Descriptions
Descriptions
Where a formula A containing a free variable — say, for example, x — means a true proposition (is true) for one and only one value of x, the notation (iota;x)A is used to mean thit value of x. The approximately equivalent English phraseology is “the x such that A” — or simply ‘the F,” where F denotes the concept (monadic propositional function) obtained from A by abstraction (q. v.) with respect to x. This notation, or its sense in the sense of Frege, is called a description.
In Principia Mathematica descriptions (or notations serving the same purpose in context) are introduced is incomplete symbols (q. v.). Russell maintains that descriptions not only may but must be thus construed as incomplete symbols — briefly, for the following reasons. The alternative is to construe a description as a proper name, so that, e.g., the description the author of Waverley denotes the man Scott and is therefore synonymous with the name Scott. But then the sentences “Scott is the author of Waverley” and “Scott is Scott” ought to be synonymous — which they clearly are not (although both are true). Moreover, such a desription as the King of France cannot be a proper name, since there is no King of France whom it may denote; nevertheless, a sentence such as “The King of France is bald” should be construed to have a meaning, since it may be falsely asserted or believed by one who falsely asserts or believes that there is a King of France.
Frege meets the same difficulties, without construing descriptions as incomplete symbols, by distinguishing two kinds of meaning, the sense (Sinn) and the denotation (Bedeutung) of an expression (formula, phrase, sentence, etc.). Scott and the author of Waverley have the same denotation, namely the man Scott, but not the same sense. The King of France has a sense but no denotation; so likewise the sentence, The King of France is bald. Two expressions having the same sense must have the same denotation if they have a denotation. When a constituent part of an expression is replaced by another part having the same sense, the sense of the whole is not altered. When a constituent part of an expression is replaced by another having the same denotation, the denotation of the whole (if any) is not altered, but the sense may be. The denotation of an (unasserted) declarative sentence (if any) is a truth-value, whereas the sense is the thought or content of the sentence. But where a sentence is used in indirect discourse (as in saying that so-and-so says that . . ., believes that . . ., is glad that . . ., etc.) the meaning is different in such a context the denotation of the sentence is that which would be its sense in direct discourse. (In quoting some one in indirect discourse, one reproduces neither the literal wording nor the truth-value, but the sense, of what he said.)
Frege held it to be desirable in a formalized logistic system that every formula should have not only a sense but also a denotation — as can be arranged by arbitrary semantical conventions where necessary. When this is done, Frege’s sense of a sentence nearly coincides with proposition (in sense (b) of the article of that title herein). — Alonzo Church
G. Frege, ber Sinn und Bedeutung. Zeitschrift fr Philosophie und philosophische Kritik, n. s., vol. 100 (1892), pp. 25-50. B. Russell, On denoting, Mind, n. s.. vol. 14 (1905). pp. 479-493.