Complete Additivity and Modal Incompleteness
Holliday WH, Litak T (2019)
Publication Language: English
Publication Type: Journal article
Publication year: 2019
Journal
Book Volume: 12
Pages Range: 487-535
Journal Issue: 3
DOI: 10.1017/S1755020317000259
Abstract
In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem (1979), "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, gamma-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's article 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 gamma-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 gamma-complete. After these results, we generalize the Blok Dichotomy (Blok, 1978) to degrees of gamma-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness.
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APA:
Holliday, W.H., & Litak, T. (2019). Complete Additivity and Modal Incompleteness. Review of Symbolic Logic, 12(3), 487-535. https://doi.org/10.1017/S1755020317000259
MLA:
Holliday, Wesley H., and Tadeusz Litak. "Complete Additivity and Modal Incompleteness." Review of Symbolic Logic 12.3 (2019): 487-535.
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