Intuitionistic Propositional Logic with Galois Negations

Studia Logica 111 (1):21-56 (2023)
  Copy   BIBTEX

Abstract

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 \(\tau (L)\).

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 89,654

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Substructural Negations.Takuro Onishi - 2015 - Australasian Journal of Logic 12 (4).
Routley-Meyer ternary relational semantics for intuitionistic-type negations.Gemma Robles & José M. Méndez - 2018 - London, United Kingdom: Elsevier, Academic Press. Edited by José M. Méndez.
The Semi Heyting–Brouwer Logic.Juan Manuel Cornejo - 2015 - Studia Logica 103 (4):853-875.
Axiomatizations of intuitionistic double negation.Milan Bozic & Kosta Došen - 1983 - Bulletin of the Section of Logic 12 (2):99-102.
Inquisitive Intuitionistic Logic.Wesley H. Holliday - 2020 - In Nicola Olivetti & Rineke Verbrugge (eds.), Advances in Modal Logic, Vol. 13. London: College Publications. pp. 329-348.
Semi-intuitionistic Logic.Juan Manuel Cornejo - 2011 - Studia Logica 98 (1-2):9-25.

Analytics

Added to PP
2022-09-26

Downloads
17 (#733,750)

6 months
5 (#242,355)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Minghui Ma
Sun Yat-Sen University

References found in this work

Heyting Algebras: Duality Theory.Leo Esakia - 2019 - Cham, Switzerland: Springer Verlag.
Star and perp: Two treatments of negation.J. Michael Dunn - 1993 - Philosophical Perspectives 7:331-357.
Intuitionistic tense and modal logic.W. B. Ewald - 1986 - Journal of Symbolic Logic 51 (1):166-179.

View all 12 references / Add more references