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• W3104000793 abstract "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." @default.
• W3104000793 created "2020-11-23" @default.
• W3104000793 creator A5058807832 @default.
• W3104000793 creator A5088813725 @default.
• W3104000793 date "2019-07-04" @default.
• W3104000793 modified "2023-09-26" @default.
• W3104000793 title "COMPLETE ADDITIVITY AND MODAL INCOMPLETENESS" @default.
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• W3104000793 doi "https://doi.org/10.1017/s1755020317000259" @default.
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