Reflective metalogical frameworks

Abstract

A metalogical framework is a logic with an associated methodology that is used to represent other logics and to reason about their metalogical properties. We propose that logical frameworks can be good metalogical frameworks when their theories always have initial models and they support reflective and parameterized reasoning.We develop this thesis both abstractly and concretely. Abstractly, we formalize our proposal as a set of requirements and explain how any logic satisfying these requirements can be used for metalogical reasoning. Concretely, we present membership equational logic as a particular metalogic that satisfies these requirements. Using membership equational logic, and its realization in the Maude system, we show how reflection can be used for different, nontrivial kinds of formal metatheoretic reasoning. In particular, one can prove metatheorems that relate theories or establish properties of parameterized classes of theories.