Abstract
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 \(\mathcal{C}\) is strong non-standard complete means that, for any countable theory Γ over \(\mathcal{C}\) and any formula φ 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 \([0,1]^\star\) of the real unit interval [0,1], e is a model for Γ, but e(φ) < 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 \([0,1]^\star\).
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Flaminio, T. Strong non-standard completeness for fuzzy logics. Soft Comput 12, 321–333 (2008). https://doi.org/10.1007/s00500-007-0184-9
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DOI: https://doi.org/10.1007/s00500-007-0184-9