In intuitionistic systems, ¬∃xφ(x) means no construction can verify ∃xφ(x), while ∀x¬φ(x) requires a uniform construction for each x, and these are not interderivable without the law of excluded middle.

?Rate how convincing each reason is below to see the overall strength.

No one has weighed in yet. Be the first to share reasons for or against this statement.

Construction (in intuitionistic logic)(what counts as 'true' in intuitionistic systems): A concrete step-by-step procedure or proof that shows something is true, rather than just asserting it—you have to actually build or demonstrate it.
Interderivable(the relationship between the two statements being compared): Two statements are interderivable if you can logically prove one from the other, and vice versa—meaning they're logically equivalent.
intuitionistic logic(Formal logic differing from classical logic in its treatment of structural rules and logical consequence): A logic that does not accept all classical structural rules, thereby invalidating certain classically valid formulas such as (p → q) ∨ p
law of excluded middle(Classical logic; shown to be incompatible with smooth infinitesimal analysis): The classical logical principle that for any proposition, either the proposition or its negation holds — applied here as: every real number is either equal to 0 or not equal to 0.

Based on where you are in your exploration