The concept of precisiation has few precursors in the literature of logic, probability theory, and philosophy of languages (Carnap, 1950; Partee, 1976). The reason is that the conceptual structure of bivalent logic, on which the literature is based, is much too limited to allow a full development of the concept of precisiation. In GTU what is used for this purpose is the conceptual structure of fuzzy logic.

Precisiation and precision have many facets. More specifically, it is expedient to consider what may be labeled A-precisiation, with A being an indexical variable whose values identify various modalities of preci- siation. In particular, it is important to differentiate between precision in value ( v-precision) and precision in meaning (m-precision). For example, proposition X = 5 is v-precise and m-precise, but proposition 2 =s X =s 6, is v-imprecise and m-precise. Similarly, proposition "X is a normally distributed random variable with mean 5 and variance 2” is v-imprecise and m-precise.

A further differentiation applies to m-precisiation. Thus, mh-precisiation is human-oriented meaning precisiation, while mm-precisiation is machine-oriented or, equivalently, mathematically based meaning precisiation (see Figure 6.5). A dictionary definition may be viewed as a form

Figure 6.5 mh- and mm-precisiation

Figure 6.6 Granular definition of a function

of mh-precisiation, while a mathematical definition of a concept, such as stability, is mm-precisiation whose result is mm-precisiand of stability.

A more general illustration relates to representation of a function as a collection of fuzzy if-then rules—a mode of representation which is widely used in practical applications of fuzzy logic (Dubois and Prade, 1996; Yen and Langari, 1998). More specifically, let f be a function from reals to reals which is represented as (see Figure 6.6).

f : if X is small then Y is small, if X is medium than Y is large, if X is large than Y is small,

where small, medium, and large are labels of fuzzy sets. In this representation, the collection in question may be viewed as mh-precisiand of f.

Figure 6.7 Granular precisiation of "approximately a." *a

When the collection is interpreted as a fuzzy graph (Zadeh, 1974; 1996) representation of f assumes the form.

which is a disjunction of Cartesian products of small, medium, and large. This representation is mm-precisiand of f.

In general, a precisiend has many precisiands. As an illustration consider the proposition "X is approximately a", or "X is *a" for short, where a is a real number. How can "X is *a" be precisiated?

The simplest precisiand of "X is *a 'is' X = a", (see Figure 6.7). This mode of precisiation is referred to as s-precisiation, with s standing for singular. This is a mode of precisiation that is widely used in science and especially in probability theory. In the latter case, most real-world probabilities are not known exactly but in practice they are frequently computed with as if they are exact numbers. For example, if the probability of an event is stated to be 0.7, then it should be understood that

0.7 is actually *0.7, that is, approximately 0.7. The standard practice is to treat *0.7 as 0.7000, that is, as an exact number.

Next in simplicity is representation of *a is an interval centering on a. This mode of precisiation is referred to as cg-precisiation, with cg standing for crisp-granular. Next is /g-precisiation of *a, with the precisiand being a fuzzy interval centering on a. Next is p-precisiation of *a, with the preci- siand being a probability distribution centering on a, and so on.

An analogy is helpful in understanding the relationship between a precisiend and its precisiands. More specifically, an mm-precisiand, p*, may be viewed as a model of precisiend, p, in the same sense as a differential equation may be viewed as a model of a physical system.

In the context of modeling, an important characteristic of a model is its "goodness of fit." In the context of NL-computation, an analogous concept is that of "cointension." The concept is discussed in the following.