222 THE SCIENCE OF LOGIC. 



which exists between two propositions which have the same subject 

 and predicate but differ both in quantity and in quality. This 

 definition is specially adapted to the propositions of the tradi 

 tional square of opposition. It coincides with the more funda 

 mental conception of contradiction as the opposition between simple 

 affirmation and simple negation, or the opposition between two 

 propositions one of which must be true and the other of which must 

 be false. This latter is the real test of contradictory opposition 

 between any two propositions : that they can neither be true to 

 gether, nor false together with a mean of truth between them, but 

 one must be true and the other false. Hence the two laws : 



1 I ) Contradictories cannot be true together ; 



(2) Contradictories cannot be false together. 



Let us illustrate these laws by applying them to the four pro 

 positions A, E, I, O. 



(1) If it be true to affirm P of all the S s (SaP) the Prin 

 ciple of Contradiction (i 3) forbids us to deny P of any of them, i.e. 

 to say some 5 is not P ; hence SoP is false. Conversely, if it be 

 true to deny P of some S (SoP), it cannot be true to affirm P of 

 them all ; hence, SaP is false. If it be true to deny P of all 

 the S s (SeP), it must be false to affirm P of any of them ; hence 

 SiP is false. Conversely, if it be true to affirm P of some of the 

 S s (SiP), it cannot be true to exclude P from all of them ; hence 

 SeP is false. 



(2) If it be false to affirm P of all the S s (SaP), it must be 

 true, by the Principle of Excluded Middle (14), that at least one 

 of them is not P ; hence SoP is true. Conversely, if it be false 

 to deny P of any even one 5 (SoP), it must be true that all 

 the S s are P ; hence SaP is true. If it be false to deny P of all 

 the S s (SeP), it must be true that some one at least of the S s is 

 P ; hence StP is true. Conversely, if it be false that any even 

 one S is P (SiP), then it must be true that none of them are 

 P ; hence SeP is true. 



From this we see that contradictories are incompatible both 

 as regards their truth and as regards their falsity ; that the truth 

 of either is inferable from the falsity of the other, and the falsity 

 of either from the truth of the other ; and that they are thus 

 perfectly correlative. These characteristics we shall find in no 

 other species of logical opposition ; hence Contradictory Opposition 

 is the most perfect of all forms of opposition. 



