## Notre Dame Journal of Formal Logic

### On the Decidability of Axiomatized Mereotopological Theories

Hsing-chien Tsai

#### Abstract

The signature of the formal language of mereotopology contains two predicates $P$ and $C$, which stand for “being a part of” and “contact,” respectively. This paper will deal with the decidability issue of the mereotopological theories which can be formed by the axioms found in the literature. Three main results to be given are as follows: (1) all axiomatized mereotopological theories are separable; (2) all mereotopological theories up to $\mathbf{ACEMT}$, $\mathbf{SACEMT}$, or $\mathbf{SACEMT}^{\prime}$ are finitely inseparable; (3) all axiomatized mereotopological theories except $\mathbf{SAX}$, $\mathbf{SAX}^{\prime}$, or $\mathbf{S\overline{B}X}^{\prime}$, where $\mathbf{X}$ is strictly stronger than $\mathbf{CEMT}$, are undecidable. Then it can also be easily seen that all axiomatized mereotopological theories proved to be undecidable here are neither essentially undecidable nor strongly undecidable but are hereditarily undecidable. Result (3) will be shown by constructing strongly undecidable mereotopological structures based on two-dimensional Euclidean space, and it will be pointed out that the same construction cannot be carried through if the language is not rich enough.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 56, Number 2 (2015), 287-306.

Dates
First available in Project Euclid: 17 April 2015

https://projecteuclid.org/euclid.ndjfl/1429277352

Digital Object Identifier
doi:10.1215/00294527-2864307

Mathematical Reviews number (MathSciNet)
MR3337381

Zentralblatt MATH identifier
1334.03011

Subjects
Primary: 03C99: None of the above, but in this section
Secondary: 06F99: None of the above, but in this section

#### Citation

Tsai, Hsing-chien. On the Decidability of Axiomatized Mereotopological Theories. Notre Dame J. Formal Logic 56 (2015), no. 2, 287--306. doi:10.1215/00294527-2864307. https://projecteuclid.org/euclid.ndjfl/1429277352

#### References

• [1] Casati, R., and A. C. Varzi, Parts and Places: The Structures of Spatial Representation, MIT Press, Cambridge, Mass., 1999.
• [2] Ciraulo, F., M. E. Maietti, and P. Toto, “Constructive version of Boolean algebra,” Logic Journal of the IGPL, vol. 21 (2012), pp. 44–62.
• [3] Clarke, B. L., “A calculus of individuals based on ‘connection,”' Notre Dame Journal of Formal Logic, vol. 22 (1981), pp. 204–18.
• [4] Düntsch, I., and E. Orłowska, “A proof system for contact relation algebras,” Journal of Philosophical Logic, vol. 29 (2000), pp. 241–62.
• [5] Enderton, H. B., A Mathematical Introduction to Logic, 2nd ed., Academic Press, Burlington, Mass., 2001.
• [6] Leśniewski, S., “Foundations of the general theory of sets, I,” (in Polish) in S. Leśniewski, Collected Works, Vol. 1, edited by S. J. Surma, J. T. Srzednicki, D. I. Barnett, and V. F. Rickey, Kluwer, Dordrecht, 1992.
• [7] Libkin, L., Elements of Finite Model Theory, Springer, Berlin, 2004.
• [8] Monk, J. D., Mathematical Logic, vol. 37 of Graduate Texts in Mathematics, Springer, New York, 1976.
• [9] Munkres, J. R., Topology, 2nd ed., Prentice Hall, London, 2000.
• [10] Pratt-Hartmann, I., “First-order mereotopology,” pp. 13–97 in Handbook of Spatial Logics, edited by M. Aiello, I. Pratt-Hartmann, and J. van Benthem, Springer, Dordrecht, 2007.
• [11] Shoenfield, J. R., Mathematical Logic, Addison-Wesley, London, 1967.
• [12] Simons, P., Parts: A Study in Ontology, Clarendon Press, Oxford, 1987.
• [13] Tarski, A., A. Mostowski, and R. M. Robinson, Undecidable Theories, North-Holland, Amsterdam, 1953.
• [14] Tsai, H., “Decidability of mereological theories,” Logic and Logical Philosophy, vol. 18 (2009), pp. 45–63.
• [15] Tsai, H., “A comprehensive picture of the decidability of mereological theories,” Studia Logica, vol. 101 (2012), pp. 987–1012.
• [16] Tsai, H., “Decidability of general extensional mereology,” Studia Logica, vol. 101 (2013), pp. 619–36.
• [17] Tsai, H., “The logic and metaphysics of part-whole relations,” Ph.D. dissertation, Columbia University, New York, 2005.
• [18] Whitehead, A. N., Process and Reality, MacMillan, New York, 1929.