Three-valued Logics in Modal Logic

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 threevalued 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.

A modal theorem-preserving translation of a class of three-valued logics of incomplete information

“…The motivation of this translation is to better understand the meaning of 3-valued connectives and formulas in the scope of handling incomplete information. Moreover, the above cited translations into S5, like (Kooi and Tamminga, 2013), focus on the separation between valid, invalid and contingent formulas only (as expected with S5). In contrast, we here deal with the issue of inference of a formula from a set of formulas in three-valued logics, and show it translates into inference from a knowledge base in MEL.…”

“…Translations of three-valued logics into modal logic are not new either. For instance, Duffy (1979), and very recently Kooi and Tamminga (2013) use S5 as a target language. Minari (2002) applies the above Tarski expression of the modal possible to Wajsberg axioms of Lukasiewicz logic, and studies the resulting modal system.…”

Game theoretical semantics for some non-classical logics

“…In a recent work, an efficient translation between three-valued logics and modal logic S5 was given (Kooi & Tamminga, 2013). A model of modal logic S5 is defined as a tuple (W, R, V ) where W is a non-empty set, R is an equivalence relation on W × W and V is the standard valuation function.…”

Vagueness, Truth and Permissive Consequence

“…Dually, for P a to hold tolerantly could be taken to mean that a is not determinately not rich. As a matter of fact, instead of introducing strict and tolerant levels for the truth values of our sentences, we could decide to work with a single notion of assertion, but to translate the strict and tolerant metalanguage distinctions into appropriate modal sentences of our object-language, enriched with determinateness operators (see [Kooi and Tamminga, 2012] for an exact statement of such a modal translation for basic LP and K3 sentences). For example, to say that the tolerance principle is tolerantly valid, in modal terms, would turn out to be equivalent to the observation that the following "gap principle" holds classically in our models: 14 (5) ∀xy(DP x ∧ xI P y ⊃ ¬D¬P y)…”