64 . THE PRINCIPLES OF SCIENCE. [CHAP 



To show this let 



A = carbon, B = metallic, 



C = capable of powerful magnetic influence. 

 The premises readily take the forms 



b = be, (i, 



A = A&, (2) 



and substitution for b in (2) by means of (i) gives the 

 conclusion 



A = Afo. _ (3) 



Our principle of inference then includes the rule of 

 negative premises whenever it is true, and discriminates 

 correctly between the cases where it does and does not 

 hold true. 



The paralogism, anciently called the Fallacy of Undis- 

 tributed Middle, is also easily exhibited and infallibly 

 avoided by our system. Let the premises be 



Hydrogen is an element, (i) 



All metals are elements. (2) 



According to the syllogistic rules the middle term "element " 

 is here undistributed, and no conclusion can be obtained ; 

 we cannot tell then whether hydrogen is or is not a metal. 

 Represent the terms as follows 



A = hydrogen, 

 B = element, 

 C = metal. 

 The premises then become 



A = AB, (0 



C = CB. (2) 



The reader will here, as in a former page (p. 62), find it 

 impossible to make any substitution. The only term which 

 occurs in both premises is B, but it is differently combined 

 in the two premises. For B we must not substitute A, 

 which is equivalent to AB, not to B. Nor must we confuse 

 together CB and AB, which, though they contain one com- 

 mon letter, are different aggregate terms. The rule of sub- 

 stitution gives us no right to decompose combinations ; 

 and if we adhere rigidly to the rule, that if two terms are 

 stated to be equivalent we may substitute one for the other, 

 we cannot commit the fallacy. It is apparent that the form 

 of premises stated above is the same as that which we 

 obtained by translating two negative premises into the 

 affirmative form. 



