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Strong non-standard completeness for fuzzy logics. (English) Zbl 1132.03332
Summary: In this paper we introduce the notion of strong nonstandard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well-known construction of ultraproducts. Roughly speaking, to say that a logic \(\mathcal C\) is strong nonstandard complete means that, for any countable theory \(\Gamma\) over \(\mathcal C\) and any formula \(\varphi\) such that \(\Gamma\not\vdash_{\mathcal C} \varphi\), there exists an evaluation \(e\) of \(\mathcal C\)-formulas into a \(\mathcal C\)-algebra \(\mathcal A\) such that the universe of \(\mathcal{A}\) is a non-Archimedean extension \([0,1]^*\) of the real unit interval \([0,1], e\) is a model for \(\Gamma\), but \(e (\varphi) < 1\). Then we 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 nonstandard measures, that is, measures taking values in \([0,1]^*\).

MSC:
03B52 Fuzzy logic; logic of vagueness
03H05 Nonstandard models in mathematics
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