Definitions
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 n. Plural form of sequent.
Etymologies
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Examples

Weakening is made an eliminable rule by letting initial sequents have the form

Troelstra mentions Ketonen's work as “an early analysis of cutfree proofs in Gentzen calculi with axioms; but he considers the form of cutfree derivations in the pure calculus where axioms are present in the antecedent of the sequents derived” (Troelstra and Schwichtenberg 2000, p. 142).

By invertibility, the question of derivability is replaced by one or two equivalent questions of derivability on simpler sequents.

Thus, there are no derivable sequents with all assumptions true and the conclusion false.

Gentzen also generalized sequents so that they have, instead of one conclusion, a list of possible cases after the arrow.

Then write each rule instance so that the premisses and conclusion have the open assumptions listed at left, with an arrow separating the conclusion, so, as sequents.

Analogous rules permit weakening and contraction in the right, succedent parts of sequents.

If we opted to use sequent rules, we would have to decide whether or not to allow “substructure” (see the entry on substructural logics) and whether to allow multiple conclusions in the sequents.

A, in which we have a collection of antecedents and a single consequent, for classical logic it is fruitful to consider sequents of the form

Axiom sequents in LK are valid, and the conclusion of a rule is valid iff its premises are.
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