Ivan di Liberti: Context, Judgement, Deduction

Abstract

We introduce judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we recover type theory and proof theory as special kinds of judgemental theories. Our analysis sheds light on both the topics, providing a new point of view. In the case of type theory, we provide an abstract definition of type constructor featuring the usual formation, introduction, elimination and computation rules. In proof theory we offer a deep analysis of structural rules, demystifying some of their properties, and putting them into context. We finish the paper discussing the internal logic of a topos, a predicative topos, an elementary 2-topos et similia, and show how these can be organized in judgemental theories.

Based on a joint work with Greta Coraglia: https://arxiv.org/abs/2111.09438.