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.

Parameterized Metareasoning in Membership Equational Logic

“…As a consequence, in a reflective logic, metatheorems involving families of theories can be represented and logically proved as theorems about its universal theory. Basin, Clavel, and Meseguer showed in [1] that logical frameworks can be good metalogical frameworks when their theories always have initial models and they support reflective and parameterized reasoning; they also showed that membership equational logic is a particular logical framework that satisfies these requirements. In this paper, we extend their ideas and apply them to the (parameterized) deduction theorem.…”

“…More concretely, a logic is reflective when there exists a universal theory in which we can represent and reason about all finitely presentable theories in the logic, including the universal theory itself [5]. As a consequence, in a reflective logic, metatheorems involving families of theories can be represented and proved as theorems about its universal theory [1]. A universal theory U MEL for membership equational logic is described in [6], along with a representation function ( ) that encodes pairs, consisting of a finitely presentable membership equational theory with nonempty kinds and a sentence in it, as sentences in U MEL .…”

“…In the previous section we have recalled an inductive principle to reason about terms in a family of theories, which constitutes both an application of the ideas introduced in [1] as well as a generalization. 2 In this section we turn our attention to parameterization, which was already studied in [1] using the deduction theorem as a case study.…”

“…2 In this section we turn our attention to parameterization, which was already studied in [1] using the deduction theorem as a case study. Here we consider a generalization of the parameter theories and of the corresponding inductive principle, and we use them to formalize two versions of the deduction theorem not expressible in the formalisms presented in [1,4].…”

“…Basin, Clavel, and Meseguer showed in [1] that membership equational logic is a good metalogical framework because of its initial models and support of reflective reasoning. A development and an application of those ideas was presented later in [4].…”