On Monday, July 31, 2017 at 11:51:18 AM UTC-4, Michael Shulman wrote:
>
> I'm not quite sure what a "strict proposition" is, but if you mean 
> having a type of propositions that doesn't include all of them, then 
> the reason that's not enough for me is that frequently in univalent 
> type theory we encounter types that we *prove* to be propositions even 
> though they are not "given as such", such as "being contractible" and 
> "being a proposition", and this is responsible for a significant 
> amount of the unique flavor and usefulness of univalent type theory. 
>
> For semantic reasons I wouldn't want to use intuitionistic set theory, 
> because it doesn't naturally occur as the internal language of 
> categories.  You can perform contortions to model it therein, but that 
> involves interpreting it into type theory rather than the other way 
> around.  I don't know what you mean by "somehow clean it up into a 
> type system", but if you can do that cleaning up and obtain a type 
> theory (a "formal type theory" in Bob Harper's sense, not a 
> "computational type theory", i.e. one that is inductively generated by 
> rules rather than assigning types to untyped terms in an "intended 
> semantics"), then I'd like to see it.
>

I'm thinking it would look a lot like OTT, but with unique choice for 
strict props. Not all subsingletons are strict props, so they would not be 
subject to propositional extensionality. So this is apparently not what you 
want.

However, I'm not very confident about all that. I wouldn't be that 
surprised if it worked out differently. (In particular, something OTT-like, 
but with unique choice may not even make sense.) But I don't think it'll be 
what you want, in any case. I've lost interest in that approach.