On elementary equivalence in fuzzy predicate logics (Q1935367)

This paper is a contribution to the model theory of fuzzy logic, building upon some previous work by \textit{P. Hájek} and \textit{P. Cintula} [J. Symb. Log. 71, No. 3, 863--880 (2006; Zbl 1111.03030)] where the basics of a theory of elementary equivalence for fuzzy predicate logics were developed. In particular, Hájek and Cintula had proved that in core fuzzy logics, a theory \(T'\) is a conservative extension of a theory \(T\) iff every exhaustive model of \(T\) can be elementarily one-to-one mapped into some model of \(T'\), and conjectured that the result continues to hold when ``exhaustive model'' is replaced by ``arbitrary model''. In this paper, the authors provide a counterexample to this conjecture. Other notable results in this article include a characterisation of elementary equivalence between models of fuzzy predicate logics that uses elementary mappings, as well as an investigation of the properties of elementary extensions in witnessed and quasi-witnessed theories.