170 PROFESSOR Boole's mathematical theory 



immediate comparison of JTand Y, then no concepts enter into the 

 argument except jf and Y, and the argument is reduced to conver- 

 sion. But if the conclusion be drawn mediately, it must be by the 

 comparison of Xand T'with some third thing: not with a plurality 

 of other things, but with some single thing. Here we have the mind 

 drawing its inference in a syllogism. What the various admissible 

 forms of conversion and syllogism may be, or whether these forms 

 have been correctly specified by particular eminent logicians, are 

 minor questions. The essential thing in a philosophical respect is, 

 that the mind, in the inferences which it draws, does and can work 

 in no other moulds than those described. All this seems to us so 

 plain that we confess ourselves utterly puzzled to comprehend how 

 men of profound and original genius have been beguiled into an 

 assertion of the contrary. 



Professor Boole himself, in summing up his assault on the Aristo- 

 telian Logic, comes very near admitting what we contend for. " As 

 Syllogism," he says, "is a species of elimination, the question before 

 u>i manifestly resolves itself into the two following ones: 1st. 

 Whether all elimination is reducible to Syllogism ; 2nd. Whether 

 deductive reasoning can, with propriety, be regarded as consisting 

 only of elimination. I believe, upon careful examination, the true 

 answer to the former question to be, that it is always theoretically 

 possible so to resolve and combine propositions that elimination may 

 subsequently be effected by the syllogistic canons, but that the pro- 

 cess of reduction would in many instances be constrained and unna- 

 tural, and would involve operations which are not syllogistic. To 

 the second question I reply, that reasoning cannot, except by an 

 arbitrary restriction of its meaning, be confined to the process of 

 elimination." With regard to this second question, we merely note 

 in passing, that we have proved in the preceding paragraph that in- 

 ference, where not immediate or of the nature of. conversion, can be 

 nothing else than elimination. It is, however, with the first ques- 

 tion, whether elimination is reducible to syllogism, that we have now 

 more particularly to do ; and we accept with satisfaction the admis- 

 sion, guarded and (to some extent) neutralised as it is, that every line 

 of argument may be thrown into a form in which the eliminations that 

 take place are effected by the syllogistic canons. It is quite irrele- 

 vant to notice, as Professor Boole does, that the process of reduction 

 would, in many instances, be constrained and unnatural ; for we are 



