We investigate the relationship between two well known formalizations of context: Propositional Logic of Context (PLC) 4, and Local Models Semantics (LMS) 11. We start with a summary of the desiderata for a logic of context, mainly inspired by McCarthy's paper on generality in AI 15 and his notes on formalizing context 16. We briefly present LMS, and its axiomatization using MultiContext Systems (MCS) 14. Then we present a revised (and simplified) version of PLC, and we show that local vocabularies - as they defined in 4 - are inessential in the semantics of PLC. The central part of the paper is the definition of a class of LMS (and its axiomatization in MCS, called MMCC), which is provably equivalent to the axiomatization of PLC as described in 4. Finally, we go back to the general desiderata and discuss in detail how the two formalisms fulfill (or do not fulfill) each of them.