We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both proof- theoretic and model-theoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quantifiers, although not present in the definition of logic and logic translation, can be recovered by their abstract (...) properties and are preserved and reflected by translations under suitable conditions. (shrink)

In this essay we analyse and elucidate the method to establish and clarify the scope of logic theorems offered within the theory of institutions. The method presented pervades a lot of abstract model theoretic developments carried out within institution theory. The power of the proposed general method is illustrated with the examples of interpolation and definability, as they appear in the literature of institutional model theory. Both case studies illustrate a considerable extension of the original scopes of the two classical (...) theorems. Our presentation is rather narrative with the relevant logic and institution theory concepts introduced and explained gradually to the non-expert reader. (shrink)

The purpose of a logical framework such as LF is to provide a language for defining logical systems suitable for use in a logic-independent proof development environment. All inferential activity in an object logic is to be conducted in the logical framework via the representation of that logic in the framework. An important tool for controlling search in an object logic, the need for which is motivated by the difficulty of reasoning about large and complex systems, is the use of (...) structured theory presentations. In this paper a rudimentary language of structured theory presentations is presented, and the use of this structure in proof search for an arbitrary object logic is explored. The behaviour of structured theory presentations under representation in a logical framework is studied, focusing on the problem of “lifting” presentations from the object logic to the metalogic of the framework. The topic of imposing structure on logic presentations, so that logical systems may themselves be defined in a modular fashion, is also briefly considered. (shrink)