392 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM, 



But we have not yet closed our invesligation ; for we have to examine the remaining syllogism 

 of universal premises, or Y)X + Y) Z= XZ. If we go through the cases, in the last order of 

 headings, we shall find as follows. 



First Form, Transformation. Description. Remarks. 



Y^X+Y)Z = XZ Y)X+Y)Z^XZ I^a Derived from /,„ 



Y^w + Y)Z= ccZ Y.X +Y)Z= Z:X Ob„ o„„ 



y).v + y)Z = xZ X) Y + y . ss = Z: X o^, o,., 



y).v + y)z = xx X)Y+Z)Y=xz i.f4 Not yet obtained 



Y) X + Y)z = XZ Y . X + Y . Z = ,vz i^E Derived from 



oB 



Y)X + Y)z = Xz Y)X+ Y.Z = X:Z O^b 0,g, 



y)X ^ y)z = Xz y . .v + Z) Y= X : Z O,^ 0„. 



y)X + y) Z = XZ y.w + y . « = XZ /„ Not yet obtained. 



The derivation here mentioned is merely strengthening a premiss. We thus obtain the only two 

 remaining forms 



ijj X)Y+Z)Y = .vz I /„ y..v + y.z=_XZ. 



These cannot be derived from the twelve previously established by strengthening a premiss, 

 though their equivalents (the other six) can. These two last syllogisms differ from all the rest in 

 having no counterparts, and may therefore be called single syllogisms. 



The old rules are of course true as to the old syllogisms: but most of them are inapplicable to 

 the new ones. Particular premises, indeed, never gave a conclusion, as yet : but premises both nega- 

 tive may, and in the case of j^^, the middle terra is universal in neither premiss. Again, both premises 

 may be negative, and may give a positive form of conclusion. The following rules, however, will be 

 found to hold good. 



From premises both particular, nothing follows. The middle term cannot be particular in both, 

 except in j^,,,: nor can its contrary be universal in both, except in /^^. One negative premiss 

 always yields a negative conclusion, and two negative premises an affirmative. When one 

 premiss is particular, the conclusion is particular. When e is in the premises the conclusion is 

 never in i. 



I now take the two cases in which particular premises may give a conclusion: namely 



Ijj XY + ZY= XZ XY ^ Y.Z^ X.Z 0,, 



on the suppositions that the Ys mentioned in both premises are in number more than all the Ys. 

 If }', and Y-i stand for the fractions of the whole number of I's mentioned or implied in the two 

 premises, and j/, and y., for the fractions of the ys implied or mentioned, we shall by a repetition of 

 the process on YX + YZ = XZ (the other being obtained in the course of the process) arrive at the 

 following results or their counterparts: remembering that Y^ + K, is greater or less than 1, accord- 

 ing as y, + j/u is less or greater. (See the Addition at the end of this paper.) 



Designation. Syllogism. Condition of its validity. 



Iij YX + YZ = XZ r, + r., greater than 1 



0,„ YX^Y.Z^X.Z 



»:„„ Y:X^Y.Z= .vz 



0„( X: Y+ yz = X Z F, 4- Y, less than 1 



iji y'" + ysr = XZ 



Ooi X:Y+ yz =X:Z 



loo X:Y+Z:Y=XZ 



