Source: J. Symbolic Logic
Volume 74, Issue 1
We examine second order intuitionistic propositional logic,
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 ⊥, ∨, ∧, →, ∃.
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