The concept of a group constraint
X is a group variable, G[A], and R is a group constraint on G[A]. More specifically, if X is a group variable of the form
then R is a constraint on the A, written in G[A is R]. To illustrate, if we have a group of n Swedes, with Name; being the name of ith Swede, and Aj being the height of Name,, then the proposition "most Swedes are tall" is a constraint on the At which may be expressed as (Zadeh, 1983a; 2004a)
or, more explicitly,
where A, = Height (Name,), i = 1,...,n, and most is a fuzzy quantifier which is interpreted as a fuzzy number (Zadeh, 1983a; 1983b).
Primary constraints, composite constraints, and standard constraints
Among the principal generalized constraints there are three that play the role of primary generalized constraints. They are:
Possibilistic constraint: X is R,
Probabilistic constraint: X isp R
Veristic constraint: X isv R.
A special case of primary constraints is what may be called standard constraints: bivalent possibilistic, probabilistic, and bivalent veristic. Standard constraints form the basis for the conceptual framework of bivalent logic and probability theory.
A generalized constraint is composite if it can be generated from other generalized constraints through conjunction, and/or projection and/or constraint propagation and/or qualification and/or possibly other operations. For example, a random-set constraint may be viewed as a conjunction of a probabilistic constraint and either a possibilistic or a veristic constraint. The Dempster-Shafer theory of evidence is, in effect, a theory of possibilistic random-set constraints. The derivation graph of a composite constraint defines how it can be derived from primary constraints.
The three primary constraints—possibilistic, probabilistic, and veristic—are closely related to a concept which has a position of centrality in human cognition—the concept of partiality. In the sense used here, partial means: a matter of degree or, more or less equivalently, fuzzy. In this sense, almost all human concepts are partial (fuzzy). Familiar examples of fuzzy concepts are: knowledge, understanding, friendship, love, beauty, intelligence, belief, causality, relevance, honesty, mountain, and, most important, truth, likelihood, and possibility. Is a specified concept, C, fuzzy? A simple test is: If C can be hedged, then it is fuzzy. For example, in the case of relevance we can say: very relevant, quite relevant, slightly relevant, and so on. Consequently, relevance is a fuzzy concept.
The three primary constraints may be likened to the three primary colors: red, blue, and green. In terms of this analogy, existing theories of uncertainty may be viewed as theories of different mixtures of primary constraints. For example, the Dempster-Shafer theory of evidence is a theory of a mixture of probabilistic and possibilistic constraints. GTU embraces all possible mixtures. In this sense, the conceptual structure of GTU accommodates most, and perhaps all, of the existing theories of uncertainty.