We prove that the problem of determining the minimum propositional proof length is NPhard to approximate within a factor of 2 1−o(1) n . These results are very robust in that they hold for almost all natural proof systems, including: Frege systems, extended Frege systems, resolution, Horn resolution, the polynomial calculus, the sequent calculus, the cut-free sequent calculus, as well as the po...

Note: This essay was commissioned and written circa 1997 for The Cambridge Companion to Frege, which has not yet appeared. Although quite a bit of good work on Frege and Dedekind’s mathematical antecedents has since appeared (by Jeremy Avigad, Jamie Tappenden and others), I think this essay still provides the student with a useful introduction to many of the central issues. I have since refined...

In this paper we ask the question whether the extended Frege proof system EF satisfies a weak version of the deduction theorem. We prove that if this is the case, then complete disjoint NP-pairs exist. On the other hand, if EF is an optimal proof system, then the weak deduction theorem holds for EF . Hence the weak deduction property for EF is a natural intermediate condition between the optima...

We give a new characterization of the strict ∀Σj sentences provable using Σk induction, for 1 ≤ j ≤ k. As a small application we show that Buss’s witnessing theorem for strict Σk formulas already holds over the relatively weak theory PV. We exhibit a combinatorial principle with the property that a lower bound for it in constant-depth Frege would imply that the narrow CNFs with short depth j Fr...

Frege’s definitions of zero, predecession, and natural number will be explained below. As for second-order Dedekind-Peano arithmetic, the axiomatization most convenient for our purposes is the following: (1) N0 (2) Nx∧Pxy→ Ny (3) ∀x∀y∀z(Nx∧Pxy∧Pxz→ y = z) (4) ∀x∀y∀z(Nx∧Ny∧Pxz∧Pyz→ x = y) (5) ¬∃x(Nx∧Px0) (6) ∀x(Nx→∃y(Pxy)) (7) ∀F(F0∧∀x∀y(Nx∧Fx∧Pxy→ Fy)→∀x(Nx→ Fx) If (slightly non-standardly) we ...

Frege, Russell and Wittgenstein were founders (although not by any means the sole founders) of the analytic tradition in philosophy; but they did not found the philosophy of language, which has roots stretching back much further. Their principal contribution, indeed, could be regarded as being in the opposite direction: it is not so much that they applied philosophical methods to the study of l...