244 THE SCIENCE OF LOGIC 



the two desired inverses. [ (&) S e P obverts to S a P, which 

 converts to P i S which obverts to P o S which cannot be con 

 verted.] 



(3) Inversion of I. [(a) S i P converts to P i S, which ob 

 verts to P o S which cannot be converted.] [(#) S i P obverts 

 to 5 o P which cannot be converted. ] 



(4) Inversion of O. [(a) S o P cannot be converted] [(&) S o P 

 obverts to 5 i P, which converts to P i 5, which obverts to 

 P o S which cannot be converted] 



Hence we see, firstly, that only universal propositions yield 

 inverses : particulars do not. We see, secondly, that in order to 

 invert A we must begin by obverting ; that in order to invert E 

 we must begin by converting ; that in all other cases we are 

 arrested by the appearance of an O proposition for conversion. 



Hence the rules for inversion : 



(1) Convert the obverted contrapositive of A. 



(2) Convert the obverted converse 0/E. 



We see, thirdly, that the full inverse comes first in inverting A, 

 that it comes after the partial inverse in inverting E ; that in 

 both cases it is of the same quality as the original proposition ; 

 and that all inversion involves a depression of quantity from uni 

 versal to particular. 



In inversion, as in contraposition, the passage of thought is often 

 through hypothetical rather than categorical judgments : from &quot; If anything 

 is 5 it is P &quot; to &quot; If anything is not 5 it may be or need not be P &quot; (137, 



139) 



Though not a very common form of inference, inversion puts us on our 

 guard against the not infrequent fallacy of inferring from &quot; Whatever is S is 

 P &quot; that &quot; Whatever is not S is not P &quot;. We are entitled to infer only that 

 &quot; Something that is not S is not P &quot;. &quot; If all triangles are plane figures, 

 what information, if any, does this proposition give us concerning things that 

 are not triangles ? &quot; a This question simply asks for the inverse of the pro 

 position &quot; All triangles are plane figures &quot;. The answer therefore is : 

 &quot; Some things that are not triangles are not plane figures &quot;. 



A glance at this example may suggest a doubt about the validity of the 

 process of inversion. How can we reach a proposition in which the term 

 &quot;plane figures&quot; is distributed, from one in which it is undistributed? Or, 

 generally, how can we validly pass from &quot; All S is /&amp;gt;,&quot; in which P is 

 undistributed, to &quot; Some non-S is not P &quot; in which P is distributed ? And, 

 moreover, how do we know from the original proposition that there are in 

 the universe of discourse any non-S s e.g. any &quot; things that are not 



1 JBVONS, Studies in Deductive Logic , p. ix. ; cf. ibid., p. 48. 



