Positive logic for reflection principles in arithmetic

Reflection principles in arithmetic are well-known generalizations of Gödel's sentence expressing the consistency of a given arithmetical theory. Similarly to Gödel's sentence, they yield stronger examples of independent arithmetical statements. From a more general point of view, these schemata can be considered as operations on arithmetical theories. The main result of this talk gives, in a sense, an exhaustive description of algebraic properties of these operations. Connections of these area with modal logic and Kripke models will be discussed.