A type-theoretic framework for formal reasoning with different logical foundations is introduced and studied. With logic-enriched type theories formulated in a logical framework, it allows various logical systems such as classical logic as well as intuitionistic logic to be used effectively alongside inductive data types and type universes. This provides an adequate basis for wider applications...

We present a simple-to-state, abstract computational problem, whose solution implies the 1consistency of various systems of predicative Analysis and offers a way of extracting witnesses from classical proofs. In order to state the problem, we formulate the concept of transfinite update procedure, which extends Avigad’s notion of update procedure to the transfinite and can be seen as an axiomati...

We advocate a pragmatic approach to constructive set theory, using axioms based solely on set-theoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories we consider all have sound and complete classes of category-theoretic mode...

We investigate possible extensions of arbitrary given Pure Type Systems with additional sorts and rules which preserve the normalization property. In particular we identify the following interesting extensions: the disjoint union P + Q of two PTSs P and Q, the PTS ∀P.Q which intuitively captures the “Q-logic of P-terms” and PPoly which intuitively denotes the predicative polymorphism extension ...