. The London, Edinburgh and Dublin philosophical magazine and journal of science. enn is inclined to think him not freefrom 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 werestrict our attention to the part ofthe page within the square: let xdenote the part included in the onecircle, y the part included in theother circle, z the part strokeddownwards from right to left, w thepart stroked downwards from leftto right. He
. The London, Edinburgh and Dublin philosophical magazine and journal of science. enn is inclined to think him not freefrom 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 werestrict our attention to the part ofthe page within the square: let xdenote the part included in the onecircle, y the part included in theother circle, z the part strokeddownwards from right to left, w thepart stroked downwards from leftto right. Here we have ocy = zw ;and the diagram is quite general inevery other respect. The principleupon 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 0 1 U w ==Wy+ I w(l-y) + £ (l-w)y + ? (1-W)(1 -*)•v 0 1 U. and The coefficients direct us to include the first term, to exclude thethird 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 62 Notices respecting New Books. mean hence (i+2)+3+ *4i-*)=(i+3)+2+ \<i-y); \*)=\w(l-y), which appears to mean 4=4. xThe most general meaning of - cannot be considered to be a class which, on restriction by z, produces x; that is its meaning onlywhen x{\ — z)=Q. It appears that we are entitled to equate thesum of the first and fourth terms on the one side to the sum of thefirst and fourth terms on the other side, and the second term to thesecond. But we are certainly not eutitled to equate, as Mr. Yennthinks we are, the first to the first, and the fourth to the let x(l —z)=0 and tt(l —y) = 0, then the state of affairs is stillrepresented by (1 + 2)+ 3=(1+3)+ 2. Another excell
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