I study definability and types in the linear fragment of continuous logic. Linear variants of several definability theorems such as Beth, Svenonus and Herbrand are proved. At the end, a partial study of the theories of probability algebras, probability algebras with an aperiodic automorphism and AL-spaces is given.

This paper is a general introduction to Positive Logic, where only what we call h-inductive sentences are under consideration, allowing the extension to homomorphisms of model-theoric notions which are classically associated to embeddings; in particular, the existentially closed models, that were primitively defined by Abraham Robinson, become here positively closed models. It accounts for recent results in this domain, and is oriented towards the positivisation of Jonsson theories.