Page:Mind (New Series) Volume 6.djvu/511

 SYMBOLIC REASONING. 495 and no longer. As a general rule we should be as conser- vative as circumstances will permit in the significations we give to our symbols of relation ( +, =, :, etc.) : these are the constants, the fixed stars, as it were, of our logical systems ; while we may deal more freely with our planetary ever-varying symbols, x, y, a, b, etc., which generally denote numbers or ratios in mathematics, and classes, properties or statements in logic. But as even the so-called fixed stars are only found to be relatively fixed, so our so-called constant symbols of relation need only be relatively constant. In my former paper in MIND (" Symbolical Reasoning," MIND, Jan., 1880), and in my papers in the Proceedings of the Mathematical Society which had preceded it, I adhered throughout to the symbol a : /3 as my representative of an implication ; but when, after several years' abandonment, I recently returned to my logical studies and began to consider the complex relations of the higher orders of implications, as in the formula (a : 0) : { (u : a) : (u : /3), I felt the necessity of further abbreviations and adopted ap as a synonym of a : 0, so that the above formula might appear as ap : (u a : % ). I also tried how the symbol a" would act as the converse of a u and meaning that a is implied in u. This led imme- diately and of necessity to the discovery that the formulae a"a' = a" + ", ("/3) u = u"/3", and a" = 1 would on this inter- pretation hold good in logic as well as in mathematics ; but I founcl that the series of analogies stopped when I tried a" = (a**/. Then I tried a" as a substitute for a + u' (the alternative implied by u a ) and found that the preceding three analogies still held good, and also this fourth analogy a"" = (a")*, but that, on the other hand, this interpretation of a" would lead to an additional logical formula a"* = a" + a", which does not generally hold good in mathematics. If we take a" as synonymous with u a it is evident that every formula with indices may be converted at once into an equivalent formula with subscripts, and vice versa. Thus a u a* = a" + corresponds to u a i> a = (u + v) a (a/3) M = a"^" corresponds to u a/3 = u a u ft a ft : (u a : up) corresponds to /8 a : (a" : /"), and so on. The last formula may be read thus: "If @ is a factor of a (that is, if the statement @ is implied in the