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- Title
GLIVENKO AND KURODA FOR SIMPLE TYPE THEORY.
- Authors
BROWN, CHAD E.; RIZKALLAH, CHRISTINE
- Abstract
Glivenko's theorem states that an arbitrary propositional formula is classically provable if and only if its double negation is intuitionistically provable. The result does not extend to full first-order predicate logic, but does extend to first-order predicate logic without the universal quantifier. A recent paper by Zdanowski shows that Glivenko's theorem also holds for second-order propositional logic without the universal quantifier. We prove that Glivenko's theorem extends to some versions of simple type theory without the universal quantifier. Moreover, we prove that Kuroda's negative translation, which is known to embed classical first-order logic into intuitionistic first-order logic, extends to the same versions of simple type theory. We also prove that the Glivenko property fails for simple type theory once a weak form of functional extensionality is included.
- Subjects
PROPOSITIONAL calculus; FIRST-order logic; TYPE theory; NEGATION (Logic); MATHEMATICAL logic
- Publication
Journal of Symbolic Logic, 2014, Vol 79, Issue 2, p485
- ISSN
0022-4812
- Publication type
Article
- DOI
10.1017/jsl.2013.10