'68 PROFESSOR Boole's mathematical theory 



The former of these questions may, with perfect confidence, be 

 answered in the affirmative. It admits of absolute demonstration, that 

 there is no chain of valid inference which the ordinary logic is incompe- 

 tent to express, or, in other words, which is not reducible to conversion 

 or syllogism. Some logicians have been of opinion that conversion 

 is nothing else than syllogism at bottom ; but, for what we have at 

 present in view, it is unnecessary to discuss this question. Suffice 

 it to say, that, whether conversion and syllogism be substantially 

 identical or not, all immediate inference is of the nature of conver- 

 sion, and all mediate inference (or reasoning proper) of the nature of 

 syllogism. Dofes Professor Boole deny this ? Formally, and in plain 

 terms. " Possibly," he writes, " it may here be said that the logic 

 of Aristotle, in its rules of syllogism and conversion, sets forth the 

 elementary processes of which all reasoning consists, and that beyond 

 these there is neither scope nor occasion for a general method. I 

 have no desire to point out the defects of the common logic, nor do I 

 wish' to refer to it any further than is necessary, in order to place in 

 its true light the nature of the present treatise. With this end alone 

 in view, I would remark : 1st. That syllogism, conversion, &c,, are 

 not the ultimate processes of logic. It will be shown in this treatise 

 that they are founded upon, and are resolvable into, ulterior and 

 more simple processes which constitute the real elements of method 

 in logic. Nor is it true that all inference is reducible to the partic- 

 ular forms of syllogism and conversion. 2nd. If all inference were 

 reducible to these processes alone (and it has been maintained that 

 it is reducible to syllogism alone), there would still exist, &c." In 

 illustration of the statement, that some inference is not reducible to 

 the forms of syllogism and conversion, Professor Boole examines the 

 case of conversion, and arrives at the result that " conversion is a 

 particular application of a much more general process in logic, of 

 which," he adds, "" many examples have been given in this work." 

 In like manner he examines the case of syllogism ; and his conclusion 

 i;i as follows : " Here, then, we have the means of definitely resolv- 

 ing the question, whether syllogism is indeed the fundamental type 

 of reasoning, — whether the study of its laws is co-extensive with the 

 study of deductive logic. For if it be so, some indication ot the fact 

 must be given in the system of equations upon the analysis of which 

 we have been engaged. No sign, however, appears that the discus- 

 sion of all systems of equations expressing propositions is involved i© 



