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On second order intuitionistic propositional logic without a universal quantifier

Published online by Cambridge University Press:  12 March 2014

Konrad Zdanowski
Konrad Zdanowski*
Affiliation:
Instytut Matematyczny Pan, Sniadeckich 8, 00-956 Warszawa, Poland, E-mail: K.Zdanowski@impan.gov.pl
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Abstract

We examine second order intuitionistic propositional logic, IPC2. Let ℱ∃ a 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 ⊥, ⋁, ⋀, →, ∃.

Keywords

second order intuitionistic propositional logicGlivenko theorem
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 1 , March 2009 , pp. 157 - 167
Copyright
Copyright © Association for Symbolic Logic 2009

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References

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