From Fuzzy Logic to Extended Fuzzy Logic--The Concept of F-validity and the Impossibility Principle
Lotfi A. Zadeh
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, equivalence, similarity, proximity, or functionality.
The centerpiece of fuzzy logic is the concept of a generalized constraint (Zadeh 1986, 2006). The set of all generalized constraints together with the rules governing combination, projection, qualification, propagation and counterpropagation, constitute the Generalized Constraint Language (GCL). The concept of a generalized constraint provides a basis for NL-Computation, that is, computation with information described in natural language. A prerequisite to computation with information described in natural language is precisiation of meaning. In NL-Computation, a proposition is precisiated through translation into GCL.
One of the important rationales for the use of fuzzy logic is what may be called the fuzzy logic gambit. More specifically, if there is a tolerance for imprecision, precisely defined information is imprecisiated in value through translation into natural language, followed by precisiation of meaning through translation into GCL. What is gained is reduction in cost. This idea is employed in many applications of fuzzy logic.
Over the years, fuzzy logic has been enriched through introduction of a long list of concepts, ideas, and techniques. The concepts of extended fuzzy logic, FL+, and f-validity which are sketched in the following represent a more radical development. In essence, extended fuzzy logic may be viewed as an attempt at legitimizing the concept of fuzzy theorem (Zadeh 1975) and fuzzy validity. In large measure, the move from fuzzy logic, FL, to extended fuzzy logic, FL+, is a move into as yet uncharted territory. For example: I hail a cab and ask the driver to take me to address A: (a) the shortest way; and (b) the fastest way. In both cases, the driver will use his/her experience to choose a route. Clearly, (a) has a provably valid solution--a solution which a GPS system could suggest to the driver. By contrast, (b) does not have a provably valid solution, and no GPS system could suggest one. The route followed by the driver has a validity based on experience, call it f-validity. A provably valid solution, call it p-valid solution, does not exist. A problem is p-valid or f-valid depending on whether it has a p-valid or f-valid solution. In this sense, (a) is p-valid and (b) is f-valid.
To clarify the meaning of f-validity and p-validity, consider a primitive world in which figures are drawn with a spray can, with no ruler or compass available. In this world, we can envisage a fuzzified version of Euclidean geometry, call it f-geometry. In f-geometry, we have f-points, f-lines, f-triangles, f-circles, f-parallel f-lines, etc. A simple example of an f-theorem in f-geometry is: f-medians of f-triangle are f-concurrent. This f-theorem can be f-proved by fuzzification of the familiar proof of the crisp version of the theorem. It should be noted that the concept of f-validity is distinct from Polya's concept of plausibility.
A conclusion which is of key importance is that there are no crisp theorems in f-geometry. This conclusion is an instance of what may be called the impossibility principle. The impossibility principle is an f-principle in the sense that it may have f-validity but can neither be proved nor disproved. Informally, the impossibility principle may be expressed as: in an environment of imprecision, uncertainty, incompleteness of information, conflicting goals, and partiality of truth, p-validity is not, in general, an achievable objective.
In many realms of science and human thought, especially in economics, decision analysis, linguistics, psychology, and legal reasoning, progress has been driven by a quest for p-validity.