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Equivalences between logics and their representing type theories

Published online by Cambridge University Press:  04 March 2009

Philippa Gardner
Philippa Gardner
Affiliation:
Department of Computer Science, University of Edinburgh, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK. E-mail: pag@dcs.ed.ac.uk
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Abstract

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

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 5 , Issue 3 , September 1995 , pp. 323 - 349
Copyright
Copyright © Cambridge University Press 1995

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