# Proof calculus

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 are under-determining 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:

- The class of Hilbert systems, of which the most famous example is the 1928 Hilbert-Ackermann system of first-order logic;
- Gerhard Gentzen's calculus of natural deduction, which is the first formalism of structural proof theory, and which is the cornerstone of the formulae-as-types correspondence relating logic to functional programming;
- Gentzen's sequent calculus, which is the most studied formalism of structural proof theory.

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

- Aristotle's syllogistic calculus, presented in the
*Organon*, readily admits formalisation. There is still some modern interest in syllogistic, carried out under the aegis of term logic. - Gottlob Frege's two-dimensional notation of the
*Begriffsschrift*is usually regarded as introducing the modern concept of quantifier to logic. - C.S. Peirce's existential graph might easily have been seminal, had history worked out differently.

Modern research in logic teems with rival proof calculi:

- Several systems have been proposed which replace the usual textual syntax with some graphical syntax. Proof nets and cirquent calculus are among such systems.
- Recently, many logicians interested in structural proof theory have proposed calculi with deep inference, for instance display logic, hypersequents, the calculus of structures, and bunched implication.