True(X) and ~Provable(X) is Impossible

[Pete Olcott]

http://liarparadox.org/index.php/2018/02/04/truex-and-unprovablex-is-impossible/

A Fact(English) is Only known to be True(Math) when it is verified(English).
Analytic** Facts(English) are Only Verified(English) by their Proof(Math).

True(L, X) in formal language L means that there is backward-chaining
inference from X to one or more self-evidently true expressions of L.

Provable(L, Y, X) means that there is back-chaining inference from X to
one of more expressions Y of language L.

** Analytic Truth is the set of statements that can be completely verified
as True entirely based on the meaning of their words.

This is a refinement to the 1997 Mathematical Mapping Theory of Truth

Copyright 1997 2018 Pete Olcott

--
*∀X True(X) ↔ ∃Γ ⊆ Axioms Provable(Γ, X) *

[Peter Percival]

Pete Olcott wrote:
> http://liarparadox.org/index.php/2018/02/04/truex-and-unprovablex-is-impossible/

Does your Provable(X) refer to some formal theory? What does your
True(X) refer to? Is X an expression in some specified language?

I'll try to say why it matters. If the language is that of classical
propositional calculus, and if True(X) means "X is a tautology", then
there are well-known calculi that insure that Provable(X). So True(X)
and ~Provable(X) is not so much impossible as indicative of a
lackadaisical approach.

Suppose, now, that the language is that of arithmetic and True(X) means
"X is true in the standard model". There are calculi (any inconsistent
one will do) such that if True(X) then Provable(X), so True(X) and
~Provable(X) won't hold. But inconsistent calculi have the disagreeable
feature that if True(~X) then Provable(X). There are consistent calculi
in which every truth is provable: take every truth as an axiom. But
such calculi have their own problems (I referred to one in another thread).

So my three opening questions have some point. Without answers to them
your Subject header means little.

> A Fact(English) is Only known to be True(Math) when it is
> verified(English).
> Analytic** Facts(English) are Only Verified(English) by their Proof(Math).
>
> True(L, X) in formal language L means that there is backward-chaining
> inference from X to one or more self-evidently true expressions of L.
>
> Provable(L, Y, X) means that there is back-chaining inference from X to
> one of more expressions Y of language L.
>
> ** Analytic Truth is the set of statements that can be completely verified
> as True entirely based on the meaning of their words.

Like 'If Jack is a bachelor, then he is not married'? Does that have a
mathematical proof?

[Pete Olcott]

"This sentence is not true".
LP ≡ ~True(LP) // HOL with self-reference semantics

"This sentence is not provable".
G ≡ ~Provable(G) // HOL with self-reference semantics

Pathological self-reference(olcott 2004) occurs when-so-ever
the assertion that an expression makes is true and this does
not make the expression itself true.

Copyright 2004 2018

[Peter Percival]

Pete Olcott wrote:
> On 2/4/2018 12:59 AM, Pete Olcott wrote:
>> http://liarparadox.org/index.php/2018/02/04/truex-and-unprovablex-is-impossible/
>>
>>
>> A Fact(English) is Only known to be True(Math) when it is
>> verified(English).
>> Analytic** Facts(English) are Only Verified(English) by their
>> Proof(Math).
>>
>> True(L, X) in formal language L means that there is backward-chaining
>> inference from X to one or more self-evidently true expressions of L.
>>
>> Provable(L, Y, X) means that there is back-chaining inference from X to
>> one of more expressions Y of language L.
>>
>> ** Analytic Truth is the set of statements that can be completely
>> verified
>> as True entirely based on the meaning of their words.
>>
>> This is a refinement to the 1997 Mathematical Mapping Theory of Truth
>>
>> Copyright 1997 2018 Pete Olcott
>>
>
> "This sentence is not true".
> LP ≡ ~True(LP)     // HOL with self-reference semantics
>
> "This sentence is not provable".
> G  ≡ ~Provable(G)  // HOL with self-reference semantics

What is self-reference semantics?

[George Greene]

Damn, you're stupid. Will you ever just LEARN what a MODEL is? Will
you ever just read a F#'in' TEXTBOOK? TRUTH HAPPENS *IN*MODELS*
in this paradigm. Almost NO interesting first-order theory is categorical.
Take ANY theory you like. REMOVE ONE of its axioms.
THE RESULTING theory has the property that the removed axiom
IS NOW TRUE IN SOME models (all the models of the original theory) AND FALSE
IN OTHERS.

Yes, it does get to be provably true in the models in which it is true if you use
the semantics-evaluation "axioms" -- that is a DIFFERENT LEVEL of proof -- but it IS NOT provable from the reduced axiom-set and THAT IS WHY it needed to be added as an axiom to get the full original theory.

It might be one thing if you KNEW this already and were advocating a change in terminology. But if you try to attack the status quo FROM IGNORANCE, your ignorance and its incurability MUST REMAIN THE ONLY things that matter.

[George Greene]

On Sunday, February 4, 2018 at 2:00:02 AM UTC-5, Pete Olcott wrote:
> A Fact(English) is Only known to be True(Math) when it is verified(English).

Even if that were the case, that wouldn't prevent the existence of unverified truths. A truth doesn't stop being true just because nobody has verified it YET.

[Renee Keller]

>
> Like 'If Jack is a bachelor, then he is not married'? Does that have a
> mathematical proof?
>
>

Marriage varies from religion to religion and culture to culture. It is
possible to be both married and a bachelor.

[Peter Percival]

True but not helpful to Pete Olcott. He believes that each word can
only have one meaning. I am confident that for him "bachelor" means
"unmarried man" because it is an example of a statement the truth of
which is "logical, necessary, based upon meaning" that I've borrowed
from Carnap's "Meaning postulates" (/Phil. Studies/, 3, 1952, pp 65-73;
also in /Meaning and necessity/, 2nd ed., U Chicago P.) The idea that
words only have one meaning is quite silly, but cranks are never
embarrassed by defending silly positions.

[peteolcott]

On Sunday, February 4, 2018 at 9:06:49 AM UTC-6, Peter Percival wrote:
> Pete Olcott wrote:
> > http://liarparadox.org/index.php/2018/02/04/truex-and-unprovablex-is-impossible/
>
> Does your Provable(X) refer to some formal theory? What does your
> True(X) refer to? Is X an expression in some specified language?

It does not refer to anything.
You can go back and see what I said that Provable(L, Y, X) refers to.

[peteolcott]

On Monday, February 5, 2018 at 9:55:13 AM UTC-6, Peter Percival wrote:
> Renee Keller wrote:
> >>
> >> Like 'If Jack is a bachelor, then he is not married'? Does that have a
> >> mathematical proof?
>
> > Marriage varies from religion to religion and culture to culture. It is
> > possible to be both married and a bachelor.
>
> True but not helpful to Pete Olcott. He believes that each word can
> only have one meaning.

Not at all. There are an immutable and unique set of meanings.
It does not matter what you call them, you can even call them
"late for dinner" and their underlying semantics remains immutable.

Ultimately this unique set of meanings is merely a unique set of
relations between other meanings. This is very difficult stuff here.

To get a glimpse of the gist of this difficult stuff try and see
how the concept of {5 > 3} remains the same across every human language.

[peteolcott]

On Sunday, February 4, 2018 at 10:21:21 AM UTC-6, Peter Percival wrote:
> Pete Olcott wrote:
> > On 2/4/2018 12:59 AM, Pete Olcott wrote:
> >> http://liarparadox.org/index.php/2018/02/04/truex-and-unprovablex-is-impossible/
> >>
> >>
> >> A Fact(English) is Only known to be True(Math) when it is
> >> verified(English).
> >> Analytic** Facts(English) are Only Verified(English) by their
> >> Proof(Math).
> >>
> >> True(L, X) in formal language L means that there is backward-chaining
> >> inference from X to one or more self-evidently true expressions of L.
> >>
> >> Provable(L, Y, X) means that there is back-chaining inference from X to
> >> one of more expressions Y of language L.
> >>
> >> ** Analytic Truth is the set of statements that can be completely
> >> verified
> >> as True entirely based on the meaning of their words.
> >>
> >> This is a refinement to the 1997 Mathematical Mapping Theory of Truth
> >>
> >> Copyright 1997 2018 Pete Olcott
> >>
> >
> > "This sentence is not true".
> > LP ≡ ~True(LP)     // HOL with self-reference semantics
> >
> > "This sentence is not provable".
> > G  ≡ ~Provable(G)  // HOL with self-reference semantics
>
> What is self-reference semantics?

It is the syntax required to allow expressions of language to refer
directly to themselves.

Without a formal way of specifying this you has otherwise very
brilliant minds such as Alfred Tarski that can never seem to
untangle their own confusion when they conflate an expression's
reference to its name as the same thing as actual self-reference.

[Peter Percival]

You might be interested in Smullyan's 'Languages in which self reference
is possible', /JSL/, 1957.

[peteolcott]

I created my own language to do this from the ground up.
I can make sure that none of it has any incoherent aspects.
If I need an existing language that would be Prolog, because only
software can check its own consistency.

https://en.wikipedia.org/wiki/Truth-bearer
An expression of language that is intended to represent a closed WFF
(or a statement of natural language) must be a truth bearer or it is semantically ill-formed.

https://en.wikipedia.org/wiki/Statement_(logic)
If the assertion that this expression makes is not provable
and the negation of this assertion is also not provable then the
then this expression is not a statement of language because
it lacks a truth bearer.

[Peter Percival]

peteolcott wrote:

>
> https://en.wikipedia.org/wiki/Statement_(logic)
> If the assertion that this expression makes is not provable
> and the negation of this assertion is also not provable then the
> then this expression is not a statement of language because
> it lacks a truth bearer.

If a theory is weak it may be unable to prove a statement and unable to
prove its negation. A stronger theory may prove one or the other. That
formal provability is relevant to theories is something you
systematically ignore.

[Peter Percival]

Peter Percival wrote:
> peteolcott wrote:
>
>>

>> https://en.wikipedia.org/wiki/Statement_(logic)

Why do you mention the above? It has nothing to do with what you write
next.

>> If the assertion that this expression makes is not provable
>> and the negation of this assertion is also not provable then the
>> then this expression is not a statement of language because
>> it lacks a truth bearer.
>
> If a theory is weak it may be unable to prove a statement and unable to
> prove its negation.  A stronger theory may prove one or the other.  That
> formal provability is relevant to theories is something you

is relative to - yuk

> systematically ignore.
>

[peteolcott]

False(L, X) in formal language L means that there is backward-chaining
inference from X to the negation of one or more self-evidently true expressions of L.

∀L ∀Y ∀X
~( Provable(L, Y, X) ∨ Provable(L, Y, ~X) ) ⊢
~( True(L, X) ∨ False(L, X) ) ⊢
~Statement(L, X)

https://en.wikipedia.org/wiki/Statement_(logic)
https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)

Therefore the following conclusion is necessarily false.

Stanford Encyclopedia of Philosophy
Gödel's Incompleteness Theorems
"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. "

Copyright 2016, 2017, 2018 Pete Olcott

[peteolcott]

https://en.wikipedia.org/wiki/Statement_(logic)
https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)

A statement is the natural language equivalent of a logic sentence.
In both cases if they do not evaluate to True or False they are wrong.

[Peter Percival]

You have switched from provable/not provable to true/false. For some
theories they coincide, for others they don't.

[Peter Percival]

peteolcott wrote:
> On Tuesday, February 6, 2018 at 10:13:05 AM UTC-6, Peter Percival wrote:
>> peteolcott wrote:
>>
>>>
>>> https://en.wikipedia.org/wiki/Statement_(logic)
>>> If the assertion that this expression makes is not provable
>>> and the negation of this assertion is also not provable then the
>>> then this expression is not a statement of language because
>>> it lacks a truth bearer.
>>
>> If a theory is weak it may be unable to prove a statement and unable to
>> prove its negation. A stronger theory may prove one or the other. That
>> formal provability is relevant to theories is something you
>> systematically ignore.
>
>
> False(L, X) in formal language L means that there is backward-chaining
> inference from X to the negation of one or more self-evidently true expressions of L.

The notion of self-evident truths is problematic. What is
self-evidently true to one person may be self-evidently false to another.

A more sensible account of falsity *in a formal language* would make
reference to models.

[peteolcott]

All theories are subsumed under the singular immutable true of the inherent nature of semantics.

[peteolcott]

On Tuesday, February 6, 2018 at 11:44:06 AM UTC-6, Peter Percival wrote:
> peteolcott wrote:
> > On Tuesday, February 6, 2018 at 10:13:05 AM UTC-6, Peter Percival wrote:
> >> peteolcott wrote:
> >>
> >>>
> >>> https://en.wikipedia.org/wiki/Statement_(logic)
> >>> If the assertion that this expression makes is not provable
> >>> and the negation of this assertion is also not provable then the
> >>> then this expression is not a statement of language because
> >>> it lacks a truth bearer.
> >>
> >> If a theory is weak it may be unable to prove a statement and unable to
> >> prove its negation. A stronger theory may prove one or the other. That
> >> formal provability is relevant to theories is something you
> >> systematically ignore.
> >
> >
> > False(L, X) in formal language L means that there is backward-chaining
> > inference from X to the negation of one or more self-evidently true expressions of L.
>
> The notion of self-evident truths is problematic. What is
> self-evidently true to one person may be self-evidently false to another.
>

All of the existing definitions of this stuff are pretty crappy, that
is why I must start from scratch and rebuild them from the ground up.

Expressions of language L that are defined to have the semantic
property of {True} are simply finite strings assigned to a
specific set: let's come of with a brand new name so there is
no confusion: TRUE_STRINGS(L).

> A more sensible account of falsity *in a formal language* would make
> reference to models.
>

No it is that screwy stuff that confuses people into believing that
there are closed WFF that are neither provable nor refutable.

An expression X of language L is true if there is a proof from
one or more elements Y of TRUE_STRINGS of L to X.

An expression X of language L is false if there is a proof from
the negation of one or more elements Y of TRUE_STRINGS of L to X.

Any closed WFF of L ( ~True(L, X) & ~False(L, X) ) |- Incorrect(X)

Copyright 2018 Pete Olcott