Kripke completeness of infinitary predicate multimodal logics. (English) Zbl 1007.03016

Tanaka, Yoshihito

Notre Dame J. Formal Logic 40, No. 3, 326-340 (1999).

Summary: Kripke completeness of some infinitary predicate modal logics is presented. More precisely, we prove that if a normal modal logic \({\mathbf L}\) above \({\mathbf K}\) is \({\mathcal D}\)-persistent and universal, the infinitary and predicate extension of \({\mathbf L}\) with \(\text{BF}_{\omega_1}\) and BF is Kripke complete, where \(\text{BF}_{\omega_1}\) and BF denote the formulas \(\bigwedge_{i\in\omega} \square p_i\supset\square \bigwedge_{i\in \omega}p_i\) and \(\forall x\square \varphi\supset \square\forall x\varphi\), respectively. The results include the completeness of extensions of standard modal logics such as \({\mathbf K}\), and its extensions by the schemata T, B, 4, 5, D, and their combinations. The proof is obtained by extending the correspondence between the representation of modal algebras and the completeness of propositional modal logic to the infinite.

MSC:

03B45	Modal logic (including the logic of norms)
03G25	Other algebras related to logic

Keywords:

Kripke completeness; infinitary predicate modal logics; representation of modal algebras

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