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Intuitionistic first-order predicate logic is not decidable for arbitrary formulas.

However, suppose that we are given a formula of first-order predicate logic that is classically valid. Is there a decision procedure that will determine whether it is also valid intuitionistically?

Adam Dingle
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1 Answers1

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Restricting to classically valid formulas doesn't do anything here. Let $\psi$ be a fixed formula which is classically valid, but not intuitionistically. Now for a given formula $\phi$, consider $\psi \vee \phi$. This formula is classically valid, and it is intuitionistically valid if and only if $\phi$ was.

Arno
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  • Indeed. Thanks for the quick and clear reply. – Adam Dingle Jan 27 '24 at 12:13
  • Worth perhaps adding that the justification of this answer is not quite obvious or trivial: it relies on the disjunction property of IFOL to conclude that if $\psi \vee \phi$ is intuitionistically valid but $\psi$ is not, then $\phi$ must be. (Classically one can of course find formulas such that $\psi \vee \phi$ is valid even though $\psi$ and $\phi$ individually are not.) – Peter LeFanu Lumsdaine Feb 03 '25 at 16:05