From Intuitionism to Brouwer's Modal Logic

Zofia Kostrzycka

doi:10.18778/0138-0680.2020.22

Authors

Zofia Kostrzycka Opole University of Technology ul. Sosnkowskiego 31 45-272 Opole, Poland Author http://orcid.org/0000-0001-9797-6306

DOI:

https://doi.org/10.18778/0138-0680.2020.22

Keywords:

intuitionistic logic, Kripke frames, Brouwer's modal logic

Abstract

We try to translate the intuitionistic propositional logic INT into Brouwer's modal logic KTB. Our translation is motivated by intuitions behind Brouwer's axiom p →☐◊p The main idea is to interpret intuitionistic implication as modal strict implication, whereas variables and other positive sentences remain as they are. The proposed translation preserves fragments of the Rieger-Nishimura lattice which is the Lindenbaum algebra of monadic formulas in INT. Unfortunately, INT is not embedded by this mapping into KTB.

References

[1] O. Becker, Zur Logik der Modalitäten, Jahrbuch für Philosophie und phänomenologische Forschung, vol. 11 (1930), pp. 497–548.

[2] P. Blackburn, M. de Rijke, Y. Venema, Modal logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge (2001), DOI: http://dx.doi.org/10.1017/CBO9781107050884 DOI: https://doi.org/10.1017/CBO9781107050884

[3] A. Chagrov, M. Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, Oxford (1997).

[4] G. Hughes, M. Cresswell, An Introduction to Modal Logic, Methuen and Co. Ltd., London (1968).

[5] C. I. Lewis, C. H. Langford, Symbolic Logic, Appleton-Century-Crofts, New York (1932).

[6] K. Matsumoto, Reduction theorem in Lewis's sentential calculi, Mathematica Japonicae, vol. 3 (1955), pp. 133–135.

[7] J. C. C. McKinsey, A. Tarski, Some Theorems About the Sentential Calculi of Lewis and Heyting, Journal of Symbolic Logic, vol. 13(1) (1948), pp. 1–15, DOI: http://dx.doi.org/10.2307/2268135 DOI: https://doi.org/10.2307/2268135

[8] V. V. Rybakov, A modal analog for Glivenko's theorem and its applications, Notre Dame Journal of Formal Logic, vol. 3(2) (1992), pp. 244–248, DOI: http://dx.doi.org/10.1305/ndj/1093636103 DOI: https://doi.org/10.1305/ndjfl/1093636103

[9] I. B. Shapirovsky, Glivenko's theorem, finite height, and local tabularity (2018), arXiv:1806.06899.

[10] A. Wroński, J. Zygmunt, Remarks on the free pseudo-boolean algebra with one-element free-generating set, Reports on Mathematical Logic, vol. 2 (1974), pp. 77–81.