One-and-a-halfth-order logic

Murdoch J. Gabbay, Aad Mathijssen
2006 Proceedings of the 8th ACM SIGPLAN symposium on Principles and practice of declarative programming - PPDP '06  
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-halfthorder 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
more » ... ivalent, 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.
doi:10.1145/1140335.1140359 dblp:conf/ppdp/GabbayM06 fatcat:sbvrcyvv7bac5h5p2wjdmbdwia