Complete Additivity and Modal Incompleteness

TL;DR

This paper investigates the limitations of complete additivity in modal logic, demonstrating that not all normal modal logics can be characterized by completely additive modal algebras, and explores implications for modal incompleteness.

Contribution

It proves that the open question about characterizing all normal modal logics by V-BAOs has a negative answer and extends the understanding of V-incompleteness and its undecidability.

Findings

Certain modal logics are incomplete with respect to V-BAOs.

It is undecidable whether a given logic is V-complete.

The results extend the Blok Dichotomy to degrees of V-incompleteness.

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.