Is Hume’s Principle Analytic?
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 F’s is the same as the number of G’s just in case there exists a one-to-one correlation between the F’s and the G’s. 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 §§82–3 and worked out in some detail in Wright  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. So 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 itself. Later I shall suggest that the most important issues here are ones which are formulable without recourse to the notion of analyticity at all—so that much of the debate between Boolos and me could have finessed the title question.
KeywordsCardinal Number Direction Equivalence Peano Arithmetic Abstraction Principle Determinate Number
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