sequents

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

Chores 2009

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

Chores 2009

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).

Chores 2009

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.

Logical Constants MacFarlane, John 2009

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.

Chores 2009

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

Chores 2009

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

Chores 2009

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

Chores 2009

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

Substructural Logics Restall, Greg 2008

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

Automated Reasoning Portoraro, Frederic 2005