Abstract

The chapter is divided in two parts. The first part gives an introduction to issues in quantified modal logic. We provide an overview of recent work in QML and we presuppose the use of a relational semantics. We discuss models for constant domains, increasing domains and varying domains and present axiomatizations for the corresponding logics. We also discuss philosophical issues related to the interpretation of the quantifiers, terms and identity and we present a first-order quantified intensional logic. A crucial issue in QML is to provide a unified semantic framework capable of accommodating a large class of logical systems including both normal and non-normal systems. The second part of the article focuses on alternatives to the standard relational semantics for modalities deriving from the work of Kripke. In particular we give an overview of recent work in neighborhood semantics and we offer a general completeness result for the class of first order classical modal logics in terms of general frames with constant domains. We also consider some extensions of current work dealing with identity.