Remarks on abstract structures of propositions and realizers

Abstract

We present here an abstract notion of structure consisting of propositions and realizers (which we call PR-structures) giving rise to set based contravariant functors taking values in the category of sets endowed with binary relations. We will characterize those PR-structures giving rise to preorderal and posetal doctrines and we will study in particular the case of a PR-structure induced by a partial applicative structure.