CHAP. V.] PRINCIPLES OF SYMBOLICAL REASONING. 67 



it is necessary to restrict the application of these symbolical laws 

 and processes by the same conditions of interpretability under 

 which the knowledge of them was obtained. If such restriction 

 is necessary, it is manifest that no such thing as a general 

 method in Logic is possible. On the other hand, if such restric- 

 tion is unnecessary, in what light are we to contemplate pro- 

 cesses which appear to be uninterpretable in that sphere of thought 

 which they are designed to aid ? These questions do not belong 

 to the science of Logic alone. They are equally pertinent to every 

 developed form of human reasoning which is based upon the 

 employment of a symbolical language. 



3. I would observe in the second place, that this apparent 

 failure of correspondency between process and interpretation does 

 not manifest itself in the ordinary applications of human rea- 

 son. For no operations are there performed of which the mean- 

 ing and the application are not seen ; and to most minds it does 

 not suffice that merely formal reasoning should connect their 

 premises and their conclusions ; but every step of the connecting 

 train, every mediate result which is established in the course of 

 demonstration, must be intelligible also. And without doubt, 

 this is both an actual condition and an important safeguard, in 

 the reasonings and discourses of common life. 



There are perhaps many who would be disposed to extend 

 the same principle to the general use of symbolical language as 

 an instrument of reasoning. It might be argued, that as the 

 laws or axioms which govern the use of symbols are established 

 upon an investigation of those cases only in which interpretation 

 is possible, we have no right to extend their application to other 

 cases in which interpretation is impossible or doubtful, even 

 though (as should be admitted) such application is employed in 

 the intermediate steps of demonstration only. Were this ob- 

 jection conclusive, it must be acknowledged that slight ad- 

 vantage would accrue from the use of a symbolical method in 

 Logic. Perhaps that advantage would be confined to the mecha- 

 nical gain of employing short and convenient symbols in the 

 place of more cumbrous ones. But the objection itself is falla- 

 cious. Whatever our d priori anticipations might be, it is an 

 unquestionable fact that the validity of a conclusion arrived at 



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