Fuzzy Logic and Science--A Novel Perspective
Lotfi A. Zadeh
Fuzzy logic has achieved maturity. And yet there are many persistent misconceptions about fuzzy logic, especially among those who do not know what it is. To begin with, fuzzy logic is not fuzzy logic. In fact, fuzzy logic is a precise logic--a precise logic of imprecision. In fuzzy logic everything is or is allowed to be graduated, that is, be a matter of degree or, equivalently, fuzzy. And 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. Graduation and granulation are human-cognition-inspired, and are concomitants of the bounded ability of human sensory organs, and ultimately the brain, to resolve detail and store information. Formalization of the concepts of graduation and granulation may be viewed as the core of fuzzy logic.
A concept which has a position of centrality in formalization of graduation and granulation is that of a generalized constraint. Briefly, a generalized constraint is an expression of the form X isr R, is which X is the constrained variable, R is the constraining relation, and r is an indexical variable which identifies this modality of the constraint, that is, its semantics. The principal constraints are possibilistic (r=blank); veristic (r=v); probabilistic (r=p); usuality (r=u); fuzzy graph (r=fg); random set (r=rs); group (r=g); and bimodal (r=bm). Generalized constraints may be combined, qualified, and propagated, forming what is called the Generalized Constraint Language (GCL). A key idea in fuzzy logic is that the meaning of a proposition may be represented as a generalized constraint. In fuzzy logic, precision is assumed to have a multiplicity of meanings. The principal meanings are: (a) precision in value (v-precision); and (b) precision in meaning (m-precision). If p is a proposition or concept, then its precisiation may be oriented toward humans (mh-precisiation) or machines (mm-precisiation). Unless stated to the contrary, precisiation should be interpreted as mm-precisiation.
One of the principal functions of fuzzy logic is that of serving as a basis for mm-precisiation. In fuzzy logic, an object of precisiation, p, is referred to as precisiend, and its result, p*, is called a precisiand of p. A precisiand of p, p*, may be interpreted as a model of the intension of p, with intension used in its logical sense as attribute-based meaning. In general, a precisiend may have a multiplicity of precisiends, just as a system or a device may have a multiplicity of models.
A concept which plays an important role in precisiation is that of cointension. More specifically, the cointension of a precisiand, p*, is the degree to which the intension of p* matches that of p. In this sense, p* is cointensive of its cointension is high. If p is a concept, then its definition may be viewed as a precisiand of p. In this sense, a definition of p is cointensive if its intension is a good fit to the intension of p. Scientific theories are based mainly on bivalent logic which implies that if C is a concept and x is an object in C's universe of discourse, then either x is an instance of C or x is not an instance of C, with no degree of instantiation allowed. For example, a system is either stable or not stable; a number is either prime or not prime; a set is either convex or not convex, etc. But, in reality, a concept, C, which is defined as if C is bivalent, is more often than not fuzzy rather than bivalent. The problem is that, in general, a bivalent definition of a fuzzy concept is not cointensive. A conclusion which emerges is that to achieve combustion, fuzzy logic rather than bivalent logic must be employed as the definition language. A shift from bivalent logic to fuzzy logic will require a profound change in the conceptual structure of scientific theories.