THE THEORY OF SYLLOGISM, ETC. 115 



between each of the pairs XY, YZ, and one (inferred) between XZ. Here each relation 

 is supposed to extend over the terms and their contraries : thus from X ) )' Y we are to 

 infer X t ).(y, and may use it. Secondly, we may dismiss the condition that the relation 

 extends over the contrary, and then we arrive at the modification of the common or Aris- 

 totelian syllogism, already discussed : in this case, the inference can be obtained without 

 any use of the contrary, or extension of the relation. 



I* now give the heads of the results which arise from the hypothesis that one of the 

 correlations is assigned or inferred, and the other not admitted. The equivalent propositions- 

 are those of the ordinary system, with the relations unaltered : thus 



X i {)'Y=X i (.<y = x)(y -*,).)' F, 



Here x (.)' Y must be read as follows; in every instance, either x gives, or Y receives, 

 or both : and x t ) (y means that there are some instances in which neither x gives, nor Y 

 receives. 



These equivalences may be easily proved by those who remember the meaning of the 

 relative symbol. The assertion X t ) )' Y is that of a relation in which every X and every 

 x is a giver, and every Y and every y is a receiver ; but so that no Y nor y can receive 

 both from a X and from a x, no X nor x can give both to a Y and to a y. Accordingly, 

 x t ({'y immediately follows: for every y receives either from a a? or from a X, but not from 

 the latter, because then a Y and a y would both receive from that X. 



The law of entrance of the relations is as follows. When the syllogism is read in an 

 Aristotelian figure (the first or the third) which begins with the middle term, all that is 

 requisite is that the major and concluding relation should be the same: when in a figure 

 (the second or the fourth) which does not begin with the middle term, all that is requisite 

 is that the major and concluding relation should be correlatives. Each syllogism therefore 

 can be read in two ways, according as the major and minor relations are the same or cor- 

 relative. 



For example, the syllogism A t A'I', or ))((=)(, read in the first figure, is 



YUZ + X))Y=X)(Z, 

 which may be either 



F(('Z + Jr))'r-=X)('Z, or Y / {CZ + X'))Y=X I )(Z. 



To prove the first, remark that since every X supplies a F, there are but the xs to 

 supply the remaining Fs and all the ys. But since the Zs are all supplied by Fs, all 

 the ys supply %s, being themselves supplied by <a?s. Hence, by the transitive character 

 of the relation, we have £",()'*> or ■^,)('^- ^° P rove tne second, observe that every y 

 supplies a x, and if any such w supply a Z, the y which supplies that x also supplies that 

 Z; but every Z is supplied by a F. Consequently «,()'*, or X)('Z, as before. 



In each syllogism, one or other of its two cases must involve the direct entrance of 

 contraries: so that the Aristotelian form never allows two cases. For instance, 



F))'Z + ^ / ))'F = X / ))'Z 



15—2 



