The form and semantics of sequents

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

1. Unconditional assertion. No antecedent formulas.
   • Example: ⊢ B
   • Meaning: B is true.
2. Conditional assertion. Any number of antecedent formulas.
   1. Simple conditional assertion. Single consequent formula.
      • Example: A₁, A₂, A₃ ⊢ B
      • Meaning: IF A₁ AND A₂ AND A₃ are true, THEN B is true.
   2. Sequent. Any number of consequent formulas.
      • Example: A₁, A₂, A₃ ⊢ B₁, B₂, B₃, B₄
      • Meaning: IF A₁ AND A₂ AND A₃ are true, THEN B₁ OR B₂ OR B₃ OR B₄ is true.

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

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

1. The symmetry of the classical inference rules for sequents with such semantics.
2. The ease and simplicity of converting such classical rules to intuitionistic rules.
3. 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 by Gentzen (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 by Huth & Ryan 2004, p. 5.

Syntax details

In a general sequent of the form

${\displaystyle \Gamma \vdash \Sigma }$

both Γ and Σ are sequences of logical formulas, not sets. 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 of sequent calculus inference rules contains rules to swap adjacent formulas on the left and on the right of the assertion symbol (and thereby arbitrarily permute the 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), uses sets of formulas in sequents instead of sequences of formulas. Consequently the three pairs of structural rules called "thinning", "contraction" and "interchange" are not required.)

The symbol ' ${\displaystyle \vdash }$ ' 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

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 the disjunction of 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 the conjunction of 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 a line. This rule indicates that if everything above the line is true, so is everything under the line.

A typical rule is:

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

This indicates that if we can deduce that ${\displaystyle \Gamma ,\alpha }$ yields ${\displaystyle \Sigma }$, and that ${\displaystyle \Gamma }$ yields ${\displaystyle \alpha }$, then we can also deduce that ${\displaystyle \Gamma }$ yields ${\displaystyle \Sigma }$. (See also the full set of sequent calculus inference rules.)

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_ν signifies, 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, Hilbert and Bernays stated likewise that a sequent has the same meaning as the corresponding implication formula.^[5]

In 1944, Alonzo Church emphasized 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 of Sequenzen by 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. Curry in 1963,^[7] Lemmon in 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 to Prawitz (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 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.

The intuitive meaning of the sequent ${\displaystyle \Gamma \vdash \Sigma }$ 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. ${\displaystyle \Gamma \vdash }$ means that Γ proves falsity and is thus inconsistent. On the other hand an empty antecedent is assumed to be true, i.e., ${\displaystyle \vdash \Sigma }$ 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, ${\displaystyle \Gamma \vdash \Sigma }$ 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. ${\displaystyle \Gamma }$ represents a set of assumptions that we begin our logical process with, for example "Socrates is a man" and "All men are mortal". The ${\displaystyle \Sigma }$ 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 ${\displaystyle \Sigma }$ side of the turnstile. In this sense, ${\displaystyle \vdash }$ means the process of reasoning, or "therefore" in English.