Topological and Multi-Topological Frames in the Context of Intuitionistic Modal Logic

Tomasz Witczak

Bulletin of the Section of Logic (2019)

  • Volume: 48, Issue: 3, page 187-205
  • ISSN: 0138-0680

Abstract

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We present three examples of topological semantics for intuitionistic modal logic with one modal operator □. We show that it is possible to treat neighborhood models, introduced earlier, as topological or multi-topological. From the neighborhood point of view, our method is based on differences between properties of minimal and maximal neighborhoods. Also we propose transformation of multitopological spaces into the neighborhood structures.

How to cite

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Tomasz Witczak. "Topological and Multi-Topological Frames in the Context of Intuitionistic Modal Logic." Bulletin of the Section of Logic 48.3 (2019): 187-205. <http://eudml.org/doc/295593>.

@article{TomaszWitczak2019,
abstract = {We present three examples of topological semantics for intuitionistic modal logic with one modal operator □. We show that it is possible to treat neighborhood models, introduced earlier, as topological or multi-topological. From the neighborhood point of view, our method is based on differences between properties of minimal and maximal neighborhoods. Also we propose transformation of multitopological spaces into the neighborhood structures.},
author = {Tomasz Witczak},
journal = {Bulletin of the Section of Logic},
keywords = {intuitionistic modal logic; neighbourhood semantics; topological semantics; Kripke frames; soundness and completeness},
language = {eng},
number = {3},
pages = {187-205},
title = {Topological and Multi-Topological Frames in the Context of Intuitionistic Modal Logic},
url = {http://eudml.org/doc/295593},
volume = {48},
year = {2019},
}

TY - JOUR
AU - Tomasz Witczak
TI - Topological and Multi-Topological Frames in the Context of Intuitionistic Modal Logic
JO - Bulletin of the Section of Logic
PY - 2019
VL - 48
IS - 3
SP - 187
EP - 205
AB - We present three examples of topological semantics for intuitionistic modal logic with one modal operator □. We show that it is possible to treat neighborhood models, introduced earlier, as topological or multi-topological. From the neighborhood point of view, our method is based on differences between properties of minimal and maximal neighborhoods. Also we propose transformation of multitopological spaces into the neighborhood structures.
LA - eng
KW - intuitionistic modal logic; neighbourhood semantics; topological semantics; Kripke frames; soundness and completeness
UR - http://eudml.org/doc/295593
ER -

References

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  3. M. J. Collinson, B. P. Hilken, D. E. Rydeheard, An adjoint construction for topological models of intuitionistic modal logic. Extended abstract, http://sierra.nmsu.edu/morandi/old%20files/TbilisiConference/Collinson.pdf 
  4. J. M. Davoren, Topological Semantics and Bisimulations for Intuitionistic Modal Logics and Their Classical Companion Logics, Logical Foundations of Computer Science 2007, Springer 2007. 
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  6. J. M. Davoren, V. Coulthard, T. Moor, R. P. Goré, A. Nerode, Topological semantics for Intuitionistic modal logics and spatial discretisation by A/D maps, [in:] Workshop on Intuitionistic Modal Logic and Applications (IMLA), Copenhagen, Denmark 2002. 
  7. D. de Jongh, F. Sh. Maleki, Two neighborhood semantics for subintuitionistic logics, http://events.illc.uva.nl/Tbilisi/Tbilisi2017/uploaded_files/inlineitem/Dick_de_Jongh_Fateme_Shirmohammadzadeh_Maleki.pdf 
  8. D. de Jongh, F. Sh. Maleki, Weak subintuitionistic logics, https://www.illc.uva.nl/Research/Publications/Reports/PP-2016-12.text.pdf 
  9. M. Moniri, F. S. Maleki, Neighborhood semantics for basic and intuitionistic logic, Logic and Logical Philosophy, Vol. 23 (2015), pp. 339–355. 
  10. E. Pacuit, Neighborhood Semantics for Modal Logic, Springer International Publishing AG 2017. 
  11. V. H. Sotirov, Modal Theories with Intuitionistic Logic, [in:] Mathematical Logic, Proceedings of the Conference on Mathematical Logic, Dedicated to the Memory of A. A. Markov (1903–1979), September 22–23, 1980, pp. 139–171, Sofia 1984. 
  12. T. Speer, A Short Study of Alexandroff Spaces, https://arxiv.org/pdf/0708.2136.pdf 
  13. T. Witczak, Generalized Topological Semantics for Weak Modal Logics, https://arxiv.org/pdf/1904.06099.pdf 
  14. T. Witczak, Intuitionistic Modal Logic Based on Neighborhood Semantics Without Superset Axiom, https://arxiv.org/pdf/1707.03859.pdf 

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