This paper addresses a recent debate as to whether logical anti-exceptionalists should understand
logical theories in syntactic or semantic terms. InWigglesworth (2017), I propose a purely semantic
approach, while Woods (2018) has argued in favor of a purely syntactic approach. Here, I argue
that neither of these approaches is satisfactory, as both treat arguably distinct logics as equivalent
logical theories. I argue instead for an approach that combines syntactic and semantic components.
The specific approach to a combined account of logical theories is based on the category-theoretic
notion of an institution.