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The modal logic of affine planes is not finitely axiomatisable

Published online by Cambridge University Press:  12 March 2014

Ian Hodkinson  and
Altaf Hussain
Ian Hodkinson
Affiliation:
Department of Computing, Imperial College London, London, SW7 2AZ, UK, E-mail: imh@doc.ic.ac.uk, URL: http://www.doc.ic.ac.uk/~imh
Altaf Hussain
Affiliation:
Department of Computing, Imperial College London, London, SW7 2AZ, UK, E-mail: E-mail: altaf.a.hussain@googlemail.com, URL: http://www.doc.ic.ac.uk/~imh
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Abstract

We consider a modal language for affine planes, with two sorts of formulas (for points and lines) and three modal boxes. To evaluate formulas, we regard an affine plane as a Kripke frame with two sorts (points and lines) and three modal accessibility relations, namely the point-line and line-point incidence relations and the parallelism relation between lines. We show that the modal logic of affine planes in this language is not finitely axiomatisable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 3 , September 2008 , pp. 940 - 952
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[1]Aiello, M., Pratt-Hartmann, I., and van Benthem, J. (editors), Handbook of spatial logics, Springer-Verlag, 2007.CrossRefGoogle Scholar
[2]Balbiani, P., del Cerro, L. Fariñas, Tinchev, T., and Vakarelov, D., Modal logics for incidence geometries, Journal of Logic and Computation, vol. 7 (1997), no. 1, pp. 59–78.CrossRefGoogle Scholar
[3]Balbiani, P. and Goranko, V., Modal logics for parallelism, orthogonality, and affine geometries, Journal of Applied Non-Classical Logics, vol. 12 (2002), no. 3-4, pp. 365–398.CrossRefGoogle Scholar
[4]Blackburn, P., de Rijke, M., and Venema, Y., Modal logic, Tracts in Theoretical Computer Science, Cambridge University Press, 2001.CrossRefGoogle Scholar
[5]Bruck, R. H. and Ryser, H. J., The nonexistence of certain finite projective planes, Canadian Journal of Mathematics, vol. 1 (1949), pp. 88–93.CrossRefGoogle Scholar
[6]Chagrov, A. and Zakharyaschev, M., Modal logic, Clarendon Press, 1997.CrossRefGoogle Scholar
[7]Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, Clarendon Press, 1979, 5th edition.Google Scholar
[8]Hughes, D. R. and Piper, F. C., Projective planes, Graduate Texts in Mathematics, Springer-Verlag, 1973.Google Scholar
[9]Hussain, A., Some applications for extended modal logics, Ph.D. thesis, University of London, 2007.Google Scholar
[10]Monk, J. D., On representable relation algebras, Michigan Mathematics Journal, vol. 11 (1964), pp. 207–210.CrossRefGoogle Scholar
[11]Sierpiński, W., Elementary theory of numbers, Państwowe Wydawnictwo Naukowe, Warszawa, 1964.Google Scholar
[12]Venema, Y., Cylindric modal logic, this Journal, vol. 60 (1995), pp. 591–623.Google Scholar
[13]Venema, Y., Points, lines and diamonds: a two-sorted modal logic for projective planes, Journal of Logic and Computation, vol. 9 (1999), pp. 601–621.CrossRefGoogle Scholar