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 Lowenheim-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.
Highlights
In practical situations, people reason and act without having complete or sufficient knowledge about the situation that they are dealingManuscrito — Rev
Whenever a circumscribed theory is equivalent to a first-order theory, it is equivalent to a finitely axiomatizable one
It follows that classes of minimal models of first-order sentences either are elementary or are not even ∆-elementary
Summary
People reason and act without having complete or sufficient knowledge about the situation that they are dealing. Besides the ability of handling the lack of information and the use of general and uncertain knowledge, another feature of a system that models practical reasoning is to be able to backtrack and drop some previous conclusions inferred from uncertain assumptions in order to stay consistent with the new information obtained. In van Benthem (2005), van Benthem introduced the MIN(FO) logic that allows one to define the minimal relation which satisfies certain syntactically defined sort of formulas: the PIA-Conditions. Whenever a circumscribed theory is equivalent to a first-order theory, it is equivalent to a finitely axiomatizable one It follows that classes of minimal models of first-order sentences either are elementary or are not even ∆-elementary. We will introduce the main concepts of Minimal Model Theory and Circumscription that will be used in Sections 3 and 4
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