Proof calculus
formal language that specifies allowed ways to prove a statement
In mathematical logic, a proof calculus or a proof system is built to prove statements.
Overview
A proof system includes the components:
Formal language: The set L of formulas admitted by the system, for example, propositional logic or first-order logic.
Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems.
Axioms: Formulas in L assumed to be valid. All theorems are derived from axioms.
A formal proof of a well-formed formula in a proof system is a set of axioms and rules of inference of the proof system that infers that the well-formed formula is a theorem of the proof system.
Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics. For example, a paradigmatic case is the sequent calculus, which 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.
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