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 . n turn attention to specific families of modal and si logics. We prove p-equivalence of EF and SF for all extensions of KB , all tabular logics, all logics of finite depth and width, and typical examples of logics of finite width and infinite depth. In most cases, we actually show an equivalence with the usual EF system for classical logic with respect to a naturally defined translation. other hand, we establish exponential speed-up of SF over EF for all modal and si logics of infinite branching, extending recent lower bounds by P. Hrubeš. We develop a model-theoretical characterization of maximal logics of infinite branching to prove this result.