Intuitionistic logic

A logic in which a proposition is established by constructing evidence for it, so the law of excluded middle is not accepted without proof.

Proposition: A declarative statement that can be treated as true or false within a logical system. Intuitionistic logic treats propositions as claims whose proofs provide evidence.
Natural deduction: A proof system organized around introduction and elimination rules for logical connectives. Its rules give intuitionistic logic a direct account of constructing and using proofs.
Classical logic: A logic that accepts principles such as unrestricted excluded middle and double-negation elimination. It validates inferences that intuitionistic logic rejects when no constructive proof is available.
Constructive mathematics: Mathematics that requires proofs of existence to provide constructions or effective procedures. It uses intuitionistic reasoning to ensure existence claims yield explicit mathematical objects.
L. E. J. Brouwer: A Dutch mathematician who founded intuitionism and challenged classical treatments of mathematical truth. His philosophy of mathematics motivated rejecting proofs that provide no construction.
Proof theory: The study of formal proofs, their structure, and the rules that generate them. Its proof-centered interpretation makes proof structure central to the logic.
Brouwer–Heyting–Kolmogorov interpretation: An informal account of logical connectives in terms of the constructions that count as their proofs. It spells out the constructive meaning of intuitionistic conjunction, disjunction, and implication.
Minimal logic: A constructive logic weaker than intuitionistic logic because it does not derive arbitrary conclusions from contradiction. It differs at the treatment of negation and contradiction while sharing a constructive proof outlook.
Type theory: A formal system that classifies expressions by types and can serve as a foundation for mathematics and programming. Intuitionistic logic underlies the propositions-as-types interpretation used in type-theoretic foundations.
Arend Heyting: A Dutch mathematician who formalized intuitionistic logic and intuitionistic mathematics. His formal systems gave precise rules to Brouwer's constructive ideas.

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