One-and-a-halfth-order logic

Authors:
Murdoch J. Gabbay;Aad Mathijssen
Affiliations:
Heriot-Watt University, Riccarton, Edinburgh;Eindhoven University of Technology (TU/e), Eindhoven, The Netherlands
Venue:
Proceedings of the 8th ACM SIGPLAN international conference on Principles and practice of declarative programming
Year:
2006

Citing 6
Cited 6

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Abstract

The practice of first-order logic is replete with meta-level concepts. Most notably there are the meta-variables themselves (ranging over predicates, variables, and terms), assumptions about freshness of variables with respect to these meta-variables, alpha-equivalence and capture-avoiding substitution. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both algebraic and sequent 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 itWe 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.