Metaphysicians of Meaning

Frege and Russell on Sense and Denotation

Gideon Makin

• Publisher: Routledge
• Year: 2013
• ISBN:
  9781134547647
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

Russell's On Denoting and Frege's On Sense and Reference are now widely held to be two of the founding papers of twentieth century philosophy and form the heart of the famous "linguistic turn". The Metaphysicians of Meaning is the first book to challenge the accepted secondary work on these two seminal papers, forcing us to reconsider the interpretation of these two vitally important works on meaning.

Part I

Russell

Chapter 1

Russell’s first theory of denoting

The present chapter introduces Russell’s first theory of denoting, namely the one he put forth in The Principles of Mathematics and endorsed up until the time he struck upon the theory of descriptions in 1905. Two intermingled tasks are involved: one is to explain the distinctive features of the theory itself – primarily to facilitate the discussion in the following chapter. This will be done just about in full. The other is to introduce the broader theoretical setting in which Russell’s discussion of denoting is couched. Here I will merely point out some of the most salient features, without attempting to answer all the questions it gives rise to. Some of them will be touched upon at various points in the following chapters, but their more direct discussion will have to wait until Part III, after the discussion of Frege.

I The elements

Russell’s first theory of denoting can be wrapped into a single sentence: ‘a concept denotes when, 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’ (PoM: § 56). But unless we know what precisely is meant by ‘proposition’, ‘term’, ‘denote’ and ‘about’ such a statement is hardly illuminating and part of our task, therefore, is to clarify these notions.

To begin with propositions, it is crucial to bear in mind that they are not, nor are they abstracted from, symbolic or linguistic or psychological entities (PoM: § 51). On the contrary, they are conceived of as fundamentally independent of both language and mind. Propositions are first and foremost the entities which enter into logical relations of implication, and hence also the primary bearers of truth. Their other role is to be the objects certain relations to which constitute belief and knowledge.

But even when the non-linguistic conception of propositions is acknowledged at the outset, it is in constant danger of being obscured when following particular discussions. Not only do ‘truth’ and ‘implication’ apply, in their primary senses, to propositions and only derivatively to the sentences expressing them, but there is a host of related notions – ‘term’ (meaning ‘entity’),¹ ‘verb’, ‘adjective’, ‘subject’, ‘concept’, ‘substitution’, ‘description’ and ‘occurrence’ – which one is inclined to regard as relating to linguistic entities, but which Russell uses in a non-linguistic sense. On this view the application of these notions to linguistic items reflects their propositional sense rather than the other way around. The matter is both delicate and fundamental: propositions are specified primarily by appeal to sentences expressing them, and though in PoM Russell often seems to be flipping between a linguistic and a non-linguistic mode of speech, he never goes back on the fundamental point that the inquiry is concerned with propositions and their constituents, not with sentences or anything linguistic. It is crucial to bear this in mind despite the discussion often appearing to be linguistic.

I come now to the problem which gave rise to the notion of denoting as it is presented in Chapter 5 of PoM. So long as we confine our attention to the propositions expressed by sentences which do not contain denoting phrases (i.e. phrases composed of ‘the’, ‘a’, ‘some’, ‘any’, ‘all’ or ‘every’, followed by a word for a class concept), the account of what it is for a proposition to be about an entity is straightforward enough. ‘Socrates preceded Plato’, for example, is the verbal expression of a proposition which has Socrates and Plato (the men), and preceding (the relation) as constituents;² and the entities such propositions are about will always be among their constituents – more specifically, those entities occupying a subject position or positions (as there may be more than one). But the case is different with the proposition expressed by, e.g. ‘I met a man’. When that proposition is true, then there is a particular man I met – let it be Jones. But what is affirmed in ‘I met a man’ is not the same as what is affirmed in ‘I met Jones’. From this it seems to follow that the two sentences express different propositions. But while we have a reasonably clear notion of the proposition expressed by ‘I met Jones’, it is not at all clear what the proposition expressed by ‘I met a man’ is. It would seem that Jones himself does not occur in it – at least not in the same manner as in ‘I met Jones’ – but what, then, occurs in his place?

The answer to this would also have to explain the familiar phenomenon whereby any given name has a privileged set of associated denoting phrases (of the ‘the …’ kind) which may be substituted for it without affecting the sentence’s truth value (and similarly for the other types of denoting phrases, e.g. ‘a man’ and ‘a featherless biped’). Obviously enough, these are just those phrases for which an identity sentence of the form ‘[proper name] = [denoting phrase]’ is true; but this can hardly be an answer, since these sentences are merely a special case of the problem. Linguistic evidence suggests that something is substituted for Jones – just as Smith might be said to have been substituted for Jones (the man) to obtain the proposition expressed by ‘I met Smith’. But it is not at all clear, at least in the first instance, what this something is.

It is worth noting the dichotomy underlying this manner of posing the problem: if the difference between two sentences is logically significant, then they express distinct propositions. If, on the other hand, we do not think that the propositions expressed are distinct, then the difference between the sentences is dismissed as merely linguistic or psychological.

Russell’s solution to this problem in PoM invokes the allied notions of a denoting concept and the relation of denoting. A denoting concept is the special kind of propositional constituent which occurs in a proposition whose expression involves a denoting phrase; the relation this concept has to the entity the proposition is about (i.e. the denotation) is the logically primitive relation of denoting.

A great deal of attention and space in the chapter on denoting in PoM are devoted to the attempt to characterize the different kinds of objects denoted (called ‘combinations’) by the different kinds of denoting concepts.³ But this problem can here be set aside, because it can be detached from our chief concern, namely the problems which give rise to the relation of denoting. The different kinds of combination need to be distinguished even when we confine our attention to finite domains – where the relation of denoting is in principle dispensable (PoM: § 61); while the relation of denoting is called for even when (as with ‘the’-phrases in the singular) the denotation is a single entity, and hence there is no question of combinations. I mention the issue of combinations only to set it aside, and in what follows attention will be focused on ‘the’-phrases in the singular.

II The mathematical context

From what has been said so far it remains unexplained why the discussion of denoting should occur in a work on the philosophy of mathematics, to which the relevance of establishing the correct account of ‘I met a man’ is hardly obvious. The answer provides us with a first glimpse of the broader theoretical context of Russell’s discussion. To begin with, we may give a rather short answer, namely that Russell’s logicist account of mathematics led him to recognize denoting as an indefinable logical relation, a logical constant, and Part I of PoM deals with the fundamental concepts of mathematics (i.e. logic); concepts which, on this account, are indefinable. This reply, though true as far as it goes, only raises further questions: is ‘logical constant’ being used in the modern sense and, if so, what led Russell to give denoting this status?

We might begin by removing one possible source of perplexity. The absence of any primitive symbol for a relation of denoting in the logical calculus is not, on Russell’s view, in any tension with recognizing it as a logical constant. For reasons shortly to be explained, Russell viewed the division between logical notions which do occur in the calculus (and hence the symbolism) and others which do not occur as a matter of mere convenience, bearing no theoretical significance.⁴ This in turn enabled him to regard a notion as indefinable for the purpose of the technical development of the calculus, and then to proceed to its logical analysis without appeal to the symbolism. The notion of formal implication, i.e. of propositions of the form ∀x (Fx → Gx), is regarded as indefinable for technical purposes, but it then gives rise to a number of indefinables in the course of its analysis (PoM: §§ 12 and 17, and Chapter 8, especially § 93), among which is denoting.

Though the fact that Russell recognized denoting as a logical constant is beyond dispute (see PoM: §§ 31, 56, 93 and 106),⁵ his rationale for so doing is not immediately clear. Making it clear brings us back to the consideration of the foundations of mathematics. Despite the fact that the examples discussed in the chapter on denoting suggest a linguistic setting, the broader context in which the whole discussion occurs, i.e. its placement immediately before the chapter on classes, as well as remarks on denoting elsewhere in PoM, make it plain that the chief consideration for recognizing denoting as a logical constant is its role in relation to infinite classes.

With regard to infinite classes, say the class of numbers, it is to be observed that the concept all numbers, though not itself infinitely complex, yet denotes an infinitely complex object. This is the inmost secret of our power to deal with infinity. An infinitely complex concept, though there may be such, can certainly not be manipulated by the human intelligence; but infinite collections, owing to the notion of denoting, can be manipulated without introducing any concept of infinite complexity.

(PoM: § 72)⁶

Because infinite classes cannot be specified by mentioning each of their members singly, a purely extensional view of classes, Russell reasons, becomes untenable, and we must assume that in dealing with classes it is class concepts rather than the classes themselves that we deal with directly – though the view as to the identity of classes remains purely extensional (PoM: §§ 71–2). Denoting is the relation which obtains between the class concept and the class itself,⁷ and is essentially the same as the ‘determining’ involved in saying that a concept determines a class.

Another consideration requiring the relation of denoting derives from the need to recognize the null class. After repeating the above consideration from infinite classes Russell continues, ‘The consideration of classes which results from denoting concepts is more general than the extensional consideration … it introduces the null concept of a class’ (PoM: § 72). This is because ‘what is merely and solely a collection of terms cannot subsist when all the terms are removed’ (§ 73).

From what has been said so far Russell’s motivation for devising the theory of denoting may appear to stem from two quite different sources: the discussion in Chapter 5 of PoM seems to be concerned primarily with locutions from natural language, while the whole context in which it occurs, as well as references to it elsewhere in the Principles, points to a consideration of mathematics and infinite classes. How are these two sources to be reconciled? The answer points us to one of the most fundamental and to my mind most interesting features of Russell’s outlook, but since its discussion belongs more properly to Chapter 7, I confine myself here to a provisional statement.

From Russell’s perspective there is no need to reconcile the two sources, because there is no tension between them in the first place. The key to this unitary view is the notion of proposition. The underlying assumption is that both in mathematics and in natural language we express propositions, and that it is of no theoretical significance what we choose to express in natural language and what in a symbolic language. It is therefore immaterial, when addressing the question of propositions expressed by means of denoting concepts (or, more to the point, propositions which cannot contain what they are about), whether the proposition is mathematical or not. The 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.

Having set out the bare essentials of Russell’s first theory it remains for us, first, to note two applications that Russell found for it; second, to highlight three of its most essential features; and third, to note some developments the theory had undergone between PoM and OD.

III Denoting, definition and identity

Although there can be little doubt that denoting concepts were introduced by Russell first and foremost to handle the problem of how we can deal with infinite totalities (we may label this the ‘primary case’ in that respect), once the device had been made available Russell found it useful in solving two further and closely related puzzles for the sake of which alone he would probably not have introduced denoting concepts (we may thus regard them as ‘secondary cases’). The same pair was revisited in OD – though there with a different solution. The first, which concerns definitions, can hardly be better introduced than by citing Russell’s own words:

It is a curious paradox, puzzling to the symbolic mind, that definitions, theoretically, are nothing but symbolic abbreviations, irrelevant to the reasoning and inserted only for practical convenience, while yet, in the development of a subject, they always require a very large amount of thought, and often embody some of the greatest achievements of analysis. This fact seems to be explained by the theory of denoting. An object may be present to the mind, without us knowing any concept of which the said object is the instance; and the discovery of such a concept is not a mere improvement in notation. The reason why this appears to be the case is that, as soon as the definition is found, it becomes wholly unnecessary to the reasoning to remember the actual object defined, since only concepts are relevant to our deductions. In the moment of discovery, the definition is seen to be true, because the object to b...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• List of abbreviations
• Introduction
• Part I: Russell
• Part II: Frege
• Part III: Russell and Frege
• Appendix A
• Appendix B
• Appendix C
• Bibliography
• Index