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A non-arithmetical Gödel logic. (English) Zbl 1086.03018
The paper focuses on two special cases of Gödel logic, namely the logics $$G_{\downarrow}$$, which is based on the set of truth values $$V_{\downarrow}= \{\frac{a}{n}\mid n= 1, 2, \ldots\}\cup \{0\}$$, and $$G_{\uparrow}$$, based on $$V_{\uparrow}= \{\frac{n}{n+1}\mid n= 0, 1, 2, \ldots\}\cup \{1\}$$. It is shown that the set of tautologies as well as the set of satisfiable formulas of $$G_{\downarrow}$$ are non-arithmetical. On the other hand, the set of tautologies of $$G_{\uparrow}$$ is proved to be $$\Pi_2$$-complete.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness
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