Complete Additivity and Modal Incompleteness

Abstract

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper 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, 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 Vcomplete. 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.

On the Logic of Belief and Propositional Quantification

“…Even with the addition of dynamic operators as in [11], semantics based on possible-worlds is still largely appropriate, and many such extensions start with possible-world semantics. While it is well known that there are Kripke incomplete logics [12], meaning that no classes of Kripke frames can validate precisely the theorems in those logics, perhaps, when studying belief operators, Kripke frames are always enough for us, and there is nothing that can "banish" us from, to borrow from David Lewis again, "a doxastic logician's paradise"?…”

“…Definition 4. 12 We say that p is restricted by a formula μ in ϕ just in case μ is substitutable for p in ϕ and ∀p(ϕ(p) ↔ ϕ(p ∧ μ)). Lemma 4.11 ∃p(ϕ ∧ ψ) is provably equivalent to ∃pϕ ∧ ∃pψ, if there are formulas μ, ν, such that p in ϕ is restricted by μ, p in ψ is restricted by ν, and -¬(μ ∧ ν) is provable.…”

Choice-Free de Vries Duality

“…By Theorem 5.10 and Remark 5.11 in [5], de Vries algebras provide a sound and complete algebraic semantics for S 2 IC, and this result can be obtained choice-free. Combining this result with Theorem 5.4, it follows that dV -spaces also provide a choice-free sound and complete semantics for S 2 IC.✷ Since dV -spaces constitute a choice-free, filter-based representation of de Vries algebras, we may think of our choice-free de Vries duality as providing a possibility semantics for the logic of region-based theories of space, just as the choice-free Stone duality through U V -spaces serves as a foundation for possibility semantics for classical and modal propositional logic [12,13,14].…”

“…It is based on the simple but powerful idea that the appeal to (BPI) could be eliminated by working with a partially-ordered set of filters rather than a set of ultrafilters and by viewing these filters as partial approximations of a classical point. This approach has strong ties to both possibility semantics in modal logic [12,13,14] and the Vietoris functor on Stone spaces [28] and provides a semi-pointfree approach, i.e., both spatial and choice-free, to the representation of algebraic objects in semi-constructive mathematics, i.e., mathematics carried out in ZF + DC [21,26].…”

Carnap’s Problem for Modal Logic

“…Here L • and L • are the two trivial normal modal logics: by a theorem of [14], every consistent normal modal logic is a sublogic of one of these, and the only proper extension of each is the inconsistent modal logic L. 10 Else is the 'logic of elsewhere' from [24]: 2ϕ is true at w in this logic iff ϕ is true everywhere else. (The axiomatization above is from [19].…”