*358 mk de morgan, on syllogisms of transposed quantity, 

 pdy usy yd? ysu 



())( 

 )•) ) ( 

 )((•( 

 )(() 

 )•) )•) 

 () )•) 



(•C() 

 {•( (•( 



)( 

 (•( 

 (•( 

 () 

 () 

 )•) 

 )■) 

 )( 



())( 

 )•))"( 

 )((•( 

 )(() 

 )■))•) 

 ())■) 

 (■( ( ) 



() 

 )•) 

 )•) 

 )( 

 )( 

 (•( 

 (•( 

 ( ) 



)(( 

 •)(( 

 ()•( 

 ()) 

 ) (•) 

 ) (•) 

 •()) 



We ■ have here given the symbols of the premises, followed by the symbol of the 

 conclusion : from which the syllogisms may be read at length. Thus the last syllogism 

 under YDY is as follows ; — For every z there is an X which is not Y ; for every X there 

 is a Y which is not Z : whence every X is Z. 



The several cases in UDY and YDU are inverted readings each of the other: those 

 in YDY and YSY are essentially different. The cases in USY and YSU are but strengthened 

 forms of those in PDY and YDP ; the particular in the second being converted into an 

 internal universal which contains it by an alteration in the quantity of the middle term, 

 without any accession to the conclusion. The forms in YDY are derived from those in PDY 

 and from those in YDP by strengthening — or at least rendering less vague, — the particular 

 proposition by giving it that quantity which makes it an external universal : and the conclusion 

 is thereby strengthened into a universal. 



The following comparison will illustrate and extend the preceding remarks. In ordinary 

 syllogism, the existence of valid inference depends upon the presence of U and D ; that is, 

 either U must be present twice ; or U once, and D. In UDU, the most powerful of valid 

 forms, the inference is U: in USU, PDU, UDP, it is only P. In USP, PSU, PDP, PSP, 

 there is no inference : of these the first three may be said to be one remove from inference, 

 and the fourth two removes. All this may be said of the cases in which the external 

 universal is allowed to enter. Thus PSP, now two removes from inference, is made valid 

 by such accessions as make it PDY, YDP, or YSY : each accession being either a change 

 from S to D, or alteration of P into Y. 



Of all the transposed syllogisms, one half, being all in the first compartment, are either 

 identical with, or contained in, syllogisms of the ordinary kind. Of those in UDY, the one 

 marked ((() is contained in, and contains, ||)); and).(() is similarly identical with ||)). Of 

 those in YI?U, ())) is identical with ((|| : and so on. Of those in YDY, ())( is contained in, 

 but does not contain, )))) ; )•))('» contained in (•))) ; and so on. Of those in YSY, ()() is 

 contained in ))(( ; and so on. Overlooking strengthened forms, it thus appears that the really 

 new cases are all contained in PDY and YDP, of which either is but an inverted reading 



of the other. 



A. DE MORGAN. 



University Coixege, London, 

 March 15, 1860. 



