iv.] DEDUCTIVE REASONING. 51 



Inference, luith Two Simple Identities. 



One of the most common forms of inference, and one to 

 which I shall especially direct attention, is practised with 

 two simple identities. From the two statements that 

 " London is the capital of England " and " London is the 

 most populous city in the world," we instantaneously draw 

 the conclusion that " The capital of England is the most 

 populous city in the world." Similarly, from the identities 

 Hydrogen = Substance of least density, 

 Hydrogen = Substance of least atomic weight, 

 we infer 



Substance of least density = Substance of least atomic 

 weight. 



The general form of the argument is exhibited in the 

 symbols 



B = A (i) 



B = C (2) 



hence A = C. (3) 



We may describe the result by saying that terms identi- 

 cal with the same term are identical with each other ; and 

 it is impossible to overlook the analogy to the first axiom 

 of Euclid that " things equal to the same thing are equal 

 to each other." It has been very commonly supposed that 

 this is a fundamental principle of thought, incapable of 

 reduction to anything simpler. But I entertain no doubt 

 that this form of reasoning is only one case of the general 

 rule of inference. We have two propositions, A = B and 

 B = C, and we may for a moment consider the second one 

 as affirming a truth concerning B, while the former one 

 informs us that B is identical with A ; hence by substitu- 

 tion we may affirm the same truth of A. It happens in 

 this particular case that the truth affirmed is identity to 

 C, and we might, if we preferred it, have considered the 

 substitution as made by means of the second identity ID 

 the first. Having two identities we have a choice of the 

 mode in which we will make the substitution, though th/ 

 result is exactly the same in either case. 



Now compare the three following formulae, 



(1) A = B = C, hence A = C 



(2) A = B - C, hence A - C 



(3) A ~ B ~ C, no inference. 



