On Implicational and Equational Logic. 41 



principle in symbolical reasoning generally, that conventional 

 symbols of abbreviation should be adopted for all expressions 

 which have to be employed frequently. On this principle a? was 

 probably used as a convenient abbreviation for aaa, a* for aaaa, 

 and so on, before the discovery of the important law expressed 

 by the equation a m x a n = a m+n . The same necessity for sym- 

 bolical abbreviation originated the useful symbols /(#), f{os, y), 

 f\x), and many others. On this principle, since I find that 

 such statements as " If u is true /3 is true," or " « implies /3," 

 are extremely common in all reasoning, I use the simple symbol 

 « : /3 as a very convenient abbreviation*. Granted that the 

 equation a=«/3 will also accurately express the statement "a 

 implies /3," it is a much less simple and suggestive expression 

 for it. Compare, again, the implication, a/3 + yS : ab + cd, 

 with its equivalent, the equation a/3 + 7S = (a/3 + <y8)(ab + cd), 

 and the superior simplicity of the implication will be still more 

 striking. But the abbreviating power of my symbol of impli- 

 cation becomes most conspicuous in what may be called im- 

 plications of the second order, as in the syllogism 



(« : /8)(/8 : y) : (« : y). 

 May I ask Prof. Jevons how he would express this syllogism 

 in his equational notation, in pure symbols and entirely with- 

 out words. I can only see one way in which he could do this 

 consistently with his views, namely by the very clumsy equa- 

 tion 



(*=*j3)(/3=/3y) = (*==*j3)(/3={3<y)(*=*y). 



This looks so exceedingly like a reductio ad. absurdum, that I 

 cannot help hoping that it will lead Prof. Jevons to reconsider 

 his opinion that the equational form alone should be employed 

 in symbolical logic. 



So far I have argued on the assumption that my a : /3 is 

 equivalent to Prof. Jevons's a = a/3 ; and both Prof. Jevons and 

 I agree, 1 believe, in the opinion that practically this is the case. 

 At the same time, it must be borne in mind that, for this 

 assumption to be strictly true, the letters a and /3 must have 

 the same meanings in the implication « : /3 as in the equation 

 a = a/3 ; and therefore either each letter must in both forms 



* The equivalence of et : [5 and «=a/3 may "be proved formally in my 

 notation as follows : — 

 From the formula 



0=/3) = (*:/3)(/3:«) 

 we get 



(« = «£) = (* : «/3)(>/3 : «)=(«: «)(« : /3)(a/3 : «) = (« : /3) ; 

 for the factors « : a and «/3 : a, are each equal to unity — that is to say, 

 always trite, whatever the statements u and /3 may be. (See formula 3 

 of this paper further on.) 



