Strong non-standard completeness for fuzzy logics

In this paper we are going to introduce the notion of strong non-standard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well known construction by ultraproduct. Roughly speaking, to say that a logic C is strong non-standard complete means that, for any countable theory Gamma over C and any formula phi such that Gamma does not prove(C) phi, there exists an evaluation e of C-formulas into a C-algebra A such that the universe of A is a non-Archimedean extension [0, 1]* of the real unit interval [0, 1], e is a model for Gamma, but e(phi) < 1. Then we will apply SNSC to prove that various modal fuzzy logics allowing to deal with simple and conditional probability of infinite-valued events are complete with respect to classes of models defined starting from non-standard measures, that is measures taking value in [0, 1]*.