The Logic in Logicism

ALEXANDER BIRD

Dialogue XXXVI (1997), 341-60

Abstract

Frege's logicism consists of two theses: (1) the truths of arithmetic are truths of logic; (2) the natural numbers are objects. In this paper I pose the question: what conception of logic is required to defend these theses? I hold that there exists an appropriate and natural conception of logic, in virtue of which Hume's principle is a logical truth. Hume's principle, which states that the number of Fs is the number of Gs iff the concepts F and G are equinumerous, is the central plank in the neo-logicist argument for (1) and (2). I defend this position against two objections (a) Hume's principle cannot both be a logical truth as required by (1) and also have the ontological import required by (2); and (b) the use of Hume's principle by the logicist is in effect an ontological proof of a kind which is not valid.