# Effective finite-valued approximations of general propositional logics

Baaz, Matthias, and Richard Zach. 2008. “Effective Finite-Valued Approximations of General Propositional Logics.” In Pillars of Computer Science: Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday, edited by Arnon Avron, Nachum Dershowitz, and Alexander Rabinovich, 107–29. LNCS 4800. Berlin: Springer. DOI:10.1007/978-3-540-78127-1_7

Propositional logics in general, considered as a set of sentences, can be undecidable even if they have “nice” representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple—at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional logics represented in various ways can be approximated by finite-valued logics. It is shown that the minimal m-valued logic for  which a given calculus is strongly sound can be calculated. It is also investigated under which conditions propositional logics can be characterized as the intersection of (effectively given) sequences of finite-valued logics.