The starting point for Bain’s specific argument is a category-theoretic version of the ‘semantic view of theories’ on which a scientific theory is identified with its category of models—indeed this will be the setup assumed in the following argument. Note two points. First, we are not using ‘model’ here in the strict sense of model theory, but rather to mean a mathematical structure that represents a physical world that is possible according to the theory. Second, this proposal is not to be confused with Lawvere’s category-theoretic formulation of algebraic theories (* Lawvere*). In the latter, models are functors between categories, whereas in the former, the models are just objects of some category (not necessarily a functor category) such as Top. Indeed, Lawvere’s proposal is much more closely related to—though not the same as—the Topological Quantum Field Theories.

S3: T2′ cannot be translated back into T1 and so its models do not contain T1 -objects.

S4: T2′ is relevant for modeling some physical scenarios.

When taken together, S1-S4 are supposed to show that:

In Bain’s example, T1 is the category of geometric models of general relativity (GTR), and T2 is the category of Einstein algebra (EA) models of GTR (Einstein algebras were first introduced as models of GR in * Geroch 1972*). Bain is working with the idea that in geometric models of GTR, the relata, or O-objects, of GTR are space-time points of the manifold.

Finally, as regards S4, various authors have taken the utility of T2′ to be e.g. the in- clusion of singularities in space-time, and as a step towards formulating quantum gravity (* Geroch*).

We now turn to considering the inference to CS. It is not entirely clear what Bain means by ‘[the relata] do not play an essential role’ – nor does he expand on this phrase – but the most straightforward reading is that T1-objects are eliminated simpliciter from T2∗.

One might compare this situation to the way that the collection of all groups (analogous to T2) is contained in the collection of all monoids (analogous to T2∗): it might be claimed that inverses are eliminated from the collection of all monoids. One could of course speak in this way, but what this would mean is that some monoids (in particular, groups) have inverses, and some do not – a ‘monoid’ is just a general term that covers both cases. Similarly, we can see that CS does not follow from S1–S3, since T2∗ contains some models that (implicitly) quantify over T1-objects, viz. the models of T2, and some that do not, viz. the models of T2′.

We have seen that the specific argument will not work if one is concerned with eliminating reference to T1-objects from the new and more general theory T2∗. However, what if one is concerned not with eliminating reference, but rather with downgrading the role that T1-objects play in T2∗, e.g. by claiming that the models of T2′ have a conceptual or metaphysical priority? And what would such a ‘downgrading’ even amount to?