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A First Order Nonmonotonic Extension of Constructive Logic

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Abstract

Certain extensions of Nelson's constructive logic N with strong negation have recently become important in arti.cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's .rst-order logic as a basis for de.ning nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for .rst-order here-and-there logics, and their minimal strong negation extensions, for both constant and varying domains. We choose the constant domain version, which we denote by QNc5, as a basis for de.ning a .rst-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of QNc5, including Skolem forms and Herbrand theorems and Interpolation, and show that the .rst-oder version of equilibrium logic can be used as a foundation for answer set inference.

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Authors and Affiliations

  1. Dept. de Informatica, Estadística y Telemática, Universidad Rey Juan Carlos, C/ Tulipán, 28933, Móstoles, Madrid, Spain

    David Pearce

  2. Department of Applied Mathematics, University of Málaga, Bvd. Louis Pasteur s/n, E.T.S.I. Informática, 29071, Málaga, Spain

    Agustín Valverde

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  1. David Pearce
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  2. Agustín Valverde
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Correspondence to David Pearce.

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Pearce, D., Valverde, A. A First Order Nonmonotonic Extension of Constructive Logic. Stud Logica 80, 321–346 (2005). https://doi.org/10.1007/s11225-005-8473-8

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