We can discern four departments of logical theory. By an analogy which is not so very remote I will call these departments or sections the arithmetic section, the algebraic section, the section of general-function theory, the analytical section. I do not mean that arithmetic arises in the first section, algebra in the second section, and so on; but the names are suggestive of certain qualities of thought in each section which are reminiscent of analogous qualities in arithmetic, in algebra, in the general theory of a mathematical function, and in the mathematical analysis of the properties of particular functions. The first section--namely, the arithmetic stage--deals with the relations of definite propositions to each other, just as arithmetic deals with definite numbers. Consider any definite proposition; call it "_p_." We conceive that there is always another proposition which is the direct contradictory to "_p_"; call it "not-_p_." When we have got two propositions, _p_ and _q_, we can form derivative propositions from them, and from their contradictories. We can say, "At last one of _p_ or _q_ is true, and perhaps both." Let us call this proposition "_p_ or _q_." I may mention as an aside that one of the greatest living philosophers has stated that this use of the word "or"--namely, "_p_ or _q_" in the sense that either or both may be true--makes him despair of exact expression. We must brave his wrath, which is unintelligible to me. We have thus got hold of four new propositions, namely, "_p_ or _q_," and "not-_p_ or _q_," and "_p_ or not-_q_," and "not-_p_ or not-_q_." Call these the set of disjunctive derivatives. There are, so far, in all eight propositions, _p_, not-_p_, _q_, not-_q_, and the four disjunctive derivatives. Any pair of these eight propositions can be taken, and substituted for _p_ and _q_ in the foregoing treatment. Thus each pair yields eight propositions, some of which may have been obtained before. By proceeding in this way we arrive at an unending set of propositions of growing complexity, ultimately derived from the two original propositions _p_ or _q_. Of course, only a few are important. Similarly we can start from three propositions, _p_, _q_, _r_, or from four propositions, _p_, _q_, _r_, _s_, and so on. Any one of the propositions of these aggregates may be true or false. It has no other alternative. Whichever it is, true or false, call it the "truth-value" of the proposition. The first section of logical inquiry is to settle what we know of the truth-values of these propositions, when we know the truth-values of some of them. The inquiry, so far as it is worth while carrying it, is not very abstruse, and the best way of expressing its results is a detail which I will not now consider. This inquiry forms the arithmetic stage.