This link has been bookmarked by 16 people . It was first bookmarked on 07 Apr 2007, by x y.
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07 Oct 14
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to recognize valid formulas, it is sufficient to consider a single Heyting algebra whose elements are the open subsets of the real line R.[4] In this algebra, the ∧ and ∨ operations correspond to set intersection and union, and the value assigned to a formula A → B is int(AC ∪ B), the interior of the union of the value of B and the complement of the value of A. The bottom element is the empty set ∅, and the top element is the entire line R. The negation ¬A of a formula A is (as usual) defined to be A → ∅. The value of ¬A then reduces to int(AC), the interior of the complement of the value of A, also known as the exterior of A. With these assignments, intuitionistically valid formulas are precisely those that are assigned the value of the entire line.[4]
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23 Feb 14
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26 Jul 13Dante-Gabryell Monson
"Intuitionistic logic, sometimes more generally called constructive logic, is a system of symbolic logic that differs from classical logic by replacing the traditional concept of truth with the concept of constructive provability. For example, in classical logic, propositional formulae are always assigned a truth value from the two element set of trivial propositions ("true" and "false" respectively) regardless of whether we have direct evidence for either case. In contrast, propositional formulae in intuitionistic logic are not assigned any definite truth value at all and instead only considered "true" when we have direct evidence, hence proof. (We can also say, instead of the propositional formula being "true" due to direct evidence, that it is inhabited by a proof in the Curry-Howard sense.) Operations in intuitionistic logic therefore preserve justification, with respect to evidence and provability, rather than truth-valuation."
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06 Nov 12
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03 Aug 11
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08 Apr 11
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15 Nov 10
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what it means for a statement to be true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either. In constructive logic, a statement is only true if there is a proof that it is true, and only false if there is a proof that it is false. Operations in constructive logic preserve justification, rather than truth. Syntactically, intuitionist logic differs from classical logic in that the law of excluded middle and double negation elimination are not axioms of the system, and cannot be proved in it.
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02 Sep 10
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21 Nov 09
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07 Apr 07
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