[Agda] Parametricity is inconsistent with classical logic

[Martin Escardo]

The hprop reflection of a type A is an hproposition A* such that A->A*,
and for every hproposition P, A->P implies A*->P. Voevodsky defined it
in the last types meeting as

  A* = Prod P : U. hprop P -> (A -> P) -> P.

In MLTT as it stands (without his resizing rules), this gives a large type.

Because P=¬¬A is an hproposition (assuming extensionality of
empty-type-valued functions), we have that A* -> ¬¬A (but hardly ever
¬¬A -> A*, of course).

The HoTT analogue of the DNS would be

 ((x : X)->  (A x)*) ->  ((x : X)->  A x)*.

I haven't thought whether this is true. But it is not quite the same as
extensionality, which has isContr rather than the hprop reflection. The
pattern is the same, though, as observed by Hank: something is pushed
out of a product.

Martin

On 10/05/12 11:38, Altenkirch Thorsten wrote:
> 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.
> 
> Thorsten
> 
> 
> 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.
>>
>> Hank
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-- 
Martin Escardo
http://www.cs.bham.ac.uk/~mhe