224 THE SCIENCE OF LOGIC. 



set up a positive counter-assertion in the direction of contrary 

 opposition. 



Two contradictories must affirm and deny the same thing 

 about the same thing under the same respect. One of them 

 must, therefore, be false by the Principle of Contradiction (13). 

 One of them must be true by the Principle of Excluded Middle 



Any pair of propositions which cannot be true together, or false together, 

 may rightly be called contradictories, even though they be not logically 

 formulated according to the A, E, I, O scheme, or even though they have not 

 exactly the same terms as subject and predicate, e.g. &quot; All virtuous people are 

 happy &quot; and &quot; Some non-happy people are virtuous &quot;. 



113. CONTRARY OPPOSITION is that which exists between two 

 universals of opposite quality : between All S s are P and No S s 

 are P. Contrary propositions thus differ in quality only, while 

 contradictories differ in quality and quantity. The rules of con 

 trary opposition are : 



(1) Contraries cannot be true together ; 



(2) Contraries may be false together. 



(1) We prove the first rule by reference to the square of op 

 position, employing the rules we have already laid down for 

 subalternation and contradiction. Thus : If A is true I is true 

 (by subalternatiori) ; but if I is true E must be false (by contradic 

 tion} ; therefore if A is true E is false. Similarly, we prove that 

 if E is true A is false. Thus, we see that if contraries were true 

 together, contradictories would also be true together which is 

 impossible. The contradictory of a true proposition must be 

 false ; but the contrary is still further removed from the original 

 than the contradictory is. Therefore, a fortiori, the contrary of a 

 true proposition must be false. Hence, contraries cannot be true 

 together. 



(2) The second rule conveys that if one of any pair of contraries 

 be given as false we can infer nothing about the other : this latter 

 may be false or it may be true : we cannot tell. Thus, by sub- 

 lating A we posit O, but by positing O we cannot either posit or 

 sublate E : from the truth of the particular we cannot infer the 

 truth of the universal nor its falsity. Similarly, by sublating E 

 we posit I, but cannot thence infer anything about A. Therefore 

 contraries may be false together. 



Or, to put the proof in another way, one of any pair of con 

 traries does much more than merely deny the entire truth of the 



