Are there Hilbert-style Pure Type Systems?

TL;DR

This paper investigates the existence and nature of Hilbert-style formulations for pure type systems, revealing that many have trivial or no equivalent Hilbert-style systems, while some, including those underlying Coq, do have nontrivial equivalents.

Contribution

It demonstrates that most pure type systems lack nontrivial Hilbert-style equivalents, but some, like lambda* and Coq's basis, do have such equivalents with axioms as type statements.

Findings

Most PTSs have trivial or no Hilbert-style equivalents.

Some PTSs, including lambda* and Coq's basis, have nontrivial Hilbert-style equivalents.

The existence of Hilbert-style equivalents varies significantly among PTSs.

Abstract

For many a natural deduction style logic there is a Hilbert-style logic that is equivalent to it in that it has the same theorems (i.e. valid judgements with empty contexts). For intuitionistic logic, the axioms of the equivalent Hilbert-style logic can be propositions which are also known as the types of the combinators I, K and S. Hilbert-style versions of illative combinatory logic have formulations with axioms that are actual type statements for I, K and S. As pure type systems (PTSs)are, in a sense, equivalent to systems of illative combinatory logic, it might be thought that Hilbert-style PTSs (HPTSs) could be based in a similar way. This paper shows that some PTSs have very trivial equivalent HPTSs, with only the axioms as theorems and that for many PTSs no equivalent HPTS can exist. Most commonly used PTSs belong to these two classes. For some PTSs however, including lambda* and the PTS at the basis of the proof assistant Coq, there is a nontrivial equivalent HPTS, with axioms that are type statements for I, K and S.