Source: J. Symbolic Logic Volume 74, Issue 1
(2009), 157-167.
We examine second order intuitionistic propositional logic,
IPC2.
Let ℱ∃ be the set of formulas
with no universal quantification.
We prove Glivenko's theorem for formulas in ℱ∃
that is, for φ∈ℱ∃, φ is a classical tautology
if and only if ¬¬φ is a tautology of IPC2.
We show that for each sentence φ∈ℱ∃ (without
free variables), φ is a classical tautology
if and only if φ is an intuitionistic tautology.
As a corollary we obtain a semantic argument that
the quantifier ∀ is not definable in IPC2
from ⊥, ∨, ∧, →, ∃.
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription.
Read more about accessing full-text
References
T. Arts, Embedding first order predicate logic in second order propositional logic, Master's thesis, University of Njimegen, 1992.
K. Fujita, Galois embedding from polymorphic types into existential types, Proceedings of TLCA 2005 (P. Urzyczyn, editor), LNCS, vol. 3461, Springer, 2005, pp. 194--208.
H. Geuvers, Conservativity between logics and typed lambda-calculi, Types for proofs and programs, international workshop TYPES '93 (H. Barendregt and T. Nipkow, editors), LNCS, vol. 806, Springer, 1994, pp. 131--154.
M. Löb, Embedding first order predicate logic in fragments of intuitionistic logic, Journal of Symbolic Logic, vol. 41(4) (1976), pp. 705--718.
Mathematical Reviews (MathSciNet):
MR441680
T. Połacik, Pitts' quantifiers are not topological quantification, Notre Dame Journal of Formal Logic, vol. 39 (1998), pp. 531--544.
S. K. Sobolev, On the intuitionistic propositional calculus with quantifiers \normalfont(in russian), Matematicheskie Zamietki AN SSSR, vol. 22 (1977), no. 1, pp. 69--76.
Mathematical Reviews (MathSciNet):
MR457155
M. H. Sørensen and P. Urzyczyn, Lectures on the Curry--Howard isomorphism, Elsevier, 2006.
