| Fuzzy logic is a generalization of classical logic. In classical logic, the evaluation scale is the two-element set {0,1}. And in fuzzy logic, the evalua-tion scale is the continuous interval [0,1]. In classical logic, the relationships and operations between propositions are represented by logical operators. The most frequently used logical operators are conjunctions, disjunctions, impli-cations and negations. In fuzzy logic, conjunctions, disjunctions, implications and negations are generalized into t-norms, t-conorms, fuzzy implications and fuzzy negations, respectively.Recent years, the functional equations containing fuzzy logical operators have become an important topic in fuzzy logic. The first reason is that the solutions to these equations can help people determine whether a tautology in classical logic remains valid in fuzzy logic, and if not, find the conditions under which the logical equation holds in fuzzy logic. This will help people distinguish between the properties of classical logic and fuzzy logic. The second reason is that the solutions to these functional equations are closely related to the complexity of fuzzy reasoning methodologies and the rule base reduction of fuzzy systems.The evaluation scale [0,1] of fuzzy logic has one undesired property in decision making:the precision of these numbers has gone far beyond the cog-nitive capabilities of evaluators. Usually, evaluators can only remember and handle a limited number of evaluation terms. As a result, the logical reasoning on finite chains has attracted many researchers’ attention.In this paper, we solve the following implicational equation on a finite chain: I(x,y)=I(x,I(x,y)), where I is an implication(or an operation) and x, y are elements in the finite chain. This equation comes from an important tautology in classical logic: pâ†'(pâ†'q)=pâ†'q. This tautology is also called contraction law and it is a special iterative boolean law. On finite chains, the equation I(x,y)=I(x,I(x,y)) does not always hold. So in this paper, we will find the conditions under which the equation I(x,y)=I(x,I(x,y)) hold on a finite chain. When y=0and I(x,y)=S(N(x),y), the equation I(x,y)=I(x,I(x,y)) becomes I(x,N(x))=N(x). This is the necessary condition for construction fuzzy entropy from inclusion grade indicators. So the solutions to the equation I(x, y)=I(x, I(x, y)) are of great theoretical importance.In Chapter1and2, we discuss the background and the significance of the study on the functional equations containing logical operators on finite chains, and we briefly review the fuzzy logical operators t-norms, t-conorms, fuzzy negations and fuzzy implications.In Chapter3, we first review the logical operators on finite chains. These operators are the corresponding fuzzy logical operators restricted to the fi-nite chains. But they differ from fuzzy logical operators in their properties and representations. Second, we solve the equation I(x,y)=I(x,I(x,y)) on finite chains when I is an S-implication, an R-implication, a QL-operation and a QL-implication, respectively. In particular, when I is a QL-operation, we divide our discussion into9cases based on different combinations of the t-norm T and t-conorm S. To guarantee the soundness of the information ag-gregation, we require the logical operators generating I to be smooth. When discussing the general solutions to the equation, we consider a general finite chain L={x0<x1<...<xm<xm+1}. In applications, however, the finite chain usually has5,7or9rating terms. So in our examples, we give the truth tables of the implications(or operations) satisfying the functional equation on the finite chain which contains7elements. In Chapter3, we fully characterize the S-implications, R-implications, QL-operations and QL-implications satis-fying I(x,y)=I(x,I(x,y)) on finite chains when the t-norms, t-conorms and negations involved are smooth.In Chapter4, we compare the solutions to the equation I(x, y)=I(x, I(x, y)) on [0,1] and the solutions on finite chains, and give the conclusions. |
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