iv.j DEDUCTIVE REASONING. 



We thus prove that one class of triangles is entirely 

 identical with another class ; that is to say, they differ 

 only in our way of naming and regarding them. 



The great importance of this process of inference arises 

 from the fact that the conclusion is more simple and general 

 than either of the premises, and contains as much informa- 

 tion as both of them put together. It is on this account 

 constantly employed in inductive investigation, as will 

 afterwards be more fully explained, and it is the natural 

 mode by which we arrive at a conviction of the truth of 

 simple identities as existing between classes of numerous 

 objects. 



Inference of a Limited from Two Partial Identities. 



We have considered some arguments which are of the 

 type treated by Aristotle in the first figure of the syllogism. 

 But there exist two other types of argument which employ 

 a pair of partial identities. If our premises are as shown 

 in these symbols, 



B = AB (i) 



B = CB, (2) 



we may substitute for B either by (i) in (2) or by (2) in 

 (i), and by both modes we obtain the conclusion 



AB = CB, (3) 



a proposition of the kind which we have called a limited 

 identity (p. 42). Thus, for example, 



Potassium = potassium metal (i) 



Potassium = potassium capable of floating on 



water ; (2) 



hence 



Potassium metal = potassium capable of float- 

 ing on water. (3) 

 This is really a syllogism of the mood Darapti in the third 

 figure, except that we obtain a conclusion of a more exact 

 character than the old syllogism gives. From the premises 

 " Potassium is a metal '' and " Potassium floats on water," 

 Aristotle would have inferred that " Some metals float on 

 water." But if inquiry were made what the " some 

 metals " are, the answer would certainly be " Metal which 

 is potassium." Hence Aristotle's conclusion simply leaves 

 out some of the information afforded in the premises ; it 



