494 HUGH MACCOLL : power within the limits of its application ; and thus it will be that the newer but more general science of symbolic logic, with a wider sweep and bolder aim, will before long develop into a still more powerful instrument of research. The problems with which our reason has to deal are of various kinds and cannot be exhaustively classified. Sometimes from certain data or premises A, known or admitted to be true, we have to prove a conclusion Q ; that is to say, we seek to convince ourselves or others that Q as well as A is true. Sometimes we have data or premises P which are not always certain or admitted to be true, and then we seek to prove not the conclusion Q that cannot be done from uncertain or unadmitted data but the proposition that P implies Q, i.e., that if P is true Q is true which is quite another matter. Some- times our reason has to deal with the inverse problem, namely, to find from what data or premises Q we can derive a conclusion P, as when we seek the possible roots of a given equation. These are only two or three of the kinds of problems of which it is the business of logic to find solutions, but as they are among the commonest we will take them first in the order of consideration. Since the conditional proposition If P is true Q is true (or P implies Q) is one of frequent recurrence we want some symbol to represent it. What symbol should we adopt ? Various logicians have adopted various symbols, each giving some reason founded on some mathematical analogy for his own special choice. Boole adopts P = ^ Q or P = v Q ; Pierce takes P < Q ; Schroder uses P ~(rzi Q ; and there are many others ; each writer, as I have said, justifying his choice on the ground of some real or fancied mathematical analogy. My own choice has been the symbol P : Q, not (as has been erroneously supposed) on the ground of any analogy to a ratio or division, but simply because a colon symbol is easily formed, occupies but little space two important considerations and though this is less important because it is not unpleasing to the eye. I hold that we may claim the same liberty of definition and interpretation for any of our symbols of relation ( + , =, :, etc.) as we claim for any letter of the alphabet, x, when in one problem (some unit of reference being understood) we say " Let x denote his gain " ; in another " Let x denote his loss " ; and in another " Let x denote the distance of the planet Neptune". So long as it suits our purpose to attach the same meaning to any symbol, so long we should adhere to that meaning so long,
