A nezv Symbolic Treatment of the Old Logic. 2 1 1 



Before we consider syllogisms of the ordinary kind, let 

 us consider a set of syllogisms whereof one premise asserts 

 ■one of the eight relations expressed above, and the other 

 asserts the relation of exclusive alternative, expressed by 

 (— i). As the symbol of the combination of the two 

 propositions into a syllogism, let us use the sign of 

 multiplication x. The forms of such syllogisms, in 

 language and in notation, are the two following : — 



All A is B ; or, 



A is enclosure of B. 



B is that which is not C ; or, 



B and C are exclusive alternatives, 



Therefore no A is C ; or, 



A and C are excludents. 



A is that which is not B ; or, 



A and Bare exclusive alternatives. 



All B is C ; or, 



B is enclosure of C. 



Therefore all that is not A is C; or, 



A and C are alternatives. 



In all such multiplications, or syllogisms, when the 

 relative (— i) comes second it transforms the other relative 

 into its opposite : — when it comes first, it transforms the 

 other relative into the contrapositive of its opposite. The 

 following is a tabular view of the sixteen possible syllogisms 

 of this kind. 



E^{-'\) = N ( 



< / 



xE=^N'' 



^^x(-i) = A^^ i 



f _ _.> 



xE = N 



Ny.{-i) = E { 



- i) 



xN=E^ 



JV^x(-i) = E^ ( 



(-1) 



xN^=^E 



ex (-i)=n { 





xe = rC 



e''x(-i) = n' \ 



(-1) 



X <?^ + 7l 



nx (- i)=e 



(-i; 



xn = e!' 



n^ X (— i)=e'' 



(-1) 



xtC = e 



The relation expressed by ( - i) is by definition negative. 

 The equation A = (— i)B is equivalent to A = b, and both 

 mean that A is defined as all which is not B. It will be 

 seen in the multiplication shown above, and it is invariably 

 true in logic as in arithmetic, that the multiplication of 



