Epistemic Necessity, Possibility, and Truth. Tools for Dealing with Imprecision and Uncertainty in Fuzzy Knowledge-Based Systems P. Magrez* and P. Smets Universitd Libre de Bruxelles Brussels, Belgium ABSTRACT A goal in approximate reasoning is the querying of fuzzy knowledge bases. This can be viewed as the estimation of the modalities of a proposition on the basis of vague propositions that can even be only partially true, partially necessary, or partially possible. In this article, we present a framework for an integrated theory of imprecision and uncertainty. First, we describe a tool to transform a proposition qualified by a partial degree of necessity (possibility) into a fuzzier, but true, proposition. Second, we show that the degree of truth of a proposition estimated from imprecise information is not necessarily unique but belongs to an interval where the lower bound is a necessity measure and the upper bound is a possibility measure. The theory of multiple-valued truths thus finds an interpretation in this integrated theory of uncertainty and imprecision. KEYWORDS: approximate reasoning, knowledge-based systems, possibil- ity theory, fuzzy sets theory, multiple-valued truths INTRODUCTION In commonsense reasoning, two types of ignorance that lead to different models can be distinguished. First, when information is certain but imprecise, one can use modalities (necessity and possibility) and multiple-valued truths. * Senior research assistant at the National Fund for Scientific Research (Belgium). Address correspondence to Paul Magrez, IRIDIA, Universit6 Libre de 8ruxelles, Av. Ft. Roosevelt, 50, CP 194/6, 1050, Brussels, Belgium. International Journal of Approximate Reasoning 1989; 3:35-57 © 1989 Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010 0888-613X/89/$3.50 35 36 P. Magrez and P. Smets Second, when information is uncertain in that it conveys the idea that something may occur, that it is probable, that we can believe in it ..... one can use classical theories of uncertainty such as probability theory, Shafer's theory of evidence [1], and Sugeno's measures [2]. The first type of ignorance will be qualified as an epistemic ignorance as it deals with the determination of the truth status of some proposition given other propositions contained in our "knowledge base" and known to be true, even though they may be imprecise. Epistemic ignorance corresponds to the conformity obtained by a semantical matching between the meaning of a proposition and the reality as known. Imprecision, be it due to vagueness or not, induces epistemic ignorance. When the truth status of a proposition is binary, either true or false, epistemic ignorance reduces itself to that part of classical logic where one tries to deduce the truth status of a proposition given a set of true propositions, and to classical modal logic when one tries to evaluate the necessity or the possibility of a proposition. This paper generalizes this approach when one accepts that the truth status, the necessity, and the possibility of propositions can admit degrees and can be defined on a bounded ordered interval such as the [0, 1] interval. Partial truth is admitted and is essentially related to the use of vague predicates (a justification of its existence and a method for its evaluation can be found in Smets and Magrez [3, 4]). Example I A proposition such as "Paul is young" will be qualified as true if we know that Paul's age is 15 but only possible if we know that Paul's age belongs to the interval 7-77 years. The second kind of ignorance is the uncertainty representing our opinion, our judgment about the truth of a proposition. Supra-epistemic ignorance fits the concepts of probability, credibility, plausibility ..... It is called supra-epistemic because it covers a type of ignorance embedded in a metalanguage that can be constructed above the language including, among other things, epistemic ignorance, hence the "supra" qualification. This paper deals only with epistemic ignorance. Its aim is to find a link between the definition of imprecision in fuzzy set theory and the definition of ignorance in modal logic and multiple-valued logics, a link that could bridge the gap between imprecision and epistemic ignorance in a general way. This link will be important in solving various problems. For example, let an expert system reasoning process yield the information: "the truth that Paul is young is .8." From this information stored in the knowledge base, the expert system must be able to evaluate, by some kind of approximate matching, the truth of such propositions as: "Paul's age belongs to [25, 35]," "Paul is not very young," .... As we will see, the resolution of this problem will be possible when the stored proposition can be transformed into a strictly true, but maybe fuzzier, proposition. This paper presents the following points. First, we define a method to Fuzzy Knowledge-Based Systems 37 transform a partially necessary (or possible) proposition into a necessary, but eventually fuzzier and thus less precise, proposition. In other words, it becomes possible to take away any modal predicate and therefore to perform computa- tions only in the imprecise environment. Second, we show that this transforma- tion verifies the fundamental properties of necessity and possibility measures. Third, an interpretation of partial truth is proposed by reference to the two previous points. In the same way, we define how the partial truth predicate can be removed from a proposition to yield a strictly true, but less precise, proposition. Classical properties of the truth measures are shown to be also satisfied. NECESSITY AND POSSIBILITY Many propositions can be expressed in the form "X is A" [5-7]. The predicate A restricts the possible values of the variable X on a universe of discourse ~. This restriction is induced by the membership function of the subset of fl represented by the meaning of A. Example 2 The proposition "Paul is young" can be translated into "Paul's age is YOUNG." The predicate YOUNG acts as an elastic constraint on the acceptable values for the variable X (Paul's age) in the universe fl of all possible ages. For each x E fl, one can define #YOUNG(X) E [0, 1] as the grade of membership of the age x in the set YOUNG; that is, as the grade of membership of someone whose age is x in the set of young men. The grade of membership #A (X) is alSO interpreted by Zadeh as the degree of possibility, ~'A (x), that the proposition p = "X takes the value x" is true, knowing that "X is A." We have the equivalence Vx E fl: /zA(X) = 7rA(X), where ~rA: fl ~ [0, 1] is the so-called possibility distribution defined by the predicate A on the universe ft. Let us suppose now that one has two predicates A and B on the same universe. Given the true proposition "X is B," what is the possibility, or the necessity, that the proposition "X is A" is true? DEFINITION 1 Let A and B be two predicates defined on a universe ft. Knowing that "X is B'" is true, the degree of possibility that the proposition "'X is A "" is true, FI(A IB), is given by [5] II(A [B)=Sup Min(~rA(x), rB(x)) (1) xEfl where ~rA (x) is the possibility distribution defined by the predicate A on the universe fl and lrB(x) is the possibility distribution defined by the predicate B on the universe ft. As in modal logic, one has a dual relation between necessity and possibility: 38 P. Magrez and P. Smets the necessity is the complement of the possibility of the complement. Let N(A IB) be the degree of necessity that "X is A" is true given that the proposition "X is B" is true. The duality implies the following definition: N(A [B)= 1 -l-I('-, A [B) (2) NOTE 1 We will write "the possibility (necessity) of A (given B)" for "the degree of possibility (necessity) that "X is A" is true (given that the proposition "X is B" is true)." Furthermore, we use the same symbol "A" for a predicate and for the proposition "X is A" where predicate A def'mes a restriction on a universe fl (where X is the variable describing the elements of f~). Therefore, if fl represents all the possible ages, A is the predicate YOUNG, "X is A" signifies "Paul's age is YOUNG" (i.e., Paul is young), and proposition A is the proposition "X is A," where X stands for Paul's age. D~.Ftr~mON 2 Let A and B be two predicates defined on a universe ft. Knowing that "'X is B'" is true, the necessity that the proposition "'X is A "" is true, N(AIB), is given by N(A ]B)=Inf MaX(rA(X), 1 -- its(x)) (3) xEfl These definitions are generalized by the use of triangular norms (Sanchez [8]): II(A [B)=Sup ® (a'A(X), rB(X)) (4) and N(AIB)=Inf • (rA(x), 1-rB(x)) (5) xEfl where @ is a T-norm and • is the corresponding T-conorm (see appendix). Example 3 Let the proposition B: "Paul's age is around 24, 25" be present in the knowledge base. Let a second proposition A: "Paul is young" be matched by the inference engine with proposition B. Propositions A and B each induce a possibility distribution on the universe fl of all possible ages. Suppose the appropriate T-norm is the Tm T-norm (Tin), with the bounded sum (BS) its corresponding T-conorm (this choice will be argued below). The possibility of "Paul's age is young," knowing "Paul's age is around 24, 25," is given by (Figure 1): II(A IB) = Sup Tm(rA (x), IrB(X)) = .73 xEfl The necessity to have "Paul's age is young," knowing "Paul's age is around 24, 25," is N(A [B) =Inf BS (~rA(x), 1 - rB(x))= .67 xEfl Fuzzy Knowledge-Based Systems 39 1"I 1°tA B ! 20 25 30 35 Figure 1.