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

Ciucci, Davide and Dubois, Didier A modal theorem-preserving translation of a class of three-valued logics of incomplete information. (2013) Journal of Applied Non-Classical Logics, 23 (4). 321-352. ISSN 1166-3081

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Official URL: http://dx.doi.org/10.1080/11663081.2013.863491

Abstract

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management.

Item Type:Article
Additional Information:Thanks to Taylor & Francis editor. The definitive version is available at http://www.tandfonline.com/doi/abs/10.1080/11663081.2013.863491#.VPbahPmG-oE
HAL Id:hal-01122841
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE)
Université de Toulouse > Université Toulouse III - Paul Sabatier - UT3 (FRANCE)
Université de Toulouse > Université Toulouse - Jean Jaurès - UT2J (FRANCE)
Université de Toulouse > Université Toulouse 1 Capitole - UT1 (FRANCE)
Other partners > Università degli Studi di Milano-Bicocca - BICOCCA (ITALY)
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Deposited On:04 Mar 2015 15:29

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