Modal Logics of Arrows

  • Dimiter Vakarelov
Part of the Applied Logic Series book series (APLS, volume 7)


There exist many formal schemes and tools for representing knowledge about different types of data. Sometimes we can gain a better understanding if our information has some graphical representation. In many cases arrows are very suitable visual objects for representing various data structures: different kinds of graphs, binary relations, mappings, categories and so on. An abstract form of this representation scheme is the notion of arrow structure, which, in this paper, is an algebraic version of the notion of directed multi-graph.


Modal Logic Binary Relation Standard Semantic Kripke Semantic Boolean Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Andréka and Mikulás, 1994]
    H. Andréka and Sz. Mikulás. Lambek Calculus and its relational semantics: completeness and incompleteness. Journal of Logic, Language and Information, 3:1–37, 1994.CrossRefGoogle Scholar
  2. [Arsov, 1994]
    A. Arsov. Basic arrow logic with relation algebraic operators. ILLC Research Report LP-94–02, University of Amsterdam, 1994.Google Scholar
  3. [van Benthem, 1989]
    J. van Benthem. Modal logic and relational algebra. Manuscript, ILLC, University of Amsterdam, 1989.Google Scholar
  4. [van Benthem, 1985]
    J. van Benthem. Modal Logic and Classical Logic. Bibliopolis, 1985.Google Scholar
  5. [Hughes and Cresswell, 1984]
    G. Hughes and M. Cresswell. A Companion to Modal Logic. Methuen, London, 1984.Google Scholar
  6. [Jónsson and Tarski, 1951]
    B. Jónsson and A. Tarski. Boolean algebras with operators, Part I. American Journal of Mathematics, 73:891–939, 1951.CrossRefGoogle Scholar
  7. [Kramer, 1989]
    R.L. Kramer. Relativized relational algebras. In H. Andréka, D. Monk, and I. Németi, editors, Algebraic Logic. North-Holland, Amsterdam, 1989.Google Scholar
  8. [Maddux, 1982]
    R. Maddux. Some varieties containing relation algebras. Transactions of the American Mathematical Society, 272:501–526, 1982.CrossRefGoogle Scholar
  9. [Németi, 1991]
    I. Németi. Algebraizations of quantifier logics. Studia Logica, 51:485–570, 1991.CrossRefGoogle Scholar
  10. [Roorda, 1991]
    D. Roorda. Resource Logics. PhD thesis, Department of Mathematics and Computer Science, University of Amsterdam, 1991.Google Scholar
  11. [Roorda, 1993]
    D. Roorda. Dyadic modalities and lambek calculus. In M. de Rijke, editor, Diamonds and Defaults, pages 215–253. Kluwer, Dordrecht, 1993.Google Scholar
  12. [Segerberg, 1971]
    K. Segerberg. An Essay in Classical Modal Logic. Filosofiska Studier 13. University of Uppsala, 1971.Google Scholar
  13. [Vakarelov, 1990]
    D. Vakarelov. Arrow logics. Manuscript, University of Sofia, 1990.Google Scholar
  14. [Vakarelov, 1991a]
    D. Vakarelov. Modal logics for reasoning about arrows: arrow logics. In Conference Proc. 9th International Congress of Logic, Methodology, and Philosophy of Sciences, Uppsala, 1991.Google Scholar
  15. [Vakarelov, 1991b]
    D. Vakarelov. Rough polyadic modal logics. Journal of Applied Non-Classical Logics, 1:9–35, 1991.CrossRefGoogle Scholar
  16. [Vakarelov, 1992a]
    D. Vakarelov. Arrow logics with cylindric operators. Abstract of a paper presented at the 1992 European Summer Meeting of the ASL, 1992.Google Scholar
  17. [Vakarelov, 1992b]
    D. Vakarelov. A modal theory of arrows: arrow logics I. In D. Pearce and G. Wagner, editors, Logics in AI, European Workshop ‘92, pages 1–24. LNAI 633, Springer-Verlag, 1992.CrossRefGoogle Scholar
  18. [Venema, 1989]
    Y. Venema. Two-dimensional modal logics for relation algebras and temporal logic of intervals. Technical Report LP-89–03, Institute for Language, Logic and Information, 1989.Google Scholar
  19. [Venema, 1991]
    Y. Venema. Many-Dimensional Modal Logic. PhD thesis, Department of Mathematics and Computer Science, University of Amsterdam, 1991.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Dimiter Vakarelov
    • 1
  1. 1.Department of Mathematical Logic with Laboratory for Applied LogicSofia UniversityUSA

Personalised recommendations