June 9, 1881] 



NATURE 



125 



On page 94 of his work Mr. Venn strikes the key-note, as it 

 seems to me, of its whole purport and spirit. "Those who 

 propose a new notation," he says, "commonly, and not un- 

 naturally, assume that it is to supersede all others. But those 

 who approach it as strangers know that the odds are decidedly 

 that it will only prove one more of those many attempts which 

 perplex and annoy the lecturer, historian, and critic. Hence we 

 may fairly u e the argument, dear to those in authority, that if 

 we loosen the sanctions of orthodoxy, heroics will multiply. 

 Only those whose professional einploymeiit compels them to 

 study a number of different works have any idea of the be- 

 wildering variety of notation which is already before the world. 

 ... No doubt it would be rank intolerance to forbid such new 

 attempts, but an atiitiuie of slight social represiion towards them 

 may serve to check too luxuriant a growth oj new proposals." 



The italics are mine. Alas, how little Mr. Venn appreciates 

 the irrepressible restlessness of that most ungovernable organ, 

 the human brain, if he really thinks that the "attitude of slij^ht 

 social repression " which he recommends would have the desired 

 effect! Amteur logicians, as well as professionals, -vill start 

 theories and invent notations of their own in happy unconscious- 

 ness that they are causing any annoyance to "lecturers, histor-ans, 

 and critics," whom indeed they not improbably picture to them- 

 selves (when he all-absorbing nature of their occupation allows 

 them to think of thein at all), as ardent devotees of scierce like 

 themselves, who will le delighted with the new instrument of 

 research which thuy hope to place in their hands. And more 

 provoking still, scientific societies ai.d editors (including a goodly 

 number of the said lecturers, historians, and critics) will print in 

 their Proceedings sxA maga/ines new pruposals which they think 

 likely to pr<jve interesting or valuable, without being influenced 

 by any motive whatever beyond a pure and simple desire to 

 f ut ther the progress of science. 



Mr. Venn professes great admiration for the late Prof. Boole's 

 genius, and I hearti y agree with him, though we admire on some- 

 what different ground'^. I ground my admiration on the fact that 

 Boole worl>ed wonders with an unnecessarily complicated and 

 otherwise defective symbolical method of his own invention. Mr. 

 Venn apparently grounds hi- admiration on the singular supposition 

 that Boole's method i- really very simple and very effective, I ut 

 that its author did not undeisland very clearly the real principles 

 of its construction, and did not by any means apply it with as 

 much ease and dextcriiy as he might have done. I am quite 

 sure that this is not the impression which Mr. Venn intended to 

 create in the minds of his readers, but I am no less sure that 

 this is the impression w hich a perusal of his book to/// create 

 in their minds — at leat in the minds of such as have not read 

 Boole himself. One remark of Mr. Venn's surprises me. lie 

 says (p. 385) that Boole "justly regarded his problems in Proba- 

 bility as the cro"ning triumph of his system." Surely I am not 

 mistaken in my impression that I have somewhere seen Mr. Venn 

 quo'ed as holding an opinion very much at variance with this 

 statement — in fact attacking as erroneous the very principle on 

 which Hoole's " General Method in Probabilities " is based. May 

 I ask Mr. Venn this plain que>tion, Does he or does he not 

 agree with Boole's solution of the question which he pro- 

 posed on pp. 321 and 336 of his "Laws of Thought " a.s a 

 decisive "lest of the sufficiency of received methods," and (by 

 implication) of the efficacy of his own General Method ? 



The main points on which Mr. Venn and I differ are the 

 follow ing : — 



I. Mr. Venn maintains that the sign -)- in such expresions as 

 X + y -V z shoitld in logic, as in ordinary algebra, be always 

 understood in an exclusive sense, so that unless we 1 now x, 

 y, s to I e mutually exclusive, the above expression should be 

 written in a different and, as he admits, a much less simplefrm. 

 I hold, on the contrary, in common with Prof. Jfvons and several 

 others, that since, (m the non exclusive plan, the simple form 

 X + y + z may, without the sligh'est risk of ambiguity, be sub- 

 stituted at any stage of an investigation for any of its exclu-ive 

 equivalents (-uch as x + x' y + x' y z), or be replaced, if neces- 

 sary, by any exclusive equivalent, the non exclusive interpreta- 

 tion of the symbol + gives us far more mastery over our sym- 

 bolical expressions, and should therefore be preferred to the 

 needlessly restrictive and hampering exclu-ive interpretation 

 which Boole attaches to this symbol. How very serious the 

 disadvantages of this interpretation really are is unwittingly 

 illustrated by Mr. Venn himself on p. 262, where he finds him- 

 self obliged to admit that certain important simplifications which 

 he discusses are "purely a matter of tact and si ill, for which no 



strict rules can be given." If he had read niy third paper in the 

 Proceedings of the London Mathematical Society a little more 

 attentively he would have found in n.y directions for reducing 

 any complex disjunctive expression to its " primitive form" that 

 thee simplifications are not at all a pure matter of tact and 

 skill, but may be obtained by a simple, never-failing, and purely 

 mechanical process, which, however, a little tact and skill may 

 do much to abbreviate. On the exclusive interpretation of the 

 symbol -f this process would be simply unmeaning. The problem 

 which Mr. Venn discusses (expressed in my notation) is this : 



Reduce the expression {f:g) (gl:/) (/:/') to its simplest 

 form. 



By inspection [since any implication o : S is equivalent to 

 oi3' : o, and any compound implication of the fcrm (o : Jr) (/3 : x) 

 (7 :x) to a single implication 0-1-3 + 7 ■■'] 'hi'i '-'" the non- 

 exclusive plan, is seen to be equivalent lofg' + glf + 1/ .0. 



Reducing the disjunctive antecedent of this implication to its 

 primitive form (a purely mechanical process, as already remarked), 

 we get/i,'' + ^/ ;0, or its equivalent, the compound implication 

 (/■'i') ^S' ^')> ^ result which Mr. Venn obtains apparently by a 

 vague tentative process "for which no strict rules can be 

 given." 



2. Mr. Venn and I also hold different opinions as to whether 

 or not symbolical logic should have signs to express relations cor- 

 responding to those of subtraction and division in mathematics. 

 His opinion is that such signs should be introduced, and at once. 

 My opinion (an opin on which I believe I .share with most 

 logicians) is that we had better not encumber ourselves with 

 those symbols till they can be proved to subserve some useful 

 purpose. The important question is — not, as Mr. Venn appears 

 to think, whether such symbols can be intelligibly interpreted, 

 but whether they will in any way help us in discovering new 

 truths ; in other w ords, whether they can be turned to any 

 practical use in the solution of logical prcbUms. If Mr. Venn 

 can adduce a single inlelligiiile logical jiroblem which can be 

 solved more simply or easily I y the help of these signs than with- 

 out them, I shall declare myself at once a convert to his view.". 

 So far I have come across no such problem, and must there- 

 fore for the present reniain in the ranks of his opponents. As 

 an illustration of what Mr. Vmn calls the inverse method [i.e. 

 division) in logic, he gives (p. 266) the equation — 



(.V -f xy\ -a = X +_ xy z, 

 in which to (which denotes the bools in a certain library) is to 

 be expressed in terms of x, y, z (which respectively denote phi- 

 lowph'cal books, divinity books, and protestant books). His 

 result is — 



, - , o - . 



w = .r -I- xyz -i- - -r;', 

 O 

 which he translates into ordinary language thus : — 



"The library must have certainly contained all philosophy 

 and proteta.it diviuily, and may possibly have contained any 

 kind of works which are neither jshilosophy nor divinity ; this 

 latter constituent being left entirely indefinite. " 



Mr. Venn's data in the non exclusive notation would be — 

 (x + y)w = X -f- y z, 

 and my result (much more simply and easily obtained) is— 



X + yz : w : X + y + z. 

 According to my definitions of my letter-symbol---, we speak 

 throughout of some one 01 iginally unclassed book, so that w, x, 

 y, z will respectively denote the statements : It is in the library ; 

 It is a philosophical work ; It tieats of divinity ; It is 3. pro- 

 testant w ork. My result may therefore 1 e read : — 



Any work on philosophy or 1 rotestant divinity will be found 

 in the library ; and every work in the library is either philo- 

 sophical, secular, or protestant. (By a secular work I mean 

 simply a work that does not treat of divinity.) 



The antecedent of w in my result is equivalent to the first 

 "constituent " in Mr. Venn's resuU ; but the c nsequent seims to 

 me to give us much more intelligible iuf'.rmaiion about the 

 library than Mr. Venn's latter c.nstitueut ("May possibly have 

 contained," &c ), which he truly descrites as "entirely in- 

 definite." 



3. Another opinion of Mr. Venn's, unless I misunderstand 

 him, is that all logical equations should, as a preliminary to their 

 solution, be expressed in the form = 0, which, of course, is 

 equivalent to my o : o. My opinion is that this, in logic as in 

 mathema'ics, is sometimes convenient and s. metimes not, and 

 that we should not in logic, any m-ie than in mathematics, tie 

 om- hands by this or any otbtr unnecessary restriction. 



