390 



PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGIS3I, 



stand for X)Y, A". r(= V.X), Xl'{=yX) and X.V. And a, e, i, o stand for the same propo- 

 sitions when both subject and predicate are changed into contraries : that is, they stand for a? ) y, 

 .V .y {= y . x), .vy(=yiv), w.y. In these last, system is sacrificed to simplicity in using Y)X 

 and 1' : X for ,r ) y and se : y. The following table shows what transformation takes place when the 

 terms and orders are successively XY, Xy, wy, a;Y, YX, yX, yx, Yx. 



Here we mean, for instance, that E of the order (X, Y) is the same thing as the A of (X, y,) or 

 the e of (a; y) &c, ; or that X . Y, X)y are the same. As to the e of {x, y) it is identically X . Y. 

 The eight operations by which the transformations of headings are made are those which I have 

 denoted by L, P, SP, S, T, PT, SPT, and ST, of which the simplest readings are 

 For inconvertibles LPTSTSLP, 

 For convertibles L P SP S L P SP P. 



Taking the terms XY, YZ, ZX, for the premises and conclusion, and the order of reference 

 XY, ZY, XZ, tlie notation given defines the syllogism. Thus, valid or not, the syllogism O,^ 

 can be nothing but x .y + Z ) Y -^ X : Z. In turning the fundamental syllogism into the form 

 X)Y + Y) Z = X) Z, I have altered Aristotle's order of the premises, which would give Y) Z 

 + X) Y = X ) Z. Reasoning direct from his dictum de otnni et nulla, namely, that what is true 

 or false of all is true or false of every some, it would seem natural first to ascertain the fact 

 relating to the whole, and then to introduce the part which is to be considered. But in another point 

 of view it may be more natural to reverse this order. If there be tiiree boxes P, Q, and R, of 

 which I want to ascertain by means of Q whether P will go into R, it seems to me more natural to 

 try first whether P will go into Q, and then whether Q will go into R. But if the question be 

 whether R will hold P, then perhaps it may be more natural to try first whether R will hold Q, 

 and then whether Q will hold P. It must be mere matter of opinion which should be taken; and 

 the idioms of the language which people speak produce the associations on which they will decide. 



The syllogisms which we have got as yet, four universal and eight particular, contain all those 

 of Aristotle. Six of them indeed are enough for this purpose : that is to say, every syllogism of 

 Aristotle is either one of these six, or one of them with a premiss converted or strengthened, or both. 

 The six really distinct syllogisms of the old system, with the order XZ established in the conclu- 

 sion, are as follows, with the scholastic names of the forms which they have or can be made to have : 



OOA 



0„„ 



X)Y+Y)Z=X)Z 

 XY + Y)Z = XZ 



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

 XY + Z . Y= X.Z 



X: Y+ Z)Y= X: Z 



Barbara, 



Darii, Darnpti, Disamis, Dntisi, Rramantip, Dimaris. 



Celarent, Cesare, Camestres, Cnnienes. 



Ferio, Festino, Felaptnn, Ferisn, Fesapii, Freaison. 



Baroko. 



Y) X+Y:Z = X: Z I Bokardn. 



