Expert Commentary: The Connection Between Permissive-Nominal Logic and Higher-Order Logic

In this article, the authors explore the connection between Permissive-Nominal Logic (PNL) and Higher-Order Logic (HOL). PNL extends first-order predicate logic by introducing term-formers that can bind names in their arguments. The semantics of PNL lies in permissive-nominal sets, where the forall-quantifier or lambda-binder are considered term-formers satisfying specific axioms.

On the other hand, HOL and its models exist in ordinary sets, specifically Zermelo-Fraenkel sets. In HOL, the denotation of forall or lambda is functions on full or partial function spaces.

The main question the authors address is how these two models of binding are connected and what kind of translation is possible between PNL and HOL, as well as between nominal sets and functions.

The authors demonstrate a translation of PNL into HOL, focusing on a restricted subsystem of full PNL. This translation is natural but partial, as it does not include the symmetry properties of nominal sets with respect to permutations. In other words, while names and binding can be translated, their nominal equivariance properties cannot be preserved in HOL or ordinary sets.

This distinction between PNL and HOL reveals that these two systems and their models have different purposes. However, they also share non-trivial and rich subsystems that are isomorphic.

Overall, this work sheds light on the relationship between PNL and HOL and highlights the limitations of translating between them. It suggests that while certain aspects can be preserved through translation, others may be lost due to the fundamental differences in their underlying structures.

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