 \(\Gamma \nc[\Xi] \phi\) if and only if \(\Xi' \cup \Gamma \vDash \phi\) for all \(\Xi' \subseteq \Xi\) that are maximally consistent with \(\Gamma\) (i.e., \(\Xi' \cup \Gamma\) is consistent and there is no \(\Xi'' \subseteq \Xi\) for which \(\Xi' \subset \Xi''\) and \(\Xi'' \cup \Gamma\) is consistent). 