The indefinability of ``one'' (Q1610608)

A proud achievement of the logicists, Frege and Russell, was their definition of the numerals. Closely connected was Russell's theory of definite descriptions, which defined ``the''. Goldstein challenges these achievements on the ground that they employ individual variables such as ``\(x \)'', that is, variables which on instantiation will be replaced by singular terms, expressions which stand only for a single object. Goldstein does not discuss medieval discussions of definite-description-forming words like praeter (``but''), solus (``alone'') and tantum (``only'') [see \textit{William of Sherwood}, Treatise on syncategorematic words, translated by N. Kretzmann. Minneapolis: University of Minnesota Press (1968), cc. x, xi, xii] to which many of the same considerations apply. But singular terms can be explained, not as expressions standing for a single object, but in terms of their role in substitutional inference. Such an explanation has been offered c. 6 (``Substitution: What are singular terms and why are there any?'') of a book he cites [\textit{R. Brandom}, Making it explicit. Cambridge, MA: Harvard University Press (1994), pp. 334-412].