Studia Logica

, Volume 101, Issue 5, pp 1061-1072

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Three-valued Logics in Modal Logic

  • Barteld KooiAffiliated withFaculty of Philosophy, University of Groningen
  • , Allard TammingaAffiliated withFaculty of Philosophy, University of GroningenInstitute of Philosophy, University of Oldenburg Email author 


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.


Three-valued logic Modal logic Conservative translations Expressivity