Self Referential Undecidability Construed as Incorrect Questions
Self Referential Undecidability Construed as Incorrect Questions
Linguistics understands that the context of who is asked a question does change the meaning of some questions. This same reasoning applies to decision problems.
When the context of who is asked a question determines whether or not a question has a correct answer then this context can never be correctly ignored.
When a yes/no question posed to a person has no correct yes/no answer from this person then this question is construed as incorrect within the full context of who is asked.
This same reasoning applies when the input to a decider has no correct accept/reject return value from this decider.
It does not matter that the question has a correct answer from someone else or the input to the decider can be decided by another decider.
In both cases we have an incorrect question because it has no correct answer within the full context of the question.
[b]Original words by PhD computer science professor:
Can Carol correctly answer no to this question?[/b]
Let's ask Carol. If she says yes, she's saying that no is the correct answer for her, so yes is incorrect. If she says no, she's saying that she cannot correctly answer no, which is her answer. So both answers are incorrect. Carol cannot answer the question correctly.
Because:
(1) Both "yes" and "no" are the wrong answer from Carol.
(2) Linguistics understands that the context of who is asked changes the
meaning of this question, thus this context cannot be correctly ignored.
(3) An incorrect yes/no question is defined as any yes/no question
lacking a correct yes/no answer.
Then the question is an incorrect question when posed to Carol
This same reasoning equally applies to a termination analyzer H that
reports on an input D that does the opposite of whatever halt status
that H returns.
When the full context is of who is asked is considered then
Does D halt on its input? is an incorrect question for H.
Linguistics understands that the context of who is asked a question does change the meaning of some questions. This same reasoning applies to decision problems.
When the context of who is asked a question determines whether or not a question has a correct answer then this context can never be correctly ignored.
When a yes/no question posed to a person has no correct yes/no answer from this person then this question is construed as incorrect within the full context of who is asked.
This same reasoning applies when the input to a decider has no correct accept/reject return value from this decider.
It does not matter that the question has a correct answer from someone else or the input to the decider can be decided by another decider.
In both cases we have an incorrect question because it has no correct answer within the full context of the question.
[b]Original words by PhD computer science professor:
Can Carol correctly answer no to this question?[/b]
Let's ask Carol. If she says yes, she's saying that no is the correct answer for her, so yes is incorrect. If she says no, she's saying that she cannot correctly answer no, which is her answer. So both answers are incorrect. Carol cannot answer the question correctly.
Because:
(1) Both "yes" and "no" are the wrong answer from Carol.
(2) Linguistics understands that the context of who is asked changes the
meaning of this question, thus this context cannot be correctly ignored.
(3) An incorrect yes/no question is defined as any yes/no question
lacking a correct yes/no answer.
Then the question is an incorrect question when posed to Carol
This same reasoning equally applies to a termination analyzer H that
reports on an input D that does the opposite of whatever halt status
that H returns.
When the full context is of who is asked is considered then
Does D halt on its input? is an incorrect question for H.
Comments (126)
[quote=Jack]Yes, "no".[/quote] (or vice versa)
It is a solution to paradox to rule it out as soon as it rears its head, on an ad hoc basis, ie.
Rule: "if it leads to paradox it is ruled out."
But this does not seem to really get to grips with the thing.
"Will Jack's answer to this question be no?" is ruled out, but
"Will Jack's answer to this question be yes?" is ruled in.
Why? Or rather, why does one lead to paradox and the other does not? is that question ruled out?
G. Spencer-Brown has the bones of a more fruitful resolution of these things that you might find interesting. https://thephilosophyforum.com/discussion/14599/reading-the-laws-of-form-by-george-spencer-brown/p1
Quoting PL Olcott
As far as I have understood your reasoning, I think the latter fails to follow the original pattern of your premises. Firstly, you were asking for truthful yes/no answers to a specific question: 'Will Jack's answer to this question be no?' Not giving any relevance to whether the question is correct or incorrect, because you focused on the result, not the beginning. But, on the second group of premises, you focus on the questions instead. Rather than switching the 'context' - as you claim in your arguments - I think you are switching the meaning.
But maybe I am wrong, and I don't have a clue about what is going on. :smile:
I changed the words to the better words of the PhD computer science professor.
I changed the words to the better words of the PhD computer science professor.
Quoting PL Olcott
I see what you mean now. It is a paradox.
Yet, I think we can get a different result if we switch the assertions. The Anti-Liar paradox teaches us that the form of a proposition can determine its own truth or falsehood only if it is either a tautology or a contradiction. Neither the Carol's answer nor the possible answer is either a tautology or a contradiction.
Because:
A) If Carol's answer is true, then what she says is true. Therefore, it is the correct answer.
B) If Carol's answer is false, then what she says is false. Therefore, it is the wrong answer.
There is a possible alternative for Carol to answer correctly. Whether it is true or not that Carol can actually answer 'no' to the question itself.
We must not change Carol's question because it is the exact same form as
the most important theorem in computer science the halting theorem.
A PhD computer science professor wrote Carol's question with this in mind.
Once we understand that any yes/no question that lacks a correct answer
(within the linguistically required context of who is asked) is an incorrect
question then
We understand that Carol's question and the halting theorem decider/input
pair are also merely incorrect questions.
The halting theorem proves that a halt decider cannot possibly return a
correct true/false value when its input does the opposite of whatever Boolean
value that it returns. https://simple.wikipedia.org/wiki/Halting_problem
Since this turns out to merely be an incorrect question it does not place
any actual limit on computation. The fact that a baker cannot bake an
angel food cake using only house bricks for ingredients place no limit
on the baker's baking skill.
I do not see it as an 'incorrect' question/answer but whether the assertions are either true or false. Can Carol correctly answer 'no' to this question? There is a true possibility that Carol could do so.
This is very tricky like the Liar Paradox: "This sentence is not true"
We have the exact same issue with Carol's question.
Carol cannot correctly answer her question yet when she says "no" then she has correctly answered her question making "no" the wrong answer.
When Carol says "yes" this means that she can correctly answer her question with "no" yet we just proved that is incorrect.
I agree. This is getting tricky, yes. I understand that it is proven that Carol cannot answer with 'yes' to claim that she can answer 'no' to the question. According to this context, it is getting more paradoxical, and I think this was the main point of your thread. What I do not understand is why you consider the question as 'wrong' when we are debating whether Carol is capable of answering the question correctly.
I mean: 'yes/no' doesn't affect the possibility of Carol answering as the question requires. At least, we can agree that the assertion of Carol answering as required is true.
Some (if not all) undecidable decision problems are only "undecidable" because there is something wrong with the problem. When zero elements of the entire solution set provide the correct answer then this indicates that there is something wrong with the problem.
If I ask you how many feet long is the color of your car? no one can provide a correct answer because the question itself is incorrect. The same thing happens when a self-contradictory question is asked.
One thing that I found in my 20 year long quest is that self-contradictory expressions are not true. As a corollary to this self-contradictory questions are incorrect.
When we add one more step that the context of who is asked a question is a mandatory aspect of the full meaning of this question then Carol's question is incorrect for Carol.
I understand it better now, thanks!
Quoting PL Olcott
Basically, you claim that Carol's answer will always be incorrect because the question itself is incorrect too. So, we are lead to a self-contradictory result endlessly. Although I agree with you, and now I understand your thread a bit better, I still do not see the correlation between 'yes/no' - or 'correct/incorrect' - and true and false.
I think that the question being incorrect doesn't mean that Carol's answer is false. Actually, it is true that she is able to answer but not with the patterns given because the question is 'wrong'.
Carol's question is incorrect for Carol, thus it is the fault of the question and not the falt of Carol for the lack of correct answer from Carol.
There no is element in the entire solution set of {yes, no} that Carol can use as her answer thus making the question incorrect on the basis of the definition of incorrect question.
An incorrect yes/no (technically polar) question is any yes/no question lacking a correct answer from the set of {yes, no} or {true, false}.
A self-contradictory (thus incorrect) question is the analog to a self-contradictory statement such as:
"This sentence is not true".
Quoting PL Olcott
Here is where I disagree with you.
Correct/incorrect are not related to the truth or false in your question to Carol. Again, if they are either true or false, there must be something about them that makes them true or false. Your objection to the truth or falsehood cannot be determined in the context. In a sense, this means that Carol ('P' or predicate) consists of nothing but p with a predication of truth (thus, she can actually answer, but the question is incorrect). We can say that the truth or falsehood of Carol's answer is 'undetermined'.
Is this sentence true or false? "This sentence is not true" is an incorrect question because the Liar Paradox is neither true nor false and the solution set is limited to {true, false}.
The same thing applies to Carol's question when posed to Carol:
Can Carol correctly answer no to this question?
Since both yes and no are an incorrect answer from Carol this conclusively proves that Carol's question meets the stipulated definition of an incorrect question when posed to Carol.
An incorrect question is defined as the case whenever a yes/no question posed to a person has no correct yes/no answer from this person then this question is incorrect when posed to this person.
I assume of course that "who" refers to a person. But does a person have a context?
I'm afraid that by this word you mean something else, e.g. background, or you apply it incorrectly, e.g. you mean the context in which one talks. Or even something else. I can't know.
And then you repeat it: "When the context of who is asked a question determines ..."
Quoting PL Olcott
1) A question cannot have a correct answer. What can be correct is the answer given to that question. So maybe you mean that the question can receive a correct answer?
Or that the question can have more than one correct answers?
2) What does "correctly ignored" mean? Do you mean "correctly answered"?
See, these things are not details. One has to be as exact and clear in ones thesis, hypothesis, proposition, etc. in philosophy as in science
My intention is not to criticize you, but to pinpoint important elements in a philosophical discussion. And I'm addressed to the general public, because I see the phenomenon of lack of clarity and misuse of terms only too often.
By extension, all this applies and is an answer to your topic itself: If the context in which a question is asked is missing or not clear, of course this question might receive not incorrect, but inappropriate answers, i.e. answers "out of context" or "off-topic", as we use to say. A classic example is an ambiguous question that can be answered with both "Yes" and "No", about which you talked in your description.
One must also add, that in philosophy, unlike in science, we cannot talk about "correct" answers. Logical schemes, fallacious arguments, etc. are an exception, since they are special elements used in argumentation, hypotheses, propositions, etc., which are borrowed from the science of Logic.
"Is the living mammal of an elephant any type of fifteen story office building?"
has the correct answer of "no".
Is the following sentence true or false: "This sentence is not true."
has no correct answer from the set of {true, false}.
Quoting Alkis Piskas
My purpose of being here is to get feedback so that I can make my words clear enough so that they can be understood as correct.
When a decision problem decider/input pair lacks a correct Boolean return value from this decider then this decision problem instance is semantically unsound.
Well, if we want to go further and make these premises even trickier, we can assert that Carol is not forced to answer in any case. So, there is a possibility for Carol to answer with an omission. Yet again, I claim that the 'incorrect' question doesn't depend whether is posed on Carol or not.
So, we haven't proved anything conclusively yet. :smile:
:up:
The PhD computer science professor that has been published in several highly esteemed computer science journals disagrees.
(a) Yes is not a correct answer from Carol.
(b) No is not a correct answer from Carol.
(c) No answer is not a correct answer from Carol.
We have exhaustively examined every possibility and thus proven every action taken by Carol does not result in a correct answer.
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.
Interesting...
Let me think about this deeply. Maybe I can come back with more substantive comments, and see other possibilities. I appreciate how you considered each feasible scenario of Carol's behaviour. I still believe that there can be a possible correct answer.
No, it is wrong to say that a question has a correct answer. It is wrong even to say that a question has any answer at all. A question is asked by a person and is addressed to anor person or persons in order to receive, to be given an answer. And then, the answer does not go to the question, it does not become a property of the question; it goes to whom asked the question.
Indeed, sometimes we say "your question includes the answer" or the "question replies to itself", etc. But these are only fiigures of speech.
Moreover, we are talking about a "correct" answer, something which even is more difficult to be attributed to a question.
So we can say that the correct answer to the question "Is the living mammal of an elephant any type of fifteen story office building?" is "No".
I hope this is clear by now.
Quoting PL Olcott
This is a known self-contraditory statement. It cannot be answered (with "true" or "false"). That's all.
This, as any other question, does not and cannot have an answer. I explained that in detail above.
Quoting PL Olcott
I appreciate this. I hope I have contributed in some way,
(BTW, what about my second question, "What does 'correctly ignored' mean? Do you mean 'correctly answered'?" Have you sorted this out?)
Quoting PL Olcott
I wouldn't state it like that myself, but I agree. :smile:
Hi friend! Long time no see!
Check this: When I visited TPF a few minutes ago, I had in mind to check about your recent activity (comments)! How can you call this (in Japanese)? :smile:
BTW, the was an spelling error in my sentence "the context in which a question is asked is [s]mission[/s] missing or not clear".
Quoting 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.
That seems to be a ridiculous statement on your part. It is like you are saying
that it is impossible to determine whether or not an elephant is a fifteen story
office building. How would you phrase the exact same idea that I am referring to?
Quoting Alkis Piskas
Most people call this an undecidable instance, yet it is not at all any
matter of the decider not being able to figure out which of true/false
is the correct return value. It is a matter of both true and false are
incorrect return values.
It is clear that self-contradictory expressions are untrue and unfalse because they are self-contradictory. Analogous reasoning applies to self-contradictory questions.
It the same way that the Liar Paradox: "This sentence is not true" is an incorrect statement self-contradictory questions are incorrect questions.
Is this sentence true or false: "This sentence is not true" is an incorrect question because zero elements of the entire solution set of {true, false} are a correct answer.
Then we apply this same reasoning to self-contradictory decision problem instances.
When neither return value of {true, false} is correct for a decider/input pair then this
decider/input pair is essentially an incorrect question.
You are not wrong. And I think you do have a clue, and a correct one. @PL Olcott is simply confused. Besides being rude.
I would check more of your recent messages but it's got late. Maybe tomorrow ...
That is great.
(a) Carol answers "no" and she is wrong.
(b) Carol answers "yes" and she is wrong.
(c) Carol does anything else and she has not provided an answer within the solution set of {yes,no}.
Quoting 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.
Quoting Alkis Piskas
That statement indicates that you have a very good understanding of what I am saying.
[b]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.[/b]
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.
Quoting Alkis Piskas
I do not know, and I must accept that I haven't taken Japanese lessons for a while. I am very busy!
Quoting Alkis Piskas
Thank you, Alkis. This tricky thread has got me thinking more than I expected.
Quoting Alkis Piskas
OK. By the way, we can speak through PM if you want to.
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.
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}.
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?
Quoting PL Olcott
Unless synonyms or omissions are allowed, yes, Carol will always fail to answer this stipulated question set ['yes/no']
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.
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 didnt say 'yes' nor 'no'. I mean, she doesn't 'express' one of the set solutions.
Quoting 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?
It
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.
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.
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...
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.
[b]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.[/b]
[b]An incorrect question is any question that lacks a corresponding correct answer
because there is something wrong with the question. © 2015 PL Olcott[/b]
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.
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.
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.
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.
So presumably you mean something else byself contradiction, but it is unclear to me what that might be.
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".
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.
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.
Again, the result is not a simple (p & -p).
[b]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.[/b]
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.
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.
"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.
Quoting 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.
[b]The most important aspect of Gödel's 1931 Incompleteness theorem
are these plain English direct quotes of Gödel from his paper[/b]
...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
[b]Antinomy It is a term often used in logic and epistemology, when describing a
paradox or unresolvable contradiction[/b]
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...
Good. I must have misread you previously.
Quoting PL Olcott
Sure. Apart from some difficulty in your saying G is a language. I take it you mean the statement G?
Quoting 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.
Yes he does and he does it in a ridiculously convoluted way because Peano Arithematic is woefully inexpressive for this task.
Quoting 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
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.
[b]The reason that there is no proof of G in F
(everyone always make sure to ignore the reason)[/b]
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.
G is not a deduction in F. That would be silly.
Rather, Gödel shows using arithmatization and the diagonalization that the structure of F is such that there must be WFF such as G. He's not using the deductive power of F to prove that G is unprovable.
I gave you a perfectly acceptable alternative interpretation of what is happening (which you did not address). You are just applying an interpretation without any context or justification of why it MUST be taken that way; what appears to you as formal logic is just an implication you see as self-evident and singular, when it is just your imposed requirement.
Carol does not need to be indicating an incorrect answer, she could be indicating there IS NO sense of correctness in this question, and thus how CAN she answer at allthe correct answer is to throw up her hands and say no, as if to say: What?. Another way to interpret this is that, of course, Carol CAN answer no, she can say whatever she wants, defying your idea of correctness with her own truth to herself, in protest. But with no world, you make the rules, so, sure, make them however youd like. What youve proven is that such a question must be in an abstract environment with closed dictated rules, as if playing with a machine you programmed. So why bring a human (poor Carol) into it?
Exactly. That's what I attempted to explain to @PL Olcott, but it is impossible to agree with him, because according to his point, there will always be an incorrect answer because the question is 'posed' to Carol. It seems that poor Carol is guilty of everything regarding this tricky dilemma!
In other words Gödel uses a convolulted mess to show THAT G IS unprovable in F
in F while carefully hiding WHY G is unprovable in F.
[b]Antinomy It is a term often used in logic and epistemology, when describing a paradox or
unresolvable contradiction.[/b] https://www.newworldencyclopedia.org/entry/Antinomy
Gödel acknowledges that his G is {a proposition which asserts its own unprovability}
and also acknowledges that any {epistemological antinomy} (self-contradictory G) will do.
...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
[b]Her answer of "no" indicates that she cannot correctly answer "no"
yet because "no" is the correct answer her answer of "no" is incorrect.[/b]
Can Carol correctly answer no to this [yes/no] question?
The revised question requires Carol to answer from the solution set.
Any lack of answer from {yes, no} or answer from {yes,no} is not a
correct answer.
Carol's question was written by a PhD computer science professor to show
that the halting problem specification is inconsistent.
I have spoken with him directly many times and he agrees with me that an
equivalent way of saying this is that input D to decider H makes this question:
"Does your input halt on its input?" a self-contradictory thus incorrect question
for H when D is defined to do the opposite of whatever H says.
Can Carol correctly answer no to this [yes/no] question?
Was written by a PhD computer science professor as an analogy to the
conventional halting problem proofs where an input D has been defined
to do the opposite of whatever program H says.
His purpose in doing this was to show that because the question
"Does your input halt on its input? is self-contradictory for some
program/input pairs that for these pairs it is an incorrect question.
We still agree with everyone else that when an input D does the
opposite of whatever program H says that H cannot correctly say
what input D will do.
The key distinction that we make is that this does not place any actual
limit on computation. That H cannot answer an incorrect question is the
same as the fact that a baker cannot bake a perfect angle food cake
using only house bricks for ingredients. It is incorrect for us to say that
her baking skills are limited on that basis.
As long as you acknowledge that, again, the solution set is YOUR requirement, not revealing anything but the answer you dictate. What you have imposed as correct suppresses any other interpretation and thus only has one set of answers.
Now of course if you are programming a computer than the terms are set and thus easy to force into a corner, but leave it as a formal logic problem or a programming issue for it says nothing about selves or human contradiction. Yes we have expectations and implications and consequences, but we still live in a culture and answer for ourselves in a specific circumstance with possibilities that we act within, or defy.
Yes this makes it exactly the same as the halting problem's input D
that does the opposite of whatever Boolean value that decider H says.
Since the analogy of Carol's question is much easier to understand
it provides great leverage in understanding the error of the halting
problem proofs
The whole purpose of Carol's question was to show that the halting
problem specification derives a self-contradictory thus incorrect
question for some program/input pairs.
When we show this then the inability of some H to say what
input D will do when D is defined to the opposite of whatever
H says is simply an incorrect question for H.
If some program/input pairs are incorrect questions then the lack
of the ability of H to provide a correct answer is the same as the
lack of the ability of a baker to bake a perfect angel food cake using
only house brick for ingredients.
Well, no. He carefully shows why G is unprovable.
Not at all. Diagonalization only shows THAT an expression
is unprovable, it abstracts away WHY. If we were to formalize
this question: "What time is it (yes or no)?" diagonalization
could show THAT it cannot be correctly answered and have
no idea that the reason WHY it cannot be answered is a type
mismatch error. Diagonalization makes sure to discard these
details.
An odd view.
You would presumably, for consistency's sake, say the same for Turing Machines, Lambda calculus, and Markov Algorithms; each of which have similar issues. Do you also reject the uncountability of the reals?
If so, we might leave this conversation here.
[b]Not at all and now I show my words are sustained by Gödel's words.
The last paragraph is proven by all that comes before it.[/b]
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.
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
...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...
[b]Thus in my simple version we can see WHY {a proposition that asserts
its own unprovability} in F cannot be proven in F. It is because such
a proposition is an {epistemological antinomy} in F.[/b]
[b]I have also showed that an input D that does the opposite of whatever
program H says is an {epistemological antinomy} for H.[/b]
We can map every real to an integer and it does seem that we have some reals left over.
If we imagine that there can be such a thing as immediately adjacent points on a
number line, then these points would map to the integers. 1.0 + infinitesimal would
be the next point on a number line.
The "Why" you are after is simply that there are more WFF, more Turing Machines and more Markov algorithms than can be counted.
Cheers.
{epistemological antinomy} is the end all be all of why for these things.
Professor Hehner calls this exact same idea {inconsistent specification} and
I call this exact same idea {self-contradictory question}. Gödel calls it
{epistemological antinomy}.
When ordinary people hear the term {epistemological antinomy} they translate
it into {some complex thing that I don't understand} in their or internal dialogue.
That is why I use the much clearer term {self-contradictory question}.
The reason why the halting problem is not solvable is that its specification does
not forbid self-contradictory questions. When we change the specification such
the self-contradictory questions cannot exist then the conventional proofs fail to
show that the halting problem is not solvable.
The reason that the halting problem persists is that the number of possible Turing machines is not enumerable; but any Turing machine designed to check for a halt can only check at most an enumerable number of Turing machines. It therefore cannot check if every Turing machine will halt.
That simply changes the subject away from an input deriving a self-contradictory
thus incorrect question for a specific decider. The most favorite rebuttal tactic of
all of my reviewers is to make sure to always change the subject before there is
ever any closure on any point.
After all, what I said above is the case; that is the reason for the halting problem. One way to treat this is as a reductio, showing that your approach has problems.
When I stopped tolerating infinite digression it ceased.
There are issues here as well, since a question is not the sort of thing that is apt to contradiction. A pair of statements can contradict; some statements can contradict themselves; but questions that are infelicitous are "inappropriate" or "ill-founded" or some such rather than contradictory.
Austin would have a field day.
So again, for consistency, mustn't you also reject Cantor's Diagonal argument as well?
Then correctly answer this question:
Is this sentence (true or false): "This sentence is not true."
Also, "This sentence is not true" is not a question. So I'm unclear as to how your reply addresses the point that a question is not apt to contradiction.
And further the liar does not play a role in the issue at hand, Gödel incompleteness and Halting. The sentence of interest is not "This sentence is not true" but "this sentence is not provable".
Quoting Banno
Well?
Not when we are mathematically mapping Carol's question to an input D to a halt decider H that does the opposite of whatever Boolean value that H returns. It is impossible for Carol to correctly answer her question for the same reason and in the same way that it is impossible for H to return the correct halt status of D.
Quoting Banno
The question is: >>>Is this sentence true: "This sentence is not true."<<<
Quoting Banno
Gödel says that it does.
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...
The Liar Antinomy
So Gödel agrees that incorrect questions are a thing.
Only because when I make a correct point you simply
ignore rather than acknowledge it.
I form a perfect incorrect question and then you change the words
and form a strawman rebuttal of those changed words.
The question is: >>>Is this sentence true: "This sentence is not true."<<<
Quoting Banno
Here's where we are up to: can you explain how you reject diagonalisation for Gödel but not for Cantor? Or do you reject Cantor's argument, too?
I am just saying that when diagonalisation is evaluating {epistemological antinomies}
it always makes sure to ignore all of the details. It never even notices that they are
{epistemological antinomies}. [b]Diagonalisation only looks for a 1 or a 0 on the
diagonal and makes sure to ignore absolutely everything else.[/b]
So {epistemological antinomies} (why the curly brackets?) are, for example, the liar. Where is there an example of the Liar being used in a diagonalization? What might that look like? OR do you mean something else?
We need to go back to more basic things.
Do you understand why this question has no correct answer?
The question is: >>>Is this sentence true: "This sentence is not true." ???<<<
If that is too difficult then do you understand why this is not true or false?
"This sentence is not true."
But that's not right - you've been given several correct answers.
In other words you do not understand that it is an incorrect question.
We go back one more step.
Do you understand why this is not true or false?
"This sentence is not true."
Well, no. Im pointing out that you only have a problem here if you restrict yourself to yes/no with no revision.
Go on. But be brief. You seem to be repeating yourself yet again.
So in other words you think that a Turing machine halt decider might
reply: "I don't know let me think about it?"
The thought experiment must stipulate that any answer besides
[yes, no] is a wrong answer to preserve the mathematical mapping
from Carol's question to a Turing machine halt decider.
The question was written by a PhD computer science professor to
make the computer science of the halting problem easier to understand.
Ah so you are totally convinced that Carol's question
posed to Carol is a self-contradictory thus incorrect question?
Although this forum has by far the very best people of any forum
that I have ever participated in I still have to put checks and balances
into the dialogue to prevent what always otherwise turns out to be
infinite digression.
If you agree we can go on to the next point if you don't agree
then you lack the mandatory prerequisites required for the next point.
Self Referential Undecidability Construed as Incorrect Questions
https://philpapers.org/archive/OLCSRU.pdf
Has been reviewed by Professor Hehner and clarifications have been
made corresponding to his review.
So check this out:
Quoting Prof Kirk Pruhs
This is a reductio argument:
First and most obvious question is where in this the thing you called the "isomorphism from Carol's question to the halting problem proof counter- example template" is located. It's not there. But we can add it: "Will Program Z loop forever if fed itself as input?"
Will Program Z loop forever if fed itself as input? The argument shows that we can't have an answer to that question. But that's exactly the point that shows that a program such as Halt cannot be written.
So sure, "the inability of a halt decider to correctly provide the halt status of an input that does the opposite of whatever halt status is provided does not place any actual limit on computation." But the impossibility of writing the program Halt does.
The argument is not that Z is impossible, but that H is.
By skimming the paper to contrive some excuse for rebuttal you missed this:
The bottom line of all of the above reasoning is that it is agreed that the halt status of some inputs to some halt deciders cannot possibly be correctly determined when the halt decider is required to report on the behavior of the direct execution of this input.
The brand new insight by the PhD computer science professor and myself (since 2004) is that the inability of a halt decider to correctly provide the halt status of an input that does the opposite of whatever halt status is provided does not place any actual limit on computation.
It is generally the case that the inability to do the logically impossible never places any actual limits on anyone of anything. That no CAD system can possibly correctly draw a square circle places no actual limits on computation.
That Carol's question contradicts every yes/no answer that Carol can provide
isomorphic to input D to decider H that does that opposite of whatever Boolean value that H returns.
It is equally a logical impossible for any CAD system to correctly draw a square circle.
The inability to do the logically impossible never places any actual limits on anyone of anything.
I have had a think in this for thousands of hours since 2004 when someone
else directly presented me with nearly the exact same question.
Quoting PL Olcott
Your question occurs with Z, not with H. Z is problematic, but Z is also a consequence of H, hence H is problematic.
The whole halting problem proof depends on some input D that does the
opposite of whatever Boolean value that H returns. Changing the names
does not change this. When you change the names I ignore them.
This is the whole point of my and Hehner's proof
That it is a logical impossibility for H to return a value corresponding
to the behavior of the direct execution of D(D) does not in any way
limit computation because the inability to do the logically impossible
is never any actual limit to anyone or anything.
Quoting Banno
IS this the equivalent of Carol's question? If not, what is?
I don't know and I don't care. Changing the subject is no form of rebuttal.
Already answered and you simply ignored.
Quoting PL Olcott
Quoting PL Olcott
SO, Z?
It shouldn't be this hard. I'm just checking that I've understood your point.
The best that I can tell is that Z is an incorrect sloppy mess that has no actual
name and no return value. Do you not know C?
On the other hand D and H have been fully operational code (for two years now) of the x86utm operating system that I created.
Sure, show me in C.
Isomorphic means
Carol's question for Carol and input D
to decider H are the exact
In both cases their question contradicts every answer.
My conclusion is that you're unable to present your thesis in a manner that is sufficiently clear to be evaluated.
If that was true then professor Hehner would not have totally agreed with me today.
I am thinking the the problem is that I assumed you had more technical knowledge
than you do. Tell me exactly how much you know about computer programming
and I can change my words to fit your knowledge level.
I thought the my original version of a halt decider that simply
[b]tries to say what another program will do when this other program
does the opposite of whatever it says is as clear as I can get.[/b]
That is the
What I am saying is much the same as you received elsewhere:
...and so on. I don't think it's just me.
Yes everyone that does not pay complete attention makes sure
that they never understand what is said. I will sum up the point
much much more concisely.
When input D to program H does that opposite of whatever program
H says that it will do it is logically impossible for program H to correctly
say what input D will do.
[b]The inability to do the logically impossible never places any actual
limits on anyone or anything.[/b]
That no CAD system can possibly correctly draw a square circle places
no actual limits on computation. Thus the halting problem proof places
no actual limit on what can be computed.
But such is absent here.
Cheers.
Once this is understood to be true
(1) The inability to do the logically impossible never places any actual limits
on anyone or anything
(2) then the logical impossibility of solving the halting problem the way it is
currently defined places no actual limit on computation.
Introduction to the Theory of Computation 3rd Edition by Michael Sipser
When we apply the MIT Professor Michael Sipser approved halt status critieria
(a) If simulating halt decider H correctly simulates its input D until H correctly determines that its simulated D would never stop running unless aborted then
(b) H can abort its simulation of D and correctly report that D specifies a non-halting sequence of configurations.
Then H does correctly determine that halt status of every input that was
defined to do the opposite of whatever Boolean value that H returns.
All of the details of this are fully elboarated on the first page of this paper:
Termination Analyzer H is Not Fooled by Pathological Input D
https://www.researchgate.net/publication/369971402_Termination_Analyzer_H_is_Not_Fooled_by_Pathological_Input_D
The key error that I and Professor demonstrated that the inability of solving the halting problem is the same as the inability for a CAD system to correctly draw a square circle both are logically impossible thus place no actual limit on computation what-so-ever.