Abstract
The notion of arrow structure /a.s./ is introduced as an algebraic version of the notion of directed multi graph. By means of a special kind of a representation theorem for arrow structures it is shown that the whole information of an a.s. is contained in the set of his arrows equipped with four binary relations describing the four possibilities for two arrows to have a common point. This makes possible to use arrow structures as a semantic base for a special polymodal logic, called in the paper BAL /Basic Arrow Logic/. BAL and various kinds of his extensions are used for expressing in a modal setting different properties of arrow structures. Several kinds of completeness theorems for BAL and some other arrow logics are proved, including completeness with respect to classes of finite models. And the end some open problems and possibilities for further development of the “arrow” approach are formulated.
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References
VAN BENTHEM J.F.A.K., Modal Logic and Classical Logic, Bibliopolis, Napoli, 1986.
VAN BENTHEM J.F.A.K. Modal Logic and Relational Algebra, manuscript, May 1989, to appear in the proceedingsof Malcev Conference on Algebra, Novosibirsk, 1997.
VAN BENTHEM J.F.A.K. Private letter, June 1990.
HUGHES G.E. & M.J.CRESSWELL, A companion to Modal Logic, Methuen, London, 1984.
JONSSON B., TARSKI A. Boolean algebras with operators. Americ. J. Math., Part I: 73 891–993; Part II: 74,127–162, 1951.
KRAMER R.L. Relativized Relational Algebras, manuskript, April 1989, to appear in the proc. of the Algebraic Logic Conference, Budapest 1988.
MADDUX R. D. Some varieties containing relational algebras, Trans. Amer. Math. Soc. Vol 272(1982), 501–526.
MIKULAÄS Sz., The completeness of the Lambek Calculus with respect to relational semantics, ITLI Prepublications, University of Amsterdam, 1992.
NÉMETI I. Algebraizations of Quantifier Logics, an introductory overview, manuscript, June1991, to appear in Studia Logica.
ROORDA D. Dyadic Modalities and Lambek Calculus, in Colloquium on Modal Logic 1991, ed. M. de Rijke, Amsterdam 1991.
ROORDA D. Resource Logics, PhD thesis, Fac. Math. and Comp. Sc., University of Amsterdam, Amsterdam 1991.
SEGERBERG K. An Essay in Classical Modal Logic, Filosofiska Studier 13, Uppsala, 1971.
VAKARELOV D. Arrow logics, Manuscript, September 1997
VAKARELOV D. Rough Polyadic Modal Logics, Journal of Applied Non-Classical Logics, v. 1, 1(1991), 9–35.
VAKARELOV D. Modal Logics for Reasoning about Arrows: Arrow Logics, in the proc. of 9-th International Congress of Logic Methodology and Philosophy of Sciences, Section 5 — Philosophical Logic, August 7–14, 1991, Uppsala.
VAKARELOV D. Arrow logics with cylindric operators, abstract of a paper submeted to the 1992 European Summer Meeting of the ASL.
VENEMA Y. Two-dimensional Modal Logic for Relational Algebras and Temporal Logic of Intervals, ITLI-prepublication series LP-89-03, University of Amsterdam, Amsterdam 1989.
VENEMA Y. Many-dimensional Modal Logic, PhD thesis, September 1991, Fac. Math. and Comp. Sc., University of Amsterdam. To appear.
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© 1992 Springer-Verlag Berlin Heidelberg
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Vakarelov, D. (1992). A modal theory of arrows. Arrow logics I. In: Pearce, D., Wagner, G. (eds) Logics in AI. JELIA 1992. Lecture Notes in Computer Science, vol 633. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023418
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DOI: https://doi.org/10.1007/BFb0023418
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