360 HUGH MACCOLL : narrow limits. If 1 and be represented by e and rj re- spectively, we get the same result. Now let me deal with points No. 1 and No. 7 and show that, as regards their valid formulae, other systems are im- plied in mine ; while mine, on the other hand, can work out problems and evolve new and fruitful ideas which their systems are unable even to express. First, as regards their logic of propositions. In my sixth paper on the " Calculus of Equivalent Statements," in the Proceedings of the Mathe- matical Society, I use a symbol "bx in the following sense. When x denotes a statement A% then *bx denotes A. Hence, when x denotes A n , ^x must denote A', for A 7 * is synonymous with (A') e . Also, when x denotes A : B, ^x must denote their statement A < B ; for A : B means (A' + B) e , and A -< B means A' + B. Thus my symbol <> (A : B) corresponds to their symbol A -< B ; and my symbol A : B would correspond to their symbol (A^cB) 6 if they adopted my notation of ex- ponents with my signification of the symbol e. On this understanding all the valid formulae of their logic of pro- positions could be transferred from their systems into mine. Also, on the understanding that all variable propositions should be left out of account, my A e would be equivalent to my (and to their) A ; my A 77 to my A' and to the corresponding symbol in their notation ; and my symbol A : B to their symbol A ^c B ; while my interpretation of the symbol = would then be the same as theirs. But this arbitrary and unnecessary restriction of our universe of admissible statements would rob logic of nearly all its utility, whether as a practical in- strument of scientific research (as in my Calculus of Limits), or as an educational instrument of mental training and culture. The inability of other systems to express the new ideas represented by my symbols A. xy , A. xi ' z , etc., may be shown by a single example. Take the statement A 09 . This (unlike formal certainties such as e r and AB : A, and unlike formal im- possibilities such as # e and 6 : rf) may, in my system, be a certainty, an impossibility, or a variable according to the special data of our problem or investigation. But how could it be expressed in other systems ? Not at all, for its recog- nition would involve an abandonment of their erroneous con- vention (assumed throughout) that true is synonymous with certain, and false with impossible. If they ceased to consider A as equivalent to (A = 1), and A' (or their corresponding symbol) as equivalent to (A - 0), and employed their (A = 1) as equivalent to my A e , and their (A = 0) as equivalent to my Aj>, they might then express my statement A. 09 in their
