Page:Mind (New Series) Volume 12.djvu/370

 356 HUGH MACCOLL : 3. They use their symbol < in one sense in their logic of class inclusion, and in quite a different sense in their logic of propositions. I always use my symbol : in one and the same sense throughout, and a sense different from each of the meanings which they attach to their symbol -c. 4. Even to the symbol of equivalence = they attach two different meanings ; and neither meaning corresponds exactly with that which the same symbol bears in my system. 5. They divide propositions into two classes, and two only> the true and the false. I divide propositions not only into true and false, but into various other classes according to. the necessities of the problem treated ; as, for example, into certain, impossible, variable ; or into known to be true, known to- be false, neither known to be true nor known to be false ; or into formal certainties, formal impossibilities, formal variables (i.e., those which are neither] ; or into probable, improbable, even (i.e., with chance even) ; and so on ad libitum. 6. They make no distinction between the true and the certain, between 'the false and the impossible ; so that, in their system, every uncertain proposition is false, and every possible proposition true. In other words, variable propositions- propositions that are possible but uncertain, propositions whose chance of being true is some proper fraction between and 1 are excluded entirely from their universe. Many of their formulae are therefore not formal certainties ; they are only valid conditionally, and this defect, if it does not wholly destroy their utility, restricts within comparatively narrow limits their ranges of application. 7. Implications and other propositions of different orders or degrees, 1 such as (A : B) : (C : D), (A : B) et, A. 99 , A**?, etc., are not recognised (at least in my sense of the words) in other systems ; so that the whole world of new ideas opened up by this exponential or predicative system of notation is a world with which they are utterly unable to deal ; the bare attempt on the part of logicians would lead to a general break-up of all the systems now taught and a recasting of the whole of logic on different principles. This would be tantamount to the universal adoption of my system in all its essentials. Human nature being what it is, and professional prejudices being what they are, and what they can hardly help being, such a general recognition of the superiority of my system is hardly to be expected just yet ; but I think it will come in 1 For example my (A : B) : (C : D) means {(AB')>?<- + (CD'Xs whereas their (A -< B) -< (C -< D) means simply AB' + (CD')', and is therefore only a statement of the first degree.