The Principles of Mathematics (1903)

§ 59

The question which first meets us in regard to denoting is this: Is there one way of denoting six different kinds of objects, or are the ways of denoting different? And in the latter case, is the object denoted the same in all six cases, or does the object differ as well as the way of denoting it? In order to answer this question, it will be first necessary to explain the differences between the six words in question. Here it will be convenient to omit the word the to begin with, since this word is in a different position from the others, and is liable to limitations from which they are exempt.(§ 59 ¶ 1)

In cases where the class defined by a class-concept has only a finite number of terms, it is possible to omit the class-concept wholly, and indicate the various objects denoted by enumerating the terms and connecting them by means of and or or as the case may be. It will help to isolate a part of our problem if we first consider this case, although the lack of subtlety in language renders it difficult to grasp the difference between objects indicated by the same form of words.(§ 59 ¶ 2)

Let us begin by considering two terms only, say Brown and Jones. The objects denoted by all, every, any, a and some[51] are respectively involved in the following five propositions. (1) Brown and Jones are two of Miss Smith's suitors; (2) Brown and Jones are paying court to Miss Smith; (3) if it was Brown or Jones you met, it was a very ardent lover; (4) if it was one of Miss Smith's suitors, it must have been Brown or Jones; (5) Miss Smith will marry Brown or Jones. Although only two forms of words, Brown and Jones and Brown or Jones, are involved in these propositions, I maintain that five different combinations are involved. The distinctions, some of which are rather subtle, may be brought out by the following considerations. In the first proposition, it is Brown and Jones who are two, and this is not true of either separately; nevertheless it is not the whole composed of Brown and Jones which is two, for this is only one. The two are a genuine combination of Brown with Jones, the kind of combination which, as we shall see in the next chapter, is characteristic of classes. In the second proposition, on the contrary, what is asserted is true of Brown and Jones severally; the proposition is equivalent to, though not (I think) identical with, Brown is paying court to Miss Smith and Jones is paying court to Miss Smith. Thus the combination indicated by and is not the same here as in the first case: the first case concerned all of them collectively, while the second concerns all distributively, i.e. each or every one of them. For the sake of distinction, we may call the first a numerical conjunction, since it gives rise to number, the second a propositional conjunction, since the proposition in which it occurs is equivalent to a conjunction of propositions. (It should be observed that the conjunction of propositions in question is of a wholly different kind from any of the combinations we are considering, being in fact of the kind which is called the logical product. The propositions are combined quâ propositions, not quâ terms.)(§ 59 ¶ 3)

The third proposition gives the kind of conjunction by which any is defined. There is some difficulty about this notion, which seems half-way between a conjunction and a disjunction. This notion may be further explained as follows. Let a and b be two different propositions, each of which implies a third proposition c. Then the disjunction a or b implies c. Now let a and b be propositions assigning the same predicate to two different subjects, then there is a combination of the two subjects to which the given predicate may be assigned so that the resulting proposition is equivalent to the disjunction a or b. Thus suppose we have if you met Brown, you met a very ardent lover, and if you met Jones, you met a very ardent lover. Hence we infer if you met Brown or if you met Jones, you met a very ardent lover, and we regard this as equivalent to if you met Brown or Jones, etc. The combination of Brown and Jones here indicated is the same as that indicated by either of them. It differs from a disjunction by the fact that it implies and is implied by a statement concerning both; but in some more complicated instances, this mutual implication fails. The method of combination is, in fact, different from that indicated by both, and is also different from both forms of disjunction. I shall call it the variable conjunction. The first form of disjunction is given by (4): this is the form which I shall denote by a suitor. Here, although it must have been Brown or Jones, it is not true that it must have been Brown, nor yet that it must have been Jones. Thus the proposition is not equivalent to the disjunction of propositions it must have been Brown or it must have been Jones. The proposition, in fact, is not capable of statement either as a disjunction or as a conjunction of propositions, except in the very roundabout form: if it was not Brown, it was Jones, and if it was not Jones, it was Brown, a form which rapidly becomes intolerable when the number of terms is increased beyond two, and becomes theoretically inadmissible when the number of terms is infinite. Thus this form of disjunction denotes a variable term, that is, whichever of the two terms we fix upon, it does not denote this term and yet it does not denote one or other of them. This form accordingly I shall call the variable disjunction. Finally, the second form of disjunction is given by (5). This is what I shall call the constant disjunction, since here either Brown is denoted, or Jones is denoted, but the alternative is undecided. That is to say, our proposition is now equivalent to a disjunction of propositions, namely Miss Smith will marry Brown, or she will marry Jones. She will marry some one of the two, and the disjunction denotes a particular one of them, though it may denote either particular one. Thus all the five combinations are distinct.(§ 59 ¶ 4)

It is to be observed that these five combinations yield neither terms nor concepts, but strictly and only combinations of terms. The first yields many terms, while the others yield something absolutely peculiar, which is neither one nor many. The combinations are combinations of terms, effected without the use of relations. Corresponding to each combination there is, at least if the terms combined form a class, a perfectly definite concept, which denotes the various terms of the combination combined in the specified manner. To explain this, let us repeat our distinctions in a case where the terms to be combined are not enumerated, as above, but are defined as the terms of a certain class.(§ 59 ¶ 5)

§ 59 n. 1. I intend to distinguish between a and some in a way not warranted by language; the distinction of all and every is also a straining of usage. Both are necessary to void circumlocution.