498 On the Algebraical Analogues of Logical Relations. [Nov. 20, 



In (d) one term on the right must = l, and each one of the rest must 

 = 0. This has no logical analogue. The ambiguity is diminished by- 

 taking some relation between x and y, as x — xy^ which by (c) gives 

 ccy' = 0, and hence, by (d) t 



l=xy+oc'y+x'y', (<?) 



showing that one of three (instead of four) terms must = 1, and each of 

 the other two must = 0. This has also no logical analogue. But in (e) 

 we find that x occurs only in the term xy, and y' only in the term x'y' } 

 whereas x' and y' both occur in two terms. This is the only relation 

 useful in logic. But further, owing to the necessity for all terms vanish- 

 ing except one, there can be only one term in which x occurs, in which 

 case neither x nor y' can occur, and there may be no such term. This 

 again has no logical analogue. 



In logic (only a primary proposition being considered for brevity) let 

 U be the things themselves in the " universe of our discourse " (ibid. 

 p. 44), X those among them having the name X n and attribute X a , and 

 similarly for T, T w , Y a . Also let X', T' be those things among them 

 which are not X and T. Let XT mean those things X which are also 

 things T, and YX those things T which are also things X. Let P = Q 

 mean that the group of things P is the same as the group of things Q. 

 Then, disregarding the order in which attributes occur, and the number 

 of times that they recur, XT=YX, and X=XX. Let vP represent the 

 number of the things P, then 



vU=kX +vX!, vTJ=vY+vY', vXX' = 0, vX=vXX, . («',&') 

 rX=vXY+vXY' vX' = vX'Y +vX!Y', .... (c') 



vU=^XY+ y XY'-f-vX'Y+vX'Y', (d') 



which are of the same form as (a, b, c, d). But in (d'), although one 

 term on the right may =vU, in which case each of the rest must = 0, 

 most generally no term on the right of (d') will=j/TJ, and any number of 

 the terms may be greater than 0. This has no analogue in the algebra 

 in question. If there is a limiting equation as X=XY, or " all X is Y," 

 giving kXY'=0 from (c'), or " there is no X which is not Y," then (d') 

 reduces to 



yU=vXY+^X'Y+ y X'Y', («') 



and all the X which exist (there may be none) are Y, and all the Y' 

 which exist (there may be none) are X'. This has the algebraical ana- 

 logue already mentioned. But there may be both X and Y', in which 

 case neither yXY nor vX'Y' either =i/U or=0, and this has no alge- 

 braical analogue. Again, whether either or neither of yX and vY'=0, 

 vX'Y may be greater than ; and this has no algebraical analogue — al- 

 though, if both yX=0 and j/Y=0, then vX'Y=rU, which has an alge- 

 braical analogue. 



The difference between the laws of such an algebra and of logical cal- 

 culation, therefore, appears to be one of principle and not of interpreta- 

 tion on]y. 



