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=== 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 ' β’ ' in a sequent to signify provability. He defined it to mean exactly the same as the implication operator ' β '. Using ' β ' instead of ' β’ ' and ' β ' instead of ' β ', he wrote: "The sequent A<sub>1</sub>, ..., A<sub>ΞΌ</sub> β B<sub>1</sub>, ..., B<sub>Ξ½</sub> signifies, as regards content, exactly the same as the formula (A<sub>1</sub> & ... & A<sub>ΞΌ</sub>) β (B<sub>1</sub> β¨ ... β¨ B<sub>Ξ½</sub>)".<ref>{{harvnb|Gentzen|1934|p=180}}. : 2.4. Die Sequenz A<sub>1</sub>, ..., A<sub>ΞΌ</sub> β B<sub>1</sub>, ..., B<sub>Ξ½</sub> bedeutet inhaltlich genau dasselbe wie die Formel ::: (A<sub>1</sub> & ... & A<sub>ΞΌ</sub>) β (B<sub>1</sub> β¨ ... β¨ B<sub>Ξ½</sub>).</ref> (Gentzen employed the right-arrow symbol between the antecedents and consequents of sequents. He employed the symbol ' β ' for the logical implication operator.) In 1939, [[David Hilbert|Hilbert]] and [[Paul Bernays|Bernays]] stated likewise that a sequent has the same meaning as the corresponding implication formula.<ref>{{harvnb|Hilbert|Bernays|1970|p=385}}. : FΓΌr die inhaltliche Deutung ist eine Sequenz ::: A<sub>1</sub>, ..., A<sub>r</sub> β B<sub>1</sub>, ..., B<sub>s</sub>, : worin die Anzahlen r und s von 0 verschieden sind, gleichbedeutend mit der Implikation ::: (A<sub>1</sub> & ... & A<sub>r</sub>) β (B<sub>1</sub> β¨ ... β¨ B<sub>s</sub>) </ref> 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, β, is not comparable to our syntactical notation, β’, 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)."<ref>{{harvnb|Church|1996|p=165}}.</ref> Numerous publications after this time have stated that the assertion symbol in sequents does signify provability within the theory where the sequents are formulated. [[Haskell Curry|Curry]] in 1963,<ref>{{harvnb|Curry|1977|p=184}}</ref> [[John Lemmon|Lemmon]] in 1965,<ref name=Lemmon1965p12 /> and Huth and Ryan in 2004<ref>{{harvtxt|Huth|Ryan|2004|p=5}}</ref> all state that the sequent assertion symbol signifies provability. However, {{harvtxt|Ben-Ari|2012|p=69}} states that the assertion symbol in Gentzen-system sequents, which he denotes as ' β ', is part of the object language, not the metalanguage.<ref>{{harvnb|Ben-Ari|2012|p=69}}, defines sequents to have the form ''U'' β ''V'' for (possibly non-empty) sets of formulas ''U'' and ''V''. Then he writes: : "Intuitively, a sequent represents 'provable from' in the sense that the formulas in ''U'' are assumptions for the set of formulas ''V'' that are to be proved. The symbol β is similar to the symbol β’ in Hilbert systems, except that β is part of the object language of the deductive system being formalized, while β’ is a metalanguage notation used to reason about deductive systems."</ref> According to [[Dag Prawitz|Prawitz]] (1965): "The calculi of sequents can be understood as meta-calculi for the deducibility relation in the corresponding systems of natural deduction."<ref>{{harvnb|Prawitz|2006|p=90}}.</ref> And furthermore: "A proof in a calculus of sequents can be looked upon as an instruction on how to construct a corresponding natural deduction."<ref>See {{harvnb|Prawitz|2006|p=91}}, for this and further details of interpretation.</ref> 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.
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