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Turnstile (symbol)

In mathematical logic and computer science the symbol ⊢ () 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 turnstile represents a binary relation. It has several different interpretations in different contexts: In epistemology, Per Martin-Löf (1996) analyzes the symbol thus: "...[T]he combination of Frege's Urteilsstrich, judgement stroke [ | ], and Inhaltsstrich, content stroke [—], came to be called the assertion sign." Frege's notation for a judgement of some content A can then be read I know A is true. In the same vein, a conditional assertion can be read as: From P, I know that Q In metalogic, the study of formal languages; the turnstile represents syntactic consequence (or "derivability"). This is to say, that it shows that one string can be derived from another in a single step, according to the transformation rules (i.e. the syntax) of some given formal system. As such, the expression means that Q is derivable from P in the system. Consistent with its use for derivability, a "⊢" 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 axioms. As such, the expression means that Q is a theorem in the system. In proof theory, the turnstile is used to denote "provability" or "derivability". For example, if T is a formal theory and S is a particular sentence in the language of the theory then means that S is provable from T. This usage is demonstrated in the article on propositional calculus. The syntactic consequence of provability should be contrasted to semantic consequence, denoted by the double turnstile symbol . One says that is a semantic consequence of , or , when all possible valuations in which is true, is also true. For propositional logic, it may be shown that semantic consequence and derivability are equivalent to one-another.

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Related concepts (10)
Substructural logic
In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are relevance logic and linear logic. In a sequent calculus, one writes each line of a proof as Here the structural rules are rules for rewriting the LHS of the sequent, denoted Γ, initially conceived of as a string (sequence) of propositions.
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