Abstract
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, Łukasiewicz, 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, Jaśkowski, 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’s 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.
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Notes
- 1.
A system is over-complete if any formula is a theorem.
- 2.
A negation \(\lnot \) is internal if “\(\Gamma , A \vdash \Delta \) iff \( \Gamma \vdash \Delta , \lnot A\)” or equivalently if “\(\Gamma \vdash \Delta , A\) iff \( \Gamma , \lnot A \vdash \Delta \).” An implication \(\rightarrow \) is internal if “\(\Gamma , A \vdash \Delta , B\) iff \(\Gamma \vdash \Delta , A \rightarrow B\).”
- 3.
Not to be confused with the MEL contrapositive form of \({\Box }a \rightarrow {\Box }b\), i.e., \(({\Box }b)' \rightarrow ({\Box }a)'\), of course equivalent to the former.
- 4.
The convention differs from the ones in the preliminary version of this paper [11] where we used \({\Box }p\) to stand for “at least one source asserts p.” The latter convention leads to a logic where \({\Box }\) has the same properties as \({\lozenge }\) in a KD system, which may be misleading.
- 5.
Note that Jaśkowski traces back this implication to Słupecki.
- 6.
So-called, as it violates the contradiction law and satisfies the excluded middle law.
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Acknowledgments
The authors wish to thank Philippe Besnard for remarks on the first draft that led us to improve the presentation.
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Ciucci, D., Dubois, D. (2015). From Possibility Theory to Paraconsistency. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2719-9_10
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