Substitution Frege and extended Frege proof systems in non-classical logics

Substitution Frege and extended Frege proof systems in non-classical logics

We investigate the substitution Frege (SF ) proof system and its relationship to extended Frege (EF ) in the context of modal and superintuitionistic (si) propositional logics. We show that EF is p-equivalent to tree-like SF , and we develop a “normal form” for SF proofs. We establish connections between SF for a logic L, and EF for certain bimodal expansions of L. We then turn attention to spe...

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.

Proof internalization in generalized Frege systems for classical logic

We present a general method for inserting proofs in Frege systems for classical logic that produces systems that can internalize their own proofs.

Frege systems for extensible modal logics

By a well-known result of Cook and Reckhow [4, 12], all Frege systems for the Classical Propositional Calculus (CPC ) are polynomially equivalent. Mints and Kojevnikov [11] have recently shown p-equivalence of Frege systems for the Intuitionistic Propositional Calculus (IPC ) in the standard language, building on a description of admissible rules of IPC by Iemhoff [8]. We prove a similar result...

Frege Systems