## 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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### References:

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