Integrating classical and intuitionistic type theory

Intuitionistic Type Theory

Intuitionistic Type Theory (also Constructive Type Theory or Martin-Löf Type Theory) is a formal logical system and philosophical foundation for constructive mathematics (link). It is a fullscale system which aims to play a similar role for constructive mathematics as Zermelo-Fraenkel Set Theory (link) does for classical mathematics. It is based on the propositions-as-types principle and clarif...

Herbrand-type Theorems: the Classical and Intuitionistic Cases

A unified approach to the construction of original forms of the famous Herbrand theorem is suggested for first-order classical and intuitionistic logics without equality. It does not explore skolemization, have wording on deducibility, and as usual, makes the reduction of deducibility in the first-order logics to deducibility in their propositional fragments. The forms use the original notions ...

Constructing Recursion Operators in Intuitionistic Type Theory

Martin-Löf’s Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of well-founded relations. Included are < on natural numbers, and rela...

Intuitionistic choice and classical logic

Combining Classical and Intuitionistic Implications

We present a simple logic that combines, in a conservative way, the implicative fragments of both classical and intuitionistic logics, thus settling a problem posed by Dov Gabbay in [5]. We also show that the logic can be given a nice complete axiomatization by adding four simple mixed axioms to the usual axiomatizations of classical and intuitionistic implications.