Expressiveness and definability in circumscription

Manuscrito 34 (1):233-266 (2011)

Authors
Ana Martins
University of São Paulo
Abstract
We investigate expressiveness and definability issues with respect to minimal models, particularly in the scope of Circumscription. First, we give a proof of the failure of the Löwenheim-Skolem Theorem for Circumscription. Then we show that, if the class of P; Z-minimal models of a first-order sentence is Δ-elementary, then it is elementary. That is, whenever the circumscription of a first-order sentence is equivalent to a first-order theory, then it is equivalent to a finitely axiomatizable one. This means that classes of models of circumscribed theories are either elementary or not Δ-elementary. Finally, using the previous result, we prove that, whenever a relation Pi is defined in the class of P; Z-minimal models of a first-order sentence Φ and whenever such class of P; Z-minimal models is Δ-elementary, then there is an explicit definition ψ for Pi such that the class of P; Z-minimal models of Φ is the class of models of Φ ∧ ψ. In order words, the circumscription of P in Φ with Z varied can be replaced by Φ plus this explicit definition ψ for Pi.
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DOI 10.1590/S0100-60452011000100011
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[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
Fixed Point Logics.Anuj Dawar & Yuri Gurevich - 2002 - Bulletin of Symbolic Logic 8 (1):65-88.
Decidability and Definability with Circumscription.John S. Schlipf - 1987 - Annals of Pure and Applied Logic 35 (2):173-191.
A Logic for Default Reasoning.Ray Reiter - 1980 - Artificial Intelligence 13:81-137.

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