[Agda] Why no impredicative prop?

[Jon Sterling]

Harley,

System F is only "intuitionistic" in the formal sense (i.e. the sense of
"intuitionistic logic" as lacking classical axioms). It is not
intuitionistic in the sense of Brouwer or Heyting, i.e. there is not a
convincing meaning explanation that would validate impredicative
principles.

Best,
Jon


On Wed, Dec 30, 2015, at 06:34 AM, Harley Eades III wrote:
> Hi, Thorsten.
> 
> > On Dec 29, 2015, at 5:11 PM, Thorsten Altenkirch <Thorsten.Altenkirch at nottingham.ac.uk> wrote:
> > 
> > More on the philosphical side: impredicativity is basically a classical idea which arises from the identification of Prop and Bool. I don’t know an intuitionistic justification of impredicativity.
> 
> I am not sure I understand what this means.  Do you have a reference
> regarding this?  System F is intuitionistic, but doesn’t this claim it
> isn’t?
> 
> Very best,
> Harley
> 
> > 
> > Thorsten
> > 
> > From: <agda-bounces at lists.chalmers.se <mailto:agda-bounces at lists.chalmers.se>> on behalf of Timothy Carstens <intoverflow at gmail.com <mailto:intoverflow at gmail.com>>
> > Date: Tuesday, 29 December 2015 17:19
> > To: "agda at lists.chalmers.se <mailto:agda at lists.chalmers.se>" <agda at lists.chalmers.se <mailto:agda at lists.chalmers.se>>
> > Subject: [Agda] Why no impredicative prop?
> > 
> > Hello,
> > 
> > I'm a relatively new Agda user, coming from a background using Coq for the past few years. As is well known, Coq has an impredicative Prop sort, while Agda does not.
> > 
> > Why not? Is it simply a matter of project goals, or is there something about Agda's underlying theory which is incompatible with having an impredicative sort at the bottom of the universe hierarchy?
> > 
> > Forgive me if this is an old topic or has an obvious answer; I'm not a logician, but I use proof assistants at work.
> > -t
> > 
> >  
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