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 ad- ditive 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 [Japaridze, 1988, Boolos, 1993]. 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 [Blok, 1978] to degrees of V-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.