Fuzzy Implications in the Mizar System

Abstract

This paper is a description of first steps towards providing the computer-supported formalization of a relatively recent textbook by Baczyński and Jarayam “Fuzzy Implications”. We present some of the issues connected with the use of Mizar-computerized proof assistant together with its extensive repository of mathematical texts checked for their logical correctness called the Mizar Mathematical Library, based on Zermelo-Fraenkel set theory and classical first order logic. As some building blocks towards fuzzy numbers were provided, we implemented some important tools needed for proper formalization work, as nine basic fuzzy implications, opening also possibility of smooth introducing virtually any operators of such kind. Together with triangular norms and conorms this development seems promising as the preliminary step towards fuzzy logic in the repository of Mizar texts.