Mb DE morgan, ON SYLLOGISMS OF TRANSPOSED QUANTITY. *357 



quantities give a particular conclusion. And, as before noted, the symbol of the conclusion 

 is derived as in the ordinary syllogism, with this difference, that the quantity of each imparting 

 term must be changed in forming the conclusion. 



Let external and internal universality be denoted by Y and U; and sameness and oppo- 

 sition of the quantities of the middle terms by S and D. The ordinary syllogism is valid 

 under the conditions expressed in UDU, USU, UDP, PDU: and each of these forms has 

 eight varieties. The syllogism of transposed quantity is valid under any one of the forms 

 YDU, UDY, YDY, YSU, USY, YSY, YDP, PDY: and each of these forms has also eight 

 varieties. There are therefore 64 varieties of transposed syllogism; twice as many as there 

 are of ordinary syllogism. 



In the forms YDU and UDY, or whenever an external and internal universal give 

 a universal, the internal universal is really a simple identity or contrariety of its terms. Thus 

 in ())) which gives (( , or 'For every Z an X is Y, every Y is Z, give every Z is X,' 

 the first premise tells us that there cannot be more Zs than Ys, and the second that 

 there cannot be more Ys than Zs. Hence the Ys and Zs exist in equal numbers ; that is, 

 since every Y is Z, Y and Z are identical. But in ())•(, or X()Y).(Z, which is X()Y))z, 

 Y and z are identical by the same reasoning, or Y and Z are contraries. 



When two external universals are conjoined, the concluding terms, or one concluding 

 term and the contrary of the other, must be arithmetically of the same number of 

 instances, though not necessarily identical and, it may be, even externals of each other. 

 Thus in ()() or in ' For every Z an X is Y, and for every X a Y is Z' we see that there 

 cannot be more Zs than Xs, nor more Xs than Zs. Hence Z and X are of the same 

 numbers of instances. Again in ())( or 'for every z an X is Y, and for every X there is 

 something neither Y nor Z' we see that there cannot be more zs than Xs, nor more Xs 

 than zs. Hence X and z are of the same number of instances, though the conclusion X))Z 

 shows that no two instances, one of each, are identical. 



The vague quantity of a particular conclusion is the vague quantity of the particular 

 premise, when there is such a thing : when there is no such thing, it is either the 

 quantity of the middle term, or of its contrary, whichever is universal in both premises. 



The following is the list of varieties of syllogism : — 

 UDY YDU YDY 



YSY 



