ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 



451 



(Book V. Prop. 7) having proved that the parallel to a certain line is the perpendicular, 

 he gives further proof, of the kind' above, that the perpendicular is the parallel. And this 

 is supposed to be the climax of rigour; proof by syllogism that if the sole A be the sole B, the 

 sole B is the sole A. But where would syllogism have been, if this had not been true ? 



It must not be forgotten, in defence of Euclid, and of geometry without logic, that the 

 above procedure may give evidence. When thought has not been analysed, and those who 

 teach are determined that it shall not be analysed, Euclid presents the perfection of the way 

 of doing without. But the time must come when his rich mass of raw thought shall be the 

 material of exercise for logical analysis ; when it shall be employed to place the forms of 

 thought in their due order of sequence ; when it shall be the ground on which it shall be 

 learnt that the conversion of identity by help of syllogism is reasoning in a circle. 



I shall proceed to connect the exemplar form with others: but there are several points 

 which it will first be desirable to notice. 



By a restrictive proposition I mean one which, of its own nature, imposes some absolute 

 condition, positive or negative, upon the quantity of one or both of its terms, or of one or 

 both of the contraries of its terms. I say absolute condition : not relative, as in ' All X is 

 Y', which demands that the Ys shall at least equal the Xs in number. The only such 

 propositions yet met with are • Some X is not some Y', which requires that, when identical, 

 X and Y are not singular: and 'Any X is any Y' which imposes on X and Y both singu- 

 larity and identity. But besides singular identity, we shall find ourselves, so soon as we 

 begin to carry every mode of enunciation into every case, obliged to recognise penultimate 

 identity, in which the contraries of our two terms are singular and identical ; also singular and 

 penultimate idetUity, in which both' our terms and their contraries are singular and iden- 

 tical ; and singular and penultimate contrariety, in which two singular terms are each iden- 

 tical with the contrary of the other. The laws of thought will produce these forms' by 

 the score, as we shall see. 



' We may laugh at the geometer establishing by syllogism 

 the conversion of identity, but such is the force of habit that 

 the logician may be a geometer without carrying away into 

 logic the illustrations which lie nearest the surface. My op- 

 ponent, Mr Mansel — out of formal logic — is a mathematician, 

 and applies psychological thought to first principles. In formal 

 logic he argues in favour of " All A is all B" being a simple 

 proposition, in opposition (iv. 116) to my assertion that it is 

 complex; and Hamilton quotes his argument with approbation. 

 Mr Mansel says " I cannot assert ' all A is B and all B is A ' 

 without having thought of A and B as coextensive, i.e. without 

 having made the judgment 'all A is all B'." Euclid (i. 6), 

 the universe being triangle, proves that " all isosceles is isogo- 

 nal", and then (i. 6), proves that "all isogonal is isosceles"; 

 and then, and not till then, does his reader become aware that 

 " all isosceles is all isogonal." Both the components are in 

 thought before the compound. Geometry is the richest field of 

 coextensive notions: it swarms with instances of coextension 

 gained by synthesis of counter-inclusions. I admit that a com- 

 pound cannot be decomposed except by those who have got it 

 to decompose: but, on the other hand, those who have hold of 



the component"! may put them together. In the dining-room 

 pudding may be treated as compound of flour and plums: 

 but if before that, in ihe kitchen, flour and plums had not been 

 treated as components of pudding, the dining-room process 

 would have been Barmecide theory. 



^ I am duly sensible of the figure which a universe of two 

 instances will cut : but I may say on my own behalf, that 

 though 1 shook it out of the pepperbox, I did not put it in. 

 The laws of thought, which did put it in, are solely responsible 

 for this contempt of established authority. Nor can I even 

 claim the invention of the mode of shaking which brought it 

 out. Hamilton had used the method, and produced, if not 

 singular identity, at least its denial : this was the first of the 

 class of restrictives. I think that here, as elsewhere, it will be 

 found that one instance is but ill understood until more 

 arrive. 



" By introducing "some X is not some Y ", the denial of a 

 restrictive, Hamilton, when nonpartitively interpreted, has 

 given a conclusion to two invalid forms (■( )) and (( ). ). It 

 will presently be pointed out that every one of the thirty-two 

 invalid forms gives a conclusion, the denial of a restrictive. 



