In this paper we give an abstract representation of propositional Horn logic (finite or infinitary) in terms of set-valued functions |${\mathcal F} : \wp (\mathsf {PR}) \rightarrow \wp (\mathsf {PR})$|⁠; here PR is the collection of atomic propositions. We investigate the properties of set-valued functions and establish that they satisfy the axioms of weak lazy Kleene algebra. These axioms and other properties are used to prove uniform left interpolation for propositional Horn logic. Then we introduce thin set-valued functions, which enable us to prove polynomial interpolation for propositional Horn logic. For the infinite case, the Axiom of Choice is used.

We investigate fragments of intuitionistic propositional logic containing implication but not disjunction. These fragments are finite, but their size grows superexponentially with the number of generators. Exact models are used to characterize the fragments.

We provide a strongly complete infinitary proof system for hybrid logic. This proof system can be extended with countably many sequents. Thus, although these logics may be non-compact, strong completeness proofs are provided for infinitary hybrid versions of non-compact logics like ancestral logic and Segerberg's modal logic with the bounded chain condition. This extends the completeness result for hybrid logics by Gargov, Passy, and Tinchev.

The dynamic modal logic DML is presented, featuring actions that change the interpretation of a propositional variable or a modality. The semantics is defined both in terms of modal structures and of labelled transition systems (Kripke models). The extension µDML with recursively defined actions aims to unify and extend dynamic epistemic logics proposed by Plaza, Gerbrandy, Baltag, Van Ditmarsch and others. The main technical result is the completeness and decidability of µDML.

Abstract We show in this paper that all fragments of intuitionistic propositional logic based on a subset of the connectives ∧, ∨, →, ¬ satisfy interpolation. Fragments containing ↔ or ¬¬ are briefly considered.