SYMBOLIC REASONING. 513 arising from the possible variation of our standard unit of time. And so with all our units of comparison. For theoretical reasoning, as well as for the practical needs of daily life, we find it convenient to assume our units constant while other things vary ; so we make this assumption the basis of all our reasoning without needlessly saddening our minds by a too logical analysis of its legitimacy. 13. Symbolic logic too has its paradoxes, that is to say, formulae which appear paradoxical till they are explained, and then cease to be paradoxes. Such is the formula 77 : e, which asserts that "an impossibility implies a certainty". As soon as we define the implication A : B, by which we symbolise the statement that " A implies B," to mean simply (AB') 17 , which asserts that the affirmation A coupled with the denial B' contradicts our data or definitions, the paradox vanishes. For then 77 : e is seen simply to mean (ye) 11 , which is a clear truism. 1 14. Another paradox at first sight is the statement that the simple affirmative A, though equivalent to A T , which asserts that A is true, is not synonymous with A T ; and that, in like manner, the denial A', though equivalent to A', which asserts that A is false, is yet not synonymous with A'. Other symbolic systems, it is true, do not draw this distinction ; but mine does, and so, I believe, do all civilised languages. The fact that they do is a prima facie presumption in favour of my opinion that the distinction is real and corresponds to a logical need. Surely it is more than a coincidence that every civilised language should have two separate expressions, one corresponding (taking an example at random) to the English statement " It rains," and another to the English statement " It is true that it rains " ; and also two separate expressions, one corresponding to " It does not rain," and another to "It is false that it rains ". The two state- ments A and A T are equivalent because neither can be true without the other being so also ; but they are not synony- mous ; otherwise we could always substitute A T for A and vice versa in no matter what expression without altering its meaning. That this cannot always be done may be shown as follows. Let A denote a variable statement 6 r , that is to say, a statement, such as "It rains," which we assume to be true now or in the case considered, but which is not neces- sarily or always true. Then, on the one hand, we get 1 Observe that though rj x : t y , like t x : t y , is a formal certainty, yet Tit .'. t y , unlike t x .'. f v , is a formal impossibility. Non-Euclideans seem to ine to forget this when they say that their systems are as " self-con- sistent " as the Euclidean.
