From Possibilistic Information to Kleene's Strong Multi-valued Logics

Possibilistic logic in general [7, 8, 9, 10] 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 possibilistic-logical 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 [5]. Let us first define the most common notations. By (L,≤), we denote a complete lattice [1] with top 1 and bottom 0, where we assume that 0 6= 1. The meet of (L,≤) is denoted by _, its join by ^. By T we denote a triangular norm on (L,≤) that is completely distributive w.r.t. supremum [6]. We also use the set T = {false, true} of truth values in classical propositional logic. On T , we define the total order relation ≤ = {(false, false), (false, true), (true, true)}. (T ,≤) is a Boolean chain of length 2, with top true and bottom false. On this chain, we may define as usual the complement ¬, called negation; the meet ∧, called conjunction; the join ∨, called disjunction and the implication ⇒. We also consider a universe X. A X − L-mapping h is called sup-normal iff supx∈X h(x) = 1. The set of all X − L-mappings is denoted by LX . With a subset A of X, we may associate its characteristic X − T -mapping χA, with, for any x in X: χA(x) = true if x ∈ A and χA(x) = false if x 6∈ A. Next, we introduce the notion of a possibilistic extension, which is related to Zadeh’s extension principle, but is here only used within a possibilistic context, without reference to fuzzy sets. By X, X1, . . . , Xn and Y we denote arbitrary universes. First of all, with a X − Y -mapping φ we can associate a LX −LY -mapping φ̃, defined as follows. For any X −L-mapping h the Y −L-mapping φ̃(h) is given by, for any y in Y : φ̃(h) · y = supφ(x)=y h(x). φ̃ is called the (L,≤)-possibilistic extension of φ. Also, with a X1 × · · · ×Xn − Y -mapping φ we can associate a LX1 × · · · × LXn − LY -mapping φ̃T , defined as follows. For any (h1, . . . , hn) in LX1 × · · · × LXn the Y − L-mapping φ̃T (h1, . . . , hn) is given by, for any y in Y : φ̃T (h1, . . . , hn) · y = supφ(x1,... ,xn)=y T n k=1hk(xk). φ̃T is called the (L,≤)-possibilistic T -extension of φ.