Reference: Makarios, S. A Model Theory for a Quantified Generalized Logic of Contexts. 2006.
Abstract: We investigate the formal semantics of a quantified version of a generalization of the logic of contexts that uses the operator $ist(c,\phi)$ introduced by Guha [] and consonant to the notion of context logic as proposed by McCarthy []. We offer a full, rigorous, model-theoretic treatment of the unrestricted quantified context-logic formalism, filling what we perceive as a gap in the formal foundation of the logic of context.
Notes:
Full paper available as pdf.