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(the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). ⊃ {\displaystyle \supset } may mean the same as ⇒ {\displaystyle \Rightarrow } (the symbol may also mean superset ).
The turnstile symbol was originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935). [9] Syntactic consequence does not depend on any interpretation of the formal system. [10]
In logical argument and mathematical proof, the therefore sign, ∴, is generally used before a logical consequence, such as the conclusion of a syllogism. The symbol consists of three dots placed in an upright triangle and is read therefore. While it is not generally used in formal writing, it is used in mathematics and shorthand.
Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is ...
A symbol is an idea, abstraction or concept, tokens of which may be marks or a metalanguage of marks which form a particular pattern. Symbols of a formal language need not be symbols of anything. For instance there are logical constants which do not refer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses ...
In logic, the symbol ⊨, ⊧ or is called the double turnstile.It is often read as "entails", "models", "is a semantic consequence of" or "is stronger than". [1]It is closely related to the turnstile symbol , which has a single bar across the middle, and which denotes syntactic consequence (in contrast to semantic).
The inference line represents syntactic consequence, [15] sometimes called deductive consequence, [45] which is also symbolized with ⊢. [46] [45] So the above can also be written in one line as ,. [h] Syntactic consequence is contrasted with semantic consequence, [47] which is symbolized with ⊧.
However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related.