A Note on Gödel-Dummet Logic LC

Gemma Robles; José M. Méndez

doi:10.18778/0138-0680.2021.15

Authors

Gemma Robles University of León, Department of Psychology, Sociology and Philosophy https://orcid.org/0000-0001-6495-0388
José M. Méndez University of Salamanca https://orcid.org/0000-0002-9560-3327

DOI:

https://doi.org/10.18778/0138-0680.2021.15

Keywords:

Intermediate logics, Gödel-Dummet logic LC

Abstract

Let \(A_{0},A_{1},...,A_{n}\) be (possibly) distintict wffs, \(n\) being an odd number equal to or greater than 1. Intuitionistic Propositional Logic IPC plus the axiom \((A_{0}\rightarrow A_{1})\vee ...\vee (A_{n-1}\rightarrow A_{n})\vee (A_{n}\rightarrow A_{0})\) is equivalent to Gödel-Dummett logic LC. However, if \(n\) is an even number equal to or greater than 2, IPC plus the said axiom is a sublogic of LC.

References

[1] A. R. Anderson, N. D. Belnap Jr., Entailment. The Logic of Relevance and Necessity, vol. I, Princeton University Press, Princeton, NJ (1975).
Google Scholar

[2] M. Dummett, A propositional calculus with denumerable matrix, Journal of Symbolic Logic, vol. 24(2) (1959), pp. 97–106, DOI: https://doi.org/10.2307/2964753
Google Scholar DOI: https://doi.org/10.2307/2964753

[3] K. Gödel, Zum intuitionistischen Aussagenkalkül, Anzeiger der Akademie der Wissenschaften in Wien, vol. 69 (1932), pp. 65–66.
Google Scholar

[4] D. D. Jongh, F. S. Maleki, Below Gödel-Dummett, [in:] Booklet of abstracts of Syntax meets Semantics 2019 (SYSMICS 2019), Institute of Logic, Language and Computation, University of Amsterdam (2019), pp. 99–102.
Google Scholar

[5] J. Moschovakis, Intuitionistic Logic, [in:] E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, winter 2018 ed. (2018), URL: https://plato.stanford.edu/archives/win2018/entries/logic-intuitionistic
Google Scholar

[6] G. Robles, J. M. Méndez, A binary Routley semantics for intuitionistic De Morgan minimal logic HM and its extensions, Logic Journal of the IGPL, vol. 23(2) (2014), pp. 174–193, DOI: https://doi.org/10.1093/jigpal/jzu029
Google Scholar DOI: https://doi.org/10.1093/jigpal/jzu029

[7] J. K. Slaney, MaGIC, Matrix Generator for Implication Connectives: Version 2.1, Notes and Guide (1995), http://users.cecs.anu.edu.au/jks/magic.html
Google Scholar