Possibility Theory, Probability Theory and Multiple-Valued Logics: A Clarification
Annals of Mathematics and Artificial Intelligence
Annals of Mathematics and Artificial Intelligence
A Simple Modal Logic for Reasoning about Revealed Beliefs
ECSQARU '09 Proceedings of the 10th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty
Generalized possibilistic logic
SUM'11 Proceedings of the 5th international conference on Scalable uncertainty management
A simple logic for reasoning about incomplete knowledge
International Journal of Approximate Reasoning
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There are several three-valued logical systems. They give the impression of a scattered landscape. The majority of the works on this subject gives the truth tables, sometimes an Hilbert style axiomatization in a basic propositional language and a completeness theorem with respect to those truth tables. We show that all the reasonable connectives in three-valued logics can be built starting from few of them. Nevertheless, the issue of the usefulness of each system in relation with the third truth value is often neglected. Here, we review the interpretations of the third truth value. Then, we focus on the unknown case, suggested by Kleene. We show that any formula in three-valued logics can be encoded as a fragment of an epistemic logic (formulae of modal depth 1, with modalities in front of literals), preserving all tautologies and inference rules. We study in particular, the translation of Kleene, Gödel, Łukasiewicz and Nelson logics. This work enables us to lay bare the limited expressive power of three-valued logics in uncertainty management.