**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.
- 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.

Read more about this topic: Proof Calculus

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