Interpretability of theories

We interpret one theory in another when we can translate all the concepts and structures of the first theory into the language and structural resources of the host theory.

Let us continue our investigation of interpretability between models and theories.

In previous posts, we had introduced the basic concept of Interpretability with numerous examples, before moving on to the Interpretation Translation and also the concept of Mutual and bi-interpretation of models.

In this post, I shall expand the interpretability concept beyond models to consider the interpretation of one theory in another. To interpret one theory in another just means to provide a uniform manner of interpreting a model of the interpreted theory inside any given model of the host theory.

Many logicians, to be sure, take this case of theory interpretation as the main case of interpretability, and these logicians focus principally or even exclusively on the interpretation of theories when investigating interpretability.

My perspective, however, in keeping with my general inclination toward semantical concepts in logic, is that it is the interpretation of models that is fundamental for understanding interpretability. And I view the interpretation of theories accordingly through the semantic lens—an interpretation of a target theory S inside a host theory T, as we said, is simply a uniform method of interpreting a model of S inside any given model of T.

So let us develop the theory of the interpretation of theories.

This is part of series of installments on interpretability drawn from my book-in-progress A Panorama of Logic, an introduction to topics in logic for philosophers, mathematicians, and computer scientists, serialized here on Infinitely More. Fresh content each week.

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The main concept is that we shall in order to interpret one theory S inside another theory T, one should provide a uniform way of interpreting the objects and structural features of the first theory S inside the host theory T.

The host theory T is thus able to tell us all about a certain simulated version of the interpreted theory S. The host theory provides us with a fully detailed account of objects and structural features that constitute a realm in which S is true, using only the language and structural features of the host theory.

Let me be more precise.