Expressiveness and Definability in Circumscription

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 ∆elementary, then it is elementary. That is, whenever the circumscription of a firstorder 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 ∆-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 firstorder sentence φ and whenever such class of P ;Z-minimal models is ∆-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.