MODELING VAGUENESS BY NONSTANDARDNESS

Authors
Citation
A. Tzouvaras, MODELING VAGUENESS BY NONSTANDARDNESS, Fuzzy sets and systems, 94(3), 1998, pp. 385-396
Citations number
10
Categorie Soggetti
Statistic & Probability",Mathematics,"Computer Science Theory & Methods","Statistic & Probability",Mathematics,"Computer Science Theory & Methods
Journal title
ISSN journal
01650114
Volume
94
Issue
3
Year of publication
1998
Pages
385 - 396
Database
ISI
SICI code
0165-0114(1998)94:3<385:MVBN>2.0.ZU;2-7
Abstract
We are concerned here with a mathematical modeling of vague predicates and vague partitions (or equivalences or similarities) by the help of nonstandard sets of integers. The modeling is faithful in that it cap tures all basic features of the notion of vagueness. Nevertheless, it is not applicable to concrete phenomena, since nonstandard numbers are too big to be used in actual counting. The properties of a vague meas urable similarity similar to which we consider as most important and w hich are captured in the present modeling are: To similar to there cor responds an assignment mu(x) of integral values to the objects of the domain of discourse, and a corresponding distance d(x,y) = \mu(x) - mu (y)\, such that: (i) If x similar to y and d(x,z) less than or equal t o d(x,y), then x similar to z. (ii) For any two similar to -similar ob jects x,y and for any two similar to -nonsimilar objects x',y',d(x,y) < d(x',y'). (iii) For any x,y such that x similar to y, we can find a z in the same class, i.e. x similar to y similar to z such that d(x,y) < d(x,z). (iv) For any similar to -nonsimilar objects x,y such that m u(x) < mu(y), there is a third object z, nonsimilar to both x,y, such that mu(x) < mu(z) < mu(y).Further we discuss the issue of what a ''ri ght'' nonstandard model of numbers would be and some related questions . (C) 1998 Elsevier Science B.V.