A mathematical characterization of interpretation between theories

Of the various notions of reduction in the logical literature, relative interpretability in the sense of Tarskiet al. [6] appears to be the central one. In the present note, this syntactic notion is characterized semantically, through the existence of a suitable reduction functor on models. The latter mathematical condition itself suggests a natural generalization, whose syntactic equivalent turns out to be a notion of interpretability quite close to that of Ershov [1], Szczerba [5] and Gaifman [2]. This author's contribution was completed during a fellowship held at The Netherlands Institute for Advanced Study, 1982–3. Both authors would like to thank Wilfrid Hodges for his valuable comments and advice concerning an earlier version of this paper.