Intuitionistic Type Theory
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Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher, in 1972. Martin-Löf has modified his proposal a few times; his early, impredicative formulations were inconsistent as demonstrated by Girard's paradox, and later formulations were predicative. He also proposed extensional and then intensional variants of intuitionistic type theory. Intuitionistic t...
Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher, in 1972. Martin-Löf has modified his proposal a few times; his early, impredicative formulations were inconsistent as demonstrated by Girard's paradox, and later formulations were predicative. He also proposed extensional and then intensional variants of intuitionistic type theory. Intuitionistic type theory is based on a certain analogy or isomorphism between propositions and types: a proposition is identified with the type of its proofs. This identification is usually called the Curry Howard isomorphism, which was originally formulated for intuitionistic logic and simply typed lambda calculus. Type Theory extends this identification to predicate logic by introducing dependent types, that is types which contain values. Type Theory internalizes the interpretation of intuitionistic logic proposed by Brouwer, Heyting and Kolmogorov, the so called BHK interpretation.
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