486 



NA rURE 



\_MarcIi2^, li 



tions in the /';'^<:i^£'rtfi>/fj of the American Academy of Arts 

 and Sciences (vol. vii. pp. 250-29S, 402-412, 416-432), 

 we have from him the wonderful investigation con- 

 tained in his " Description of a Notation for the Logic of 

 Relatives, resulting from an Amplification of the Concep- 

 tions of Boole's Calculus of Logic " {Miinoirs of the 

 American Academy, vol. ix. Cambridge, L^.S., 1870, 4to). 

 The contents of this remarkable treatise, which fills sixty- 

 two quarto pages, demand the most careful study, but it 

 would be quite impossible in this article to enter upon 

 such study. Prof. Peirce has however quite recently 

 interpreted his own views in a new memoir " On the 

 Algebra of Logic," of which the first part, completed by 

 the author in April last, was printed in the Amen'cdii 

 Jounial of Mathematics, vol. iii., and issued in September 

 (4to> J 7 PP-)- After noticing the beautiful typography in 

 which the Anierkan Jonnial rejoices, we find in this 

 memoir a very careful inquiry as to what is really the 

 form and nature of logical inference. 



Prof. Peirce treats in succession of the Derivation of 

 Logic, of Syllogism and Dialogism (a new name for a 

 form of argument), of Forms of Propositions, the Algebra 

 of the Copula, the Internal Multiplication and the Addition 

 of Logic, the Resolution of Problems in Non-relative 

 Logic, with a further chapter on the Logic of Relatives. 

 The fundamental point, however, which is under discussion 

 in the first two chapters touches the nature of the copula. 

 There is abundance of evidence to show that given a few 

 elementary forms, it is possible to spin out logical or 

 mathematical formulae simply without limit. But the 

 superstructure rests entirely upon the basis of elementary 

 truth contained in the first axioms. In logical science it 

 is emphatically true that " C'est le premier pas qui coite. ' 

 There is a momentous choice to be made at the outset, 

 and if we then take a wrong view of the nature of the 

 logical copula, we can never come right again by any 

 amount of development or formulisation. 



Prof. Peirce after mentioning that four different alge- 

 braic methods of solving problems in the logic of non- 

 relative terms have been proposed by recent English and 

 German logicians, adopts a fifth, which he thinks is 

 perhaps simpler and certainly more natural than any of 

 the others. Peirce conmiences by expressing all the 

 premises by means of the copulas — < and H^, "re- 

 membering that A = B is the same as A — < B and 

 B — < A" (p. 37). These new symbols are to be inter- 

 preted so that -A — < B means {.\ implies B), in the way 

 that water implies liquidity, or all water is liquid. The 

 symbol — < is the negative of the above, so that C— < D 

 means that C does not imply D. He then lays down five 

 other processes which give the elementarv theorems of 

 the calculus, showing how to develop, simplify, transpose, 

 and infer equivalency by these symbols. As however 

 these processes occupy two quarto pages in their first 

 statement, it is evident that they cannot be reproduced 

 here. The question which really emerges is not as to the 

 power and originality shown by Prof Peirce, about which 

 no reader of his memoirs can entertain the slightest 

 doubt, but as to the wisdom of the first step, the selection 

 of the relation expressed by the symbol — < instead of 

 that expressed by the famihar sign of equality =■■. Prof. 

 Peirce begins by remarking that A = B is the same as 

 A — < B with B — < A. For instance, all equilateral 

 triangles are equiangular, and all equiangular triangles 

 are equilateral. But though these two assertions are 

 equivalent to " equilateral triangle = equiangular triangle," 

 Prof. Peirce elects to treat the two parts of the apparently 

 compound proposition separately, his reasons being given 

 partially on p. 21. This is not the first time that the 

 same choice has been made ; for, not to speak of Aristotle 

 and the Aristotelians generally, De Morgan elected to 

 base his systems of logic upon inclusion and e.xclusion, 

 instead of upon equality. In his symbols X i] Y is com- 



pounded of X ) ) Y and X ( ( Y (Syllabus, p. 24), that is 

 to say all Xs are all Ys is made of all Xs are Ys and all 

 Ys are Xs. Now without going far afield, I beheve that 

 a .sufficient reason may be given for holding that both De 

 Alorgan and Peirce have chosen wrongly. A class is 

 made up of individuals, and the very conception of a class 

 thus implies the relation of identity expressed in A = B. 

 If I say the colour of glacier ice is identical with the 

 colour of pure rain water, it is impossible to break this 

 assertion up into "The colours of glacier ice are among 

 those of pure rain water," and " The colours of pure rain 

 water are, &c." The colour is one indivisible and identical. 

 Now if there is at the basis of all reasoning an elementary 

 assertion of the form A = B, which is incapable of reso- 

 lution into anything simpler, this sufficiently proves that 

 Peirce's A — < B, or De Morgan's A ) ) B cannot be the 

 original elementary form of assertion. Moreover, when 

 we say that all equiangular triangles = all equilateral 

 triangles, the real basis of assertion is that- each possible 

 equiangular triangle is identical with one possible equi- 

 lateral triangle. The plural is made up of the singular, and 

 the singular is incapable of logical decomposition. You 

 may decompose A = B into As are Bs, and Bs are As, 

 but ultimate decomposition gives us A' = B', A" = B", 

 A'" = B'", A', A", &c., being individuals. 



It is highly curious, however, that this very question 

 arises again with reference to the so-called Calculus of 

 Equivalent Statements recently published by Mr. Hugh 

 MacColl, B.A., in the P)-ocecdings of the London Mathe- 

 matical Society (First paper, November 1877, vol. ix. pp. 

 9-20; Second paper, June 13, 1S78, vol. ix. pp. 177-186; 

 "Third paper, vol. x. pp. 16-28 ; Fourth paper, vol. xi. ; 

 see also Mind, January 1880, pp. 45-60, and the Philo- 

 sop/iical Matcaaine iox September 1880). 



There can be no doubt that Mr. MacColl has shown 

 much skill in devising neat symbolic forms, and much 

 power in using them. Comparing his processes with 

 those of De Morgan, for instance, it is impossible not to 

 admire their symmetry and lucidity. But when we touch 

 the real point, the nature of assertion and inference, I am 

 obliged to hold that Mr. MacColl has, like De Morgan 

 and Peirce, elected wrongly. What De Morgan e.xpressed 

 by X ) ) Y, and Peirce by X — < Y, MacColl puts in the 

 form .1" :y, calling the assertion an implication. Curiousl)' 

 enough, he professes never to treat of things, but only of 

 assertions, so that with him .r :_;' means that the assertion 

 .r implies the assertion _)', or whenever .r is true, y is true. 

 Having carefully considered Mr. MacColl's proposals, I 

 felt obliged to write of them in a recent publication as 

 follows : — " It is difficult to believe that there is any 

 advantage in these innovations ; certainly, in preferring 

 implications to equations, Mr. MacColl ignores the neces- 

 sity of the equation for the application of the Principle of 

 Substitution. His proposals seem to me to tend towards 

 throwing Formal Logic back into its Ante-Boolian con- 

 fusion." 



In a paper printed in the Philosophical Maga::inc for 

 January 1S81, Mr. MacColl takes me to task and invites 

 me to make good the charge about .\nte-Boohan con- 

 fusion, by entering into a friendly contest in the problem 

 columns of the Educational Times. Having just recently 

 spent the better part of fifteen months in solving other 

 people's problems, and in inventing some two or three 

 hundred new ones, published in " Studies in Deductive 

 Logic," I certainly do not feel bound to sacrifice my 

 peace of mind for the next few years by engaging to 

 solve any problems which the ingenuity and^leisure of 

 Mr. MacColl or his friends may enable them to devise. 

 I therefore decline his proposal with thanks. But I can 

 easily explain what I mean by ante-Boolian, or what 

 comes to much the same thing, anti-Boolian confusion. 

 The great reform effected by Boole was that of making 

 the equation the corner-stone of logic, as it had always 

 been that of mathematical science. Not only did this 



