DEDUCTIVE REASONING. 69 



A = some crystals. 



B = bodies equally elastic in all directions 



C = doubly refracting light 



c = not doubly refracting light. 



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

 i 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 = ABC, 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 j 



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 



