# Proof calculus

In mathematical logic, a proof calculus or a proof system is built to prove statements.

## Overview

A proof system includes the components:

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.

## Examples of proof calculi

The most widely known proof calculi are those classical calculi that are still in widespread use:

Many other proof calculi were, or might have been, seminal, but are not widely used today.

Modern research in logic teems with rival proof calculi: