IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 /y 
 
 
 1.0 
 
 I.I 
 
 Ui|28 |2.5 
 
 ■50 *^^ RIBH 
 
 It Vi 12.2 
 
 m 
 
 ■ 
 
 1 
 
 1.25 |||.4 III 1.6 
 
 
 < — i 
 
 6" 
 
 ► 
 
 <m 
 
 r 
 
 / 
 
 
 ^J> 
 
 v/ ^ 
 
 ^:^* 
 ■^ 
 
 f> 
 
 !> 
 
 ?^ 
 
 '/ 
 
 Hiotographic 
 
 Sdences 
 
 Corporation 
 
 23 WiST MAIN STREIT 
 
 WEBSTER, N.Y. US80 
 
 (716)872-4303 
 
 ^ 
 
 '^V^ 
 
) 
 
 M 
 
 4^ 4^ 
 
 CIHM/ICMH 
 
 Microfiche 
 
 Series. 
 
 CIHIVI/ICIVIH 
 Collection de 
 microfiches. 
 
 i» 
 
 Canadian Institute for Historical MIcroreproductions / Institut Canadian de microreproductions historiques 
 
Technical and Bibliographic Notes/Notes techniques et bibliographiques 
 
 The Institute has attempted to obtain the best 
 original copy available for filming. Features of this 
 copy which may be bibliographically unique, 
 which may alter any of the images in the 
 reproduction, or which may significantly change 
 the usual method of filming, are checked below. 
 
 D 
 
 D 
 
 D 
 
 D 
 
 
 □ 
 
 D 
 
 Coloured covers/ 
 Couverture de couleur 
 
 I I Covers damaged/ 
 
 Couverture endommagte 
 
 Covers restored and/or laminated/ 
 Couverture restaurie et/ou pelliculte 
 
 □ Cover title missing/ 
 Le 
 
 titre de couverture manque 
 
 l~~| Coloured maps/ 
 
 Cartes g6ographiques en couleur 
 
 Coloured ink (i.e. other than blue or black)/ 
 Encre de couleur (i.e. autre que bleue ou noire) 
 
 r~~| Coloured plates and/or illustrations/ 
 
 Planches et/ou illustrations en couleur 
 
 Bound with other material/ 
 Relii avec d'autres documents 
 
 Tight binding may cause shadows or distortion 
 along interior margin/ 
 
 La re liure serrde peut causer de I'ombre ou de la 
 distortion le long de la marge intirieure 
 
 Blank leaves added during restoration may 
 appear within the text. Whenever possible, these 
 have been omitted from filming/ 
 II se peut que certaines pages blanches ajouttes 
 lors d'une restauration apparaissent dans le texte, 
 mais, iorsque cela itait possible, ces pages n'ont 
 pas 6t6 filmAes. 
 
 Additional comments:/ 
 Commentaires suppl6mentaires; 
 
 L'Institut a microfilm^ le meilleur exempiaire 
 qu'il lui a iti possible de se procurer. Les details 
 de cet exempiaire qui sont peut-Atre uniques du 
 point de vue bibliographique, qui peuvent modifier 
 une image reproduite, ou qui peuvent exiger une 
 modification dans la mithode normale de filmage 
 sont indiqute ci-dessous. 
 
 □ Coloured pages/ 
 Pages de couleur 
 
 □ Pages damaged/ 
 Pages endommagdes 
 
 I I Pages restored and/or laminated/ 
 
 The 
 toti 
 
 Pages restaurdes et/ou peiliculAes 
 
 Pages discoloured, stained or foxei 
 Pages d^color^es, tachetdes ou piqudes 
 
 Pages detached/ 
 Pages d6tach6es 
 
 Showthrough/ 
 Transparence 
 
 Quality of prir 
 
 Quality in^gaie de I'impression 
 
 Includes supplementary materii 
 Comprend du matiriel supplimentaire 
 
 Only edition available/ 
 Seule Edition disponible 
 
 r~^ Pages discoloured, stained or foxed/ 
 
 I I Pages detached/ 
 
 r~71 Showthrough/ 
 
 n~| Quality of print varies/ 
 
 I I Includes supplementary material/ 
 
 I I Only edition available/ 
 
 D 
 
 Pages wholly or partially obscured by errata 
 slips, tissues, etc., have been ref limed to 
 ensure the best possible image/ 
 Les pages totalement ou partiellement 
 obscurcies par un feuillet d'errata, une pelure, 
 etc., ont ix6 film6es i nouveau de facon d 
 obtenir la meilleure image possible. 
 
 The 
 post 
 ofti 
 film 
 
 Ori| 
 
 beg 
 
 the 
 
 sion 
 
 oth( 
 
 first 
 
 slor 
 
 oril 
 
 The 
 shal 
 TIN 
 whi 
 
 Mai 
 diff 
 enti 
 beg 
 righ 
 reqi 
 met 
 
 This item is filmed at the reduction ratio checked below/ 
 
 Ce document est filmi au taux de rMuction indiquA ci-dessous. 
 
 10X 
 
 
 
 
 14X 
 
 
 
 
 18X 
 
 
 
 
 22X 
 
 
 
 
 26X 
 
 
 
 
 aox 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 12X 
 
 
 
 
 10X 
 
 
 
 
 aox 
 
 
 
 
 24X 
 
 
 
 
 28X 
 
 
 
 
 32X 
 
 
The copy filmad here hat bMn raproducad thanka 
 to tha ganarosity of: 
 
 Douglas Library 
 Quaan's Univarsity 
 
 L'axamplaira fllmA f ut raproduit grAca A la 
 gAnAroaltA da: 
 
 Douglas Library 
 Quaan's Univarsity 
 
 Tha imagas appaaring hara ara tha bast quality 
 possibia conaidaring tha condition and laglbility 
 of tha original copy and in kaaping with tha 
 filming contract spacifications. 
 
 Original copias in prinwd papar covars ara filmad 
 baginning with tha front covar and anding on 
 tha last paga with a printad or illuatratad impras- 
 sion, or tha back covar whan approprlata. All 
 othar original copias ara filmad baginning on tha 
 first paga with a printad or illuatratad impras- 
 sion, and anding on tha last paga with a printad 
 or illuatratad imprassion. 
 
 Tha last racordad frama on aach microficha 
 shall contain tha symbol ^^ (moaning "CON- 
 TINUED"), or tha symbol ▼ (moaning "END"), 
 whichavar applias. 
 
 IVIaps, platas, charts, ate, may ba filmad at 
 diffarant raduction ratioa. Thosa too larga to ba 
 entiraly includad in ona axposura ara filmad 
 baginning in tha uppar laft hand cornar, laft to 
 right and top to bottom, as many framaa as 
 raqulrad. Tha following diagrams illustrata tha 
 mathod: 
 
 Laa imagas suivantas ont 4t4 raproduitas avac la 
 plua grand aoin, compta tanu da la condition at 
 da la nattatA da I'axampialra film*, at an 
 conformity avac las conditions du contrat da 
 filmaga. 
 
 Laa axamplairas originaux dont la couvartura en 
 papiar ast imprimte aont filmte an commandant 
 par la pramiar plat at an tarminant soit par la 
 darniAra paga qui comporta una amprainta 
 d'impraaaion ou d'illuatration, soit par la "^acond 
 plat, aalon la cas. Toua las autras axamplairas 
 originaux sont filmte an commanpant par la 
 pramlAra paga qui comporta una amprainta 
 d'imprassion ou d'illustration at an tarminant par 
 la darnlAra paga qui comporta una talla 
 amprainta. 
 
 Un das symbolas suivants apparaftra sur la 
 darnidra imaga da chaque microficha, salon la 
 cas: la symbols -^ signifie "A SUIVRE", la 
 symbols V signifia "FIN". 
 
 Las cartas, planchas, tableaux, ate, pauvent Atra 
 filmto A das taux da rAduction diffArents. 
 Lorsqua la document ast trop grand pour Atra 
 raproduit en un seul clichA, il est filmA A partir 
 de I'angia supArieur gauche, de gauche A droite, 
 et de haut an bas, en prenant la nombre 
 d'imagas nAcessaira. Las diagrammes suivants 
 illustrant la mAthoda. 
 
 ': 
 
 1 2 3 
 
 t 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
L 
 
 From the Comadian Journal for May, 1865. 
 
 P 
 
 EEMAEKS Oi^ PROFESSOR BOOLE'S MATHEMATICAL 
 THEORY OF THE LAWS OF THOUGHT. 
 
 BT GEOEGB PAXTON TjJ^UKa, M. A., 
 INBPBCTOB 07 OBAMMAB SCHOOLS POB UFPBB CAVAOA. 
 
 In a recent issue we announced the death of Professor George 
 Boole, of Queen's College, Cork, a man of varied and profound ac< 
 quirements, and of singular originality of mind. The work on vrhich 
 his fame will mainly rest is undoubtedly his ''Investigation of the 
 Laws of Thought, on which are founded the Mathematical Theories 
 of Logic and Probabilities." We have long purposed to call atten- 
 tion to this remarkable production, though various circumstances 
 have hitherto prevented us from doing so. The present seems a 
 suitable occasion for testifying our admiration of the genius of the 
 deceased philosopher, and, at the same time, endeavouring to give a 
 brief account, inadequate as it must necessarily be, of what may be 
 termed his Mathematico*logical speculations. 
 
 The primary, though not the exclusive, design of the '< Investiga- 
 tion," is to express in the symbolical language of a Calculus, the 
 fundamental Laws of Thought, and upon this foundation to establish 
 the science of Logic and construct its method. 
 
 The elemertary symbols of Professor Boole's Calculus are of three 
 kinds : 1st. Literal symbols, as x, y, &c., representing the objects of 
 our conceptions ; 2nd. Signs of operation, as +» — i X; and Srd, 
 
 11908 
 
2 
 
 PROFESSOR BOOLK S MATHEMATICAL THEORY 
 
 the sign of identity, =. The si^ + is used to express the mental 
 operation by which parts (of eztensiye quantity) are collected into a 
 whole. For instance) if x represent anitnah, and y ve^etdblet, x + y 
 will represent the class made up of animaU and vegetahlet together. 
 On the other hand) the sign — is used to express the mental operation 
 of separating a whole (of extensive quantity) into its parts. Thu8» 
 X representing human heinga, and g representing negroes, x — g will 
 represent all human beings except negroes. With regard to the sign 
 XfX X y or d; y (as it may be written) is used to denote those ob- 
 jects which belong at once to the class x and to the class y; just as, 
 in common language, the expression darhwaters denotes those objects 
 which are at once dark and waters. Hence we obtain a method of 
 representing^ a concept taken particularly. For, if x denote men, 
 then, since some men may be viewed as those who besides belonging 
 to the class 'x belong also to some other class v, some men will be 
 denoted by i) ik^ In general, 
 
 vx s= some x. (1) 
 
 It can easily be shown, that, as in Algebra, so in the logical sys- 
 tem which we are describing, the literal symbols, a?, y, &c., are com- 
 mutative ; that is, 
 
 o?y - y «; (2) 
 
 and that they are also distributive ; that is. 
 
 z (x+g)=s zx-^zg. 
 
 (3) 
 
 Another relation between Algebra and the Logical System under 
 eoniriddration is, that, in the latter as well as in the former, a literal 
 symbol may be transposed from one side of an equation to the other 
 by changing the sign of operation, + or — . But there is an im- 
 portant relation which subsists in the science of Thought, and not 
 generally in Algebra, namely, 
 
 x*=:x .;.... .(4) 
 
 That this is true in the Logical system, is plain ; for x*, which is 
 another form of x x, denotes (by definition) those things whifih belong 
 at once to the class x and to the class x; that is, it denotes simply 
 those things which belong to the class x; and it is therefore identi- 
 cal with X. But though the equation (4) does not generally subsist 
 in Algebra, it subsists when x is unity or zero. If, therefore, we 
 take the science of Algebra with the limitation that its unknown 
 
OP TBX LAWS or TBOU6RT. 
 
 # 
 
 the mental 
 cted into a 
 
 €8 together. 
 d operation 
 rts. Thus, 
 a? — y will 
 to the sign 
 e those ob- 
 y; jttst as, 
 lose objects 
 method of 
 enote men, 
 1 belonging 
 'Hen will be 
 
 logical sys- 
 2., are corn- 
 
 stem uttd^r 
 er, a literal 
 o the other 
 'e is an im- 
 
 [it, and not 
 
 e*, which is 
 hi&h belong 
 otes simplj 
 fore identi- 
 ■ally iubsist 
 ereforOi we 
 8 unknown 
 
 qnuititiOB can rwseive no tiiIiin distinct from unity and zero, the 
 analogy between the two soienees will still be preserved. 
 
 It is necessary to observe that unity and zero (1 and' 0) ore virtually 
 included by Professor Boole among 'his literal symbols. Of course 
 we con give 1 and any meaning we please, provided the meaning 
 once imposed on them be rigidly adhered to. By 0, then. Professor 
 Boole understands Nothing — a class (if the et^cpression may be per- 
 mitted) in which no object whatever is found. On the other hand, 
 by 1 he understands the universe of conceivable objects. Thus 1 
 and are at two opposite poles ; the former including every thing dn 
 its extension; the latter, nothing. The meaning which has been 
 affixed to 1 and preserves, in the Logical system as in Algebra, the 
 equations, 
 
 I X z = X, ) .gv 
 
 and,0 X X = 0} ) ^ ^ 
 
 for, the meaning of the former is, that objects which are common to 
 the univeme and to the class x are identical with those which con- 
 stitute the class x ; and the latter means, that there are no objects 
 which are common to a class in which nothing is found and to a 
 class X : both of which propositions are self-evident. From the 
 meaning affixed to 1, we see what the meaning of 1 — as must be. 
 In £Etct, X aHid 1 — a; are logical contradictories, the latter denoting 
 all conceivable objects except those which belong to the former ; so 
 
 that 
 
 1 — X = not X (6) 
 
 This value of the symbol 1 being admitted, we can, by the principles 
 
 of transposition and distribution [see (8)] reduce equation (4) to 
 
 the form, 
 
 x{l-'x)=0 (7) 
 
 The law here expressed, which is termed the Law of Duality, plays 
 a most important part in the development of logical functions, and 
 in the elimination of symbols. In fact, it may be described as the 
 germ out of which Professor Boole^s whole system is made to unfold 
 itself. 
 
 Having shown how concepts, whether taken universally or parti- 
 cularly, are represented, and also how the contradictory of a concept 
 is represented, we have next to notice the manner of expressing 
 judgments. All judgments are regarded by our author as affirm- 
 ative ; the negation, in those which are commonly called negative. 
 
4 PROFESSOR BOOLE 8 MATHEMATICAL THEORY 
 
 being attached by him to the predicate. But an affirmative judg- 
 ment is nothing else than an assertioui through immediate comparison, 
 of the identity of concepts. Suppose, therefore, that we are required 
 to express the judgment, " Some stones are precious." Let x denote 
 itone»; «ad ift preoiout. The proposition means, that some stones 
 are identical with some precious things. Consequently, its symbolical 
 expression [see (l)j js, 
 
 vx = vy. 
 
 If the judgment to be represented had been, " Some stones are not 
 precious," its expression would [see (6)] have been 
 
 vx =v (1 — y). 
 
 These examples in the meantime may suffice. More complicated 
 forms will present themselves afterwards. 
 
 With the few simple preliminary explanations which have been 
 given, and which were necessary to render intelligible some of the 
 criticisms presently to be offered, we are now prepared to state the 
 view which our author takes of the science of Logic. Logic he re- 
 gards as the science of Inference ; and the problem which it seeks to 
 solve is this : Given certain relations among any number of concepts 
 (Xf y, Zf &c.), it is required to find what inferences can be drawn regard, 
 ing any one of these or regarding a given function of any one of them. 
 A properly constructed science of Logic would require to solve this 
 problem adequately, and by a definite and invariable method. Now, 
 Professor Boole claims that the view which he presents of the prob- 
 lem which Logic has to solve, is both deeper and broader than that 
 commonly taken ; and he claims at the same time that he has devised 
 an adequate method, different from all existing methods, for solving 
 this problem, and that his method is one of definite and invariable 
 application. 
 
 The objections brought against the logic of the schools, that it is 
 neither sufficiently deep nor sufficiently broad, will probably take 
 our readers by surprise. It is not difficult to understand how a 
 question might be raised as to the practical utility of the scholastic 
 logic ; but most persons who have examined the subject will be ready 
 to admit, both that the scholastic logic is well founded, and that, 
 when properly developed from its first principles, it formo a complete 
 and perfect system. In the opinion of our author, however, it is so 
 defective in its foundation, and so incomplete in its superstructure, 
 as not to be entitled to the name of a science. " To what final con- 
 
 clusions 
 of the I 
 
OF THE LAWS OF THOUGHT. 
 
 lative judg- 
 iompariioxi, 
 re required 
 letx denote 
 lome stones 
 I symbolical 
 
 nes are not 
 
 iomplicated 
 
 haye been 
 ome of the 
 state the 
 iOgic he re- 
 it seeks to 
 >f concepts 
 iwn regard, 
 le of them. 
 solve this 
 lod. Now, 
 f the prob- 
 than that 
 bas devised 
 for solving 
 invariable 
 
 9, that it is 
 >babl7 take 
 ;and how a 
 scholastic 
 ill be ready 
 I, and that, 
 a complete 
 rer, it is so 
 rstructure, 
 b final con- 
 
 clusions," he say 8) " are we then led respecting the nature and extent 
 of the scholastic logic P I think to the following : that it is not a 
 science, but a collection of scientific truths, too incomplete to form 
 a system of themselves, and not sufficiently fundamental to serve as 
 the foundation upon which a perfect system may rest." 
 
 In order that it may be understood in what sense it is held that 
 the foundation of the scholoitio logic is defective, we make two other 
 quotations. " That which may be regarded as essential in the spirit 
 and procedure of the Aristotelian, and of all cognate systems of 
 logic, is the attempted classification of the allowable forms of infer- 
 ence, and the distinct reference of those forms, collectively or indi- 
 vidually, to some general principle of an axiomatic nature, such as 
 the Dictum of Aristotle." Again : " Aristotle's Dictum de omni 
 et nullo is a self-evident principle, but it is not found among those 
 ultimate laws of the reasonin^^ faculty to which all other laws, how- 
 ever plain and self-evident, admit of being traced, and from which 
 they may in strictest order of scientific evolution be deduced. For 
 though of every science the fundamental truths are usually the most 
 simple of apprehension, yet is not that simplicity the criterion by 
 which their title to be regarded as fundamental must be judged. 
 This must be sought for in the nature and extent of the structure 
 which they are capable of supporting. Taking this view, Leibnitz 
 appears to me to have judged correctly when he assigned to the 
 principle of contradiction a fundamental place in logic ; for we have 
 seen the consequences of that law of thought of which it is the 
 axiomatic expression." The sum of what is contained in these pas- 
 sages, in so far as they bear on the point before us, is, 1st, That the 
 foundation of thn Aristotelian, and of all cognate systems of logic, is 
 some such canon as the Dictum ; 2nd, That that canon, and other 
 maxims of a like description, though self-evident, are not deep 
 enough to serve as a basis for a science of logic in which all the 
 forms of thought are to be exhibited ; and, 3rd, That the only prin- 
 ciple sufficiently fundamental to form the basis of a complete science 
 of logic is the principle of contradiction. Now what is the real 
 state of the case P Nothing is more certain than that the Dictum 
 was not considered by Aristotle as either the exclusive or the ulti- 
 mate foundation of his logical system. Not the exclusive foundation j 
 for, as a matter of fact, many of the forms of thought embraced in 
 the Aristotelian logic receive no direct warrant from the Dictum, 
 
PROriMdR BOOLs'l MATHBMATICAI. THEORY 
 
 hat urn. 1m dMriv^d Ijnnb ifc only by tke aid of tb« principle of oontani- 
 tUotion. Not the ultimate fbimdation ; for what ia the Diotum, but 
 apartioiilar oaie of a more eomprehenBire, and (in thia senae) mora 
 fundameiital, law P Ariatotle aaw tbia, and has ezpreased it aa elear^ 
 as any man that oyer Uved. " It ia stanifest,** he iays» '< that w> on* 
 can conceive to himself that the same thing can at once be and not 
 be, for thus he would hold repugnant opinions, and subvert the 
 reaUty of truth. Wherefore, all who attempt to demonatrate, reduce 
 everything to thia aa the ultimate doctrine ; for this is by nature 
 the principle of all otlier axioms." 
 
 Professor Boole's acceptance of the Leibnitzian maxim (though it 
 was much older than Leibnitz) that the true foundation of the sci- 
 ence of logic is the principle of contradiction, has the appearance of 
 being at variance with some extraordinary statementa which he else- 
 where makesi to the effect that the principle of contradiction is a 
 consequence of the law of duality. We may remind our readera 
 that the law of duality [see (4) and (7)] is substantially the prin* 
 ciple out of which all the details of Professor Boole's own doctrine 
 are evolved. Now, under the influence of what was, perhaps, uot 
 an unnatural desire to vindicate for bis system a peculiar depth of 
 foundation. Professor Boole has been betrayed into observations by 
 which his fame as a philosophic thinker must be seriously affected. 
 For instance : " that axiom of metaphysicians which is termed the 
 principle of contradiction, and which affirms that it is impossible for 
 any being to possess a quality and at the same time not to possess it, 
 is a consequence of the fundamental law of thought, whose expres- 
 sion is «* = w" And again : " the above interpretation has been 
 introduced, not on account of its immediate vdue in the present 
 system, but as an illustration of a significant fact in the philosophy 
 of the intellectual powers, viz., that what has commonly been re- 
 garded as the fundamental axiom of metaphysics is but the conse- 
 quence of a law of thought) mathematical in its form." In thus 
 speaking of the principle of contradiction as a consequence of the 
 law of duality, Professor Boole seems to take away the fundamental 
 character of the principle of contradiction ; for» if that principle be, 
 in the proper sense of the term, a consequence of something else» it 
 cannot be itself truly fundamental. Yet, as we have seen. Professor' 
 Boole admits that it is the real and deepest foundation of the science 
 of logic* What, then, does he mean ? On the one hand, he cer< 
 
of eentam- 
 
 ictoniy but 
 
 nBe)]nofo 
 
 M dearly 
 
 Ult IM OS* 
 
 )e and not 
 ibvert thia 
 ite, nduce 
 by nature 
 
 [though ifc 
 )f the 8Qi« 
 earanee of 
 9h he else* 
 ictioii if a 
 ur readers 
 r the prin- 
 u doctrine 
 rhaps, not 
 ir depth of 
 vations by 
 f affected, 
 termed the 
 oBsible for 
 posiess it, 
 so ezpres- 
 has been 
 le present 
 shilosopbj 
 r been re- 
 bhe conse- 
 In thus 
 ace of the 
 ndamental 
 iuciple be, 
 Qg else, it 
 Professor' 
 be science 
 id, he cer< 
 
 
 or TBK LAWS or TBOUOBT. I 
 
 tainly does not intend to deny that the principle of contradiction is 
 self-evident. On the other hand> it is plain that he L^ea hold that 
 the principle of contradiction can be deduced from the lav of duality. 
 But (we ask) how? Can the principle of contradiction be deduced 
 from the law of duality* without our assuming the principle of con- 
 tradiction itself as the basis of the deduction? This would be 
 absurd ; for a conclusion can be established in no other way than by 
 pointing out that the supposition of its being ialse involres a contra- 
 diction. In the particular caae before us» the equation « (1 — «) = 0, 
 which is that expression of the law of duality in which the principle 
 of contradiction is regarded as being brought to light, is only reached 
 by a process of reasoning, every step of which takes the principle of 
 contradiction for granted. The only interpretation, therefore, which 
 Professor Boole's words can bear, unless we give them a meaning 
 palpably absurd, is, that a formula, which we are enabled to state by 
 assuming the law of contradiction, contains a symbolic representa- 
 tion of that law. This hardly seems to us a very significant &ct in 
 the philosophy of the intellectual powers. If indeed the formula in 
 question could be shown to represent some law of thought of wider 
 application than the law of contradiction, that would be a very sig- 
 nificant fact. But such is not the case. The equation « (1 — <v) == 
 is just the law of contradiction symbolic^y expressed : neither n^oro 
 nor less. 
 
 The Aristotelian logic is charged with being incomplete^ as well as 
 with being not suiEciently fuudaiMc:.tML By this our author does 
 not mean that Aristotle and his followers have casually omitted some 
 forms of thought which their system ought to have embraced : bad 
 they done so, the fault would have been chargeable — not upon the 
 system, but upon its expounders ; but he means, that, from the very 
 nature of the system, there is an indefiuite variety of problems 
 belonging to the science of inference, which their system is incapaUe 
 of solving, or for the solution of which at all events it furnishes no 
 definite and certain method. 
 
 It will be observed that there are two questions here, which, as 
 radically distinct from one another, require to be considered sepa- 
 rately : the one being, whether the Aristotelian logic is capable of 
 solving all the problems belonging to the science of inference ; and 
 the other, whether it furnishes a definite and certain method for the 
 solution of these. 
 
8 
 
 PROFESSOR BOOLK 8 MATHEMATICAL THEORY 
 
 The former of these questions may, with perfect confidence, be 
 answered in the affirmative. It admits of absolute demonstration, that 
 there is no chain of valid inference which the ordinary logic is incompe- 
 tent to express, or, in other words, which is not reducible to conversion 
 or syllogism. Some logicians have been of opinion that conversion 
 is nothing else than syllogism at bottom ; but, for what we have at 
 present in view, it is unnecessary to discuss this question. Suffice 
 it to say, that, whether conversion and syllogism be substantially 
 identical or not, all immediate inference is of the nature of conver- 
 sion, and all mediate inference (or reasoning proper) of the nature of 
 syllogism. Does Professor Boole deny this ? Formally, and in plain 
 terms. " Possibly," he writes, ** it may here be said that the logic 
 of Aristotle, in its rules of syllogism and conversion, sets forth the 
 elementary processes of which all reasoning consists, and that beyond 
 these . there is neither scope nor occasion for a general method. I 
 have no desire to point out the defects of the common logic, nor do I 
 wish to refer to it any further than is necessary, in order to place in 
 its true light the nature of the present treatise. With this end alone 
 in view, I would remark : Ist. That syllogism, conversion, &c., are 
 not the ultimate processes of logic. It will be shown in this treatise 
 that they are founded upon, and are resolvable into, ulterior and 
 more simple processes which constitute the real elements of method 
 in logic. Nor is it true that all inference is reducible to the partic- 
 ular forms of syllogism and conversion. 2nd. If all inference were 
 reducible to these processes alone (and it has been maintained that 
 it is reducible to syllogism alone), there would still exist, &c." In 
 illustration of the statement, that some inference is not reducible to 
 the forms of syllogism and conversion, Professor Boole examines the 
 case of conversion, and arrives at the result that " conversion is a 
 particular application of a much more general process in logic, of 
 which," he adds, " many examples have been given in this work." 
 In like manner he examines the case of syllogism ; and his conclusion 
 is as follows : " Here, then, we have the means of definitely resolv- 
 ing the question, whether syllogism is indeed the fundamental type 
 of reasoning, — whether the study of its laws is co-extensive with the 
 study of deductive logic. For if it be so, some indication oi the fact 
 must be given in the system of equations upon the analysis of which 
 we have been engaged. No sign, however, appears that the discus- 
 sion of all systems of equations expressing propositions is involved in 
 
 that of 1 
 
 vrriterso 
 
 merely 
 
 tic inferc 
 
 version a 
 
 piave of c 
 
 more gei 
 
 If reduc 
 
 we take 
 
 ing, whi 
 
 an instai 
 
 are prep 
 
 his "mo 
 
 Nay, we 
 
 lutely d 
 
 cannot 1 
 
 To mi 
 
 argumei 
 
 .is an ii 
 
 proposit 
 
 exactly 
 
 of termi 
 
 two. 1 
 
 courage 
 
 Boole's 
 
 mind, ai 
 
 jmd heri 
 
 in the I 
 
 . <M0 eelj 
 
 but sin 
 
 fraetise 
 
 eompar: 
 
 of two ( 
 
 a plural 
 
 eompar 
 
 ment e: 
 
 must I 
 
 immedi 
 
 compai 
 
ifidenoe, be 
 kratioD, that 
 is incompc- 
 > conversion 
 i conversion 
 ; we have at 
 on. Suffice 
 ubstantially 
 ) of conver- 
 le nature of 
 and in plain 
 at the logic 
 ts forth the 
 that beyond 
 method. I 
 pc, nor do I 
 ' to place in 
 is end alone 
 on, &o., are 
 bhis treatise 
 ilterior and 
 I of method 
 the partic- 
 rence were 
 itained that 
 ;, &C." In 
 educible to 
 amines the 
 'ersion is a 
 in logic, of 
 this work." 
 conclusion 
 bely resolv- 
 lental type 
 re with the 
 ot the fact 
 is of which 
 the discus- 
 involved in 
 
 OP THE LAWS or TBOVGHT. 9 
 
 that of the particular system examined in this chapter. • And yet 
 writers on logic have been all but unanimous in their assertion, not 
 merely of the supremacy, but of the universal sufficiency of syllogis- 
 tic inference in deductive reasoning.'* These statements, that con- 
 version and syllogism are branches of a much more general process, 
 have of course no meaning except on the supposition that the "much 
 more general process" is not reducible to conversion and syllogism, 
 f reducible to these, it would not be a more general process. Now 
 e take our stand firmly on the position, that a chain of valid reason- 
 g, which cannot be broken into parts, every one of which shall be 
 instance either of conversion or of syllogism, is not possible. We 
 are prepared to show this in the case of every one of the examples of 
 his "more general process" which Professor Boole gives in his work. 
 Nay, we go farther, and as was intimated above, hold it to be abso- 
 lutely demonstrable, that, from the nature of the case, inference 
 cannot be of any other description than conversion or syllogism. 
 
 To make this out, let it be remarked that the conclusion of an 
 argument exhibits a relation between two terms, say JTand T. It 
 . is an important assumption in Professor Boole's doctrine, that a 
 proposition may exhibit a relation between many terms. This is not 
 exactly true. A proposition may involve a relation between a variety 
 of terms implicitly ; but explicitly exhibits a relation only between 
 two. Take, for instance, the proposition — " Men who do not possess 
 courage and practise self-denial are not heroes." Here, on Professor 
 Boole's method, a variety of concepts are supposed to be before the 
 mind, as, men, those who practise self-denial, those who possess cowage, 
 and heroes. But in reality, when we form the judgment expressed 
 in the proposition given, the separate concepts, men, those who prac' 
 Use self-denial, those who possess courage, are not before the mind ; 
 but simply the two concepts, men who do not possess courage and 
 fraetise self-denial, and heroes. What is a judgment but an act of 
 comparison? And the comparison is essentially a comparison 
 of two concepts, each of which may no doubt involve in its expression 
 a plurality of concepts, but these necessarily bound together by the 
 comparing mind into a unity. Now, if the conclusion of an argu- 
 ment exhibits a relation between two terms ^and Y, this conclusion 
 must be drawn (what other way is possible?) either through an 
 immediate comparison of X and J' with one another, or by a mediate 
 comparison of them through something else. If it be drawn by an 
 
 •^^ffmmmm 
 
m 
 't 
 
 m 
 
 10 
 
 PROFESSOR l^OpLK 8 MATHEMATICAL THEORY 
 
 immediate comparison of JTand T, then uo concepts enter into the 
 argument ezoei^t Xand Tt and the argument is reduced to ooavar- 
 sion. Bat if the conclusion be drawn mediately, it must he by the 
 comparison of Jfand Ywith sonie third thing: not with a plurality 
 of other things* but with some single thing. Here we have the mind 
 drawing its inference in a syllogism. What the various admissible 
 forma of conversion and syllogism may be, or whether these forms 
 have been correctly specified by particular eminent logieiaos* are 
 minor questions. The essential thing in a philosophical respect is* 
 that the mind, in the inferences which it dra-vs, does and can work 
 in no other moulds than those described. All this seems to us so 
 plain that we confess ourselves utterly puzzled to comprehend how 
 men of profound and original genius have been beguiled into an 
 assertion of the contrary. 
 
 Professor Boole himself* in summing up his assault on the Aristo- 
 telian Logic, comes very near admitting "vhat we contend for. " As 
 Syllogism*" he says, " is a species of elimination* the question before 
 us manifestly resolves itself into the two following ones: 1st. 
 Whether all elimination is reducible to Syllogism ; 2nd. Whether 
 deductive reasoning can, with propriety, be regarded as consisting 
 only of elimination. I believe, upon careful examination, the true 
 answer to the former question to be, that it is always theoreticaUy 
 possible so to resolve and combine propositions that elimination may 
 subsequently be effected by the syllogistic canons, but that the pro» 
 cess of reduction would in many instances be ccmstrained and unna- 
 tural* and would involve operations which are not syllogistic. To 
 the second question I reply, that reasoning cannot* except by an 
 arbitrary restriction of its meaning, be confined to the process of 
 elimination.*' With regard to this second question* we merely note 
 in passing* that we have proved in the preceding paragraph that in- 
 ference, where not immediate or of the nature of conversion, can be 
 nothing else than elimination. It is* however* with the first ques- 
 tion* whether elimination is reducible to syllogism, that we have now 
 more particularly to do ; and we accept with satisfaction the admis- 
 sion* guarded and (to some extent) neutralised as it is* that every line 
 of argument may be throTm into a form in whieh the eliminations that 
 take place are effected by the syllogistic canons. It is quite irrele- 
 vant to notice* as Professor Boole does* that the process of reduction 
 would, in many instances, be constrained and unnatural ; for we are 
 
 not here i 
 eharge* tl 
 «re not s^ 
 Vf^ the " E 
 4athor h 
 ^egrouE 
 H^e prodi: 
 Sliced to 
 &back 
 ||lte impc 
 ion or 8] 
 
 In stat 
 against i 
 i}iat* froi 
 #y of pi 
 Klfttem. i 
 fvents* i1 
 trust* ful 
 of infere 
 But Pro 
 4ucible 
 lamenec 
 |p deteri 
 IP well a 
 lAiould bi 
 which, ii 
 qf the pi 
 ]|iany d 
 IM^tical 
 lo^own, 
 iipd disti 
 fiance* t 
 aasign tl 
 their 8U( 
 complex 
 form a I 
 probabil 
 can sup] 
 Now, 
 
 r 
 
OF THE LAWS OF THOUGHT. 
 
 tl 
 
 iter into the 
 4 to ooBvar- 
 ist be by the 
 h ft plurality 
 ave the mind 
 18 ftdmissihie 
 the^e fpnns 
 ogieianft are 
 ftl respept is« 
 nd can work 
 ems to U9 so 
 >rehend how 
 iled into an 
 
 I the Aristo- 
 dfor. <*Ab 
 istion before 
 
 ones: lit. 
 id. Whether 
 18 oonsiBting 
 on, thetrae 
 theoretically 
 lination may 
 that the pro* 
 \A and unna- 
 iogistic. To 
 izcept by an 
 le process of 
 merely note 
 raph that in- 
 rsion, can be 
 >e first ques- 
 ve have now 
 I the admis- 
 fit eyery line 
 [nations that 
 quite irrele- 
 of reduction 
 
 for we are 
 
 not here in the province of Bhetoric*. Much more to the purposie is the 
 ebarge, that the process of reduction would inrolve operations which 
 ■ mte not syllogistic. The operations referred^ to are those embraced 
 in the " much more general process " in which, as we have seen, our 
 Anther holds conversion and syllogism to be contained. Of course, 
 %e ground which we take in reply is, on the one hand, to challenge 
 ^e production of an instance of valid inference, which cannot be re- 
 Sliced to either conversion or syllogism ; and on the other hand, to 
 ihll back upon the demonstration which we have given of the abso- 
 lute impossibility of valid inference being anything else than conver- 
 lion or syllogism. 
 
 In stating the charge of incompleteness brought by our Author 
 Ugainst the Aristotelian system* we explained his meaning to be, 
 ||iat, from the very nature of the system, there is an indefinite vari- 
 ity of problems belonging to the science of inference, which the 
 igfstem is incapable of solving, or for the solution of which, at all 
 ^ents, it furnishes no definite and certain method. We have, we 
 trust, fully refuted the opinion that there are problems in the science 
 of inference which the Aristotelian logic is incapable of solving. 
 But Professor Boole urges, that, even if all inference were re- 
 4ucible to conversion and syllogism, "there would still exist the 
 imne necessity for a general method. For it would still be requisite 
 |p determine in what order the processes should succeed each other, 
 IP well as their particular nature, in order that the desired relation 
 diould be obtained. By the desired relation I mean that full relation 
 which, in virtue of the premises, connects any elements selected out 
 gf the premises at will, and which, moreover, expresses that relation 
 ip any desired form and order. If we may judge from the mathe- 
 Ipatical sciences, which are the most perfect examples of method 
 Ijaown, this directive function of method constitutes its chief office 
 ipd distinction. The fundamental processes of arithmetic, for in- 
 stance, are in themselves but the elements of a possible science. To 
 ipsign their nature is the first business of its method, but to arrange 
 their succession is its subsequent and higher function. In the more 
 complex examples of logical deduction, and especially in those which 
 form a basis for the solution of difficult questions in the theory of 
 probabilities, the aid of a directive method, such as a Calculus alone 
 can supply, is indispensable.*' 
 
 Now, we at once admit that the Aristotelian logic neither has, nor 
 

 ■ 
 
 
 
 * 
 
 
 • 
 
 
 
 12 
 
 PROFBSSOR BOOLE S MATHEMATICAL THEORY 
 
 ■'I 
 
 H 
 
 professes to have, any sucli method as that here described. But can 
 it justly, on that account, be charged with incompleteness? A 
 science must not, because it does not teach everything, be therefore 
 reckoned incomplete : enough, if it teaches the whole of its own 
 proper circle of truths. The special question which the scholastic 
 logic proposes to itself is: what are the ultimate abstract forms 
 according to which all the exercises of the discursive faculty pro- 
 ceed ? The science is complete, because it furnishes a perfect answer 
 to this question. 
 
 But, it may be said, is it not desirable to have a method enabling 
 us certainly to determine, in every case, the relation which any of 
 the concepts explicitly or implicitly entering into a group of premi- 
 ses bear to the others ? Most desirable. And herein consists the 
 real value of Professor Boole's labours. He has devised a brilliantly 
 original Calculus by which he can, through processes as definite as 
 those which the Algebraist applies to a system of equations, solve 
 the most complicated problems in the science of inference — problems 
 which, without the aid of some such Calculus, persons most thoroughly 
 versed in the ordinary logic might have no idea how to treat. In 
 expressing our dissent, as we have been obliged very strongly to do, 
 from much that is contained in Professor Boole's treatise, we have 
 no desire to rob that eminent writer of the credit justly belonging 
 to him. Our wish has been simply to separate the chaif from the 
 wheat, and to point out accurately what constitutes, as far as the 
 *' Investigation " is concerned. Professor Boole's claim to renown. 
 
 Our readers will, however, be now anxious to obtain some fuller 
 information regarding the method about which so much has been 
 said, and which is the same with " the more general process " under 
 which the processes of the scholastic logic are held by Professor Boole 
 to be comprehended. This part of our article must necessarUy be 
 altogether technical ; and we shall require to ask our readers to take 
 a few things on trust ; but we hope to be able to present the sub- 
 ject in such a manner as to give at least some idea of the system 
 we are to endeavour to describe. Those who desire to become 
 thoroughly acquainted with it will of course study the " Investiga- 
 tion " for themselves. 
 
 We begin by referring to the development of logical functions. 
 An expression which in any manner involves the concept a?, is called 
 a function of the concept, and is written/ (a?). Now there is one 
 
 I 
 
RY 
 
 OF THE LAVrS OF THOUGHT. 
 
 13 
 
 )ed« But can 
 leteness ? A 
 ^t be therefore 
 ►le of its own 
 the scholastic 
 ibstract forms 
 e faculty pro- 
 }erfect answer 
 
 ithod enabling 
 which any of 
 •oup of premi- 
 :n consists the 
 id a brilliantly 
 as definite as 
 [uations» solve 
 Lce — problems 
 )st thoroughly 
 to treat. In 
 trongly to do, 
 itise, we have 
 itly belonging 
 ;haif from the 
 as far as the 
 to renown. 
 in some fuller 
 luch has been 
 •ocess" under 
 rofessor Boole 
 necessarily be 
 waders to take 
 ;sent the sub- 
 of the system 
 :e to become 
 Investiga- 
 
 te 
 
 ;al functions, 
 pt CC) is called 
 7 there is one 
 
 standard form to which functions of every kind maybe reduced. 
 This form is not an arbitrary one, but is determined by the circumi- 
 •tance that every conceivable object must rank under one or other 
 of the two contradictory classes x and 1 — x. Hence every con- 
 ceivable object is included in the expression, 
 
 h ua; + » (1 — a?); (8) 
 
 f>per values being given to u and v. For, if a given concept belong 
 the class Xt then, by making v = 0, the expression (8) becomes ux, 
 ,#hich, by (1), means some x ; and if the given concept belong to the 
 rgiass 1 — X, then, by making tt = 0, the expression (8) becomes 
 f (1 — x), which, by (1) and (6), means some not x. Therefore, 
 jf (x) being any concept depending on x, we may put 
 
 f{x) = Ma? + t> (1 — x) (9) 
 
 It has been shown that one of the coefficients, u, o, must al- 
 %ays be zero ; but the forms of these coefficients may be determined 
 more definitely. Por, by making a? = in (9), the result is v =/ (0) ; 
 Indby making a? = 1, there results m =/(!); by substituting which 
 Values of u and v in (9), we get 
 
 /Or) =/(l)a; +/(0)(l-x) (10) 
 
 This is tl)e expansion or development of the function x. The ex- 
 pressions x,l — X, are called the constituents of the expansion ; 
 IUid/(l) and/(0) are termed the coefficients. The same phrase- 
 ology is employed when a function of two or more symbols is de- 
 veloped. 
 
 ^ Any one in the least degree acquainted with mathematical processes 
 jrill understand how the development of functions of two or more 
 itymbols can be derived from equation (10). In fact, by (10), we 
 liave 
 
 /(25, y) =/(i. y) '^ +/(0, y) (1 - a:). 
 
 But again, by (10), 
 
 /(i,y)=/(M)y + /'(i,o)(i -y), 
 and 
 
 /(0,y)=/(0, l)y+/(0,0)(l-y). 
 "/(*.y)=/(l. l):ry +/(l,0)a?(l - y) 
 
 +/(0.1)y(l~a?)+/(0,0)(l-a;)(l-y) (n) 
 
 The development of a function of three symbols may be written 
 down, as we shall have occasion in the sequel to refer to it : 
 
 i 
 
m 
 
 PROFESSOR Boole's hatbematical theory 
 
 'igt 
 
 + /(l,0,l)«xr(l-y)+/(l,0,0)a:(l-y)(l-*y 
 +/(0. 1, 1) y« (I - «) +/(0, 1, 0) y (1 - a?) (1 - ty 
 H-/(0,0, 1)»(1 -»)(! -y) 
 + /(0.0,0) (1 -«) (1 -y)(l -^) (12) 
 
 As the object of the expansion of logical symbols may not be eyi* 
 dent at first sight, and as the process may consequently be regarded 
 by some as barbarous, we may observe that not only is there a defi- 
 nite aim in the development, but the thing aimed flit, has, in our 
 opinion, been most felicitously accomplished. Of this our readers 
 will probably be satisfied when they are introduced to some sped' 
 mens of the use which is made of the formul® obtained ; in ibe 
 meantime it may throw some light on the character of these formula 
 if we notice that the constituents of an expansion represent the 
 several exclusive divisions of what our author terms the universe of 
 discourse, formed by the predication and denial in every possible 
 way of the qualities denoted by the literal symbols. In the simplest 
 case, that in which the function is one of a single concept, it will be 
 teen by a glance at (10) that there are only two such possible ways, 
 g and 1 — X. In the case of a function of two symbols, there are 
 [see (11)] four such ways, ay, x (1 — y), y (1 — «), (1 — as) (1 — y). 
 In a function of three symbols there are eight such ways ; and so on. 
 A development in which the constituents are of this kind prepares 
 the way for ascertaining all the possible conclusions, in the way 
 either of affirmation or denial, that can be deduced, regarding any 
 concept, from any given relations between it and the other concepts. 
 
 If S be the sum of the constituents of an expansion, and P the 
 product of any two of them, then 
 
 S=h .,(13) 
 
 andP = 0. (14) 
 
 The truth of these beautiful and important propositions will easily 
 be gathered by an intelligent reader from an inspection of the for- 
 mulae, (10), (11), (12). Another important proposition is involved 
 in (14), namely, that, if /(x) = 0, either the constituent or the co- 
 efficient in every term of the expansion of/ («) must be zero. For, 
 let 
 
 /{x) = Q + AX+A^X^ + +JnXn; 
 
 where J, A^ , &c., are the coefficients which are not zero, their corre»> 
 ponding constituents being X, Xi, ^c. ; while Q represents the sum 
 
 of those 
 Vor, sin 
 
 lut, by 
 
 ut A i 
 V These 
 lably n( 
 fhem su 
 iiedure ^ 
 
 Ouri 
 ii given 
 fbew th 
 
 .Then, t 
 «^r, by t 
 ^hia pr 
 
 ^ce sb 
 
 fuence, 
 fow, t 
 Jliea, V 
 
 !• (16) 
 then, 
 
 iind so 
 
IRY 
 
 OF THB LAWS OF TBOVGHT^ 
 
 15 
 
 
 
 -y)0~*> 
 
 (12) 
 
 lay not be eyi* 
 J be reggrded 
 is there a defi- 
 it, has, in our 
 is our readers 
 ;o some speci> 
 ained; in ibe 
 these formula 
 represent the 
 he universe of 
 9vei7 possible 
 n the simplest 
 lept, it will be 
 )ossible ways, 
 ols, there are 
 - a;) (1 - y). 
 ^ ; and so on. 
 kind prepares 
 s, in the way 
 regarding any 
 l;her concepts, 
 n, and P the 
 
 
 
 
 
 ms will easily 
 
 on of the for- 
 
 )n is inyolred 
 
 ent or the co- 
 
 e zero. For, 
 
 >y their corre»> 
 lents the sum 
 
 of those terms in which the coe£E[cients are zero. Then we say that 
 
 X=0 (16) 
 
 Vor, since Q = 0, and f(x) is supposed to yanish, 
 - AX + A^ X^ + &e. =0 
 
 :. AX^ + A^ XX^ + &c. = 
 
 |ut, by (14), XXi = ATXg = = JT^ = 0. Therefore 
 
 "^ AX^ =0. 
 
 llut A is not zero. Therefore X must be zero. 
 These principles having been laid down, our best course will pro- 
 ,bly now be to take a few exampV^s, and to offer in connection with 
 em such explanations as may seem necessary of the mode of pro- 
 cedure which they are intended to illustrate. 
 
 Our first example shall be one in which but a single proposition 
 ia given : " clean beasts are , those which both divide the hoof and 
 fbew the cud." Let 
 
 2 = clean beasts, 
 y = beasts dividing the hoof, 
 z — beasts chewing the cud. 
 Then, the given proposition, symbolically expressed, is, 
 
 X =1 y z, 
 f^T, by transposition, 
 
 w x--yz=zO (16). 
 
 Jf his premiss contains a relation between three concepts ; and, ac> 
 ^rding to Professor Boole, a properly constructed science of infer- 
 llnce should enable us, by some defined process, to show what conao' 
 
 fuence, as respects any one of these, follows from the premiss. 
 I'ow, the definite and invariable process which Professor Boole ap> 
 lilies, with the design which has been indicated, to an equation such 
 |0 (16), is to develop the first member of the equation. Writings 
 ♦hen, 
 
 f\x, y, 2) = a? — y 4f, 
 
 we have, /(1, 1, 1) = 0, 
 /(0,0,0) = 0, 
 tmd so on. Hence [see (L2)] the developement required is 
 « — ya=:a;y(l— r) + a;ar(l-.y) 
 
 + a;(l-y)(l - «)-yz(l -a;) 
 + a; y « + y (I - «) (1 - 2r) 
 H-0 2(l-a5)(l-y) 
 + 0(1-^) (l-y)(l-«). 
 
16 
 
 PROFESSOR BOOLE 8 MATHEMATICAL THEORY 
 
 (17) 
 
 Therefore, by (16), 
 
 «y (l-ar) + aj«(l-y) + « (1-y) (l-a)-y» (l-ar)=0: 
 4uui therefore, by (15), 
 
 a; y (1 — «) = 0. 
 
 03 2r (1 — y) = 0, 
 
 X (I - y) (1 - «) = 0, 
 
 y « (1 — a?) = 0. J 
 
 Still farther, since, by (13), the sum of the constitutents of an ex- 
 pansion is unity ; and since four of the constituents in the expan- 
 sion otx — y z have been shewn to be zero ; it follows that the sum 
 of the remaining constituents in the expansion of or — y ^ is unity. 
 That is, 
 
 a; y 2? + y (1 - cc) (1 - 2r) + a (1 - a;) (1 -. y) 
 
 + (i-«')(i-y)(i-^) = i (18) 
 
 It is obvious that this method can be applied in every case. To 
 what then does it lead ? First of all, in the group of equations (17), 
 we have brought before us all the different classes (if the expression 
 may be permitted) to which the given proposition warrants us in 
 saying that nothing can belong ; and next, in equation (18) we have 
 brought before us those different classes to one or other of which 
 the given proposition warrants us in asserting that everything must 
 belong. For instance, the first of equations (17) denies the exis- 
 tence of beasts which are clean (jc) and divide the hoof (y) but do 
 not «hew the cud (1 — z); the second denies the existence of beasts 
 which are clean (x) and chew the cud («) but do not divide the 
 hoof (1 — y) i and so on. Equation (18), again, informs us that 
 the universe, which is represented by 1, is made up of four classes, 
 in one or other of which therefore every thing must rank ; the first 
 denoted hj xy z, the second by y (1 — x) (1 — a-) ; and so on. As 
 an example of the interpretation of the expressions by which these 
 classes are denoted, we may take the last, (1— a;)(l— y)(l — z). 
 This represents things which are neither clean beasts, nor beasts 
 chevnng the cud, nor beasts dividing the hoof. 
 
 By the method en^loyed, we have been able to indicate certain 
 classes which do not exist, and also to indicate certain classes in one 
 or other of which every thing existing is found. But this, it may be 
 said, is not a solution of the most general problem of inference. 
 The most general problem is : to express (speaking mathematically) 
 any one of the symbols entering into the given premiss, or any func' 
 
 But tho 
 multipli 
 belegit 
 oifects 
 
 those CO 
 
 ii does ] 
 
 H«Qce( 
 froin(l< 
 
 that CO 
 
 aarived 
 
 tHieexp 
 
 ment. 
 t0 2;, w( 
 
 Here v 
 yet bee 
 Algebx 
 logical 
 and, aE 
 
EORY 
 
 OF THE LAWS OF THOUGHT. 
 
 17 
 
 «(1— af)=0: 
 
 .. (17) 
 
 t 
 
 utents of an ex- 
 its in the expan- 
 ws that the sum 
 ^ — 1/ sis unity. 
 
 -y) 
 
 . (18) 
 
 svery case. To 
 equations (17), 
 f the expression 
 warrants us in 
 m (18) we have 
 other of which 
 iverything must 
 lenies the exis- 
 hoof (y) but do ^ 
 itenco of beasts 
 not divide the 
 iforms us that 
 of four classes, 
 rank; the first 
 and so on. As 
 by which these 
 
 - y) (1 - z). 
 its, nor beasts 
 
 adicate certain 
 classes in one 
 
 this, it may be 
 of inference. 
 
 athematically) 
 
 h or any func- 
 
 liou thereof, as an explicit function of the others. To the problem 
 •■put even thus in its widest generality. Professor Boole's processes 
 eitend. It would make our article too lengthened were wo to go 
 iii|0 minute details ; but we must endeavour to give some idea of the 
 oonrse here followed, as it both is extremely interesting as a matter 
 of pure speculation, and forms an important part of the system under 
 OMisideration. 
 
 Take the equation in (16), a; — y 2; = ; and, as a simple instance 
 will serve the purpose of illustration as well as a complicated one, 
 let the inquiry be: how can c be expressed in terms of x and y/ 
 Inordinary Algebra we should have 
 
 
 y 
 
 .(19) 
 
 Bnt though both sides of an equation may, in Logic as in Algebra, be 
 multiplied (so to speak) by the same quantity, they cannot, in Logic, 
 Im legitimately divided by the same quantity. For instance, let the 
 dlfscts common to the class X and to the class U be identical with 
 tilose common to the class Y and to the class U; in other words, lei 
 
 if does not follow that Xis identical with F, or symbolically, that 
 
 X=T. 
 
 Hiuce equation (19) could not, in Logic, be legitimately deduced 
 frqm (16), even if y were an explicit factor of x. But still further, 
 
 Ullen X has not y as one of its factors, the expression - is not, in the 
 
 tl|;ical system, interpretable. Nevertheless, Professor Boole shows 
 tliftt conclusions both interpretable and correct will ultvmalely be 
 ti^ved at, if the value of z be deduced Algebraically, as in (19), and 
 
 iitt expression - be then, as a logical function, subjected to develop- 
 
 ■ - „!. J 
 
 BOBnt. 
 
 X 
 
 Now, if - be developed by (11), and the expansion equated 
 
 to Zi we get 
 
 zz=.xy + ^aj(l-y) + 0(l-a^)y + £(l-a:)(l-y) (20) 
 
 Here we have two symbols, % and -J, the meaning of which has not 
 yet been determined. Our author shows that the former, which in 
 Algebra denotes an indefinite numerical quantity, denotes in the 
 logical system an indefinite class. In Algebra \ denotes infinity ; 
 and, as is well known, when it occurs as the co-efficient in a term in 
 
18 
 
 PROFESSOR Boole's mathematical theory 
 
 an equation all of whose other terms are finite, this indicates that 
 
 the quantityvf which it is the co-e£Scient is zero. So, in the logical 
 
 system, if, in any term of an equation obtained in the manner in 
 
 which equation (20) has been obtained, the co-efficient be i, the 
 
 corresponding constituent must be 0* These are certainly very 
 
 remarkable analogies. But let us see what follows. We have first, 
 
 from (20), 
 
 x{l-if) = 0. 
 
 Hence as the equation (20) describes the separate olasaes of which 
 a consists, and as there is no such class as 2 (1 — ^) in existence, 
 the second term on the right hand side of equation (20) may be 
 rejected. The third term also may be omitted, its co-efficient being 
 zero. This reduces the equation to the form> 
 
 « = « y + § (1 — a) (1 —y) : 
 which means, that beasts which chew the cud consist of the class xy, 
 together with an indefinite remainder of beasts common to the classes 
 1 — X and 1 — y. 
 
 Before leaving the subject of inference from a single premiss, we 
 must say a few words regarding elimination ; for though, in Algebra, 
 elimination is possible only when two or more equations are given, 
 Professor Boole, shows that, in Logic, a class symbol may be elimi- 
 nated from a single equation. In fact, elimination from two or more 
 premises is ultimately reduced by our author to elimination from a 
 single premiss. And yet, as if to preserve the analogy between 
 Algebra, and Logic, even where the two sciences seem to differ most 
 widely from one another, the possibility of eliminating x from a sin- 
 gle premiss in the latter science, arises from the circumstance, that, 
 in that science the equation previously referred to as expressing the 
 Law of Duality always subsists ; and it is by the combination of that 
 equation with the given proposition that the elimination of x from 
 the given proposition is efi*ected. For let the given proposition be 
 
 /(*)=0 (21) 
 
 Then, by (10), 
 
 /(l)«'+/(0)(l--^) = 0. 
 
 •• ^{/(0)-/(l)} =/(0), 
 and, {l-x) j/(0) -/(I)} = -/(I). 
 
 ••• a^Cl-^') 1/(0) -/(1)}2 = -/(0)/(l). 
 But, bj the Law of Duality, sc (1 — a;) 5r 0. Therefore 
 
or THE LAWS OF THOUGHT. 
 
 t§ 
 
 /(0)/(l)=0: (22) 
 
 which is the result of the elimination of x from equatitn (21). We 
 cannot pause to give examples of the use of the formula (22) ; hut 
 we must quote an interpretation of it, viewed as the result of the 
 elimination of » from (21), which strikes us as extremely elegant. The 
 formula implies that either /(O) == 0, or/(l) = 0. Now the latter 
 equation/ (1) s expresses what the given proposition /(a;) = 
 would become if x made up the universe; and the former/ (0) = 
 expresses what the given proposition would become if x had no 
 existence. Hence, (22) being derived from (21), it foUows that what 
 is equally true whether a given class of objects embraces the whole 
 universe or disappears from existencCf is independent of that class 
 altogether. 
 
 The principle of elimination is extended by our author to groups 
 of equations, by the following process. Let 
 
 F=0. 
 Z7=0, 
 
 .(23) 
 
 be a series of equations, in which T, U, V, &c., are functions of the 
 concept X. Then 
 
 T* + F" 4- 17« 4- &c. = (24) 
 
 It is shown by Professor Boole that the combined interpretation of 
 the system of equations (23) is involved in the single equation (24). 
 Indeed, had all the terms in the developments of T, P, U^ &c., been 
 sueh as to satisfy the Law of Duality, it would have been sufficient 
 to have written 
 
 2» + F + Z7 + &c. = 0. 
 
 In order now to eliminate x from the group (23), it is sufficient to 
 eliminate it, by the method described in the preceding paragraph^ 
 from the single equation (24) ; and, if the result be 
 
 this equation itvill involve all the conclusions that can legitimately be 
 derived from the series of equations (28) with regard to the mutual 
 relations of tno concepts, exclusive of x, which enter into these 
 equations. 
 
 We do not see how it is possible for any one not blinded by pre- 
 judice against every thing like an alliance of Logic with formula and 
 
20 
 
 PROPK880R BOOLB'8 MATRBMATICAL THEORY 
 
 processea of a mathematical aipeot to deny that these are yery 
 remarkable principles. By way of initance, we select from the work 
 under review the following problem* in which two premises are giyen. 
 Let it be granted, first, that the annelida are soft-bodied, and either 
 naked or enclosed in a tube ; and, next, that they consist of all inver- 
 tebrate animals having red blood in a double system of circulating 
 vessels. Put 
 
 A = annelida, s =s soft-bodied animals, 
 
 n = naked, t = enclosed in a tube, 
 
 i = invertebrate, r = having red blood in &c. 
 
 Then tbo given premises are 
 
 A=svt\n(l—i)^t{l—n)\ (26) 
 
 A=sir (26) 
 
 Suppose the problem then to be: to find the relation in which 
 •oft bodied animals enclosed in tubes stand to the following elements, 
 viz., the possession of red blood, of an external covering, and of a 
 vertebral column. Professor Boole would doubtless have granted 
 that this problem admits of being solved by what he calls the ordi- 
 nary logic ; but he would probably have contended that the ordinary 
 logic does not possess any definite and invariable method of solution. 
 A skilful thinker may be able to find out how syllogisms may be 
 formed so as ultimately to give him the relation which soft bodied 
 animals enclosed in tubes bear to the elements specified; but what 
 of thinkers who are not very skilful ? How are they to proceed ? 
 In Professor Boole's system, the process is as determinate, and as 
 certain of leading to the desired result, as the rules for solving a 
 group of simple equations in Algebra. Eliminate v, the symbol of 
 indefinite quantity, from (25). Beduce (25), thus modified, and (26), 
 to a single equation, by the method described in a previous paragraph. 
 The equation is 
 
 i<{l-*n(l-^)-,^(l-«)j + J(l-i>) + i>(l-.^+n^=0. 
 
 Then, since the annelida are not to appear in the conclusion, we must 
 eliminate A, by (22), from this equation. This will be found to give 
 us 
 
 »> { 1 - sn {\ " t) - at(l - n)] + nt =i 0. 
 And ultimately we get 
 
 . »/=tr(l -n)-f ^, (1-0(1 -n)+$(l -0(1 - n) ; 
 the interpretation of which is : Soft bodied animaU enclosed in tuhea 
 
OP THE LAWS OF THOUGHT. 
 
 (tt) consist of all invertebrate animals having red blood (ir) and not 
 naked (I — • n), and an indefinite remainder {%) of invertebrate ani' 
 mats (i) not having red blood (i — r) and not naked (I — w) and of 
 vertebrate animals (I — i) which are not naked (I — n). 
 
 We have entered bo fully into the explanation of Professor Boole's 
 system in its bearing on what he terms Primary (virtually equivalent 
 to Categorical) Propositions, that we cannot follow him into the field 
 of Secondary (virtually equivalent to Conditional, that is, Disjunctive 
 and Hypothetical) Propositions. Nor is it necessary that we should 
 do 80 ; for our object is not to give a synopsis of the " Investigation," 
 but simply to make the nature of the work understood ; and, for 
 that purpose, what has been said is sufficient. The application 
 of the Calculus to Secondary Propositions is exceedingly similar, in 
 respect not only of the general method followed, but even of the 
 particular formulee obtained, to its application to Primary. All that 
 is peculiar in the treatment of Secondary Propositions arises from 
 the introduction of the idea of Time. Por instance, the proposition, 
 " If XIb Yt A is B" is held to be not substantially different in 
 meaning from this: "the time in which X is J, is time in which 
 A is B." Such being the fundamental view taken, symbols like x and 
 y are used to represent the portions of time in which certain pro- 
 positions (e.g., XiaYfA is B) are true. Then, the symbol 1 denot- 
 ing the universe of Time, or Eternity, the expressions, 1 — or, 1 — y, 
 will denote those portions of time respectively in which the propo- 
 sitions, XiB Yf A is Bf are not true ; and so on. 
 
 The extension of his method, by Professor Boole, to the theory of 
 Probabilities, is a splendid effort of genius on the part of the author, 
 and furnishes a most convincing illustration of the capabilities of 
 the method. The part of the " Investigation " which is devoted to 
 this subject, is much too abstruse to admit of being here more par- 
 ticularly considered ; but, to show what the method can accomplish 
 — though the bow of Ulysses perhaps needs the arm of Ulysses to 
 bend it — we may simply state one of the problems of which Pro- 
 fessor Boole gives the solution. " If an event can only happen as a 
 
 consequence of one or more of certain causes, Ai, A^t , A^^ 
 
 and if generally O^ represents the probability of the cause A^, and 
 Pi the probability that, if the cause A^ exist, the event JS will occur, 
 then the series of C, and pi being given, required the probability of 
 the event H." 
 
22 
 
 boolb'i thbory or the laws or thought. 
 
 To thoie who hftve followed ui thni ftr, it will b« evident what 
 fiiud judgment we are to paaa on the work under review. On the 
 one hand, as a contribution to philosophy, in the strict sense of that 
 term, it does not possess any value. Professor Boole distinctly, 
 though modestly enough, avows the opinion, that, in his " Investi- 
 gation," he has gone deeper than any previous inquirers into the 
 principles of discursive thinking, and that he has thus thrown new 
 light on the constitution of the human mind. We are sorry to be 
 unable to accept this view. But, on the other hand. Professor 
 Boole is entitled to the praise of having devised a Method, according 
 to which, through definite processes, it can be ascertained what con- 
 clusions, regarding any of the concepts entering into a system of 
 premises, admit of being drawn from these premises. This Method 
 depends on a Calculus, original, ingenious, singularly beautiful both 
 in itself and in its relations to the science of Algebra, and capable 
 (in hands like those of its inventor) of striking and important appli- 
 cations. In a word, the merit of the Treatise lies in that part of it 
 which has nothing to do with the Laws of Thought, bnt which is 
 devoted to showing how inferences, from data however numerous 
 and complicated, and whatever be the matter of the discourse, can 
 be reached through definite mathematical processes. 
 
what 
 Q the 
 
 that 
 ictlj, 
 resti- 
 ) the 
 
 new 
 to be 
 essor 
 rding 
 con- 
 m of 
 thod 
 both 
 )able 
 ppli- 
 of it 
 ;h is 
 rous 
 
 can