Eliminating undecidability in formal systems

[PeteOlcott]

You already specified that it was an integer. That it might be a cat
is specified as impossible.

Because I am trying to make my point gradually,

1. for all x: x = x, x ∈ Integers

You decided it's true in 5 seconds, even though the set of Integers is infinite, because you made some assumptions about the meaning of "="

Now I drop the Integer constraint:

2. for all x: x = x, x ∈ ALL ( https://en.wikipedia.org/wiki/ALL_(complexity) )

What would you say about the truth-value now?

What are the semantics of "=" in a universal context?

∀x ∈ Thing (x = x) // A thing is itself
Apologize for losing patience with you.

I spent 22 years on pathological self-reference and I am only a few minutes
away from finishing it. As soon as other people understand what I am saying I am done.

Anyone have a very firm grasp on Tarski Undefinability should be able to validate
my work in ten minutes.

[Logik]

∀x ∈ Thing (x = x) // A thing is itself
Apologize for losing patience with you.

OK, now I want you to rationalise/formalize/explain this.

Your THEORETICAL grounding tells you that: ∀x ∈ Thing (x = x) ⇔ TRUE.

You have before you a logical system which is:
A. Consistent (guaranteed by reality e.g a physical computer)
B. X = X is false (contradicting your axiom)

https://repl.it/repls/MinorWavySale

Code: Select all

class thing:
  def __eq__(self, other):
    return False

x = thing()
print(x == x)
=> False

Since your lived experience disagrees with your axiom, would you say that your axiom is universal?

If your axiom is not universal, then would you say any argument based on that axiom is sound?

[PeteOlcott]

∀x ∈ Thing (x = x) // A thing is itself
Apologize for losing patience with you.

OK, now I want you to rationalise/formalize/explain this.

Your THEORETICAL grounding tells you that: ∀x ∈ Thing (x = x) ⇔ TRUE.

You have before you a logical system which is:
A. Consistent (guaranteed by reality e.g a physical computer)
B. X = X is false (contradicting your axiom)

https://repl.it/repls/MinorWavySale

Code: Select all

class thing:
  def __eq__(self, other):
    return False

x = thing()
print(x == x)
=> False

Since your lived experience disagrees with your axiom, would you say that your axiom is universal?

If your axiom is not universal, then would you say any argument based on that axiom is sound?

How could my lived experience disagree with my axiom?
How could a thing not be itself?
My axiom says that a thing is always necessarily itself.

[Logik]

How could my lived experience disagree with my axiom?

Because there is living proof before you.

You merely ASSUME x = x ⇔ True.
I have given you a deductive type-system which DECIDES x = x ⇔ False

By Curry-Howard correspondence it's valid proof. The axiom is satisfied.

How could a thing not be itself?

Wrong question.

The right question is "What does it even MEAN for a thing to be itself?"
As far as I am concerned it's a meaningless expression in English.
You are welcome to convince me otherwise by translating the semantics of "a thing is itself" into formal logic.

My axiom says that a thing is always necessarily itself.

And my axiom says that it isn't.

Observe that you haven't solved undecidability because the axiom of choice (AC) remains

Which axiom should I choose?

A. x = x ⇔ False
B. x = x ⇔ True

[PeteOlcott]

How could my lived experience disagree with my axiom?

Because there is living proof before you.

A functioning deductive system (a.k.a a computer) which DECIDES ( x = x ) ⇔ False
You merely ASSUME ( x = x ) ⇔ True.

That you can make a computer program that is a liar
only proves that you can make a computer program that is a liar,
it says nothing at all about axiomatic truth itself.
Why would you think otherwise?

[Logik]

That you can make a computer program that is a liar
only proves that you can make a computer program that is a liar,
it says nothing at all about axiomatic truth itself.
Why would you think otherwise?

Nonsense. Hanlon's razor applies.

It's not the computer's fault that you overload/equivocate the meaning of "=".

This should be a deja vu. You can't explain what you mean by "the same" in an exact, precise sense (which supports Tarski's undefinability).

Let me demonstrate to you.

A is the same as А formalizes as 'A' = 'А'

Is the above true or false?

Hint: https://repl.it/repls/BasicFamousEnterprise

Are you "lying" when you said it's true? Is the computer "lying" when it said it's false?
No - you just made a mistake!

That is the error of all logicians/mathematicians. You ASSUME your axioms, and assumption is the mother of all f-ups ...

[PeteOlcott]

That you can make a computer program that is a liar
only proves that you can make a computer program that is a liar,
it says nothing at all about axiomatic truth itself.
Why would you think otherwise?

Nonsense. Hanlon's razor applies.

It's not the computer's fault that you overload/equivocate the meaning of "=".

I did no such thing. You merely provided a computer program
that is a liar as an example where the axiomatization of truth fails.

Truth is merely the mathematical mapping between abstract representations
of actuality and actuality itself. Your computer program ignored this and lied.

[Logik]

I did no such thing. You merely provided a computer program
that is a liar as an example where the axiomatization of truth fails.

*sigh*

Not quite, it is two different ways, but you did not quite anchor them correctly. What is known to be true with justifiable 100% complete certainty on the basis of sound deductive inference and what is estimated to be true on the basis of inductive inference.

Doesn't that make you a liar by your own criterion? Using the same word two different ways within the same argument? Aristotle called that equivocation but I'll go with your term. Liar.

According to you a "liar" is anybody who CHOOSES different axioms to you.

How and why did you DECIDE x = x ⇔ True instead of x = x ⇔ False ?

[Logik]

Truth is merely the mathematical mapping between abstract representations
of actuality and actuality itself. Your computer program ignored this and lied.

More *sigh*

There can be no difference anywhere that doesn't make a difference elsewhere—no difference in abstract truth that doesn't express itself in a difference in concrete fact and in conduct consequent upon that fact, imposed on somebody, somehow, somewhere and some-when. The whole function of philosophy ought to be to find out what definite difference it will make to you and me, at definite instants of our life, if this world-formula or that world-formula be the true one. --WIlliam James

What is the mapping between the abstract sentence "a thing is the same as itself" and reality?

You are merely rooting for the correspondence theory of truth, ignoring the 20 other truth-theories.

[PeteOlcott]

What is the mapping between the abstract sentence "a thing is the same as itself" and reality?

You are merely rooting for the correspondence theory of truth, ignoring the 20 other truth-theories.

My mathematical mapping theory of truth combines the correspondence theory with the coherence theory:
Truth is the mathematical mapping from abstract representations of actuality to actuality itself:
(1) Abstract to Empirical is correspondence theory.
(2) Abstract to Abstract is coherence theory.

Your computer program conflicts with (contradicts) other axioms and is rejected on that basis.

Within the pre-existing order of the set of all knowledge truth itself already has its own formal
specification to be discovered rather than created.

[Logik]

Your computer program conflicts with (contradicts) other axioms and is rejected on that basis.

Yeah. About that... There are no such things as contradictions.

Here is a logical system in which A & ~A evaluates to True (contrary to that which the LNC prescribes)

https://repl.it/repls/LinenAnnualDistributeddatabase

The LNC is an arbitrary choice - like all axioms.

You keep ignoring the point I keep making. Tell me what your premises are , and what conclusion you want the logic-system to produce and I will construct it for you.

[PeteOlcott]

Your computer program conflicts with (contradicts) other axioms and is rejected on that basis.

Yeah. About that... There are no such things as contradictions.

OK then you are saying that ((5 > 3) ∧ (3 > 5)) is true within the
conventional semantic meaning of all of the symbols as specified
by an algorithm such as a "C" compiler.

int main()
{
int x = 5;
int y = 3
if (x > y && y > x)
printf("There is no such thing as contradiction!");
else
printf("Logik is a fibber!");
}

[Logik]

OK then you are saying that ((5 > 3) ∧ (3 > 5)) is true within the
conventional semantic meaning of all of the symbols as specified
by an algorithm such as a "C" compiler.

int main()
{
int x = 5;
int y = 3
if (x > y && y > x)
printf("There is no such thing as contradiction!");
else
printf("Logik is a fibber!");
}

Observe how you keep dragging the discussion down to integers. The only place where your argument works.
Why do x and y need to be integers? They can be functions.

It's both raining (P) and not-raining outside (~P). Because it's raining only on one side of the street.

Further observe how you have ignored my point (yet again). That given any input I can construct you any output.

The proof I gave you is written in C. Because Python is written in C. Because Turing-completeness anything I wrote in Python, I can translate to C.

There is no such thing as "conventional semantic meaning" in formal logic. Isn't that the implication of Tarski?
Truth is semantic and cannot be defined.

[Logik]

int main()
{
int x = 5;
int y = 3
if (x > y && y > x)
printf("There is no such thing as contradiction!");
else
printf("Logik is a fibber!");
}

The rules of arithmetic are arbitrary...
All logical rules are arbitrary.
All operators and symbols are arbitrary.

https://repl.it/repls/ExcitedSunnyVolume

Appealing to "convention" is a bandwagon fallacy.

[PeteOlcott]

int main()
{
int x = 5;
int y = 3
if (x > y && y > x)
printf("There is no such thing as contradiction!");
else
printf("Logik is a fibber!");
}

The rules of arithmetic are arbitrary...
All logical rules are arbitrary.
All operators and symbols are arbitrary.

https://repl.it/repls/ExcitedSunnyVolume

Appealing to "convention" is a bandwagon fallacy.

Bullshit.