Sequent

Inmathematical logic,asequentis a very general kind of conditional assertion.

A_{1},\,\dots,A_{m}\,\vdash \,B_{1},\,\dots,B_{n}.

A sequent may have any numbermof conditionformulasA_i(called "antecedents") and any numbernof asserted formulasB_j(called "succedents" or "consequents"). A sequent is understood to mean that if all of the antecedent conditions are true, then at least one of the consequent formulas is true. This style of conditional assertion is almost always associated with the conceptual framework ofsequent calculus.

Introduction

The form and semantics of sequents

Sequents are best understood in the context of the following three kinds oflogical judgments:

Unconditional assertion.No antecedent formulas.
- Example: ⊢B
- Meaning:Bis true.
Conditional assertion.Any number of antecedent formulas.
1. Simple conditional assertion.Single consequent formula.
  - Example:A₁,A₂,A₃⊢B
  - Meaning: IFA₁ANDA₂ANDA₃are true, THENBis true.
2. Sequent.Any number of consequent formulas.
  - Example:A₁,A₂,A₃⊢B₁,B₂,B₃,B₄
  - Meaning: IFA₁ANDA₂ANDA₃are true, THENB₁ORB₂ORB₃ORB₄is true.

Thus sequents are a generalization of simple conditional assertions, which are a generalization of unconditional assertions.

The word "OR" here is theinclusive OR.^[1]The motivation for disjunctive semantics on the right side of a sequent comes from three main benefits.

The symmetry of the classicalinference rulesfor sequents with such semantics.
The ease and simplicity of converting such classical rules to intuitionistic rules.
The ability to prove completeness for predicate calculus when it is expressed in this way.

All three of these benefits were identified in the founding paper byGentzen (1934,p. 194).

Not all authors have adhered to Gentzen's original meaning for the word "sequent". For example,Lemmon (1965)used the word "sequent" strictly for simple conditional assertions with one and only one consequent formula.^[2]The same single-consequent definition for a sequent is given byHuth & Ryan 2004,p. 5.

Syntax details

In a general sequent of the form

\Gamma \vdash \Sigma

both Γ and Σ aresequencesof logical formulas, notsets.Therefore both the number and order of occurrences of formulas are significant. In particular, the same formula may appear twice in the same sequence. The full set ofsequent calculus inference rulescontains rules to swap adjacent formulas on the left and on the right of the assertion symbol (and thereby arbitrarilypermutethe left and right sequences), and also to insert arbitrary formulas and remove duplicate copies within the left and the right sequences. (However,Smullyan (1995,pp. 107–108), usessetsof formulas in sequents instead of sequences of formulas. Consequently the three pairs ofstructural rulescalled "thinning", "contraction" and "interchange" are not required.)

The symbol ' $\vdash$ ' is often referred to as the "turnstile","right tack "," tee "," assertion sign "or" assertion symbol ". It is often read, suggestively, as" yields "," proves "or" entails ".

Properties

Effects of inserting and removing propositions

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. This is one of the symmetry advantages which follows from the use of disjunctive semantics on the right hand side of the assertion symbol, whereas conjunctive semantics is adhered to on the left hand side.

Consequences of empty lists of formulas

In the extreme case where the list ofantecedentformulas of a sequent is empty, the consequent is unconditional. This differs from the simple unconditional assertion because the number of consequents is arbitrary, not necessarily a single consequent. Thus for example, ' ⊢B₁,B₂' means that eitherB₁,orB₂,or both must be true. An empty antecedent formula list is equivalent to the "always true" proposition, called the "verum",denoted" ⊤ ". (SeeTee (symbol).)

In the extreme case where the list ofconsequentformulas of a sequent is empty, the rule is still that at least one term on the right be true, which is clearlyimpossible.This is signified by the 'always false' proposition, called the "falsum",denoted" ⊥ ". Since the consequence is false, at least one of the antecedents must be false. Thus for example, 'A₁,A₂⊢ ' means that at least one of the antecedentsA₁andA₂must be false.

One sees here again a symmetry because of the disjunctive semantics on the right hand side. If the left side is empty, then one or more right-side propositions must be true. If the right side is empty, then one or more of the left-side propositions must be false.

The doubly extreme case ' ⊢ ', where both the antecedent and consequent lists of formulas are empty is "not satisfiable".^[3]In this case, the meaning of the sequent is effectively ' ⊤ ⊢ ⊥ '. This is equivalent to the sequent ' ⊢ ⊥ ', which clearly cannot be valid.

Examples

A sequent of the form ' ⊢ α, β ', for logical formulas α and β, means that either α is true or β is true (or both). But it does not mean that either α is a tautology or β is a tautology. To clarify this, consider the example ' ⊢ B ∨ A, C ∨ ¬A '. This is a valid sequent because either B ∨ A is true or C ∨ ¬A is true. But neither of these expressions is a tautology in isolation. It is thedisjunctionof these two expressions which is a tautology.

Similarly, a sequent of the form ' α, β ⊢ ', for logical formulas α and β, means that either α is false or β is false. But it does not mean that either α is a contradiction or β is a contradiction. To clarify this, consider the example ' B ∧ A, C ∧ ¬A ⊢ '. This is a valid sequent because either B ∧ A is false or C ∧ ¬A is false. But neither of these expressions is a contradiction in isolation. It is theconjunctionof these two expressions which is a contradiction.

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 aline.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 $\Sigma$ .(See also the full set ofsequent calculus inference rules.)

Interpretation

History of the meaning of sequent assertions

The assertion symbol in sequents originally meant exactly the same as the implication operator. But over time, its meaning has changed to signify provability within a theory rather than semantic truth in all models.

In 1934, Gentzen did not define the assertion symbol ' ⊢ ' in a sequent to signify provability. He defined it to mean exactly the same as the implication operator ' ⇒ '. Using ' → ' instead of ' ⊢ ' and ' ⊃ ' instead of ' ⇒ ', he wrote: "The sequent A₁,..., A_μ→ B₁,..., B_νsignifies, as regards content, exactly the same as the formula (A₁&... & A_μ) ⊃ (B₁∨... ∨ B_ν) ".^[4](Gentzen employed the right-arrow symbol between the antecedents and consequents of sequents. He employed the symbol ' ⊃ ' for the logical implication operator.)

In 1939,HilbertandBernaysstated likewise that a sequent has the same meaning as the corresponding implication formula.^[5]

In 1944,Alonzo Churchemphasized that Gentzen's sequent assertions did not signify provability.

"Employment of the deduction theorem as primitive or derived rule must not, however, be confused with the use ofSequenzenby Gentzen. For Gentzen's arrow, →, is not comparable to our syntactical notation, ⊢, but belongs to his object language (as is clear from the fact that expressions containing it appear as premisses and conclusions in applications of his rules of inference). "^[6]

Numerous publications after this time have stated that the assertion symbol in sequents does signify provability within the theory where the sequents are formulated.Curryin 1963,^[7]Lemmonin 1965,^[2]and Huth and Ryan in 2004^[8]all state that the sequent assertion symbol signifies provability. However,Ben-Ari (2012,p. 69) states that the assertion symbol in Gentzen-system sequents, which he denotes as ' ⇒ ', is part of the object language, not the metalanguage.^[9]

According toPrawitz(1965): "The calculi of sequents can be understood as meta-calculi for the deducibility relation in the corresponding systems of natural deduction."^[10]And furthermore: "A proof in a calculus of sequents can be looked upon as an instruction on how to construct a corresponding natural deduction."^[11]In other words, the assertion symbol is part of the object language for the sequent calculus, which is a kind of meta-calculus, but simultaneously signifies deducibility in an underlying natural deduction system.

Intuitive meaning

A sequent is aformalizedstatement ofprovabilitythat is frequently used when specifyingcalculifordeduction.In the sequent calculus, the namesequentis used for the construct, which can be regarded as a specific kind ofjudgment,characteristic to this deduction system.

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 interpretedconjunctivelywhile the formulae on the right can be considered as adisjunction.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 alogical 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 classicalintuitionistic logic,such asGlivenko's theorem).

In any case, these intuitive readings are only pedagogical. Since formal proofs in proof theory are purelysyntactic,themeaningof (the derivation of) a sequent is only given by the properties of the calculus that provides the actualrules 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 theturnstile.In this sense, $\vdash$ means the process of reasoning, or "therefore" in English.

Variations

The general notion of sequent introduced here can be specialized in various ways. A sequent is said to be anintuitionistic sequentif there is at most one formula in the succedent (although multi-succedent calculi for intuitionistic logic are also possible). More precisely, the restriction of the general sequent calculus to single-succedent-formula sequents,with the same inference rulesas for general sequents, constitutes an intuitionistic sequent calculus. (This restricted sequent calculus is denoted LJ.)

Similarly, one can obtain calculi fordual-intuitionistic logic(a type ofparaconsistent logic) by requiring that sequents be singular in the antecedent.

In many cases, sequents are also assumed to consist ofmultisetsorsetsinstead of sequences. Thus one disregards the order or even the numbers of occurrences of the formulae. For classicalpropositional logicthis does not yield a problem, since the conclusions that one can draw from a collection of premises do not depend on these data. Insubstructural logic,however, this may become quite important.

Natural deductionsystems use single-consequence conditional assertions, but they typically do not use the same sets of inference rules as Gentzen introduced in 1934. In particular,tabular natural deductionsystems, which are very convenient for practical theorem-proving in propositional calculus and predicate calculus, were applied bySuppes (1999)andLemmon (1965)for teaching introductory logic in textbooks.

Etymology

Historically, sequents have been introduced byGerhard Gentzenin order to specify his famoussequent calculus.^[12]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.

Kleene^[13]makes the following comment on the translation into English: "Gentzen says 'Sequenz', which we translate as 'sequent', because we have already used 'sequence' for any succession of objects, where the German is 'Folge'."

Notes

^The disjunctive semantics for the right side of a sequent is stated and explained byCurry 1977,pp. 189–190,Kleene 2002,pp. 290, 297,Kleene 2009,p. 441,Hilbert & Bernays 1970,p. 385,Smullyan 1995,pp. 104–105,Takeuti 2013,p. 9, andGentzen 1934,p. 180.
^^a ^bLemmon 1965,p. 12, wrote: "Thus a sequent is an argument-frame containing a set of assumptions and a conclusion which is claimed to follow from them. [...] The propositions to the left of '⊢' become assumptions of the argument, and the proposition to the right becomes a conclusion validly drawn from those assumptions."
^Smullyan 1995,p. 105.
^Gentzen 1934,p. 180.
2.4. Die Sequenz A₁,..., A_μ→ B₁,..., B_νbedeutet inhaltlich genau dasselbe wie die Formel
(A₁&... & A_μ) ⊃ (B₁∨... ∨ B_ν).
^Hilbert & Bernays 1970,p. 385.
Für die inhaltliche Deutung ist eine Sequenz
A₁,..., A_r→ B₁,..., B_s,

worin die Anzahlen r und s von 0 verschieden sind, gleichbedeutend mit der Implikation
(A₁&... & A_r) → (B₁∨... ∨ B_s)
^Church 1996,p. 165.
^Curry 1977,p. 184
^Huth & Ryan (2004,p. 5)
^Ben-Ari 2012,p. 69, defines sequents to have the formU⇒Vfor (possibly non-empty) sets of formulasUandV.Then he writes:
"Intuitively, a sequent represents 'provable from' in the sense that the formulas inUare assumptions for the set of formulasVthat are to be proved. The symbol ⇒ is similar to the symbol ⊢ in Hilbert systems, except that ⇒ is part of the object language of the deductive system being formalized, while ⊢ is a metalanguage notation used to reason about deductive systems. "
^Prawitz 2006,p. 90.
^SeePrawitz 2006,p. 91, for this and further details of interpretation.
^Gentzen 1934,Gentzen 1935.
^Kleene 2002,p. 441

References

Ben-Ari, Mordechai(2012) [1993].Mathematical logic for computer science.London: Springer.ISBN 978-1-4471-4128-0.
Church, Alonzo(1996) [1944].Introduction to mathematical logic.Princeton, New Jersey: Princeton University Press.ISBN 978-0-691-02906-1.
Curry, Haskell Brooks(1977) [1963].Foundations of mathematical logic.New York: Dover Publications Inc.ISBN 978-0-486-63462-3.
Gentzen, Gerhard(1934)."Untersuchungen über das logische Schließen. I".Mathematische Zeitschrift.39(2): 176–210.doi:10.1007/bf01201353.S2CID 121546341.
Gentzen, Gerhard(1935)."Untersuchungen über das logische Schließen. II".Mathematische Zeitschrift.39(3): 405–431.doi:10.1007/bf01201363.S2CID 186239837.
Hilbert, David;Bernays, Paul(1970) [1939].Grundlagen der Mathematik II(Second ed.). Berlin, New York: Springer-Verlag.ISBN 978-3-642-86897-9.
Huth, Michael; Ryan, Mark (2004).Logic in Computer Science(Second ed.). Cambridge, United Kingdom: Cambridge University Press.ISBN 978-0-521-54310-1.
Kleene, Stephen Cole(2009) [1952].Introduction to metamathematics.Ishi Press International.ISBN 978-0-923891-57-2.
Kleene, Stephen Cole(2002) [1967].Mathematical logic.Mineola, New York: Dover Publications.ISBN 978-0-486-42533-7.
Lemmon, Edward John(1965).Beginning logic.Thomas Nelson.ISBN 0-17-712040-1.
Prawitz, Dag(2006) [1965].Natural deduction: A proof-theoretical study.Mineola, New York: Dover Publications.ISBN 978-0-486-44655-4.
Smullyan, Raymond Merrill(1995) [1968].First-order logic.New York: Dover Publications.ISBN 978-0-486-68370-6.
Suppes, Patrick Colonel(1999) [1957].Introduction to logic.Mineola, New York: Dover Publications.ISBN 978-0-486-40687-9.
Takeuti, Gaisi(2013) [1975].Proof theory(Second ed.). Mineola, New York: Dover Publications.ISBN 978-0-486-49073-1.

External links

"Sequent (in logic)",Encyclopedia of Mathematics,EMS Press,2001 [1994]

[1] The disjunctive semantics for the right side of a sequent is stated and explained byCurry 1977,pp. 189–190,Kleene 2002,pp. 290, 297,Kleene 2009,p. 441,Hilbert & Bernays 1970,p. 385,Smullyan 1995,pp. 104–105,Takeuti 2013,p. 9, andGentzen 1934,p. 180.

[Lemmon1965p12-2] Lemmon 1965,p. 12, wrote: "Thus a sequent is an argument-frame containing a set of assumptions and a conclusion which is claimed to follow from them. [...] The propositions to the left of '⊢' become assumptions of the argument, and the proposition to the right becomes a conclusion validly drawn from those assumptions."

[3] Smullyan 1995,p. 105.

[4] Gentzen 1934,p. 180.
2.4. Die Sequenz A₁,..., A_μ→ B₁,..., B_νbedeutet inhaltlich genau dasselbe wie die Formel
(A₁&... & A_μ) ⊃ (B₁∨... ∨ B_ν).

[5] Hilbert & Bernays 1970,p. 385.
Für die inhaltliche Deutung ist eine Sequenz
A₁,..., A_r→ B₁,..., B_s,

worin die Anzahlen r und s von 0 verschieden sind, gleichbedeutend mit der Implikation
(A₁&... & A_r) → (B₁∨... ∨ B_s)

[6] Church 1996,p. 165.

[7] Curry 1977,p. 184

[8] Huth & Ryan (2004,p. 5)

[9] Ben-Ari 2012,p. 69, defines sequents to have the formU⇒Vfor (possibly non-empty) sets of formulasUandV.Then he writes:
"Intuitively, a sequent represents 'provable from' in the sense that the formulas inUare assumptions for the set of formulasVthat are to be proved. The symbol ⇒ is similar to the symbol ⊢ in Hilbert systems, except that ⇒ is part of the object language of the deductive system being formalized, while ⊢ is a metalanguage notation used to reason about deductive systems. "

[10] Prawitz 2006,p. 90.

[11] SeePrawitz 2006,p. 91, for this and further details of interpretation.

[12] Gentzen 1934,Gentzen 1935.

[13] Kleene 2002,p. 441

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