Proof calculus

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. ==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 syllogisms, carried out under the aegis of term logic. • Gottlob Frege's two-dimensional notation of the Begriffsschrift (1879) is usually regarded as introducing the modern concept of quantifier to logic. • C.S. Peirce's existential graph easily might have been seminal, had history worked out differently. Modern research in logic teems with rival proof calculi: • Several systems have been proposed that 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. == See also ==