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1. This property is typically used when proving completeness viacanonical model construction.

2. This entry will take neighborhood semantics and non-normal modallogics as the reference formalisms. The reason for this simply lies inthe intuitive and straightforward connection between coalitionalchoices and subsets of outcomes in a relational structures. Everythingin this entry could be done only using normal modal logics (and Kripkemodels) however we felt that the neighborhood approach could offer anadded value in terms of clarity. For a mathematical treatment of theconnection between the two approaches see for instance Hansen, Kupke,& Pacuit 2009 or the classic Kracht & Wolter 1999.

3. Let \(A\) be a set. Afilter on \(A\) is a set\(X\subseteq 2^A\), such that:

1. \(\emptyset \not\in X\)
2. \(A_1 \in X\) and \(A_2 \in X\) imply that \(A_1 \cap A_2 \inX\)
3. \(A_1 \in X\) implies that \(A_2\in X\) whenever \(A_1\subseteqA_2\).

Aprincipal filter is a filter with a leastelement.

4. We adopt a rough version of hybrid logic here, starting from theassumption that there exists an atomic propositionnamingeach world. For a technical discussion on hybrid logic see Blackburnet al. 2001.

5. Using the same argument one can show that Playable Coalition Logic issound and complete with respect to Truly Playable Coalition Models(see Blackburn et al. 2001 for the definition of soundness andcompleteness of a set of axioms with respect to a class offrames/models).