Sequent

foundations

# Sequent

## Idea

In formal logic a sequent (Gentzen 35, Martin-Löf 83) or hypothetical judgement (Pfenning, Davies 00) is an expression in the metalanguage which is a string of symbols of the form

$Antecedent \vdash Succedent$

where

1. $Antecedent$ are symbols indicating judgements called the antecedents or context,

2. $Succedent$ are symbols indicating judgements called the succedents or (if it is just a single judgement) the consequent

3. the consequence sign or turnstile-symbol “$\vdash$” expresses that $Succedent$ is a consequence of $Antecedent$:

$Antecedent$ yields $Succedent$.”

or

“With $Antecedent$ the $Succedent$ can be proven.”

or

$Antecedent$, con-sequent-ly $Succedent$.”

Or similar.

Historically the “consequence” here was early on transmuted to “sequenz” (Gentzen) and then later to “sequent”. See the section History below.

In systems of formal logic such as natural deduction/type theory such sequents express rules for explicit symbol manipulation admitted in the system rather than formal implications within the system. The latter instead are expressed by terms of function type $t : \phi \to \psi$. But the term introduction rule for terms of function types say that given one, one is allowed to get the other.

Typically one allows a list of expressions on both sides of the turnstile-symbols as in

$Antec_1, \cdots, Antec_k \vdash Succ_1, \cdot, Succ_l$

often abbreviated as

$\vec Antec \vdash \vec Succ$

in which case on the left a conjunction of the antecedents and on the right a disjunction of succedents is understood, as in

“If $Antec_1$ and $Antec_2$… are given then one of $Succ_1, \cdots, Succ_l$ is a consequence”.

An intuitionistic sequent has at most one succedent, which is then called the consequent. Often “intuitionistic sequent” is used to mean one with exactly one succedent, although technically it would make more sense to call those minimal sequents.

Another variant is that in some frameworks the antecedent and succedent displayed are required to be propositions and the free variables of the context are instead displayed beneath the turnstile as in

$\vec \phi(x)\, true \vdash_{\vec x} \vec \psi(x) \, true \,.$

If the framework regards propositions as types then this is the same as writing

$\vec x. \vec \phi(x) \vdash \vec \psi(x) \,.$

Finally one can of course consider sequences of sequents

$(\Gamma_1 \vdash \Delta_1), \cdots, (\Gamma_n \vdash \Delta_n)$

and if these are read disjunctively it is like a higher-order sequent without antecedent and called a hypersequent.

Rules for formal manipulation of sequents are called sequent calculi or deduction calculi. See there for more details.

## Definition

The precise nature of sequents has been formalized differently in different systems of formal logic. We discuss a few

### Intuitionistic sequents

(…) (Martin-Löf) (…)

### Gentzen’s sequents

(…) (Gentzen) (…)

(…) Simmons (…)

## Semantics

We discuss aspects of the categorical semantics of sequents, hence their interpretation when the ambient formal logic is regarded as the internal language of a category.

### In homotopy type theory

Under the categorical semantics of homotopy type theory sequents in the type theory pretty accurately correspond to morphisms in the (∞,1)-topos. We indicate how this works, first for type declarations, then for terms of dependent types.

Let $\mathbf{H}$ be an (∞,1)-topos. Write $Type \in \mathbf{H}$ for the internal universe of small objects of $\mathbf{H}$, called the object classifier.

This is defined as the representing object for the core of the small codomain fibration, exhibited by an equivalence of ∞-groupoids

$name : Core(\mathbf{H}_{/X})^{small} \stackrel{\simeq}{\to} \mathbf{H}(X,Type) \,.$

This equivalence sends an $X$-family $(E \to X) \in \mathbf{H}_{/X}$ to its “name”, denoted

$X \stackrel{\vdash E}{\to} Type \,,$

which is the morphism characterized by fitting into an ∞-pullback square of the form

$\array{ E &\to& \widehat{Type} \\ \downarrow &\swArrow& \downarrow \\ X &\underset{\vdash E}{\to}& Type } \,,$

If here we simply replace, notationally, the arrow “$\to$” by the turnstile “$\vdash$”, display a generic generalized element $x$ of $X$ and then write $E(x)$ to highlight the dependence of the fibers of $E$ on $x$ in $X$, then the symbols “$X \stackrel{\vdash E}{\to} Type$” become the sequent “$x : X \vdash E(x) : Type$”. This sequent is the syntax of which the morphism is the categorical semantics.

Similarly, if $X \stackrel{t}{\to}_X E$ is a section of $E$ over $X$, hence a generalized element in the slice (∞,1)-topos $\mathbf{H}_{/X}$, then by replacing the arrow-symbol by a turnstile-symbol we get “$x : X \vdash t(x) : E(x)$”. This is the sequent for the term $t$ of the dependent type $E$.

In summary we have under the relation between category theory and type theory the notational correspondence:

morphisms to sequents.

$\,$typesterms
∞-topos theory$\;\;\;\;X \stackrel{\vdash \;\;\;\;E}{\to} \;\;\Type$$\;\;\;\;X \stackrel{\vdash \;\;\;t}{\to} {}_X \;\;E$
homotopy type theory$x : X \vdash E(x) : Type$$x : X \vdash t(x) : E(x)$

## History

The notion of sequent was introduced in section 2.3 of (Gentzen 1935) (called Sequenz there). In (Martin-Löf 1984, pages 29-30) it says

The forms of hypothetical judgement $[$ have $]$ the form

$A_1 true, \cdots, A_n true \vdash A prop$

which says that $A$ is a proposition under the assumptions that $A_1, \cdots, A_n$ are all true, and, on the other hand, the form

$A_1 true, \cdots, A_n true \vdash A true$

which says that the proposition A is true under the assumptions that $A_1, \cdots, A_n$ are all true. Here I am using the vertical bar for the relation of logical consequence, that is, for what Gentzen expressed by means of the arrow $\to$ in his sequence calculus, and for which the double arrow $\Rightarrow$ is also a common notation. It is the relation of logical consequence, which must be carefully distinguished from implication. What stands to the left of the consequence sign, we call the hypotheses, in which case what follows the consequence sign is called the thesis, or we call the judgements that precede the consequence sign the antecedents and the judgement that follows after the consequence sign the consequent. This is the terminology which Gentzen took over from the scholastics, except that, for some reason, he changed consequent into succedent and consequence into sequence, Ger. Sequenz, usually improperly rendered by sequent in English.

$\phantom{-}$symbol$\phantom{-}$$\phantom{-}$in logic$\phantom{-}$
$\phantom{A}$$\in$$\phantom{A}$element relation
$\phantom{A}$$\,:$$\phantom{A}$typing relation
$\phantom{A}$$=$$\phantom{A}$equality
$\phantom{A}$$\vdash$$\phantom{A}$$\phantom{A}$entailment / sequent$\phantom{A}$
$\phantom{A}$$\top$$\phantom{A}$$\phantom{A}$true / top$\phantom{A}$
$\phantom{A}$$\bot$$\phantom{A}$$\phantom{A}$false / bottom$\phantom{A}$
$\phantom{A}$$\Rightarrow$$\phantom{A}$implication
$\phantom{A}$$\Leftrightarrow$$\phantom{A}$logical equivalence
$\phantom{A}$$\not$$\phantom{A}$negation
$\phantom{A}$$\neq$$\phantom{A}$negation of equality / apartness$\phantom{A}$
$\phantom{A}$$\notin$$\phantom{A}$negation of element relation $\phantom{A}$
$\phantom{A}$$\not \not$$\phantom{A}$negation of negation$\phantom{A}$
$\phantom{A}$$\exists$$\phantom{A}$existential quantification$\phantom{A}$
$\phantom{A}$$\forall$$\phantom{A}$universal quantification$\phantom{A}$
$\phantom{A}$$\wedge$$\phantom{A}$logical conjunction
$\phantom{A}$$\vee$$\phantom{A}$logical disjunction
symbolin type theory (propositions as types)
$\phantom{A}$$\to$$\phantom{A}$function type (implication)
$\phantom{A}$$\times$$\phantom{A}$product type (conjunction)
$\phantom{A}$$+$$\phantom{A}$sum type (disjunction)
$\phantom{A}$$0$$\phantom{A}$empty type (false)
$\phantom{A}$$1$$\phantom{A}$unit type (true)
$\phantom{A}$$=$$\phantom{A}$identity type (equality)
$\phantom{A}$$\simeq$$\phantom{A}$equivalence of types (logical equivalence)
$\phantom{A}$$\sum$$\phantom{A}$dependent sum type (existential quantifier)
$\phantom{A}$$\prod$$\phantom{A}$dependent product type (universal quantifier)
symbolin linear logic
$\phantom{A}$$\multimap$$\phantom{A}$$\phantom{A}$linear implication$\phantom{A}$
$\phantom{A}$$\otimes$$\phantom{A}$$\phantom{A}$multiplicative conjunction$\phantom{A}$
$\phantom{A}$$\oplus$$\phantom{A}$$\phantom{A}$additive disjunction$\phantom{A}$
$\phantom{A}$$\&$$\phantom{A}$$\phantom{A}$additive conjunction$\phantom{A}$
$\phantom{A}$$\invamp$$\phantom{A}$$\phantom{A}$multiplicative disjunction$\phantom{A}$
$\phantom{A}$$\;!$$\phantom{A}$$\phantom{A}$exponential conjunction$\phantom{A}$

## References

In section 2.3

• Gerhard GentzenUntersuchungen über das logische Schließen I Mathematische Zeitschrift 39(1), Springer-Verlag 1935. <http://dx.doi.org/10.1007/BF01201353>.

what today is called a sequent is introduced under Sequenz (Ger: sequence), apparently derived from Konsequenz (Ger: consequence) and denoted by a single arrow “$\to$”.

In the lectures

• Per Martin-Löf, On the meaning of logical constants and the justifications of the logical laws, leture series in Siena (1983) (web)

where the author provides a modern foundation for logic based on a clear separation of the notions of judgment and proposition (see at Martin-Löf dependent type theory) the author says (pages 29-30) that “the forms of hypothetical judgements that I shall need” are Gentzen‘s sequents without the symmetry between antecedent and succedent that Gentzen used.

Referring explicitly to these lectures, these are are then just called hypothetical judgements in section 3 of

• Frank Pfenning, Rowan Davies, A judgemental reconstruction of modal logic (2000) (pdf)

In section D1.1 of

sequents are introduced in the context of a basic introduction to formal logic. There the the notation $\phi \vdash_{\vec c} \psi$ is used where $\phi$ is required to be a proposition and the context variables $\vec x$ are typeset below the turnstile. From the categorical semantics in section D1.2 it is clear that in the sense of dependent type theory these variables are to stand to the left of the turnstile.