Type theory is a family of formal systems ranging from programming language semantics to the foundations of mathematics. In practice, type theories are defined by means of “inference rules”. Everyone in the community understands them to some extent, but they do not have any commonly accepted rigorous interpretation. Or, rather, they have several interpretations, none of which is entirely satisfactory.
In this work, after a brief overview of the literature, we introduce a rigorous, semantic notion of inference rule, our thesis being that most syntactic inference rules written in practice may be directly interpreted in this framework. If time permits, we will sketch how this covers quantitative type theories.
This is joint work in progress with André Hirschowitz and Ambroise Lafont.