Self Referential Undecidability Construed as Incorrect Questions

PL Olcott

PL Olcott is simply confused. Besides being rude. — Alkis Piskas

No, it is wrong to say that a question has a correct answer. — Alkis Piskas

I honestly can't see how your above statement can possibly be correct and you have not provided a correct version of my statement to contrast with. You must be using some obscure idiomatic (term of the art) meaning that 99% of the population never heard of.

Well, depending on the question-statement, I would rather say ambiguous or circular or self-contradictory or --if it refers to an argument-- a fallacious argument.
I think that the attributes "correct" and "incorrect" are too general and/or ambiguous themselves. — Alkis Piskas

That statement indicates that you have a very good understanding of what I am saying.
I use the term "incorrect question" so that the question gets the blame for the lack of a
correct answer. Conventionally the question is always considered correct and the decider
gets the blame.

javi2541997

Check this: When I visited TPF a few minutes ago, I had in mind to check about your recent activity (comments)! — Alkis Piskas

Ha! This is funny because when I saw this thread I thought you would dive in, because I am aware that you like logic and tricky questions.

How can you call this (in Japanese)? :smile: — Alkis Piskas

I do not know, and I must accept that I haven't taken Japanese lessons for a while. I am very busy!

You are not wrong. And I think you do have a clue, and a correct one. — Alkis Piskas

Thank you, Alkis. This tricky thread has got me thinking more than I expected.

I would check more of your recent messages but it's got late. Maybe tomorrow ... — Alkis Piskas

OK. By the way, we can speak through PM if you want to.

javi2541997

↪PL Olcott

Well, after having a reasoning with myself, I came to the conclusion that omission cannot be an incorrect answer from Carol. This is due to the premise one: you are expecting from her an ambiguous answer: 'yes/no'. Either of each is wrong, but her silence doesn't. We are not on a duty, but a simple question. But she is not forced to answer at all, right?
Ability and possibility of answering are the key factors in Carol. Because:

A) Carol is capable of answering, but there is no possibility with the patters given.

B) Carol is able to answer, but she remains in silence and doesn't say 'no' nor 'yes' to not fall into the trap of the 'incorrect' question.

C) The only possible correct answer is the omission of Carol because the 'incorrect' result is posed to her only if she answers in any case.

If Carol doesn't want to get blamed for answer incorrectly, then she remains silent.

PL Olcott

Well, after having a reasoning with myself, I came to the conclusion that omission cannot be an incorrect answer from Carol. — javi2541997

Someone did find a loophole in Carol's question, it is corrected below:
(Carol could answer with a word that is synonymous with no)

Can Carol correctly answer “no” to this [yes/no] question?
is proven to be "no" on the basis that anything that Carol can say or fail to say cannot possibly provide a correct answer to that question from the stipulated solution set of {yes, no}.

javi2541997

(Carol could answer with a word that is synonymous with no) — PL Olcott

Exactly. Good point, but it is difficult to find a synonym with 'no' because this is an adverb used to give negative answers. I did research on Google and in the Cambridge Dictionary and I found 'none', a pronoun. Does this could be a correct answer if it is used by Carol?

Carol can say or fail to say cannot possibly provide a correct answer to that question from the stipulated solution set of {yes, no}. — PL Olcott

Unless synonyms or omissions are allowed, yes, Carol will always fail to answer this stipulated question set

PL Olcott

Unless synonyms or omissions are allowed, yes, Carol will always fail to answer this stipulated question set — javi2541997

So Carol's question when posed to Carol meets the definition of an incorrect question
in that both answers from the solution set of {yes, no} are the wrong answer.

Simplified Halting Problem Proof
Likewise no computer program H can say what another computer program D will do
when D does the opposite of whatever H says.

javi2541997

So Carol's question when posed to Carol meets the definition of an incorrect question
in that both answers from the solution set of {yes, no} are the wrong answer. — PL Olcott

Well, yes. If there is only a set binary solution: 'yes/no'. What I disagree with, is that an omission from Carol is not necessarily an incorrect answer, since she didn’t say 'yes' nor 'no'. I mean, she doesn't 'express' one of the set solutions.

Likewise no computer program H can say what another computer program D will do
when D does the opposite of whatever H says. — PL Olcott

OK. I don't get this. I thought we were debating about Carol and now two computer programs have arisen. It is not clear to me why 'computer program D' will always do the opposite of 'computer program H'. You are assuming that Carol will always do the opposite as well, and then this is why there will be an incorrect answer continuously, right?

PL Olcott

What I disagree with, is that an omission from Carol is not necessarily an incorrect answer — javi2541997

It <is> the lack of a correct answer thus
Can Carol correctly answer “no” to this [yes/no] question?
has (a) yes (b) no (c) anything else as not a correct answer to Carol's question
thus proving that anything that Carol can say or fail to say is not a correct answer
when posed to Carol.

PL Olcott

OK. I don't get this. I thought we were debating about Carol — javi2541997

Carol's question was written by a PhD computer science professor as
a simple analogy to the halting problem proofs. It was written to provide
a rebuttal to these proofs.

javi2541997

It <is> the lack of a correct answer thus
Can Carol correctly answer “no” to this [yes/no] question?
has (a) yes (b) no (c) anything else as not a correct answer to Carol's question
thus proving that anything that Carol can say or fail to say is not a correct answer
when posed to Carol. — PL Olcott

OK. But why is the question being asked to Carol if it will end up in an incorrect answer? It is difficult to see the logic of the question at all. So our answer to both paradoxes may be that they actually have no meaning and so cannot have a truth value. Your proposition is Universal Negative. Carol's answer is not correct - "No S is P" -. Yet, the relation with other logical propositions can connect contradictions, where the truth of one implies the falsehood of the other, and vice versa. Thus, it is false that Carol's answer will always be incorrect because there could be a possibility - as we debated before - but, at the same time, due to the set of preferences, it is also true that Carol cannot answer correctly. We end up in a loop where it is impossible to leave...

PL Olcott

OK. But why is the question being asked to Carol if it will end up in an incorrect answer? — javi2541997

Because is has the exact same form as the halting problem: decider/input
pair it proves that the most important computer science theorem that exists
is incorrect.

Simplified as this:
No computer program H can correctly say what another computer program
D will do when D does the opposite of whatever H says.

As long as it is understood that Carol's question is self-contradictory for Carol and
it is also understood that input D for program H is self-contradictory for H then
it can be understood that the only reason that the halting problem proofs can
show that the halting problem cannot be solved is that the input D to H derives
a self-contradictory thus incorrect question.

When presented with a self-contradictory thus incorrect question the blame
for not answering this question must go to the question and not the answerer.

PL Olcott

When the solution set is restricted to {yes, no} and no element of this solution set is a correct answer from Carol then the question posed to Carol is incorrect.
— PL Olcott
Well, depending on the question-statement, I would rather say ambiguous or circular or self-contradictory or --if it refers to an argument-- a fallacious argument.
I think that the attributes "correct" and "incorrect" are too general and/or ambiguous themselves. — Alkis Piskas

An incorrect question is any question that lacks a corresponding correct answer
because there is something wrong with the question. © 2015 PL Olcott

A self-contradictory question is a type of incorrect question that lacks a correct
answer because the question contradicts both elements of the solution set: {yes, no}

Can Carol correctly answer “no” to this [yes/no] question?
Is a self-contradictory (thus incorrect) question when posed to Carol.

When Carol says "no" indicating that "no" is an incorrect answer
this makes "no" the correct answer thus not incorrect thus Carol is wrong.

When Carol says "yes" indicating that "no" is a correct answer this makes
"yes" the wrong answer.

Banno

The original article seems to be

https://www.cs.toronto.edu/~hehner/OSS.pdf?fbclid=IwAR2uE4I_faeh_MPXAom8fl7FyTtwqi_Ll7VjxSqabll6zjGQ2kCJMDOz9wI

The supposed outcome is that no computer program A can say what another computer program B will do when B does the opposite of whatever A says

But what if A just prints "B will do the opposite of whatever I say it will do"?

So I'm unconvinced.

PL Olcott

The supposed outcome is that no computer program A can say what another computer program B will do when B does the opposite of whatever A says

But what if A just prints "B will do the opposite of whatever I say it will do"?

So I'm unconvinced. — Banno

Yes that is the article that I am basing this on. Professor Hehner
totally agrees with my understanding of his work. We discussed
it as recently as yesterday and many other times.

There was a nearly identical version of Carol's question that was
addressed directly to me in 2004 long before Professor Hehner
wrote about these things. This one was called Jack's question.

Hehner's version corrected some loopholes. Then someone else
pointed out another loop hole recently. Carol could correctly answer
the Hehner version with a word that is synonymous with "no".
Thus I added a restriction on the solution set.

Can Carol correctly answer “no” to this [yes/no] question?

Most people familiar with the theorem of computation halting
problem proofs understand that the halt decider is only allowed
to answer with something equivalent to yes or no.

Banno

↪PL Olcott

Sure. What's unclear to me is what it is you think this tells us about the halting problem.

PL Olcott

↪PL Olcott Sure. What's unclear to me is what it is you think this tells us about the halting problem. — Banno

As I and Professor Hehner have said the halting problem specification is essentially
a self-contradictory thus incorrect question for some decider/input pairs, thus places
no actual limit on computation. It is the same as the inability of CAD systems to correctly
draw square circles.

Banno

↪PL Olcott

I saw that earlier. Problem is that a self contradiction is something of the form (p & -p); and it’s clear the halting problem is not of this sort.

So presumably you mean something else by”self contradiction”, but it is unclear to me what that might be.

Antony Nickles

↪PL Olcott

As you say, it does depend on whom we are asking. But the failure of this paradox shows even more. Even if we just ask Carol (and not ask someone else about Carol's abilities, as I take you to be saying as the alternative), the answer can be either yes or no because the interpretation of the question is not grounded by a context or custom (as Wittgenstein pointed out with our use of rules. PI # 198). There are not implications and assumptions here that would constrain our possible answer, unless we imagine a fixed context so the question is interpreted along predictable lines.

Can Carol correctly answer “no” to this question?
Carol answers no without this being a paradox because there is no possibility of being "correct". As in: can she [answer correctly]? no, she cannot.
Carol answers yes without paradox because the correct answer is; no, she cannot answer correctly (for the same reason as above), but here (with yes) "the correct answer" would be that it is true, as in: the fact of the matter is.

Without any context of what the circumstance is in which this question is asked, either interpretation can apply. This is why abstracted thought experiments, moral puzzles, and paradoxes only show that when, and to whom, and in what circumstances, doing what activities, etc. all matter and are not internalized into language, as if in its "meaning".

PL Olcott

So presumably you mean something else by”self contradiction”, but it is unclear to me what that might be. — Banno

When input D is defined to do the opposite of whatever value that decider
H returns then "Does your input halt on its input?" becomes a self-contradictory
question for this decider/input pair.

PL Olcott

Carol answers no without this being a paradox because there is no possibility of being "correct". As in: can she [answer correctly]? no, she cannot. — Antony Nickles

When Carol says "no" indicating that "no" is an incorrect answer
this makes "no" the correct answer thus not incorrect thus Carol is wrong.

The exact same thing equally applies to input D to decider H
where D does the opposite of whatever Boolean value that H returns.

The halting problem specification is a self-contradictory thus an
incorrect question for some decider/input pairs.

The HP proofs do not limit what can be computed any more than the
fact that CAD systems cannot draw square circles limits computation.

Banno

↪PL Olcott

perhaps that is not as clear as you seem to think. My guess is that a much more formal account is needed. The problem is that “self” is ambiguous.

Again, the result is not a simple (p & -p).

PL Olcott

↪PL Olcott perhaps that is not as clear as you seem to think. My guess is that a much more formal account is needed. The problem is that “self” is ambiguous. — Banno

A more formal account is probably beyond the technical capability of most here.
This D is defined to do the opposite of whatever Boolean value that H returns.

// The following is written in C
//
01 typedef int (*ptr)(); // pointer to int function
02
03 int D(ptr x)
04 {
05   int Halt_Status = H(x, x);
06   if (Halt_Status)
07     HERE: goto HERE;
08   return Halt_Status;
09 }

Banno

↪PL Olcott

Sure, nice. So whereabouts in such a coding are we going to see the equivalent of (p & ~p)? Where's the demonstration?

PL Olcott

↪PL Olcott Sure, nice. So whereabouts in such a coding are we going to see the equivalent of (p & ~p)? Where's the demonstration? — Banno

I never said anything like that. That is merely contradictory and thus not at all self-contradictory.
"This sentence is not true." is self contradictory. If it is true that it is not true that makes it true.

My unique take on Gödel 1931 Incompleteness (also self-referential)
Any expression of the language of formal system F that asserts its
own unprovability in F to be proven in F requires a sequence of
inference steps in F that prove they themselves do not exist.

It is not at all that F is in any way incomplete.
It is simply that self-contradictory statements cannot be proven
because they are erroneous.

Banno

↪PL Olcott

Mmm. I'm just attempting to help bring out your usage. So for you, (p & ~p) is a contradiction, and false, but also for you, (this sentence is not true) is a self-contradiction, and also false. Is that right?

I hope not, and that I've misunderstood, because (this sentence is not true) cannot be false.

But now I'm also not following your rendering of Gödel.

Self-reference itself is not problematic. So, for instance the following sentence is true and self-referential. This sentence contains five words. Hence, further, "This statement is not provable in F" may be self-referential but true.

PL Olcott

I hope not, and that I've misunderstood, because (this sentence is not true) cannot be false. — Banno

"This sentence is not true." is not a truth bearer and thus cannot be true
or false. "What time is it?" is also not a truth bearer.

Self-reference itself is not problematic. So, for instance the following sentence is true and self-referential. This sentence contains five words. Hence, further, "This statement is not provable in F" may be self-referential but true. — Banno

Yes that is the same example that I use of self-reference that is not problematic.
If you understand the basics about how mathematical proofs work then you know
that a proof is a sequence of inference steps that ends in a conclusion.

When an expression of language G asserts that it is not provable in F
G := (F ⊬ G) then to be proven in F requires a sequence of inference
steps in F.

Everything that is provable in F always requires some sequence of
inference steps in F that reach a conclusion.

Since we are proving that G is unprovable in F then these steps must
prove that they themselves do not exist. It may be intially difficult
to understand. It took me quite a few years to explain how this
is self-contradictory.

The most important aspect of Gödel's 1931 Incompleteness theorem
are these plain English direct quotes of Gödel from his paper
...there is also a close relationship with the “liar” antinomy,14 ...
...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...
(Gödel 1931:43-44)

Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And Related Systems

https://mavdisk.mnsu.edu/pj2943kt/Fall%202015/Promotion%20Application/Previous%20Years%20Article%2022%20Materials/godel-1931.pdf

Antinomy It is a term often used in logic and epistemology, when describing a
paradox or unresolvable contradiction
https://www.newworldencyclopedia.org/entry/Antinomy

Quoted from above indicates that Gödel knews that he relied on self-contradiction
...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...

Banno

"This sentence is not true." is not a truth bearer and thus cannot be true
or false. — PL Olcott

Good. I must have misread you previously.

When an expression of language G asserts that it is not provable in F
G := (F ⊬ G) then to be proven in F requires a sequence of inference
steps in F. — PL Olcott

Sure. Apart from some difficulty in your saying G is a language. I take it you mean the statement G?

Since we are proving that G is unprovable in F then these steps must
prove that they themselves do not exist — PL Olcott

Unclear.

Gödel does not prove in F that some statement in F is not provable. Rather he numbers all the provable statements on F, the shows via a diagonal argument that there is a statement G that is not amongst them and yet is true in F.

PL Olcott

Gödel does not prove in F that some statement in F is not provable. — Banno

Yes he does and he does it in a ridiculously convoluted way because Peano Arithematic is woefully inexpressive for this task.

...We are therefore confronted with a proposition which asserts its own unprovability. 15 ... (Gödel 1931:43-44) — PL Olcott

G := (F ⊬ G) is a propostion in F that asserts its own unprovability in F stripped of the extraneous mess of Gödel numbers.

F ⊢ GF ↔ ¬ProvF (┌GF┐). // This one is similar to mine
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom

Banno

↪PL Olcott

Hmm.

To be sure, G is a statement in F (is that what you are saying?)

But there is no proof of G in F. That's the point of G.

The arithmatization of F - the assigning of Gödel numbers - does not take place within the language F. Rather it is about the language F.

PL Olcott

But there is no proof of G in F. That's the point of G. — Banno

The reason that there is no proof of G in F
(everyone always make sure to ignore the reason)

is that to prove there is no proof G in F requires a sequence of
inference steps that prove that they themselves do not exist.

Gödel makes sure to hide the reason behind Gödel numbers
and diagonalization.

Self Referential Undecidability Construed as Incorrect Questions

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