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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 to 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 in a first-order language rather than conditional tautologies.

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rdfs:label	Sequent calculus (en)
rdfs:comment	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 to 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 in a first-order language rather than conditional tautologies. (en)
sameAs	Sequent calculus
dbp:wikiPageUsesTemplate	dbt:Citation_needed dbt:Cite_book dbt:Cite_journal dbt:Main_article dbt:Reflist dbt:Sfn dbt:Springer
Subject	Automated theorem proving Proof theory Logical calculi
gold:hypernym	Tautology
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Sequent_calculus?oldid=1071298584&ns=0
Wikipage page ID	252329 (xsd:integer)
page length (characters) of wiki page	52977 (xsd:nonNegativeInteger)
Wikipage revision ID	1071298584 (xsd:integer)
has abstract	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 to 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 in a first-order language rather than conditional tautologies. Sequent calculus is one of several extant styles of proof calculus for expressing line-by-line logical arguments. * Hilbert style. Every line is an unconditional tautology (or theorem). * Gentzen style. Every line is a conditional tautology (or theorem) with zero or more conditions on the left. * Natural deduction. Every (conditional) line has exactly one asserted proposition on the right. * Sequent calculus. Every (conditional) line has zero or more asserted propositions on the right. In other words, natural deduction and sequent calculus systems are particular distinct kinds of Gentzen-style systems. Hilbert-style systems typically have a very small number of inference rules, relying more on sets of axioms. Gentzen-style systems typically have very few axioms, if any, relying more on sets of rules. Gentzen-style systems have significant practical and theoretical advantages compared to Hilbert-style systems. For example, both natural deduction and sequent calculus systems facilitate the elimination and introduction of universal and existential quantifiers so that unquantified logical expressions can be manipulated according to the much simpler rules of propositional calculus. In a typical argument, quantifiers are eliminated, then propositional calculus is applied to unquantified expressions (which typically contain free variables), and then the quantifiers are reintroduced. This very much parallels the way in which mathematical proofs are carried out in practice by mathematicians. Predicate calculus proofs are generally much easier to discover with this approach, and are often shorter. Natural deduction systems are more suited to practical theorem-proving. Sequent calculus systems are more suited to theoretical analysis. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Sequent_calculus
is Wikipage redirect of	System LJ System LK Gentzen-style system Gentzen calculus Gentzen system Gentzen systems Reduction tree Reduction trees Inference line Inference lines
is Link from a Wikipage to another Wikipage of	Focused proof Formal proof Logic Logic: The Laws of Truth Noncommutative logic Abductive reasoning Admissible rule Arnon Avron Logic gate Monotonicity of entailment Bunched logic Calculus (disambiguation) Cedent Frege system Geometric logic Geometry of interaction Gerhard Gentzen Giorgi Japaridze Jape (software) Judgment (mathematical logic) Hilbert system History of logic Metalogic Method of analytic tableaux Curry–Howard correspondence Cut-elimination theorem Cut rule Deep inference Horizontal bar Linear logic List of functional programming topics List of mathematical logic topics Hypersequent Completeness of atomic initial sequents Computability logic Craig interpolation Term algebra Hoare logic Minimal logic Analytic proof First-order logic Gaisi Takeuti Idempotency of entailment Index of logic articles Index of philosophy articles (R–Z) KeY Gentzen-style system Gentzen calculus Gentzen system Gentzen systems Extensions of First Order Logic Kripke semantics LJ LK Modal logic Modus ponens Ivan Orlov (philosopher) Mathematical logic Multiple-conclusion logic Natural deduction Nested sequent calculus Inference line Inference lines

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