How does intuitionistic logic differ from classical logic?

In the philosophy of logic, the primary distinction between classical logic and intuitionistic logic lies in their approach to truth and mathematical existence. While classical logic assumes that every proposition is either true or false, intuitionism—pioneered by L.E.J. Brouwer—contends that mathematical statements are only meaningful if they can be constructively established or "known" through mental construction.123

The most significant formal difference is the rejection of the law of excluded middle in intuitionistic systems, particularly when applied to infinite sets.14

• Classical Logic: Accepts the principle that for any proposition $p$, either $p$ or $~p$ must be true, regardless of whether a proof for either exists.14
• Intuitionistic Logic: Refuses to accept "Either $p$ or not-$p$" as a universal logical truth. Because intuitionists require a constructive proof to assert a truth, they reject this principle in cases where no such proof is currently available, such as with certain properties of infinite sets.24

Consequently, intuitionistic logic also rejects the principle of double negation elimination ($\neg \neg p\supset p$), which is a standard theorem in classical systems.4