[ 61 ] 



Fig. 1. 



VII. Notices respec ting New Books. 



Symbolic Logic. By John Veishst, M.A., Fellow and Lecturer in the 

 Moral Sciences, Gonville and Caius College, Cambridge. London : 

 Macmillan and Co. 



"JY|~B. VENN'S work forms an important contribution to the 

 XtX Modern Logic, and is destined to give a great impetus to 

 the study and development of that science. 



One excellent feature of the work is the amount of information 

 it contains about the results reached by the early mathematical 

 logicians. These logicians nourished before logic had been divorced 

 from mathematics ; and some of them had got a good way towards 

 establishing the science on an analytical basis — Lambert in parti- 

 cular. Indeed, he appears to me to be quite sound on a nice point 

 with respect to which Mr. Venn is inclined to think him not free 

 from error (p. 272). The question is as follows : — Given xy = zw, 

 where each of the symbols denotes a class of things, is it correct to 



conclude that - = - ? I illustrate the question by the accompany- 

 ing diagram (fig. 1). Suppose we 

 restrict our attention to the part of 

 the page within the square : let x 

 denote the part included in the one 

 circle, y the part included in the 

 other circle, z the part stroked 

 downwards from right to left, w the 

 part stroked downwards from left 

 to right. Here we have ocy = zw ; 

 and the diagram is quite general in 

 every other respect. The principle 

 upon which I should proceed is as 



„ „ 00 U< . , , t 



follows : - = — is a legitimate de- 



z y 



duction, because the symbols are commutative in aty and in zw ; 



X w 



and the meaning of - and of — must be fixed so as to suit this 



& s y 



deduction. By the Boolian process of development, 



^^ z+ K(l-z)+° T (l-x)z+°-(l-x)(l-z); 

 z 1 U 



w ==W y+ I w(l-y) + £ (l-w)y + ? (1- W )(1 -*)• 

 v 1 U 



and 



The coefficients direct us to include the first term, to exclude the 

 third term, and (we will assume) to take a part of the fourth term. 



- = >r may be taken to 



x y 



Looking at the diagram, we then find that 



