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GLIVENKO AND KURODA FOR SIMPLE TYPE THEORY

Published online by Cambridge University Press:  25 June 2014

CHAD E. BROWN  and
CHRISTINE RIZKALLAH
CHAD E. BROWN
Affiliation:
PROGRAMMING SYSTEMS LAB UNIVERSITÄT DES SAARLANDES CAMPUS E1 3, 66123 SAARBRÜCKEN GERMANYE-mail: cebrown@ps.uni-saarland.de
CHRISTINE RIZKALLAH
Affiliation:
MAX-PLANK INSTITUT FÜR INFORMATIK CAMPUS E1 4, 66123 SAARBRÜCKEN GERMANYE-mail: crizkall@mpi-inf.mpg.de
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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.

Keywords

F.4.1I.2.3.simple type theoryelementary type theoryhigher-order logicGlivenko’s Theorem
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 2 , June 2014 , pp. 485 - 495
Copyright
Copyright © The Association for Symbolic Logic 2014 

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