# Sequent

In proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction. In the sequent calculus, the name sequent is used for the construct which can be regarded as a specific kind of judgment, characteristic to this deduction system.

## Explanation

A sequent has the form

$\Gamma\vdash\Sigma$

where both Γ and Σ are sequences of logical formulae (i.e., both the number and the order of the occurring formulae matter). The symbol $\vdash$ is usually referred to as turnstile or tee and is often read, suggestively, as "yields" or "proves". It is not a symbol in the language, rather it is a symbol in the metalanguage used to discuss proofs. In a sequent, Γ is called the antecedent and Σ is said to be the succedent of the sequent.

## Intuitive meaning

The intuitive meaning of the sequent $\Gamma\vdash\Sigma$ is that under the assumption of Γ the conclusion of Σ is provable. Classically, the formulae on the left of the turnstile can be interpreted conjunctively while the formulae on the right can be considered as a disjunction. This means that, when all formulae in Γ hold, then at least one formula in Σ also has to be true. If the succedent is empty, this is interpreted as falsity, i.e. $\Gamma\vdash$ means that Γ proves falsity and is thus inconsistent. On the other hand an empty antecedent is assumed to be true, i.e., $\vdash\Sigma$ means that Σ follows without any assumptions, i.e., it is always true (as a disjunction). A sequent of this form, with Γ empty, is known as a logical assertion.

Of course, other intuitive explanations are possible, which are classically equivalent. For example, $\Gamma\vdash\Sigma$ can be read as asserting that it cannot be the case that every formula in Γ is true and every formula in Σ is false (this is related to the double-negation interpretations of classical intuitionistic logic, such as Glivenko's theorem).

In any case, these intuitive readings are only pedagogical. Since formal proofs in proof theory are purely syntactic, the meaning of (the derivation of) a sequent is only given by the properties of the calculus that provides the actual rules of inference.

Barring any contradictions in the technically precise definition above we can describe sequents in their introductory logical form. $\Gamma$ represents a set of assumptions that we begin our logical process with, for example "Socrates is a man" and "All men are mortal". The $\Sigma$ represents a logical conclusion that follows under these premises. For example "Socrates is mortal" follows from a reasonable formalization of the above points and we could expect to see it on the $\Sigma$ side of the turnstile. In this sense, $\vdash$ means the process of reasoning, or "therefore" in English.

## Example

A typical sequent might be:

$\phi,\psi\vdash\alpha,\beta$

This claims that either $\alpha$ or $\beta$ can be derived from $\phi$ and $\psi$.

## Property

Since every formula in the antecedent (the left side) must be true to conclude the truth of at least one formula in the succedent (the right side), adding formulas to either side results in a weaker sequent, while removing them from either side gives a stronger one.

## Rules

Most proof systems provide ways to deduce one sequent from another. These inference 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:

$\frac{\Gamma,\alpha\vdash\Sigma\qquad \Gamma\vdash\alpha}{\Gamma\vdash\Sigma}$

This indicates that if we can deduce that $\Gamma,\alpha$ yields $\Sigma$, and that $\Gamma$ yields $\alpha$, then we can also deduce that $\Gamma$ yields $\Sigma$.

## Variations

The general notion of sequent introduced here can be specialized in various ways. A sequent is said to be an intuitionistic sequent if there is at most one formula in the succedent. This form is needed to obtain calculi for intuitionistic logic. Similarly, one can obtain calculi for dual-intuitionistic logic (a type of paraconsistent logic) by requiring that sequents be singular in the antecedent.

In many cases, sequents are also assumed to consist of multisets or sets instead of sequences. Thus one disregards the order or even the number of occurrences of the formulae. For classical propositional logic this does not yield a problem, since the conclusions that one can draw from a collection of premises does not depend on these data. In substructural logic, however, this may become quite important.

## History

Historically, sequents have been introduced by Gerhard Gentzen in order to specify his famous sequent calculus. In his German publication he used the word "Sequenz". However, in English, the word "sequence" is already used as a translation to the German "Folge" and appears quite frequently in mathematics. The term "sequent" then has been created in search for an alternative translation of the German expression.