One-and-a-halfth-order Logic

Murdoch Gabbay, Aad Mathijssen

Research output: Contribution to journalArticlepeer-review

33 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)521-562
Number of pages42
JournalJournal of Logic and Computation
Issue number4
Publication statusPublished - Aug 2008


  • α-conversion
  • First-order logic
  • Meta-variables
  • Nominal techniques
  • Substitution


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