# Logical Reations for an Impredicative System in a Predicative MetaTheory

Logical Relations for Impredicative languages like System F seem to rely critically on impredicativity of the ambient logic. Specifically, the interpretation for the forall-type will be defined in terms of all typed relations. In an impredicative system (like CiC/Coq) that's fine, but it seems to be impossible in a predicative system (like Agda).

How can this be done? For example, how would you prove normalization for System F in Agda? Do you have to build your own impredicative universe?

In general, what we usually call the logical relations argument isn't really linked to impredicativity: the main idea is simply to interpret terms in some abstract algebra $\cal A$, and to represent types as a ($n$-ary) relation $R \subseteq \cal A^n$.
This works perfectly fine for all kinds of type theories, including dependently typed theories, see e.g. Shürmann and Sarnat: Structural Logical Relations where a predicative logic (that of Twelf) is used to prove a certain property (decidability of equality) for a predicative calculus (simply typed $\lambda$-calculus) using logical relations.
As you may have suspected, however, it is not possible to prove normalization of system F in Agda (if Agda isn't secretly stronger than expected, i.e. about the strength of Martin-Löf type theory with a bunch of universes). This is because normalization of system F implies consistency of 2nd order arithmetic ($\mathrm{PA}_2$) which is stronger than M-L type theory with any number of (predicative) universes.