Cambridge Philosophical Society. 131 



names), the propositions ' Every X is Y ' and ' no X is y ' are sim- 

 ply identical. In the same manner, the particular and universal 

 proposition are only accidentally distinct. If in ' some Xs are Ys ' 

 the Xs there specified had had a name belonging to them only, say 

 Z, then the preceding proposition would have been identical in mean- 

 ing with * every Z is Y,' 



From the above it is made to follow, that every legitimate syllo- 

 gism can be reduced to one of universal affirmative premises, either 

 by introduction of contrary terms, or invention of subgeneric names. 



In considering the nature of the simple proposition, Mr. De Mor- 

 gan uses a notation proposed by himself. Thus — 

 Every X is Y is denoted by X)Y A 

 NoXisY . . . . X.Y E 



Some Xs are Ys . . . XY I 



Some Xs are not Ys . . X:Y O 

 and names which are contraries are denoted by large and small let- 

 ters. Aristotle having excluded the contrary of a name from formal 

 logic, and having thereby reduced the forms of proposition to four, 

 these forms (universal affirmative, universal negative, particular affir- 

 mative, particular negative) the writers on logic in the middle ages 

 represented by the letters A, E, I, O. Thus X)Y and Y)X are 

 equally represented by A. When contraries are expressly intro- 

 duced, all the forms of assertion or denial which can obtain between 

 two-terras and their contraries, are eight in number ; and the most 

 convenient mode of representing them is as follows : — Let the letters 

 A, E, I, O have the above meaning, but only when .the order of sub- 

 ject and predicate is XY. Then let a, e, i, o stand for the same 

 propositions, after x and y, the contraries, are written for X and Y. 

 The complete system then is — 



A=X)Y a=x)y~Y)X 



0=X:Y o=a^:y=Y:X 



E:=X.Y e-=-x.y 



I=XY i-=-xy 



and every form in which subject and predicate are in any manner 

 chosen out of the four X, Y, x, y, so that one shall be either X or x, 

 and the other either Y or y, is reducible to one or other of the pre- 

 ceding. 



The propositions e and i, which are thus newly introduced, are 

 only expressible as follows, with reference to X and Y. 



(t.) There are things which are neither X nor Y. 



(e.) There is nothing but is either X. or Y or both. 



The connexion of these eight forms is fully considered, and the 

 various syllogisms to which they lead. Rejecting every form of syl- 

 logism in which as strong a conclusion can be deduced from a weaker 

 premise ; rejecting, for instance, 



Y)X+Y)Z=XZ 



because XZ equally follows from Y)X4-YZ, in which YZ is weaker 

 than Y)Z — all the forms of inference are reduced to three sets. 

 1 . A set of two, called single because the interchange of the terms 



K2 



