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Journal of Logic and Computation Advance Access published online on November 22, 2007
Journal of Logic and Computation, doi:10.1093/logcom/exm064
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original papers

One-and-a-halfth-order Logic

Murdoch J. Gabbay

Department of Computer Science, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK. E-mail: murdoch.gabbay{at}gmail.com

Aad Mathijssen

Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: aad.mathijssen{at}gmail.com


  Abstract

The practice of first-order logic is replete with meta-level concepts. Most notably there are meta-variables ranging over formulae, variables, and terms, and properties of syntax such as alpha-equivalence, capture-avoiding substitution and assumptions about freshness of variables with respect to meta-variables. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of one-and-a-halfth-order logic derivability, show them equivalent, show that the derivations satisfy cut-elimination, and prove correctness of an interpretation of first-order logic within it. We discuss the technicalities in a wider context as a case-study for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation for future implementation.

Keywords: first-order logic; nominal techniques; substitution; {alpha}-conversion; meta-variables


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