94 MR. JOSEPH JOHN MURPHY ON AN ' 



we cannot infer either 



A=ABC, 

 or 



A=AC. 



But though in logic a relative term with zero index is 

 not necessarily equal to unity, yet every such term has 

 two important properties which belong to unity and are 

 not combined in any other number — namely, that it is 

 equal to its own reciprocal, and equal to its own second 

 power. Thus if 



A = L°B, 

 it follows that 



B=iv°A. 



That is to say, if A is a fellow pupil o£ B, then B is a 

 fellow pupil of A. And if 



A=L°M and M=X°B 



it follows that 



A=(i:°)^B = £°B. 



That is to say, if A is a fellow pupil of B and B of C, 

 then A is a fellow pupil of C. This inference is a syl- 

 logism, the middle term, M, being eliminated. It must 

 be observed that its validity depends on the relation of 

 fellow pupils being understood in relation to the same 

 teacher throughout. With this convention, the axiom 

 that " fellow pupils of the same teacher are fellow pupils 

 of each other •'"' has the same self-evidence as the axiom 

 that '^ equals of the same thing are equals of each other /' 

 and both are cases of Jevons's principle of the substitution 

 of similars, that " what is true of any thing is true o£ its 

 like.^^ 



We now proceed to apply these principles to the old 

 logic. The proposition of the old logic, " all A is B/' is 



