On the difference between bridge rules and lifting axioms

In a previous paper, we proposed a first formal and conceptual comparison between the two important formalizations of context in AI: Propositional Logic of Context (PLC) and Local Models Semantics/MultiContext Systems (LMS/PLC) The result was that LMS/MCS is at least as general as PLC, as it can be embedded into a particular class of MCS, called MPLC. In this paper we go beyond that result, and prove that, under some important restrictions (including the hypothesis that each context has finite and homogeneous propositional languages), MCS/LMS can be embedded in PLC with generic axioms. To prove this theorem, we prove that MCS cannot be embedded in PLC using only lifting axioms to encode bridge rules. This is an important result for a general theory of context and contextual reasoning, as it proves that lifting axioms and entering context are not enough to capture all forms of contextual reasoning that can be captured via bridge rules in LMS/PL