Gentzen's LK Calculus
2025-07-14

Contents

Rules
Structural Rules
Logical Rules
Bibliography

LK calculus is Gentzen's standard sequent calculus (Sequenzenkalkul, literally "calculus of sequences") for classical first-order logic. The letter K stands for the German word klassisch [1].

Definition A (Sequent).

A sequent is an expression of the form

where and are finite (possibly empty) sequences of formulas (of some language). is called the antecedent, while is the succedent.

Intuitively, the idea behind a sequent is that if all of the formulas in the antecedent hold, then at least one of the formula in the succedent holds. That is, if and , then holds iff

holds.

1 Rules

The inference rules

1.1 Structural Rules

Weakening

Contraction

Exchange

1.2 Logical Rules

Negation

Conjunction

Disjunction

todohttps://builds.openlogicproject.org/content/first-order-logic/sequent-calculus/sequent-calculus.pdf

Bibliography