Deriving Logical Relations from Interpretations of Predicate Logic

This paper extends the results of Hermida's thesis about logical predicates to more general logical relations and a wider collection of types. The extension of type constructors from types to logical relations is derived from an interpretation of those constructors on a model of predicate logic. This is then further extended to n-ary relations by pullback. Hermida's theory shows how right adjoints in the category of fibrations are composed from a combination of Cartesian lifting and a local adjunction. This result is generalised to make it more applicable to left adjoints, and then shown to be stable under pullback, deriving an account of n-ary relations from standard predicate logic. A brief discussion of lifting monads to predicates includes the existence of an initial such lifting, generalising existing results.