100 THE PKINCIPLES OF SCIENCE. [CHAP. 



or by the process of Ellipsis before described (p. 57) 

 A = Ac. 



Third Example. 



As a somewhat more complex example I take the 

 argument thus stated, one which could not be thrown into 

 the syllogistic form : 



" All metals except gold and silver are opaque ; there- 

 fore what is not opaque is either gold or silver or 

 is not-metal." 



There is more implied in this statement than is dis- 

 tinctly asserted, the full meaning being as follows : 



All metals not gold or silver are opaque, (i) 



Gold is not opaque but is a metal, (2) 



Silver is not opaque but is a metal, (3) 



Gold is not silver. (4) 



Taking our letters thus 



A = metal C = silver 



B = gold D = opaque, 



we may state the premises in the forms 



Abe = AbcD (i) 



B = ABrf (2) 



C = ACd (3) 



B = Be. (4) 



To obtain a complete solution of the question we take 

 the sixteen combinations of A, B, C, D, and striking out 

 those which are inconsistent with the premises, there remain 

 only 



AbCd 

 AbcD 

 abcD 

 abed. 



The expression for not-opaque things consists of the 

 three combinations containing d, thus 



d = ABcd -|. AbCd + abed, 

 or d = Ad (Be -|- JC) -|- abed. 



In ordinary language, what is not-opaque is either metal 

 which is gold, and then not-silver, or silver and then not 

 gold, or else it is not-metal and neither gold nor silver. 



