From possibility theory to paraconsistency

The significance of three-valued logics partly depends on the interpretation of the third truth-value. When it refers to the idea of unknown, we have shown that a number of three-valued logics, especially Kleene, Ł eene, e s, and Nelson, can be encoded in a simple fragment of the modal logic KD, called MEL, containing only modal formulas without nesting. This is the logic of possibility theory, the semantics of which can be expressed in terms of all-or-nothing possibility distributions representing an agent’s epistemic state. Here we show that this formalism can also encode some three-valued paraconsistent logics, like Priest, Jawe have, and Sobociński’s, where the third truth-value represents the idea of contradiction. The idea is just to change the designated truth-values used for their translations.We show that all these translations into modal logic are very close in spirit to Avron t early work expressing natural three-valued logics using hypersequents. Our work unifies a number of existing formalisms and the translation also highlights the perfect symmetry between three-valued logics of contradiction and three-valued logics of incomplete information, which corresponds to a swapping of modalities in MEL.