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&_160;[disambiguation needed] 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. The types of Type Theory play a similar role to sets in set theory but functions definable in Type Theory are always computable.
In the context of Type Theory a connective is a way of constructing types, possibly using already given types. The basic connectives of Type Theory are
Of special importance are 0 or ? (the empty type), 1 or ? (the unit type) and 2 (the type of Booleans or classical truth values). Invoking the Curry-Howard isomorphism again, ? stands for False and ? for True.
