Is Hume's Principle Analytic?

One recent ‘neologicist’ claim is that what has come to be known as “Frege’s Theorem”—the result that Hume’s Principle, plus second-order logic, suffices for a proof of the Dedekind-Peano postulate—reinstates Frege’s contention that arithmetic is analytic. This claim naturally depends upon the analyticity of Hume’s Principle itself. The present paper reviews five misgivings that developed in various of George Boolos’s writings. It observes that each of them really concerns not ‘analyticity’ but either the truth of Hume’s Principle or our entitlement to accept it and reviews possible neologicist replies. A two-part Appendix explores recent developments of the fifth of Boolos’s objections—the problem of Bad Company—and outlines a proof of the principle Nq, an important part of the defense of the claim that what follows from Hume’s Principle is not merely a theory which allows of interpretation as arithmetic but arithmetic itself. 1 It was George Boolos who, following Frege’s somewhat charitable lead at Grundlagen §63, first gave the name, “Hume’s Principle,” to the constitutive principle for identity of cardinal number: that the number of Fs is the same as the number of Gs just in case there exists a one-to-one correlation between the Fs and the Gs. The interest—if indeed any—of the question whether the principle is analytic is wholly consequential on what has come to be known as Frege’s Theorem: the proof, prefigured in Grundlagen §§82–83 [5] and worked out in some detail in Wright [21]1 that second-order logic plus Hume’s Principle as sole additional axiom suffices for a derivation of second-order arithmetic—or, more cautiously, for the derivation of a theory which allows of interpretation as second-order arithmetic. (Actually I think the caution is unnecessary—more of that later.) Analyticity, whatever exactly it is, is presumably transmissible across logical consequence. If second-order consequence is indeed a species of logical consequence, the analyticity of Hume’s Principle would ensure the analyticity of arithmetic—at least, provided it really is second-order arithmetic, and not just a theory which merely allows interpretation as such, which is a second-order consequence of Hume’s Principle. What significance that finding would have would then depend, of course, on the significance of the notion of analyticity Received April 16, 1998