This paper is an investigation into what could be a goodexplication of ``theory S is reducible to theory T''''. Ipresent an axiomatic approach to reducibility, which is developedmetamathematically and used to evaluate most of the definitionsof ``reducible'''' found in the relevant literature. Among these,relative interpretability turns out to be most convincing as ageneral reducibility concept, proof-theoreticalreducibility being its only serious competitor left. Thisrelation is analyzed in some detail, both from the point of viewof the reducibility axioms and of modal logic.

A definition of elementary interpretation, equivalent (up to isomorphisms) to the ones of [3] and [4], is given. The defining condition, used here, seems to confirm that intuitions agree with the choice of the class of elementary interpretations, which was done in [3].

Glymour and Quine propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria.