Complete Additivity and Modal Incompleteness

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem, `Syntactic aspects of modal incompleteness theorems,' and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness

Model Theory and Proof Theory of Coalgebraic Predicate Logic

We propose a generalization of first-order logic originating in a neglectedwork by C.C. Chang: a natural and generic correspondence language for any typesof structures which can be recast as Set-coalgebras. We discuss axiomatizationand completeness results for several natural classes of such logics. Moreover,we show that an entirely general completeness result is not possible. We studythe expressive power of our language, both in comparison with coalgebraichybrid logics and with existing first-order proposals for special classes ofSet-coalgebras (apart from relational structures, also neighbourhood frames andtopological spaces). Basic model-theoretic constructions and results, inparticular ultraproducts, obtain for the two classes that allowcompleteness---and in some cases beyond that. Finally, we discuss a basicsequent system, for which we establish a syntactic cut-elimination result

Model Theory and Proof Theory of Coalgebraic Predicate Logic

We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and completeness results for several natural classes of such logics. Moreover, we show that an entirely general completeness result is not possible. We study the expressive power of our language, both in comparison with coalgebraic hybrid logics and with existing first-order proposals for special classes of Set-coalgebras (apart from relational structures, also neighbourhood frames and topological spaces). Basic model-theoretic constructions and results, in particular ultraproducts, obtain for the two classes that allow completeness---and in some cases beyond that. Finally, we discuss a basic sequent system, for which we establish a syntactic cut-elimination result