in.] PROPOSITIONS. 47 



A = B and B = A 



to express exactly the same identity differently written. 

 All need for rules of conversion disappears, and there will 

 be no single proposition in the system which may not be 

 written with either end foremost. Thus A = AB is the 

 same as AB = A, aC = bC is the same as &C = aC, and so 

 forth. 



The same remarks are partially true of differences and 

 inequalities, which are also reciprocal to the extent that 

 one thing cannot differ from a second without the second 

 differing from the first. Mars differs in colour from 

 Venus, and Venus must differ from Mars. The Earth differs 

 from Jupiter in density ; therefore Jupiter must differ from 

 the Earth. Speaking generally, if A - B we shall also 

 have B - A, and these two forms may be considered ex- 

 pressions of the same difference. But the relation of 

 differing things is not wholly reciprocal. The density of 

 Jupiter does not differ from that of the Earth in the same 

 way that that of the Earth differs from that of Jupiter. 

 The change of sensation which we experience in passing 

 from Venus to Mars is not the same as what we experience 

 in passing back to Venus, but just the opposite in nature. 

 The colour of the sky is lighter than that of the o/;ean ; 

 therefore that of the ocean cannot be lighter than that of 

 the sky, but darker. In these and all similar cases we gain 

 a notion of direction or character of change, and results of 

 immense importance may be shown to rest on this notion. 

 For the present we shall be concerned with the mere fact 

 of identity existing or not existing. 



Twofold Interpretation of Propositions. 



Terms, as we have seen (p. 25), may have a meaning 

 either in extension or intension ; and according as one or 

 the other meaning is attributed to the terms of a proposi- 

 tion, so may a different interpretation be assigned to the 

 proposition itself. When the terms are abstract we must 

 read them in intension, and a proposition connecting such 

 terms must denote the identity or non-identity of the 

 qualities respectively denoted by the terms. Thus if we 

 say 



Equality = Identity of magnitude, 



