Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either. In constructive logic, a statement is 'only true' if there is a constructive proof that it is true, and 'only false' if there is a constructive proof that it is false. Operations in constructive logic preserve justification, rather than truth.
Syntactically, intuitionistic logic is a restriction of classical logic in which the law of excluded middle and double negation elimination are not axioms of the system, and cannot be proved. There are several semantics commonly employed. One semantics mirrors classical Boolean-valued semantics but uses Heyting algebras in place of Boolean algebras. Another semantics uses Kripke models.
Constructive logic is practically useful because its restrictions produce proofs that have the existence property, making it also suitable for other forms of mathematical constructivism. Informally, this means that given a constructive proof that an object exists, then that constructive proof can be turned into an algorithm for generating an example of it.
Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism.
Read more about Intuitionistic Logic: Syntax, Semantics, Relation To Other Logics
Famous quotes containing the word logic:
“...some sort of false logic has crept into our schools, for the people whom I have seen doing housework or cooking know nothing of botany or chemistry, and the people who know botany and chemistry do not cook or sweep. The conclusion seems to be, if one knows chemistry she must not cook or do housework.”
—Ellen Henrietta Swallow Richards (1842–1911)