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Journal of Logic and Computation 2001 11(1):71-83; doi:10.1093/logcom/11.1.71
© 2001 by Oxford University Press
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Preferential Logics are X-logics

Lionel Forget1, Vincent Risch2 and Pierre Siegel3

1 LIM - ESA CNRS 6077, Centre de Mathématiques et d'Informatique, 39, rue Joliot-Curie, 13453 Marseille Cedex 13 France. E-mail: forget{at}lim.univ-mrs.fr 2 LIM - ESA CNRS 6077, Parc Scientifique et Technologique de Luminy, 163 avenue de Luminy - Case 901, 13288 Marseille Cedex 09 France. E-mail: risch{at}lim.univ-mrs.fr 3 LIM - ESA CNRS 6077, Centre de Mathématiques et d'Informatique, 39, rue Joliot-Curie, 13453 Marseille Cedex 13 France. E-mail: siegel{at}lim.univ-mrs.fr

This paper shows how to define nonmonotonic logics from any classical logics L and any set X of formulas of L. In this context, the nonmonotonic inference relation {vdash}X is defined by A {vdash}X B if every classical theorem of A {cup} B which is in X is a theorem of A. The properties of the relation {vdash} X are studied. We show, in particular, that the elementary properties (supraclassicity, or, left logical equivalence, cut, etc.) are verified for any X. Moreover, we prove that cumulativity is verified if the set of formulas of the language, which are not in X, is deductively closed. Then we prove a representation theorem, i.e. in the finite case every preferential nonmonotonic logic is an X -logic. We also study a particular form of the set X for general propositional circumscription.

Keywords: Nonmonotonic logic; preferential model approach; representation theorem; circumscription

Accepted 28 July 1998.


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