Generalized Theory of Uncertainty (GTU)--Principal Concepts and Ideas
Lotfi A. Zadeh
Uncertainty is an attribute of information. There is a universal acceptance of the thesis that information is statistical in nature. Concomitantly, existing theories of uncertainty are based on probability theory. The generalized theory of uncertainty (GTU) departs from existing theories. First, the thesis that information is statistical in nature is replaced by a much more general thesis that information is a generalized constraint, with statistical uncertainty being a special, albeit important, case. Equating information to a generalized constraint is the fundamental thesis of GTU.
Second, the foundation of GTU is shifted from bivalent logic to fuzzy logic. As a consequence, in GTU everything is or is allowed to be a matter of degree or, equivalently, fuzzy. Concomitantly, all variables are, or are allowed to be granular, with a granule being a clump of values drawn together by a generalized constraint.
And third, one of the principal objectives of GTU is achievement of NL-capability, that is, the capability to operate on information described in natural language. NL-capability has high importance because much of human knowledge, including knowledge about probabilities, is described in natural language.
The centerpiece of GTU is the concept of a generalized constraint. The concept of a generalized constraint is motivated by the fact that most real-world constraints are elastic rather than rigid, and have a complex structure even when simple in appearance. Briefly, if X is a variable taking values in a universe of discourse, U, then a generalized constraint on X, GC(X), is an expression of the form X isr R, where R is a constraining relation, and r is an indexical variable which defines the modality of the constraint, that is, its semantics. The principal constraints are possibilistic (r=blank); veristic (r=v); probabilistic (r=p); random set (r=r); fuzzy graph (r=fg); usuality (r=u); bimodal (r=bm); and group (r=g). Generalized constraints may be combined, qualified, propagated and counterpropagated. A generalized constraint may be a system of generalized constraints. The collection of all generalized constraints constitutes the generalized constraint language, GCL.
A prerequisite to computation with information described in natural language is precisiation of meaning. More specifically, if p is a proposition or a system of propositions drawn from a natural language, then the meaning of p is precisiated by translating p into the generalized constraint language GCL. The object of precisiation, p, and the result of precisiation, p*, are referred to as the precisiend and precisiand, respectively. The degree to which the intension, that is, the attribute-based meaning of p* matches the intension of p is referred to as the cointension of p* and p. A precisiend, p*, is cointensive if cointension of p* and p is in some specified sense, high.
In GTU, deduction of an answer: ans(q), to a query, q, involves modules. The Precisiation module, P, operates on the initial information set, p, expressed as INL, and results in a cointensive precisiend, p*. The Protoform module, Pr, serves as an interface between the Precisiation module and the Deduction/Computation module, D/C. The input to Pr is a generalized constraint, p*, and its output is a protoform of p*, that is, its abstracted summary, p**. The Deduction/Computation module is basically a database (catalog) of rules of deduction which are, for the most part, rules which govern generalized constraint propagation and counterpropagation. The principal deduction rule is the Extension Principle. The rules are protoformal, with each rule having a symbolic part and a computational part. The protoformal rules are grasped into modules, with each module comprising rules which are associated with a particular class of generalized constraints, that is, possibilistic constraints, probabilistic constraints, veristic constraints, usuality constraints, etc.