From Intuitionism to Brouwer's Modal Logic

Zofia Kostrzycka

Bulletin of the Section of Logic (2020)

  • Volume: 49, Issue: 4, page 343-358
  • ISSN: 0138-0680

Abstract

top
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.

How to cite

top

Zofia Kostrzycka. "From Intuitionism to Brouwer's Modal Logic." Bulletin of the Section of Logic 49.4 (2020): 343-358. <http://eudml.org/doc/296794>.

@article{ZofiaKostrzycka2020,
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.},
author = {Zofia Kostrzycka},
journal = {Bulletin of the Section of Logic},
keywords = {intuitionistic logic; Kripke frames; Brouwer's modal logic},
language = {eng},
number = {4},
pages = {343-358},
title = {From Intuitionism to Brouwer's Modal Logic},
url = {http://eudml.org/doc/296794},
volume = {49},
year = {2020},
}

TY - JOUR
AU - Zofia Kostrzycka
TI - From Intuitionism to Brouwer's Modal Logic
JO - Bulletin of the Section of Logic
PY - 2020
VL - 49
IS - 4
SP - 343
EP - 358
AB - 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.
LA - eng
KW - intuitionistic logic; Kripke frames; Brouwer's modal logic
UR - http://eudml.org/doc/296794
ER -

References

top
  1. [1] O. Becker, Zur Logik der Modalitäten, Jahrbuch für Philosophie und phänomenologische Forschung, vol. 11 (1930), pp. 497–548. 
  2. [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 
  3. [3] A. Chagrov, M. Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, Oxford (1997). 
  4. [4] G. Hughes, M. Cresswell, An Introduction to Modal Logic, Methuen and Co. Ltd., London (1968). 
  5. [5] C. I. Lewis, C. H. Langford, Symbolic Logic, Appleton-Century-Crofts, New York (1932). 
  6. [6] K. Matsumoto, Reduction theorem in Lewis's sentential calculi, Mathematica Japonicae, vol. 3 (1955), pp. 133–135. 
  7. [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 
  8. [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 
  9. [9] I. B. Shapirovsky, Glivenko's theorem, finite height, and local tabularity (2018), arXiv:1806.06899. 
  10. [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. 

NotesEmbed ?

top

You must be logged in to post comments.