# Effective Finite-Valued Approximations of General Propositional Logics

## TL;DR

This paper explores how complex propositional logics can be approximated by simpler finite-valued logics, providing methods to compute minimal such logics and conditions for their effective characterization.

## Contribution

It introduces a way to compute the minimal finite-valued logic for which a calculus is strongly sound and investigates conditions for representing propositional logics as intersections of finite-valued logics.

## Key findings

- Minimal m-valued logic for strong soundness can be calculated.
- Propositional logics can be characterized as intersections of finite-valued logics under certain conditions.
- Finite-valued semantics offer computationally simpler alternatives for propositional logic reasoning.

## Abstract

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.