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http://en.wikipedia.org/wiki/Intuitionistic_logic
Incidentally (a further parallel to the Axiom of Choice) Intuitionist Logic is also referred to as Constructive Logic.
http://en.wikipedia.org/wiki/Constructivist_mathematics
One branch of constructivist mathematics eschews the axiom of choice, and all do not allow proof of existence by RAA on non-existence.
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http://en.wikipedia.org/wiki/Intuitionistic_logic
Incidentally (a further parallel to the Axiom of Choice) Intuitionist Logic is also referred to as Constructive Logic.
http://en.wikipedia.org/wiki/Constructivist_mathematics
One branch of constructivist mathematics eschews the axiom of choice, and all do not allow proof of existence by RAA on non-existence.