Henkin and Hybrid Logic

Abstract

Leon Henkin was not a modal logician, but there is a branch of modal logic that has been deeply influenced by his work. That branch is hybrid logic, a family of logics that extend orthodox modal logic with special propositional symbols (called nominals) that name worlds. This paper explains why Henkin’s techniques are so important in hybrid logic. We do so by proving a completeness result for a hybrid type theory called HTT, probably the strongest hybrid logic that has yet been explored. Our completeness result builds on earlier work with a system called BHTT, or basic hybrid type theory, and draws heavily on Henkin’s work. We prove our Lindenbaum Lemma using a Henkin-inspired strategy, witnessing ◊-prefixed expressions with nominals. Our use of general interpretations and the construction of the type hierarchy is (almost) pure Henkin. Finally, the generality of our completeness result is due to the first-order perspective, which lies at the heart of both Henkin’s best known work and hybrid logic.