96 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



The opponents of the strengthened syllogisms are universals weakened in their conclusions. 



The change of the figure from XY YZ XZ to ZY YX ZX is merely the writing of the 

 symbol from the other end : it turns )))) into (((( and ()).( into ).((). Those syllogisms 

 which agree in their parentheses and differ only in their affirmations and negations are the con- 

 comitants of my work (pp. 88, 89, 94), or the coexistents in a complex syllogism. Thus A X A X A X , 

 O A x O x and A x ] 1 are )))) ).))) and ))).). These syllogisms coexist in my complex syllo- 

 gism X),i)iX)| ; and perhaps, then, the best notation for this last might be derived from ).)).), 

 indicative of the only one of the four combinations which is not valid as a simple syllogism. 



I need not enter further into this subject, as what is here given on notation may be easily 

 applied throughout my work. 



Section IV. 



ON THE SYMBOLIC FORMS OF THE SYSTEM IN WHICH ALL THE COMBINATIONS 

 OF QUANTITY ARE INTRODUCED BY ARBITRARY INVENTION OF FORMS 



OF PREDICATION. 



This system, which belongs to Sir William Hamilton, has not yet been published in detail 

 by its learned author, except in lectures; in which, I believe, it was first published in 1840 or 

 1841. There is some account of its forms, communicated by him, and printed with his sanc- 

 tion, in Mr. Thomson's Outlines, already cited. See also F. L. pp. 300 — 302. 



The modes of predication in this system, are, by hypothesis, as follows, at least when the 

 language of extent, preferred by Sir William Hamilton, is changed into that of numeration of 

 instances. The symbols attached are dictated by the quantities of the terms, with reference to 

 the order XY. 



All Xs are all Ys 



Some Xs are some Ys 



All Xs are some Ys 



Some Xs are all Fs 



Some authors had gone so far (F. L. Appendix n.) as to adapt expressed quantity to the 

 predicate, for the purpose of procuring convertible forms : and Mr. Thomas Solly (Syllabus of 

 Logic, 1839, p. 47) gave the above eight forms, with his reasons for reducing them to four. 

 But Sir William Hamilton is the first who published the idea of taking all phases of usual 

 quantification, and making them the basis of a system of syllogism. 



It will be observed that I have not called this system an extension of that of Aristotle. 

 That it is more extensive, in one sense, I admit ; namely, in so far as it includes all which 

 Aristotle included, and more. But a mathematician cannot therefore call it an extension, 

 accustomed as he is to a very precise use of that term. With him enlargement is not exten- 

 sion, unless the wider extent be governed by the laws of the narrower one. The name multi- 

 plication, conceived by aid of integer numbers, is properly allowed to be extended to fractional 



