Journal of Symbolic Logic

On second order intuitionistic propositional logic without a universal quantifier

Abstract

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 ⊥, ∨, ∧, →, ∃.

Article information

Source
J. Symbolic Logic, Volume 74, Issue 1 (2009), 157-167.

Dates
First available in Project Euclid: 4 January 2009

https://projecteuclid.org/euclid.jsl/1231082306

Digital Object Identifier
doi:10.2178/jsl/1231082306

Mathematical Reviews number (MathSciNet)
MR2499424

Zentralblatt MATH identifier
1163.03010

Citation

Zdanowski, Konrad. On second order intuitionistic propositional logic without a universal quantifier. J. Symbolic Logic 74 (2009), no. 1, 157--167. doi:10.2178/jsl/1231082306. https://projecteuclid.org/euclid.jsl/1231082306