Three-valued Logics in Modal Logic

Barteld Kooi,Allard Tamminga

doi:10.1007/s11225-012-9420-0

Barteld Kooi, Allard Tamminga

Open Access

https://doi.org/10.1007/s11225-012-9420-0

Copy DOI

Abstract

Abstract Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises the question whether there are three-valued logics for which there is a shorter translation into S5. The answer is affirmative: we present an elegant linear translation of the Logic of Paradox and of Strong Three-valued Logic into S5.

Highlights

Translations of one logic into another logic might serve as a bridge to carry over technical results and philosophical insights
We present a Translation Manual that translates any three-valued logic defined into S5
We prove our theorem that every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5: Theorem 3.4

Summary

Introduction

Translations of one logic into another logic might serve as a bridge to carry over technical results and philosophical insights. The translation of intuitionistic propositional logic into the modal logic S4, based on Godel’s [9], strongly supports the provability interpretation of intuitionistic logic. A conservative mapping of a logic L1 into a logic L2 is a function that preserves valid formulas in both ways:. The notion of a conservative mapping is too blunt an instrument for studying translations of three-valued logics into modal logic. The function (φ) = p ∧ ¬p (where p is some atomic formula) conservatively maps Strong Three-valued Logic (K3) [10, 11] into classical propositional logic, because K3 does not have any valid formulas. We show that because for every S5-model there is a translationally equivalent three-valued valuation and vice versa, every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We present a linear translation that conservatively translates both LP and K3 into S5

Three-valued Logics

A Translation Manual for Three-valued Logics

A Linear Translation of LP and K3 into S5

Linear X-translations

Conclusion