Noetherian Gödel logics

We introduce Noetherian Gödel logics, Gödel logics where the set of truth values is a closed subset of $[0,1]$ containing $0$ and $1$ and without any infinite ascending sequences. There are infinitely many such logics, including the well-known logic $\textsf {G}^\downarrow $ whose set of truth value...

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Bibliographic Details
Published in:Journal of logic and computation 2022-12, Vol.32 (8), p.1487-1503
Main Authors: Aguilera, Juan Pablo, Bydžovský, Jan, Fernández-Duque, David
Format: Article
Language:English
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Summary:We introduce Noetherian Gödel logics, Gödel logics where the set of truth values is a closed subset of $[0,1]$ containing $0$ and $1$ and without any infinite ascending sequences. There are infinitely many such logics, including the well-known logic $\textsf {G}^\downarrow $ whose set of truth values is $T_\downarrow = \{0\}\cup \{1/n:n\in \mathbb {N}\setminus \{0\}\}$. We compute the complexity of satisfiability and validity for each Noetherian Gödel logic and, in particular, in the logic $\textsf {G}^\downarrow $. This yields optimal strengthening of the results of Baaz–Leitsch–Zach and Hájek
ISSN:0955-792X
1465-363X
DOI:10.1093/logcom/exac064