Proof calculus

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

Overview

A proof system includes the components:[1][2]

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.[2]

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.

  • 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:

See also

References

  1. Anita Wasilewska. "General proof systems" (PDF).
  2. "Definition:Proof System - ProofWiki". proofwiki.org. Retrieved 2023-10-16.
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