Page:Mind (New Series) Volume 6.djvu/516

 500 HUGH MACCOLL: I think I may predict that synonyms are destined to play an important part in the future development of symbolic logic, as they undoubtedly have done in the natural evolu- tion of ordinary language. As new needs and new ideas arise with the growth of civilisation and the general advance of humanity, do we not often find that two words which were at first synonyms gradually differentiate and, while still remaining synonymous in some combinations, cease to a be so in others ? just as a^ and -5- are interchangeable in the equivalent statements a e @ e ap and o^o, but not in the non-equivalent statements a e j3'a,p and a e yS 5, of which the second, but not the first, is always an impossibility. And does not this principle of evolution powerfully contribute to the precision and utility of a language, both as an instru- ment of research and as a medium for communicating our ideas to others ? In connexion with these remarks it will not be irrelevant to mention that before the idea of differ- a, entiating between the symbols a$ and -~ occurred to me I was in the habit of using sometimes the one and sometimes the other as a synonym for the general implication a : /3 ; so that when I afterwards found it convenient to give symbolic expression to the idea of a causal implication I at the same time found a suitable symbol for the purpose ready to my hand. In ordinary speech not only do we find different words with the same meaning, but also different meanings to the same word; yet it seldom happens that this leads to real ambiguity : the context nearly always removes all danger of miscomprehension. The same rule holds good in symbolic logic, with this difference that here, from our complete liberty to define our symbols as we please, we can guard against the danger with absolute certainty. In my first paper in the Proceedings of the Mathematical Society a paper which deals almost exclusively with the limits of multiple integrals and with probability I employed each of the symbols x ly x 2, x 3 , etc., y lt y 2 , 3/3, etc. (and so on for any number of variables x, y, z, etc.), in two different senses in the same argument; the symbol y 8, for example, denoting the eighth limit of the variable y registered in an accompany- ing table of reference, and also denoting the statement that this limit y s is positive ; yet this double signification of the