1873.] On the Algebraical Analogues of Logical Relations. 497 



II. " On the Algebraical Analogues of Logical Relations." By 

 Alexander J. Ellis, B.A., F.R.S., F.S.A., F.C.P.S., F.C.P. 

 Received March 19, 1873. 



(Abstract.) 



The object of this paper is to examine the " mathematical theory of 

 logic," thus laid down by Dr. George Boole in his ' Laws of Thought/ 

 p. 37: — "Let us conceive of an Algebra iu which the symbols x, y, z, 

 &c. admit indifferently of the values and 1, and of these values alone. 

 The laws, the axioms, and the processes of such an algebra will be 

 identical in their whole extent with the laws, the axioms, and the pro- 

 cesses of an Algebra of Logic. Difference of interpretation will alone 

 divide them." For this purpose, first the laws of such an algebra have 

 been investigated independently of logic ; and secondly the laws of primary 

 and secondary logical propositions as laid down by Dr. Boole have been 

 developed in an algebraical form, and compared with the former. The 

 main results presumed to be established are : — 



1. That there is a fundamental difference between such an algebra 

 and logic, inasmuch as the algebra admits of only two phases, and 1, 

 and logic admits of three phases, namely, not only none and all, corre- 

 sponding to and 1, but also some, " which, though it may include in its 

 meaning all, does not include none " (ibid. p. 124), and hence has no 

 analogue in such an algebra ; that is, an algebra of and I can corre- 

 spond only to a logic of none and all. 



2. That, notwithstanding this difference, there are certain formal re- 

 lations of equations which allow the algebra of and 1 to be used as an 

 algorithm for the purpose of arriving at certain logical forms, which, 

 however, have then to be interpreted on a basis which has not even any 

 analogy to the algebraical. 



3. That the introduction of this algorithm introduces theoretical diffi- 

 culties, adds to the amount of work, and is entirely unnecessary even 

 for the purposes of the theory of probabilities founded upon it by Dr. 

 Boole. 



In the paper the general case is examined, and a solution is pro- 

 posed for the difficulties arising from the interpretation of the symbols 

 {} and ^, which Boole says " must be established experimentally " (ibid. 

 p. 92). The following simple case will show the nature of the investi- 

 gation ; but, for brevity, even this case is not fully considered. 



In algebra, let x and y have either one of the values or 1, and no 

 other. And let x', y be connected with them by the equations 



l=cc-\-x' and l=y-\-y'. (a) 



Then xx' = 0, and x=x 2 ; (b) 



and by simple multiplication 



x=xy + xy', x' = x'y + x'y', . . . \ . (c) 



l=xy+xy' + afy+oc'y' (d) 



2 e 2 



