Propositional Structure and B. Russell's Theory of Denoting in The Principles of Mathematics

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Acknowledgments

I would like to thank Mike Byrd for his invaluable help and encouragement with this manuscript. I am also very grateful for the comments I received at different stages from Dennis Stampe, Martha Gibson, Alan Sidelle, Genoveva Martí, and an anonymous referee.

Notes

A word about quotation marks and italics: I will use single quotation marks to mention a word or a string of words and double quotation marks to indicate that a word is being used in a more or less peculiar sense. I will use italics for two purposes: to emphasize, and to mention propositional constituents (see note 16 about the latter use). The double use should cause no confusion.

See a standard introduction to Government and Binding theory, for example Haegemann 1994 .

See PoM (1903, § 46, p. 42). From now on I will make references to PoM by writing the number of the section followed by the number of the page (46/42 in the present case).

See also Cocchiarella ( 1980 , p. 25): ‘denoting concepts themselves, it is clear, are the conceptual counterparts of (surface) grammatical subjects of English declarative sentences. [they] are taken by Russell as occurring in propositions in a manner that at least semantically is completely analogous to the way that (surface) grammatical subjects occur in the English declarative sentences that express those propositions’. Furthermore, see Hylton ( 1990 , pp. 267–268); and Geach ( 1962 , §§ 40–41, pp. 56–58), where the point is implicit in Geach's discussion of the contrast between Russell's PoM theory and the Fregean analysis of denoting phrases as quantifiers.

For example, see Hylton ( 1990 , ch. 6).

See, for example, Geach ( 1962 , §§ 37–38, pp. 51–55).

See Dowty et al. ( 1981 , especially ch. 7, for an accessible presentation).

Another way to challenge the contrast is by rejecting the view that the 1905 theory ‘butchers’ the surface structure. Cf. Neale 1990 .

Throughout this paper I will use the phrase ‘theory of denoting’ mostly to refer to the views Russell put forward in 1903 in PoM. When I need to refer to the later views proposed in ‘On denoting’ I will mark the reference as clearly as possible.

This entails that ‘Plato admired Socrates’ and ‘Socrates admired Plato’ express propositions that differ in structure, and thus that the structure is partly determined by the order of the constituents. Russell's conception of structure in PoM is quite fine-grained.

No standards of simplicity are explicitly provided by Russell, however.

As we will see, Russell is forced, by his theory of denoting, to introduce combinations of terms, to be thought of as objects that are not terms. Some of these objects are inherently plural while others do not have number (they cannot be conceived as being either one or many). Russell notes that this new ontological category raises ‘grave logical properties’ (p. 55, note *). More about these objects below.

Notice that ‘predicate’ applies to propositional constituents and not linguistic entities (which would be things if they occurred in a proposition).

Russell also uses ‘standing for’ as a synonym of ‘indicating’ (51/47). The entity that a word stands for is the one with which we need to be acquainted in order to grasp propositions expressed by sentences that contain the word.

Russell's assertion is what many would call ‘predication’ and is not to be confused with Frege's, or the ordinary, notion of assertion. To repeat a familiar point, there is predication in the antecedent of ‘If Tom is smart, he'll leave town for a while’, but there is no (Fregean) assertion.

I use ‘A is greater than B’ (Russell's own example) as a singular expression that designates the proposition that A is greater than B, or the denotation of ‘the proposition expressed by “A is greater than B”’. In general, I will indicate whatever belongs to the propositional level by writing the corresponding expression in italics. The attentive reader will notice that Russell's example raises a subtle complication because it contains metavariables; I am going to ignore this complication (it is independent of the convention of using italics). The reader will also notice that my convention is not iterative (one cannot italicize twice). Therefore, it does not enable one to construct an expression that (1) refers to a propositional constituent that itself refers to the propositional constituent of another proposition, and (2) makes this fact explicit (of course one could still name the first propositional constituent by using an italicized expression, but it will not be apparent that this expression names something that itself names another propositional constituent). Compare this with the case in which we iterate quotation marks—let us say single quotation marks. This gives us an expression that names something that itself names an expression, and it is obvious that this is so. The impossibility of repeating the application of italics does not matter for my purposes because I do not need an iterative device. (For other purposes this may be inconvenient; for example, an iterative naming device is often used to simplify the discussion of the Grey's elegy argument at the center of ‘On Denoting’. Cf. P. Hylton's discussion of it in 1990, pp. 249-51. Of course, everything could be said by means of descriptions, forgoing naming devices altogether.)

As we saw above, every constituent of the proposition which occurs as a term-of-the-proposition (i.e. in such a way that we could substitute any other term for it without “destroying” the proposition) is something that the proposition is about: A is greater than B is both about A and about B. This idea seems to apply to the constituents of a relational proposition independently of the proposition's analysis into a subject and an assertion, thus A is greater than B is about two constituents, regardless of how it is analyzed. Russell is not too clear on this, but he seems to presuppose that when analysis divides the proposition into A and is greater than B, B still occurs as a term-of-the-proposition, even though it is part of what Russell calls the ‘assertion’. Otherwise what the proposition is about would be analysis-dependent and no analysis would show A is greater than B to be about more than one thing.

Russell's view could be, and has been challenged. See Ramsey 1925 and the discussions in Strawson ( 1959 , ch. V) and Dummett ( 1973 , pp. 61–67).

See the beginning of 43/39, where Russell makes the same point about Socrates is a man. See also 48/45.

Russell does recognize a deep difference between things and concepts but does not locate the difference in the things and the concepts themselves, as Frege does. The difference for Russell, only emerges within the propositional context as the inability of things to provide a proposition with unity (a collection of things cannot be a proposition). Frege and Russell agree that not everything can be in predicate position because they both see a connection between occupying that position and ensuring unity, but they disagree as to what can occupy subject position. Frege bars concepts from subject position while Russell does not. See 49/46.

Michael Byrd pointed out to me that, since not everything can stand in this kind of relation to something else (only concepts can), one is naturally led to think that there must be something about their nature that enables them to stand in the is relation to a subject. I will not pursue the matter here. It is important to note, however, that the difference in the manner of occurrence is signalled linguistically. One and the same concept is indicated by ‘humanity’ and ‘human’, but the two words signal different manners of occurrence.

Russell's claim that a relation between Socrates and humanity is implied by ‘Socrates is human’ is based on the claim that any s–p proposition is logically equivalent to a proposition of a different form, namely one that asserts that an individual (the indication or denotation of the grammatical subject of the s–p proposition) belongs to a class (57/54).

See Russell 1907 for a later but very clear presentation of the matter.

See (1898, p. 167).

In 1899, a paper delivered but never published.

In 1900 he writes: ‘Hence any identical content which can be ascribed to a number of terms must have a being distinct from that of the terms to which it can be ascribed … . But when once we admit the identical content to be something distinct from the terms of which it is predicated, predication becomes a relation between subject and predicate’ (pp. 229–230). See also (1901a, p. 252, passage repeated in 1901b, p. 275).

See also (54/50), where, in making the point about ordinary relations (called ‘verbs’) Russell quite explicitly says something that is true of assertions of the form is F as well: ‘The verb, when it is used as a verb, embodies the unity of the proposition’. Here ‘used as a verb’ means predicated (cf. ‘… the concept in question is used as a concept, that is, it is actually predicated of a terms or asserted to relate two or more terms’ (49/46). Notice that the relation of predication is a sui generis one, and it should not be identified with that of instantiation. The difference is hard to articulate and was a constant source of difficulties for Russell, since denying the difference leads to the impossibility of false propositions. For an account of the difficulties and all the changes they inspired in Russell's thinking see Candlish 1996 .

See for example Aristotle: ‘A single [connective-free] affirmation or negation is one which signifies one thing about one thing’ (1963, 18a13).

In listing the core notions that can be found within our intuitions about predication I do not mention unity. The view that predication is responsible for the unity of the proposition, important as it is, is more of a philosophical observation than a part of our intuition.

However, see section 6.

Russell does not elaborate on this notion of ‘giving rise to’ and sometimes simply says that the various denoting concepts are ‘closely allied’ (58/56) to the predicate. It is worth noticing also that words like ‘a’, ‘some’, ‘every’, ‘all’, ‘the’, are said to have ‘some definite meaning’ but this meaning remains for Russell exceedingly vague (indeed, to the point of being evanescent).

In PoM, Russell does not make explicit reference to any principle according to which in order to grasp a proposition we need to be acquainted with all its constituents, but he presupposes a similar (albeit a little less restrictive) principle when he says that all the propositions we can know are of finite complexity (141/145–146). This is, of course, sufficient to motivate the introduction of denoting concepts.

Russell does this with reluctance. In a footnote on page 55 he points out: ‘The fact that a word can be framed with a wider meaning than term raises grave logical problems.’ Among the problems Russell might have had in mind is the question of whether the completely unrestricted range of the variables in mathematics (7/7) contains such objects.

Russell characterizes an aggregate as a whole that is definite as soon as its constituents are given. It is the simplest notion of whole of which we can think. Given any class with more than one member, the aggregate is to be identified with the whole composed of all the members, that is, with the class as one. See 135/139.

The problem of null class-concepts, i.e. those that give rise to empty classes, and the problem of class-concepts having more than one instance are not addressed. Russell seems to think that improper definite descriptions are just ill-formed descriptions: ‘The word the, in the singular, is correctly employed only in relation to a class-concept of which there is only one instance’ (63/62).

I say ‘logically connected’ because the connection is created by the relation of denotation, and the latter is a logical relation, according to Russell.

However, two consequences of the strategy must be noted. One is that the admission of denotationless denoting concepts would entail the admission of propositions that involve no predication whatsoever, even though they can be conceived as structured entities containing elements of the type that in general form propositions. An example would be The largest prime number is odd. Another consequence regards attitudes. If we assent to (4) we have some sort of fix on the predication involved in it (by ‘the predication involved in a proposition’ I mean the predicating, or saying, something of some entity performed by the component is + concept of the proposition). This should be independent of our grasping Elizabeth II is unhappy. Therefore our cognitive relation to the predication involved in (4) is not grounded in grasping a proposition—Elizabeth II is unhappy—where the relata of the relation of predication do co-occur. So, we are not aware of the predication involved in (4) as something that only exists in the confines of a proposition (again, if we can grasp (4) without grasping Elizabeth II is unhappy—but I take this to be uncontroversial). However, this does not rule out that we need to ground our grasp of this, and presumably any, predication in the grasp of some proposition.

In this proposition, the denoting concept All men occurs in subject position. The proposition must be distinguished from the formal implication for every value of x, if x is a man, then x is mortal. According to Russell the latter is equivalent ‘But it seems highly doubtful whether it is the same proposition’ (40/36); see also (73/74). Russell had two reasons to distinguish propositions like (5) from formal implications. The first emerges in an unpublished note from May 1901: ‘It seems all must be taken as an indefinable: for a formal implication is the assertion of all implications of a certain class, so that x ∈ a .⊃_x . x ∈ b cannot be taken to define all, though it may define “a is part of b”’ (1993, p. 567). The same view is repeated about every term in (44/40). Another reason is that in propositions like (5) the quantification is restricted, whereas it is not in formal implications. In Russell's words: ‘Peano held … that what is asserted [in All men are mortal] is the formal implication “x is a man implies x is mortal.” This is certainly implied, but I cannot persuade myself that it is the same proposition. For in this proposition … it is essential that x should take all values, and not only such as are men. But when we say “all men are mortals,” it seems plain that we are only speaking of men, and not of all other imaginable terms.’ (77/79); see also (41/38).

I am interpreting the quoted remark as saying: and yet of that one any assertion is true which is true of any a. The use of ‘proposition’ instead of ‘assertion’ is a minor mistake, I believe.

How does any a differ from all a's and every a in this respect? The observation that, whatever particular a_i we may pick, it is not true that any a denotes it, seems to apply to all a's and every a. None of the latter two seems to denote any selected a_i . We cannot point to a₂ and say: all a's (or every a) denotes it. Yet, if all a's are F (or Every a is F) is true, F will be true of a₂. This, however, should be understood correctly. Russell thinks that the answer to the question: Does all a's denote a₂? actually should be ‘Yes’, and the same goes for every a. The distinctive fact about these two concepts is that they do not denote only a₂. Russell puts it as follows (61/50): all a's denotes a₁ and a₂ (so it denotes a₂, but not only a₂) and every a denotes a₁ and it denotes a₂ (so, again, it denotes a₂, but not only a₂). Since the concepts do not denote only a₂, we cannot use them to make an assertion about just a₂. Every assertion we make through them is also about a₂, but not only about it. With any a, on the other hand, a certain singularity of denotation is introduced. Nonetheless, in section 7 we will see that the peculiarity of any a raises a sharp difficulty.

In saying ‘in some way’ I do not mean to suggest that there is more than one relation of denoting. In the passage from 61/59 each occurrence of the word ‘denotes’ has the same meaning.

This view takes terms, not combinations, to be what is denoted, but in 62/62, where Russell first expresses the view, the word ‘object’, not ‘term’ is used. This is probably a simple mistake.

Thus, there would be a precise sense in which the unofficial account includes various ways of denoting, but this does not seem to be the sense in which Russell asked whether there is one way of denoting several different kinds of (paradoxical) objects or whether there are various ways of denoting

Putting things in terms of denoting an ambiguous term is not very helpful, on the other hand. (This suggests that talk of an ambiguous term is only a sloppy variant of talk of ambiguous denotation). Parsons shows how the problem can be solved with the introduction of lambda-abstracts, but he correctly points out that the theory of denoting itself, understood as Dau's official account, does not do any work in the solution of the problem (1988, p. 28).

This may suggest either that it is up to us to decide which one is denoted, or that there is a fact of the matter about which one is the one denoted, and therefore that we could express a contradictory proposition by means of that sentence. Perhaps we inevitably do so when the extension of the class-concept a has only one member. This might be contrasted (unfavorably) with the contemporary account of phrases like ‘some philosopher’. Since on the latter account they do not denote, we are not even tempted to think that ‘some philosopher is F and some philosopher is not F’ is contradictory when there exists only one philosopher. That one philosopher is not denoted at all and thus we cannot elicit a contradiction. I think a few comments about all this are in order here. First, consider the case in which there is only one philosopher in the world: Russell's account is not at a huge disadvantage relative to the contemporary one. According to the latter, the sentence ‘some philosopher is F and some philosopher is not F’ is not contradictory in that sense of ‘contradictory’ which applies when one thing is designated and of it is it said both that it is F and that it is not F. However, the sentence is false, and inevitably so. In other words, with restriction to the interpretations in which there is only one philosopher in the domain, the sentence is logically false (it is ‘semi’-logically false). The only difference with Russell's account is that according to the latter the sentence, besides being inevitably false, is contradictory in the above sense, in virtue of the particular way in which one of its components (‘some philosopher’) functions semantically: it denotes something—of which F is asserted and denied. The difference, then, regards only the internal semantic functioning of the sentence which is responsible for the inevitable falsity. Second, Russell's account and the contemporary one diverge more sharply if we consider the case in which there is more than one philosopher in the world. In this case, as I said, it seems that ‘some a is F and some a is not F’ could express a contradictory proposition. About this, I am inclined to make two points. One is that the possibility envisioned is not very clear. If it is not the world itself that determines which a_i is denoted by some a in virtue of containing only one a, what could? Russell's theory says that one term is denoted, but nothing is said about any way to determine which one, and the notion of ‘determining which one’ is not very clear to me. The other point is that we can retreat to the claim that the sentence ‘some a is F and some a is not F’ is not a contradiction because it does not inevitably express one.

I presume that it is possible to argue that All numbers are numbers is connected in the relevant way with each proposition of the form n is a number and that its unity is “overdetermined”. But I am not sure I understand what this means. The issue becomes nebulous. Again, one is far from any intuitive understanding.