Requiring the logically impossible is always an invalid requirement
Anything that is required to have (simultaneous mutually exclusive properties) cannot exist.
Requiring a square to be round or a circle to have four equal length sides: a {Square Circle}
is an example of this.
It is clear that the impossibility of creating a CAD system that can correctly draw square circles places no limits on what computers can do.
It is less clear that requiring a program H to report on the behavior of another program D that does the opposite of whatever H says is a logical impossibility when we see that program H1 can correctly say what D will do.
When we get back to the original {halting problem} we can see that no program H can ever always say what every program D will do because some D will do the opposite of whatever H says.
So the when the {halting problem} requires a program H to always say whatever program D will do includes programs that do the opposite of whatever H says this is requiring the logically impossible, thus the same as requiring a CAD system to correctly draw square circles.
The {halting problem} as defined requires the logically impossible therefore it places no actual limits on what can be computed.
Requiring a square to be round or a circle to have four equal length sides: a {Square Circle}
is an example of this.
It is clear that the impossibility of creating a CAD system that can correctly draw square circles places no limits on what computers can do.
It is less clear that requiring a program H to report on the behavior of another program D that does the opposite of whatever H says is a logical impossibility when we see that program H1 can correctly say what D will do.
When we get back to the original {halting problem} we can see that no program H can ever always say what every program D will do because some D will do the opposite of whatever H says.
So the when the {halting problem} requires a program H to always say whatever program D will do includes programs that do the opposite of whatever H says this is requiring the logically impossible, thus the same as requiring a CAD system to correctly draw square circles.
The {halting problem} as defined requires the logically impossible therefore it places no actual limits on what can be computed.
Comments (88)
Are you referring to Squaring the circle? Are you sure about your opening statement?
Computer programs can actually do this to any preassigned degree of accuracy. But using only a string compass available to the ancients and a straight edge is impossible in a finite period of time.
I am talking about creating a perfectly round thing that
cannot be round because it has four equal length sides,
thus a logically impossible square circle.
Relativity theory ran into that. It proposed something logically impossible, except to somebody who is willing to drop some obvious but unstated premises.
The inability to do an impossible thing isn't a limit? What is 'impossible' if not a limit?
Godel showed that the program H cannot solve the task to which it is put. You seem to know this since you reference the halting problem. I was just discussing this very thing in another thread.
Again, I don't find it logically impossible. All you need is to prevent access of H to the inputs of D. This again illustrates the fact that you've presuming conditions which have not explicitly been stated, same as the square circle.
Quoting PL Olcott
Agree, squaring the circle is an exercise in what can and cannot be constructed with a compass and straight edge. But the thing you list as impossible can be done, even without drawing a rhombus, which fits your definition of a square.
Just cut the circle (without moving anything) into four 90 degree arcs. Make them a different color if that helps to visualize it. There you go. Perfectly round figure that is also a figure consisting of four equal length sides. That was really trivial. Point is (the important part): Don't be so hasty to declare something impossible.
I am not talking about squaring a circle I am talking about drawing a circle that
"all points on a two dimensional surface that are equidistant from the center" and these exact same points form four straight sides of equal length in the same two dimensional plane.
My notion of {Square-Circle} is the epitome of logically impossible.
On the surface of the Earth, imagine drawing squares centered on the North Pole with increasing length sides until the sides coincide with the circle of the equator. As said ,what can be done is all about unstated presumptions.
I am trying to form a concrete example of the logically impossible
changing the subject to the not logically impossible is a strawman rebuttal.
I can still do it despite these new definitions.
Quoting magritteTip O' the hat to magritte, who actually paid attention to my point, and similarly did not see a restriction to Euclidean space, which is interesting because our universe is not Euclidean, so it's not an obvious assumption to make. One can draw a square circle on the 2D closed plane of the surface of (an idealized) Earth. It would have four equal length straight sides.
Quoting PL Olcott
I think only a mod can change the subject of a topic, so it is safe.
So yes, just be more careful when selecting your logically impossible thing, because so far none of them has qualified. Then, having found some suitable impossible requirement, in order to make your point in the subject line, you need to come up with a scenario that lists it as a requirement. That was also missing in the OP. So let's say for argument sake that a square circle (suitably defined) is impossible. What argument would plausibly list that as a requirement?
I did not provided 100% of all of the details of the mathematical definition of a square and a circle so that people having zero knowledge of math would be able to get the gist of my idea. Its really is not that hard to understand that a thing: (that must be round) and (cannot be round) cannot exist.
Anything that is required to have (simultaneous mutually exclusive properties) cannot exist.
Yes, you can't have a plane euclidian shape that is both a square and a circle. And yes, this is because these words do not describe or correspond to anything.
The sides of 's supposed square do not meet at right angles. Nor is it a plane euclidean shape.
When such a contradiction is met, one ought go back and check one's assumptions. The assumption that must be rejected in your work is that there can be an algorithm that correctly predicts whether any Turing Machine will halt.
This thread is a continuation of Self Referential Undecidability Construed as Incorrect Questions.
When the definition of the halting problem results in requirement that cannot be met because this requirement is a logical impossibility it is this problem definition that must be rejected. The inability to do the logically impossible never derives any limitation on anyone or anything.
The logical impossibility of solving the halting problem (within its current definition) is exactly the same as the logical impossibility of creating a CAD system that correctly draws square circles.
When a problem definition requires a logical impossibility then it is the problem definition that must be rejected.
When an assumption leads to a contradiction, the assumption must be rejected.
Yes and likewise for any problem definition that requires the logically impossible.
When we simply drop the requirement that termination analyzer H report on the behavior
of the directly executed D(D) and instead H reports on D correctly simulated by H then
the conventional halting problem proofs fail to prove that halting is undecidable.
When it is understood that requiring the logically impossible is an invalid requirement then the whole notion of undecidability is shown to be incoherent.
All decision problems that require the logically impossible are thus rejected as incorrect.
and yet see the argument as valid.
Do you agree that the argument is a reductio? If not, what structure does it have?
An agument cannot possibly be valid if it contains a fatal flaw.
When-so-ever any decision problem requires the logically impossible
this decision problem must be rejected as incorrect.
You keep saying that. Sure. Turing's argument is not an example of that. It is a reductio.
Reductio ad absurdum.
At least acknowledge that.
[b]The whole concept of decision problem undecidability is fatally flawed
because it requires satisfying a logically impossible requirement.[/b]
Instead of determining that some input is undecidable for some decision
problem we reject the decision problem itself.
Here it is again:
Quoting Prof Kirk Pruhs
I've bolded the assumption for you. It leads directly to the contradiction I've italicised.
Where's the flaw?
The flaw is that the whole notion of decision problem undecidability
is inherently flawed in that it requires the logically impossible.
Every undecidable decision problem is simply wrong.
When I ask you What time is it (yes or no)?
Does the lack of your correct answer indicate that you are stupid
or is there something wrong with the question?
Quoting PL Olcott
Does this apply generally? Are all supposed reductio arguments so flawed? They all contain a logical impossibility...
This by way of pointing out that your argument is not well-formed.
Yes this applies generally. The undecidability of all undecidable decision
problems is always anchored a the requirement to satisfy a logical
impossibility.
Thus the halting problem is analogous determining that computation
is limited by the failure to satisfy any other logical impossibility such as
requiring a CAD system to correctly draw a square circle.
To all reductio arguments?
To all decision problem definitions.
https://en.wikipedia.org/wiki/Reductio_ad_absurdum
Finds the logical impossibility thus is fine.
When it finds a contradiction is derived by a decision problem
then it is this decision problem that must be rejected.
When it is contradicted that some H can correctly determine
the halt status of the direct execution of every D, then this
definition of the problem is rejected as incorrect.
Why? Isn't that just special pleading?
Quoting PL Olcott
Yes, in the example argument above, Z is shown to imply a contradiction, and so is to be rejected.
That's the reason for rejecting the assumption.
I dunno. This all appears pretty basic. It still looks to ma as if you have not followed the structure of the argument.
When I ask you the question: What time is it (yes or no)?
it is dead obvious that your inability to provide a correct
answer is not your fault it is the fault of the question that
requires a logically impossible answer.
It is the same for all questions that require logically impossible
answers and their analog of
decision problems that have instances of questions that require
logically impossible answers.
Seems to me to be in Z; the contradiction that is used to deny the assumption.
But I don't know what you think here.
In the other thread I suggested that the analogue would be "Will Program Z loop forever if fed itself as input?"
So the problem must be H.
So you can't see that the fact that Z does the opposite of whatever
H says makes saying the correct thing a logical impossibility for H?
That is all that it takes to determine that the halting problem as defined is invalid.
Another logically impossible problem is making a CAD system that can draw square
circles. In other words it must draw a thing that
[b]All undecidable decision problems are simply invalid because their problem definition
requires the logically impossible.[/b]
When a human is asked a question that is defined to have no correct answer
such as: What time it is (yes or no)? we know to reject the question and not
blame the human.
When a computer program is presented with data that has been defined to
have no correct answer (Boolean return value) we blame the program and
not the data.
This is inconsistent. Professor Hehner and I both agree that any program
specification that lacks a correct (Boolean return value) for some inputs
proves that this specification is unsatisfiable thus incorrect.
I keep trying to make my words more clear. That a PhD computer science professor perfectly agrees with my exact words provides a subtantial weight of evidence that I am not simply a crackpot.
Most people start with the idea that I must be a crackpot thus pay no attenton to what I actually say and put ALL of their energy into pointing out mistakes that turn out to not exist.
That's why I've asked you to show as explicitly as you can where Carol's question occurs.
Quoting Banno
That depends on what your H does and you didn't say.
I have spent two full time years making the x86utm operating system so that I could make a real H and Z so I have complete proof what they do and why and how they do it. My H is called H and my current Z is called D.
Any expert C programmer can follow this proof by examining all of its source-code.
The key essence of this source-code is on page 1 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
:smile:
My H simulates its D until it can see that D keeps calling H in recursive
simulation. Then it aborts this simulation and returns 0 for non-halting.
Page 1 of my paper shows all of the details of this to any expert C programmer.
These words may be a little too technical for you.
It seems that everyone agrees with this:
(a) When the halting problem is defined with a program
specification that requires an H to report on the behavior
of the direct execution of D(D) that does the opposite of
whatever Boolean value that H returns then this is an
unsatisfiable program specification.
(b) An unsatisfiable program specification is merely
the inability to do the logically impossible thus places
no actual limit on anyone or anything.
:wink: You don't have to be here. No need to respond if my comments are not useful.
Or is it that what you have to say cannot be made sufficiently clear?
You started this with Carol's question, went on to claim that Gödel's theorem was wrong, backtracked to Turing and now obfuscate.
That it is possible to write a program which undermines H shows that H cannot be applied to all programs. Quod Erat Demonstrandum.
I just totally proved my whole point if and only if you fully understand all of its words.
I have to write them so that computer programmers and computer scientists can
most easily understand them.
This key point mostly uses ordinary words.
The inability to do the logically impossible places no actual limit on anyone or anything.
Anyone that agrees with the above point agrees with the complete essence of our proof.
:rofl:
If you would prove Turing wrong, you will need more than mere assertion.
You did not pay attention to the words that were targeted for you:
This key point mostly uses ordinary words.
The inability to do the logically impossible places no actual limit on anyone or anything.
Anyone that agrees with the above point agrees with the complete essence of our proof.
Hadn't we?
That is, a logical impossibility places a limit on the viability of our assumptions... to adopt your odd wording.
If one's assumptions lead to contradiction, then at least one is in error. Assuming we can produce H leads to contradiction. Hence we cannot produce H.
We also cannot correctly determine the square root of a stack of pancakes.
[b]Logical impossibilities never create actual limits.
Logical impossibilities never create actual limits.
Logical impossibilities never create actual limits.
Logical impossibilities never create actual limits.[/b]
Yep. Quite agree. If your conclusion is a logical impossibility, there is something amiss with your assumptions.
Or with your process.
When the halting problem is defined such that solving it is logically impossible
then we reject this problem definition as unsound for the same reason that
we reject this question as unsound: What time is it (yes or no)?
Every question that is defined to have no correct answer is an incorrect
question.
Like always there is no sense publishing anything until some people understand
that the words are correct. There are now two people in the world that understand
this and everyone else seems to be so sure that we are wrong that they can't
even pay complete 100% attention to a single sentence.
I have a completely different proof that shows every single detail of
how H does correctly determine the halt status of the impossible input.
That a PhD computer science professor of decades has his own
proof of this and agrees that my proof is correct provides sufficiently
compelling evidence that this is not the case. He has been published
several times in the two most prestigious computer science journals.
Impossibility in philosophy is used in the physical event -- i.e. it's impossible to be in two places at the same exact time. Another one, it's impossible for humans to fly in the air.
Illogical is what you mean when you say a circle cannot be a square. Of course you would object to my description as you might think, but squares and circles are physical objects. Actually, they are conceptual objects, hence, logical in the sense of "it makes sense by definition that a circle has 360 degree rotation while the sum of the degree of the square is also 360".
Logically impossible is the strongest kind of impossible.
A thing that is required to have simultaneous mutually exclusive properties
like a square circle that must be round and must not be round is logically
impossible. Making a perfect angel food cake using only house bricks for
ingredients might be possible by rearranging the atoms of the bricks.
Yeah, it is. (p & ~p). Contradiction.
Quoting Impossible Worlds (Stanford Encyclopaedia, my bolding)
Quoting L'éléphant
Tell that to an electron in a double-slit experiment.
That was a very apt cite.
However, you might be thinking of "logical possibility" which the likes of Chalmers are prone to use. But we can't state the opposite: logically impossible. It's nonsense.
Contradictions are not the same as logically impossible.
Impossible worlds is the exact same concept as logically impossible.
[logically impossible] is contrast wth [physically impossible] and
covers every expression of language that is proved to be impossible
entirely on the basis of the meaning of its words.
There are zero possible worlds where someone can correctly draw a
a square circle.
Logical possibilities are all the statements that cannot be proven false entirely on the basis of the meaning of their words. Here is a very strange logical possibility.
Although this is attributed to Bertrand Russell, I remember creating this myself in 1993 and then reading that Bert came up with the same thing. Bert's version lacks the key detail that this creation event must have been instantaneous otherwise subconsious memory would have a record of it.
https://neurochatter.com/continuity-of-self-was-the-world-put-into-place-five-minutes-ago/#:~:text=The%20%E2%80%9Cfive%20minute%20hypothesis%E2%80%9D%20by,into%20existence%20five%20minutes%20ago.
Here's the Ngram.
that require all expressions to be either satisfiable or their negation satisfiable.
...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)
Antinomy
...term often used in logic and epistemology, when describing a paradox or unresolvable contradiction. https://www.newworldencyclopedia.org/entry/Antinomy
Thus when we plug the formalized {epistemological antinomy} of the Liar Paradox into
a similar undecidability proof, we find that this semantically unsound expression "proves"
that the formal system that contains it is incomplete.
Gödel does not use the liar. The sentence of interest is not "This sentence is not true" but "This sentence cannot be proved".
But we've covered this, earlier.
I created Minimal Type Theory that spits out the directed graph of its own WFF.
This is the only system that I know of where the Liar Paradox can be formalized correctly,
every other system cheats and knowingly formalizes self-reference incorrectly.
https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
LP := ~True(LP)
Quoting Banno
G := (F ? G)
Prolog rejects both of the above expressions as semantically unsound.
It detects that same cycle in their directed graph that I call pathological
self reference.
Quoting PL Olcott
When you started off in your OP, you wanted to make a statement that is necessarily false. Which is fine. But now I think this whole thread is just nonsense.
Do the properties of a circle hold necessarily? And do the properties of a square hold necessarily? Then it goes without saying that the circle and square have asymmetrical relations. It is necessarily false that a circle can be drawn as a square.
Quoting PL Olcott
Thank god that "incompleteness" is not accepted as one of the logical status of a statement.
Incompleteness
If that happens, we don't judge it as incomplete -- we judge it as contingently false in this system, but not in all possible worlds. A proposition is non-contingent only if, necessarily, it cannot be the case (that is, in all possible worlds, it is false).
Quoting L'éléphant
That is factually incorrect. As soon as any WFF of any formal system is determined to neither be provable nor refutable in that formal system then that formal system
Gödel himself said that this does include self-contradictory expressions.
...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)
Antinomy
...term often used in logic and epistemology, when describing a paradox or unresolvable contradiction. https://www.newworldencyclopedia.org/entry/Antinomy
You're applying something like Gödel's theorem to something like modal logic. No wonder we can't understand each other. Logic uses a lot of propositions that aren't theorems. The "logical status" of a statement does not need a "complete theorem" in order to be .. a logical conclusion.
In effect, we aren't claiming a "complete theorem" when we say that, to say "It is raining and it is not raining at the same time" is a contradictory statement. We also aren't claiming a complete theorem, or even an incomplete theorem when we say that "if Paul is older than Tom, then Paul must have been born earlier than Tom".
Think. Do you really need a theorem to say that a square can't be drawn like a circle? No. While it is true that the definition of the square and the definition of the circle are both theorems themselves, when we make a determination that a circle cannot be drawn like a square, our own statement is not, or does not require a formulation of a theorem itself. We make a decision based on the existing theorems.
Mathematical Incompleteness determines that a formal system
WFF x of the language L of a formal system F can neither be proved nor refuted in F.
Tarski even uses the actual Liar Paradox as the key basis of his whole Undefinability Theorem:
(3) x ? Provable if and only if x ? True. https://liarparadox.org/Tarski_275_276.pdf
Here is the Liar Paradox as a WFF of Minimal Type Theory LP := ~True(LP)
The ":=" operator is like macro substitution and provides the means for an expression to directly refer to its actual self.
Prolog rejects the same expression when encoded in Prolog:
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Yet mathematical incompleteness still blames the formal system and not the semantically unsound expression.
I have never been talking about that. I am talking about making single geometric object that
But your thread is called: 'Requiring the logically impossible is always an invalid requirement'.
According to this premise, why should we demand from 'God' to make a single geometric object that is entirely a square and, simultaneously, is entirely a circle on the same two-dimensional plane then?
Maybe you were suggesting that a deity could do the logically impossible, because 'God' tends to be beyond human understanding. But again, we are in a paradox, because we are accepting that it is invalid to require logically impossible tasks...
Your threads are always very knotty!
That is not what I meant. I want to define a task that is logically impossible. Most people don't know what logically impossible means.
Ah, I see.
Quoting PL Olcott
Myself included, I am not going to lie to you. What you explain and write in your threads is very interesting, but I admit that I don't usually understand what it really means.
Logically impossible is the maximum of all impossibilities.
Things that the creator of the universe cannot do are logically impossible things. God could make
a real live two-dimensional Bugs Bunny, this only require rewriting the laws of nature. God
cannot make a square circle, because it is contradictory, it must have (mutually exclusive)
properties that it cannot have.
My original example of an impossible task was to bake a perfect angel food cake using only house bricks for ingredients. Someone pointed out the rearranging the molecules of the bricks could make this possible. He suggested that I use the term logically impossible. Then people here complained that they never heard of this and had no idea what it means. The key example that I knew was the existence of a thing that required simultaneous mutually exclusive properties.
Well, logically impossible means something that is self-contradictory. I understand clearly your example of the 'square circle'. This is logically impossible because the concepts of reality contradict each other. So, we can assume that a 'square circle' is both actually impossible and logically impossible.
Nonetheless, to bake a cake using only house bricks is something which is logically impossible but actually possible. Because depending on the concepts of my - or your - reality, that cake can eventually be cooked using only house bricks. Maybe it is an impossible task for you, but not for me. Agree?
If one can rearrange the molecules of a house brick to become an angel food cake then it is not logically impossible to make an angel food cake from house bricks.
There is nothing that anyone can do to make an object that has four equal length sides and simultaneously has zero equal length sides.
Yes, because there is simply nothing that a round square could be. I think this is the main point after all.
No the actual main point is that the halting problem proofs are incorrect because they require a computer program to provide a correct answer to a self-contradictory question that has no correct answer because it is a self-contradictory question.
When we correct the halting problem so that it is no longer asking a self-contradictory question we get a different answer.
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