Transforming formal proof into sound deduction (rewritten)

[PeteOlcott]

But that doesn't get you any closer to solving the problem of narrative. The subject of discussion (the signified) may be an actual cat, but for the purpose of the conversation the signifiers we use to speak about it could be: cat, animal, pet, fucking meowing nuisance.

I am not and have not been ever working on the problem of narrative.

I am starting with simply refuting the incompleteness theorem to establish
the analytic notion of truth as it applies to pure abstractions. It seems that
within my stipulative specification of a formal system I have done that.

When we include my work on work on the halting problem, Tarski Undefinability
and the Liar Paradox this has taken me 22 years and 12,000 hours.

[Logik]

I am starting with simply refuting the incompleteness theorem to establish
the analytic notion of truth as it applies to pure abstractions. It seems that
within my stipulative specification of a formal system I have done that.

Obviously. All language/symbolism is 'pure abstraction'. We understand that. The word "cat" is of no resemblance to an actual cat.

When you treat language as a closed system (a system of finite volume) then sure. You can make any empirical observation about the system's behaviour because you control all the variables.

But human/natural languages are not finite-volume systems. The volume of "meaning" (e.g semantics) varies immensely.

[Logik]

When we include my work on work on the halting problem, Tarski Undefinability
and the Liar Paradox this has taken me 22 years and 12,000 hours.

You keep saying that. And you still can't produce a system (actual programming languge) which can decide whether any linguistic expression is valid or invalid.

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

[PeteOlcott]

When we include my work on work on the halting problem, Tarski Undefinability
and the Liar Paradox this has taken me 22 years and 12,000 hours.

You keep saying that. And you still can't produce a system (actual programming languge) which can decide whether any linguistic expression is valid or invalid.

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

Stipulating this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

Stipulating this specification of True and False:
(1) True(F, x) ↔ (F ⊢ x).
(2) False(F, x) ↔ (F ⊢ ¬x).

Within the above stipulations the following expression asserts that there
are expressions G of the language of formal system F that are neither
true nor false: ∃F∃G (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G))).

Within the assumption that formal systems are Boolean
that logic sentence now is simply false.

Making the following paragraph simply false:
The first incompleteness theorem states that in any consistent
formal system F within which a certain amount of arithmetic can
be carried out, there are statements of the language of F which
can neither be proved nor disproved in F. (Raatikainen 2018:1)

Thus refuting REFUTING Gödel's 1931 Incompleteness Theorem

[Logik]

Stipulating this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

Sure. That's neither my conception nor my definition of an axiom.

Thus refuting REFUTING Gödel's 1931 Incompleteness Theorem

OK, but just about every computer scientist in 2019 agrees that even though Gödel's theorem was historically prior, Turing's halting problem is conceptually prior.

From the computational paradigm Gödel incompleteness is a corollary of Turing.

And I don't have to go any further than this claim:

¬(True⇔False) ⇔ True
¬(True⇔False) ⇔ False

You get to DECIDE which one is your axiom.

[PeteOlcott]

Stipulating this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

Sure. That's neither my conception nor my definition of an axiom.

Thus refuting REFUTING Gödel's 1931 Incompleteness Theorem

OK, but just about every computer scientist in 2019 agrees that even though Gödel's theorem was historically prior, Turing's halting problem is conceptually prior.

From the computational paradigm Gödel incompleteness is a corollary of Turing.

And I don't have to go any further than this claim:

¬(True⇔False) ⇔ True
¬(True⇔False) ⇔ False

You get to DECIDE which one is your axiom.

I also have a refutation of the halting problem totally worked out.
I only have to complete the C++ programming so that I can provide a
full execution trace of the Peter Linz H deciding halting for the Peter Linz
input pair: (Ĥ, Ĥ). Every single detail of the virtual machine code for
H and Ĥ is fully specified. I merely have to finish encoding the UTM
interpreter. http://liarparadox.org/Peter_Linz_HP(Pages_315-320).pdf

The only possible way that I can decide this previously undecidable problem
is that I noticed a tiny little detail that no one has ever noticed before.

If we are free to choose between the two axioms that you specified then
that would enable me to correctly say that I am talking to a badminton
racket instead of a human being. Because I see the second axiom as really
nothing more than permission to lie, I choose the first axiom.

[Logik]

If we are free to choose between the two axioms that you specified then
that would enable me to correctly say that I am talking to a badminton
racket instead of a human being. Because I see the second axiom as really
nothing more than permission to lie, I choose the first axiom.

No. You keep mistaking the complex for the simple.

The choice is not between two axioms. The choice is between VERY MANY axioms. So many that it's hard to choose.

You have stated it over and over again that you aren't talking about computational complexity. Fine, but then your work amounts to nothing more than literature as far as I am concerned.

In my world "abstractions" make contact with the ground and model errors have real-world consequences.
Reality is really complex. Models are really incomplete and model-errors really results in harm.

If you think think incompleteness (failing to state all the information available at my disposal) is lying, rather than a physical limitation of human minds (bandwidth) you are really out of touch with reality.

[PeteOlcott]

If we are free to choose between the two axioms that you specified then
that would enable me to correctly say that I am talking to a badminton
racket instead of a human being. Because I see the second axiom as really
nothing more than permission to lie, I choose the first axiom.

The choice is not between two axioms. The choice is between VERY MANY axioms. So many that it's hard to choose.

The choice between these two axioms is the pinnacle choice of the decision tree:
(1) True(F, x) ↔ (F ⊢ x).
(2) False(F, x) ↔ (F ⊢ ¬x).

You act as though its arbitrary like the difference between kissing someone and
killing them is only swapping "ss" for "ll", otherwise there is no difference at all.

You act this way I do not. My whole rule of life it to treat others. with as much
empathy as I possibly can. I know there is a huge difference between kissing and
killing and likewise true and False are opposites.

[PeteOlcott]

If we are free to choose between the two axioms that you specified then
that would enable me to correctly say that I am talking to a badminton
racket instead of a human being. Because I see the second axiom as really
nothing more than permission to lie, I choose the first axiom.

In my world "abstractions" make contact with the ground and model errors have real-world consequences.
Reality is really complex. Models are really incomplete and model-errors really results in harm.

It is not that the above is not very important, it is that I have been trying for 12,000 hours
over 22 years going as fast as I can to prove my other point and cannot afford to be distracted
by side issues until this point if fully made. With endless digression I would fail to make
my point working 24/7 for a billion years. With zero digression it has already taken 22 years.

So it seems that I have proved my point and correctly refuted
the 1931 Incompleteness Theorem by the following reasoning.
It does not matter if you don't think that refuting the Incompleteness Theorem
is of no consequence the people that would hire me would think otherwise.

Stipulating this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

Stipulating this specification of True and False:
Axiom(1) True(F, x) ↔ (F ⊢ x).
Axiom(2) False(F, x) ↔ (F ⊢ ¬x).

Stipulating that formal systems are Boolean:
Axiom(3) ∀F ∈ Formal_System ∀x ∈ Closed_WFF(F) (True(F,x) ∨ False(F,x))

Within the above stipulations the following logic sentence:
∃F∃G (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
(1) Asserts there exists expressions G of formal system F that are neither true nor false.
(2) Is decided to be false on the basis of Axiom(3).

Making the following paragraph false:
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried
out, there are statements of the language of F which can neither be
proved nor disproved in F. (Raatikainen 2018)

[Logik]

The choice between these two axioms is the pinnacle choice of the decision tree:
(1) True(F, x) ↔ (F ⊢ x)
(2) False(F, x) ↔ (F ⊢ ¬x)

No, it isn't. The first choice is what I want/expect to accomplish with this particular logic-system.

What is the purpose of that which I am doing? Your axioms above make probabilistic reasoning impossible.

Everything seems rather black-and-white in your world.

You act this way I do not. My whole rule of life it to treat others. with as much
empathy as I possibly can't. I know there is a huge difference between kissing and
killing and likewise true and False are opposites.

Oi vei, you have definitely spent 22 years and 12000 hours deluding yourself.

I act as though I recognize that kissing and killing are useful towards different purposes and in different contexts.

[PeteOlcott]

The choice between these two axioms is the pinnacle choice of the decision tree:
(1) True(F, x) ↔ (F ⊢ x)
(2) False(F, x) ↔ (F ⊢ ¬x)

No, it isn't. The first choice is what I want/expect to accomplish with this particular logic-system.

What is the purpose of that which I am doing? Your axioms above make probabilistic reasoning impossible.

Everything seems rather black-and-white in your world.

https://en.wikipedia.org/wiki/Truth-con ... _semantics

Truth-conditional semantics is an approach to semantics of natural
language that sees meaning (or at least the meaning of assertions)
as being the same as, or reducible to, their truth conditions.

[PeteOlcott]

Everything seems rather black-and-white in your world.

You act this way I do not. My whole rule of life it to treat others. with as much
empathy as I possibly can't. I know there is a huge difference between kissing and
killing and likewise true and False are opposites.

Oi vei, you have definitely spent 22 years and 12000 hours deluding yourself.

I act as though I recognize that kissing and killing are useful towards different purposes and in different contexts.

This is not contextual, it is foundational: ¬(True ↔ False)
It is stipulated in the same way that {cats} are not any type of {dog} is stipulated.

[A_Seagull]

Stipulating this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

Stipulating this specification of True and False:
(1) True(F, x) ↔ (F ⊢ x).
(2) False(F, x) ↔ (F ⊢ ¬x).

Then formal proofs to theorem consequences specify sound deduction.

In sound deductive inference an argument is unsound unless there
is a connected chain of deductive inference steps from true premises
to a true conclusion.

Applying the law of the excluded middle (P ∨ ¬P) to the above we derive:
(3) Sound(F, x) ↔ (True(F, x) ∨ False(F, x)).

When we transform formal proof to theorem consequences into the
sound deductive inference model then truth and provability can
no longer diverge. Any closed WFF that was previously undecidable
is decided to be unsound.

I don't think you need all those references to true or truth. They are surplus to requirements.

If you have a connected chain of deductive inference from the axioms to the conclusion then your case is proven, albeit only within that axiomatic system.

What has not been proven within that system cannot be clamed to be proven within that system.

[PeteOlcott]

Stipulating this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

Stipulating this specification of True and False:
(1) True(F, x) ↔ (F ⊢ x).
(2) False(F, x) ↔ (F ⊢ ¬x).

Then formal proofs to theorem consequences specify sound deduction.

In sound deductive inference an argument is unsound unless there
is a connected chain of deductive inference steps from true premises
to a true conclusion.

Applying the law of the excluded middle (P ∨ ¬P) to the above we derive:
(3) Sound(F, x) ↔ (True(F, x) ∨ False(F, x)).

When we transform formal proof to theorem consequences into the
sound deductive inference model then truth and provability can
no longer diverge. Any closed WFF that was previously undecidable
is decided to be unsound.

I don't think you need all those references to true or truth. They are surplus to requirements.

If you have a connected chain of deductive inference from the axioms to the conclusion then your case is proven, albeit only within that axiomatic system.

What has not been proven within that system cannot be clamed to be proven within that system.

Within the above stipulations the following logic sentence:
∃F∃G (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
(1) Asserts there exists expressions G of formal system F that are neither true nor false.
(2) Is decided to be false on the basis of Axiom(3).

Making the following paragraph false:
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried
out, there are statements of the language of F which can neither be
proved nor disproved in F. (Raatikainen 2018)

[PeteOlcott]

Stipulating this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

So you have axiomatically assumed that axioms are Boolean. Why?

If that is how you conceptualize an "axiom" (expression of language that has the semantic value of Boolean True).

Then you are necessarily dismissing all of probability theory.

It is 100% true that XYZ news reported that there is an 80% chance of rain tomorrow.

It is 100% true that according to probability theory a fair toss of an unbiased coin has an average of a 50% chance of heads.