Equivalences between Logics and Their Representing Type Theories

We propose a new framework for representing logics, called LF + and based on the Ed-inburgh Logical Framework. The new framework allows us to give, apparently for the rst time, general deenitions which capture how well a logic has been represented. These deen-itions are possible since we are able to distinguish in a generic way that part of the LF + entailment which corresponds to the underlying logic. This distinction does not seem to be possible with other frameworks. Using our deenitions, we show that, for example, natural deduction rst-order logic can be well-represented in LF + , whereas linear and relevant logics cannot. We also show that our syntactic deenitions of representation have a simple formulation as indexed isomorphisms, which both connrms that our approach is a natural one and provides a link between type-theoretic and categorical approaches to frameworks.