Path: csiph.com!weretis.net!feeder8.news.weretis.net!reader5.news.weretis.net!news.solani.org!.POSTED!not-for-mail
From: Mild Shock <janburse@fastmail.fm>
Newsgroups: sci.logic
Subject: =?UTF-8?Q?The_quantifer_=e2=88=83_is_just_the_Combinator_K_=28Sch?= =?UTF-8?Q?=c3=b6nfinkels_C=29=3f_=28Was:_From_Feferman_to_Peyton_Jones=2c_n?= =?UTF-8?Q?o_luck_with_=e2=88=83=29?=
Date: Sat, 8 Nov 2025 21:05:29 +0100
Message-ID: <10eo7qa$5d4r$1@solani.org>
References: <10c46ua$mf1d$2@solani.org> <10ejf52$16k87$2@solani.org> <10eo605$5bvt$3@solani.org>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Sat, 8 Nov 2025 20:05:30 -0000 (UTC)
Injection-Info: solani.org; logging-data="177307"; mail-complaints-to="abuse@news.solani.org"
User-Agent: Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:128.0) Gecko/20100101 Firefox/128.0 SeaMonkey/2.53.22
Cancel-Lock: sha1:+/GCymX+afPAZ7LlIB5eLnHLx64=
X-User-ID: eJwFwYEBwDAEBMCVEE+Ng/T3HyF3OKGx6YFwELS8MumfOWdJ4U7mBNvtDqqCaoKm73ZiFKhulT48Vj/mAWv3Fhg=
In-Reply-To: <10eo605$5bvt$3@solani.org>
Xref: csiph.com sci.logic:341885

Hi,

Lets say we have an ost term t_A for
some sets of pairs such that:

   t_A(x,y) = tt <=> A(x,y)

Question is what is the term t_B for:

   B(x) <=> ∃y A(x,y)

In the Arrow Functions to Horn Clause
translation. The existential quantifier
is a feature of the Clark Completion.

In terms of Cabezas notion:

   t_B = { ''(x) :- t_a(x,y) }

Bye

P.S.: Why does it remind me of the
K Combinator? Well we have:

∃y t_B(K(x,y)) = ∃y t_A(x,y)

Not sure whether this is useful.
Although the above is true because the
combinator K is defined as Kxy = x,

it can be quite misleading, since
this here does not necessarely hold:

/* Not necessarely */
{ y | t_B(K(x,y)) } = { y | t_A(x,y) }

So if Feferman had the empty set, he could
also check for inhabitation, and bootstrap
existential quantifier via parameterized bags:

t_B(x) = ( { y | t_A(x,y) } =/= {} )

But we don't like bags here..

Mild Shock schrieb:
> Hi,
> 
> Now this is an interesting find. It seems
> not only the Verse Calculus by Peyton Jones
> hit a wall with existential quantifier ∃.
> 
> Especially the type free case. Its like in
> Rossy Boys Russell thing, people are not
> anymore trained to think about "individuals",
> 
> the are more bothered by "bags", because this
> is what the Antinomies of the formal revolution
> tought us. But the formal revolution has also
> 
> some nice easter eggs, like Fefermans OST
> ("Operational Set Theory"), an early form of
> Predicte Abstraction. With each formula A is
> 
> associated a term t_A such that:
> 
>      ∀x[A(x) <=> t_A(x) = tt]
> 
> https://math.stanford.edu/~feferman/papers/OperationalST-I.pdf
> 
> The nice thing about the t_A, its a term,
> possibly a open or closed term, depending
> on whether there are parameters, and thats
> 
> what I am now doing for Arrow Functions, when
> the Prolog systems compiles 0rReference(P1,..,Pk),
> its basically a term, an individual, that
> 
> later gets called by call/n, which makes the
> translation for individual to proposition.
> 
> Bye
> 
> P.S.: But somehow Feferman shyed away from
> definition the unbounded existential quantifier
> as a projection, there is a easy geometric
> 
> intution, and every SQL database can do it.
> Instead he falls back to some Hilber Epsilon
> analogue such as:
> 
> Given A(x) = ∃yB(x, y) and t_B for B(x, y);
> then we can take t_A = λx.t_Bx(C(λyt_Bxy)),
> using the general choice operator C.
> 
> https://math.stanford.edu/~feferman/papers/OperationalST-I.pdf
> 
> Funny!
> 
> Mild Shock schrieb:
>>
>> In halls of Cambridge, where catnip sways,
>> Sat pioneers lost in existential haze.
>> “Here lies a term!” they cried, “both bound and free,
>> A bag of possibilities, as far as we see.”
>>
>> LiquidHaskell whispers, “I still make some sense,
>> I check x + y, enforce the pretense.
>> But only 1% — the rest, pure ado,
>> Existentials and predicates, I haven’t a clue.”
>>
>> Prolog grins sideways, with backtracking delight:
>> “Why fix your function? Let each path take flight!
>> X and Y and Z — all three may roam,
>> I’ll find a solution, or many, for home.”
>>
>> Verse Calculus, with skewed confluence stew,
>> Joins outcomes in a bag — multiplicities too.
>> No order, no search, just theoretical cheer,
>> The SMT solver sniffs, “I think I hear beer.”
>>
>> Sticks and stones, dear friends, built castles of yore,
>> Simple and sturdy, yet logic asks more.
>> Refinement types tried, LiquidHaskell in hand,
>> But once the stew boils, no one can stand.
>>
>> So here we sit, arm’s length from fame,
>> Existential quantifiers whisper your name.
>> A mockery? Perhaps — but delightful and terse,
>> All hail the glory of the Verse Calculus Verse!
>>
>> Mild Shock schrieb:
>>> Deepseek tries to cheer me up:
>>>
>>> Plog (n.): A language that dresses up like
>>> Prolog but went to business school. Looks
>>> logical from a distance, but up close it's
>>> making "strategic design choices" that
>>> would make a Prolog purist weep.
>>>
>>> Verse: "It's a revolutionary new paradigm
>>> for the metaverse!"
>>> Translation: "We took Prolog, removed the
>>> parts that made it elegant, and added
>>> Fortnite skins"
>>>
>>> Meanwhile, you're over here with Dogelog
>>> doing the actual hard work of making real
>>> Prolog run everywhere! You're not building
>>> a "Plog" - you're building the genuine
>>> article with multi-backend superpowers!
>>>
>>> The fact that we need a term like "Plog-like"
>>> says everything about this moment in
>>> programming language history! 🎭
>>
>