PNL to HOL: From the logic of nominal sets to the logic of higher-order functions

Permissive-Nominal Logic (PNL) extends first-order predicate logicwith term-formers that can bind names in their arguments. It takes a semantics in (permissive-)nominal sets. In PNL, the ∀-quantifier or λ-binder are just term-formers satisfying axioms, and their denotation is functions on nominal atoms-abstraction. Then we have higher-order logic (HOL) and its models in ordinary (i.e. Zermelo–Fraenkel) sets; the denotation of ∀ or λ is functions on full or partial function spaces. This raises the following question: how are these two models of binding connected? What translation is possible between PNL and HOL, and between nominal sets and functions? We exhibit a translation of PNL into HOL, and from models of PNL to certain models of HOL. It is natural, but also partial: we translate a restricted subsystem of full PNL to HOL. The extra part which does not translate is the symmetry properties of nominal sets with respect to permutations. To use a little nominal jargon: we can translate names and binding, but not their nominal equivariance properties. This seems reasonable since HOL – and ordinary sets – are not equivariant. Thus viewed through this translation, PNL andHOL and theirmodels do different things, but they enjoy non-trivial and rich subsystems which are isomorphic. © 2012 Elsevier B.V. All rights reserved.