In mathematical logic, a proof calculus corresponds to a family of formal systems that use a common style of formal inference for its inference rules. The specific inference rules of a member of such a family characterize the theory of a logic.
Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi can be used for radically different logics. For example, a paradigmatic case as the sequent calculus can be used to express the consequence relations of both intuitionistic logic and relevance logic. Thus, loosely speaking, a proof calculus is a template or design pattern, characterized by a certain style of formal inference, that may be specialized to produce specific formal systems, namely by specifying the actual inference rules for such a system. There is no consensus among logicians on how best to define the term.
Reference:
http://en.wikipedia.org/wiki/Proof_calculus
