We present a notion of classical pure type system, which extends the formalism of pure type system with a double negation operator. 1 Introduction It is an old idea that proofs in formal logics are certain functions and objects. The Brower-Heyting-Kolmogorov (BHK) interpretation [15,51,40], in the form stated by Heyting [40], states that a proof of an implication P ! Q is a "construction " which transforms any proof of P into a proof of Q. This idea was formalized independently by Kleene's realizability interpretation [46,47] in which proofs of intuitionistic number theory are interpreted as numbers, by the Curry-Howard (CH) isomorphism [21,43] in which proofs of intuitionistic implicational propositional logic are interpreted as simply typed -terms, and by the Lambek-Lawvere (LL) isomorphism [52,55] in which proofs of intuitionistic positive propositional logic are interpreted as morphisms in a cartesian closed category. In the latter cases, the interpretations have an inverse, in th...