Abstract
Craig's interpolation theorem fails for the prepositional logicsE of entailment,R of relevant implication andT of ticket entailment, as well as in a large class of related logics. This result is proved by a geometrical construction, using the fact that a non-Arguesian projective plane cannot be imbedded in a three-dimensional projective space. The same construction shows failure of the amalgamation property in many varieties of distributive lattice-ordered monoids.
Similar content being viewed by others
References
Anderson, A. R. and Belnap, N. D.Entailment. Vol. 1, Princeton University Press, New Jersey, 1975.
Birkhoff, G.Lattice Theory. Third edition. Amer. Math. Soc., Providence, R.I. 1967.
Blumenthal, L. M.A Modern View of Geometry. Freeman, San Francisco, 1961. Reprinted Dover, New York, 1980.
Brady, R. T. ‘Natural deduction systems for some quantified relevant logics’,Logique et Analyse 27 (1984), pp. 355–377.
Comer, S. D. ‘Classes without the amalgamation property’,Pacific J. Math. 28 (1969), pp. 309–318.
Garner, L. E.An Outline of Projective Geometry. North Holland, New York and Oxford, 1981.
Grätzer, G.General Lattice Theory. Academic Press, New York and San Francisco, 1978.
Grätzer, G., Jónsson, B., and Lakser, H. ‘The amalgamation property in equational classes of modular lattices’,Pacific J. Math. 45 (1973), pp. 507–524.
Hall, M. Jr.Combinatorial Theory. Second edition. John Wiley, New York, 1986.
Jónsson, B. ‘Representation of modular lattices and of relation algebras’,Trans. Amer. Math. Soc. 92 (1959), pp. 449–464.
Lyndon, R. C. ‘Relation algebras and projective geometries’,Michigan Math. J. 8 (1961), pp. 21–28.
McKenzie, R.The representation of relation algebras. Doctoral dissertation, University of Colorado, 1966.
McRobbie, M. A. ‘Interpolation theorems for some first-order distribution-free relevant logics’ (Abstract),J. Symbolic Logic (1983), pp. 522–523.
Meyer, R. K. ‘Relevantly interpolating inRM’. Research Paper No. 9, Logic Group, Department of Philosophy, R.S.S.S., Australian National University, 1980.
Von Neumann,J. Continuous Geometries. Princeton University Press, Princeton, New Jersey, 1960.
Robinson, A. ‘A result on consistency and its application to the theory of definition’,Indag. Math. 18 (1956), pp. 47–58.Selected Papers, Vol. 1, pp. 87–98, Yale U.P., New Haven and London, 1979.
Routley, R. and Meyer, R. K. ‘The semantics of entailment I’, inTruth, Syntax and Semantics, ed. H. Leblanc, pp. 194–243, North-Holland, Amsterdam, 1973.
Stevenson, F. W.Projective Planes. W. H. Freeman, San Francisco, 1972.
Urquhart, A. ‘The undecidability of entailment and relevant implication’,J. Symbolic Logic 49 (1984), pp. 1059–1073.
Urquhart, A. ‘Relevant implication and projective geometry’,Logique et Analyse, 103–104 (September 1983), pp. 345–357.
Veblen, O. and Young, J. W.Projective Geometry, Volume 1, Ginn, Boston, 1910.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Urquhart, A. Failure of interpolation in relevant logics. J Philos Logic 22, 449–479 (1993). https://doi.org/10.1007/BF01349560
Issue Date:
DOI: https://doi.org/10.1007/BF01349560