THE THEORY OF SYLLOGISM, ETC. 87 



ment ; let an odd number, usually one, denote negation or non-agreement. Thus X )) Y means 

 that all Xs are Fs ; X(.( Y means that some Xs are not Fs : but )) and (.( specify the charac- 

 ters of the propositions ; as do also (( and ).). We must conceive ourselves at liberty to read 

 either way : thus X)) Y and Y((X both denote that every X is F. 



A syllogism may be denoted by juxtaposition of the symbols of the premises, taking the 

 'order XY, YZ, XZ. Thus 'Every AT is F, some Zs are not Fs, therefore some Zs are not 

 Xs ' may be stated thus, 



X))Y).)Z gives X).)Z or ))).) gives).). 



In the Aristotelian system, and in my extension, the canon of formation of the inference, 

 when there is one, is ; — Erase the symbols of the middle term, the remaining symbols shew 

 the inference. Thus, if there be a valid inference from ).( (.) it is ). .) or )). Thus also 

 X()Y))Z expresses the premises 'Some Xs are Fs, and every F is Z: 1 erase F and its 

 accompaniments, and we have XQZ for the conclusion, or ' Some Xs are Zs.' In Sir William 

 Hamilton's system, this law is not quite universal in the symbolic deduction of the inference ; 

 but a certain variation, which I shall presently suggest, will make it so. 



If we wish to read by distinction of figure, that is, by Aristotelian figure, we may contrive 

 it thus. Let the subject of each proposition have its quantity denoted by a thicker or larger 

 parenthesis. Then the first figure, in which we read through the concluding terms, would 

 present the appearance | ] ; remembering that when the distinction of major and minor term of 

 conclusion is preserved, we read the second premise first. Thus 



X))Y))Z = X))Z, which is now symbolized in the first figure, is Y))Z + X)) Y = X))Z. . 



The second figure, in which we read to the middle term, is as seen in | |; the third, in which 

 we read from the middle term, is as seen in |||| ; and the fourth, in which we read through the 

 middle term, is as seen in ||||. 



All notation, no doubt, is both pictorial and arbitrary : nevertheless there are cases in 

 which one or the other character decidedly predominates. The arbitrary character decidedly 

 predominates in the preceding notation : but the syllogism admits of a graphical representation 

 which is as suggestive as a diagram of geometry. This was partially adopted by Lambert 

 and Euler (F. L. p. 323), and may be more completely shewn, and without new types or wood- 

 cuts, in the following way. 



Let all the instances in the universe of the syllogism be represented by the points of a 

 definite straight line : but to avoid confusion, let this straight line be repeated as often as it is 

 necessary to introduce a name. Let the division of this straight line into a continuous and 

 a dotted portion signify the distribution of the universe into a name and its contrary. When 

 a proposition is asserted, let a second line run over so much of the extent of each name as is 

 declared by the proposition to be in agreement or disagreement with the whole or part of the 

 other : extents which fall under one another being taken as in agreement. Thus in the follow- 

 ing diagrams we see the propositions ' Every X is F,' and ' Some things are neither Xs nor Fs, 1 



X L= X 



Y U== Y 



