
Logical foundations of fuzzy mathematics
Běhounek, Libor ; Jirků, Petr (advisor) ; Gottwald, Siegfried Johannes (referee) ; Dvořák, Antonín (referee)
The dissertation consists of the author's published papers on logicbased fuzzy mathe matics. It is accompanied with a cover study (Part I of the thesis), which introduces the area of logicbased 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 logicbased 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 logicbased 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. Logicbased fuzzy mathematics is part of a broader area of nonclassical mathematics (i.e., mathematical disciplines axiomatizable in nonclassical logics), as well as a specic subeld of general fuzzy methods. Following earlier isolated developments in logicbased fuzzy set...
