In 1903 Russell’s paradox came over the mathematical world with a double stroke. Bertrand Russell himself published it under the heading “The Contradiction” in chapter 10 of his Principles of Mathematics (Russell 1903 ). Almost at the same time Gottlob Frege (1848–1925) referred to Russell’s paradox in the postscript of the second and final volume of his Grundgesetze der Arithmetik (Frege 1903 ...

Frege is celebrated as an arch-Platonist and an arch-realist. He is renowned for claiming that truths of arithmetic are eternally true and independent of us, our judgments and our thought; that there is a ‘third realm’ containing nonphysical objects that are not ideas. Until recently, there were few attempts to explicate these renowned claims, for most philosophers thought the clarity of Frege’...

The complexity of simple stochastic games (SSGs) has been open since they were defined by Condon in 1992. Despite intensive effort, the complexity of this problem is still unresolved. In this paper, building on the results of [4], we establish a connection between the complexity of SSGs and the complexity of an important problem in proof complexity–the proof search problem for low depth Frege s...

We prove a quasi-polynomial lower bound on the size of bounded-depth Frege proofs of the pigeonhole principle PHPm n where m 1 1 polylog n n. This lower bound qualitatively matches the known quasi-polynomial-size bounded-depth Frege proofs for these principles. Our technique, which uses a switching lemma argument like other lower bounds for boundeddepth Frege proofs, is novel in that the tautol...

We want to encourage researchers to investigate the potential of proof systems that modify a given set of formulas (e.g., a set of clauses in propositional logic) in a way that preserves satisfiability but not necessarily logical equivalence. We call such modifications interferences, because they can change the models of a given set of formulas. Interferences differ from classical inferences, w...

We prove lower bounds of the form exp (n " d) ; " d > 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.