242 THE SCIENCE OF LOGIC 



For the partial or simple contrapositive, first obvert, then con 

 vert; for the full or ob verted contrapositive, obvert again. 



Applying this process to A, E, I, O, respectively, we have the 

 following table : 



Original proposition S a P S e P S i P S o P 



Obverse S e P S aP S oP S i P 



Partial contrapositive P e S P i S None P i S 



Full contrapositive P a S P o S None P o S 



We saw that the O proposition could not be converted : we 

 now note the interesting fact that it can be contraposed, whereas 

 the I proposition, which can be converted, cannot be contraposed. 



We note, secondly, that the contraposition of A and of O is 

 simple, i.e. unaccompanied by a loss of quantity; whereas E, 

 losing its quantity, may be said to suffer contraposition per ac- 

 cidens or by limitation. 



The comparative scarcity of formally negative terms makes the expres 

 sion of the contrapositive somewhat awkward in the case of categorical pro 

 positions. We simplify matters somewhat by substituting for &quot;not-/ 3 &quot;. 

 &quot; Whatever, (whoever) is (are) not P &quot;. Thus, &quot; All S s are P &quot; : &quot; Whatever 

 is not P is not S &quot;. This mode of inference is of very common occurrence ; 

 but it is more usually made in the hypothetical (139) than in the categorical 

 form. The strict categorical universal &quot; All 5 is P,&quot; involves that &quot; If any 

 thing is S it is /*,&quot; from which we infer that &quot; If anything is not P it is not 

 5,&quot; which is another and better way of stating that &quot; All not-/&quot;s are not-5,&quot; 

 or &quot; No not-P s are S &quot;. From &quot; All ruminants are cloven-footed,&quot; it 

 follows by contraposition that &quot; If an animal is not cloven-footed it is not a 

 ruminant &quot;. The corresponding inference from the E proposition, &quot; No S is 

 P &quot; (&quot; If anything is S it is not P &quot;), would be &quot; If anything is not P it (may be 

 or) need not be 6&quot;,&quot; which is the modal hypothetical equivalent of &quot; Some 

 not-/ 3 is not 5 &quot;. From &quot; No clergymen are members of Parliament,&quot; it would 

 follow by contraposition merely that &quot; If anyone is not a member of Parlia 

 ment he (may be or) need not be a clergyman &quot;. The O proposition gives a 

 similar contrapositive. 



The recognition that &quot; Whatever is S is P &quot; formally involves &quot; What 

 ever is not P is not S&quot; independently of the matter or meaning of the 

 terms of these propositions &quot; renders unnecessary the special proofs that 

 Euclid gives of certain of his theorems. ... It will be found that taking 

 Euclid s first book, proposition 6 is obtainable by contraposition from pro 

 position 1 8, and 19 from 5 and 18 combined ; or that 5 can be obtained by 

 contraposition from 19, and 18 from 6 and 19. Similar relations subsist 

 between propositions 4, 8, 24, and 25, and again, between axiom 12 and 

 propositions 16, 28, and 29.&quot; 1 



In physical induction, in which we seek to establish general physical 



1 KEYNES, op, cit., p. 136. 



