Fuzzy Logic

Let S be a nonempty set and denote by P(S) the class of all subsets of S. Then a closure operator in S is any map J : P(S) → P(S) such that, for X, Y ∈ P(S), (see, e.g., Cohn [1965]). The theory of closure operators is a powerful and elegant tool for (monotone) logics. Indeed, given any deduction apparatus whose set of well-formed formulas is &1D53D;, we can consider the related deduction operator D: P(F) → P(F), i.e., the operator defined by setting, for any X ∈ P(F), D(X) equal to the set of formulas we can derive from X. Then D is a closure operator. This led several authors to propose a general approach in which an abstract logic is a pair (F, D) where F is the set of formulas in a given language and D a closure operator in F (see Tarski [1956], Brown and Suszko [1973], Wójcicki [1988]). The extension of such an approach to fuzzy logic is straightforward. It is enough to substitute the lattice of all subsets of F with the lattice of all fuzzy subsets of F. This is in accordance with the definition of deduction operator given in Pavelka [1979].

Let P be a “well defined” property in a set S, i.e., a property such that given any element x in S, either x satisfies P or not. Then, the axiom of separation in classical set theory enables us to assert that the elements of S satisfying P define a subset of S we denote by {x ∈ S : x satisfies P}. As an example, if S is the set of natural numbers and P is the property “odd”, then the set {x ∈ S : x is an odd number} is defined. Now, there are properties that are “vague” and therefore not well defined. This since they can be satisfied with several different degrees. For instance, “short”, “big”, “near”, are vague properties. Then, the question arises whether these properties isolate some type of subset and therefore whether we can give a precise meaning to intuitive notions as “the set of short men”, “the set of big numbers”, “the set of shops near to the station”, and so on.

Several notions in crisp mathematics can be translated into the corresponding notions in fuzzy mathematics in a uniform way by Zadeh Extension Principle (see Zadeh [1975]b). So, it is natural to ask the following question:

Given a crisp logic, does there exist a canonical way to extend it in a fuzzy logic?

In accordance with Hilbert’s approach, in this chapter we will show that it is possible to define the deduction operator of a fuzzy logic by a suitable extension of the notions of inference rule and proof. To justify such an extension we start from the famous paradox, the “Heap Paradox”, involving the vague predicate “is small”. A similar paradox is the “Bald Man Paradox”. An interesting treatment of such paradoxes by means of fuzzy logic can be found in Goguen [1968/69]. The paradox runs as follows:

Let U be any complete lattice. Then, as observed in Chapter 4, in fuzzy logic it would be misleading to consider an initial valuation v: F→ U as a fuzzy subset of F As a matter of fact, for any formula α, the number v(α) is not the truth value of α but a constraint on its actual truth value, namely a constraint like

“the truth value of a is greater than or equal to v(α)”.

As observed in Section 6 in Chapter 3, any crisp (abstract) logic can be extended to a fuzzy (abstract) logic in a canonical way. In this chapter we will consider an extension principle which fits the Hilbert systems well (see Gerla [1995]). Namely, given any crisp H-system S = (A, R) and a crisp rule r ∈ R, we define the fuzzy rule r* = (r′r″) by setting where ∧ denotes the minimum operation. We call r* the canonical extension of r.

Formula (6.1) in Chapter 3 for the canonical extension of a closure operator enables us to apply a crisp deduction apparatus to fuzzy information, i.e., information “stratified” at several levels of validity. Now, it is also possible to apply a “stratified” deduction apparatus (i.e., various deductive instruments each with a related degree of validity) to a crisp information. We can represent such a state of affairs assuming that, for every λ ∈ U, a crisp deduction operator D _λ is defined. Given a set X of formulas, we interpret D _λ (X) as the set of formulas that we can derive from Xby using arguments which are “reliable” to degree A.

Generally speaking, all fuzzy logics considered so far arise from the fiizzification of notions having a metalogic character (see Section 10 in Chapter 3). By contrast, in this chapter we will investigate several fuzzy logics directly arising from “fuzzy worlds”, i.e., worlds whose properties can be vague and therefore whose truth-value assignments can be graded. This leads to truth-functional semantics, i.e., semantics in which the truth value of a composite sentence is determined by the truth-values of its constituents. More precisely, we consider a language by starting from

The notion of “vagueness” must be sharply distinguished from the notion of “uncertainty”. Accordingly, fuzzy logic must be sharply distinguished from probabilistic logic. Indeed, a graded truth value for a formula α mustn’t be confused with a measure of our degree of belief in α. To emphasize this difference, consider the following example:

Let α be the claim “the rose on the table is red” and imagine two different situations. In the first one we can freely examine the rose but, as a matter of fact, the color looks not exactly red. Then α is neither fully true nor fully false and we can express that by assigning to α a truth value, as an example 0.8, different from 0 and 1. This number is the truth degree of the claim that the color of the rose is red (fuzziness). In the second situation, imagine a world in which all the roses are either clearly red or clearly yellow. In such a world the claim α is either true or false but, inasmuch as we cannot examine the rose on the table, we are not able to know what is the case. Nevertheless, we have an opinion about the possible color of that rose and we could assign to α a number, as an example 0.8, as a subjective measure of our degree of belief that α is true (probability). This opinion could result from some general experience, from the percentage of the red roses in the world, from the taste of the possessor of the rose and so on, but not necessarily from a complete knowledge of that particular rose.

Traditional control techniques are possible only in the case of complete understanding of the physical nature of the problem and only after a suitable mathematical treatment leading to a usable model. This enables us to obtain a numerical function f whose intended meaning is that f(x) is the correct control given x. Unfortunately, this is not the case for a majority of real systems. Difficulties can arise, for instance, from poor understanding of the underlying phenomena (and therefore from a lack of theory), or from the complexity of the resulting mathematical model. In such cases fuzzy control, as devised in Zadeh [1965], [1975]a, [1975]b and in Mamdani [1981], is a very useful tool. To explain the idea, we can distinguish two phases in the building of a fuzzy controller.

The concepts of a decidable subset and a recursively enumerable subset are crucial for first order classical logic. In particular, they are basic tools for the proof of the famous limitative theorems about the undecidability and incompleteness of first order logic (see, for example, Shoenfield [1967]). Then, the question of a suitable extension of such concepts to fuzzy set theory arises. A first proposal in such a direction was made by E. S. Santos in an interesting series of papers. Indeed, Santos, starting from an idea of L. Zadeh (Zadeh [1968]), proposed the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program. Santos proved that all these definitions determine the same notion of computability for fuzzy maps (see Santos [1970] and Santos [1976]). As in the classical case, a corresponding definition of recursively enumerable fuzzy subset is obtained by calling recursively enumerable any fuzzy subset which is the domain of a computable fuzzy map. Successively, a notion of recursive enumerability was proposed in Harkleroad [1984] where a fuzzy subset s is said to be recursively enumerable if the restriction of s to its support is a partial recursive function.