The concept of cointensive precisiation

Precisiation is a prerequisite to computation with information described in natural language. To be useful, precisiation of a precisiend, p, should result in a precisiand, p*, whose meaning, in some specified sense, should be close to that of p. Basically, cointension of p* and p is the degree to which the meaning of p* fits the meaning of p.

In dealing with meaning, it is necessary to differentiate between the intension or, equivalently, the intensional meaning, i-meaning, of p, and the extension, or, equivalently, the extensional, e-meaning of p. The concepts of extension and intension are drawn from logic and, more particularly, from modal logic and possible world semantics (Cresswell, 1973; Lambert and van Fraassen, 1970; Belohlavek and Vychodil, 2006). Basically, e-meaning is attribute-free and i-meaning is attribute-based. As a simple illustration, if A is a finite set in a universe of discourse, U, then the e-meaning of A, that is, its extension is the list of elements of A, {u1,...,un}, щ being the name of ith element of A, with no attributes associated with u. Let a (щ) be an attribute-vector associated with each ui. Then the intension of A is a recognition algorithm which, given a (ui), recognizes whether щ is or is not an element of A. If A is a fuzzy set with membership function mA then the e-meaning and i-meaning of A may be expressed compartly as

where mA(u)/u means that mA(u) is the grade of membership of ut in A; and

with the understanding that in the г-meaning of A the membership function, mA is defined on the attribute space. It should be noted that, when A is defined through exemplification, it is said to be defined ostensively. Thus, о-meaning of A consists of exemplars of A. An osten- sive definition may be viewed as a special case of extensional definition. A neural network may be viewed as a system which derives г-meaning from о-meaning.

Clearly, г-meaning is more informative than e-meaning. For this reason, cointension is defined in terms of intensions rather than extensions of precisiend and precisiand. Thus, meaning will be understood to be г-meaning, unless stated to the contrary. However, when the precisiend is a concept, which plays the role of definiendum and we know its extension but not its intension, cointension has to involve the extension of the definiendum (precisiend) and the intension of the definiens (precisiand).

As an illustration, let p be the concept of bear market. A dictionary definition of p—which may be viewed as a mh-precisiand of p—reads "A prolonged period in which investment prices fall, accompanied by widespread pessimism." A widely accepted quantitative definition of bear market is: We classify a bear market as a 30-per cent decline after 50 days, or a 13-per cent decline after 145 days (Shuster). This definition may be viewed as a mm-precisiand of bear market. Clearly, the quantitative definition, p*, is not a good fit to the perception of the meaning of bear market which is the basis for the dictionary definition. In this sense, the quantitative definition of bear market is not cointensive.

Intensions are more informative than extensions in the sense that more can be inferred from propositions whose meaning is expressed intensionally rather than extensionally. The assertion will be precisiated at a later point. For the present, a simple example will suffice.

Consider the proposition p: Most Swedes are tall. Let U be a population of n Swedes, U = (u1,...,un), u1 = name of ith Swede.

A precisiand of p may be represented as

where most is a fuzzy quantifier which is defined as a fuzzy subset of the unit interval (Zadeh, 1983a; Zadeh, 1983b). Let mtallг), i = (1,...,n) be the grade of membership of иг in the fuzzy set of tall Swedes. Then the e-meaning of tall Swedes may be expressed in symbolic form as

"most" and antonym of "most"

Figure 6.8 "most" and antonym of "most"

Accordingly, the i-precisiand of p may be expressed as Similarly, the i-precisiand of p may be represented as

where ht is the height of u.

As will be seen later, given the e-precisiend of p we can compute the answer to the query: How many Swedes are not tall? The answer is 1-most (see Figure 6.8). However, we cannot compute the answer to the query: How many Swedes are short? The same applies to the query: What is the average height of Swedes? As will be shown later, the answers to these queries can be computed given the /-precisiand of p.

The concept of cointensive precisiation has important implications for the way in which scientific concepts are defined. The standard practice is to define a concept within the conceptual structure of bivalent logic, leading to a bivalent definition under which the universe of discourse is partitioned into two classes: objects which fit the concept and those which do not, with no shades of gray allowed. This definition is valid when the concept that is defined, the definiendum, is crisp, that is, bivalent. The problem is that in reality most scientific concepts are fuzzy, that is, are a matter of degree. Familiar examples are the concepts of causality, relevance, stability, independence, and bear market. In general,

Stability in a fuzzy concept

Figure 6.9 Stability in a fuzzy concept

when the definiendum (precisiend) is a fuzzy concept, the definiens (pre- cisiand) is not cointensive, which is the case with the bivalent definition of bear market. More generally, bivalent definitions of fuzzy concepts are vulnerable to the Sorites (heap) paradox (Sainsbury, 1995).

As an illustration, consider a bottle whose mouth is of diameter d, with a ball of diameter D placed on the bottle (see Figure 6.9). When D is slightly larger than d, based on common sense the system is stable. As D increases, the system becomes less and less stable. But Lyapounov's definition of stability leads to the conclusion that the system is stable for all values of D so long as D is greater than d. Clearly, this conclusion is counterintuitive. The problem is that, under Lyapounov's bivalent definition of stability, a system is either stable or unstable, with no degrees of stability allowed.

What this example points to is the need for redefinition of many basic concepts in scientific theories. To achieve cointension, bivalence must be abandoned.

 
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