8. Conclusion

The notion of fuzzy relation or fuzzy multi-dimensional distribution which has been studied in the paper is certainly not new and has been paid a lot of attention in fuzzy literature. On the other hand, one traditional direction of fuzzy research has consisted in fuzzifying classical logics. These two approaches have being developed in great extent independently. Fuzzy relations are usually described in algebraic terms (e.g., as fuzzy relational algebra) while fuzzy logics are usually obtained from some classical logic by introducing fuzzy parameters (in fact, there are two big approaches to fuzzifying classical logics: (i) fuzzifying interpretations (e.g., [L71, L72]) and fuzzifying formulas themselves, e.g., introducing weights to propositions (e.g., [DLP91])). One general result of this paper is that we have established a connection between these two directions. Now we know that any local (fuzzy) distribution can be viewed as a proposition in logical sense and we can combine them just as ordinary propositions by means of connections AND and OR to build more complex propositions [S93b], particularly, fuzzy CNF and fuzzy DNF. Of course, it is not enough to declare that the distribution (relation) is a proposition and in the paper we have shown that the whole behavior of our formal system is analogous to and even more general then that of the propositional logic.

Traditionally logical inference has been thought of as applying inference rules to axioms and theorems which have been already proved and obtaining new theorems (deduction process). As a result we could infer logical statements which express in an explicit form different properties of the formal system hidden in the original representation by means of axioms. Although we have showed in the paper that our approach to logical inference is analogous to this one, we also give another interpretation for it. According to this view inference process is considered as an equivalent transformation of our representation of the semantics by means of axioms to some other representation which is more appropriate in the sense of explicit representation of necessary properties. In this case a consequence relation is only one of many possible equivalent transformations which allows us to remove unnecessary statements. This interpretation of logical inference seems more general especially when considering non-minimax operations for composing distributions.

One general result of the approach described in the paper is that we clearly distinguish two notions of values of variables and values of distributions which are often mixed in traditional formalisms. The values of individual logical variables can be associated with the syntax or objective part of the problem domain. They define the matter of propositions, i.e., what the proposition is about, e.g., it can be some states space. On the other hand, the values of distributions are associated with the semantics or subjective part of the problem domain. They define the proposition itself, e.g., what we think or know about the possible states. However in the case of superpositions the same set of values can represent both syntactic and semantic values. For example, when we negate some proposition we in essence apply the proposition (negation) to the set of values which are semantical for the negated proposition but syntactic for the negation.

Suppose we know that inference process is an equivalent transformation of our representation to some form, i.e., to infer something we have to change the form of representation in such a way that the semantics remain the same. Then the question arises: What form of representation we have to seek for, and why it is better than other forms, i.e., what is the goal of inference process? An answer is the following. Our general goal is to reveal interesting in some sense (global) properties of a multi-dimensional distribution, i.e., to find the form of representation where these properties would be explicit. Usually explicit form of representation assumes that the property is expressed in one statement. The property which is looked for in most cases is the projection of the whole distribution on some variable(s) or the proposition about one variable. Although there may be also other properties (e.g., correlations between individual variables) this one is supposed the most important and is considered to be the goal of general inference process.

Perhaps the most important result described in the paper is new fuzzy resolution operation. It generalizes traditional consensus operation and resolution in logic in two directions: (i) values of variables are supposed to be many-valued [Zk89] and even continuos, and (ii) distributions are supposed to take values from the interval [0,1] [S91]. The criterion of adjacency formulated in the paper enforces the analogy with crisp case since it allows us to determine when the resolvent is not trivial. To define correctly the resolution operation and the criterion of adjacency we had to introduce such new notions as reduced forms, degree of incomparability, constant of disjunction.

Although it is not described in this paper, it can be easily shown that we do not need an operation of negation. Instead of it we can use more general operation of superposition (proposition about proposition), a particular case of which represents negation. More about this can be read in [S93b].

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