[Agda] Parametricity is inconsistent with classical logic

[Martin Escardo]

I am not talking about is_prop, but rather about the hprop reflection, 
which is analogous to double negation (which is_prop is not). Martin

On 10/05/12 12:35, Bas Spitters wrote:
> In Hlevel.v:
>
> Definition forall_isprop {X} (P : X ->  Type) :
>    (forall x, is_prop (P x)) ->  is_prop (forall x, P x).
>
> Bas
> On Thu, May 10, 2012 at 1:16 PM, Martin Escardo<m.escardo at cs.bham.ac.uk>  wrote:
>> 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
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-- 
Martin Escardo
http://www.cs.bham.ac.uk/~mhe