Notre Dame Journal of Formal Logic

Ontologies for Plane, Polygonal Mereotopology

Oliver Lemon and Ian Pratt

Abstract

Several authors have suggested that a more parsimonious and conceptually elegant treatment of everyday mereological and topological reasoning can be obtained by adopting a spatial ontology in which regions, not points, are the primitive entities. This paper challenges this suggestion for mereotopological reasoning in two-dimensional space. Our strategy is to define a mereotopological language together with a familiar, point-based interpretation. It is proposed that, to be practically useful, any alternative region-based spatial ontology must support the same sentences in our language as this familiar interpretation. This proposal has the merit of transforming a vague, open-ended question about ontologies for practical mereotopological reasoning into a precise question in model theory. We show that (a version of) the familiar interpretation is countable and atomic, and therefore prime. We conclude that useful alternative ontologies of the plane are, if anything, less parsimonious than the one which they are supposed to replace.

Article information

Source
Notre Dame J. Formal Logic Volume 38, Number 2 (1997), 225-245.

Dates
First available in Project Euclid: 12 December 2002

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1039724888

Mathematical Reviews number (MathSciNet)
MR1489411

Digital Object Identifier
doi:10.1305/ndjfl/1039724888

Zentralblatt MATH identifier
0897.03014

Subjects
Primary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35]
Secondary: 03C65: Models of other mathematical theories

Citation

Pratt, Ian; Lemon, Oliver. Ontologies for Plane, Polygonal Mereotopology. Notre Dame J. Formal Logic 38 (1997), no. 2, 225--245. doi:10.1305/ndjfl/1039724888. http://projecteuclid.org/euclid.ndjfl/1039724888.


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References

  • [1] Allen, J. F., ``An interval-based representation of temporal knowledge,'' pp. 221--26 in The Seventh International Joint Conference on Artificial Intelligence (IJCAI), 1981.
  • [2] Asher, N., and L. Vieu, ``Toward a geometry of common sense: a semantics and a complete axiomatization of mereotopology,'' pp. 846--52 in International Joint Conference on Artificial Intelligence (IJCAI), 1995.
  • [3] Balbiani, P., L. F. del Cerro, T. Tinchev, and D. Vakarelov, ``Geometrical structures and modal logic,'' pp. 43--57 in Practical Reasoning. Lecture Notes in Artificial Intelligence, 1085, edited by D. Gabbay and H.-J. Ohlbach, Springer-Verlag, Berlin, 1996.
  • [4] Basri, S. A., A Deductive Theory of Space and Time. Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1966.
  • [5] Biacino, L., and Gerla, G., ``Connection structures,'' Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 242--47.
  • [6] Borgo, S., N. Guarino, and C. Masolo, ``A pointless theory of space based on strong connection and congruence,'' pp. 220--29 in Principles of Knowledge Representation and Reasoning. Proceedings of the Fifth International Conference (KR), edited by L. C. Aiello, J. Doyle, and S. C. Shapiro, Morgan Kaufmann, San Francisco, 1996.
  • [7] Carnap, R., Introduction to Symbolic Logic and its Applications, Dover, New York, 1958.
  • [8] Casati, R., and A. Varzi, Holes and Other Superficialities, The MIT Press, Cambridge, 1994.
  • [9] Casati, R., and A. Varzi, ``The structure of spatial localization,'' Philosophical Studies, vol. 82 (1996), pp. 205--39.
  • [10] Chang, C. C., and H. J. Keisler, Model Theory, 3d edition, North-Holland, Amsterdam, 1990.
  • [11] Clarke, B. L., ``A calculus of individuals based on `connection','' Notre Dame Journal of Formal Logic, vol. 23 (1981), pp. 204--18.
  • [12] Clarke, B. L., ``Individuals and points,'' Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 61--75.
  • [13] Davis, E., Representing and Acquiring Geographic Knowledge. Research Notes in Artificial Intelligence, Morgan Kaufmann, Los Altos, 1986.
  • [14] Davis, E., Representations of Commonsense Knowledge, Morgan Kaufmann, San Mateo, 1990.
  • [15] Dudek, G., P. Freedman, and S. Hadjres, ``Using local information in a non-local way for mapping graph-like worlds,'' pp. 1639--45 in Thirteenth International Joint Conference on Artificial Intelligence (IJCAI), Morgan Kaufmann, San Mateo, 1993.
  • [16] Goldblatt, R., Mathematics of Modality. CSLI Lecture Notes, 43, CSLI, Stanford, 1993.
  • [17] Gotts, N., J. Gooday, and A. Cohn, ``A connection based approach to commonsense topological description and reasoning,'' Monist, vol. 79 (1996), pp. 51--75.
  • [18] Haarslev, V., ``Formal semantics of visual languages using spatial reasoning,'' pp. 156--63 in IEEE Symposium on Visual Languages, IEEE Computer Society Press, Los Alamitos, 1995.
  • [19] Henkin, L., P. Suppes, and A. Tarski, editors, The Axiomatic Method, with Special Reference to Geometry and Physics, North-Holland, Amsterdam, 1959.
  • [20] Hodges, W., Model Theory. Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.
  • [21] Knight, J. F., A. Pillay, and C. Steinhorn, ``Definable sets in ordered structures II,'' Transactions of the American Mathematical Society, vol. 295 (1986), pp. 593--605.
  • [22] Koppelberg, S., Handbook of Boolean Algebras, vol. 1, North-Holland, Amsterdam, 1989.
  • [23] Kuipers, B., ``Modeling spatial knowledge,'' Cognitive Science, vol. 2 (1978), pp. 129--53.
  • [24] Lemon, O., ``Review of Logic and Visual Information by E. M. Hammer,'' Journal of Logic, Language, and Information, vol. 6 (1997), pp. 213--16.
  • [25] Lemon, O., and I. Pratt, ``Spatial logic and the complexity of diagrammatic reasoning,'' Machine Graphics and Vision. Special Issue on Diagrammatic Representation and Reasoning, vol. 6 (1997), pp. 89--108.
  • [26] Pillay, A., and C. Steinhorn, ``Definable sets in ordered structures I,'' Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565--92.
  • [27] Pratt, I., ``Map semantics,'' pp. 77--91 in Spatial Information Theory: A Theoretical Basis for GIS, edited by A. Frank and I. Campari, vol. 716 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1993.
  • [28] Pratt, I., and D. Schoop, ``A complete axiom system for polygonal mereotopology of the real plane,'' Technical Report UMCS97-2-2, University of Manchester, Manchester, 1997.
  • [29] Rescher, N., and J. Garson, ``Topological logic,'' The Journal of Symbolic Logic, vol. 33 (1968), pp. 537--48.
  • [30] Rescher, N., and A. Urquhart, Temporal Logic. Library of Exact Philosophy, vol. 3, Springer-Verlag, New York, 1971.
  • [31] Roeper, P., ``Region-based topology,'' Journal of Philosophical Logic, vol. 26 (1997) pp. 251--309.
  • [32] Segerberg, K., ``Two dimensional modal logic,'' Journal of Philosophical Logic, vol. 2 (1973) pp. 77--96.
  • [33] Shanahan, M., ``Default reasoning about spatial occupancy,'' Artificial Intelligence, vol. 74 (1995), pp. 147--63.
  • [34] Shehtman, V. B., ``Modal logics of domains on the real plane,'' Studia Logica, vol. 42 (1983) pp. 63--80.
  • [35] Tarski, A., ``Foundations of the geometry of solids,'' pp. 24--29 in Logic, Semantics, Metamathematics, Clarendon Press, Oxford, 1956.
  • [36] Varzi, A., ``Spatial reasoning in a holey world,'' pp. 326--36 in Advances in Artificial Intelligence, edited by P. Torasso, vol. 728 of Lecture Notes in Artificial Intelligence, Springer-Verlag, Berlin, 1993.
  • [37] Vieu, L., ``A logical framework for reasoning about space,'' pp. 25--35 in Spatial Information Theory: A Theoretical Basis for GIS, edited by A. Frank and I. Campari, vol. 716 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1993.
  • [38] von Wright, G. H., ``A modal logic of place,'' pp. 65--73 in The Philosophy of Nicholas Rescher: Discussion and Replies, edited by E. Sosa, Reidel, Dordrecht, 1979.
  • [39] Whitehead, A. N., Process and Reality, MacMillan, New York, 1929.