed. Again, we find that 
 the only term in the conclusion corresponding 
 to a D, is S, which also corresponds to a D in 
 the premises. We are therefore certain that 
 no term is distributed in the conclusion that is 
 not also distributed in the premises, or, in other 
 words, that there is no illicit process; and as 
 the middle is distributed, it follows that the 
 syllogism A A A in the first figure is sound. 
 
 If however we compare the form of the 
 mood A A A, with the form of the third figure, 
 we find that S the subject of the conclusion 
 corresponds to D, and is therefore distributed ; 
 but that S the predicate of the minor premiss 
 corresponds to U, and is therefore undistri- 
 buted : whence it follows that A A A in the 
 third figure has an illicit process of the minor. 
 
 (61.) Several laws may be obtained for each 
 figure, by an examination of their peculiar 
 forms. 
 
 In the first figure, the minor premiss must 
 be affirmative : for if it were negative, the 
 major premiss must be affirmative, and there- 
 fore have its predicate, which is the major 
 term, undistributed. But the conclusion must 
 be negative, and therefore its predicate would 
 be distributed, which would accordingly give 
 rise to an illicit process of the major. 
 
92 A SYLLABUS OF LOGIC. 
 
 As the minor premiss must be affirmative, 
 the middle term, which is its predicate, must 
 be undistributed in that premiss, and therefore 
 distributed in the other premiss, in which it 
 holds the place of subject. Hence it follows 
 that the major premiss must always be uni- 
 versal in the first figure. 
 
 In the second figure, the middle term is the 
 predicate in each premiss ; and as it must be 
 distributed in one of them, one premiss must 
 be negative, and therefore the conclusion also. 
 In the third figure the minor premiss must 
 always be affirmative for the same reasons as 
 in the first figure ; and therefore its predicate 
 (which is the minor term), being undistributed, 
 the conclusion must be particular, for were it 
 universal, there would be an illicit process of 
 the minor. There are several other rules 
 which may be derived from the primary laws, 
 but as all of them may be more simply evolved 
 by means of algebraical symbols, we shall post- 
 pone their consideration for the present. 
 
 Transformation of the figures of Syllogism. 
 
 (62.) It has been already observed, that 
 Aristotle's dictum (or the first figure) is the 
 form in which our reasoning appears the most 
 natural, and is most easily comprehended. 
 This is the reason why the other three 
 
SYLLOGISM. 93 
 
 figures have been considered as unnatural 
 deviations from it, and that laws have been 
 laid down for the reduction of all syllogisms 
 to the original form of the first. But as this 
 peculiarity of the dictum is based on the laws 
 of an understanding-conception, and as Formal 
 Logic can pay no attention to a distinction 
 which originates in grounds that lie out of its 
 field, it will be more correct to consider the 
 laws for transforming a syllogism from any 
 one figure into another, especially as they will 
 include the laws for reduction (or transforma- 
 tion to the first) as a particular case. As the 
 distinction of figure depends entirely on the 
 position of the terms, conversion, by which 
 alone this can be altered, is the only method 
 of transformation. But this operation is not 
 always possible, as in many of the moods the 
 propositions are not of a convertible form : 
 e. g. it is impossible to transform the mood 
 A E O from the second figure into the first ; 
 for the major premiss is not simply convertible, 
 and limited conversion would give the mood 
 I E O, which contains an illicit process of the 
 major. The following table can therefore only 
 give those conditions which must be answered 
 by any syllogism in each figure for its trans- 
 formation into any other; but in order to 
 know if a syllogism is capable of answering 
 
A SYLLABUS OF LOGIC. 
 
 these conditions, it will be necessary to ex- 
 amine its particular mood. 
 
 TRANSFORMATION TABLE. 
 
 Figure to be 
 transformed. 
 
 Propositions to be converted for transformation into the 
 
 First figure 
 
 Second figure 
 
 Third figure 
 
 -s 
 
 Fourth figure 
 
 First 
 
 
 Major 
 
 Minor 
 
 Conclusion * 
 or 
 Major & Minor 
 
 Second 
 
 Major 
 
 
 Major and 
 
 Minor 
 
 Minor or 
 Major & Con- 
 clusion 
 
 Third 
 
 Minor 
 
 Major and 
 Minor 
 
 
 Major 
 
 Fourth 
 
 Conclusion 
 or 
 
 Major & Minor 
 
 Minor 
 
 Major 
 
 
 (63.) It appears from the table at Art. 60, 
 that there are six moods in each figure which 
 answer all the required conditions for a sound 
 categorical syllogism. Five of these have 
 been very generally rejected by logicians, on 
 the grounds of their having particular conclu- 
 sions, although universal are warranted by the 
 premises.f But this regard to a more prac- 
 
 * " It is hardly necessary to observe, that whenever the conclu- 
 sion is converted, the major and minor terms, which are respec- 
 tively its predicate and subject, must be interchanged, and 
 therefore the premises that contain them. 
 
 f These five are the moods A A I in the first figure, E A O 
 in the first and second, and A E O in the second and fourth. 
 
SYLLOGISM, 95 
 
 tical utility is always very unphilosophical in 
 an a priori science, and especially if it be at 
 all destructive of the symmetry of our results. 
 The mathematician will readily acknowledge 
 his obligations to symmetry for the light it 
 throws upon truths already known, as well as 
 for its efficiency as an organum for the dis- 
 covery of new ones : and the same regard 
 should be paid to it in every pure science, but 
 more particularly in one which is intended to 
 increase the accuracy and rigid strictness of 
 our thought. In the present case there are 
 six syllogisms in each of the four figures, and 
 this reduction would have only nineteen, four 
 in the first and second, six in the third, and 
 five in the fourth. Some of the old schoolmen 
 carried their veneration for Aristotle so high, 
 as to reject the fourth or Galenic * figure 
 entirely, and thus reduced the whole number 
 to fourteen. But this reduction is nearly as 
 unphilosophical as the other, for the only 
 rational ground upon which it can rest will 
 apply to the second and third figures also, 
 though not perhaps to the same extent. We 
 shall therefore submit to the reader the whole 
 
 * The fourth figure was first recognised by Galen, the great 
 medical philosopher, who flourished in the second century. 
 
96 A SYLLABUS OP LOGIC. 
 
 twenty-four syllogisms, collected under their 
 respective fig u res . * 
 
 Fig. 1. A A A, A A I, All, E A E, E A O, E I O. 
 
 Fig. 2. A E E, A E O, A O O, E A E, E A O, E I O. 
 
 Fig. 3. I A I, A A I, A I I, O A O, E A O, E I O. 
 
 Fig. 4. A E E, A A I, I A I, A E O, E A O, E I O. 
 
 Hypothetical Syllogism. 
 (64.) One premiss of a hypothetical syllogism 
 is a hypothetical proposition ; the other premiss 
 is a categorical proposition, and either asserts 
 
 * There is a barbarous practice of naming the various forms of 
 a categorical syllogism by certain words which constitute mnemo- 
 nic hexameters. A subject which should always be treated in 
 a rational point of view ('. e. as appertaining to the reason), is in 
 this manner degraded to a mere historical record of the deduc- 
 tions of others, and draws upon the memory alone. Having 
 entered my protest against these lines, I still think it proper to 
 subjoin them, in order that the student may understand the allu- 
 sion when he hears such phrases as ' a syllogism in barbara,' &c. 
 The vowels contained in the words of these lines give the names 
 of the moods, and the consonants refer to their other peculiari- 
 ties, such, for instance, as the methods of reducing them to the 
 first figure. The two first lines give the nature of the four cate- 
 gorical propositions. 
 
 Asserit A, negat E, verum generaliter ambae. 
 Asserit I, negat O, sed particulariter amba?. 
 
 Barbara, Celarent, Darii, Ferioque, prioris 
 Cesare, Camestres, Festino, Baroko, secundse. 
 Tertia, Darapti, Disamis, Datisi, Felapton, 
 Bokardo, Ferison habet : Quarta insuper addit 
 Bramantip, Camenes, Dimaris, Fesapo, Fresison : 
 Quinque Subalterni, totidem Generalibus orti, 
 Nomen habent nullum, nee, si bene collegis, usum. 
 
SYLLOGISM. 97 
 
 the antecedent, or denies the consequent. In 
 the former case, which is called the modus 
 ponens, the conclusion infers the truth of the 
 consequent ; in the latter case, which is called 
 the modus tollens, the conclusion infers the 
 falsity of the antecedent. The general forms 
 of these two cases are, ' If A is, B is ; but A is ; 
 therefore B is ;' and ' If A is, B is ; but B is 
 not ; therefore A is not/ Example, ' If what 
 we learn from the bible is true, we ought not 
 to do evil that good may come ; but what we 
 learn from the bible is true ; therefore we ought 
 not to do evil that good may come.' 
 
 These are the only two forms which a 
 hypothetical syllogism can assume. For no 
 variation can enter on the side of the hypo- 
 thetical (which is usually styled the major) 
 premiss, as there is but one form of such 
 propositions ; neither can there be any other 
 form for the categorical (which is usually 
 styled the minor) premiss than those already 
 mentioned, for nothing can be inferred by 
 denying the antecedent or asserting the con- 
 sequent. As, moreover, the latter premiss 
 only concerns the truth or falsity of the 
 members of the hypothetical, all variations in 
 its form, when considered merely as a cate- 
 gorical proposition, must affect its matter and 
 not its form when considered as the minor 
 premiss. H 
 
98 A SYLLABUS OF LOGIC. 
 
 Disjunctive Syllogism. 
 
 (65.) In this syllogism, we commence with 
 a disjunctive judgment, and proceed either by 
 asserting the truth of one member of the 
 division, and thence inferring the falsity of 
 all the rest, which is called the ' modus ponens,' 
 or else by asserting the falsity of all the 
 members but one, and hence inferring the 
 truth of that one, which latter method is called 
 the "modus tollens/ The general form of 
 these two cases will be, ' Either A is, or B is, 
 or C is ; but A is ; therefore neither B is, nor 
 C is/ And ' Either A is, or B is, or C is ; but 
 neither B is, nor C is; therefore A is.' We 
 may take as an example ' Either the Pope is 
 infallible, or there is at least one great error 
 in the Romish church ; but the Pope is not 
 infallible ; therefore there is at least one great 
 error in the Romish church/ 
 
 These may be shewn to be the only forms 
 of a disjunctive syllogism by reasoning very 
 similar to that employed in the case of the 
 
 hypothetical. 
 
 Dilemma, fyc. 
 
 (66.) Besides the three simple forms of syl- 
 logism already mentioned, there are several 
 others,of which perhaps the Dilemma is the most 
 important. This is an hypothetical syllogism, 
 whose consequent is divided into members by 
 
SYLLOGISM. 99 
 
 a disjunctive judgment. Thus, 'if A is B, 
 either C is D, or E is F ' is the general form 
 of the major premiss of a dilemma ; and as it 
 is hardly ever used except in the modus tol- 
 lens, the minor premiss and conclusion will 
 be 'But neither C is D, nor E is F, there- 
 fore A is not B.' 
 
 (67.) The following are examples of com- 
 binations of premises which differ from those 
 already given in respect of the moments of 
 relation. Thus, two hypotheticals will give 
 
 ' If A is B, C is D ; but if C is D, E is F ; 
 therefore if A is B, E is F.' 
 
 Or, ' If A is B, C is D ; but if E is F, C is 
 not D ; therefore if E is F, A is not B/ 
 
 ' If A is B, C is D ; and if A is not B, E is F; 
 therefore either C is D, or E is F.' 
 
 In the same manner two disjunctives will 
 give 
 
 ' A is either B or C ; but B is either D or E ; 
 therefore A is either C or D or E. J 
 
 Or a categorical and disjunctive 
 
 A is either B or C ; D is A ; therefore D 
 is either B or C. 9 
 
 Enthymeme. 
 
 (68.) The Enthymeme is a syllogism abridged 
 by the suppression of one of its premises, which 
 is nevertheless understood, as the argument 
 
 H 2 
 
100 A SYLLABUS OF LOGIC. 
 
 would not be valid without it. For, as the 
 conclusion and either premiss are sufficient to 
 indicate what the other premiss must be, we 
 rarely express both premises in practice, but 
 generally leave one of them to the hearer or 
 reader to supply. The following is an example 
 of a categorical enthymeme : 
 
 f The science of Logic is very useful, as it 
 enables us to detect the formal fallacies in 
 the arguments of our adversaries'. Here the 
 major premiss is suppressed. The completed 
 syllogism will stand thus 
 
 * Whatever enables us to detect the formal 
 fallacies in the arguments of our adversaries 
 is very useful ; the science of Logic enables 
 us, &c.; therefore the science of Logic is 
 very useful.' Had the minor premiss been 
 suppressed, the enthymeme would have been 
 
 ' The science of Logic is very useful, for 
 any thing is useful that enables us to de- 
 tect the formal fallacies in the arguments of 
 our adversaries.' 
 
 The following is an example of an hypo- 
 thetical enthymeme : 
 
 ' It will certainly rain, for the sky looks 
 very black/ 
 
 In this case the major premiss is suppressed. 
 The syllogism when completed would stand 
 thus 
 
SYLLOGISM. 101 
 
 ' If the sky looks black, it will certainly rain. 
 
 * The sky does look black. 
 
 c Therefore it will certainly rain.' 
 Had the minor premiss been suppressed, the 
 enthymeme would have been of the following 
 form : 
 
 ' It will certainly rain, for it always rains 
 if the sky looks black.' 
 
 The following examples are two disjunc- 
 tive enthymemes in which the major and minor 
 premises of the same syllogism are respec- 
 tively suppressed : 
 
 ' He must be in York, for he is not in 
 London.' 
 
 The suppressed premiss is, ' he must be 
 either in London or York.' 
 
 If the minor premiss is suppressed the en- 
 thymeme will become 
 
 ' He must be in York, for he must be either 
 in London or York.' 
 
 Sorites. 
 
 (69.) In a Sorites the conclusion of a syllogism 
 is not expressed, but made the suppressed 
 premiss of an enthymeme whose conclusion 
 may be made the suppressed premiss of ano- 
 ther, and similarly for any number of enthy- 
 memes. Thus, ' B is A, C is B, D is C, E is D ; 
 
102 A SYLLABUS OF LOGIC. 
 
 therefore E is A* is a categorical Sorites. 
 Again, ' If A is, B is ; if B is, C is ; if C is, 
 D is ; but D is not ; therefore A is not/ is a 
 specimen of an hypothetical Sorites. 
 
( 103 ) 
 
 SECTION V. 
 
 THE DEDUCTIO AD ABSURDUM, OR INDIRECT 
 PROOF. 
 
 (70.) THIS name is given to a circuitous method 
 of proving one proposition from two or more 
 others, by means of at least three syllogisms. 
 As, however, the principle of the proof is quite 
 independent of the number of given proposi- 
 tions, our present object will be fully answered 
 by an investigation of that case, in which they 
 are limited to two. And, moreover, as the 
 forms of syllogism contain all the principles 
 or functions of the reason, by which one pro- 
 position can be thought as necessarily con- 
 nected with several others, it follows, that 
 any conclusion at which we can arrive by the 
 deductio ad absurdum, might also have been 
 obtained by the direct application of one of 
 the regular syllogistic forms of ratiocination. 
 We may therefore consider the given proposi- 
 tions, and the one to be deduced from them, 
 as the premises and conclusion of a syllo- 
 gism, and proceed to shew how this conclu- 
 
104 A SYLLABUS OF LOGIC. 
 
 sion may be obtained by a different chain of 
 reasoning. 
 
 The method, then, consists first, in assum- 
 ing the falsity of the conclusion and truth of 
 one premiss, and deducing from these propo- 
 sitions as premises, the falsity of the other 
 premiss, as a conclusion ; secondly, in taking 
 this conclusion as the consequent, and the 
 premises of the last syllogism as the antece- 
 dent in a hypothetical syllogism, and inferring 
 the falsity of the antecedent by the modus 
 tollens ; thirdly, in dividing this falsity in a 
 disjunctive syllogism into its three members, 
 viz. the falsity of each proposition separately, 
 or of both together, and inferring the falsity 
 of that member of the division which is the 
 contradictory of the original conclusion by 
 the modus ponens. But the reader will under- 
 stand this method more easily by examining 
 the following general form : 
 
 Let A and B represent the premises, and 
 C the conclusion of any syllogism. In order 
 to prove C by the indirect method, we com- 
 mence with assuming that C is not true. 
 The three syllogisms may be then stated as 
 follows : 
 
 First syllogism : ' A is ; C is not ; therefore 
 B is not. 
 
 Second syllogism : ' If A is, and C is not, 
 
INDIRECT PROOF. 105 
 
 it follows that B is not ; but B is ; there- 
 fore it is false that "A is and C is not/" 
 
 Third syllogism : ' Either both propositions 
 " A is" and " C is not" are false, or else one 
 of them is false ; but that " A is " is not false ; 
 therefore that "C is not" is false, (i.e. C 
 
 is'). 
 
 The hypothetical syllogism is rarely if ever 
 expressed in practice ; the disjunctive, perhaps, 
 never. But, in an analysis of the indirect 
 proof, it would be just as unreasonable to 
 neglect the consideration of these syllogisms 
 on the ground of their being rarely expressed, 
 as it would be in an analysis of syllogism 
 generally to neglect the consideration of one 
 of the premises on the ground of our usually 
 reasoning in enthymemes. 
 
 Although the dcductio ad alsurdum requires 
 premises from which the conclusion might 
 have been deduced by the direct method, yet 
 is it frequently very useful when these premises 
 are of such a nature as not to admit of a very 
 convenient syllogistic form. It is on this account 
 not unfrequently used in the propositions of 
 geometry, though Euclid has occasionally intro- 
 duced it when the direct proof would have been 
 equally simple. An instance of this will be 
 found in the fourth proposition of the third book, 
 in which it is required to prove that ' If in a 
 
106 A SYLLABUS OF LOGIC. 
 
 circle two straight lines cut one another which 
 do not both pass through the centre, they do 
 not bisect each the other/ The indirect proof 
 assumes that they bisect each other, and then 
 shews that they must both be perpendicular 
 to a line joining the centre and the point of 
 their intersection, which is absurd ; therefore, 
 &c. Whereas the direct proof states that all 
 chords which are bisected by the line drawn 
 from the centre to the point of intersection 
 must be perpendicular to that line ; but both 
 of these chords are not perpendicular to that 
 line; therefore both of these chords are not 
 bisected by the line drawn from the centre 
 to the point of their intersection, which can- 
 not therefore be the point of their bisection.* 
 But every proposition which admits the deduc- 
 tio ad absurdum, will also admit of a direct 
 proof from the same data, though in a great 
 many cases the direct proof would be exceed- 
 ingly clumsy, and not nearly as simple as the 
 indirect. There is a very striking instance 
 of this in the usual proof of the falsity of 
 a hypothetical or disjunctive proposition from 
 two categorical premises. The reasoning is 
 rather abstruse, but it bears too completely 
 upon the present subject to be entirely 
 
 * This syllogism is the mood A O O in the second figure. 
 
INDIRECT PROOF. 107 
 
 overlooked, and is therefore subjoined in a 
 note.* 
 
 * Hypothetical and disjunctive differ from categorical proposi- 
 tions in not possessing the quality of negation objectively. For 
 if the cognition upon which any one of them is grounded (t. e. the 
 dependence of one particular cognition upon another, or its divi- 
 sion into members) does not hold in nature, there is no other 
 hypothetical or disjunctive proposition which can simply assert 
 this want of objectivity. This remark has been made already 
 (Art. 42 44) where it was observed that a proposition which 
 simply denies the dependence of a particular cognition on another, 
 or the completeness of the division of a cognition, in neither case 
 gives any new dependence of cognitions, or any new exhausting 
 division, and consequently cannot be either hypothetical or dis- 
 junctive, but must be simply categorical. If then two categorical 
 propositions are granted as premises, from which we are to deduce 
 as conclusion the falsity of an hypothetical or disjunctive proposi- 
 tion, these two premises must be referred to the thinking subject as 
 the only middle term by which they can be united in an act of 
 reason, for all common objective grounds are denied them by the 
 very nattfre of the case. This necessity for a subjective reference 
 renders the reasoning so much more abstruse, that the mind 
 naturally chooses the other method of deductio ad absurdum, 
 which, from its objective nature, is much easier of comprehension. 
 In this method we assume the falsity of our desired conclusion 
 (which conclusion is, in this case, the contradictory of an hypo- 
 thetical or disjunctive proposition) and therefore assume the truth 
 of the hypothetical or disjunctive, and conjoining it with one of 
 our given premises, deduce the falsity of our other premiss ; in 
 this manner we entirely avoid the necessity of any reference to 
 the thinking subject. We will exemplify this reasoning as 
 follows : 
 
 Let the given premises be, * A is ; B is not,' from which we 
 are to deduce the falsity of the hypothetical proposition, 
 
 ' If A is, B is.' 
 
 If I would deduce this falsity (which being the contradictory of 
 a hypothetical must be contained in a categorical proposition) 
 by the direct method, I must refer the two premises to the 
 thinking subject ' I' in some such manner as the following : 
 
108 A SYLLABUS OF LOGIC. 
 
 (71.) The deductio ad dbsurdum supplies the 
 principle upon which may be founded a rather 
 pretty and symmetrical arrangement of the 
 twenty-four categorical syllogisms. It also 
 suggests a method of proving the necessary 
 equality of the moods true in the first three 
 figures without the aid of mathematical 
 analysis, and of shewing the reason why the 
 number of negative syllogisms is exactly double 
 the number of the affirmative. And although 
 we do not propose to derive any particular 
 practical advantage from its consideration, yet 
 anything that tends to give- additional order 
 and theoretical completeness must always have 
 a sufficient value in a pure science to warrant 
 its insertion. 
 
 As in every categorical syllogism we may 
 
 ' Whatever is in ray present consciousness is conjoined in it 
 with my assent to the conception of B not being. 
 
 4 My assent to the conception of A being is in my present con- 
 sciousness. 
 
 ' Therefore my assent to the conception of A being is conjoined 
 in my present consciousness with my assent to the conception of 
 B%eing.' 
 
 This conclusion is the categorical statement of the falsity of the 
 hypothetical which can only be shewn directly by this or some 
 other equally clumsy method. The indirect proof however, is 
 simple enough, and may be stated thus : 
 
 Let us assume that the hypothetical is true, or that ' if A is, 
 B is ; but A is; therefore B is ; but B is not, therefore,' &c. 
 
 As we have already stated, the simplicity of the proof in this 
 case arises from every premiss contained in it having an objective 
 reference, which the contradictory of a hypothetical has not. 
 
INDIRECT PROOF. 
 
 109 
 
 either employ the major premiss and contra- 
 dictory of the conclusion to disprove the minor 
 premiss, or the minor premiss and contradic- 
 tory of the conclusion to disprove the major 
 premiss, it follows that there must be two dis- 
 tinct syllogisms, with either of which we may 
 commence an indirect proof. The forms which 
 they respectively assume will readily appear 
 from the following table : 
 
 
 Major Premiss and 
 Contradictory of 
 Conclusion being 
 Premises. 
 
 Minor Premiss and 
 Contradictory of 
 Conclusion being 
 Premises. 
 
 First Figure. 
 (M-P) 
 < S M > becomes 
 
 U - pj 
 
 Second Figure. 
 (M- P) 
 
 \ s ~ p f 
 
 U -Mj 
 
 Third Figure. 
 
 f S - P ) 
 { S -MV 
 
 \M-PJ 
 
 Second Figure. 
 (P-M) 
 < S M > becomes 
 \S-Pj 
 
 First Figure. 
 (P- M) 
 
 1 S ~ P f 
 (S -M/ 
 
 Third Figure. 
 (S - M) 
 
 1 S ~ P ^ 
 U M) 
 
 Third Figure. 
 I'M- P) 
 < M S V becomes 
 \S- PJ 
 
 Second Figure. 
 ( S - P) 
 1 M - P \ 
 
 (M-SJ 
 
 First Figure. 
 (S -P) 
 
 {M - si 
 
 \M-PJ 
 
 Fourth Figure. 
 ( P -M) 
 < M S V becomes 
 
 Is-pJ 
 
 Fourth ligure. 
 (S- P) 
 fP-Ml 
 
 (M- s 
 
 Fourth Figure. 
 
 rM- s S 
 
 \ s ~ p t 
 IP-MJ 
 
 From this table it is immediately evident 
 that for every sound mood in the first figure, 
 
110 
 
 A SYLLABUS OF LOGIC. 
 
 there must also be a sound mood in the second 
 and third ; that for every sound mood in the 
 second figure, there must also be one in the 
 first and third ; and for every sound mood in 
 the third figure, one in the first and second. 
 Hence it follows, that the number of moods 
 that are sound in each of the first three figures 
 must be the same. But the form of the fourth 
 figure is such, that it only admits of an indirect 
 proof by syllogisms in the same figure; and 
 the equality, therefore, of the number of its 
 moods to that of the moods in the other three 
 figures, is not susceptible of this method of 
 proof. 
 
 This table will supply the grounds of a 
 division of categorical syllogisms into eight 
 systems, containing three each. Six of these 
 systems have one mood in each of the first 
 three figures. The other two are contained 
 entirely in the fourth. The systems are as 
 follows: 
 
 Figure 
 
 1 
 
 2 
 A A I 
 
 3 
 
 4 
 
 5 
 
 6 
 
 First 
 
 A A A 
 
 A I I 
 
 E A E 
 
 E AO 
 
 E I O 
 
 Second 
 
 A O O 
 
 A E O 
 
 A E E 
 
 E I O 
 
 E A O 
 
 E AE 
 
 Third 
 
 A O 
 
 E A O 
 
 E 10 
 
 I A I 
 
 A A I 
 
 A I I 
 
INDIRECT PROOF, 
 
 Fourth Figure. 
 
 Ill 
 
 7 
 
 8 
 
 A A I 
 
 I A I 
 
 A EO 
 
 AEE 
 
 E A 
 
 E I O 
 
 Each system must contain one affirmative 
 and two negative syllogisms. For the contra- 
 dictory of the conclusion of every affirmative 
 syllogism must constitute one of the premises 
 in its two complementary syllogisms, or those 
 contained in the same system with itself; and 
 as this contradictory must be a negative pro- 
 position, the two syllogisms that contain it 
 must be negative also. Hence, every affirma- 
 tive syllogism must have two complementary 
 negative syllogisms. And as the affirmative 
 contradictory of the conclusion of a negative 
 syllogism may be combined with either its 
 negative or its affirmative premiss, its two 
 complementary syllogisms must themselves 
 also be the one negative, and the other affir- 
 mative. As, then, every system contains one 
 affirmative, and two negative syllogisms, it is 
 evident that the whole number of negative 
 must double the number of affirmative syllo- 
 gisms. 
 
SECTION VI. 
 
 SYMBOLICAL EXPRESSION FOR THE LAWS OF 
 CATEGORICAL SYLLOGISM. 
 
 (72.) THE conditioning laws of categorical 
 syllogism admit of a very simple analytical 
 expression from which all its properties may 
 be readily obtained. But the more especial 
 object in treating this subject mathematically, 
 is the exhibition of that symmetry, from which, 
 the equality of the number of moods that are 
 true in the first three figures, may be derived 
 a priori to all consideration of the moods 
 themselves. Before, however, we proceed to 
 make any assumptions, it is necessary to re- 
 mind the reader, that the usual arithmetical 
 interpretation of the symbols employed has no 
 natural connection with the subject under con- 
 sideration, but can merely be useful as an index 
 to the laws of their combination. If, for in- 
 stance, any sets of symbols be juxtaposed, and 
 equated to zero, the equation will indicate that 
 at least one of these factors may itself be singly 
 equated to zero ; and, in that case, all its con- 
 
SYMBOLICAL. 113 
 
 stituent symbols must receive whatever inter- 
 pretations may arise from the transposition of 
 terms, or any other operations, that are for- 
 mally analogous to those of arithmetical alge- 
 bra, and have been admitted in our particular 
 application of the science. In the present case 
 the conception of homogeneity is not intro-" 
 duced, and it will therefore follow, that, if 
 several symbols are equated to several others, 
 each of those on one side of the equation must 
 be considered as equated to one on the other, 
 but no division of a symbol into parts will be 
 considered admissible. 
 
 The laws of categorical syllogism have been 
 already once stated, but are nevertheless re- 
 peated here, as a trifling alteration in the 
 manner of expressing those respecting quality 
 will be necessary for their reduction to a 
 mathematical form. As the latter will be re- 
 presented in a single equation, it will be more 
 convenient to throw them into one rule, which 
 may be stated as follows : 
 
 (1.) At least one of the premises must have 
 an affirmative copula, and the quality of the 
 conclusion will be the same as that of the re- 
 maining premiss. 
 
 As this law respects the quality of the copu- 
 las, which will always determine the quantity 
 of the predicates, it may be written thus : 
 
114 A SYLLABUS OF LOGIC. 
 
 1. The predicate of at least one of the 
 premises is undistributed, and the predicate of 
 the conclusion will be of the same quantity as 
 the predicate of the other premiss. 
 
 The other three laws are 
 
 2. The middle term must be distributed 
 in at least one of the premises, 
 
 3. The major term must not be distributed 
 in the conclusion, unless it has been distributed 
 in the major premiss. 
 
 4. The minor term must not be distri- 
 buted in the conclusion, unless it has been dis- 
 tributed in the minor premiss. 
 
 Let u represent an undistributed term. 
 
 Let d represent a distributed term. 
 
 Also, let p l9 p z , p 3 , and s l9 s 2 , s 3 , represent 
 the predicates and subjects respectively of the 
 three propositions as expressed below. 
 
 SUBJECTS. PREDICATES. 
 
 Major premiss s v pi 
 
 Minor premiss s z p 2 
 
 Conclusion s 3 p 3 
 
 where each of these symbols must = d or u. 
 
 Now, observing from rule (1), that at least 
 one of the predicates of the premiss (i. e. p l9 p 2 ) 
 must be undistributed or = u 9 we may express 
 the other predicate as = p } -f p^ - u. 
 
 But by the latter clause of law (1) we also 
 find that whatever is the quantity of that pre- 
 
SYMBOLICAL. 115 
 
 dicate, must also be that of the predicate of 
 the conclusion, and we shall therefore have 
 
 ^3=jPi+A-^ .... law (1); 
 and this equation will be a correct expression 
 for law (1). 
 
 Now from law (2) we find that the middle 
 term must be distributed (i.e.=d), in at least 
 one premiss, and as s l and p 2 are the middle 
 term in the first figure, one or both of the 
 following equations must be true in that 
 
 figure, 
 
 (* l - d) = 
 
 (ft- J) = 0,- 
 and we shall therefore have 
 
 (*i - <0 (ft - = ^w (2) 
 
 as an equation expressing the second law for 
 the first figure ; and as the same reasoning will 
 apply to all the four, we shall have 
 
 Fig. (1).... (*, _rf)(p a ~rf) = 0} 
 2)....<ft-<*)(ft-rf)=0f 
 3)....(* I -i)(* a -0 = or 
 
 Fig. (4) .... (p, - d) (s, _ d) = J 
 
 Again, from law (3) we know that either the 
 predicate of the conclusion, i.e. p 3 , must be 
 undistributed or = u y or else the major term 
 which is either s l or p L according to the figure 
 must be distributed or = d; and by reasoning 
 similar to the preceding we shall have 
 
 I 2 
 
116 A SYLLABUS OF LOGIC. 
 
 Fig. (1) .... ( Pl - d) ( Ps - ) = 
 
 Fig. (3) ; .' .' .' (p\ ~ 2 ( 1 3 1 S j: law ' 3 '- 
 
 Fig. (4) (s l - d} (p 3 u} = 
 
 In precisely the same manner we may obtain 
 the following expressions for law (4). 
 
 Fig. 
 
 Fig.(2)....(, 2 -^)(, 3 - w) = 0^^ law( ^ 
 
 We shall now assume the symbols p' 3 and * 3 
 of such a nature that 
 
 ^'3 +^3 = d + u (5). 
 
 and s' 3 + 5 3 = d + u (6). 
 
 And as it has been already stated that homo- 
 geneity is not introduced in the conception of 
 our symbols, it will follow from these equations 
 that p' 3 and *' 3 are each = d or u. They will 
 also be respectively complementary to p 2 and 
 s 3 in such a manner that when one = u, the 
 other = d, and vice versa ; and they will there- 
 fore be of precisely the same nature as the 
 other six symbols, p l9 p. i9 &c. 
 
 From (5) and (6) we obtain for the values of 
 p 3 , * 3 respectively 
 
 P 3 = d + ti - p' 3 
 
 s 3 = d 4- u s' y 
 
 And substituting these values of p 3 and s 3 in 
 the equations (1), (3), and (4), we shall get 
 
SYMBOLICAL. 117 
 
 the following sets of equations for the four 
 figures of categorical syllogism. 
 
 V 5 i - (Pz - I) v Fi rs t figure. 
 
 (P\ d} O' - d) = (3) ' 
 
 (4) 
 
 (Pi d) (P2 d) = (2) . 
 
 V ^ J ' > Second Figure 
 
 (s l a) (p a d} = (3) ' 
 
 (s 2 d) (s' 3 d) = (4) 
 
 Pi "I- P 2 +*P' 3 = d -f 2 w... (1) 
 
 (.,-,.,(. f --) = (2) 
 
 (Pi - d) (P 3 (3) ' 
 
 (p 2 J. d) (/ 3 d} == (4) 
 
 Pi + P 2 + P'a = ^ + 2 (0 
 
 f FourthFigure 
 C^y 
 (p a - ^) (*' 3 - <0 =0 (4) 
 
 (73). If we examine the equations to the 
 first three figures, we shall find each set per- 
 
 * The following is the most general symbolical expression for 
 the syllogistic laws. 
 
 Let x and y represent the middle term in the major and minor 
 premises respectively, and m and n the major and minor terms. 
 The equations will then be equally applicable to all the four 
 figures, and will assume the following forms : 
 
 (a: d) ( y d) = (2). 
 
 (m d) O 3 - d) = (3). 
 
 (n c?) (s' 3 d) = (4). 
 
 where x 4- m = p\ + s } (5). 
 
 and y -\~ n =>jp 2 + 5 2 ( 6 ) 
 
118 A SYLLABUS OF LOGIC. 
 
 fectly symmetrical with respect to the other two. 
 For in equation (1), which is common to all 
 the three figures, the symbols p l9 p^ p 3 are 
 perfectly symmetrically involved, and the only 
 assumptions that have been made respecting 
 $ * 2 , $ 3 are that each of them must either 
 d or u. Hence any interchange among the 
 three symbols *,, $ 2 , * 3 or the three p l9 p z , p' 3 
 will not in the least affect the form of the 
 equations. But if we interchange p 2 and p in 
 the equations to the first figure, we shall 
 obtain the equations to the second ; and if we 
 interchange s l and s' 3 in the first figure, we 
 shall obtain the equation to the third ; or lastly, 
 if we interchange p 2 an'dj0' 3 , also 5, and $ 3 ', in the 
 equations to the third figure, we shall obtain 
 those to the second ; and of course the same 
 interchanges will reproduce the first from the 
 second, the first from the third, &c. : hence it 
 follows that the sets of equations to the first 
 three figures are perfectly symmetrical with 
 regard to each other.* 
 
 If however it were required to obtain the 
 
 * Although the sets of equations to the first three figures are 
 perfectly symmetrical with respect to each other, yet does it by 
 no means follow that the corresponding equations represent the 
 same laws. The equation that contains the law against illicit 
 major in the first figure, corresponds to the similar equation in 
 the third ; but in the second figure it corresponds to the equation 
 containing the law for the distribution of the middle term ; and 
 similarly of the others. 
 
SYMBOLICAL. 119 
 
 equations to the fourth figure from those to 
 either of the other three, it would be neces- 
 sary to interchange one or more of the three 
 symbols s l9 s 2 , sj with one or more of the 
 symbols p l9 p^ p^\ and this cannot be admitted, 
 as the three latter symbols are all involved in 
 equation (1), whereas the three former are 
 neither involved in that or any other cor- 
 responding equation ; and hence it follows 
 that the equations to the fourth figure are 
 not symmetrical with those to the other three. 
 
 (74.) As the symbols are symmetrically in- 
 volved in the equations to the first three 
 figures, we can know a priori to all other 
 considerations that there must be the same 
 number of solutions, and therefore the same 
 number of true moods for all of them. But as 
 the equations to the fourth figure do not 
 involve the symbols symmetrically with the 
 equations to the other three, we cannot say at 
 once that the number of their solutions must 
 be the same as in those to the other figures, 
 but can only shew that it is so by determining 
 the number of solutions in each case. For 
 this purpose it will be sufficient to investigate 
 separately the number of solutions for the first 
 and fourth figures. 
 
 We will commence by examining the first 
 figure. It is evident from equation (I), that 
 
120 A SYLLABUS OF LOGIC. 
 
 one of the symbols p l9 /? 2 , /r must = d, and 
 the other two each = u ; and from equation 
 (3) that either p l or p^ must = d. Hence 
 there will be two solutions for equation (3) 
 (accordingly as p l or p 3 d), which, will cor- 
 respond to two solutions for equation (1). 
 
 But as equation (3) requires that either p { or 
 p' 3 should = d, it follows from equation (1) 
 that p 2 must always = u, and therefore from 
 equation (2) that s l = d l9 or otherwise neither 
 factor in equation (2) would vanish. Hence 
 there is but one solution for equation (2), and 
 only two solutions for the equations (1 ) and (3), 
 as the latter are mutually dependent on each 
 other. Equation (4) will have three solutions, 
 accordingly as both together or either sepa- 
 rately of the symbols s z and * 3 ' = d. And 
 as the solutions of equation (4) are quite in- 
 dependent of the symbols involved in the 
 equations (1), (2), (3), it follows that any one 
 of the three solutions of the former may be 
 combined with either of the two solutions of 
 the other three equations, and thus produce 
 six sets of solutions which will correspond to 
 the six moods that are'' sound in the first 
 figure.* 
 
 * Great care must be taken, in the interpretation of these equa- 
 tions into their corresponding moods, not to confound p' 3 with p y 
 or s' 3 with 53. The values of p' 3 , and s' 3 are first determined 
 
SYMBOLICAL. 121 
 
 In the fourth figure we shall have three solu- 
 tions for equation ( 1 ), accordingly as either of 
 the three symbols p l9 p 2 , or p' 3 may d. Of 
 the other three equations (2), (3), and (4), that 
 which contains the particular symbol of 
 the three p }) /? 2 , p^ which = d, will admit of 
 two solutions ; but the two equations which 
 respectively contain those two of the three 
 symbols p l9 p^ p 3 that are = u, will admit but 
 of one solution. Hence, each of the three 
 solutions of equation (1) may be combined 
 with the two solutions of one of the other 
 three equations, and thus produce six com- 
 binations which will respectively answer to 
 the six moods of the fourth figure. 
 
 Although we have only been able to prove 
 that the number of moods in the fourth figure 
 is equal to that in the other three, by the 
 numerical tentative method already given, yet 
 can we at least shew that no other method 
 is possible. For it is evident that the only 
 method by which such an equality can be 
 shewn independently of numbers, is that of 
 form, which has been already employed in be- 
 
 by the four equations to the figure under consideration, and 
 from them the values of p 3 and s 3 are deduced by means of the 
 equations 
 
 and will indicate the peculiar categorical form of the conclusion. 
 
122 A SYLLABUS OF LOGIC. 
 
 hoof of the first three figures. But the equa- 
 tions to those three figures completely deter- 
 mine two of their six symbols (s l and p 2 in the 
 first ; s. and p' in the second ; and p and s' 
 
 l * 3 * 2 3 
 
 in the third), whereas the equations to the 
 fourth do not determine a single one, but 
 admit variations in all. Hence it follows 
 that no artifice can bring the equations to 
 the fourth figure under the same form as 
 the equations to the other three, and conse- 
 quently that all formal proof of the equality 
 of the number of the solutions is absolutely 
 impossible. 
 
 (75.) Among other advantages in the sym- 
 bolical expression of the laws of categorical 
 syllogism, we may mention the facility with 
 which the derived secondary laws may be ob- 
 tained, and the peculiar fallacies exposed which 
 their violation entails. For instance, we know 
 from equation (1), that one alone of the sym- 
 bols p l9 p z , p' 3 can = d ; and moreover from 
 equation (3) in the first figure, that either 
 p l or p^ must = d. We may therefore con- 
 clude that /? 2 = u, that is to say, the predi- 
 cate of the minor premiss in the first figure 
 is undistributed, and the quality of its copula 
 affirmative. But if p 2 = n, it will follow from 
 equation (2) in the first figure that s x = d, or 
 in other words that the subject of the major 
 
SYMBOLICAL. 123 
 
 premiss is distributed, and that premiss an 
 universal proposition. Should either of these 
 secondary laws be violated, the conditions of 
 one of the equations (2) or (3) cannot be ful- 
 filled, and there must either be a fallacy of 
 undistributed middle, or illicit process of the 
 major. In the same manner it may be proved, 
 that in the second figure s } = d, and p' 3 = u, 
 (or p 3 = d,) in other words, that the major 
 premiss must be universal, and the conclu- 
 sion negative. But these examples are suf- 
 ficient to enable the reader to derive the other 
 laws for himself. 
 
 Truth of Premises. 
 
 (76.) To assert that ' if the premises are true, 
 the conclusion that is deduced from them must 
 be true likewise/ is a mere tautology; for 
 the very definition of a conclusion is ' that 
 proposition, the truth of which follows neces- 
 sarily from the truth of the premises.' But the 
 converse of this proposition is by no means 
 true, for it does not follow that if the conclu- 
 sion is true, the premises from which it is 
 deduced must be true also. A more satisfac- 
 tory explanation of this subject will be given 
 in a future section : the following, however, is 
 
124 A SYLLABUS OF LOGIC. 
 
 an unobjectionable proof by the method of the 
 deductio ad absurdum.* 
 
 Let us assume that if the conclusion is true, 
 the premises must be true also ; it will follow 
 that if the premises are not true, the conclu- 
 sion will not be true either. Let any sound 
 mood in any figure be represented by the 
 general symbols x y %, and let a?, y, %', represent 
 the formal contradictories of the propositions 
 x, y, % respectively. Let one or both of the 
 premises x and y be false ; it follows from our 
 present hypothesis, that % must be false also. 
 If then, for example, x is false, and its falsity 
 is sufficient to ensure the falsity of *, and if 
 we substitute for x a proposition which merely 
 states that falsity, we may also substitute for * 
 a proposition which merely states its falsity, 
 and thus change the form without affecting 
 the soundness of the reasoning. But the con- 
 tradictories of x and % respectively state the 
 falsity of those propositions, and consequently 
 a syllogism of the form x y *' must represent 
 
 * This proof will only strictly apply to categorical syllogisms. It 
 may, however, be extended to the hypothetical and disjunctive by 
 converting their form into the matter of a categorical. Thus the 
 hypothetical, ' if A is, B is ; A is, therefore B is,' may have its form 
 put into the matter of a categorical in the usual way, commencing 
 with ' all the cases of A being,' &c,, and as this categorical syl- 
 logism may have false premises and true conclusion, it is evident 
 the hypothetical may also. 
 
SYMBOLICAL. 125 
 
 an unobjectionable mood in the same figure as 
 that of x y %. And as we might have assumed 
 that y was false instead of x, or that both were 
 false together, we shall have the three moods 
 x 1 y %', x y *', and x y ' all sound in the same 
 figure as that of x y *. 
 
 But contradictories differ from each other in 
 the quantity both of their predicates and sub- 
 jects. If, then, we recur to the symbolical 
 expression of the syllogistic laws (Art. 73), we 
 must find that for every solution of the equa- 
 tions there given, three corresponding solu- 
 tions may be obtained by exactly reversing the 
 values of * 3 and p' 3 together with the values 
 of one or both pairs of symbols ^, p l and s^ p 2 
 in such a manner that those which = u should 
 = d, and vice versd. But it was shewn in Art. 
 74, that in the first figure the values of s l and 
 p 2 are determined by the equations (1), (2), 
 and (3), and consequently neither of the pairs 
 of symbols s } , p { or s 2 , p 2 can have their values 
 reversed in that figure. In the second figure 
 we find the symbol p' 3 determined by the equa- 
 tions (1) and (2), and in the third figure the 
 symbol s' 3 determined by the equations (1), 
 (3), and (4), and consequently the changes of 
 values cannot take place in these two figures 
 any more than in the first. Again, it appears 
 from the equations to the fourth figure, that 
 
126 A SYLLABUS OF LOGIC. 
 
 when * i can change its value, p { is determined, 
 and when s 2 can change its value, p 2 is deter- 
 mined, and consequently in neither case can 
 one of the pairs ^, p l or * 2 , p 2 both change their 
 values simultaneously. Rente it follows that 
 the conclusions at which we have arrived are 
 all false, and consequently that the hypothesis 
 which we assumed must be false also ; and that 
 true conclusions may be deduced from false 
 premises in every figure of categorical syllo- 
 gism. 
 
FORMAL LOGIC. 
 
 BOOK II. 
 SECTION I. 
 
 LIMITATIONS UPON THE FORM OF 
 JUDGMENTS, &C. 
 
 (77.) In the present section, it is proposed to 
 trace the limitations upon the form of judgments, 
 the superiority of the first figure of categorical 
 syllogism, and the possibility of true conclu- 
 sions from false premises, to a common a priori 
 ground in the very constitution of the under- 
 standing itself. The ground in question is a 
 simple property of the understanding-concep- 
 tion, Substance, and may be stated as follows. 
 Substance, or the substratum of phenomena, 
 (i.e. the thing that is, but is not phenomenon,) 
 can never become a predicate. 
 
 For the conception of Substance may be 
 defined as that which is thought as remaining 
 when all possible predicates have been ab- 
 
128 A SYLLABUS OF LOGIC. 
 
 stracted from it. It is therefore impossible to 
 make it a predicate, for nothing is left which 
 can become its subject ; and were we to at- 
 tempt to predicate it of phenomena, we must 
 previously think a substratum for these phe- 
 nomena, and should therefore only be predicat- 
 ing the simple conception-substance of itself, 
 which is absurd.* 
 
 Law between the predicate and copula of 
 categorical propositions. 
 
 (78.) This property of the understanding-con- 
 ception, Substance, will immediately explain the 
 reason of the law between the predicate and 
 copula of categorical propositions. This law, 
 
 * It is necessary to put the reader on his guard against a cer- 
 tain species of categorical judgments which appear to militate 
 against the observations in the text, and to contain the conception 
 of substance in the predicate as well as subject. The predicate 
 in these judgments contains matter of such a nature as to deter- 
 mine it to a particular object, e. g. ' that man travelled with me 
 yesterday.' In this judgment, taken alone, I really only think of 
 the existing man before me, and predicate of him all that is con- 
 tained in my conception of his having travelled with me yesterday. 
 But the instant after making such a judgment as this, I might very 
 probably convert it in my own mind, and think first of the exist- 
 ing man with whom I travelled yesterday, and afterwards predi- 
 cate of him that he then stood before me. And the case with 
 which the understanding can thus at pleasure make either term a , 
 subject containing the conception of substance, very naturally 
 produces a false semblance of that conception being in the pre- 
 dicate. 
 
LIMITATIONS, &C. 129 
 
 which has already been stated in Art. 33, is 
 repeated here : 
 
 Affirmative copulas have undistributed pre- 
 dicates ; negative copulas have distributed pre- 
 dicates. 
 
 For, inasmuch as the predicate cannot con- 
 tain the conception of substance, it is not 
 thought in respect of its sphere, or the things 
 that are contained under it, but in respect of 
 its matter, or the representations contained in 
 it. This term is accordingly a mere concep- 
 tion, which is never formally determined as to 
 its quantity, and the precise limits of its 
 sphere must always remain unknown. The 
 limits of the sphere of the subject cannot, 
 therefore, be exactly compared with those of 
 the sphere of the predicate, but the former 
 term must either be placed wholly in or wholly 
 out of the latter. If it is placed wholly in, it 
 is compared with only a part of the sphere of 
 the predicate ; if it is placed wholly out, it is 
 compared with the whole. From these con- 
 siderations it is immediately evident, that the 
 quantity of the predicate is always undistri- 
 buted in affirmative, and distributed in nega- 
 tive judgments. 
 
130 A SYLLABUS OF LOGIC. 
 
 There is no formal proposition of identity. 
 
 (79.) For if the subject is single, it evidently 
 cannot be identical with the predicate which is a 
 conception. But if the subject is a conception, 
 it must be considered either as to its partial 
 representations, i. e. what is thought in it, or 
 else as to its sphere, i. e. what is thought under 
 it. In the first case the conception in the 
 subject must be literally the same as the con- 
 ception in the predicate, and the result would 
 be a tautology, but not a judgment : in the 
 second case the subject, which refers to the 
 sphere or aggregate of individuals, cannot of 
 course be identical with the predicated con- 
 ception, the quantum of whose sphere is 
 formally indeterminate. In neither case there- 
 fore could there be a formal proposition of 
 identity. It need hardly be observed, that in 
 the proposition ' A is identical with B,' the 
 identity is expressed in the matter, and not in 
 the form. 
 
 Figures of Categorical Syllogism- 
 
 (80.) The great advantages in elegance and 
 perspicuity that the first possesses over the 
 other three figures, have been already men- 
 tioned in the first book ; and the ground of 
 this superiority may now be derived from the 
 
LIMITATIONS, &C. 131 
 
 considerations introduced at the commence- 
 ment of this section. 
 
 It has been shewn in Art. 77, that the con- 
 ception substance is never placed in the 
 predicate of a categorical proposition. Hence 
 a conception, which in the subject is considered 
 as to its sphere, (or the aggregate of indi- 
 viduals that are contained under it,) in the 
 predicate is considered as to its contents, (or 
 the representation that is contained in it as 
 a mere conception). Any change then in the 
 situation of a term introduces the necessity of 
 a change in the manner in which it is thought. 
 Now in the first figure, both subject and pre- 
 dicate of the conclusion retain the same 
 position which they held in the premises, and 
 consequently no such change in the manner of 
 thinking them is necessary. But in the second 
 figure, the major term changes its place from 
 subject in the major premiss to predicate in 
 the conclusion ; and in the third figure the 
 minor term changes its place from predicate in 
 the minor premiss to subject in the conclusion, 
 and consequently a change takes place in the 
 manner of thinking one extreme in each of 
 these figures. Again, in the fourth figure 
 neither of the extremes hold the same position 
 in the conclusion which they hold in the 
 premises, and consequently two changes in the 
 
 K2 
 
132 A SYLLABUS OF LOGIC. 
 
 nature of the terms take place. In this man- 
 ner, then, it is sufficiently easy to account for 
 the superiority of the first, and inferiority of 
 the fourth, to all the other figures. For all but 
 the first involve the necessity of some change 
 in the manner of our thinking one at least of 
 the extremes; and the fourth involves two 
 such changes. If the reader will make the 
 experiment of different moods in different 
 figures, he will become immediately conscious 
 that their comparative merits entirely depend 
 on the cause alleged. Perhaps the mood 
 E I O is the fittest for the experiment, as it is 
 sound in all figures. 
 
 True Conclusions from False Premises. 
 
 (81.) True" conclusions may be logically 
 deduced from false premises in every correct 
 form of categorical syllogism. 
 
 This proposition has been formally de- 
 monstrated in the first book, by means of 
 the symbolical expression of the syllogistic 
 laws. 
 
 The following is another rather preferable 
 proof, which could not with propriety have 
 been introduced there, as it rests upon certain 
 considerations from Transcendental Logic. 
 
 As no categorical proposition is either for- 
 mally identical, or formally exhaustive, (which 
 
LIMITATIONS, &C, 133 
 
 latter form would really be negatively identical, 
 since it would assert the identity of not B with 
 A,) it follows that no categorical proposition 
 completely determines the whole of one term 
 with respect to the whole of the other. For 
 instance, in the proposition ' No A is B,' each 
 term is placed in the external sphere of the 
 other, but whereabouts in it, is quite undeter- 
 mined. Again, in the proposition ' All A is B,' 
 B is a conception, the excess of whose sphere 
 above that of A is also undetermined, as it may 
 vary from nothing to infinity. Now the inde- 
 terminate parts of the premises can never be 
 introduced in the conclusion, which must 
 follow from them necessarily, and therefore 
 depend on the determinate alone. Hence it is 
 evident that the materially necessary parts of 
 the premises are invariably less than those 
 which are formally necessary. If then we 
 suppose an error in the indeterminate, and 
 therefore unavailable parts of the premises, they 
 will of course be false, although the conclusion, 
 which is materially dependent on the remaining 
 parts of those premises, is itself true. 
 
 It ought to be, if it is not, an axiom in an 
 a priori science, that the general proof should 
 invariably precede all reference to particular 
 examples. But the reasoning in the last para- 
 graph may now be elucidated, by taking as an 
 
134 A SYLLABUS OF LOGIC. 
 
 example the mood A A A. The general form 
 of a syllogism in this mood is the following : 
 All B is A, 
 All C is B, 
 Therefore All C is A. 
 
 In this syllogism, the indeterminate parts of 
 the terms are the excess of A above B, and 
 of B above C, and these parts are accordingly 
 unavailable in the conclusion. For it is only 
 so far as C agrees with B, that it agrees with 
 what B agrees ; and in the same manner it 
 is only so far as B agrees with A, that C, which 
 agrees with B, can be concluded to agree with 
 A. Let us then assume an error in the com- 
 parison of that part of B, which exceeds C, 
 with A. Still A will be predicated of so much 
 B as agrees with C, and consequently the con- 
 clusion will be true, though the major premiss 
 is false. 
 
 The following is an example of a sound syl- 
 logism, of the form A A A in the first figure, in 
 which the major premiss is false, though the 
 conclusion is true : 
 
 ' All animals are quadrupeds/ 
 ' All horses are animals/ 
 Therefore ' All horses are quadrupeds.' 
 
 We subjoin three diagrams of this form 
 of syllogism, in which the conclusion is true, 
 though one or both premises are false. In 
 
LIMITATIONS, &C. 135 
 
 Fig. 1, both premises are false ; in Fig. 2, the 
 major premiss is false ; and in Fig. 3, the minor 
 premiss is false. 
 
 Fig. 1. 
 
 False, < All B is A.' 
 False, < All C is B.' 
 True, < All C is A.' 
 
 False, < All B is A.' True, ' All B is A.' 
 
 True, ' All C is B.' 
 True, < All C is A.' 
 
 False, < All C is B.' 
 True, All C is A.' 
 
 These diagrams will also represent many 
 other forms of syllogism, in which the con- 
 clusion is true, though one or both premises 
 are false. For instance, if A be taken as the 
 middle term, B as the major, and C the minor, 
 Fig. 1 will represent the mood A E E in the 
 second figure with a true conclusion, though 
 both its premises are false. Again, if C be 
 taken as middle term, A as major, and B as 
 minor, Fig. 2 will represent the mood E I O 
 in the first figure having its conclusion and 
 minor premiss true, but its major premiss false. 
 And similarly a great many other cases might 
 be adduced, to which the above diagrams would 
 be equally applicable. 
 
 This manner of proof may easily be extended 
 to the other forms of syllogism. For instance, 
 
136 A SYLLABUS OF LOGIC. 
 
 in the hypothetical syllogism the antecedent 
 contains the grounds of the truth of the con- 
 sequent ; but as it may also contain much 
 more, their exact limits cannot be determined. 
 The truth therefore of the consequent will be 
 only formally, and not materially dependent on 
 some part of the antecedent ; and if an error 
 is introduced in the minor premiss of a syllo- 
 gism in the modus ponens, and in that part 
 of it which does not constitute one of the 
 grounds of the consequent, the result will be 
 a true conclusion, though one of the premises 
 is false : e. g. 'If the whole of the regi- 
 ment were going to Canada, Captain A. would 
 
 go ; but the whole of the regiment are 
 
 going to Canada ; therefore Captain A. will go/ 
 Now the antecedent contains all the necessary 
 grounds for the truth of the consequent, and 
 a great deal more besides ; and consequently 
 our minor premiss, which states that the whole 
 regiment is going, might be false, and yet 
 the conclusion that ' Captain A. is going' true. 
 
 As the conclusion must always be true if 
 the premises are true, and will sometimes be 
 true when they are false, it follows that the 
 probability of the truth of the conclusion must 
 always be rather greater than the probability 
 of the truth of the premises. 
 
LIMITATIONS, &C. 137 
 
 Conclusions of the Reason. 
 
 (82.) The limitation which the understanding 
 imposes upon the actual use of the reason, is 
 very well calculated to place in a clear and 
 strong light the distinct functions of the two 
 faculties in every categorical syllogism ; it is 
 therefore briefly re-considered here. 
 
 The reason can deduce a conclusion from 
 premises of the form I O in the first figure, 
 by precisely the same mental operation as from 
 the premises A A, or premises of any other 
 legitimate form. These propositions are of 
 the form 
 
 ' Some B is A ; 
 ' Some C is not B :' 
 
 from which the conclusion may be logically 
 deduced, that ' Some C is not some A.' 
 
 But this conclusion, though derived by 
 the same function of the reason as any 
 other legitimate conclusion (i. e. by a con- 
 junction in the consciousness of two acts of 
 the understanding), is nevertheless absolutely 
 worthless, as it may be predicated a priori 
 of any objects of which the understanding can 
 think. For whatever be the nature of the 
 hypothesis respecting the relation of C and A, 
 (e. g. let 'All C be identical with all A/) still 
 will it be true that ' Some C is not some 
 
138 A SYLLABUS OF LOGIC. 
 
 A.' And here therefore we find the logical 
 reason performing its regular office in com- 
 plete blindness, and quite independently of 
 the nature of the result when considered in 
 reference to the understanding. 
 
( 139 
 
 SECTION II. 
 
 MODALITY OF SYLLOGISM. 
 
 (83.) Modality has been already defined as 
 the determination of a judgment in respect of 
 its relation to truth. If the matter of the judg- 
 ment is merely in accordance with the a priori 
 laws of thinking, the judgment is, in respect 
 of its modality, problematical, or of the lowest 
 degree : if it is in accordance not only with 
 those laws, but also with the matter of the 
 senses, it is assertive, or of the second degree ; 
 but if one of these a priori laws becomes the 
 matter of the judgment, the latter is, in respect 
 of its modality, necessary, or of the highest 
 degree.* 
 
 As the formal use of the reason is quite in- 
 dependent of the matter of judgments, which 
 at the same time determines their modality, it 
 follows that this point of judging cannot have 
 any influence on the nature of the conclusion 
 in respect of its categorical form. But this 
 remark does not extend to the modality of 
 
 * Vide Arts. 25, 26, on this subject. 
 
140 A SYLLABUS OF LOGIC. 
 
 the conclusion, which may in some measure be 
 determined by that of the premises, and the 
 extent to which one is a criterion of the other 
 will supply the topic of the present section. 
 
 (84.) In order to place this subject in as clear 
 a light as possible, it will be necessary to say 
 a few words on the different sides from which 
 the modality of a conclusion may be viewed. 
 And in the first place, great care must be 
 taken never to confound the modality of this 
 proposition, 'that such a conclusion follows 
 from such premises/ with the modality of the 
 conclusion considered merely as to its own 
 matter, and quite independently of its grounds. 
 
 The former modality is always of the highest 
 degree, as it constitutes the very essence of a 
 conclusion that it follows necessarily from its 
 premises ; whereas the real modality of the 
 conclusion depends on the peculiar nature of 
 its matter, and is determinable in the same 
 manner as that of any other judgment. 
 
 But this distinction will become more ap- 
 parent, when the above proposition assumes 
 its appropriate form of a hypothetical, in 
 which the conclusion is the consequent, and 
 the premises the antecedent. It may then be 
 stated thus : If such and such premises are 
 both true, it will follow that such a conclusion 
 is true. Now the relation between these 
 

 MODALITY OF SYLLOGISM. 141 
 
 judgments is unquestionably necessary, as it 
 consists in those laws of the reason by which 
 the syllogism is known to be a correct one ; 
 but this is no criterion whatever for the moda- 
 lity of the judgments themselves, which consti- 
 tute the matter of the hypothetical, and are 
 therefore only propounded problematically. 
 Before proceeding any further, an example 
 will throw some light upon the foregoing 
 remarks. 
 
 'All men are liars;' 
 
 ' Obadiah is a man ;' 
 Therefore ' Obadiah is a liar/ 
 Now the modality of this conclusion, con- 
 sidered in reference to the premises, is of the 
 highest degree, or necessary ; for the propo- 
 sition that ' Obadiah is a liar* follows ne- 
 cessarily from the two propositions that 'all 
 men are liars,' and that ' Obadiah is a man.' 
 But the modality of this conclusion considered 
 by itself is by no means necessary, since no 
 necessity is contained in the proposition that 
 ' Obadiah is a liar,' for no contradiction is 
 involved in the supposition that he always tells 
 the truth. 
 
 As we shall have occasion to recur to these 
 two views of the modality of a conclusion, we 
 shall designate them by the terms ( proper,' or 
 that of any judgment considered merely as to 
 
142 A SYLLABUS OF LOGIC. 
 
 its own nature, and 'consequential,' or that 
 which depends on its reference to the judg- 
 ments from which it is deduced. 
 
 (85.) The proper modality of a conclusion 
 cannot be fully determined by that of the pre- 
 mises, for the modality of the premises depends 
 on their matter, and therefore on the middle 
 term which occurs in each of them. But the 
 conclusion will be unaffected as to matter, and 
 therefore as to modality, by any change in the 
 matter of the middle term, provided its formal 
 quantity is preserved. Hence it follows that 
 a change is possible in the modality of the 
 premises, while that of the conclusion remains 
 the same, and therefore that the modality of 
 the one can never be fully determined by that 
 of the other. This will appear from the 
 following example. Let the two premises be - 
 
 ' All animals are organic beings ;' 
 
 ' All horses are animals.' 
 
 These propositions are analytical, and there- 
 fore necessary ; and the conclusion derived 
 from them, that ' All horses are organic beings/ 
 is also analytical and necessary. But provided 
 the formal quantity of the middle term ' ani- 
 mals, be retained in each premiss, any other 
 matter may be substituted for it without 
 affecting the legitimacy of the reasoning. 
 Thus let the term ' cows' be substituted for 
 
MODALITY OF SYLLOGISM. 143 
 
 the middle term e animals/ ' All cows are 
 organic beings ; all horses are cows ; therefore 
 all horses are organic beings.' Now the same 
 conclusion has been logically deduced from 
 these as from the original premises, and con- 
 sequently its modality is the same in each case. 
 But in the former syllogism the modality of 
 both the premises was necessary ; whereas in 
 the latter, one of them is so far from being 
 necessary that it is not even true. Hence it is 
 evident that the nature of the premises can 
 never be a complete criterion for the proper 
 modality of the conclusion. There is, how- 
 ever, one law which will enable us to arrive at 
 some sure knowledge on this subject, and may 
 be stated thus : The modality of the conclu- 
 sion is never of a lower degree than the lowest 
 in either premiss. 
 
 (86.) Previous, however, to the consideration 
 of this law, it will be necessary to shew why the 
 case of two problematical judgments must be 
 entirely put out of the question : for two 
 judgments of this modality always leave the 
 possibility of such a disjunctive relation exist- 
 ing between them, that they are never asser- 
 tively true simultaneously.* No conclusion, 
 
 * It is necessary to distinguish between events that are * simul- 
 taneously possible,' and 'possible simultaneously.' The first 
 signifies that ' each is at the same time possible to happen;' the 
 
144- A SYLLABUS OF LOGIC. 
 
 therefore, can be deduced from them : but the 
 truth of this remark will become more evident 
 by an example. Let us suppose that a box 
 contains only black and white balls, of which 
 I abstract several, and put them in a bag 
 without observing their colour : T may then 
 state as problematical judgments 
 
 ' All the balls in the bag may be white ; 
 several black balls may be in the bag.' But I 
 cannot deduce from these judgments even the 
 problematical conclusion that ' several black 
 balls may be white/ although such a syllogism 
 in respect of its mere rational ft qm (i. e. A I I 
 in the first figure) would be perfectly legitimate. 
 It is evident in this example that a disjunctive 
 relation exists between the two premises, and 
 consequently that they can never be true simul- 
 taneously, though either may be true when 
 taken alone. As however the assertive modality 
 of either premiss would destroy the possibility 
 of such a relation, the objection will only 
 apply to those cases in which both are proble- 
 matical. 
 
 other, that both may possibly happen at the same time.' If I put 
 my hand into a bag and draw out only one ball, I may draw out 
 either a black one or a white one, and these events are therefore 
 simultaneously possible. But as one ball cannot be both a black 
 ball and a white ball too, these events are not possible simultane- 
 ously. The judgments in the example in the text, and also in 
 every disjunctive proposition, are simultaneously possible, but not 
 possible simultaneously. 
 
MODALITY OF SYLLOGISM. 145 
 
 (87.) After the exclusion of this particular 
 combination of premises, five others will remain 
 to which no such objection can be offered. 
 These will accordingly come under the law, 
 that the modality of the conclusion is never 
 of a lower degree than that in the weakest* 
 premiss. 
 
 To attempt a strict demonstration of any 
 law respecting the operations of the Reason, 
 involves the absurdity of making that faculty 
 both judge and defendant in its own case : for 
 in every proof the reason must tacitly assume 
 the validity of its own laws, and any pretend- 
 ed demonstration of them from themselves is 
 open to the objection of reasoning in a circle. 
 It is useless therefore to sue the reason in 
 its own court. All that can be done in a 
 question respecting its functions, is to place 
 it in as axiomatic a light as possible, and 
 admit as final the decision of the faculty 
 itself. In the present instance it is perhaps 
 sufficient to observe, that inasmuch as the 
 connection between the major arid minor 
 terms in the conclusion is entirely dependent 
 on their previous comparison with the middle 
 term in their respective premises, this connec- 
 tion must be of at least as high a degree as 
 
 * By the term ' weakest' that premiss is intended, whose mo- 
 dality is of at least as low a degree as that of the other premiss. 
 
 L 
 
A SYLLABUS OF LOGIC. 
 
 the lowest between the middle term and either 
 extreme. And the same reasoning will also 
 make evident the impossibility of logically 
 deducing from the premises a higher degree 
 of modality for the conclusion, * though 
 such may perhaps exist. It will therefore 
 be necessary to introduce the distinction of 
 'proper' and 'derived' modality, for the pur- 
 pose of avoiding those long periphrases which 
 would otherwise be perpetually recurring. 
 
 (88.) Perhaps the only great logical error 
 of which Aristotle is guilty in his Analytics, 
 refers to this very point of the modality of 
 syllogism ; and as an exposure of the fallacy 
 will supply the best explanation of the last few 
 paragraphs, we shall briefly consider it here. 
 
 The following passage will be found in the 
 ninth chapter of the first book of the Former 
 Analytics : 
 
 I $ 7TOTE KOI TTJQ ETEQCLQ TTQOTQ.<T(i)Q dvayKCLldC, 
 
 dvayKaiiov yivEaQai TOV avXXoyifffjLov, 7rXr)i/ oi/^ oiroTEpag 
 
 /, CtAXtt TTJQ TTpOQ TO H~ioV CtfCjOOV, OlOV El TO JJ.EV A TV 
 
 B f dvdyKrjQ et'Xr/Trrai virdpypv 77 JJ.T) VTrdp-^or, TO SE B rw 
 F vTrdpxpv fjiovov ovT&g yap etXij/jjucvwv rJy TrporaVcwv e^ 
 <}vdyKr)G TO A T<ij F vTrap^et rj oJ^ vTrap^Et. Eret yap iravrl 
 
 TWV B eort, (j>avEpoy OTI KOI TO F e aVay^e eorat QO.TEOOV 
 rovrwv. 
 
 * This law is precisely analogous to the mechanical axiom 
 that every chain is of the same strength as its weakest link. 
 
MODALITY OF SYLLOGISM. 147 
 
 (Translation*} It happens in some cases 
 that if one premiss is necessary the syllogism 
 becomes necessary (i. e. the conclusion of the 
 syllogism), not however either premiss at ran- 
 dom, but that which involves the major ex- 
 treme; just as if, for instance, B is assumed 
 to be A or not A, of necessity, and C is assumed 
 merely to be B (i. e. assertively, not necessarily); 
 for the premises being thus assumed, C will be 
 A, or will not be A, of necessity. For since 
 ' All B is A, or is not A,' of necessity, and ' C is 
 a part of B/ it is evident that C will be one of 
 these things (i. e. ' A or not A') of necessity. 
 
 Now as merely assertive judgments can only 
 have empirical grounds for their truth, it 
 follows that their predicates are stated of the 
 sphere or individuals contained under the 
 conception of the subject, and not of the 
 conception itself. For a conception can only 
 be predicated of another conception, (prior at 
 least to the empirical consideration of the 
 sphere of the latter,) either by being a superior 
 conception, in which case the judgment is 
 analytical, or else by some a priori law ; but 
 in either case the judgment is necessary, and 
 therefore more than merely assertive. 
 
 Now, by the hypothesis, the minor premiss 
 in the above syllogism is assertive, and con- 
 sequently the middle term B is only predicated 
 
 L 2 
 
148 A SYLLABUS OF LOGIC. 
 
 of the minor term C in respect of its sphere, 
 and not of its matter. But the truth of the 
 conclusion (considered as such) depends en- 
 tirely on this minor premiss, and in that 
 judgment, therefore, A must be predicated of C 
 in respect of its sphere and not its matter. 
 But when a conception is predicated only of 
 the sphere of another, the judgment cannot 
 be necessary, as it involves no a priori law of 
 thinking, and conveys no certainty that an 
 exception may not be found on some future 
 occasion. Hence the derived modality of the 
 conclusion that ' all C is A, or is not A,' can 
 never exceed the assertive or second degree, 
 which is that of the weakest premiss. Let us 
 take the following example : 
 
 ' All members of Caius College are members 
 of the University. Several Norfolk men are 
 members of Caius College. Therefore several 
 Norfolk men are members of the University/ 
 
 Now the major premiss of this syllogism is 
 analytical, and therefore necessary, for it con- 
 stitutes a part of the conception of a member of 
 a college that he should be a member of the 
 University. But the minor premiss on the 
 contrary is merely assertive, as it is by no 
 means necessary to our conception of Norfolk 
 men that several of them should be members 
 of Caius College. And the conclusion is also 
 
MODALITY OF SYLLOGISM. 149 
 
 merely assertive, as it does not necessarily 
 enter into our conception of Norfolk men that 
 they must be members of the University. 
 
 It is evident from this example, that only 
 an assertive conclusion can be deduced from 
 a necessary major and an assertive minor 
 premiss. But this law regards the derived 
 modality alone, and leaves the proper moda- 
 lity quite undetermined. For a trifling alter- 
 ation in the example just given will render 
 the proper modality of the conclusion neces- 
 sary, and leave that of the premises unaltered. 
 Let the minor term be changed into ' Several 
 members of the University who reside in 
 Norfolk ;' the minor premiss will still be only 
 assertive, as it is not necessary that any one 
 of these men should be a member of Caius 
 College ; but the conclusion will be analytical, 
 and therefore its proper modality will be ne- 
 cessary. 
 
 But, that Aristotle never intended to say 
 that the conclusion from premises so assumed 
 might sometimes* be necessary in its own 
 nature, and sometimes only assertive, appears 
 from his giving this proof in the general 
 symbols A, B, C ; and also from his predi- 
 
 * The restriction contained in the word TTOTS, evidently alludes 
 to the limitation of the law to those cases in which the major 
 premiss is necessary, and the minor assertive. 
 
150 A SYLLABUS OF LOGIC. 
 
 eating necessity of the syllogism, and not 
 merely of the conclusion. Neither is he merely 
 alluding to the consequential modality of the 
 conclusion, for he distinguishes between the 
 proper and consequential modalities more 
 than once, and particularly in the middle of 
 the following chapter. So that his evident 
 meaning is this : if a necessary major premiss 
 and an assertive minor premiss be assumed 
 in the first figure, that modality which can 
 be derived from that of the premises for the 
 conclusion as a judgment considered by itself, 
 will be necessary and not merely assertive. 
 
 (89.) A fallacy involving a breach of the 
 same law is to be found at the commencement 
 of the nineteenth chapter of the first book 
 of the Former Analytics.* Aristotle here lays 
 down as a general rule, that if one premiss 
 is necessary and negative, and the other 
 problematical and affirmative, an assertive 
 conclusion may be deduced from them. 
 
 " For let ' All B be necessarily not A/ and 
 ' All C be possibly A :' if the negative premiss 
 is converted, ' All A will be necessarily not B ;' 
 but 'All C may possibly be A;' there will 
 therefore be a syllogism in the first figure to 
 the effect that ' All C is possibly not B.' But 
 
 * This passage is at page 97 of Bekker's 8vo. Edition, Oxford 
 
 reprint: 'Ei 6' j /ULIV 1% ai/ay/ojs TeOgftj TO <rT/ot}Ti/c<)j/. 
 
MODALITY OF SYLLOGISM. 151 
 
 it is evident, moreover, that ' All C will be not 
 B.' For let us assume that ' Some C is B :' if 
 then ' All B is necessarily not A/ and ' Some 
 C is B,' it follows that ' Some C cannot be A,' 
 which is absurd, as by the original hypothesis, 
 'All C may be A.' Therefore, &c." 
 
 This reasoning is extremely ingenious, but 
 not less sophistical. It will be found to hinge 
 upon the very fallacy which has just been 
 exposed in the last paragraph. For the con- 
 clusion which Aristotle wishes to deduce, is 
 assertive, viz. ' All A is B :' he accordingly 
 assumes the assertive contradictory of this 
 conclusion as the minor premiss of the first 
 syllogism in an indirect proof, and from this 
 assertive minor, and the original necessary 
 major premiss, fallaciously deduces a necessary 
 conclusion, to the effect that, ' Some C cannot 
 be A.' And as this conclusion is at variance 
 with the other premiss that ' All C may be A,' 
 he infers that his assumption of the contra- 
 dictory of the proposition ' All A is B' was 
 unwarranted, and therefore concludes that 
 ' All A is B.' Now the necessity of the con- 
 clusion that ' Some C cannot be A* was merely 
 consequential and not proper, and the legiti- 
 mate conclusion that ' Some C is not A' is not 
 at all opposed to the original minor premiss that 
 ' All C is possibly A/ For this, as well as 
 
152 A SYLLABUS OF LOGIC. 
 
 every other problematical conclusion, contains 
 the conception of the possibility of the con- 
 trary. The proposition ' All C is possibly A* 
 is therefore perfectly compatible with the 
 proposition that 'Some C is possibly not A/ 
 And it must be observed that the assertive 
 modality of the conclusion that * Some C is 
 not A/ arises from the assumption of the 
 assertive premiss that ' Some C is B/ which 
 latter proposition states more than the pre- 
 mises would warrant in the conclusion of the 
 original syllogism, though it is by no means 
 contradictory to it. We will bring the question 
 to an experimentum crucis, which will render 
 the fallacy immediately evident. This will be 
 most readily effected by taking a particular 
 example, and carrying out the whole argu- 
 ment in precisely the same form as that in 
 which it is given by Aristotle. 
 
 Let a bag contain several balls whose colour 
 is unknown, and may therefore possibly be white 
 or black, or any other colour. The necessary 
 and negative premiss may then be, ' No white 
 balls are black balls/ and the problematical 
 affirmative, 'All the balls in the bag may be 
 white balls ;' from which Aristotle's reason- 
 ing would deduce as conclusion, that ( none 
 of the balls in the bag are black balls.' His 
 proof would run as follows : ' Let some of the 
 
MODALITY OF SYLLOGISM. 153 
 
 balls in the bag be black balls; as no black 
 balls can be white balls, it follows that some 
 of the balls in the bag cannot be white balls. 
 But this is absurd, for by the hypothesis all 
 the balls in the bag may be white. There- 
 fore, &c.' 
 
 This species of fallacy will become yet 
 more apparent by stripping it of the syllo- 
 gism with which it is connected, and proving 
 by its means that every judgment which is 
 true problematically, must also be true asser- 
 tively. Let the judgment be, ' All A may pos- 
 sibly be B.' It will follow from this that ' All 
 A is B.' For if it is false that ' All A is B,' it 
 must necessarily be true that ( Some A is 
 not B.' But this is absurd, as by the hypo- 
 thesis ' All A may be B.' Hence it follows, that 
 the assumed falsity of the proposition ' All A 
 is B,' must itself be false, and therefore that 
 < All A is B.' 
 
 Both of these fallacies of Aristotle are of 
 considerable importance, as they violate this 
 fundamental principle of the modality of syl- 
 logism, namely, that the derived modality of 
 the conclusion is the same as that of the 
 weakest premiss. 
 
 (90.) There is however another fallacy, very 
 near the end of the twenty-eighth chapter of the 
 same book, which is introduced here rather 
 
154 A SYLLABUS OF LOGIC. 
 
 as a curiosity than for any other reason, as 
 in all probability it is the only instance of a 
 false mood and figure in the whole work. It 
 is stated in this passage that the denying B of 
 H is exactly equivalent to identifying B with 
 some T, where T has been previously assumed 
 to represent the whole external sphere of a 
 certain term E, which is itself a predicate of 
 H. This paralogism is of the form A E E in 
 the first figure, which has an illicit major, or 
 E A E in the fourth figure, which has an illicit 
 minor.* 
 
 It is rather remarkable, that, notwithstand- 
 ing Aristotle's entire rejection of the fourth 
 figure, he has introduced it once in this very 
 chapter, in the mood A A I.f 
 
 * Page 115, To yap /urj ti^t'xo-flcu TO B, K. T. \. 
 f Page 112, El 8s TW H TO B TOLVTOV, K. T. \. 
 
APPENDIX. 
 
 THE following are a few examples in which the reader 
 can try his skill in detecting fallacies, determining the 
 peculiar form of syllogisms, and supplying the sup- 
 pressed premises of enthymemes. The arguments alone 
 have been adopted from the different authors whose 
 names are attached, as alterations in the mode of ex- 
 pressing them have invariably been found necessary to 
 bring them a little nearer the simple syllogistic forms. 
 Several of the examples contain more than one syllo- 
 gism. 
 
 (1.) None but those who are contented with their 
 lot in life can justly be considered happy. But the 
 truly wise man will always make himself contented with 
 his lot in life, and therefore he may justly be consi- 
 dered happy. 
 
 (2.) All nations, whose commerce has been very ex- 
 tensive, have reached a great height in refinement and 
 luxury. The Romans reached a great height in refine- 
 ment and luxury. Therefore the Romans must have 
 had a very extensive commerce. 
 
156 APPENDIX. 
 
 (3.) A really terrible enemy would never excite so 
 small a degree of fear that any other passion could 
 master the feeling. But there is no passion so weak 
 that it cannot master the fear of death. Therefore 
 death is no very terrible enemy. Bacon. 
 
 (4.) None but the contented are happy; the good 
 are happy; therefore they are contented. 
 
 (5.) If there is a possibility of the existence of God, 
 nothing can be more evident than that men ought to 
 live virtuously and piously, and that vice is the most 
 absurd thing in nature. But that God exists is per- 
 fectly possible, as no demonstration can be given of 
 the contrary. Therefore men ought to live virtuously 
 and piously, and vice is the most absurd thing in 
 nature. Clarke. 
 
 (6.) If all cats are animals, and all animals are organic 
 beings, it follows that all cats are organic beings. But 
 all cats are organic beings. Therefore all cats are ani- 
 mals, and all animals are organic beings. 
 
 (7.) A statesman should particularly avoid anything 
 that tends to bring him into contempt. Idle and 
 frivolous duels will probably have this effect, and he 
 should therefore particularly avoid them. Taylor's 
 Statesman. 
 
 (8.) All intelligible propositions must be either true or 
 false. The two propositions ' Caesar is living still,' and 
 * Caesar is dead,' are both intelligible propositions ; there- 
 fore they are both true, or both false. 
 
 (9.) God acts according to laws because he knows 
 them, he knows them because he has made them, he has 
 made them because they bear a certain relation to his 
 wisdom and power; therefore God acts according to 
 
APPENDIX. 157 
 
 laws because they bear a certain relation to his wisdom 
 and power. Montesquieu. 
 
 (10.) Blessed are the pure in heart, for they shall see 
 God. 
 
 (11.) None but the good are really great, and all the 
 good are happy. The slaves of passion are never really 
 great, and therefore they are never happy. 
 
 (12.) No man who is in London can be in York. As 
 no person has told me where Mr. A is, for all I know to 
 the contrary he may be in York. Therefore I conclude 
 that he is not in London. For if this conclusion is false, 
 it must be true that he is in London/ But no man that 
 is in London can be in York. Therefore Mr. A is not 
 in York, which is contrary to the original hypothesis 
 that he may be in York ; and therefore the second 
 hypothesis, that he is in London, must be false, and 
 Mr. A is not in London. 
 
 (13.) All God's gifts are intended for use. Our 
 foresight and power over the future are God's gifts. 
 Therefore it is intended that we should use our fore- 
 sight and power over the future. But Slavery pre- 
 cludes all possibility of the exercise of these powers ; 
 it is therefore opposed to the intentions of Providence 
 Channing. 
 
 (14.) Many things are more difficult than to do no- 
 thing. Nothing is more difficult to do than to walk 
 on one's head. Therefore many things are more dif- 
 ficult than to walk on one's head. 
 
 (15.) If God does not grant his grace upon the 
 same conditions to all mankind, he is a respecter of 
 persons. But God is no respecter of persons. There- 
 
158 APPENDIX. 
 
 fore God grants his grace upon the same conditions 
 to all mankind. 
 
 (16.) He that is of God heareth God's words: 
 ye therefore hear them not, because ye are not of God. 
 John, c. viii, v. 47. Quoted from Whately^s Logic. 
 
 (17.) It is highly probable that all persons who 
 have established a new religion entirely subversive of 
 the old, have suffered persecution. The first preachers 
 of Christianity were such persons. It is therefore highly 
 probable that they suffered persecution. Paley's Evi- 
 dences. 
 
 (18.) The men who barter their eternal welfare for 
 temporary gratifications are very deficient in real wis- 
 dom. But the men who are thus deficient are by no 
 means few. Therefore the men who barter their eternal 
 welfare for temporary gratifications are by no means few. 
 
 (.19.) None but Whigs vote for Mr. B. All who vote 
 for Mr B. are ten-pound householders. Therefore none 
 but Whigs are ten-pound householders. 
 
 (20.) The waking state succeeds the sleeping, and the 
 sleeping succeeds the waking ; things become cold from 
 having been hot, and hot from having been cold ; men 
 can only become taller from having been shorter, and 
 shorter from having been taller. Thus all contraries 
 mutually produce, and are produced from, each other. 
 But the states of life and death are contrary to each 
 other. Therefore the state of life succeeds that of death, 
 as that of death does that of life. Plato's Phcedo. 
 
 (21.) If the Mosaic account of the cosmogony is 
 strictly correct, the sun was not created till the fourth 
 day. And if the sun was not created till the fourth day, 
 it could not have been the cause of the alternation of day 
 
APPENDIX. 159 
 
 and night for the first three days. But either the word 
 ' day' is used in Scripture in a different sense to that in 
 which it is commonly accepted now, or else the sun must 
 have been the cause of the alternation of day and night 
 for the first three days. Hence it follows that either 
 the Mosaic account of the cosmogony is not strictly 
 correct, or else the word e day' is used in Scripture in 
 a different sense to that in which it is commonly ac- 
 cepted now. 
 
 (22.) Men who would peril their own lives and those 
 of their fellow-creatures on no better pretext than that of 
 maintaining a reputation for courage, would unquestion- 
 ably engage in duels upon very slight provocation. 
 But none except very weak men would ever peril any 
 human life with such an object alone, and we therefore 
 conclude that none but very weak men would engage in 
 duels upon very slight provocation, 
 
 (23.) Laws, in the widest acceptation of the word, are 
 the necessary relations which derive their origin from 
 the nature of things. The Deity, therefore, has his laws, 
 for he bears a certain relation to the universe as its 
 creator and preserver. Montesquieu. 
 
 (24.) Suffering is a title to an excellent inheritance; 
 for God chastens every son whom he receives. (Quoted 
 literally from Jeremy Taylor's Holy Living and Dying.) 
 This sentence may be put under the following form : All 
 whom God receives, he chastens ; all who suffer, God 
 chastens ; therefore, all who suffer, God receives. 
 
 (25.) To do a certain evil for a problematical good is 
 contrary to the spirit of Christianity. But to hurry 
 a great criminal into the presence of his Creator, and this 
 for the sake of the very questionable advantage which 
 
160 APPENDIX. 
 
 capital may have over other severe punishments in 
 deterring others from crime, is to do a certain evil for 
 a problematical good. Therefore capital punishment is 
 contrary to the spirit of Christianity. 
 
 ADDITION TO NOTE p. 117. 
 
 The complete elimination of the indeterminate symbols x, y, m, n 
 from the equations on note to p. 117, will furnish additional equa- 
 tions representing the general laws of Categorical Syllogism in 
 their simplest form : each of whose solutions will correspond to a 
 possible mood. 
 
 It is unnecessary to insert the process of elimination ; but it 
 will be seen without difficulty that the equations thus obtained 
 are the following : 
 
 Quality p l + p 2 + p'a = d + 2 u (1). 
 
 Laws of particular ((s l d) (s 2 d) = CH 
 
 premises and par- < (s t d) (s' 3 d) = I (2). 
 
 ticular conclusion. [(s 2 d) (s' 3 d) = OJ 
 
 I E O excluded as f 
 
 leading to illicit 1 (s } d) (pi _ d) (p' 3 d) = 0... (3). 
 
 major. [ 
 
 (1.) Implies that two of the quantities p t p 2 p'3 must be = u, 
 and a third = d, and admits therefore of 3 solutions. 
 
 (2.) Imply that any two or all three of the quantities s { s 2 s' 3 
 must = d . . and admit therefore of 4 solutions. 
 
 The whole number of solutions therefore admissible from (1) 
 and (2) is 3 X 4 = 12. 
 
 Of these one is rendered inadmissible by (3). The whole num. 
 ber of possible moods is therefore 1 1 . 
 

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