Logica Universalis

, Volume 11, Issue 2, pp 153–175 | Cite as

Topological Representation of Intuitionistic and Distributive Abstract Logics

  • Andreas Bernhard Michael Brunner
  • Steffen LewitzkaEmail author


We continue work of our earlier paper (Lewitzka and Brunner in Log Univers 3(2):219–241, 2009) where abstract logics and particularly intuitionistic abstract logics are studied. Abstract logics can be topologized in a direct and natural way. This facilitates a topological study of classes of concrete logics whenever they are given in abstract form. Moreover, such a direct topological approach avoids the often complex algebraic and lattice-theoretic machinery usually applied to represent logics. Motivated by that point of view, we define in this paper the category of intuitionistic abstract logics with stable logic maps as morphisms, and the category of implicative spectral spaces with spectral maps as morphisms. We show the equivalence of these categories and conclude that the larger categories of distributive abstract logics and distributive sober spaces are equivalent, too.


Abstract logics topology spectral spaces with implication duality intuitionistic logics 

Mathematics Subject Classification

Primary 03B22 03G10 Secondary 03B20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We would like to thank the anonymous referee for many helpful comments and suggestions improving this work. The first author would like to thank the support from MaToMUVI Project with Number 247584 and the hospitality of University of Salerno (UNISA), where a part of this work was finished in the fall of 2014. The second author thanks the hospitality of the University of Potsdam, where part of this work was finished in the fall of 2016.


  1. 1.
    Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A.: Bitopological duality for distributive lattices and Heyting algebras. Math. Struct. Comput. Sci. 20, 359–393 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bezhanishvili, G., Mines, R., Morandi, P.J.: Topo-canonical completions of closure algebras and Heyting algebras. Algebra Univers. 58, 1–34 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bloom, S.L., Brown, D.J.: Classical abstract logics. Diss. Math. 102, 43–51 (1973)zbMATHGoogle Scholar
  4. 4.
    Blok, W., Pigozzi, D.: Algebraizable logics, vol. 396. Memoirs of the American Mathematical Society, Providence (1989)zbMATHGoogle Scholar
  5. 5.
    Brown, D.J., Suszko, R.: Abstract logics. Diss. Math. 102, 9–42 (1973)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brunner, A.B.M., Lewitzka, S.: Topological Representation of Intuitionistic and Abstract Logics, Abstract Published in XVI. Encontro Brasileiro de Lógica, Petrópolis, Rio de Janeiro (2011)Google Scholar
  7. 7.
    Caleiro, C., Gonçalves, R.: Equipollent logical systems. In: Beziau, J.Y. (ed.) Logica Universalis: Towards a General Theory of Logic, 2nd edn. Birkhaeuser Verlag, Basel (2007)Google Scholar
  8. 8.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  9. 9.
    Eiben, Á.E., Janossy, A., Kurucz, Á.: Combining algebraizable logics. Notre Dame J. Form Log 37(2), 366–380 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fiorentini, C.: Kripke completeness for intermediate logics. PhD. Thesis (2000)Google Scholar
  11. 11.
    Fitting, M.: Intuitionistic logic, model theory and forcing. North Holland, Amsterdam (1969)zbMATHGoogle Scholar
  12. 12.
    Font, J.M., Verdú, V.: A first approach to abstract modal logics. J. Symb. Log. 54, 1042–1062 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Goguen, J.A., Burstall, R.M.: Introducing institutions. Lecture Notes in Computer Science, vol. 164, pp. 221–256 (1984)Google Scholar
  14. 14.
    Hochster, M.: Prime ideal structure in commutative rings. Trans. AMS 142, 43–60 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jansana, R.: Propositional consequence relations and algebraic logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Spring 2011 edn.
  16. 16.
    Johansson, I.: Der Minimalkalkül, ein reduzierter intuitionistischer formalismus. Compos. Math. 4, 119–136 (1937)zbMATHGoogle Scholar
  17. 17.
    Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)zbMATHGoogle Scholar
  18. 18.
    Lewitzka, S.: Abstract logics, logic maps and logic homomorphisms. Log. Univers. 1(2), 243–276 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lewitzka, S.: \(\in _{4}\): A \(4\)-valued Truth Theory and Metalogic, preprint (2007)Google Scholar
  20. 20.
    Lewitzka, S., Brunner, A.B.M.: Minimally generated abstract logics. Log. Univers. 3(2), 219–241 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mendes, C.A., Mariano, H.L.: Towards a good notion of categories of logics. arXiv:1404.3780v2 (2016)
  22. 22.
    Miraglia, F.: An Introduction to Partially Ordered Structures and Sheaves, Contemporary Logic Series, vol. 1, Polimetrica International Scientific Publisher, Milan, Italy (2006)Google Scholar
  23. 23.
    Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rasiowa, H.: An Algebraic Approach to Non-Classical Logic. North-Holland Publ. Co., Amsterdam (1974)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Andreas Bernhard Michael Brunner
    • 1
  • Steffen Lewitzka
    • 2
    Email author
  1. 1.Departamento de Matemática, Instituto de MatemáticaUniversidade Federal da Bahia - UFBASalvadorBrazil
  2. 2.Departamento de Ciência da Computação Instituto de MatemáticaUniversidade Federal da Bahia - UFBASalvadorBrazil

Personalised recommendations