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 named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal Logic, a version of first-order many-sorted logic with equality containing primitives for renaming via name-swapping and for freshness of names, from which a notion of binding can be derived. Its axioms express key properties of these primitives, which are satisfied by the FM-sets model of syntax introduced in [11]. Nominal Logic serves as a vehicle for making two general points. Firstly, nameswapping has much nicer logical properties than more general forms of renaming while at the same time providing a sufficient foundation for a theory of structural induction/recursion for syntax modulo α-conversion. 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.