Skip to main content
SpringerLink
Log in

Bi-Heyting algebras, toposes and modalities

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

The aim of this paper is to introduce a new approach to the modal operators of necessity and possibility. This approach is based on the existence of two negations in certain lattices that we call bi-Heyting algebras. Modal operators are obtained by iterating certain combinations of these negations and going to the limit. Examples of these operators are given by means of graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Curry, H. B.: Introduction to Mathematical Logic, McGraw-Hill, New York, San Francisco, Toronto, London, 1963.

    Google Scholar 

  2. Ghilardi, S. and Meloni, G. C.: Modal and tense predicate logic: Models in presheaves and categorical conceptualization, in: F. Borceux (ed.), Categorical Algebra and Its Applications (Proceedings, Louvain-la-Neuve, 1987), Lecture Notes in Mathematics 1348, Springer-Verlag, 1988.

  3. Johnstone, P. T.: Conditions related to De Morgan's law, in: M. P. Fourman, C. J. Mulvey, and D. S. Scott (eds), Applications of Sheaves (Proceedings, Durham 1977) Lecture Notes in Mathematics 753, Springer-Verlag, 1979.

  4. Kock, A. and Reyes, G. E.: Doctrines in categorical logic, in: J. Barwise (ed.), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, New York, Oxford, 1977.

    Google Scholar 

  5. Lavendhomme, R., Lucas, Th., and Reyes, G.E.: Formal systems for topos-theoretic modalities, Bull. Soc. Math. Belgique (série A), XLI, Fascicule 2, 1989.

  6. Lawvere, F. W.: Introduction, in: F. W. Lawvere and S. Schanuel (eds), Categories in Continuum Physics (Buffalo 1982), Lecture Notes in Mathematics 1174, Springer-Verlag, 1986.

  7. Lawvere, F. W.: Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes, in: A. Carboni, M. C. Pedicchio and G. Rosolini (eds), Category Theory (Proceedings, Como 1990). Lecture Notes in Mathematics 1488, Springer-Verlag, 1991, 279–297.

  8. Lawvere, F. W.: Categories of spaces may not be generalized spaces as exemplified by directed graphs, in: J. H. Pérez and X. Caicedo (eds), Memorias Del Seminario-Taller En Teoría De Categorías, Revista Colombiana de Matemáticas, septiembrediciembre 1986, números 3-4.

  9. Makkai, M. and Reyes, G. E.: First-Order Categorical Logic, Lecture Notes in Mathematics 611, Springer-Verlag, 1977.

  10. Reyes, G. E.: A topos-theoretic approach to reference and modality, Notre Dame Journal of Formal Logic 32 (1991), 359–391.

    Google Scholar 

  11. Reyes, G. E. and Zolfaghari, H.: Topos-theoretic approaches to modalities, in: A Carboni, M. C. Pedicchio and G. Rosolini (eds), Category Theory (Proceedings, Como 1990), Lecture Notes in Mathematics, 1488, Springer-Verlag, 1991.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Université de Montréal, C.P. 6128 Succursale A, H3C 3J7, Montréal, Canada

    Gonzalo E. Reyes & Houman Zolfaghari

Authors
  1. Gonzalo E. Reyes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Houman Zolfaghari
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Reyes, G.E., Zolfaghari, H. Bi-Heyting algebras, toposes and modalities. Journal of Philosophical Logic 25, 25–43 (1996). https://doi.org/10.1007/BF00357841

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00357841

Keywords

Search

Navigation