Transforming formal proof into sound deduction (rewritten)

[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:
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 formal proofs to theorem consequences
now express the sound deductive inference model eliminating incompleteness,
undecidability and inconsistency from the notion of formal systems.

The following logic sentence is refuted on the basis of Axiom(3)
∃F∃G (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
Because it asserts there are sentences G of formal system F that are not true or false.

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]

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.

[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?

I have assumed nothing. I have stipulated

Because it seemed to make much more sense than assuming that they are rattlesnakes.

[Logik]

Because it seemed to make much more sense than assuming that they are rattlesnakes.

Were those your only options?

A. Boolean True/False.
B. Rattlesnakes

That sounds like a false dichotomy....

How about a many-valued logic ?

[PeteOlcott]

Because it seemed to make much more sense than assuming that they are rattlesnakes.

Were those your only options?

A. Boolean True/False.
B. Rattlesnakes

That sounds like a false dichotomy....

How about a many-valued logic ?

I am stipulating that the whole system is Boolean because that is
how my point is made.

I am also stipulating that formal proof corresponds to the sound
deductive inference model because that is how my point is made.

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.

[Logik]

I am stipulating that the whole system is Boolean because that is how my point is made.

I am also stipulating that formal proof corresponds to the sound deductive inference model because that is how my point is made.

So you have axiomatically ASSUMED that Truth is boolean and that proofs are deductive?

Isn't that a truism?

[PeteOlcott]

I am stipulating that the whole system is Boolean because that is how my point is made.

I am also stipulating that formal proof corresponds to the sound deductive inference model because that is how my point is made.

So you have axiomatically ASSUMED that Truth is boolean and that proofs are deductive?

Isn't that a truism?

If you want to assume that Truth pertains to rattlesnakes (or anything else)
feel free yet not in the thread. The sound deductive inference model
eliminates incompleteness and inconsistency from formal systems. That
is not an assumption it is provable.

[Logik]

If you want to assume that Truth pertains to rattlesnakes (or anything else)
feel free yet not in the thread. The sound deductive inference model
eliminates incompleteness and inconsistency from formal systems. That
is not an assumption it is provable.

OK, but incompleteness and inconsistency are removed only IF your axioms are accepted.

The question "Should I accept those axioms?" is a decision problem.

The answer is "Yes. You should accept the axioms IF you want to eliminate incompleteness and inconsistency within your system'"

But that is a long-winded way to argue for the Coherence theory of truth

[PeteOlcott]

If you want to assume that Truth pertains to rattlesnakes (or anything else)
feel free yet not in the thread. The sound deductive inference model
eliminates incompleteness and inconsistency from formal systems. That
is not an assumption it is provable.

OK, but incompleteness and inconsistency are removed only IF your axioms are accepted.

The question "Should I accept those axioms?" is a decision problem.

The answer is "Yes. You should accept the axioms IF you want to eliminate incompleteness and inconsistency within your system'"

But that is a long-winded way to argue for the Coherence theory of truth

Heh that is great. We are finally totally agreeing.
In my other post I simplified this ever more.

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).

We may not longer need Boolean/¬Boolean or Sound/¬Sound.

Within the above stipulations this logic sentence says that there are some
logic sentences that are neither True nor False: ∃F∃G (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))

Now seems to be simply false, with no need for ¬Boolean or ¬Sound.

[Logik]

Heh that is great. We are finally totally agreeing.
In my other post I simplified this ever more.

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

OK, but you have ended up where I don't want to go.

I don't care about linguistic truth.
I care about semantic truth.

[PeteOlcott]

Heh that is great. We are finally totally agreeing.
In my other post I simplified this ever more.

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

OK, but you have ended up where I don't want to go.

I don't care about linguistic truth.
I care about semantic truth.

There is a morphism between them, AKA coherence theory of truth.
For the correspondence theory of truth it would be a bijection.

[Logik]

There is a morphism between them.

Sure. The morphism is self-expression.

I can trivially express a semantic truth as "This grobmunf is nice from far but far from nice".
You can't extract my semantics from that, but that doesn't mean it's not semantically valid and sound.

[PeteOlcott]

There is a morphism between them.

Sure. The morphism is self-expression.

I can trivially express a semantic truth as "This grobmunf is nice from far but far from nice".
You can't extract my semantics from that, but that doesn't mean it's not semantically valid and sound.

There is a morphism between them, AKA coherence theory of truth
applies from abstraction to abstraction for purely analytical truth.

For the correspondence theory of truth it would be a bijection from
abstraction to a model of the physical world for both empirical truth
and the hybrid combination of analytical and empirical truth.

I am not sure that we can really have purely empirical truth, that would
be things such as the direct experience of the actual taste of strawberries.

[Logik]

There is a morphism between them, AKA coherence theory of truth.

Sure. This is colloquially called metaphysics.

For the correspondence theory of truth it would be a bijection.

Yes. This is why I pointed you to the field of semiotics a while back. The signifier vs signified distinction is isomorphic to the bijection you speak of.

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.

You are trying to prescribe language. 1:1 mapping between words and real-world objects.
You are going to have a hard time navigating around phenomenological bracketing and eidetic reduction.

[PeteOlcott]

replaced