A sequent represents a formal step in a proof. Typically it consists of two lists of formulasMathworldPlanetmathPlanetmath, one representing the premises and one the conclusions. A typical sequent might be:


where ϕ and ψ are the premises and α and β are the conclusions.

This claims that, from premises ϕ and ψ either α or β must be true. Note that is not a symbol in the languagePlanetmathPlanetmath, rather it is a symbol in the metalanguage used to discuss proofs. Also, notice the asymmetry: everything on the left must be true to conclude only one thing on the right. This does create a different kind of symmetry, since adding formulas to either side results in a weaker sequent, while removing them from either side gives a stronger one.

Some systems allow only one formula on the right.

Most proof systems provide ways to deduce one sequent from another. These rules are written with a list of sequents above and below a line. This rule indicates that if everything above the line is true, so is everything under the line. A typical rule is:


This indicates that if we can deduce Σ from Γ, we can also deduce it from Γ together with α.

Note that the capital Greek letters are usually used to denote a (possibly empty) list of formulas. [Γ,Σ] is used to denote the contraction of Γ and Σ, that is, the list of those formulas appearing in either Γ or Σ but with no repeats.

Title sequent
Canonical name Sequent
Date of creation 2013-03-22 13:05:11
Last modified on 2013-03-22 13:05:11
Owner Henry (455)
Last modified by Henry (455)
Numerical id 10
Author Henry (455)
Entry type Definition
Classification msc 03F03
Related topic GentzenSystem
Defines contraction
Defines premise
Defines conclusion