XPost: sci.logic

On 4/19/24 2:04 PM, olcott wrote:

On 4/18/2024 8:58 PM, Richard Damon wrote:

On 4/18/24 9:11 PM, olcott wrote:

On 4/18/2024 5:31 PM, Richard Damon wrote:

On 4/18/24 10:50 AM, olcott wrote:

On 4/17/2024 10:13 PM, Richard Damon wrote:

On 4/17/24 10:34 PM, olcott wrote:

...14 Every epistemological antinomy can likewise be used for a
similar
undecidability proof...(Gödel 1931:43-44)

*Parphrased as*
Every expression X that cannot possibly be true or false proves
that the
formal system F cannot correctly determine whether X is true or
false.
Which shows that X is undecidable in F.

Nope.

Just more of your LIES and STUPIDITY.

Which shows that F is incomplete, even though X cannot possibly be a >>>>>>> proposition in F because propositions must be true or false.

But that ISN'T the definition of "Incomplete", so you are just LYING. >>>>>>
Godel showed that a statment, THAT WAS TRUE, couldn't be proven in F. >>>>>>
You don't even seem to understand what the statement G actually
is, because all you look at are the "clift notes" versions, and
don't even understand that.

Remember, G is a statement about the non-existance of a number
that has a specific property. Until you understand that, your
continued talking about this is just more LIES and DECIET, proving >>>>>> your absoulute STUPIDITY.

A proposition is a central concept in the philosophy of language, >>>>>>> semantics, logic, and related fields, often characterized as the >>>>>>> primary
bearer of truth or falsity.
https://en.wikipedia.org/wiki/Proposition

Right, and if you don't know what the proposition is that you are
arguing about, you are just proven to be a stupid liar.

If you are going to continue to be mean and call me names I will stop >>>>> talking to you. Even if you stop being mean and stop calling me names >>>>> if you continue to dogmatically say that I am wrong without pointing >>>>> out all of the details of my error, I will stop talking to you.

This is either a civil debate and an honest dialogue or you will
hear nothing form me.

I say you are WRONG, because you ARE.

You say Godel's statement that is unprovable, is unprovable because
it is an epistimalogical antinomy, when it isn't.

It is a statement about the non-existance of a number that satisfies
a particular property, which will be a truth bearing statement (The
number must either exist or it doesn't)

THAT MAKES YOU A LIAR.

*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*

Well, Godel wasn't talking about "undecidability", but incompletenwss,
which is what the WORDS you used talked about. (Read what you said
above).

INCOMPLETENESS is EXACTLY about the inability to prove statements that
are true.

*That is an excellent and correct foundation for what I am saying*

When we create a three-valued logic system that has these
three values: {True, False, Nonsense} https://en.wikipedia.org/wiki/Three-valued_logic

IF you want to work with a Three Value logic system, then DO SO.

But, remember, once you make you system 3-values, you immediately loose
the ability to reference to anything proved in the classical two-value

Then "This sentence is not true" has the semantic value of {Nonsense}
This sentence is not true: "This sentence is not true" has the semantic
value of {True}.

Although it may be difficult to understand that is exactly the
difference between Tarski's "theory" and "metatheory" simplified
as much as possible.

And, once you add that third value to logic, you can't USE Tarski, or
even talk about what he did, as it is OUTSIDE your frame of logic.

This is Tarski's Liar Paradox basis https://liarparadox.org/Tarski_247_248.pdf

That he refers to in this paragraph of his actual proof
  "In accordance with the first part of Th. I we can obtain
   the negation of one of the sentences in condition (α) of
   convention T of § 3 as a consequence of the definition of
   the symbol 'Pr' (provided we replace 'Tr' in this convention
   by 'Pr')." https://liarparadox.org/Tarski_275_276.pdf

Allows his original formalized Liar Paradox:

x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x

Right, He shows that this statement is EXPRESSABLE in the meta-theory (something I don't think you understand)

to be reverse-engineered from Line(1) of his actual proof:
(I changed his abbreviations of "Pr" and "Tr" into words)

Note, "Th I" was established without reference to the meaning of the class.

Here is the Tarski Undefinability Theorem proof
(1) x ∉ Provable if and only if p    // assumption

NOT ASSUMPTION, he has shown that such an x must exist in the theory (if
it meets the requirements)

(2) x ∈ True if and only if p        // assumption

NOT ASSUMPTION, but from the DEFINITION of what Truth is, the statement
x is true if and only if it is true (since p is the whole statement x)

(3) x ∉ Provable if and only if x ∈ True. // derived from (1) and (2)
(4) either x ∉ True or x̄ ∉ True;     // axiom: True(x) ∨ ~True(~x)
(5) if x ∈ Provable, then x ∈ True;  // axiom: Provable(x) → True(x) (6) if x̄ ∈ Provable, then x̄ ∈ True;  // axiom: Provable(~x) → True(~x)
(7) x ∈ True
(8) x ∉ Provable
(9) x̄ ∉ Provable

Right.

Thus proving that there exists and x where x must be true, and x must be unprovable.

You just don't understand what an "assumption" is and what is an
application of a proven statement.

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XPost: sci.logic

On 4/20/24 2:05 AM, olcott wrote:

On 4/19/2024 6:20 PM, Richard Damon wrote:

On 4/19/24 2:04 PM, olcott wrote:

On 4/18/2024 8:58 PM, Richard Damon wrote:

On 4/18/24 9:11 PM, olcott wrote:

On 4/18/2024 5:31 PM, Richard Damon wrote:

On 4/18/24 10:50 AM, olcott wrote:

On 4/17/2024 10:13 PM, Richard Damon wrote:

On 4/17/24 10:34 PM, olcott wrote:

...14 Every epistemological antinomy can likewise be used for a >>>>>>>>> similar
undecidability proof...(Gödel 1931:43-44)

*Parphrased as*
Every expression X that cannot possibly be true or false proves >>>>>>>>> that the
formal system F cannot correctly determine whether X is true or >>>>>>>>> false.
Which shows that X is undecidable in F.

Nope.

Just more of your LIES and STUPIDITY.

Which shows that F is incomplete, even though X cannot possibly >>>>>>>>> be a
proposition in F because propositions must be true or false.

But that ISN'T the definition of "Incomplete", so you are just >>>>>>>> LYING.

Godel showed that a statment, THAT WAS TRUE, couldn't be proven >>>>>>>> in F.

You don't even seem to understand what the statement G actually >>>>>>>> is, because all you look at are the "clift notes" versions, and >>>>>>>> don't even understand that.

Remember, G is a statement about the non-existance of a number >>>>>>>> that has a specific property. Until you understand that, your
continued talking about this is just more LIES and DECIET,
proving your absoulute STUPIDITY.

A proposition is a central concept in the philosophy of language, >>>>>>>>> semantics, logic, and related fields, often characterized as >>>>>>>>> the primary
bearer of truth or falsity.
https://en.wikipedia.org/wiki/Proposition

Right, and if you don't know what the proposition is that you
are arguing about, you are just proven to be a stupid liar.

If you are going to continue to be mean and call me names I will >>>>>>> stop
talking to you. Even if you stop being mean and stop calling me
names
if you continue to dogmatically say that I am wrong without pointing >>>>>>> out all of the details of my error, I will stop talking to you.

This is either a civil debate and an honest dialogue or you will >>>>>>> hear nothing form me.

I say you are WRONG, because you ARE.

You say Godel's statement that is unprovable, is unprovable
because it is an epistimalogical antinomy, when it isn't.

It is a statement about the non-existance of a number that
satisfies a particular property, which will be a truth bearing
statement (The number must either exist or it doesn't)

THAT MAKES YOU A LIAR.

*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*
*That is NOT how undecidability generically works and you know it*

Well, Godel wasn't talking about "undecidability", but
incompletenwss, which is what the WORDS you used talked about. (Read
what you said above).

INCOMPLETENESS is EXACTLY about the inability to prove statements
that are true.

*That is an excellent and correct foundation for what I am saying*

When we create a three-valued logic system that has these
three values: {True, False, Nonsense}
https://en.wikipedia.org/wiki/Three-valued_logic

IF you want to work with a Three Value logic system, then DO SO.

But, remember, once you make you system 3-values, you immediately
loose the ability to reference to anything proved in the classical
two-value

Then "This sentence is not true" has the semantic value of {Nonsense}
This sentence is not true: "This sentence is not true" has the semantic
value of {True}.

Although it may be difficult to understand that is exactly the
difference between Tarski's "theory" and "metatheory" simplified
as much as possible.

And, once you add that third value to logic, you can't USE Tarski, or
even talk about what he did, as it is OUTSIDE your frame of logic.

For teaching purposes it is easier to think of it as
a third semantic value. In actuality it would be
rejected as invalid input.

So make up your mind!!!

The problem is that the DEFINITION of a Halt Decider, or a Truth
Predicate is that NO INPUT is "invalid". For a Halt Decider, IT IS
DEFINED that if the input doesn't represent a Halting Computation, the
answer is NO, and for a Truth Predicate, if the statement is not True,
then the Truth Predicate says No, be it a false statement, or a
statement that is not a Truth Bearer.

Thus there is not option to "reject".

This is Tarski's Liar Paradox basis
https://liarparadox.org/Tarski_247_248.pdf

That he refers to in this paragraph of his actual proof
   "In accordance with the first part of Th. I we can obtain
    the negation of one of the sentences in condition (α) of
    convention T of § 3 as a consequence of the definition of
    the symbol 'Pr' (provided we replace 'Tr' in this convention
    by 'Pr')." https://liarparadox.org/Tarski_275_276.pdf

Allows his original formalized Liar Paradox:

x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x

Right, He shows that this statement is EXPRESSABLE in the meta-theory
(something I don't think you understand)

I do. I understand it better than most.
This sentence is not true: "This sentence is not true" is true.

SO, the truth predicate could s

to be reverse-engineered from Line(1) of his actual proof:
(I changed his abbreviations of "Pr" and "Tr" into words)

Note, "Th I" was established without reference to the meaning of the
class.

Here is the Tarski Undefinability Theorem proof
(1) x ∉ Provable if and only if p    // assumption

NOT ASSUMPTION, he has shown that such an x must exist in the theory
(if it meets the requirements)

That is an adaptation of his Liar Paradox: x ∉ Tarski if and only if p

So, His Th I proves that there exists a statement that can be expressed
of that form.

To reject that, you need to find the error in the proof of TH I.

(2) x ∈ True if and only if p        // assumption

NOT ASSUMPTION, but from the DEFINITION of what Truth is, the
statement x is true if and only if it is true (since p is the whole
statement x)

Convention T

He gets to use the convention he wants to use.

(3) x ∉ Provable if and only if x ∈ True. // derived from (1) and (2) >>> (4) either x ∉ True or x̄ ∉ True;     // axiom: True(x) ∨ ~True(~x)
(5) if x ∈ Provable, then x ∈ True;  // axiom: Provable(x) → True(x) >>> (6) if x̄ ∈ Provable, then x̄ ∈ True;  // axiom: Provable(~x) → True(~x)
(7) x ∈ True
(8) x ∉ Provable
(9) x̄ ∉ Provable

Right.

Thus proving that there exists and x where x must be true, and x must
be unprovable.

You just don't understand what an "assumption" is and what is an
application of a proven statement.

Tarski assumes the Liar Paradox and finds out that this
assumption does not work out.

Nope. You just can't read what he says because you shut your eyes to
truth that you can not accept.

The fact you don't understand it doesn't make it wrong.

--- SoupGate-Win32 v1.05
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XPost: sci.logic

On 4/20/24 11:20 AM, olcott wrote:

On 4/20/2024 2:54 AM, Mikko wrote:

On 2024-04-19 18:04:48 +0000, olcott said:

When we create a three-valued logic system that has these
three values: {True, False, Nonsense}
https://en.wikipedia.org/wiki/Three-valued_logic

Such three valued logic has the problem that a tautology of the
ordinary propositional logic cannot be trusted to be true. For
example, in ordinary logic A ∨ ¬A is always true. This means that
some ordinary proofs of ordinary theorems are no longer valid and
you need to accept the possibility that a theory that is complete
in ordinary logic is incomplete in your logic.

I only used three-valued logic as a teaching device. Whenever an
expression of language has the value of {Nonsense} then it is
rejected and not allowed to be used in any logical operations. It
is basically invalid input.

In other words, you admit that you are being inconsistant about what you
are saying, because your whole logic system is just inconsistant.

You don't seem to understand that predicates, DEFINED to be able to work
on ALL memebers of the input domain, must IN FACT, work on all members
of that domain.

For a Halt Decider, that means the decider needs to be able to answer
about ANY machine given to it as an input, even a machine that uses a
copy of the decider and acts contrary to its answer.

If you are going to work on a different problem, you need to be honest
about that and not LIE and say you are working on the Halting Problem.

And, if you are going to talk about a "Truth Predicate", which is
defined to be able to take ANY "statement" and say if it is True or not,
with "nonsense" statements (be they self-contradictory statements, or
just nonsense) being just not-true.

ANY statement means any statement, so if we define this predicate as
True(F, x) to be true if x is a statement that is true in the field F,
then we need to be able to give this predicate the statemet:

In F de define s as NOT True(F, s)


If you claim that your logic is ACTUALLY "two-valued" then if True(F,s)
returns false, because s is a statement without a truth value, then we
have the problem that the definition of s now says that s has the value
of NOT false, which is True.

So, the True predicate was WRONG, as True of a statement that IS true,
must be true.

If True(F,s) is true, then we have that s is not defined as NOT true,
which is false, so the True predicate is again WRONG.

The predicate isn't ALLOWED to say "I reject this input" as that isn't a
truth value (since you claimed you are actually useing a two-valued
logic) and this predicate is defined to ALWAYS return a truth value.

So, it seems you have a two-valued logic system with three logical values.

Which is just A LIE!

You are just proving you are too stupid to understand what you are
talking about as you don't understand the meaning of the words you are
using, as you just studied the system by Zero order principles.

--- SoupGate-Win32 v1.05
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XPost: sci.logic

On 4/21/24 11:26 AM, olcott wrote:

On 4/20/2024 10:39 AM, Richard Damon wrote:

On 4/20/24 11:20 AM, olcott wrote:

On 4/20/2024 2:54 AM, Mikko wrote:

On 2024-04-19 18:04:48 +0000, olcott said:

When we create a three-valued logic system that has these
three values: {True, False, Nonsense}
https://en.wikipedia.org/wiki/Three-valued_logic

Such three valued logic has the problem that a tautology of the
ordinary propositional logic cannot be trusted to be true. For
example, in ordinary logic A ∨ ¬A is always true. This means that
some ordinary proofs of ordinary theorems are no longer valid and
you need to accept the possibility that a theory that is complete
in ordinary logic is incomplete in your logic.

I only used three-valued logic as a teaching device. Whenever an
expression of language has the value of {Nonsense} then it is
rejected and not allowed to be used in any logical operations. It
is basically invalid input.

In other words, you admit that you are being inconsistant about what
you are saying, because your whole logic system is just inconsistant.

Not at all.
An undecidable sentence of a theory K is a closed wf ℬ of K such that neither ℬ nor ¬ℬ is a theorem of K, that is, such that not-⊢K ℬ and not-⊢K ¬ℬ. (Mendelson: 2015:208)

The notion of incompleteness and undecidability requires non truth
bearers to be construed as truth bearers.

Nope, and you stating that just proves your stupidity.

A Theory K will define its "language" and what statements it accepts
within it. Normally that "language" excludes non-truth-bearers. This
seems to be something outside your understanding, as you don't seem to understand anything about the nature of actual FORMAL logic systems, but
seem to be stuck

Yes, non-truth bearing statements will be undecidable,

A proposition is a central concept in the philosophy of language,
semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. https://en.wikipedia.org/wiki/Proposition

When we quit construing expressions that cannot possibly be true or
false as propositions then incompleteness and undecidability cease to
exist.

Nope. There exist statements that are True, in that they have an
(infinite) sequence of connections from the truth makers of the system
to the statement, but are not provable, as there is no FINITE sequence
of connections that do so.

Godel's G is an example of this, stating that there does not exist a
number that matches a specific property. Since the property is
computable for all numbers, we know that G must be a truth bearer, as
either such a number exists, or it doesn't exist.

This fact can be established in F, as either it is false, because we CAN
find such a number, and the checking of the number with the relationship provides a definite proof that G is false, or no such number exists, and
this is established by the INFINITE chain of checking every number, and
seeing that none satisfies it.

We happen to be able to reduce that infinite chain to be finite in a partitulare meta-theory of F that understands a hidden meaning in the relationship, and allows us to PROVE that no such number exists.

This PROVES that G is a true statement. While the proof is in Meta-F,
the proof also establishes that G is true in F.

On 4/18/2024 8:58 PM, Richard Damon wrote:

INCOMPLETENESS is EXACTLY about the inability to prove statements that
are true.

Truth_Bearer(F, x) ≡  ∃x ∈ F ((F ⊢ x) ∨ (F ⊢ ¬x))

Nope, not PROVES, but ESTABLISHES.

Truth_Bearer(F, x) ≡ ∃x ∈ F ((F ⊨ x) ∨ (F ⊨ ¬x))

Truth Bearing allows for the INFINTE sequence to establish the fact,
even if that can not be a proof of it.

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...(Gödel 1931:43-44)

Yep, you can build another proof just like the one presented based on
any epistemological antinomy. Note, the proof USES the antinomy, but
does not "derive" from it, in that its validity and soundness are not
based on the truth of the antinomy.

You don't seem to understand the syntactic transformation that was done
on the statement at the beginning, that created a NEW PROPOSITION, that
turns out to be a Truth Bearer.

"X says that X is not True in F", is an epistemological antinomy.

"X says that X is not Provable in F" is not, as the logical valuation of
X being True but not Provable is a possible valid combination of states.

Gödel is essentially saying that expressions that are not propositions
prove that a formal system of propositions has undecidable propositions.

Nope.

Since you don't understand what Godel did, you are just showing you
stupidity by making your claim.

That you persist is making the claim after being shown to be wrong, make
you just a stupid pathological liar, and proves you just don't
understand what Truth actually is.

You don't seem to understand that predicates, DEFINED to be able to
work on ALL memebers of the input domain, must IN FACT, work on all
members of that domain.

For a Halt Decider, that means the decider needs to be able to answer
about ANY machine given to it as an input, even a machine that uses a
copy of the decider and acts contrary to its answer.

If you are going to work on a different problem, you need to be honest
about that and not LIE and say you are working on the Halting Problem.

And, if you are going to talk about a "Truth Predicate", which is
defined to be able to take ANY "statement" and say if it is True or
not, with "nonsense" statements (be they self-contradictory
statements, or just nonsense) being just not-true.

ANY statement means any statement, so if we define this predicate as
True(F, x) to be true if x is a statement that is true in the field F,
then we need to be able to give this predicate the statemet:

In F de define s as NOT True(F, s)


If you claim that your logic is ACTUALLY "two-valued" then if
True(F,s) returns false, because s is a statement without a truth
value, then we have the problem that the definition of s now says that
s has the value of NOT false, which is True.

So, the True predicate was WRONG, as True of a statement that IS true,
must be true.

If True(F,s) is true, then we have that s is not defined as NOT true,
which is false, so the True predicate is again WRONG.

The predicate isn't ALLOWED to say "I reject this input" as that isn't
a truth value (since you claimed you are actually useing a two-valued
logic) and this predicate is defined to ALWAYS return a truth value.

So, it seems you have a two-valued logic system with three logical
values.

Which is just A LIE!

You are just proving you are too stupid to understand what you are
talking about as you don't understand the meaning of the words you are
using, as you just studied the system by Zero order principles.

--- SoupGate-Win32 v1.05
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