Sequent calculus

Sequent calculus is a formal proof system whose judgments, called sequents, express entailment between collections of formulas. Its inference rules derive sequents from simpler ones.

Sequent: A sequent is a formal judgment expressing that a context of assumptions entails a conclusion. Sequents are the judgments that sequent calculus derives.
Classical logic: Classical logic is a system of reasoning that accepts principles such as excluded middle and unrestricted double negation elimination. Classical sequent calculus formalizes this familiar account of entailment.
Natural deduction: Natural deduction is a proof system organized around introduction and elimination rules for logical connectives. It emphasizes reasoning from and discharging assumptions rather than manipulating sequents.
Proof theory: Proof theory studies formal proofs as mathematical objects, including their structure, transformations, and strength. Sequent calculi are central tools for analyzing proof structure.
Gerhard Gentzen: Gerhard Gentzen was a German mathematician who introduced natural deduction and sequent calculus in the 1930s. Gentzen developed sequent calculus as part of his foundational work on proofs.
Inference rule: An inference rule specifies how judgments may be derived from previously established judgments. Each calculus step applies a rule to derive a new sequent.
Intuitionistic logic: Intuitionistic logic treats a proof of a proposition as evidence for it and rejects unrestricted excluded middle. Its sequent calculus restricts conclusions to capture constructive reasoning.
Hilbert system: A Hilbert system is a proof system with axiom schemas and a small set of inference rules, typically modus ponens. Its long axiom-based derivations contrast with sequent calculus's local rule structure.
Automated theorem proving: Automated theorem proving uses algorithms to construct or verify proofs in formal logical systems. Sequent rules can be implemented as proof-search steps.
Investigations into Logical Deduction: Investigations into Logical Deduction is Gerhard Gentzen's 1935 paper presenting natural deduction and sequent calculus. This paper introduced the principal sequent-calculus formulations.

Linked from 28 pages