172 PROFESSOR Boole's mathematical theory 



professes to have, any sucli method as that here described. But can 

 it justly, on that account, be charged with incompleteness? A 

 science must not, because it does not teach everything, be therefore 

 reckoned incomplete : enough, if it teaches the whole of its own 

 proper circle of truths. The special question which the scholastic 

 logic proposes to itself is : what are the ultimate abstract forms 

 according to which all the exercises of the discursive faculty pro- 

 ceed ? The science is complete, because it furnishes a perfect answer 

 to this question. 



But, it may be said, is it not desirable to have a method enabling 

 us certainly to determine, in every case, the relation which any of 

 the concepts explicitly or implicitly entering into a group of premi- 

 ses bear to the others ? Most desirable. And herein consists the 

 real value of Professor Boole's labours. He has devised a brilliantly 

 original Calculus b}'' which he can, through processes as definite as 

 those which the Algebraist applies to a system of equations, solve 

 the most complicated problems in the science of inference — problems 

 which, without the aid of some such Calculus, persons most thoroughly 

 versed in the ordinary logic might have no idea how to ti'eat. In 

 expressing our dissent, as we have been obliged very strongly to do, 

 from much that is contained in Professor Boole's treatise, we have 

 no desire to rob that eminent writer of the credit justly belonging 

 to him. Our wish has been simply to separate the chaff from the 

 wheat, and to point out accurately what constitutes, as far as the 

 " Investigation " is concerned. Professor Boole's claim to renown. 



Our readers will, however, be now anxious to obtain some fuller 

 information regarding the method about which so much has been 

 said, and which is the same with " the more general process " under 

 which the processes of the scholastic logic are held by Professor Boole 

 to be comprehended. This part of our article must necessarily be 

 altogether technical ; and we shall require to ask our readers to take 

 a few things on trust ; but we hope to be able to present the sub- 

 ject in such a manner as to give at least some idea of the system 

 we are to endeavour to describe. Those who desire to become 

 thoroughly acquainted with it will of course study the "Investiga- 

 tion " for themselves. 



We begin by referring to the development of logical functions. 

 An expression which in any manner involves the concept x, is called 

 a function of the concept, and is written/" (a;). Now there is one 



