441. Let us now inquire what is involved in the conception of two, and in particular by what features a pair is distinguished from a single one, on the one hand, and from three, or any larger set, on the other.
442. A mathematician will be inclined to pronounce this the most ridiculously trifling question to be called a problem that could well be imagined. A pair, he may say, is just an object and an object, and that is all that is involved in this puffed-category of the dyad. But any logician will tell him that that statement, at any rate, is inaccurate. For the purposes of the logic of mathematics it is fatally inaccurate. A married couple is not a man. Neither is it a woman, and a fortiori it is not, at once, a man and a woman. Nor is it disjunctively either a man or a woman. It is a third object, to whose constitution, which is its nature, and therefore to its existence, too, a man is requisite and a woman is requisite. A pair is an object to whose constitution a subject and another subject are necessary and sufficient. This corresponds to a part of feature number eight of fact.
443. But accepting this amendment, which to his customary way of thinking is microscopic, the mathematician will be inclined to say, here is a perfect definition; and excepting a few little corollaries, there is nothing more to be said of the dyad. It behooves me, then, to clearly state what the inquiry is which I propose to institute. It is not a mathematical inquiry; because the business of the mathematician is to frame an arbitrary hypothesis, which must be perfectly distinct at the outset, so far, at least, as concerns those features of it upon which mathematical reasoning can turn, and then to deduce from this hypothesis such necessary consequences as can be drawn by diagrammatical reasoning. The present problem is one of logical analysis. Instead of setting out with a distinct hypothesis of a diagrammatic kind, we have the confused fact that a dyad is a conception of the highest utility, though we are not prepared to say exactly what its nature is, nor even, in all cases, whether a given case should properly be reckoned as a duality or not. We are somewhat in the position of a naturalist who knows that whales are large swimming animals, which spout water, and yield blubber, spermaceti, and whalebone, but knows little else about them, and who proposes to himself to examine the anatomy and physiology of whales so as to assign them their place in the system of the animal kingdom. He does not intend to preserve the popular description nor delimitation of the class of whales. He will perhaps see reason to extend the name to some animals not popularly called whales and to refuse it to others that are so called. He will also subdivide the group, and classify it according to the facts. So far as our inquiry is a logical analysis, the greatest difference between it and that of a taxonomic biologist consists in the circumstance that we are not forced to institute special observations, because all the facts are either well known or can be ascertained by careful reflection upon those that are known.
444. But besides being logical in the sense of demanding a logical analysis, our inquiry also relates to two as a conception of logic. The term »logic« is unscientifically by me employed in two distinct senses. In its narrower sense, it is the science of the necessary conditions of the attainment of truth. In its broader sense, it is the science of the necessary laws of thought, or, still better (thought always taking place by means of signs), it is general semeiotic, treating not merely of truth, but also of the general conditions of signs being signs (which Duns Scotus called grammatica speculativa1)), also of the laws of the evolution of thought, which since it coincides with the study of the necessary conditions of the transmission of meaning by signs from mind to mind, and from one state of mind to another, ought, for the sake of taking advantage of an old association of terms, be called rhetorica speculativa, but which I content myself with inaccurately calling objective logic, because that conveys the correct idea that it is like Hegel's logic. The present inquiry is a logical one in the broad sense. It is a study of dyads in the necessary forms of signs.
Our method must be to observe how logic requires us to think and especially to reason, and to attribute to the conception of the dyad those characters which it must have in order to answer the requirements of logic.
445. We can at once see that a pair, having a structure, must present a variety of features; and this is a character in which the dyad differs markedly from the monad, which having no structure nor parts in any sense, is bare of all features except that each one is something peculiar. This corresponds to feature number one of fact.