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| {{distinguish|ㅏ|ト|卜}}
| | Yoshiko is her title but she doesn't like when people use her full name. Managing people is how I make cash and it's something I really enjoy. One of his favorite hobbies is taking part in crochet but he hasn't produced a dime with it. Delaware is the only place I've been residing in.<br><br>Here is my site; [http://Vendorportal.citypower.Co.za/Activity-Feed/My-Profile/UserId/18066 Vendorportal.citypower.Co.za] |
| In [[mathematical logic]] and [[computer science]] the symbol <math>\vdash</math> has taken the name '''turnstile''' because of its resemblance to a typical [[turnstile]] if viewed from above. It is also referred to as '''tee''' and is often read as "yields", "proves", "satisfies" or "entails". The symbol was first used by [[Gottlob Frege]] in his 1879 book on logic, ''[[Begriffsschrift]]''.<ref>[[Gottlob Frege]], Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle, 1879.</ref>
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| In [[TeX]], the turnstile symbol <math>\vdash</math> is obtained from the command <tt>\vdash</tt>. In [[Unicode]], the turnstile symbol (<big>{{unicode|⊢}}</big>) is called '''right tack''' and is at code point U+22A2.<ref>[http://unicode.org/charts/PDF/U2200.pdf Unicode standard]</ref> On a [[typewriter]], a turnstile can be composed from a [[vertical bar]] (|) and a [[dash]] (–). In [[LaTeX]] there is a turnstile package which issues this sign in many ways, and is capable of putting labels below or above it, in the correct places.<ref>http://www.ctan.org/tex-archive/macros/latex/contrib/turnstile</ref>
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| == Interpretations ==
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| The turnstile represents a [[binary relation]]. It has several different [[interpretation (logic)|interpretations]] in different contexts:
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| * In [[metalogic]], the study of [[formal language]]s; the turnstile represents [[Logical consequence|syntactic consequence]] (or "derivability"). This is to say, that it shows that one string can be [[formal proof|derived]] from another in a single step, according to the [[rule of inference|transformation rules]] (i.e. the [[syntax (logic)|syntax]]) of some given [[formal system]].<ref>http://dingo.sbs.arizona.edu/~hammond/ling178-sp06/mathCh6.pdf</ref> As such, the expression
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| :<math>P \vdash Q</math>
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| :means that <math>Q</math> is derivable from <math>P</math> in the system.
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| :Consistent with its use for derivability, a "<math>\vdash</math>" followed by an expression without anything preceding it denotes a [[theorem]], which is to say that the expression can be derived from the rules using an [[empty set]] of [[axiom]]s. As such, the expression
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| :<math>\vdash Q</math>
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| :means that <math>Q</math> is a theorem in the system.
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| *In [[proof theory]], the turnstile is used to denote "provability". For example, if <math>T</math> is a [[Theory (mathematical logic)|formal theory]] and <math>S</math> is a particular sentence in the language of the theory then
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| :<math>T \vdash S</math>
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| :means that <math>S</math> is [[formal proof|provable]] from <math>T</math>.<ref>[[A. S. Troelstra]] and [[H. Schwichtenberg]], ''Basic Proof Theory'', second edition, [[Cambridge University Press]], 2000, ISBN 978-0-521-77911-1.</ref> This usage is demonstrated in the article on [[propositional calculus]].
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| * In the [[typed lambda calculus]], the turnstile is used to separate typing assumptions from the typing judgment.<ref>http://www.mscs.dal.ca/~selinger/papers/lambdanotes.pdf</ref><ref>David A. Schmidt, The Structure of Typed Programming Languages, [[MIT Press]], 1994, ISBN 0-262-19349-3</ref>
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| * In [[category theory]], a reversed turnstile ([[⊣]]), as in <math>F \dashv G</math>, is used to indicate that the [[functor]] <math>F</math> is [[adjoint functors|left adjoint]] to the functor <math>G</math>.
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| * In [[APL_(programming_language)|APL]] the symbol is called "right tack" and represents the ambivalent right identity function where both ''X⊢Y'' and ''⊢Y'' are ''Y''. The reversed symbol "⊣" is called "left tack" and represents the analogous left identity where ''X⊣Y'' is ''X'' and ''⊣Y'' is ''Y''. <ref>http://www.jsoftware.com/papers/APLDictionary.htm</ref><ref>[[Kenneth E. Iverson]]. A Dictionary of APL. 1987</ref>
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| ==See also==
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| *[[Double turnstile]]
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| *[[Sequent]]
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| *[[Sequent calculus]]
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| *[[List of logic symbols]]
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| *[[List of mathematical symbols]]
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| == Notes ==
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| {{reflist}}
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| {{logic}}
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| [[Category:Mathematical symbols]]
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| [[Category:Mathematical logic]]
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| [[Category:Logic symbols]]
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| [[Category:Deductive reasoning]]
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| [[Category:Proof theory]]
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| [[Category:Logical consequence]]
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Yoshiko is her title but she doesn't like when people use her full name. Managing people is how I make cash and it's something I really enjoy. One of his favorite hobbies is taking part in crochet but he hasn't produced a dime with it. Delaware is the only place I've been residing in.
Here is my site; Vendorportal.citypower.Co.za