[Agda] Parametricity is inconsistent with classical logic

Altenkirch Thorsten psztxa at exmail.nottingham.ac.uk
Thu May 10 12:38:49 CEST 2012

This looks interesting.

I was thinking that ¬ ¬ turns a Set into a proposition, i.e. a type whose
equalities are contractible.

Actually, on a related topic - when carrying out the setoid construction
in Type Theory (related to my LICS 99 paper) I needed that Prop is closed
under Pi, I.e.

((x : X)-> isProp(F x)) -> isProp((x : X)-> F x)

but as far as I can see this doesn't imply extensionality.

To put it in another way: extensionality is equivalent to H-level 0
(contractible types) is closed under Pi, and it implies that H-level 1
(propositional types) is closed under Pi but the latter is weaker.


On 10/05/2012 10:59, "Peter Hancock" <hancock at spamcop.net> wrote:

>>>>> ((x : X) → ¬ ¬ F x) → ¬ ¬ ((x : X) → F x)
>I just wanted to say that DNS in this form resembles the
>extensionality axiom in Voevodsky's form.
>    ((x : X)-> isContr(F x)) -> isContr((x : X)-> F x)
>A double negation seems to be a crude way of turning an
>inhabited set into a singleton.
>Or is that nonsense?  It looks like an extensionality axiom.
>Agda mailing list
>Agda at lists.chalmers.se

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