Proofs in Context

When reasoning about reasoning, one often needs to carry out some proof steps within a diierent theory from the current one and then to lift the conclusions back into the original theory. The deductive rules required to carry out these steps involve either reeec-tion principles or some other notion of proof in context. Reeection principles have to be carefully restricted in order to avoid paradoxes. Such restrictions however limit significantly their usefulness and also defeat intuition while building proofs in context. Standard formulations of the reeection rules also assume explicit knowledge of the theory one reasons about. This is not always the case for theories representing agents or for theories which refer to each other, as those required for expressing common knowledge. A careful analysis of proofs dealing with multiple theories which appeared in recent literature reveals steps which are not properly accounted for, improper use of lifting, extra logical constructions or unnecessary assumptions: problems often due to the involved mechanisms of reeection. We show that proof in context can replace the most common uses of reeec-tion principles. Proofs can be carried out by switching to a context and reasoning within it. Context switching does not correspond to reeection or reiication but involves a further level of nesting of theory within another theory. We introduce a generalised rule for proof in context and a convenient notation to express nesting of contexts, which allow us to carry out reasoning about reasoning in a safe and natural way.