[Haskell-cafe] Relationship between ((a -> Void) -> Void) and (forall r. (a -> r) -> r)

[David Feuer]

Classically, the excluded middle is an axiom, not a theorem. There is no
code/proof.

On Sun, May 17, 2020, 9:02 PM Kim-Ee Yeoh <ky3 at atamo.com> wrote:

> Very cool to see the constructive code for the proof of double negation in
> intuitionistic logic.
>
>
> But what about the Curry-Howard correspondence for classical logic?
>
>
> What would the classical code for the classical proof of excluded middle
> look like?
>
> On Fri, May 15, 2020 at 11:09 PM Chris Smith <cdsmith at gmail.com> wrote:
>
>> This was indeed a fun puzzle to play with.  I think this becomes easier
>> to interpret if you factor out De Morgan's Law from the form you posted at
>> the beginning of your email.
>>
>> https://code.world/haskell#PYGDwpaMZ_2iSs74NnwCUrg
>>
>>
>> On Fri, May 15, 2020 at 5:23 AM Ruben Astudillo <ruben.astud at gmail.com>
>> wrote:
>>
>>> On 13-05-20 09:15, Olaf Klinke wrote:
>>> > Excersise: Prove that intuitionistically, it is absurd to deny the law
>>> > of excluded middle:
>>> >
>>> > Not (Not (Either a (Not a)))
>>>
>>> It took me a while but it was good effort. I will try to explain how I
>>> derived it. We need a term for
>>>
>>>    proof :: Not (Not (Either a (Not a)))
>>>    proof :: (Either a (Not a) -> Void) -> Void
>>>
>>> A first approximation is
>>>
>>>    -- Use the (cont :: Either a (Not a) -> Void) to construct the Void
>>>    -- We need to pass it an Either a (Not a)
>>>    proof :: (Either a (Not a) -> Void) -> Void
>>>    proof cont = cont $ Left <no a to fill in>
>>>
>>> Damn, we can't use the `Left` constructor as we are missing an `a` value
>>> to fill with. Let's try with `Right`
>>>
>>>    proof :: (Either a (Not a) -> Void) -> Void
>>>    proof cont = cont $ Right (\a -> cont (Left a))
>>>
>>> Mind bending. But it does make sense, on the `Right` constructor we
>>> assume we are have an `a` but we have to return a `Void`. Luckily we can
>>> construct a `Void` retaking the path we were gonna follow before filling
>>> with a `Left a`.
>>>
>>> Along the way I had other questions related to the original mail and
>>> given you seem knowledgeable I want to corroborate with you. I've seen
>>> claimed on the web that the CPS transform *is* the double negation [1]
>>> [2]. I don't think that true, it is almost true in my view. I'll
>>> explain, these are the types at hand:
>>>
>>>     type DoubleNeg a = (a -> Void) -> Void
>>>     type CPS a = forall r. (a -> r) -> r
>>>
>>> We want to see there is an equivalence/isomorphism between the two
>>> types. One implication is trivial
>>>
>>>     proof_CPS_DoubleNeg :: forall a. CPS a -> DoubleNeg a
>>>     proof_CPS_DoubleNeg cont = cont
>>>
>>> We only specialized `r ~ Void`, which mean we can transform a `CPS a`
>>> into a `DoubleNeg a`. So far so good, we are missing the other
>>> implication
>>>
>>>     -- bind type variables: a, r
>>>     -- cont   :: (a -> Void) -> Void
>>>     -- absurd :: forall b. Void -> b
>>>     -- cc     :: a -> r
>>>     proof_DoubleNeg_CPS :: forall a. DoubleNeg a -> CPS a
>>>     proof_DoubleNeg_CPS cont = \cc -> absurd $ cont (_missing . cc)
>>>
>>> Trouble, we can't fill `_missing :: r -> Void` as such function only
>>> exists when `r ~ Void` as it must be the empty function. This is why I
>>> don't think `CPS a` is the double negation.
>>>
>>> But I can see how people can get confused. Given a value `x :: a` we can
>>> embed it onto `CPS a` via `return x`. As we saw before we can pass from
>>> `CPS a` to `DoubleNeg a`. So we have *two* ways for passing from `a` to
>>> `DoubleNeg a`, the first one is directly as in the previous mail. The
>>> second one is using `proof_CPS_DoubleNeg`
>>>
>>>     embed_onto_DoubleNeg :: a -> DoubleNeg
>>>     embed_onto_DoubleNeg = proof_CPS_DoubleNeg . return
>>>       where
>>>         return :: a -> CPS a
>>>         return a = ($ a)
>>>
>>> So CPS is /almost/ the double negation. It is still interesting because
>>> it's enough to embed a classical fragment of logic onto the constructive
>>> fragment (LEM, pierce etc). But calling it a double negation really
>>> tripped me off.
>>>
>>> Am I correct? Or is there other reason why CPS is called the double
>>> negation transformation?
>>>
>>> Thank for your time reading this, I know it was long.
>>>
>>> [1]: http://jelv.is/talks/curry-howard.html#slide30
>>> [2]:
>>>
>>> https://www.quora.com/What-is-continuation-passing-style-in-functional-programming
>>>
>>> --
>>> -- Rubén
>>> -- pgp: 4EE9 28F7 932E F4AD
>>>
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>>
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>
> --
> -- Kim-Ee
> _______________________________________________
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