The logical error of the Liar Paradox

[PeteOlcott]

The Tarski Undefinability proof depends upon the Liar Paradox, without it this proof fails.

When we assume this Truth predicate: ∀x True(x) ↔ ⊢x
(An expression of language x is only true when it is
the consequence of a formal proof from an empty set of premises).
Then the Liar Paradox: "This sentence is not true."
Would be formalized as this symbolic logic: LP ↔ ~⊢LP.

If a logician hypothesizes that the symbolic logic (including its truth
predicate basis) formalization of the Liar Paradox is a precise translation
of its English form then it is very easy for them to see its logical error.

Because of "↔" LP has the same truth value as its unprovability, LP can
only be true when it is unprovable. Because the assumed truth predicate
requires LP to be provable LP can only be true when it is not true.

Now that it has been shown that the Liar Paradox can be construed as
having the same truth value as the non existence of its formal proof
from an empty set of premises, we can see that it is self-contradictory
in that the lack of such a proof (construed within the truth predicate)
would directly contradict an LP truth value of true.

Refuting this formalization of the essence of the 1931 Incompleteness Theorem
∀F (F ∈ Formal_Systems & Q ⊆ F) → ∃G ∈ L(F) (G ↔ ~(F ⊢ G)) (Panu Raatikainen 2018)
Raatikainen is the author of the SEP article and many related papers:
https://plato.stanford.edu/entries/goed ... pleteness/

As long as the following sentence is unsatisfiable:
∃F ∈ Formal_Systems ∃G (WFF(F,G) ∧ (G ↔ ~(F ⊢ G)))
Then the Raatikainen sentence would also be unsatisfiable.
The same reasoning applied to the formalized Liar Paradox applies equally to the above sentence.

Significant rewrites after many of the comments below were written

[wtf]

If a logician hypothesizes that the symbolic logic is a precise translation of
the English sentence ...

Who claims that? Surely most of the philosophy of language is an attempt to deal with the obvious failure of symbolic logic to capture the meaning and usage of natural language.

[PeteOlcott]

If a logician hypothesizes that the symbolic logic is a precise translation of
the English sentence ...

Who claims that? Surely most of the philosophy of language is an attempt to deal with the obvious failure of symbolic logic to capture the meaning and usage of natural language.

If we hypothesize that it is correct and see where this hypothesis leads, that adds
to the evidence that it is correct. I have spent 7000 hours on this stuff in the last
three years, and at least another 7000 hours in the preceding 20 years.

More very significant evidence that this formalization is correct is that this
formalization makes the exact same sentence that the Tarski Undefinability
Theorem depends on in that this formal LP can be neither proven nor disproven.

[wtf]

If we hypothesize that it is correct and see where this hypothesis leads, that adds
to the evidence that it is correct.

Is that your core thesis? That symbolic logic exactly and perfectly represents natural language? Do you know a lot of languages? Are you perhaps a radical Chomskyite? (Referring to his technical work in linguistics, not necessarily his politics)? Have you read much philosophy of language? Use language much in daily life? Isn't natural language terribly ambiguous and almost never to be taken literally?

Example: Suppose you tell me you've refuted Gödel, Turing, Tarski; and also found the missing dollar.

And I respond: "Yeah, right!"

How would you logically symbolize my meaning?

I have spent 7000 hours on this stuff in the last
three years, and at least another 7000 hours in the preceding 20 years.

Of what evidentiary value is that? Time spent is an input, not an output. I don't doubt your commitment. Only the soundness of your claims.

[PeteOlcott]

I updated the original post with new key details.

[PeteOlcott]

I have spent 7000 hours on this stuff in the last
three years, and at least another 7000 hours in the preceding 20 years.

Of what evidentiary value is that? Time spent is an input, not an output. I don't doubt your commitment. Only the soundness of your claims.

Noted. The amount of effort only indicates the degree of persistence, not that
anything that I say is correct.

The strongest evidence that my formalization of the Liar Paradox is correct
is that this formalization provides the Tarski Undefinability proof exactly what it
needs and already claims the Liar Paradox is, neither provable nor disprovable.

[wtf]

The strongest evidence that my formalization of the Liar Paradox is correct
is that this formalization provides the Tarski Undefinability proof exactly what it
needs and already claims the Liar Paradox is, neither provable nor disprovable.

Ok you only meant to say that symbolic logic matches English for this one sentence. My misunderstanding.

I can't say I've ever taken the Liar paradox seriously and don't actually understand all the learned words that have been written about it. So if you've proved it or disproved or resolved it in some way, I won't dispute you. Not only because I can't, but also because I lack the interest. I don't regard it as a sensible problem. It's just word play.

But on the Gödel stuff, well you just didn't answer my question and the question I asked was one that would occur to virtually anyone familiar with the incompleteness theorems. (Which one did you refute, by the way? The first or the second?)

If Gödel's first incompleteness theorem (if that's the one you meant) is wrong, then there is a formal system in which every well-formed statement of mathematics has either a proof or a disproof, but not both. Do you at least understand that much about it?

[PeteOlcott]

I can't say I've ever taken the Liar paradox seriously and don't actually understand all the learned words that have been written about it. So if you've proved it or disproved or resolved it in some way, I won't dispute you. Not only because I can't, but also because I lack the interest. I don't regard it as a sensible problem. It's just word play.

But on the Gödel stuff, well you just didn't answer my question and the question I asked was one that would occur to virtually anyone familiar with the incompleteness theorems. (Which one did you refute, by the way? The first or the second?)

If Gödel's first incompleteness theorem (if that's the one you meant) is wrong, then there is a formal system in which every well-formed statement of mathematics has either a proof or a disproof, but not both. Do you at least understand that much about it?

The Tarski Undefinability proof crucially depends upon the Liar Paradox, so showing
that the Liar Paradox is erroneous causes Tarski's proof to fail.

I only refuted the simplified essence of the first incompleteness theorems conclusion.
The reasoning that I provided above shows that some expressions of formal language
are neither provable nor disprovable because they are erroneous.

Yes I agree with these consequences. These are the exact same consequences that I
have been aiming for all along. Additionally I also aim to apply these same consequences
for formalized natural language, ala a system like Doug Lenat's cyc:
https://en.wikipedia.org/wiki/Cyc

[wtf]

I only refuted the simplified essence of the first incompleteness theorems conclusion.

I don't know what that means. You refuted the "simplified essence" of X but you did not necessarily refute X? So is Gödel's incompleteness theorem refuted or not?

Yes I agree with these consequences. These are the exact same consequences that I
have been aiming for all along.

So you claim, to be clear, that there exists a formal system in which every statement of mathematics has either a proof or a disproof but not both?

Why is it so difficult to pin you down on this simple question? Do you understand why the first incompleteness theorem logically implies what I've said?

[Logik]

So you claim, to be clear, that there exists a formal system in which every statement of mathematics has either a proof or a disproof but not both?

It depends.

In an infinitist paradigm: No. (because halting problem)
In a finitist paradigm: Yes (just brute force the FINITE state-space)

Recursively - observe that finitism/infinitism is just a choice e.g itself an Entscheidungsproblem

https://en.wikipedia.org/wiki/Entscheidungsproblem

[PeteOlcott]

I only refuted the simplified essence of the first incompleteness theorems conclusion.

I don't know what that means. You refuted the "simplified essence" of X but you did not necessarily refute X? So is Gödel's incompleteness theorem refuted or not?

Yes I agree with these consequences. These are the exact same consequences that I
have been aiming for all along.

So you claim, to be clear, that there exists a formal system in which every statement of mathematics has either a proof or a disproof but not both?

Why is it so difficult to pin you down on this simple question? Do you understand why the first incompleteness theorem logically implies what I've said?

This is the formalization of the essence of the 1931 Incompleteness Theorem
∀F (F ∈ Formal_Systems & Q ⊆ F) → ∃G ∈ L(F) (G ↔ ~(F ⊢ G)) (Panu Raatikainen 2018)

He is the author of this article and has many published papers in the field:
https://plato.stanford.edu/entries/goed ... pleteness/

There exists a formal system in which every statement of mathematics has either a proof or a disproof but not both?
Exactly.

[PeteOlcott]

So you claim, to be clear, that there exists a formal system in which every statement of mathematics has either a proof or a disproof but not both?

It depends.

In an infinitist paradigm: No. (because halting problem)
In a finitist paradigm: Yes (just brute force the FINITE state-space)

Recursively - observe that finitism/infinitism is just a choice e.g itself an Entscheidungsproblem

https://en.wikipedia.org/wiki/Entscheidungsproblem

All undecidable decision problems are only undecidable because they are stated incorrectly.
It took me at least 3000 hours since 2004 studying the same two pages of the Peter Linz text
to notice a key detail that everyone else had been missing.

http://liarparadox.org/Peter_Linz_HP(Pages_318-319).pdf

[PeteOlcott]

So you claim, to be clear, that there exists a formal system in which every statement of mathematics has either a proof or a disproof but not both?

When we have things such as the Goldbach conjecture things get a little fuzzy.
Whether or not we have currently enough knowledge to decide provability one
way or the other is not the same thing as whether or not provability is possible.

I can commit that to the best of my knowledge and understanding that every
statement of mathematics has either a proof or a disproof but not both.

A refutation of the 1931 Incompleteness Theorem would not necessarily directly
have this as its consequence. I would not bet my soul that there is not another
undiscovered Incompleteness proof. To the best of my current understanding
any Incompleteness proof is analogous to proving that 3 > 5, categorically impossible.

I don't know or care about any other consequences because this is too much
like counting my chickens before they hatch. I have already addressed this one
key consequence before and you did not notice.

It seems like your key consequence is at the root of Hilbert's program:
https://en.wikipedia.org/wiki/Hilbert%2 ... 's_program
For that reason it makes perfect sense that it would be very important to you.

Because I come from a computer science background I think of all of these things
from a formalist perspective.

In foundations of mathematics, philosophy of mathematics, and philosophy of logic,
formalism is a theory that holds that statements of mathematics and logic can be
considered to be statements about the consequences of the manipulation of strings.

I appreciate your reviews.

[Logik]

All undecidable decision problems are only undecidable because they are stated incorrectly.

We know this. Any algorithm without exit criteria is non-halting (infinite loops).
Scanning through an infinite set is one such scenario (e.g proving all the integers exist).

And the original Turing machine (with an infinite ticker tape) suffers from the same problem: infinitism

http://liarparadox.org/Peter_Linz_HP(Pages_318-319).pdf

It will take me a while to parse your formalization, but intuitively it seems to me you are doing what I described above.

A non-halting finite state machine (infinite loop).

[Logik]

Here's the fundamental problem: you don't know what an "error" is. let alone a "logical error".
Compilers/interpreters are reasonably good at detecting grammatical, semantic or syntactic errors. But they can't guard against intent.

Any proposition you furnish - I can construct a logical system (with a set of axioms) to assert it true OR false.

If you expect input X to produce "TRUE" - I'll make it say "FALSE"
If you expect input Y to produce "FALSE" - I'll make it say "TRUE"

It's a bit of a nail in that coffin. Garbage in - Garbage out etc...

Any set of rules you produce, I can probably navigate them in a way so that I can control the output of your system.

You can spend another 30000 hours trying to understand it from a book, or you can just go to DEF CON ( https://en.wikipedia.org/wiki/DEF_CON ) and have a 13 year old demonstrate it to you.

Mathematical or logical "proof" is not worth shit in the real world. Only proof-of-work is.