Descriptions in Nonextensional Contexts

Descriptions in Nonextensional Contexts Gustav Bergmann Philosophy of Science, Vol. 15, No. 4. (Oct., 1948), pp. 353-355. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28194810%2915%3A4%3C353%3ADINC%3E2.0.CO%3B2-K Philosophy of Science is currently published by The University of Chicago Press.

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DESCRIPTIONS I N NONEXTENSIONAL CONTEXTS GUSTAV BERGMANN

1. I n his paper "On Denoting" Russell1 points out what he calls a puzzle every satisfactory theory of denoting will have to solve. Of the three sentences George IV. wished to know whether Scott mas the author of Waverley (A) Scott is the author of Waverley (B) George IV. wished to know whether Scott was Scott (C') the first two, (A) and (B), are presumably true, while the third, (C), is presumably false. The reason given for considering this arrangement of truth values a puzzle is (1) that the 'is' in (B) signifies identity; (2) that the proper name 'Scott' and the descriptive phrase 'the author of Waverley' denote the same object; and that, therefore, (3) if the latter is replaced by the former in the true sentence (A), the resulting sentence, (C), should again be true. Yet (C) is false, or, as one also says, the context considered is nonextensional with respect t o this replacement. As Carnap2 who recently examined Russell's argument understands it, and as I understand it, Russell's solution of the puzzle is this. According to his analysis of descriptions, the descriptive phrase, being a so-called incomplete symbol, does not denote anything. Thus premise (2) is denied and the conclusion (3)whose disagreement with the actual truth value of (C) constitutes the puzzlecannot be drawn. In other words, Russell asserts, or seems to assert, that if one accepts his analysis of descriptions, descriptive phrases may occur in nonextensional contexts without giving rise to any dificulties. The purpose of this note is to show that if "analysis of descriptions" is taken t o refer to the language constructedin Principia iMathenzatica (PM), then the italicized statement, though not literally false, is rather misleading. To achieve this purpose, I shall present in two different ways what is essentially the same argument. 2. The first argument starts from the observation that the whole development of PM, with the exception of some asides, is syntactical. Consider the calculus generated from that of PRI by adding to its primitive signs classes of proper names and constant predicates ('a7,'b', . . . ,%','gl', . - , . . - ) . The point is that there is no means of expressing, either in this calculus or in the metalanguage that is, in fact, used in PM, what is expressed by

(Dl

'a' and ‘(72) fl(x)' denote the same object.

The only formula one could conceivably mistake for a transcription of (D) is

But then one must remember that the identity sign of PM, which occurs in M i n d , S I V , 1905,479-93.

~ l f e a n i n ga nd Necessity, Chicago, 1917.

(p. 136 ff.)

353

354

GUSTAV BERGMANN

(D'), is a syntactical identity. By calling it so I wish to call attention to the circumstance that the formula is a theorem of the calculus3. The dot symbol indicates in the customary manner an arbitrary context. Hence, if one takes (D') t o be what it is not, namely, an equivalent of (D), then it follows that if a description and a proper name denote the same object, the calculus does not contain contexts that are nonextensional with respect t o this name and this description. 2a. The second argument uses, not the (semantical) notion of denoting, but the notion of truth; and again, this use of a semantical concept is merely an expository device in the semiformal presentation of a syntactical situation. ACcording to the familiar analysis of PM, the statement 'The so-and-so is a suchand-such' is t o be rendered by the formula

Properly expanded, (F) becomes a conjunction one term of which is Thus, for a statement (F) to be true, (G) must be true. But if (G) is true, and if the language contains a proper name 'a' that fulfills the operand of the existential quantifier in (G), then one can also assert '(x)lfi(x) = (x = a)]' and, furthermore, specialize this formula for '(ix)fl(x). One obtains

Since, if (G) is true, the left side of this formula can be proved t o be true, its right side, which is identical with (D'), is also true. It follows from (E) that if (G) is true, the language does not contain contexts which are nonextensional with respect to 'a' and '(~x)fl(x)'. If (G) is false, then all statements (F) are false and the language does not contain a proper name that fulfills the operand of the existential quantifier in (G). 3. It is instructive t o inquire why Carnap (1.c.) erroneously accepts Russell's "solution" as one possible solution of the puzzle-though, it is true, not as the The reason, I believe, is twofold. First, Carnap's main one he ~ ~ o u prefer. ld interest in his recent book is t o eliminate certain difficulties that arise if one attempts t o formalize the use of such terms as 'denoting'. And it is true, of course, that if descriptive phrases are said not to denote at all, one may get rid of some types of this sort of difficulty. Second, Carnap's own contextual definition of descriptive phrases-or what he considers to be one-is different from that of Russell4. For his definiens does not, like Russell's, contain the sentence By applying directly what Quine calls the substitutivity feature of identity t o t h e incomplet,e symbol I have not violated the precautions necessary in such cases. I t is easy t o prove (E). This does of course not refer t o the rather minor difference due t o the fact t h a t Carnap adopts Frege's proposal of a conventional referent if ( G )is false.

DESCRIPTIONS IN NONEXTENSIONAL CONTEXTS

355

(G), but, instead, a similar sentence that has in place of the identity sign between 'x' and 'y' an equivalent of the predicate which occurs in (D). No matter how this latter predicate ('denote the same object') is correctly transcribed in a formal language, there is no reason to believe that the transcription must have the form of a sentence with a syntactical identity, either between the two expressions that denote or, perhaps, between their names. So the difficulty I wished to point out arises only in connection with Russell's notion of description. To discuss the relative merits of Russell's and Carnap's procedures or to inquire how the predicate of (D) should be rendered in a formal language is not the purpose of this paper.

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