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Toward a generalized theory of uncertainty (GTU) – an outline. (English) Zbl 1074.94021

Summary: It is a deep-seated tradition in science to view uncertainty as a province of probability theory. The generalized theory of uncertainty (GTU) which is outlined in this paper breaks with this tradition and views uncertainty in a much broader perspective.

Uncertainty is an attribute of information. A fundamental premise of GTU is that information, whatever its form, may be represented as what is called a generalized constraint. The concept of a generalized constraint is the centerpiece of GTU. In GTU, a probabilistic constraint is viewed as a special–albeit important–instance of a generalized constraint.

A generalized constraint is a constraint of the form $X$ isr $R$, where X is the constrained variable, $R$ is a constraining relation, generally non-bivalent, and $r$ is an indexing variable which identifies the modality of the constraint, that is, its semantics. The principal constraints are: possibilistic ($r$ = blank); probabilistic $\left(r=p\right)$; veristic $\left(r=v\right)$; usuality $\left(r=u\right)$; random set $\left(r=rs\right)$; fuzzy graph $\left(r=fg\right)$; bimodal $\left(r=bm\right)$; and group $\left(r=g\right)$. Generalized constraints may be qualified, combined and propagated. The set of all generalized constraints together with rules governing qualification, combination and propagation constitutes the generalized constraint language (GCL).

The generalized constraint language plays a key role in GTU by serving as a precisiation language for propositions, commands and questions expressed in a natural language. Thus, in GTU the meaning of a proposition drawn from a natural language is expressed as a generalized constraint. Furthermore, a proposition plays the role of a carrier of information. This is the basis for equating information to a generalized constraint.

In GTU, reasoning under uncertainty is treated as propagation of generalized constraints, in the sense that rules of deduction are equated to rules which govern propagation of generalized constraints. A concept which plays a key role in deduction is that of a protoform (abbreviation of prototypical form). Basically, a protoform is an abstracted summary – a summary which serves to identify the deep semantic structure of the object to which it applies. A deduction rule has two parts: symbolic – expressed in terms of protoforms – and computational.

GTU represents a significant change both in perspective and direction in dealing with uncertainty and information. The concepts and techniques introduced in this paper are illustrated by a number of examples.

##### MSC:
 94D05 Fuzzy sets and logic in connection with communication 68T37 Reasoning under uncertainty