DEDUCTIVE REASONING. 69 



A = some crystals. 



B = bodies equally elastic in all directions 



C = doubly refracting light 



c = not doubly refracting light. 



Our argument is of the same form as before, and may 

 be concisely stated in one line 



A = AB = ABc. 

 If we take PQ for the indefinite some crystals, we have 



PQ = PQB = PQBc. 



The only difference is that the negative term c occurs 

 instead of C in the mood Darii (p. 68). 



On the Ellipsis of Terms in Partial Identities-. 



The reader will probably have noticed that the conclu 

 sion which we obtain from premises is often more full 

 than that drawn by the old Aristotelian processes. Thus 

 from Sodium is a metal/ and Metals conduct electricity, 

 we inferred (p. 66) that Sodium = sodium metal, con 

 ducting electricity/ whereas the old logic simply concludes 

 that Sodium conducts electricity. Symbolically, from 

 A = AB, and B = BC, we get A = ABC, whereas the old 

 logic gets at the most A = AC. It is therefore well to 

 show that without employing any other principles of 

 inference than those already described, we may infer 

 A = AC from A = ABO, though we cannot infer the 

 latter more full and accurate result from the former. 

 We may show this most simply as follows : 



By the first law of thought it is evident that 



AA = AA ; 



and if we have given the proposition A = ABC, we may 

 substitute for both the A s in the second side of the above, 

 obtaining 



AA = ABC . ABC. 

 But from the property of logical symbols expressed in the 



