Extension Of Classical Set Theory Research Articles

Formula omitted]- Sets as a possibilistic set theory

The motivation of possibility theory is to represent the imprecision intrinsic in natural language, in particular in expressions of the form x is R. A suitable topos of presheaves, i.e. an extension of classical set theory, is presented where sets of the form {x : x is R} can be constructed with x is R having the meaning given in possibility theory. The aim of the paper is to give an explicit mathematical description of the context sensitivity of linguistic imprecision. This can be done by defining sets that vary over the possible contexts, i.e. sets in a topos of presheaves defined over a category that represents the flow of information.

Sets as singularities in the intensional universe

This paper is motivated by the search for a natural and deductively powerful extension of classical set theory. A theory of properties U is developed, based on a system of relevant logic related to RQ. In U the set {a, b, c,...} is identified with the property [x: x=a ∨ x=b ∨ x=c...]. The universe of all sets V, is identified with the property of being a hereditary set. The main result is that relevant implication → collapses to material implication ⊃ for sentences with quantifiers restricted to V. This demonstrates the naturalness of the system. However, an aparent lack of deductive power leads to the conclusion that the best extension of classical set theory is to be found in intensional theories with the unrestricted comprehension schema based on weak relevant logics. The author has obtained similar collapses of → to ⊃ for these systems.

Harry C. Bunt. Mass terms and model-theoretic semantics. Cambridge studies in linguistics, no. 42. Cambridge University Press, Cambridge etc. 1985, xiii + 325 pp.

'Mass terms', words like water, rice and traffic, have proved very difficult to accommodate in any theory of meaning since, unlike count nouns such as house or dog, they cannot be viewed as part of a logical set and differ in their grammatical properties. In this study, motivated by the need to design a computer program for understanding natural language utterances incorporating mass terms, Harry Bunt provides a thorough analysis of the problem and offers an original and detailed solution. An extension of classical set theory, Ensemble Theory, is defined, and this provides the conceptual basis of a framework for the analysis of natural language meaning which Dr Bunt calls Two-level model-theoretic semantics. The validity of the framework is convincingly demonstrated by the formal analysis of a fragment of English including sentences with quantified and modified mass terms. Separate chapters of the book are devoted to an axiomatic definition of Ensemble Theory and a detailed discussion of its status as a mathematical formalism.

Ordination on the basis of fuzzy set theory

Fuzzy set theory is an extension of classical set theory where elements of a set have grades of membership ranging from zero for non-membership to one for full membership. Exactly as for classical sets, there exist operators, relations, and mappings appropriate for these fuzzy sets. This paper presents the concepts of fuzzy sets, operations, relations, and mappings in an ecological context. Fuzzy set theory is then established as a theoretical basis for ordination, and is employed in a sequence of examples in an analysis of forest vegetation of western Montana, U.S.A. The example ordinations show how site characteristics can be analyzed for their effect on vegetation composition, and how different site factors can be synthesized into complex environmental factors using the calculus of fuzzy set theory.