In this essay I will consider two theses that are associated with Frege, and will investigate the extent to which Frege “really” believed them. Much of what I have to say will come as no surprise to scholars of the historical Frege. But Frege is not only a historical figure; he also occupies a site on the philosophical landscape that has allowed his doctrines to seep into the subconscious water...

We show that the Gaussian Elimination algorithm can be proven correct with uniform Extended Frege proofs of polynomial size, and hence feasibly. More precisely, we give short uniform Extended Frege proofs of the tautologies that express the following: given a matrix A, the Gaussian Elimination algorithm reduces A to row-echelon form. We also show that the consequence of this is that a large cla...

Three-valued logics belong to a family of nonclassical logics that started to flourish in the 1920s and 1930s, following the work of ( Lukasiewicz, 1920), and earlier insights coming from Frege and Peirce (see (Frege, 1879), (Frege, 1892), (Fisch and Turquette, 1966)). All of them were moved by the idea that not all sentences need be True or False, but that some sentences can be indeterminate i...

Propositional proof complexity is a field of theoretical computer science which concerns itself with the lengths of formal proofs in various propositional proof systems. Frege systems are an important class of propositional proof systems. Extended Frege augments them by allowing the introduction of new variables to abbreviate formulas. Perhaps the largest open question in propositional proof co...

We associate a CNF-formula to every instance of the mean-payoff game problem in such a way that if the value of the game is non-negative the formula is satisfiable, and if the value of the game is negative the formula has a polynomial-size refutation in Σ2-Frege (i.e. DNF-resolution). This reduces mean-payoff games to the weak automatizability of Σ2-Frege, and to the interpolation problem for Σ...

Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional-calculus (i.e., Frege) proof is any proof starting from a set of axioms and deriving new Boolean formulas using a fixed set of sound derivation rules. Establishing any superpolynomial size lower bound on Frege proofs (in terms of the size of the formula proved) is a major open problem in proof complexi...