Are interpretation and formalization inverse to each other?

Question

I am not sure if I understand this correctly. Please correct me.

In a formal system,

an interpretation is a mapping from its formal language to one of its structures ie models.

an formalization is a mapping from one of its models to its formal language?

So are they inverse processes to each other, in the sense that their domains and codomains are exchanged?

Thanks!

Answer 1

I suspect that the structure that you seek is the Galois connection between theories and models. A web search on such should turn up many expositions, e.g. in a slightly more general context, see Section 2.2 in Goguen and Burstall: INSTITUTIONS: Abstract Model Theory for Specification and Programming.