On the classification of first order Gödel logics
The representation of Gödel logics via their set of truth values has the disadvantage that uncountably many truth value sets have indistinguishable logical properties. In this paper, the representation of Gödel logics via their set of valid sentences is considered. This has to deal with the insufficiencies with respect to satisfiability and entailment. It is not obvious how to describe the sets of 1-satisfiable sentences of these logics. Furthermore, while there are uncountably many entailment relations, there are only countably many sets of validities [\textit{A. Beckmann} et al., Order 25, No. 4, 281--298 (2008; Zbl 1171.03027)]. One of the corollaries of this article deals with the former insufficiency by providing a functional relation between the set of sets of valid formulas and the set of sets of 1-satisfiable formulas. This is achieved by giving a complete characterization of axiomatizability of validity and (the complements of) two types of satisfiabilities, based on the topological properties of the underlying truth value set. The absoluteness operator $\Delta$, introduced in [Interdisciplinary Syst. Res. 88, 291--301 (1986; Zbl 0593.03031)] by \textit{G. Takeuti} and \textit{S. Titani} as ``necessity operator'' $\square$ is considered. Recursively enumerable first order Gödel logics with and without $\Delta$ are characterized with respect to validity and non-satisfiability.
scientific article; zbMATH DE number 6974478