In mathematical logic, a proof calculus or a proof system is built to prove statements.
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👉 Proof calculus in the context of Hilbert system
In logic, more specifically proof theory, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style system, Hilbert-style proof system, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of formal proof system attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well.
It is defined as a deductive system that generates theorems from axioms and inference rules, especially if the only postulated inference rule is modus ponens. Every Hilbert system is an axiomatic system, which is used by many authors as a sole less specific term to declare their Hilbert systems, without mentioning any more specific terms. In this context, "Hilbert systems" are contrasted with natural deduction systems, in which no axioms are used, only inference rules.
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Proof calculus in the context of Sequent calculus
In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than David Hilbert's earlier style of formal logic, in which every line was an unconditional tautology. More subtle distinctions may exist; for example, propositions may implicitly depend upon non-logical axioms. In that case, sequents signify conditional theorems of a first-order theory rather than conditional tautologies.
Sequent calculus is one of several extant styles of proof calculus for expressing line-by-line logical arguments.
Proof calculus in the context of Natural deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.