Further remarks on the dual negation in team logics

The dual or game-theoretical negation $\lnot$ of independence-friendly logic (IF) and dependence logic (D) exhibits an extreme degree of semantic indeterminacy in that for any pair of sentences $\phi$ and $\psi$ of IF/D, if $\phi$ and $\psi$ are incompatible in the sense that they share no models, there is a sentence $\theta$ of IF/D such that $\phi\equiv \theta$ and $\psi\equiv \lnot \theta$ (as shown originally by Burgess in the equivalent context of the prenex fragment of Henkin quantifier logic). We show that by adjusting the notion of incompatibility employed, analogues of this result can be established for a number of modal and propositional team logics, including Aloni's bilateral state-based modal logic, Hawke and Steinert-Threlkeld's semantic expressivist logic for epistemic modals, as well as propositional dependence logic with the dual negation. Together with its converse, a result of this type can be seen as an expressive completeness theorem with respect to the relevant incompatibility notion; we formulate a notion of expressive completeness for pairs of properties to make this precise.