THE THEORY OF SYLLOGISM, ETC. 97 



ones, because, among other reasons, in every problem in which integers demand integer multi- 

 plication, fractional data demand what is thence called fractional multiplication. Botany was 

 once agriculture, but in its present state it cannot properly be called an extension of agricul- 

 ture : the union of England and Scotland was not an extension of England. This refusal to 

 use the word extension, in the present case, is not the assertion of any defect in the system, but 

 rather the contrary : it is quite open to inquiry whether the best form of syllogism be what I 

 call an extension of Aristotle, or contain the incorporation of new fundamental principles. 



Perhaps some may ask why I have called my own system an extension of Aristotle. I 

 answer that no new laws are propounded, though the application of the old ones to an enlarged 

 subject-matter of predication introduces some new forms of expression, and some striking points 

 of view from which to look at the old ones. Every one of my syllogisms can be reduced to an 

 Aristotelian form, without any addition except that of contraries to the matters of predication. 

 For example, one of my new syllogisms, ))(( = )(> or ' -All As and all Zs are Ys, therefore 

 some things (namely, all that are not Ys) are neither As nor Zs — is reducible to ordinary 

 form. With X Y Z it is new : but with X y % it is Fesapo, being 



X).(y))x = X).)z. 



The syllogism ))(( = )( can thus be made Aristotelian : but in my system, a plain man 

 who sees clearly that some things are proved to be neither men nor mice, were it only because 

 they do not eat cheese, may rest content that his knowledge, even in the form of the light of 

 nature, can be made science, without the necessity of having recourse to the following very 

 venerable, but very unsatisfactory, form : 



No man is a non-eater of cheese 

 All non-eaters of cheese are other things than mice 

 Therefore some other things than mice are also not men. 

 Now take one of Sir William Hamilton's peculiar syllogisms ; — 

 Some men are soldiers, Some animals are not men, 

 Therefore some soldiers are not some animals. 



This syllogism cannot in any way be made Aristotelian, either with the terms as they stand, or 

 with any others derived from them by a method independent of the syllogism itself; for instance, 

 the derivation of the contrary from the direct term. Sir William Hamilton's system is there- 

 fore an independent addition to that of Aristotle ; and the addition must be discussed on its 

 own merits. 



The forms )) (( ).) (.( ( ) ).( exist in the old system, in that of contraries, and in that 

 of invention of predicates. The peculiar propositions of the second and third may therefore be 

 compared as follows, and under the same* symbols : 



* Looking at the fact that the system of contraries admits 

 premises both negative, and that of invention of predicates 

 admits premises both particular, with other analogies which I 

 do not describe here, — I strongly suspect that the two systems 

 have some correlative formation in which the distinction of 

 affirmative and negative appears in the first with the same laws 



of form under which that of universal and particular appears 

 in the second; and vice versa. If this be the case, then (•) 

 though it have a particular form in common language, will be 

 a universal, and )( a particular. This is a hint for the conside- 

 ration of the reader : I have not been able to make anything 

 out of it, as yet. 



Vol. IX. Part I. 13 



