  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.
