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# A First Order Nonmonotonic Extension of Constructive Logic

David Pearce and Agustín Valverde
Studia Logica: An International Journal for Symbolic Logic
Vol. 80, No. 2/3, Negation in Constructive Logic (Jul. - Aug., 2005), pp. 321-346
Stable URL: http://www.jstor.org/stable/20016720
Page Count: 26
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## Abstract

Certain extensions of Nelson's constructive logic N with strong negation have recently become important in artificial 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 first-order logic as a basis for defining 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 first-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 $\text{QN}_{5}^{c}$, as a basis for defining a first-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of $\text{QN}_{5}^{c}$, including Skolem forms and Herbrand theorems and Interpolation, and show that the first-oder version of equilibrium logic can be used as a foundation for answer set inference.

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