From Possibilistic Information to Kleene’s Strong Multi-Valued Logics

Abstract

Possibilistic logic in general [Dubois, Lang and Prade, 1989; Dubois, Lang and Prade, 1994; Dubois and Prade, 1990; Esteva, Garcia and Godo, 1994] investigates how possibilistic uncertainty about propositions is propagated when making inferences in a formal logical system. In this paper, we look at a very particular aspect of possibilistic logic: we investigate how, under certain independence assumptions, the introduction of possibilistic uncertainty in classical propositional logic leads to the consideration of special classes of multi-valued logics, with a proper set of truth values and logical functions combining them. First, we show how possibilistic uncertainty about the truth value of a proposition leads to the introduction of possibilistic truth values. Since propositions can be combined into new ones using logical operators, possibilistic uncertainty about the truth values of the original propositions gives rise to possibilistic uncertainty about the truth value of the resulting proposition. Furthermore, we show that in a limited number of special cases there is truth-functionality, i.e. the possibilistic truth value of the resulting proposition is a function of the possibilistic truth values of the original propositions. This leads to the introduction of possibilisticlogical functions, combining possibilistic truth values. Important classes of such functions, the possibilistic extension logics, result directly from this investigation Finally, the relation between these logics and Kleene’s strong multi-valued systems is established. This paper is intended as a brief summary of the much more detailed account that can be found in [de Cooman, 1995].