Abstract: The problem of co-hyperintensionality asks us to find a (maximally coarse-grained) relation between the sentences of a given language such that applying any operator to sentences in that relation yields equivalent statements. For example, which relation has to hold between the sentences φ and ψ such that ‘A knows φ’ is equivalent to ‘A knows ψ’? The aim is to thus find the adequate notion of synonymy inside hyperintensional contexts like knowledge, belief, or explanation.

The problem presupposes a language with a notion of sentence equivalence. We define the ‘hyperintensionalization’ of such an equivalence and show that it is the unique co-hyperintensionality relation—thus, reducing hyperintensionality to equivalence (or validity). We obtain various results about the operation of hyperintensionalization concerning stability, fixed points, congruences, and compositionality. Moreover, ordering operators by their ability to differentiate sentences is not linear (contrary to what is often tacitly assumed). These results elucidate the structure of hyperintensionality independent of particular semantic frameworks.

However, this also demonstrates the limits of hyperintensionality: It cannot provide a universal notion of content identity, since fine-graining semantics has no unique stopping point. It arguably cannot capture the everyday notion of (defeasible) synonymy. And, on the standard conception, it cannot account for the non-transitive aspects of certain intentional contexts, but we provide a revised counterfactual conception that can.