Define an entailment relation for quantitative predicates

Develop an explicit entailment relation for the sets of quantitative predicates defined in the paper—compatible with the intended quantitative semantics—so that the syntax can be quotiented by rewrites or governed by a coherent proof system.

Background

The paper specifies a syntax for multiplicative and additive quantitative predicates and provides their semantics via p-means and integrals. To turn this syntax into a usable logic, an entailment relation (or proof-theoretic framework) is required.

The authors highlight that it is unclear how such an entailment should be formulated. A later section attempts categorical semantics via enriched hyperdoctrines, but those attempts fail, reinforcing the need for an appropriate entailment framework.