Sequent
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For other uses, see Sequent (disambiguation).
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.
ExplanationA sequent has the form
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 meaningThe intuitive meaning of a sequent such as the one given above is that under the assumption of Γ the conclusion of Σ is provable. In a classical setting, the formulae on the left of the turnstile are interpreted conjunctively while the formulae on the right are 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. Failed to parse (Missing texvc executable; please see math/README to configure.): \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., Failed to parse (Missing texvc executable; please see math/README to configure.): \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. The above interpretation, however, is 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. Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma side of the turnstile. In this sense, Failed to parse (Missing texvc executable; please see math/README to configure.): \vdash means the process of reasoning, or "therefore" in English. ExampleA typical sequent might be:
or Failed to parse (Missing texvc executable; please see math/README to configure.): \beta can be derived from Failed to parse (Missing texvc executable; please see math/README to configure.): \phi and Failed to parse (Missing texvc executable; please see math/README to configure.): \psi . PropertySince 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. RulesMost 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:
from Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma , we can also deduce it from Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma together with Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha. Note that the capital Greek letters are usually used to denote a (possibly empty) list of formulae. Failed to parse (Missing texvc executable; please see math/README to configure.): [\Gamma,\Sigma] is used to denote the contraction of Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma and Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma , that is, the list of those formulas appearing in either Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma or Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma but with no repeats. VariationsThe 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 premisses does not depend on these data. In substructural logic, however, this may become quite important. HistoryHistorically, 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.
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