Model Theory
Section titled “Model Theory”The formalism for the instrumental view.
What it is
Section titled “What it is”Model theory is the branch of mathematical logic that studies the relationship between formal languages and the structures that satisfy them. A model is a structure — a set of objects plus relations and functions — that makes a given set of sentences true. A theory is a set of sentences closed under logical consequence.
The key idea: the same theory can have multiple models, and the same structure can satisfy multiple theories. Model theory studies when two structures are equivalent and when a theory uniquely determines a structure.
See Model theory on Wikipedia.
How the instrumental view uses it
Section titled “How the instrumental view uses it”Under the instrumental view, a mechanism is a predictive model of the network’s behavior — not a claim about internal structure. Model theory provides the natural formalism because it already distinguishes between a theory (the predictions) and its models (the structures that could generate those predictions).
Two instrumental descriptions are the same mechanism when they are elementarily equivalent: they satisfy the same first-order sentences about input-output behavior. This is predictive equivalence formalized.
The instrumental view deliberately avoids claims about what is inside the network. Model theory is the right tool because it is designed to reason about what follows from observations without committing to a specific underlying structure.