The predicative Frege hierarchy

The predicative Frege hierarchy

In this paper, we characterize the strength of the predicative Frege hierarchy, PV, introduced by John Burgess in his book [Bur05]. We show that PV and Q + con(Q) are mutually interpretable. It follows that PV := PV is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [Gan06] using a different proof. Another consequence of the our main result is that PV is mutually i...

Predicative fragments of Frege Arithmetic

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity in...

The Curse of Frege

I mention two methodological problems with AI's study of inference that have to do with excessive faith in classical logic. The first is the familiar bias toward formalisms that incorporate deductive patterns of argument: a bias against inductive patterns of reasoning. The second is the pretension that inference can be studied with no regard for one's habits of representation. By implicating Fr...

Frege Systems

Propositional Proofs in Frege and Extended Frege Systems (Abstract)

We discuss recent results on the propositional proof complexity of Frege proof systems, including some recently discovered quasipolynomial size proofs for the pigeonhole principle and the Kneser-Lovász theorem. These are closely related to formalizability in bounded arithmetic.