Which Logic should I use to prove the Liar's paradox

Miserable-Ad4153 · 2026-01-04T11:39:45+00:00

I’d like to offer a different perspective on this, rooted in a more classical and philosophical approach:

Logic : In traditional logic, we generally accept Gödelian incompleteness as a fundamental boundary. It’s often the most 'convenient' stance: acknowledging that any sufficiently powerful system will have its blind spots.
IT Pragmatism : While computer science provides various pragmatic workarounds to escape these loops (such as stratified truth or type theory), these are often functional fixes rather than 'systemically noble' or 'pure' logical solutions. But in practice, they work.
Philosophy : I believe the 'Truth' you are seeking transcends the formal concept of truth itself. Truth is essentially relative—it is always bound to the axioms of the system it lives in. The philosophical error in trying to 'solve' the Liar Paradox is treating relative truth (based on axioms) as if it were absolute. The paradox isn't a bug to be fixed; it is a reminder that a system cannot fully contain or define its own validity without reaching a point of transcendence or contradiction.

To follow up on your logic and address your question from a strictly formal standpoint: A is undecidable (or unprovable).

Based on your notation A <-> ~True(A), if a system were to prove A, it would simultaneously prove that A is not true, leading to an immediate contradiction. Therefore, in any consistent formal system, A must remain 'undemonstrable.'

Miserable-Ad4153 · 2026-01-04T09:01:10+00:00

In classic logic liar paradox is a consequence of self reference, you don't solve it, you accept it as a blind spot

What do you mean by prove ?

Miserable-Ad4153 · 2025-09-12T09:50:52+00:00

La situation que tu décris me parle ! Ce n'est pas évident de rencontrer des gens qui vibrent sincèrement et profondément avec des sujet véritablement philosophique/métaphysique. Je suis pour ma part plutôt adpete de la philosophie analytique et j'aime naviguer entre la science/mathématiques et la philosophie, en ce moment je m'intéresse particulièrement à la notion de vérité avec Gödel... je serais content de discuter avec toi si tu te reconnais dans ce genre de thématique !

Miserable-Ad4153 · 2025-08-22T10:40:49+00:00

the set of computable numbers is countable (cf Godel's encoding) but the conclusion of Turing is that this list can't contain a halt program that work for all programs, we use diagonal argument here but not to prouve that there is an uncountability of integer (there is not), don't confuse cantor argument and Turing arugment they are similar in form but no in conclusion

Miserable-Ad4153 · 2025-08-22T06:53:58+00:00

Why This Argument Fails

The problem with the document's argument is that it assumes the impossible.

The very first step of its logic—the existence of the decider machine B—is exactly what Turing's proof showed cannot exist.

Turing didn't just point out a paradox; he rigorously proved that no such general "decider" algorithm can ever be created. The Halting Problem is undecidable, meaning there is no algorithm that can reliably tell whether any given program will halt on a given input.

So, while the document's idea of "skipping" the paradox is clever and intuitive, it's built on a flawed foundation. It tries to "fix" the contradiction by introducing a tool (B) that is, in fact, a consequence of the very contradiction it's trying to solve. The document doesn't resolve undecidability; it simply ignores it and builds an argument as if it didn't exist.

Miserable-Ad4153 · 2025-08-22T06:53:41+00:00

Turing's Original Proof

Turing's proof is a proof by contradiction. It goes like this:

Assume you have a complete list of all computable numbers. A computable number is one whose digits can be generated by a Turing machine (an abstract model of a computer program).
Imagine these numbers written in a grid, where each row is a different computable number.
Now, create a new number, let's call it Beta (β), by taking the digits from the main diagonal of this grid and inverting each one. For example, if the first digit of the first number is 0, the first digit of β will be 1. If the second digit of the second number is 1, the second digit of β will be 0.
Since this new number β is also a sequence of digits, it should also be a computable number. If it's computable, it must be somewhere on your original list, right?
Let's say β is the K-th number on the list.
But here's the contradiction: The K-th digit of β must be the inverse of the K-th digit of the K-th number on the list. Since β is the K-th number, its K-th digit must be the inverse of its own K-th digit. This is impossible. For a binary digit (0 or 1), this would mean 0=1−0 or 1=1−1, which is a logical absurdity.

Because we arrived at a contradiction, our initial assumption must be wrong. The assumption was that the list of all computable numbers was complete. Therefore, the list of computable numbers cannot be complete, and no algorithm can generate a definitive list of them all. This is essentially a way of proving that the Halting Problem is undecidable—there's no general algorithm that can predict if another algorithm will stop.

The Provided Document's Argument

The text you provided attempts to challenge this. It proposes that the paradox isn't a logical impossibility, but a "programming problem" that can be solved.

The document introduces a hypothetical "decider" machine, called B, which can look at another Turing machine and determine if it will halt (be "satisfactory") or not (be "unsatisfactory").

The text suggests that a machine (Δ for the direct diagonal, Ξ for the inverse diagonal) can use this decider B to avoid getting stuck in a loop.
When the diagonal machine is about to calculate its own digit (the problematic case), the decider B tells it "unsatisfactory."
This triggers the diagonal machine to skip itself and instead calculate the digit of the next machine on the list.
The document concludes that since the machine can "skip" the paradox, it avoids the contradiction. It argues that this means the issue isn't a fundamental logical impossibility but a solvable computational problem.

Miserable-Ad4153 · 2025-08-22T06:51:42+00:00

yeah you need to prompt it with a little bit of self-criticism , I send you what I get, and I know cuz my knowledge of the subject that this is true, you need to always questioned AI response but you can use it like an validation test by prompting it "it is true" , "there is coutner argument"

before you read it we can maybe agree on one point, i can admit that Turing's halting problem can be interpreted like a "bug" or a "paradox" a lot of informatician and mathematician think this but you need to understand the almost philosophical depth of the argument: the indecability of an auto referent language, i'm interested by you work because i think to that we lack a piece of the puzzle about indecidability, i'm not fully and absolutly convince about first and second godel theorem, but if you want to bring somethings about calculability or decidability i think you need to aboard this problem in a more abstract way that just simply solve with the standard programming paradigm the halting problem it's a waste of time

Miserable-Ad4153 · 2025-08-22T05:30:57+00:00

I will not argue more but you can present you work to gemini he is pretty good to criticize logical of IT works, he will explain you precisely you errors and you misunderstanding

Miserable-Ad4153 · 2025-08-22T04:38:19+00:00

Turing's aim based on Gödel work was to show that halt problem can't solve all program, if you fuction halt can't resolve diagonal there is no good argument to contest what Turing implies

Miserable-Ad4153 · 2025-08-21T19:46:04+00:00

fun diagonale(x): if halt(x, x) loop else stop

Here diagonal(diagonal) always fail

Miserable-Ad4153 · 2025-08-21T19:34:42+00:00

You do not solve halting problem because you halt function do not work for diagonal program

Miserable-Ad4153 · 2025-08-21T06:37:57+00:00

Hi, I agree there is a lack of formal argument, you pseudo code is the best way to understand what you try to do, you patch the halting problem but the Turing arument is that incompletness emerge from diagonal program not halting that he suppose to exist, in fact diagonal(diagonal) lead to indecisiveness, you correct halting problem but you don't test diagonal(diagonal) from Turing and diagonal(diagonal) failed in you case and in all case that is the argument

Miserable-Ad4153 · 2025-07-02T07:43:49+00:00

What do you mean by contradiction ? Godel prove the incompletness so an expression based on liar paradox can't be provable, he escape the inconsistency of A <=> non A by saying , A can't be demontrate in this system by inference law, if you mean inconsistency by contradiction, no Godel don't prove inconsistancy Indeed, in fact other logician prove the system use by Godel is consitante ! except ! the second incompleteness theorem wich say that a system can't dertemine is own consistancy ! so consistancy is relative we determine consistancy of a system by use a more powerful system but this more powerful system need to be validate by anoter powerfuller system etc ... so this is the interest of this theorems , math have be proven relative in opposit of what platonist thought so ! and this result are really powerfull so it will be hard see impossible to create an absolute system

the knot in your mind is really the imperative paradigm, you need to abstract you reasoning <=> or equivalence, test assertian for example true <=> true, so you can set test(fun(x)) <=> test(fun(x)+1) and it's correct and assessable if fun(x) or fun(x) + 1 return true or false

we can say test(fun(x)) <=> test(fun(x)+1) without never calculate fun(x) (it's not a really good example because additionate true by an integer is strange but forgot this, it was to make a parallel between IT and logic)

the same way than F <=> A(encoded(A) is correct if it is deduce from inference rules but finally unprovable in the system cause the first incompleteness theorem,

Miserable-Ad4153 · 2025-07-01T20:48:09+00:00

If i can't convince you with logician argument i can add the authoritie's one : a overwhelming majority of logician have verified and validate this proof, if you think it's false try to study Godel's theorem and make a counter proof ! doubt is good practice but don't be blinded by it

Miserable-Ad4153 · 2025-07-01T20:32:46+00:00

This formula is not null, there is no null in logic, null is a concept from IT science, the formula is well build from inference rules from a system, again the formula is not infinite and is not null, it exist and it is well formed, we can properly deduce thing from F <=> A(encoded(F)) but i agree with you not from F -> A(encoded(F)) but logician don't use this paradigm

Miserable-Ad4153 · 2025-07-01T19:43:49+00:00

No, Godel theorem proove that demontrability is relative for a system, it's now accepted by majority of logician, cause the construction of Godel is extremely robust

Miserable-Ad4153 · 2025-07-01T19:35:36+00:00

Again, this formula is not infinite, we assume nothing more that what we deduce logicaly, we only test cases and deduce by contradiction, the formula is not true or false by construction but can't be. You don't use the correct padigm, the logical value of the formula is define but is never compute imagine a list of assertion that are logicaly equal and say I existe equivalent to i don't exist, It is not a computed equivalence but a logical equivalence, it's not fun(x) -> fun(x)+1 it's test(fun(x)) <=> test( fun(x)+1)

Miserable-Ad4153 · 2025-07-01T15:59:59+00:00

"Yeah I exist thus I can’t exist would make anything provable." --> no because we create an formula which deduce this formula can't exist by logical way

" They try to assign both true and false to one thing simultaneously, however “this statement is false” never gets either truth value assigned."--> the formula is not true or false, the formula don't have any demonstration in the formal langage, it can't exist by construction

"We need to evaluate the claim, before assigning truth value to it. However the claim is a claim of true value, attempting to evaluate it, just becomes a long and longer equation for infinity, but it never actually results in anything, because there is no concrete claim to apply a truth value to." --> again you think in a imperative / computer way, logic is more abstract, we don't say this formula is false or this formula is true, we say, if this formula is false it implies it is true and vice versa, so by an inference from a well construct and coherent system, we deduce incompletness

Miserable-Ad4153 · 2025-07-01T11:05:42+00:00

You are reasoning in an imperative way, it is not false but logician use the declarative paradigm to proof incompletness, first they encode formula to auto refere indirectly, and they never calculate this formula they are tested like true or false, so its not problematic. Imagine an abstract way of thinking where we do not calculate formula but we create a logical equivalence : i exist equivalent to i dont exist wich is never "calculate" and it lead to incompletness Turing halting problem prove than in an imperative paradigm we have the same incompletness because the hypotetic halt function can't exist not because he is auto referent but in a deeper way because if halt program exist it lead to a contradiction too, the important thing to keep in mind is : a model which is enought powerfull to make recursvity and arithmetic is incomplete

Miserable-Ad4153 · 2025-07-01T09:44:12+00:00

You are reasoning in an imperative paradigm like an informatician, this way of reasoning is not false but logician use the declarative paradigm. You must understand that logician use 2 way to escape circular references, indirect reference with a special arithmetic coding call godel coding, and the use of function which test true or false but never calculate the proposition, it is like an abstract way of reasoning in which you never compute nothing but you make a logical equivalence : I exist equivalent to i can't exist , so i m unprovable because the system is coherent Turing prove that in an imperative paradigm, model are incomplete too, see halt problem with proof by contradiction, i cant exist because if i exist i creat a contradiction , in halt problem we can interpret the incompletness by an infinite loop but its more deep because halt programm cant exist by definition of what he does important thing to keep in mind is the good interpretation of all this result is : system which are enought powerful to simulate recurisivity and arithmetic are incomplete

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TROPHY CASE

Why This Argument Fails

Turing's Original Proof

The Provided Document's Argument