Re: topos logic arising nicely

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

Toby Bartels wrote

> That may be the only way that one can *construct* such a type,
> hence the only way that one can *prove* that such a type exists,
> but how do you know that some unspecified type variable \sigma
> doesn't refer to an uninhabited type to begin with?
> The answer will depend on the interpretation, I suppose;
> some logics simply aren't applicable to some semantics.
>
> >Of course, even if type sigma is inhabited from
> >    (1)   \forall x:\sigma. A(x)->B
> >one must not conclude
> >    (2)   \exists x:\sigma. A(x) /\ B
> >BUT only
> >    (3)  \exists x:\sigma. A(x) -> B
>
> Of course.  Only with \exists x:\sigma. A(x) will (2) follow.

Certainly, if you allow type variables then the problem crops up. I don't
really see the point why one would like to have type variables unless one
can perform constructions with types in a uniform way, e.g. when considering
logic on top of system F, system F\omega or even on top of a dependent type
theory (possibly with an impredicative universe). I guess that in case of HOL
type variables were rather motivated by the analogy to functional languages
with their "shallow" polymorphism.
The point of my mail rather was that so-called topos logic admits subtype
formation, i.e. that {x:A|phi(x)} is a type whenever \phi is a predicate on A.
This, of course, allows one to introduce types with undecided inhabitedness.
See W.Phoa's Edinburgh lecture notes for a calculus extending HOL with subtypes
(or Bart Jacob's book).
But I would be surprised if HOL has subtype formation as from a logical
point of view subtypes are neither necessary nor convenient. Adding subtypes
is only necessary for getting a topos out of a model of HOL.

Thomas