Logical Foundations of Quantitative Equality

In quantitative reasoning one compares objects by distances, instead of equivalence relations, so that one can measure how much they are similar, rather than just saying whether they are equivalent or not. In this paper we aim at providing a logical ground to quantitative reasoning with distances in Linear Logic, using the categorical language of Lawvere's doctrines. The key idea is to see distances as equality predicates in Linear Logic. We use graded modalities to write a resource sensitive substitution rule for equality, which allows us to give it a quantitative meaning by distances. We introduce a deductive calculus for (Graded) Linear Logic with quantitative equality and the notion of Lipschitz doctrine to give it a sound and complete categorical semantics. We also describe a universal construction of Lipschitz doctrines, which generates examples based for instance on metric spaces and quantitative realisability.