From Bivalent Logic to Multivalued Logic to Fuzzy Logic and Beyond
Lotfi A. Zadeh
In large measure, science is based on bivalent logic. The brilliant successes of science are visible to all. But alongside the successes we see underachievements and outright failures. We can send men to the moon but we cannot automate driving a car in moderately heavy city traffic. We can build jumbo jets but we cannot build tennis playing robots. We can build computers that can process billions of bits of information per second, but we cannot write a program that can summarize a book. Why?
Humans, but not machines, have a remarkable capability to perform a wide variety of physical and mental tasks without any measurements or any computations. In performing such tasks, humans employ perceptions--perceptions of time, distance, direction, speed, intent, likelihood, and other attributes of physical and mental objects. Perceptions are intrinsically imprecise, reflecting the bounded ability of sensory organs, and ultimately the brain, to resolve detail and store information.
Bivalent logic, with its intolerance of imprecision and partial truth, is not the right logic for dealing with perceptions. Mutlivalued logic is a generalization of bivalent logic, allowing truth to take more than two values. Mutlivalued logic is a step in the right direction but it does not go far enough. What is needed to deal with perceptions is fuzzy logic, FL, a system of concepts and techniques in which everything is, or is allowed to be, graduated, that is, 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 values drawn together by indistinguishability, similarity, proximity, or functionality. One of the principal objectives of fuzzy logic is formalization/mechanization of the human ability to reason and make decisions in an environment of imprecision, uncertainty, partiality of information, and partiality of truth. A natural language is basically a system for describing perceptions. This suggests a key idea, namely, dealing with perceptions not directly but through their descriptions in a natural language. This idea is the point of departure in the fuzzy-logic-based computational theory of perceptions (CTP).
The centerpiece of fuzzy logic is the concept of a generalized constraint. A generalized constraint, GC(X), is an expression of the form X isr R, where X is the constraint variable, R is a constraining relation and r identifies the modality of the constraint, that is, the way in which R constrains X. Generalized constraints may be combined, qualified, propagated, and counterpropagated. The set of all generalized constraints together with the rules which govern combination, qualification, propagation, and counterpropagation, is the Generalized Constraint Language (GCL). The fundamental thesis of fuzzy logic is that information may be represented as a generalized constraint. The traditional view that information is statistical in nature is a special, albeit important, case. A proposition is a carrier of information. As a consequence, the meaning of a proposition may be represented as a generalized constraint. A representation of meaning as a generalized constraint is a meaning postulate of fuzzy logic.