Logical foundations of fuzzy mathematics

Abstract

Předložená disertační práce sestává z autorových publikovaných článků o logických základech fuzzy matematiky, doplněných strhující studií (tvoricich úvodní část disertace), ve které je představen formálně logický přístup k fuzzy matematice. Dále je v ní dokládána důležitost výzkumu v tomto oboru a charakterizován jeho současný stav, popsán autorův příspěvek k oboru a podány komentare k jednotlivým článkům, z nichž disertace skládá.

The dissertation consists of the author's published papers on logic-based fuzzy mathe- matics. It is accompanied with a cover study (Part I of the thesis), which introduces the area of logic-based fuzzy mathematics, argues for the signicance of the area of re- search, presents the state of the art, indicates the author's contribution to the eld, and comments on the papers comprising the thesis. Fuzzy mathematics can be characterized as the study of fuzzy structures, i.e., math- ematical structures in which the two values 0, 1 are at some points replaced by a richer system of degrees. Under the logic-based approach, fuzzy structures are formalized by means of axiomatic theories over suitable systems of fuzzy logic, whose rules replace the rules of classical logic in formal derivation of theorems. The main advantages of the logic-based approach are the general gradedness of dened notions, methodological clarity provided by the axiomatic method, and the applicability of a foundational architecture mimicking that of classical mathematics. Logic-based fuzzy mathematics is part of a broader area of non-classical mathematics (i.e., mathematical disciplines axiomatizable in non-classical logics), as well as a specic subeld of general fuzzy methods. Following earlier isolated developments in logic-based fuzzy set...