Skip to main content
Log in

Elementary Polyhedral Mereotopology

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

A region-based model of physical space is one in which the primitive spatial entities are regions, rather than points, and in which the primitive spatial relations take regions, rather than points, as their relata. Historically, the most intensively investigated region-based models are those whose primitive relations are topological in character; and the study of the topology of physical space from a region-based perspective has come to be called mereotopology. This paper concentrates on a mereotopological formalism originally introduced by Whitehead, which employs a single primitive binary relation C(x,y) (read: “x is in contact with y”). Thus, in this formalism, all topological facts supervene on facts about contact. Because of its potential application to theories of qualitative spatial reasoning, Whitehead's primitive has recently been the subject of scrutiny from within the Artificial Intelligence community. Various results regarding the mereotopology of the Euclidean plane have been obtained, settling such issues as expressive power, axiomatization and the existence of alternative models. The contribution of the present paper is to extend some of these results to the mereotopology of three-dimensional Euclidean space. Specifically, we show that, in a first-order setting where variables range over tame subsets of R 3, Whitehead's primitive is maximally expressive for topological relations; and we deduce a corollary constraining the possible region-based models of the space we inhabit.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Borgo, S., Guarino, N. and Masolo, C.: A pointless theory of space based on strong connection and congruence, in L. C. Aiello, J. Doyle, and S. C. Shapiro (eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR '96), 1996, pp. 220-229.

  2. Chang, C. C. and Keisler, H. J.: Model Theory, 3rd edn, North Holland, Amsterdam, 1990.

    Google Scholar 

  3. Clarke, B. L.: A calculus of individuals based on "connection", Notre Dame J. Formal Logic 22(3) (1981), 204-218.

    Google Scholar 

  4. Clarke, B. L.: Individuals and points, Notre Dame J. Formal Logic 26(1) (1985), 61-75.

    Google Scholar 

  5. de Laguna, T.: Point, line, and surface as sets of solids, J. Philos. 19 (1922), 449-461.

    Google Scholar 

  6. Dornheim, C.: Undecidability of plane polygonal mereotopology, in A. G. Cohn, L. K. Schubert, and S. C. Schubert (eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the Sixth International Conference (KR '98), 1998, pp. 342-353.

  7. Galton, A.: Qualitative Spatial Change, Oxford University Press, Oxford, 2000.

    Google Scholar 

  8. Gotts, N. M., Gooday, J. M. and Cohn, A. G.: A connection based approach to commonsense topological description and reasoning, Monist 79(1) (1996), 51-75.

    Google Scholar 

  9. Hodges, W.: Model Theory, Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  10. Koppelberg, S.: Handbook of Boolean Algebras, Vol. 1, North-Holland, Amsterdam, 1989.

    Google Scholar 

  11. Massey, W. S.: Algebraic Topology: An Introduction, Harcourt, Brace & World, New York, 1967.

    Google Scholar 

  12. Newman, M. H. A.: Elements of the Topology of Plane Sets of Points, Cambridge University Press, Cambridge, 1964.

    Google Scholar 

  13. Papadimitriou, C. H., Suciu, D. and Vianu, V.: Topological queries in spatial databases, in Proceedings of PODS'96, 1996, pp. 81-92.

  14. Pratt, I. and Lemon, O.: Ontologies for plane, polygonal mereotopology, Notre Dame J. Formal Logic 38(2) (1997), 225-245.

    Google Scholar 

  15. Pratt, I. and Schoop, D.: A complete axiom system for polygonal mereotopology of the real plane, J. Philos. Logic 27(6) (1998), 621-658.

    Google Scholar 

  16. Pratt, I. and Schoop, D.: Expressivity in polygonal, plane mereotopology, J. Symbolic Logic 65(2) (2000), 822-838.

    Google Scholar 

  17. Randell, D. A., Cui, Z. and Cohn, A. G.: A spatial logic based on regions and connection, in B. Nebel, C. Rich, and W. Swartout (eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the Third International Conference (KR '92), 1992, pp. 165-176.

  18. Simons, P.: Parts: A Study in Ontology, Clarendon Press, Oxford, 1987.

    Google Scholar 

  19. van den Dries, L.: O-minimal structures, in W. Hodges, M. Hyland, C. Steinhorn, and J. Truss (eds.), Logic: From Foundations to Applications, Oxford University Press, Oxford, 1996, pp. 137-186.

    Google Scholar 

  20. Whitehead, A. N.: Process and Reality, Macmillan, New York, 1929.

    Google Scholar 

  21. Wilson, R. J.: Introduction to Graph Theory, Longman, London, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pratt-Hartmann, I., Schoop, D. Elementary Polyhedral Mereotopology. Journal of Philosophical Logic 31, 469–498 (2002). https://doi.org/10.1023/A:1020184007550

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020184007550

Navigation