Frege and Russell Combined

OSWALDO CHATEAUBRIAND

DESCRIPTIONS: FREGE AND RUSSELL COMBINED

My aim in this paper is to sketch an account of descriptions that combines Russell’s main intuition about descriptions with Frege’s main intuition about descriptions. The basic idea is that Russell’s intuition was an intuition about descriptions in predicate position, whereas Frege’s intuition was an intuition about descriptions in subject position. Since neither of them distinguished explicitly subject positions from predicate positions, they generalized their treatment to all positions. My approach will be to preserve these intuitions by distinguishing notationally between subject positions and predicate positions.

1.

Russell’s basic intuition seems to me to be the following. A statement of the form (1)

a is the F

means (2)

a is F and nothing else is.

This analyzes (1) as involving two predications about a, namely: (3)

a is an F

and (4)

nothing other than a is an F .

(1) is true just in case both of these predications are true. Hence, if either a is not an F or something other than a is an F , (1) is false. This analysis is what I shall refer to as Russell’s predicative intuition about descriptions, for it analyzes ‘is the F ’ in (1) as a predication about a.1 I agree that this is, in general, the correct analysis of statements of form (1). Synthese 130: 213–226, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Frege’s basic intuition seems to me to be the following. A statement of the form (5)

The F is G

is a subject-predicate statement where ‘the F ’ is a singular term. If there is a unique thing that is F , then the singular term ‘the F ’ denotes this thing and (5) is true if it (the denotation of ‘the F ’) is G and false if it isn’t G. If there isn’t a unique thing that is F , then the singular term ‘the F ’ does not denote and (5) is neither true nor false. It is neither true nor false because G is predicated of the denotation of ‘the F ’, and if ‘the F ’ does not denote, then there is nothing that is or isn’t G.2 This analysis is what I shall refer to as Frege’s singular term intuition about descriptions. Just as I agree that Russell’s is in general the correct analysis of (1), I agree that, in general, Frege’s is the correct analysis of (5). Russell extended his analysis of (1) to (5) essentially as follows. Consider statements of the form (6)

An F is G.

Russell argued, reasonably, that in such a statement the indefinite description ‘an F ’ is not a singular term denoting an ambiguous F , say, but that the correct analysis of (6) is (7)

∃x(x is an F and x is G).

But, similarly, we can analyze (5) as (8)

∃x(x is the F and x is G),

which, given the analysis of ‘x is the F ’ as (3)–(4), becomes (9)

∃x(x is F and nothing other than x is F and x is G).3

Although I agree that statements of form (8), or (9), are perfectly legitimate, I do not agree that they give a correct analysis of (5) in general.4 Frege extended his analysis of (5) to (1) by interpreting (1) as the identity statement (10)

a = the F ,

where ‘the F ’ is in subject position and is properly treated as a singular term. I agree that we do make such identity statements, and that Frege’s

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is the correct analysis of them, but I do not agree that (10) is the correct analysis of (1) in general.5

2.

I shall now introduce a notation to distinguish subject positions from predicate positions. The way I do this is based on Frege’s distinction between function and argument(s) and on his idea that quantification is a higher order predication. Consider the quantified form (11)

∀x(F x → Gx).

We can think of this as a predication in several different ways. For instance: (12)

[∀x(Zx → W x)](F, G),

(13)

[∀x(Zx → Gx)](F ),

(14)

[∀x(F x → W x)](G),

(15)

[∀xZx]([F x → Gx](x)).

In each case the expression within square brackets on the left is the predicate, followed by the arguments within parentheses.6 In (12) the predicate denotes the second order binary relation subordination and the arguments are the first order properties F and G. In (13) the predicate denotes the second order property subordination to G and the argument is the property F . In (14) the predicate denotes the second order property superordination to F and the argument is the property G. In (15) the predicate denotes the second order property universality and the argument is the first order property is not an F or is a G. Although (12) is the most natural reading of (11) as a subject-predicate form, the other readings are also used in specific contexts. Thus, in doing the usual definition of truth for quantified sentences one uses reading (15). I shall follow Frege in maintaining that when a non-denoting singular term occurs in subject position in a statement, then the statement is neither true nor false. The qualification “in subject position” is important because a non-denoting singular term may occur as part of a predicate in a statement without resulting in lack of truth value for the statement. Consider (16)

John reasons like Sherlock Holmes,

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for example. If interpreted as the relational statement (17)

[x reasons like y](John, Sherlock Holmes),

the statement is neither true nor false, for Frege’s reasons. If, however, we interpret (16) as the predication (18)

[x reasons like Sherlock Holmes](John),

then the statement may be true or false, assuming that ‘John’ denotes, because we may consider the predicate ‘[x reasons like Sherlock Holmes](x)’ a legitimate predicate whose conditions of applicability are determined by the Connan Doyle stories.

3.

Going back now to Frege’s and Russell’s intuitions about descriptions, we may distinguish two operators corresponding to the predicate ‘is the F ’ and to the singular term ‘the F ’. Russell’s analysis of (1) as the predication (2) can be formulated as (19)

[F x & ∀y(Fy → y = x)](a),

which predicates ‘is the F ’ of a. I shall abbreviate this as (20)

[!xF x](a),

where ‘!’ can be thought of as an operator which for any first order predicate ‘F x’ yields a first order predicate (21)

[!xF x](x),

i.e., (22)

[F x & ∀y(Fy → y = x)](x),

which applies to a thing if ‘F x’ applies uniquely to that thing, and which does not apply to anything otherwise. Frege’s analysis of (5) can be formulated as (23)

[Gx](ιxF x),

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where ‘ι’ is an operator which for any first order predicate ‘F x’ yields a singular term (24)

ιxF x

which denotes a thing if ‘F x’ applies uniquely to that thing and which does not denote otherwise. Frege’s analysis of (1) can be formulated as (25)

[x = y](a, ιxGx).

Although I agree that sometimes we use (1) to make an identity statement, and that Frege’s is the correct analysis of such identity statements, I do not agree that this is the correct interpretation of (1) in general. We can see this from the fact that we deny statements of form (1) both on the grounds that a is not an F and on the grounds that a is not the only F . If someone claims that Russell is the author of Principia Mathematica, I can deny this on the grounds that Russell wrote Principia Mathematica jointly with Whitehead. Similarly, if someone claims that Gödel is the author of Principia Mathematica, I can deny this on the grounds that Gödel was not an author of Principia Mathematica. Thus, whereas the identity statements (26)

Russell = the author of Principia Mathematica

and (27)

Gödel = the author of Principia Mathematica

are neither true nor false according to Frege’s analysis, the predications (28)

Russell is the author of Principia Mathematica

and (29)

Gödel is the author of Principia Mathematica

are both false.7 This was Russell’s predicative intuition, although Russell also ended up analyzing statements of form (1) as a sort of identity statement. For his actual analysis was (30)

∃x(x is the F & a = x),

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which becomes (31)

∃x(F x & ∀y(Fy → y = x) & a = x).

This parallels the analysis of (5) as (32)

∃x(x is the F & Gx),

which becomes (33)

∃x(F x & ∀y(Fy → y = x) & Gx).

There are many different interpretations of these forms in subjectpredicate terms, depending on what we consider to be asserted of what. Two natural readings of (33) are (34)

[∃x(Zx & ∀y(Zy → y = x) & W x)](F, G)

and (35)

[∃x(Zx & W x)]([!xF x](x), G),

where (34) is a predication about the properties F and G and (35) is a predication about the properties is the F and G. Similarly, we can have several readings for (31). An interesting case where the singular term intuition and the predicative intuition get combined are statements of the form (36)

The F is the G.

I think that the most natural analysis of this form is (37)

[!xGx](ιxF x),

corresponding to the reading (38)

The F is G and nothing else is.

But (36) can also be meant as the identity statement (39)

The F is the same as the G,

in which case the correct analysis is Frege’s (40)

[x = y](ιxF x, ιxGx).

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Russell also analyzes (36) as an identity statement, but his analysis comes out as (41)

∃x(F x & ∀z(F z → z = x) & ∃y(Gy & ∀z(Gz → z = y) & x = y)),

which corresponds to a third reading of (36) as (42)

Something is the F and something is the G and they are the same thing.

(41) can be interpreted as a predication about the properties F and G (43)

[∃x(Zx & ∀z(Zz → z = x) & ∃y(Wy & ∀z(W z → z = y) & x = y)](F, G),

or can be interpreted as a predication about the properties is the F and is the G (44)

[∃x(Zx & ∃y(Wy & x = y)]([!xF x](x), [!xGx](x)),

as well as in other ways. Although I agree that we can make such statements, I do not agree that Russell’s is a correct analysis of (36) in general. Another possible interpretation of a statement of form (36) which has a close relation to Russell’s analysis is (45)

∀x(x is the F ↔ x is the G).

If we analyze ‘x is the F ’ and ‘x is the G’ predicatively, then (45) can be formulated in my notation as (46)

[∀x(Zx ↔ W x)]([!xF x](x), [!xGx](x)),

and in ordinary notation as (47)

∀x((F x & ∀y(F z ↔ y = x) ↔ (Gx & ∀y(Gy → y = x))),

which is essentially equivalent to (41) minus the existential claims. If I say that the governor of California is the chairman of the Board of Regents of the University of California, for instance, I may not be referring to anyone in particular, nor stating an identity, nor implying that there is (at present, say) either a governor of California or a chairman of the Board of Regents of the University of California. I may be stating something about the offices

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(properties) of (being the) governor of California and (being the) chairman of the Board of Regents of the University of California. What I am stating is that whoever occupies one of these offices occupies the other.

4.

It is important to notice that when we allow for statements that are neither true nor false, two notions of logical equivalence that coincide for twovalued statements do not coincide anymore. These notions are: (48)

Two forms ϕ and ψ are c-logically equivalent if and only if each is a logical consequence of the other,

where, in terms of interpretations, ϕ is a logical consequence of ψ if there is no interpretation in which ψ is true and ϕ is not true. (49)

Two forms ϕ and ψ are tv-logically equivalent if and only if they have the same truth value in every interpretation,

where sameness of truth value in an interpretation means both true or both false or both truth valueless in that interpretation. It is well known that Frege’s and Russell’s analyses (25) and (33) of a statement of form (1) are not generally materially equivalent. For if there isn’t a unique F , (25) is truth valueless and (33) is false. This means that (25) and (33) are not tv-logically equivalent, because tv-logical equivalence is a generalization of material equivalence over the totality of interpretations. Nevertheless, (25) and (33) are c-logically equivalent, for there is no interpretation in which one of them is true and the other not true. The analysis (19) that I suggested for (1), following Russell’s predicative intuition, is also c-logically equivalent to (25) and (33) – and it is tv-logically equivalent to (33). In fact, it holds generally that Frege’s analysis, Russell’s analysis and my analysis are c-logically equivalent. And in some cases my analysis is tv-logically equivalent to one or the other (but not to both). Thus, my analysis (23) of (5), which is the same as Frege’s, is clogically equivalent (but not tv-logically equivalent) to Russell’s analysis (31). For (36) my analysis (37), Frege’s analysis (40) and Russell’s analysis (41) are c-logically equivalent, though none are tv-logically equivalent. ((47), on the other hand, which is an additional analysis appropriate to certain contexts, is neither tv- nor c-logically equivalent to the others.)

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5.

There are some cases where the distinction between descriptions in subject position and descriptions in predicate position may not seem adequate. One such case that was very important for Russell is that of statements of the form (50)

The F exists,

and, especially, (51)

The F does not exist.

As Frege before him, Russell maintained that existence cannot be predicated of objects, and that if names stand for objects, then it wouldn’t make sense to say such things as ‘Scott exists’, ‘France exists’, etc. And if ‘the author of Waverley’ is used as a name, then it wouldn’t make sense to say that the author of Waverley exists. But this does make sense, and it is perfectly true according to Russell, because what it means is (52)

∃!x(x is an author of Waverley).

Hence, in general, the proper analysis of (50) is (53)

∃!xF x

and the proper analysis of (51) is (54)

¬∃!xF x,

where (53) and (54) are abbreviations for (55)

∃x(F x & ∀y(Fy → y = x))

and (56)

¬∃x(F x & ∀y(Fy → y = x)).

Although I disagree with the view that ‘exists’ cannot be used as a predicate, I agree that Russell’s analysis of (50)–(51) is very natural. It seems to me that there are two subject-predicate readings of (50). One has ‘exists’ as predicate and the singular term ‘the F ’ as subject; the other has

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the existential quantifier ‘there is’ as predicate and the property is the F as subject. Thus: (57)

[x exists](ιxF x)

(58)

[∃xZx]([!xF x](x)).

(57) is true if ‘ιxF x’ denotes (i.e., if there is a unique F ) and is truth valueless otherwise. (58) is true if there is a unique F and is false otherwise. (57) and (58) are c-logically equivalent but not tv-logically equivalent. That the existential quantifier reading is a very natural reading can be seen from the fact that we often paraphrase statements of form (50) as (59)

There is such a thing as the F ,

which formulated with a variable is (60)

There is an x such that x is the F .

Analyzing ‘x is the F ’ as in (19)–(20) we get (58).8 For (51) we have four natural readings depending on whether we treat the negation as predicate negation or as statement negation (‘it is not the case that’). Thus: (61)

[¬(x exists)](ιxF x)

(62)

¬([x exists](ιxF x))

(63)

[¬∃xZx]([!xF x](x))

(64)

¬([∃xZx]([!xF x](x))).

(61) is never true; it is false if ‘ιxF x’ denotes and it is truth valueless otherwise. (62) is true if ‘ιxF x’ does not denote and is false otherwise. (63)–(64) are true if there isn’t a unique F and false otherwise. (62)–(64) are all c-logically equivalent and tv-logically equivalent, but (61) is neither tv- nor c-logically equivalent to the others. The distinction between predicate negation and statement negation is also a very natural distinction if one has an analysis in terms of subjects and predicate. Russell himself uses the distinction in his analysis of descriptions conceptualizing it as a distinction of scope. Thus a statement of the form (65)

The F is not G

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has the two analyses (66)

∃x(F x & ∀y(Fy → y = x) & ¬Gx)

and (67)

¬(∃x(F x & ∀y(Fy → y = x) & Gx)).9

I see the difference between the two kinds of negation as a change of subject that involves a change in truth conditions. In predicate negation the subject remains the same, and a predicate negation is true (false) if and only if the negated statement is false (true). Thus predicate negations of truth valueless statements are also truth valueless. In statement negation we are denying that something or other is the case, and the subject now becomes the content of the statement that is denied. A statement negation is true if and only if the statement denied is not true – i.e., is either false or truth valueless. Hence a statement negation is always either true or false. I think that our intuitions about the truth or falsity of statements involving non-denoting names and descriptions can often be justified as intuitions based on statement negation.

6.

There are many other contexts that one may want to analyze, but I will conclude with just one additional example. Consider statements of the form (68)

a is an R of the S of b.

For instance: (69)

Frege was a teacher of the author of Meaning and Necessity.

A natural Fregean reading of (68) may be, in subject-predicate notation, (70)

[Rxy](a, ιxSxb),

which is a relational statement with two singular terms as subjects. It seems clear to me that we do make such relational statements. Russell’s reading, on the other hand, is (71)

∃z(Raz & (Szb & ∀w(Swb → w = z))),

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which seems much less natural. Nevertheless, Russell’s reading is very natural if we consider (71) as a relational statement with a and b as subjects (72)

[∃z(Rxz & (Szy & ∀w(Swy → w = z)))](a, b),

where the predicate denotes the relational product of the relations x is an R of z and z is the S of y. For just as we take a statement of the form (73)

a is an R of an S of b

as a relational product (74)

[∃z(Rxz & Szy)](a, b),

we can take (68) as the relational product (72) where instead of S we have the relation [!zSzy](z, y).10 And, as in earlier cases, Frege’s analysis (70) and Russell’s analysis (71) are c-logically equivalent but not tv-logically equivalent.

7.

Although I believe that the account of descriptions that I suggested above combines the best intuitive features of Frege’s account and of Russell’s account, it may be argued that from a formal point of view it combines the worst features of these accounts. For it involves working with terms that do not denote, and statements that are neither true nor false, deriving from Frege’s account, and working with subject-predicate distinctions that are akin to Russell’s scope distinctions. I acknowledge the complications but I do not see them as defects. For, on the one hand, I do not know any language, ordinary or scientific, where the occurrence of non-denoting terms (names, descriptions) and truth valueless statements can be ruled out. And, on the other hand, it seems to me that to disregard the distinction between what is said and what it is said about leads to many confusions in our account of logical matters. I have tried to illustrate this last point above in connection with Frege’s and Russell’s accounts of descriptions.11

NOTES 1 This predicative intuition is expressed quite clearly by Russell in ‘Knowledge by Ac-

quaintance and Knowledge by Description’ (p. 215) and in The Problems of Philosophy (p. 53). In the latter he says:

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“The proposition ‘a is the so-and-so’ means that a has the property so-and-so, and nothing else has. Mr. A. is the unionist candidate for this constituency’ means ‘Mr. A. is a unionist candidate for this constituency, and no one else is’.” The same intuition is present in Russell’s initial account of definite descriptions in ‘On Denoting’, where he says (p. 44): “Thus when we say ‘x was the father of Charles II’ we not only assert that x had a certain relation to Charles II, but also that nothing else had this relation.” This predicative interpretation of descriptions in statements of form (1) is also considered by Lyons in Semantics 1 (p. 185) in connection with the example ‘Giscard d’Estaing is the President of France’. 2 “On Sense and Reference”, pp. 32–33 (original pagination). 3 “On Denoting”, p. 44. Russell does not do this explicitly through step (8), but after analyzing ‘x is the father of Charles II’ predicatively he moves on to the example ‘the father of Charles II was executed’ – where ‘the father of Charles II’ appears in subject position – and analyzes it as in (9). 4 In “On Referring” Strawson makes many important points against Russell’s analysis of statements of form (5). I generally agree with his considerations and the points I am making in this paper are quite compatible with his. 5 Although in “The Philosophy of Logical Atomism” (p. 245) Russell states emphatically that “[i]n ‘Scott is the author of Waverley’ the ‘is’, of course, expresses identity, i.e., the entity whose name is Scott is identical with the author of Waverley”, there is no suggestion in the passages quoted in note 1, or in the surrounding context, that ‘a is the so-and-so’ must be analyzed as an identity rather than as a predication. In “On Denoting” (p. 55) Russell makes the following interesting remark: “The usefulness of identity is explained by the above theory. No one outside of a logic-book ever wishes to say ‘x is x’, and yet assertions of identity are often made in such forms as ‘Scott was the author of Waverley’ or ‘thou art the man’. The meaning of such propositions cannot be stated without the notion of identity, although they are not simply statements that Scott is identical with another term, the author of Waverley, or that thou art identical with another term, the man. The shortest statement of ‘Scott is the author of Waverley’ seems to be ‘Scott wrote Waverley; and it is always true of y that if y wrote Waverley, y is identical with Scott’. It is in this way that identity enters into ‘Scott is the author of Waverley’; and it is owing to such uses that identity is worth affirming.” Since this analysis of ‘Scott is the author of Waverley’ is again the predicative analysis, what Russell is saying is that in the analysis of (1) identity enters in the second predication (4) of (3)–(4) – see (19) in the text. 6 This notation is used in model theory, especially in connection with definability. 7 It is important to emphasize that these are statements of form (1) and that statements of form (5) cannot be denied on the grounds that there isn’t a unique F . The latter is what Frege argued in “On Sense and Reference” (p. 40) in connection with the example ‘whoever discovered the elliptic form of the planetary orbits died in misery’, and his argument applies equally well to ‘the discoverer of the elliptic form of the planetary orbits died in misery’. 8 Thus in The Problems of Philosophy (p. 53) Russell says that “[w]hen we say ‘the soand-so exists’, we mean that there is just one object which is the so-and-so’.” Given his predicative analysis of ‘is the so-and-so’, the ‘just one’ is superfluous and can be replaced

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by ‘an’. In fact, later in the paragraph he analyzes ‘The Unionist candidate for this constituency exists’ as meaning ‘some one is a Unionist candidate for this constituency, and no one else is’. 9 This distinction is important for Russell in order to maintain the principle of excluded middle, among other things. 10 In Principia Mathematica ∗ 30, Whitehead and Russell call such functional relations “descriptive functions”. 11 This paper derives from a chapter in a more general unpublished manuscript concerned with issues in the philosophy of logic. In writing the paper I followed my oral presentation at the Congress.

REFERENCES

Frege, G.: 1960, ‘On Sense and Reference’, in M. Black and P. Geach (eds), Translations from the Philosophical Writings of Gottlob Frege, Blackwell, Oxford. Lyons, J.: 1977, Semantics 1, Cambridge University Press, Cambridge. Russell, B.: 1956, ‘On Denoting’, in R. C. Marsh (ed.), Logic and Knowledge. Essays 1901–1950, Allen and Unwin, London. Russell, B.: 1917, ‘Knowledge by Acquaintance and Knowledge by Description’, in Mysticism and Logic and Other Essays, Allen and Unwin, London. Russell, B.: 1912, The Problems of Philosophy, Williams and Norgate, London. Russell, B.: 1956, ‘The Philosophy of Logical Atomism’, in Logic and Knowledge. Strawson, P.: 1971, ‘On Referring’, in Logico-Linguistic Papers, Methuen, London. Whitehead, A. N. and Russell, B.: 1913, Principia Mathematica 1, Cambridge University Press, Cambridge. Department of Philosophy Pontifícia Universidade Catolica de Rio de Janeiro Rio de Janeiro Brazil

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