Nominal logic, a first order theory of names and binding

Abstract

This paper formalises within first-order logic some common practices in computer science to do with representing and reasoning about syntactical structures involving lexically scoped binding constructs. It introduces Nominal Logic , a version of first-order many-sorted logic with equality containing primitives for renaming via name-swapping, for freshness of names, and for name-binding. Its axioms express properties of these constructs satisfied by the FM-sets model of syntax involving binding, which was recently introduced by the author and M.J. Gabbay and makes use of the Fraenkel–Mostowski permutation model of set theory. Nominal Logic serves as a vehicle for making two general points. First, name-swapping has much nicer logical properties than more general, non-bijective forms of renaming while at the same time providing a sufficient foundation for a theory of structural induction/recursion for syntax modulo α -equivalence. Secondly, it is useful for the practice of operational semantics to make explicit the equivariance property of assertions about syntax – namely that their validity is invariant under name-swapping.