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Intuitionistic Propositional Logic with Galois Negations


Intuitionistic propositional logic with Galois negations (\(\mathsf {IGN}\)) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair \((\lnot ,{\sim })\) and dual Galois pair \((\dot{\lnot },\dot{\sim })\) of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for \(\mathsf {IGN}\) are developed. A Hilbert-style axiomatic system \(\mathsf {HN}\) is given for \(\mathsf {IGN}\), and Galois negation logics are defined as extensions of \(\mathsf {IGN}\). We give the bi-tense logic \(\mathsf {S4N}_t\) which is obtained from the minimal tense extension of the modal logic \(\mathsf {S4}\) by adding tense operators. We give a new extended Gödel translation \(\tau \) and prove that \(\mathsf {IGN}\) is embedded into \(\mathsf {S4N}_t\) by \(\tau \). Moreover, every Kripke-complete Galois negation logic L is embedded into its tense companion \(\tau (L)\).

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The work of the authors was supported by Chinese National Funding of Social Sciences (18ZDA033). Thanks are given to the referees for their comments which are helpful for revising the manuscript.

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Correspondence to Minghui Ma.

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Ma, M., Li, G. Intuitionistic Propositional Logic with Galois Negations. Stud Logica 111, 21–56 (2023).

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  • Heyting algebra
  • Galois negations
  • Intuitionistic logic
  • Tense logic