Derivability in Locally Quantified Modal Logics via Translation in Set Theory

Abstract

Two of the most active research areas in automated deduction in modal logic are the use of translation methods to reduce its derivability problem to that of classical logic and the extension of existing automated reasoning techniques, developed initially for the propositional case, to first-order modal logics. This paper addresses both issues by extending the translation method for propositional modal logics known as □ -as-Pow (read “box-as-powerset”) to a widely used class of first-order modal logics, namely, the class of locally quantified modal logics. To do this, we prove a more general result that allows us to separate (classical) first-order from modal (propositional) reasoning. Our translation can be seen as an example application of this result, in both definition and proof of adequateness.

The first author was partially supported by the MURST project Software Architectures and Languages to Coordinate Distributed Mobile Components. The second author was partially supported by the CNR project log(S.E.T.A.): Specifiche Eseguibili e Teorie degli Aggregati. Metodologie e Piattaforme.