Expressiveness and definability in circumscription

• Autor(a) principal: Ferreira,Francicleber Martins
• Data de Publicação: 2011
• Outros Autores: Martins,Ana Teresa
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Manuscrito (Online)
• Texto Completo: http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0100-60452011000100011
• topic: Minimal models; Circumscripition; Expressiveness; Definability

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.