Geometrical structures and modal logic
Although, in natural language, space modalities are used as frequently as time modalities, the logic of time is a well-established branch of modal logic whereas the same cannot be said of the logic of space. The reason is probably in the more simple mathematical structure of time: a set of moments together with a relation of precedence. Such a relational structure is suited to a modal treatment. The structure of space is more complex: several sorts of geometrical beings as points and lines together with binary relations as incidence or orthogonality, or only one sort of geometrical beings as points but ternary relations as collinearity or betweeness. In this paper, we define a general framework for axiomatizing modal logics which Kripke semantics is based on geometrical structures: structures of collinearity, projective structures, orthogonal structures.
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- 1.P. Balbiani, V. Dugat, L. Fariñas del Cerro and A. Lopez. Eléments de géométrie mécanique. Hermès, 1994.Google Scholar
- 2.P. Balbiani, L. Fariñas del Cerro, T. Tinchev and D. Vakarelov. Modal logics for incidence geometries. Journal of Logic and Computation, to appear.Google Scholar
- 3.J. van Benthem. The Logic of Time. Reidel, 1983.Google Scholar
- 4.D. Gabbay. An irreflexivity lemma with applications to axiomatizations of conditions on tense frames. U. Mönnich (editor), Aspects of Philosophical Logic. 67–89, Reidel, 1981.Google Scholar
- 5.D. Gabbay, I. Hodkinson and M. Reynolds. Temporal Logic: Mathematical Foundations and Computational Aspects. Volume I, Oxford University Press, 1994.Google Scholar
- 6.R. Goldblatt. Orthogonality and Spacetime Geometry. Springer-Verlag, 1987.Google Scholar
- 7.Heyting. Axiomatic Protective Geometries. North-Holland, 1963.Google Scholar
- 8.D. Hilbert. Foundations of Geometry. Second english edition, Open Court, 1971.Google Scholar
- 9.G. Hughes and M. Cresswell. A Companion to Modal Logic. Methuen, 1984.Google Scholar
- 10.M. de Rijke. The modal logic of inequality. Journal of Symbolic Logic, Volume 57, Number 2, 566–584, 1992.Google Scholar
- 11.L. Szczerba and A. Tarski. Metamathematical properties of some affine geometries. Y. Bar-Hillel (editor), Logic, Methodology and Philosophy of Science. 166–178, North-Holland, 1972.Google Scholar
- 12.D. Vakarelov. A modal theory of arrows. Arrow logics I. D. Pearce and G. Wagner (editors), Logics in AI, European Workshop JELIA '92, Berlin, Germany, September 1992, Proceedings. Lecture Notes in Artificial Intelligence 633, 1–24, Springer-Verlag, 1992.Google Scholar
- 13.D. Vakarelov. Many-dimensional arrow structures. Arrow logics II. Journal of Applied Non-Classical Logics, to appear.Google Scholar