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Consider the concept of a module. This can be understood a multisorted algebraic theory $\mathsf{Mod}$ on two sorts, a scalarsort $S$ and a vectorsort $V$. An interpretation of $\mathsf{Mod}$ in $\mathbf{Top}$ assigns to the scalarsort a topological field, to the vectorsort a topological Abelian group, and to the scalar multiplication arrow $S \times V \rightarrow V$ a continuous function satisfying appropriate compatibility constraints.

But we can also imagine partial interpretations of $\mathsf{Mod}$ in $\mathbf{Top}.$ E.g. Assign to the scalarsort a topological field $K$, but leave the action and vectorsort unspecified. The result is "a partially interpreted theory" that we might call $K\text{-}\mathsf{Mod}.$ Define that a model of $K\text{-}\mathsf{Mod}$ is just a model of $\mathsf{Mod}$ in $\mathbf{Top}$ such that the vectorsort is assigned the topological field $K$. It follows that a model of $K\text{-}\mathsf{Mod}$ is just a topological $K$-module.

Is anyone talking about this kind of thing, where we actually deal with partial interpretations of theories?

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  • $\begingroup$ Sure, in essence you are admitting a lot of constants into the theory, one for each element of $K$, or alternatively a lot of operations -- one scalar multiplication for each element of $K$. In categorical logic this is a natural thing to do. $\endgroup$
    – Andrej Bauer
    Commented Jul 3, 2014 at 15:38
  • $\begingroup$ @Andrej, wouldn't it be more a kind of Skolemization, especially if the vectors were assigned first? Also, isn't this the kind of thing occurring in CS, possibly in type resolution (or whatever magic happens) for classes in object-oriented programming? Gerhard "Partially Interpreting The Concepts Here" Paseman, 2014.07.03 $\endgroup$
    – Gerhard Paseman
    Commented Jul 3, 2014 at 16:14
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    $\begingroup$ I guess you're not just adding constant symbols, you're also restricting attention to models which omit certain types (e.g., field elements not named by a constant) - but that's still a natural thing to do. $\endgroup$
    – Noah Schweber
    Commented Jul 3, 2014 at 16:21
  • $\begingroup$ @AndrejBauer, the problem with adding constant symbols is that you lose all the structure of $K$ that cannot be expressed in the language of interest, such as the topological structure. I've changed the question a bit to reflect this concern. $\endgroup$
    – goblin GONE
    Commented Jul 4, 2014 at 3:21
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    $\begingroup$ The difference between adding constants for $K$ and having a partial interpretation is that with the partial interpretation you can reliably quantify over $K$, but with the constants you can only quantify over the sort that includes $K$, but might possibly include much more. $\endgroup$
    – Joel David Hamkins
    Commented Jul 4, 2014 at 3:26

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One common example where this arises is in the consideration of $\omega$-models of set theory. These are precisely the models that interpret their natural numbers as the standard natural numbers $\omega$, or $\mathbb{N}$. Thus, the class of $\omega$-models of a given set theory are essentially the partial interpretations of set theory, where we have fixed a sort for the natural numbers to be interpreted only as the standard natural numbers.

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