On The Consistency of Second-Order Contextual Definitions

Here, "Nx:ox" is a functional expression which may be read "The number of Os". The definition says that the number of Fs is equal to the number of Gs just when the Fs and the Gs can be correlated one-to-one. Crispin Wright and Neil Tennant, among others, have recently shown that it is possible to derive the standard axioms of second-order Arithmetic from Hume's Principle, if that Principle is taken as the sole non-logical axiom within a standard second-order logic (Wright 1983, 158-69; see also Boolos 1987). Our attitude towards this result is a consequence of our attitude towards Hume's Principle. If we understand the Principle merely as stating a necessary and sufficient condition for numerical equality, the result is of much mathematical interest, but it hasn't the specifically philosophical consequences Frege would have wished to attach to it. Wright's resurrected version of Frege's logicistic project, on the other hand, depends upon taking Hume's Principle as a contextual definition: Hume's Principle is to be understood as defining names of numbers, which are, in turn, to be thought of as (able to be) introduced into our ontology by means of this very definition (Wright 1983, 104-117, 140-42, 145-49). Hume's Principle is an instance of the general form of second-order contextual definitions: