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



danger of carrying a false meaning to the last named. The consequence of the distinction is that 

 these four syllogisms, 



X)Y Z.Y 



Z.Y X)Y 



Figure 

 Name 



z.x x.z 



2 2 



Camestres Cesare 



which are identical in sense and effect, are made separate objects of study. In fact, as is known, all 

 the figures are in the first, the fourth being occasionally brought into it by a reduction which deserves 

 the name of Barbara in every case, and which the simplest use of contraries avoids. 



For syllogisms I shall adopt svich notation as 



X)Y+ Y)Z = X)Z. 



Or if a weaker conclusion be taken, 



X) Y + Y)Z>XZ. 



As there are eight modes of predicating between X and Y, and between Y and Z, it follows 

 that there are sixty-four combinations which may give conclusions. Of these, all but eight are 

 distributable two and two into counterparts, in which X and Z are interchanged, everything else 

 remainino- the same ; giving eight single, and twenty-eight pairs of counterparts. Of these, exactly 

 half, (four simple, and fourteen pairs) are wholly inconclusive. Of the conclusive cases, two single 

 ones and two pairs are rejected, because as strong a conclusion can be obtained from a weaker pre- 

 miss. There remain two single ones and twelve pairs, to which a systematic classification is to be 

 given. Instead of enumerating, I shall state a mode of deriving all the cases from a common 

 principle. 



Since every proposition is, but for accidents of language, a universal affirmative, as before 

 noticed, it will follow that there are really no forms of syllogism except those in which the 

 premises and conclusion are universal affirmatives, or can be made so by use of contraries and 

 invention of subgeneric terms. Now the only universal affirmative syllogism is 



X)Y +Y)Z = X)Z 



considering the counterpart Z)Y+Y)X=Z)X as identical in form. If we take universal 

 affirmative premises only, we have one which will have a particular conclusion, with respect to 

 the names X and Z, 



1')^+ Y)Z = XZ, 



which must be used in discovery of forms, (and will in fact give the two single syllogisms of 

 this system) though it will only ultimately enter as Y)X + YZ = XZ. Now if we change one 

 or more of the terms X, Y, Z into their contraries, we have eight modes of transformation, 

 according as we use 



XZY, XZy, .vzy, j,!sY, XzY, Xmy, vZY, x^y. 

 First take the syllogisms 



X)Y+ Y)Z = X)Z and XY+ Y)Z = XZi 



the latter of which is only the first, with the premiss X) Y weakened, and would be reduced 

 to the first form by inventing a subgeneric name for the Xs there spoken of Apply each of 

 these to the eiMit varieties just named, transforming premises and conclusion, when necessary, to 

 one of the eight standard forms of predication : 



X)Z X.Z Z)X Z.X X.Z XZ x.y xx. 



