Editing Sequent (section)

=== Intuitive meaning ===
{{Refimprove section|date=June 2014|talk=Assertion symbols in sequents do not signify provability.}}
A sequent is a [[Formalism (mathematics)|formalized]] statement of [[Proof theory|provability]] that is frequently used when specifying [[proof calculus|calculi]] for [[deductive reasoning|deduction]]. In the sequent calculus, the name ''sequent'' is used for the construct, which can be regarded as a specific kind of [[Judgment (mathematical logic)|judgment]], characteristic to this deduction system.

The intuitive meaning of the sequent <math>\Gamma\vdash\Sigma</math> 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 [[logical conjunction|conjunctively]] while the formulae on the right can be considered as a [[logical disjunction|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. <math>\Gamma\vdash</math> means that Γ proves falsity and is thus inconsistent. On the other hand an empty antecedent is assumed to be true, i.e., <math>\vdash\Sigma</math> 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, <math>\Gamma\vdash\Sigma</math> 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 translation|Glivenko's theorem]]).

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

Barring any contradictions in the technically precise definition above we can describe sequents in their introductory logical form. <math>\Gamma</math> represents a set of assumptions that we begin our logical process with, for example "Socrates is a man" and "All men are mortal". The <math>\Sigma</math> 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 <math>\Sigma</math> side of the ''turnstile''. In this sense, <math>\vdash</math> means the process of reasoning, or "therefore" in English.