256 THE SCIENCE OF LOGIC 



that might be made, we will examine just a few, seeing in the 

 first place what influence they would have on the ordinary doctrine 

 regarding Opposition and Eductions. 



(1) We have seen already that if every proposition be taken to 

 imply the existence of its subject, predicate, and their contradictories 

 (S, P, S, and P), all the ordinary eductions will be valid ; but 

 the forms given as contradictories and subcontraries in the square 

 of opposition will be no longer valid 1 (125). 



(2) If every proposition be taken to imply the existence of its sub 

 ject merely : the forms given as contradictories and subcontraries in 



the square of opposition will, as in (i), be no longer valid (125). 

 With regard to eductions, though the conversion of A and I is 

 valid, since the existence of 5 involves, in an affirmative proposi 

 tion, the existence of P, the conversion of E is invalid for the 

 opposite reason. &quot;No women are now hanged for theft in Eng 

 land&quot; must therefore be converted to a hypothetical: &quot; No people 

 now hanged for theft in England, if there be any such, are women &quot;. 

 As a consequence of the invalidity of converting E, the contraposi- 

 tives of A, and the inverses of A and E, are also invalid, if stated 

 categorically, since these involve the conversion of an E proposi 

 tion. But the contrapositives of E and O remain valid, for the 

 existence of 5 guarantees the existence of P, which is the subject 

 of these contrapositives. 



(3) V no proposition implies the existence either of its subject or 

 of its predicate, then (a) as regards opposition: S a P denies the 

 existence of 5 P s, whereas S o P merely asserts fast if there are 

 S s there are 5 Fs 2 a statement which does not deny the truth 



1 These forms will, of course, remain valid if the &quot;existence&quot; of the classes in ques 

 tion be not taken as contained in the import of the proposition, but as assumed 

 independently. This latter is what is probably understood by writers who, like 

 Professor Welton, assume the existence of S as given with the categorical proposi 

 tion, and yet hold the ordinary contradictories and subcontraries of the square of 

 opposition as valid. 



2 C/&quot;. Dr. KEYNES (op. cit., pp. 221, 225, 229), whose treatment of the whole 

 question is largely followed in the present chapter. In connexion with the supposi 

 tion in (3) above, it should be clearly borne in mind that if we regard a proposition 

 as not implying the existence of S or P, but take for granted independently that S, 

 P, S and P exist actually in the sphere to which the proposition refers us, then all 

 the traditional laws of opposition will hold good of the forms set down in the square 

 of opposition, and all the ordinary eductions will be valid. This seems to have been 

 the assumption acted on in the traditional Aristotelean logic : and wherever a doubt 

 arose as to the propriety of making such assumption about any class, such doubt was 

 expected to find expression in the form of the proposition. 



Indeed, the supposition made in (3) above i/ it be understood to exclude the 



