Editing Proof calculus (section)

==Overview==
A proof system includes the components:<ref>{{Cite web| url=http://www3.cs.stonybrook.edu/~cse541/chapter7.pdf| title=General proof systems |author=Anita Wasilewska}}</ref><ref name=":0">{{Cite web |title=Definition:Proof System - ProofWiki |url=https://proofwiki.org/wiki/Definition:Proof_System |access-date=2023-10-16 |website=proofwiki.org}}</ref>
* [[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 [[Theorem|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 proof system that infers that the well-formed formula is a theorem of proof system.<ref name=":0" />

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 relation]]s 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.