On (Russell’s Theories of) Denoting

In The Principles of Mathematics, Russell spends a chapter developing his first theory of denoting. This and his second theory of denoting both concern themselves with explaining denoting phrases, which consist of “class-concept[s] preceded by [‘all’, ‘every’, ‘any’, ‘a’, ‘some’, ‘the’,] or some synonym of one of them.”[1] So to use the class-concept of trees as an example, we have, among others, ‘all trees’, ‘any tree’, and ‘the tree’ as denoting phrases. According to the first theory, each of these phrases expresses (means) a kind of concept for which, “if it occurs in a proposition, the proposition is not about the concept, but about a term connected in a certain peculiar way with the concept.”[2] Russell unsurprisingly calls these denoting concepts, and he thinks (in PoM) that they are required to explain how denoting phrases can have meaning at all. But a few years later, he comes to the conclusion that denoting phrases don’t have any meaning at all, at least not when taken in isolation. In “On Denoting”, Russell develops a second theory of denoting in order to solve the problems that arise when the first theory is applied to certain propositions or sets of propositions.

Given where his first theory appears, it is to be expected that Russell sees denoting as somehow particularly related to mathematics. Gideon Makin, in The Metaphysicians of Meaning, writes, “the chief consideration for recognizing denoting as a logical constant is its role in relation to infinite classes.”[3] Indeed, while his theory happens to account for phrases like “any US citizen” the same way as phrases like “any finite number”, it is only in the second case that a theory of denoting concepts seems strictly necessary to Russell. When a class-concept determines a finite class, propositions (expressed in part by a denoting phrase) including (a word for) that class-concept could be understood as simple conjunctions or disjunctions of a finite number of propositions, or as propositions involving conjunctions or disjunctions of a finite number of terms. For instance, the statement, “every living Smith was born after 1900,” could be taken to simply mean, “Abel Smith was born after 1900, and Adam Smith was born after 1900, and…and Zachary Smith was born after 1900.” This complex proposition might take a very long time to express, but it is, at least in principle, comprehensible because it is finite. However, the statement, “every positive integer is odd or even,” cannot be reasoned about by a human mind if it is understood as having the same meaning as the infinitely long statement, “One is odd or even, and two is odd or even, and three is odd or even, and…”. Russell believes that “an infinitely complex concept…can certainly not be manipulated by the human intelligence.”[4]

The fact that we can, at least according to Russell, manipulate and reason perfectly well with statements about all numbers must mean that something else is going on with denoting phrases. In PoM, Russell suggests that denoting phrases express denoting concepts, which in turn denote some combination of terms. The denoted combination need be neither finite, nor anything we can know by acquaintance, because what occurs in the proposition, what is actually reasoned about, is the single denoting concept. In particular, “all integers” collectively denotes 1 and 2 and 3 and…; “every integer” severally denotes 1 and denotes 2 and denotes 3 and…; “any integer” denotes 1 or 2 or 3 or…, and does so in a way that it is irrelevant which one we pick; “an integer” denotes 1 or 2 or 3 or…, and does so in a way that we cannot pick any particular one; and “some integer” denotes 1 or denotes 2 or denotes 3 or…, and does so in a way that requires some particular one to be taken.

Unlike these five cases, phrases such as “the F” cannot be correctly employed when F is an infinite class-concept. Instead, “[the word] the is correctly employed only in relation to a class-concept of which there is only one instance.”[5] In addition to allowing us to reason about things, such as the center of mass of the solar system, of which we can have no direct knowledge by acquaintance, treating “the F” as a denoting phrase allows us to see immediately how one identity statement can be trivial while another about the same terms is informative. Saying, “Bush is the US President” is more informative than saying, “Bush is Bush,” because the propositions expressed by each statement are different. One contains a denoting concept in one and the entity named by “Bush” in the other of its term-accessible positions, and thus involves two distinct things, while the other contains the term named by “Bush” in both its term-accessible positions, and is thus tautological. It should be noted that this is similar to, though not identical with, the distinction Frege would draw between the two propositions: The referents of corresponding parts of each statement are the same, but the sense of “Bush” is different from that of “the US President”, and so the statements express different cognitive contents. With this theory of denoting concepts, we can make propositions about complexes of terms that we could never manipulate directly, either because they are infinite or because they are otherwise inaccessible to direct knowledge. We are furthermore able to manipulate these propositions, and thus use logic to deduce new truths from those which are true, because each denoting concept is a single thing with which we do have a direct mental acquaintance.

Unfortunately, there are problems that arise for this theory when it is applied to certain sets of propositions. For instance, take, “The President of the United States is a man, not a denoting concept,” and “<The President of the United States> is a denoting concept, not a man.”[6] In the subject position of the first proposition, the denoting concept <The President of the United States> occurs. What occurs in the subject position of the second proposition is uncertain, given that <The President of the United States> is supposed to denote the meaning of the phrase contained within angled brackets. It cannot be identical with <The President of the United States>, because it must denote that concept (owing to the fact that the second proposition is about that concept, and no other part of it can do the denoting), which means that the concept denotes itself. But <The President of the United States> denotes a man and yet is not itself a man. Furthermore, what is in the subject position of the second proposition cannot contain that concept as a constituent. For if it did, then what it denoted would be a function of what is denoted by <The President of the United States>, and yet it is supposed to denote that concept itself. So we would have that <The President of the United States> is a function of what it denotes, which is a particular person. But there can be no single function that always takes an entity to the concept under consideration that denotes that entity, because many concepts can denote any given thing. If we had some function that mapped (the man) George W. Bush to (the concept) <The President of the United States>, we could not appeal to that same function if the propositions in question had “the son of former President George H. W. Bush” instead of “The President of the United States.” So every possible concept that denotes a particular entity would have to be a different function of that entity, and it appears very much as though we are in a worse state than before, when it comes to having any understanding of what the meaning of a denotation really is.

It is hard to see any way to resolve this problem while maintaining that there are things at all resembling the denoting concepts Russell proposes in PoM. If denoting phrases are to have meaning in isolation, the meaning must be distinct from what is denoted, or else “Bush is Bush” would mean the same as “Bush is the US President”, which is precluded by their different effects on the truth of propositions in which they have a secondary occurrence.[7] But the meaning of a phrase occupying the subject position of the expression of a proposition is what occurs in the subject position of the proposition itself, and we just saw how problematic it is to unravel the mess that can create. Russell thus drops altogether the idea that there is some thing a denoting phrase means. Instead, such phrases are incomplete symbols that, when they form part of the expression of a proposition, can be eliminated by replacing that expression with a sentence expressing the same proposition, but from which the denoting phrase is completely absent. The theory put forth in “On Denoting” seeks to account for how each type of denoting phrase can be paraphrased out of existence. If this theory is true, then the argument of the previous paragraph will no longer present a problem, because what occurs in the subject position of each proposition will be quite different in the paraphrased versions than what it appeared to be before.

The scheme for paraphrasing away denoting phrases, which Russell gives in OD pp. 43-44, is as follows: Consider a propositional function expressed by “C(x)”, and a class-concept F, where “Fx” means “x is F”. Then we have:

“C(all F’s)” and “C(every F)” both mean “"x(Fx®C(x))”.

“C(no F)” means “"x(Fx®ØC(x))”.

“C(some F’s)” and “C(an F)” mean “Ø("xØ(C(x)ÙFx))”, which is “$x(C(x)ÙFx)”.

“C(the F)” means “$x(FxÙ"y(Fy®y = x)ÙC(x))”.

Nowhere in this account does Russell explain “C(any F)”, but it is likely he would take it to mean the same as “C(all F’s)” and “C(every F)”. It is worth noting that these different ways of understanding propositions containing denoting phrases no longer track the linguistic differences between “all”, “every”, and “any”, or between “a” and “some”. Russell clearly believed such differences existed when he wrote PoM, so it may be that he could think of no straightforward way in which that difference could be reflected after paraphrasing away the denoting phrase.

Though the process gets somewhat tedious, the above identifications of meaning allow us to now unravel fairly complex propositions that contain denoting phrases. For instance, “The current US President is the eldest son of a former US president,” means

“$x(CurrentPres(x) Ù "y(CurrentPres(y) ® y = x) Ù

$z(FormerPres(z) Ù $w(w EldestSonOf z Ù "y(y EldestSonOf z ® y = w) Ù w=x)))”

It is perhaps problematic for Russell that there are so many different ways these “unpackings” of meaning can be nested. In the above translation, for instance, there is one and only one current president, and there is at least one former president, of whom the current president is the one and only eldest son. But one could just as easily have taken the original proposition to mean there is at least one former president, who has one and only one eldest son, who is the one and only current president. Russell argues that, because of the difference between primary and second occurrence of denoting phrases, this is a matter of linguistic ambiguity that can be straightened out upon analysis into a more disambiguated language.[8] Determining exactly what truth conditions one intends for a proposition upon stating it can disambiguate the propositions Russell addresses, so the results will not be logically equivalent. However, in the above case, it seems that each interpretation is logically equivalent, so how do we know which particular proposition is the one we mean? Even supposing there were some way, perhaps based on context or on the mental state of the person speaking, that we could determine which denoting phrase in a given proposition should be considered first, there is some question as to why Russell’s set of translations should be the right ones. Why, for instance, cannot “C(the F)” be taken instead to mean “$x("y(Fy « y = x) Ù C(x))”, which is a logically equivalent? That they are distinct propositions can be seen by stating them in more standard English. Russell’s interpretation starts with “there at least one x such that x is F and anything else that is F must be identical with x,” whereas this second one begins, “there is at least one x such that, whatever entity we may consider, that entity is F if and only if it is identical with x.”

Leaving this question for a moment, let us at least credit Russell’s second theory of denoting for being able to solve the previously discussed problem of the meaning of phrases meant to denote the meanings of other denoting phrases. By “The President of the United States is a man, not a denoting concept,” we mean, “there is one and only one entity that is the President of the United States, and that entity is a man, and not a denoting concept.” In “<The President of the United States> is a denoting concept, not a man,” we now have something very different. Perhaps this proposition should be understood as, “there is one and only one entity that is the meaning of (the phrase) ‘The President of the United States’ and that entity is a denoting concept, and not a man.” But because that phrase no longer has meaning in isolation, there is not some entity that is its meaning, so the second proposition is now false because it implies the existence of something which does not exist. What we have is no longer a proposition with some potentially difficult thing in its subject position, but rather a statement as unproblematic as “there is exaclty one entity that is round and square, and…”, where whatever occurs after the “and” is irrelevant to the truth of the proposition.

So even if it does appear somewhat arbitrary to understand “the F” in one way rather than any number of other equally concise, logically equivalent ways, Russell’s second theory does keep the basic strengths of the first theory and eliminates most problematic weakness. We are still able to work meaningfully with propositions (that appear to be) about class-concepts that determine massively infinite classes, and we can still describe and reason about entities we assert to exist without having any hope to become mentally acquainted with those entities. And while a theory of denoting in the first place may only have been an absolute requirement for dealing with infinite classes, we are also able to deal with finite but cumbersomely large classes very simply, as well. Rather than dealing with something like, “Abel Smith was born after 1900, and Adam Smith was born after 1900, and…and Zachary Smith was born after 1900,” we just have to reason with, “it is always true of x that, if x is a living Smith, then x was born after 1900.” For this reason, the entire “discussion of denoting is thus pitched at a level of generality which transcends the distinction between the philosophy of mathematics and the philosophy of language.”[9] A theory necessary for studying the logical foundations of mathematics turns out to provide a general and far-reaching method for understanding certain types of propositions that would otherwise be very cumbersome to analyze logically.

[1] Russell, Principles of Mathematics (hereafter PoM), §58. In §57, he states, “A class is a certain combination of terms, a class-concept is closely akin to a predicate, and the terms whose combination forms the class are determined by the class-concept.”

[3] Makin, The Metaphysicians of Meaning, p. 14.

[5] Id., §63. Russell’s account of the other five types of phrases takes place in §§60-62.

[6] The use of < and > to distinguish the use of the meaning of a denoting phrase (which is the denoting concept) from the mention of the phrase itself, as well as the gist the argument below, are taken from Makin, pp. 24-27. Makin there seeks to clarify what Russell is attempting to show with his Gray’s Elegy argument, which occurs in “On Denoting”, pp. 48-50 (of the coursepack version).

[7] “John doesn’t know if Bush is the US President” can be true even if “John doesn’t know if Bush is Bush” is false, so the two propositions, and thus the two secondary propositions which constitute the only difference between them, cannot mean the same thing.

[8] Because there is no present King of France, Russell points out that “‘the present King of France is not bald’ is false if it means, ‘There is an entity which is now King of France and is not bald’, but is true if it means ‘It is false that there is an entity which is now King of France and is bald’,” (OD, p. 53).