446. A monad has no units. This sounds paradoxical, and seems to the mathematician an aperçu from an arbitrary point of view; but we soon see that it is the suitable point of view for logical purposes. In the pair there are unit parts; and so there are in all higher sets. Let us inquire, then, what is the function of the units of a set in the constitution of that set. We must first remark that in logic a set cannot generally be adequately represented by a diagram of a promiscuous collection of dots. Of the multitudinous examples of this in mathematics it will be sufficient to call to mind the constituents of a determinant, and how they have to be arrayed in a square block. As a general rule, the form of connection (or a part of it, at least) must be considered in logic in case a set has to be considered as such. This form of connection belongs to the set and not to its units. Now reasoning is formal. That is to say, whatever inference is sound concerning one thing or one character is sound in regard to any other thing or character whose form of connection (so far as it need be considered) is strictly analogous to that of the former. All that has to be represented, then, for the purposes of logic, is the characters of the sets themselves; and the units need exhibit nothing except what is requisite to the exhibition of the characters belonging to sets. What, then, is the use of the units, at all? And how can they, when thus stripped of all qualities, contribute to the representation of characters of sets? The answer is that if all that were desired was to present for contemplation the character of a set, the statement of the mode of its connection in abstract terms, with no particular reference to the units, would be sufficient; and in point of fact, this is the general form which metaphysicians give to their statements, so far as the usages of speech render it convenient. But when, one set having been represented, it is desired to attach to it the representation of another set, and there is a unit or units which belong to both sets, then in order to show how the total set is composed of those two sets, it is necessary to take account of the identities of their common units. Now identity is a relation which cannot be implied by a general description of the identical things; and the descriptions of the sets, so far as they leave out the individual things, are general. Hence, it follows that the only purpose in indicating the units in the representation of the set, is in order that each of them may signify its identity with an individual of another set. The identity of different units of the same set might be similarly represented. Hence, passing from the representation of the set, to the set itself, as it is logically conceived, the only function of the units in it is to establish possible identities with the units of other sets. A unit, therefore, is something essential to a set whose existence consists in its possible identity with another unit of the same or another set. Now, identity is essentially a dual relation. That is, it requires two subjects and no more. If three objects are identical, this fact is entirely contained in the fact that the three pairs of objects are identical. Hence a unit is something whose existence consists in a possible dyad of which it is the subject. Thus, there is an element of twoness in every set. So I was right in saying that the monad has no unit, since the monad in no wise involves the dyad.
447. There are certain truths about quality not considered in Section 2, for the reason that they were considered as belonging under the head of the dyad. They do not concern the monad in its aspect as one, but are dyads of monads. One of these is that whatever is a possible aspect irrespective of parts has possible parts. I mean that any object presenting a quality in its purity might be further determined. Every quality is, in itself, general. Given any possible determination, there is a possible further determination. In the beginning was nullity, or absolute indetermination, which, considered as the possibility of all determination, is being. A monad is a determination per se. Every determination gives a possibility of further determination. When we come to the dyad, we have the unit, which is, in itself, entirely without determination, and whose existence lies in the possibility of an identical opposite, or of being indeterminately over against itself alone, with a determinate opposition, or over-againstness, besides.
It follows that a set considered apart from its units is a monad. In fact, in not considering the units, we allow all sets of the same general character to collect before us, and regard those sets as a monad without parts.
But a set considered as made up of units in a peculiar connection is a dyad if its units are two, a triad if they are three, etc. A part of the above corresponds to feature number eight of fact.
448. Let us further examine the dyads of monads just mentioned. But before doing so, this designation has to be defended. It may be objected, with apparent force, that the truth just mentioned about quality is general, and that, as general, it applies to innumerable qualities and not to a pair. This is perfectly true; but then, all that we are inquiring into concerning the monad, the dyad, and plural sets, is general, and therefore, from that point of view we are not studying the monad and the dyad at all, but polyads of monads and dyads. This is true. Our thought is rational and, as such, general, or of the plural nature. But it relates to the monad and the dyad in part. So the truth just mentioned is general, but it relates to a single monad, any monad, and declares of it that a monad exists which is, for thought, equivalent to that monad further determined.
449. This is one of three regulative laws of logic of high importance which were enunciated by Kant in the Critic of the Pure Reason.1) The other two are that there is a determination less than and included in any possible determination, and that between any two determinations, one included in the other, a third may be found. Besides these dyads, both whose subjects are monads, there are also certain dyads, one of whose subjects is a monad and the other a possible dyad, that is, a unit. And there are general laws connected with these.
450. The first of these is that any unit (or units) whatsoever contemplated in itself without conscious regard to its parts would, were our sense to respond to it, be seen to embody a monad. De Morgan propounded this law, so far as it is pertinent to formal logic, affirming that any collection of objects whatsoever possess universally some character which belongs to no other object at all. For, said he, they at least possess the character of being units of that collection. Considered as a proof, this begs the question; but considered as another way of formulating the same phenomenon, and as a way which throws some light upon it, it has its value. This coincides with the principle of excluded middle. Those objects of the universe which do not possess a given character possess another character which, in reference to that universe, is in the relation of negation to the first. Hence, it is impossible to form a single class of dyads; two classes of dyads must be formed at once. Hence, considering all the monads which can appear on the contemplation of sets of units of the universe in their monadic aspect, every single unit is determined to be one subject of a dyad which has any one of those monads as its second subject, namely it is either such a dyad as determines it to have the character of being one of the units which made up the object of the contemplation in which that monad appeared, or it is such a dyad as determines the unit to have the character belonging to all the other units of the universe.
451. What has been here affirmed of collections of units is equally true of collections of monads. Namely, any monads may be contemplated together, and in their monadic aspect without regard to the single monads are seen to be one monad. There is thus a relation between monads similar to the relation of a unit to a monad. But there is this difference in the two cases: a monad thus embraced under another monad is so embraced in its very mode of being, while that a unit should be embraced under a monad has no concern with the mode of existence of the unit, which lies in its brute self-identity and otherness from all the rest. It is on the contrary an adventitious circumstance that this particular unit is embraced under this monad.
This corresponds to feature number seven and in part to feature number twelve of fact.