1. This property is typically used when proving completeness via canonical model construction.
2. This entry will take neighborhood semantics and non-normal modal logics as the reference formalisms. The reason for this simply lies in the intuitive and straightforward connection between coalitional choices and subsets of outcomes in a relational structures. Everything in this entry could be done only using normal modal logics (and Krypke models) however we felt that the neighborhood approach could offer an added value in terms of clarity. For a mathematical treatment of the connection between the two approaches see for instance Hansen, Kupke, & Pacuit 2009 or the classic Kracht & Wolter 1999.
3. Let \(A\) be a set. A filter on \(A\) is a set \(X\subseteq 2^A\), such that:
- \(\emptyset \not\in X\)
- \(A_1 \in X\) and \(A_2 \in X\) imply that \(A_1 \cap A_2 \in X\)
- \(A_1 \in X\) implies that \(A_2\in X\) whenever \(A_1\subseteq A_2\).
A principal filter is a filter with a least element.
4. We adopt a rough version of hybrid logic here, starting from the assumption that there exists an atomic proposition naming each world. For a technical discussion on hybrid logic see Blackburn et al. 2001.
5. Using the same argument one can show that Playable Coalition Logic is sound and complete with respect to Truly Playable Coalition Models (see Blackburn et al. 2001 for the definition of soundness and completeness of a set of axioms with respect to a class of frames/models).