[Agda] Parametricity is inconsistent with classical logic

Altenkirch Thorsten psztxa at exmail.nottingham.ac.uk
Fri Apr 20 15:55:01 CEST 2012

On 20/04/2012 12:46, "Arseniy Alekseyev" <arseniy.alekseyev at gmail.com>

>I've noticed that parametricity postulates are inconsistent with
>classical logic, that is if you postulate both parametricity and the
>law of excluded middle, you can derive a contradiction. Although it is
>obvious in retrospect (lem, polymorphic in A, can not possibly know
>whether to produce "yes" or "no" while being parametric!), I've never
>heard it before, so i'm asking:
>Is this a known fact?

It is certainly worth noticing. But as you say it is not too surprising.
There cannot be a natural transformation (X : Set) -> X -> Bool (.e.e
between the identity functor and the constant bool functor) which is not
constant and parametricity implies that all such functions arennatural
(hence dinaturality isn't really needed here).

You are saying that dinaturality and parametricity are equivalent. If I
remember correctly then parametricity implies dinaturality but not vice


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