# Fuzzy Logic

Post-publication activity

Humans have a remarkable capability to reason and make decisions in an environment of uncertainty, imprecision, incompleteness of information, and partiality of knowledge, truth and class membership. The principal objective of fuzzy logic is formalization/mechanization of this capability.

## Four principal facets

There are many misconceptions about fuzzy logic. To begin with, fuzzy logic is not fuzzy. In large measure, fuzzy logic is precise. Another source of confusion is the duality of meaning of fuzzy logic. In a narrow sense, fuzzy logic is a logical system. But in much broader sense which is in dominant use today, fuzzy logic, or FL for short, is much more than a logical system. More specifically, fuzzy logic has many facets (Figure 1). There are four principal facets:

1. The fuzzy-set-theoretic facet, FLs;
2. The logical facet, FLl;
3. The epistemic facet, FLe; and
4. The relational facet, FLr.

The basic concepts of graduation and granulation form the core of FL and are the principal distinguishing features of fuzzy logic. More specifically, in fuzzy logic everything is or is allowed to be graduated, that is, be a matter of degree or, equivalently, fuzzy. Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a clump of attribute-values drawn together by indistinguishability, similarity, proximity or functionality. For example, Age is granulated when its values are described as young, middle-aged and old (Figure 2). A linguistic variable may be viewed as a granulated variable whose granular values are linguistic labels of granules. In a qualitative way, graduation and granulation play pivotal roles in human cognition.

The distinguishing features of the four principal facets are the following.

### FLs

The fuzzy-set-theoretic facet, FLs, is focused on fuzzy sets, that is, on classes whose boundaries are unsharp, e.g., the class of beautiful women, the class of honest men and the class of tall mountains. The concept of a fuzzy set was introduced in (Zadeh 1965). The theory of fuzzy sets is central to fuzzy logic (Pedrycz and Gomide 1998).

In more detail, fuzzy sets are graduated in the sense that membership in a fuzzy set is a matter of degree. A fuzzy set, A, in a universe of discourse, U, is defined by a membership function which associates with each object, u, in U, the degree to which u is a member of A. A fuzzy set is basic if its membership function takes values in the unit interval. More generally, the membership function may take values in a partially ordered set. There are many types of fuzzy sets, among them fuzzy sets of Type 2 (Zadeh 1975; Mendel 2000), L-Fuzzy sets (Goguen 1967), bipolar fuzzy sets (Zhang 1998; Benferhat, Dubois, Kaci and Prade 2005) and intuitionistic fuzzy sets (Atanassov 1986). In a general setting, intersection and union of fuzzy sets are defined in terms of t-norms and t-conorms (Klement, Mesiar and Pap 2000).

### FLl

The logical facet of FL, FLl, is fuzzy logic in its narrow sense. FLl may be viewed as a generalization of multivalued logic. The agenda of FLl is similar in spirit to the agenda of classical logic (Hajek 1998; Novak, Perfilieva and Mockor 1999). Truth values in FLl are allowed to be fuzzy sets.

### FLe

The epistemic facet of FL, FLe, is concerned with knowledge representation, semantics of natural languages and information analysis. In FLe, a natural language is viewed as a system for describing perceptions. An important branch of FLe is possibility theory (Zadeh 1978; Dubois and Prade 1988). Another important branch of FLe is the computational theory of perceptions (Zadeh 1999, 2000). Figure 3: Granulation of a function. $$S$$ (small), $$M$$ (medium) and $$L$$ (large) are fuzzy sets. $$f*$$ may be viewed as a summary of $$f\ .$$

### FLr

The relational facet, FLr, is focused on fuzzy relations and, more generally, on fuzzy dependencies. In FLr, a granulated function, $$f*\ ,$$ is described as a collection of fuzzy if-then rules of the form: if $$X$$ is $$A$$ then $$Y$$ is $$B\ ,$$ where $$A$$ and $$B$$ are fuzzy sets carrying linguistic labels like small, medium, and large (Figure 3). In this sense, $$X$$ and $$Y$$ are linguistic variables (Zadeh 1973). The concept of a linguistic variable and the associated calculi of fuzzy if-then rules (Zadeh 1973, 1974; Mamdani and Assilian 1975; Bardossy and Duckstein 1995) play pivotal roles in almost all applications of fuzzy logic. A granulated function, $$f*\ ,$$ may be viewed as a summary of $$f\ ,$$ with $$f*$$ being a granular value of $$f\ .$$ An important special case of a granular function is a granular probability density function (Figure 4). In this perspective, perception of a probability distribution may be described as a granular probability distribution.

## Generalized constraint

The centerpiece of fuzzy logic is the concept of a generalized constraint (Zadeh 1986, 2006). Constraints are ubiquitous. In scientific theories, representation of constraints is generally oversimplified. Oversimplification of constraints is a necessity because existing constraint definition languages have a very limited expressive power. The concept of a generalized constraint is intended to provide a basis for construction of a maximally expressive constraint definition language--a language which can also serve as a meaning representation/precisiation language for natural languages.

Formally, a generalized constraint is expressed as GC($$X$$)$X$ isr $$R\ ,$$ where $$X$$ is the constrained variable, $$R$$ is the constraining relation, and $$r$$ is an indexical variable which serves to identify the modality of the constraint. The principal modalities are:

• possibilistic ($$r$$=blank);
• veristic ($$r$$=v);
• probabilistic ($$r$$=p);
• usuality ($$r$$=u);
• random set ($$r$$=rs);
• fuzzy graph ($$r$$=fg);
• granular ($$r$$=gr); and
• group ($$r$$=g).

The primary constraints are possibilistic, veristic and probabilistic. The standard constraints are bivalent possibilistic, bivalent veristic and probabilistic. Standard constraints have a position of centrality in existing scientific theories. A generalized constraint, GC($$X$$), is open if $$X$$ is a free variable, and is closed if $$X$$ is instantiated. A proposition is a closed generalized constraint. For example, "Lily is young," is a closed possibilistic constraint in which $$X$$=Age(Lily); $$r$$=blank; and $$R$$=young is a fuzzy set. Unless indicated to the contrary, a generalized constraint is assumed to be closed. A generalized constraint may be generated by combining, projecting, qualifying, propagating and counterpropagating other generalized constraints. The set of all generalized constraints together with the rules governing combination, projection, qualification, propagation and counterpropagation, constitutes the Generalized Constraint Language (GCL).

There is an important relationship between the concept of a generalized constraint and information. More specifically, a key idea in fuzzy logic is that of representing the information about a variable $$X\ ,$$ $$I(X)\ ,$$ as a generalized constraint on $$X\ ,$$ GC($$X$$).The symbolic equation $$I(X)$$=GC($$X$$) is the fundamental thesis of fuzzy logic.

## Propositions

A proposition is a carrier of information. A consequence of the fundamental thesis is that the meaning of a proposition, $$p\ ,$$ is expressible as a generalized constraint. This is the meaning postulate of fuzzy logic. More specifically, the meaning of a proposition is expressible as a closed generalized constraint, while the meaning of a predicate is expressible as an open generalized constraint. Equivalently, if $$p$$ is a proposition or a predicate then the meaning postulate may be stated as an assertion that the meaning of $$p$$ may be represented/precisiated through translation of $$p$$ into GCL. A very simple example of annotated translation is

Lily is young $$X$$/Age(Lily) is $$R$$/young,

implying that the constrained variable $$X$$ is Age(Lily), the constraining relation, $$R\ ,$$ is young, and the constraint is possibilistic ($$r$$=blank). Equivalently, the meaning postulate implies that the meaning of a proposition or a predicate is defined by identifying the constrained variable, $$X\ ,$$ the constraining relation, $$R\ ,$$ and the modality of the constraint, $$r\ .$$

## Granular computing

There is a close connection between the concept of a generalized constraint and the concept of a granular value. More specifically, if $$X$$ is a variable taking values in a universe of discourse, $$U\ ,$$ then $$a$$ is a singular value of $$X$$ if $$a$$ is a singleton, implying that there is no uncertainty or imprecision about the value of $$X\ .$$ If this is not the case, then a granular value of $$X\ ,$$ $$A\ ,$$ may be viewed as a representation of the state of knowledge about the value of $$X$$ (Figure 5). For example, if $$X$$ is unemployment, then 9.3 is a singular value of $$X\ ,$$ and "high" is a granular value of $$X\ .$$

The concept of a generalized constraint on $$X$$ serves to define the meaning of a granular value. Symbolically, $$A$$=GC($$X$$). In the unemployment example, "high" is the label of a generalized constraint on unemployment - more specifically, a possibilistic constraint. In granular computing, the objects of computation are granular values which are defined as generalized constraints. Granular computing is rooted in (Zadeh 1979, 1986, 1997, 1998, 1999). The term Granular Computing was suggested by T.Y. Lin (Lin 1997). The text "Granular Computing" by A. Bargiella and W. Pedrycz is the first book on granular computing (Bargiela and Pedrycz, 2002).

Granular computing provides a basis for computing with words, (CW) or, more concretely, NL-Computation, that is, computation with information described in natural language (Zadeh 2006). Since a natural language is a system for describing perceptions, NL-Computation is closely related to computation with perception-based information. As an illustration, if my perception is that most Swedes are tall, then what is the average height of Swedes? Another example: Robert usually leaves office at about 6 pm. Usually it takes him about an hour to get home. What is the probability that Robert is home after about 7 pm? NL-capability is the capability of a theory to operate on information described in natural language or, equivalently, to operate on perception-based information. The importance of NL-capability derives from the fact that much of human knowledge is expressed in natural language.

NL-Computation involves two stages. In the first stage, the information which is described in a natural language is precisiated through translation into the Generalized Constraint Language. The result is granular information expressed as system of generalized constraints. The second stage involves granular computing. Finally, the result of granular computing is retranslated into natural language.

## Extension principle

Deduction in fuzzy logic is governed by a collection of rules of deduction which, in the main, are rules that govern propagation and counter-propagation of generalized constraints. The principal rule is the extension principle. Extension principle has many versions. The simplest version (Zadeh 1965) is the following. Let $$f$$ be a function from reals to reals, $$Y=f(X)\ .$$ What we know is that $$X$$ is $$A\ ,$$ where $$A$$ is a fuzzy subset of the real line. Equivalently, what we know about $$X$$ is its granular value, that is, its possibility distribution, $$A\ .$$ What can be said about $$Y\ ,$$ that is, what is its granular value or, equivalently, its possibility distribution? In a more general form, (Zadeh 1975) $$X$$ is $$A$$ is replaced by $$f(X)$$ is $$A$$ (Figure 6). It is this form that is used in most practical applications. In a form that is used in fuzzy control, what is granulated is $$f\ ,$$ resulting in a granular function, $$f^*\ ,$$ which is defined by a collection of fuzzy-if-then rules. A simple example is$f^*\ :$ if $$X$$ is small then $$Y$$ is small

if $$X$$ is medium then $$Y$$ is large
if $$X$$ is large then $$Y$$ is small

More generally, the extension principle may be viewed as follows. Let $$Y=f(X)\ ,$$ where $$X$$ is a real-valued variable. Assume that we can compute $$Y$$ for singular values of $$f$$ and $$X\ .$$ Basically, the extension principle serves to extend the definition of $$Y$$ to granular values of $$f$$ and $$X\ .$$

## History

During much of its early history, fuzzy logic has been an object of skepticism and derision, in part because fuzzy is a word which is usually used in a pejorative sense. Today, fuzzy logic has an extensive literature and a wide variety of applications ranging from consumer products and fuzzy control to medical diagnostic systems and fraud detection (Zadeh 1990; Novak and Perfilieva 2000).

Existing scientific theories are almost without exception based on classical, bivalent logic. What is widely unrecognized is that many scientific theories can be enriched through addition of concepts and techniques drawn from fuzzy logic. In particular, fuzzy logic can add to existing theories NL-capability, that is, the capability to operate on information described in natural language or, equivalently, on perception-based information. In coming years, the issue of NL-capability is likely to grow in visibility and importance, especially in such fields as economics, law, medicine, search, question-answering and, above all, probability theory and decision analysis.