This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abashidze, M. (1987). Algebraic Analysis of the Gödel-Löb Modal System. PhD thesis, Tbilisi State University. In Russian.
Abashidze, M. and Esakia, L. (1987). Cantor’s scattered spaces and the provability logic. In Baku International Topological Conference. Volume of Abstracts. Part I, page 3. In Russian.
Aiello, M. (2002a). Spatial Reasoning: Theory and Practice. PhD thesis, ILLC, University of Amsterdam. DS-2002-02.
Aiello, M. (2002b). A spatial similarity based on games: Theory and practice. Journal of the Interest Group in Pure and Applied Logic, 10(1):1–22.
Aiello, M. and Ottens, B. (2007). The mathematical morphological view on reasoning about space. In International Joint Conference on Artificial Intelligence (IJCAI-07), pages 205–211. Morgan Kaufmann.
Aiello, M. and van Benthem, J. (2002a). Logical patterns in space. In Barker-Plummer, D., Beaver, D., van Benthem, J., and di Luzio, P. Scotto, editors, Words, Proofs, and Diagrams, pages 5–25. CSLI, Stanford.
Aiello, M. and van Benthem, J. (2002b). A modal walk through space. Journal of Applied Non-Classical Logics, 12(3–4):319–364.
Aiello, M., van Benthem, J., and Bezhanishvili, G. (2003). Reasoning about space: The modal way. J. Logic Comput., 13(6):889–920.
Allen, J. (1983). Maintaining knowledge about temporal intervals. Communications of the ACM, 26:832–843.
Allen, J. and Hayes, P. (1985). A common sense theory of time. In Joshi, A., editor, IJCAI85, volume 1, pages 528–531. International Joint Conference on Artificial Itelligence, Morgan Kaufmann.
Andréka, H., van Benthem, J., and Németi, I. (1998). Modal logics and bounded fragments of predicate logic. Journal of Philosophical Logic, 27(3):217–274.
Anger, F., van Benthem, J., Guesgen, H., and Rodriguez, R. (1996). Editorial of the special issue on ‘Space, Time and Computation: Trends and Problems’. International Journal of Applied Intelligence, 6(1):5–9.
Areces, C. and ten Cate, B. (2006). Hybrid logics. In Handbook of Modal Logic.
Artemov, S. (2006). Modal logic in mathematics. In Handbook of Modal Logic.
Awodey, S. and Kishida, K. (2006). Topological semantics for first-order modal logic. In preparation.
Balbiani, Ph. (1998). The modal multilogic of geometry. Journal of Applied Non-Classical Logics, 8:259–281.
Balbiani, Ph., Fariñas del Cerro, L., Tinchev, T., and Vakarelov, D. (1997). Modal logics for incidence geometries. Journal of Logic and Computation, 7:59–78.
Barwise, J. (1988). Three views of common knowledge. In Proceedings of the Second Conference on Theoretical Aspects of Reasoning about Knowledge (Pacific Grove, CA, 1988), pages 365–379, Los Altos, CA. Morgan Kaufmann.
Ben-Or, M., Kozen, D., and Reif, J. (1986). The complexity of elementary algebra and geometry. Journal of Computer and System Sciences, 32: 251–264.
Bennett, B. (1995). Modal logics for qualitative spatial reasoning. Bulletin of the IGPL, 3:1–22.
Bezhanishvili, G., Esakia, L., and Gabelaia, D. (2005). Some results on modal axiomatization and definability for topological spaces. Studia Logica, 81(3): 325–355.
Bezhanishvili, G. and Gehrke, M. (2005). Completeness of S4 with respect to the real line: revisited. Ann. Pure Appl. Logic, 131(1-3):287–301.
Bezhanishvili, G., Mines, R., and Morandi, P. (2003). Scattered, Hausdorff-reducible, and hereditary irresolvable spaces. Topology and its Applications, 132:291–306.
Bezhanishvili, N. and Kupke, C. (2006). Spatio-temporal logics of the real line. In preparation.
Binmore, K. (1994). Game Theory and the Social Contract. MIT Press, Cambridge.
Blackburn, P., de Rijke, M., and Venema, Y. (2001). Modal Logic. Cambridge University Press.
Blass, A. (1990). Infinitary combinatorics and modal logic. J. Symbolic Logic, 55(2):761–778.
Bloch, I. (2000). Using mathematical morphology operators as modal operators for spatial reasoning. In ECAI 2000, Workshop on Spatio-Temporal Reasoning, pages 73–79.
Blumenthal, L. (1961). A Modern View of Geometry. Dover.
Boolos, G. (1993). The Logic of Provability. Cambridge University Press, Cambridge.
Burgess, J. (1984). Basic tense logic. In Gabbay, D. and Guenthner, F., editors, Handbook of Philosophical Logic, volume II, chapter 2, pages 89–133. Reidel.
Burgess, John P. (1979). Logic and time. J. Symbolic Logic, 44(4):566–582.
Carnap, Rudolf (1998). Der logische Aufbau der Welt, volume 514 of Philosophische Bibliothek [Philosophical Library]. Felix Meiner Verlag, Hamburg. Reprint of the 1928 original and of the author’s preface to the 1961 edition.
Chagrov, A. and Zakharyaschev, M. (1997). Modal Logic, volume 35 of Oxford Logic Guides. Clarendon Press, Oxford.
Chellas, B. (1980). Modal Logic: An Introduction. Cambridge University Press.
Dabrowski, A., Moss, A., and Parikh, R. (1996). Topological reasoning and the logic of knowledge. Annals of Pure and Applied Logic, 78:73–110.
de Jongh, D. and Veltman, F. (1985). Lecture Notes on Modal Logic. ILLC, Amsterdam.
de Rijke, M. (1993). Extended Modal Logic. PhD thesis, ILLC, University of Amsterdam.
Doets, Kees (1996). Basic Model Theory. Studies in Logic, Language and Information. CSLI Publications, Stanford, CA.
Engelking, R. (1989). General Topology. Heldermann Verlag.
Esakia, L. (1981). Diagonal constructions, Löb’s formula and Cantor’s scattered spaces. In Studies in Logic and Semantics, pages 128–143. Metsniereba. In Russian.
Esakia, L. (2001). Weak transitivity-restitution. In Study in Logic, volume 8, pages 244–254. Nauka. In Russian.
Esakia, L. (2002). The modal version of Gödel’s second incompleteness theorem and the McKinsey system. In Logical Investigations. Vol. IX, pages 292–300. In Russian.
Esakia, L. (2004). Intuitionistic logic and modality via topology. Annals of Pure and Applied Logic, 127:155–170.
Fagin, R., Halpern, J., Moses, Y., and Vardi, M. (1995). Reasoning About Knowledge. MIT Press, Cambridge, MA.
Fontaine, G. (2006). Axiomatization of ML and Cheq. Master’s thesis, ILLC, University of Amsterdam.
Gabbay, D., Kurucz, A., Wolter, F., and Zakharyaschev, M. (2003). Many-Dimensional Modal Logics: Theory and Applications. Elsevier, Uppsala. Studies in Logic and the Foundations of Mathematics, Volume 148.
Gabbay, D. and Shehtman, V. (1998). Products of modal logics. I. Log. J. IGPL, 6(1):73–146.
Gabelaia, D. (1999). Modal logics GL and Grz: semantical comparison. In Proceedings of the ESSLLI Student Session, pages 91–97.
Gabelaia, D. (2001). Modal Definability in Topology. Master’s thesis, ILLC, University of Amsterdam.
Gabelaia, D. (2004). Topological, Algebraic and Spatio-Temporal Semantics for Multi-Dimensional Modal Logics. PhD thesis, King’s College, London.
Gabelaia, D., Kurucz, A., Wolter, F., and Zakharyashev, M. (2005). Products of ‘transitive’ modal logics. Journal of Symbolic Logic, 70:993–1021.
Gabelaia, D. and Sustretov, D. (2005). Modal correspondence for topological semantics. In Abstracts of Algebraic and Topological Methods in Non-Classical Logics II (Barcelona 2005), pages 80–81.
Gärdenfors, P. (2000). Conceptual Spaces. MIT Press.
Gerhardt, S. (2004). A Construction Method for Modal Logics of Space. Master’s thesis, ILLC, University of Amsterdam.
Goldblatt, R. (1980). Diodorean modality in Minkowski space-time. Studia Logica, 39:219–236.
Goldblatt, R. (1987). Orthogonality and Spacetime Geometry. Springer-Verlag.
Goranko, V. and Passy, S. (1992). Using the universal modality: Gains and questions. J. Logic Comput., 2(1):5–30.
Harel, D., Kozen, D., and Tiuryn, J. (2000). Dynamic Logic. Foundations of Computing Series. MIT Press, Cambridge, MA.
Helly, E. (1923). Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresber. Deutsch. Math. Verein, 32:175–176.
Hintikka, J. (1962). Knowledge and Belief. Cornell University Press.
Hodkinson, I. and Reynolds, M. (2006). Temporal logic. In Handbook of Modal Logic.
Kamp, J. (1968). Tense Logic and the Theory of Linear Order. PhD thesis, University of California, Los Angeles.
Kelley, J. (1975). General Topology. Springer-Verlag, New York.
Konev, B., Kontchakov, R., Wolter, F., and Zakharyaschev, M. (2004). On dynamic topological and metric logics. In Proceedings of AiML 2004, pages 182–196. King’s College Publications.
Kuratowski, K. (1966). Topology. Vol. I. Academic Press, New York.
Kuratowski, K. and Mostowski, A. (1976). Set Theory. North Holland, Amsterdam-New York-Oxford.
Kurtonina, N. (1995). Frames and Labels. A Modal Analysis of Categorial Inference. PhD thesis, ILLC, Amsterdam.
Kurtonina, N. and de Rijke, M. (1997). Bisimulations for temporal logic. Journal of Logic, Language and Information, 6:403–425.
Leitgeb, H. (2007). ‘A new analysis of quasianalysis’. Journal of Philosophical Logic.
Lewis, D. (1969). Convention. Harvard University Press.
Litak, T. (2004). Some notes on the superintuitionistic logic of chequered subsets of R∞. Bull. Sect. Logic Univ. Łódź, 33(2):81–86.
Marx, M. (1995). Algebraic Relativization and Arrow Logic. PhD thesis, ILLC.
Marx, M. and Venema, Y. (1997). Multi Dimensional Modal Logic. Kluwer.
Matheron, G. (1967). Eléments pur Une Theorie des Milieux Poreaux. Masson.
McKinsey, J. and Tarski, A. (1944). The algebra of topology. Annals of Mathematics, 45:141–191.
Mikulas, S. (1995). Taming Logics. PhD thesis, ILLC.
Mints, G. (1998). A completeness proof for propositional S4 in Cantor Space. In E. Orlowska, editor, Logic at work : Essays dedicated to the memory of Helena Rasiowa. Physica-Verlag, Heidelberg.
Moerdijk, I. (1982). Some topological spaces which are universal for intuitionistic predicate logic. Indagationes Mathematicae, 44(2):227–235.
Pauly, M. (2001). Logic for Social Software. PhD thesis, ILLC, University of Amsterdam.
Peleg, D. (1987). Concurrent dynamic logic. Journal of the ACM, 34(2): 450–479.
Pratt, V. (1999). Chu spaces. In School on Category Theory and Applications (Coimbra, 1999), volume 21 of Textos Mat. Sér. B, pages 39–100. Univ. Coimbra, Coimbra.
Preparata, F. and Shamos, M. (1985). Computational Geometry: An Introduction. Springer-Verlag.
Randell, D., Cui, Z., and Cohn, A. (1992). A spatial logic based on regions and connection. In Proc. of Int. Conf. on Principles of Knowledge Representation and Reasoning (KR’92), pages 165–176. San Mateo.
Randell, D., Witkowski, M., and Shanahan, M. (2001). From images to bodies: Modelling and exploiting occlusion and motion parallax. In Proc. of Int. Joint Conference on Artificial Intelligence (IJCAI-01).
Rasiowa, H. and Sikorski, R. (1963). The Mathematics of Metamatematics. Panstwowe Wydawnictwo Naukowe.
Segerberg, K. (1970). Modal logics with linear alternative relations. Theoria, 36:301–322.
Segerberg, K. (1973). Two-dimensional modal logic. J. Philos. Logic, 2(1): 77–96.
Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press.
Shehtman, V. (1983). Modal logics of domains on the real plane. Studia Logica, 42:63–80.
Shehtman, V. (1990). Derived sets in Euclidean spaces and modal logic. Technical Report X-1990-05, Univ. of Amsterdam.
Shehtman, V. (1993). A logic with progressive tenses. In Diamonds and defaults (Amsterdam, 1990/1991), volume 229 of Synthese Lib., pages 255–285. Kluwer Acad. Publ., Dordrecht.
Shehtman, V. (1999). ‘Everywhere’ and ‘here’. Journal of Applied Non-Classical Logics, 9(2–3):369–379.
Shehtman, V. (2006). Derivational modal logics. Moscow Mathematical Journal. submitted.
Sheremet, M., Tishkovsky, D., Wolter, F., and Zakharyaschev, M. (2006). Comparative similarity, tree automata, and diophantine equations. LPAR. to appear.
Spaan, E. (1993). Complexity of Modal Logics. PhD thesis, University of Amsterdam, Institute for Logic, Language and Computation.
Stebletsova, V. (2000). Algebras, Relations and Geometries. PhD thesis, University of Utrecht.
Stebletsova, V. and Venema, Y. (2001). Undecidable theories of Lyndon algebras. J. Symbolic Logic, 66(1):207–224.
Steinsvold, C. (2005). Personal communication.
Szczerba, L. and Tarski, A. (1965). Metamathematical properties of some affine geometries. In Bar-Hillel, Y., editor, Int. Congress for Logic, Methodolog, and Philosophy of Science, pages 166–178. North-Holland.
Tarski, A. (1938). Der Aussagenkalkül und die Topologie. Fund. Math., 31: 103–134.
Tarski, A. (1959). What is elementary geometry? In L. Henkin and P. Suppes and A. Tarski, editor, The Axiomatic Method, with Special Reference to Geometry ad Physics, pages 16–29. North-Holland.
ten Cate, B., Gabelaia, D., and Sustretov, D. (2006). Modal languages for topology: expressivity and definability. Preprint.
Troelstra, A. (1992). Lectures on Linear Logic. CSLI.
van Benthem, J. (1983a). Correspondence theory. In Gabbay, D. and Guenthner, F., editors, Handbook of Philosophical Logic, volume II, pages 167–247. Reidel Publishing Company.
van Benthem, J. (1983b). The Logic of Time, volume 156 of Synthese Library. Reidel, Dordrecht. [Revised and expanded, Kluwer, 1991].
van Benthem, J. (1991a). Language in Action. Categories, Lambdas and Dynamic Logic. North-Holland, Amsterdam.
van Benthem, J. (1991b). Logic and the flow of information. In Prawitz, D., Skyrms, B., and Westertal, D., editors, Proceedings of the 9th International Conference of Logic, Methodology and Philosophy of Science, pages 693–724. Elsevier.
van Benthem, J. (1992). Logic as programming. Fundamenta Informaticae, 17(4):285–317.
van Benthem, J. (1995). Temporal logic. In Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 4, Oxford Sci. Publ., pages 241–350. Oxford Univ. Press, New York.
van Benthem, J. (1996). Exploring Logical Dynamics, volume 156. CSLI Publications, Stanford/Cambridge University Press.
van Benthem, J. (1999). Temporal patterns and modal structure. Log. J. IGPL, 7(1):7–26. Special issue on Temporal Logic. A. Montanari, A. Policriti, and Y. Venema eds.
van Benthem, J. (2000a). Information transfer across Chu spaces. Log. J. IGPL, 8(6):719–731.
van Benthem, J. (2000b). Logical structures in mathematical morphology. Available at http://www.science.uva.nl/johan/MM-LL.ps.
van Benthem, J. (2002). Invariance and definability: two faces of logical constants. In Sieg, W., Sommer, R., and Talcott, C., editors, Reflections on the Foundations of Mathematics. Essays in Honor of Sol Feferman, ASL Lecture Notes in Logic, pages 426–446. ASL.
van Benthem, J., Bezhanishvili, G., Cate, B. ten, and Sarenac, D. (2005). Modal logics for products of topologies. Studia Logica, 84(3):369–392.
van Benthem, J., Bezhanishvili, G., and Gehrke, M. (2003). Euclidean hierarchy in modal logic. Studia Logica, 75(3):327–344.
van Benthem, J. and Blackburn, P. (2006). Basic modal model theory. In Handbook of Modal Logic.
van Benthem, J. and Sarenac, D. (2004). The geometry of knowledge. In Aspects of universal logic, volume 17 of Travaux Log., pages 1–31. Univ. Neuchâtel, Neuchâtel.
van Dalen, Dirk (2005). Mystic, Geometer, and Intuitionist. The Clarendon Press Oxford University Press, Oxford. The life of L. E. J. Brouwer, 1881–1966. Vol. 2, Hope and disillusion.
Venema, Y. (1996). A crash course in arrow logic. In Marx, M., Masuch, M., and Pólos, L., editors, Arrow Logic and Multimodal Logic. CSLI.
Venema, Y. (1999). Points, lines and diamonds: A two-sorted modal logic for projective planes. Journal of Logic and Computation, 9(5):601–621.
Venema, Y. (2007). Algebras and Co-algebras. In Blackburn, P., van Benthem, J. and Wolter, F., editors, Handbook of Model Logic, pages 331–426. Elsevier, Amsterdam.
von Plato, J. (1995). The axioms of constructive geometry. Annals of Pure and Applied Logic, 76(2):169–200.
Wooldridge, M. (2002). An Introduction to Multiagent Systems. J. Wiley, New York.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this chapter
Cite this chapter
van Benthem, J., Bezhanishvili, G. (2007). Modal Logics of Space. In: Aiello, M., Pratt-Hartmann, I., Van Benthem, J. (eds) Handbook of Spatial Logics. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5587-4_5
Download citation
DOI: https://doi.org/10.1007/978-1-4020-5587-4_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-5586-7
Online ISBN: 978-1-4020-5587-4
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)