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V. Tmplicational and Equational Logic. 

 By Hugh McColl, B.A., University of London*. 



PROF. JEVONS, in his new work, ' Studies in Deductive 

 Logic/ of which he has kindly sent me a copy, refers 

 to my papers in 'Mind ' and in the 'Proceedings of the London 

 Mathematical Society' in terms which might give rise to 

 some misapprehension as to the real nature of my symbo- 

 lical method. He says that " I reject equations in favour of 

 implications," and in so doing " ignore the necessity of the 

 equation for the application of the Principle of Substitution." 



Now, it is quite true that I reject equations in favour of 

 implications in those classes of logical problems (and they are 

 very numerous) in which implications lead to the simplest, 

 shortest, and most elegant solutions ; but there are other 

 classes of problems, especially in mathematics, which necessi- 

 tate the equational form of statement ; and in these I do not 

 hesitate to adopt it. The simple truth is, that my method 

 admits of both forms ; and, as a matter of fact, I employ both, 

 sometimes even in the same problem. In my first paper in 

 the Proceedings of the London Mathematical Society (which 

 treats of the limits of multiple integrals) I adopt the equational 

 form throughout ; in my second and third papers, which relate 

 entirely to questions of pure logic, I generally adopt the im- 

 plicational form, as the simplest and most effective ; while in 

 my fourth paper, which treats of probability, I mainly adopt 

 the equational form. 



As to the statement that "I ignore the necessity of the 

 equation in the application of the Principle of Substitution," 

 I am not quite sure that I understand what it means. I cer- 

 tainly recognize the principle that if a = j3 i then/(«)=/(/3), 

 or, as the rule may be expressed symbolically in my notation, 

 (a = /3) : {/(«) =/(/5) J- ; but I cannot in the least understand 

 what bearing this has upon the advantages or disadvantages 

 of my system of implications. 



The question whether the implication a : (S, or its equiva- 

 lent, the equation u=aj3, should be preferred in a symbolical 

 system of logic, must be decided on the broad grounds of prac- 

 tical convenience. I believe it may be taken as a useful 



clearly kept in view, viz. that this assumption or theory, by opening out 

 an absolutely limitless field of speculative hypothesis, completely annihi- 

 lates all method or rational system in physical inquiry, and. therefore that 

 all progress or insight into the physical processes underlying phenomena 

 is absolutely brought to a standstill so long as this theory is adhered to. 

 * Communicated by the Author. 



