Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
Changing the foundation to proof theoretic semantics where
truth is well-founded provability blocks TarskirCOs diagonal
step most clearly seen on line (3)
Here is the Tarski Undefinability Theorem proof
(1) x ree Provable if and only if p
(2) x ree True if and only if p
(3) x ree Provable if and only if x ree True. // (1) and (2) combined
(4) either x ree True or x|a ree True;-a-a-a-a // axiom: ~True(x) re? ~True(~x)
(5) if x ree Provable, then x ree True;-a // axiom: Provable(x) raA True(x) (6) if x|a ree Provable, then x|a ree True;-a // axiom: Provable(~x) raA True(~x)
(7) x ree True
(8) x ree Provable
(9) x|a ree Provable
https://liarparadox.org/Tarski_275_276.pdf
A proof theoretic prover rejects expressions that
do not have "a well-founded justification tree within
Proof theoretic semantics".
The same way that Prolog does
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
Changing the foundation to proof theoretic semantics where
truth is well-founded provability blocks TarskirCOs diagonal
step most clearly seen on line (3)
Here is the Tarski Undefinability Theorem proof
(1) x ree Provable if and only if p
(2) x ree True if and only if p
(3) x ree Provable if and only if x ree True. // (1) and (2) combined
(4) either x ree True or x|a ree True;-a-a-a-a // axiom: ~True(x) re? ~True(~x)
(5) if x ree Provable, then x ree True;-a // axiom: Provable(x) raA True(x) (6) if x|a ree Provable, then x|a ree True;-a // axiom: Provable(~x) raA True(~x)
(7) x ree True
(8) x ree Provable
(9) x|a ree Provable
https://liarparadox.org/Tarski_275_276.pdf
A proof theoretic prover rejects expressions that
do not have "a well-founded justification tree within
Proof theoretic semantics".
The same way that Prolog does
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
On 2/5/2026 10:55 AM, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
Changing the foundation to proof theoretic semantics where
truth is well-founded provability blocks TarskirCOs diagonal
step most clearly seen on line (3)
Here is the Tarski Undefinability Theorem proof
(1) x ree Provable if and only if p
(2) x ree True if and only if p
(3) x ree Provable if and only if x ree True. // (1) and (2) combined
(4) either x ree True or x|a ree True;-a-a-a-a // axiom: ~True(x) re? ~True(~x)
(5) if x ree Provable, then x ree True;-a // axiom: Provable(x) raA True(x) >> (6) if x|a ree Provable, then x|a ree True;-a // axiom: Provable(~x) raA True(~x)
(7) x ree True
(8) x ree Provable
(9) x|a ree Provable
https://liarparadox.org/Tarski_275_276.pdf
A proof theoretic prover rejects expressions that
do not have "a well-founded justification tree within
Proof theoretic semantics".
The same way that Prolog does
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
With actual competent human review
x ree Provable rco x ree True // proof theoretic semantics
is changed to
x ree Provable rcA x ree True // proof theoretic semantics
This is Tarski's line (5)
This overrules anything that contradicts it because
it has now attained axiom status.
Below I show how this overrules Tarski line (3)
thus overcoming Tarski Undefinability when we
change its foundation from truth conditional semantics
to proof theoretic semantics. PTS was not available
at the time That he wrote
"The Concept of Truth in Formalized Languages"
(3) x ree Provable if and only if x ree True.
can be divided into
(3)(a) if x ree Provable, then x ree True
(3)(b) if x ree True, then x ree Provable
(5) if x ree Provable, then x ree True
(5) combined with (3)(b) becomes
if x ree ProvablerCarCethen rCex ree Provable
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really define.
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really define.
It is an axiom: reCx (Provable(x) rcA True(x))
There are dozens of papers needed to verify this.
It will take me quite a while to form proper citations
of these papers. It is anchored in proof theoretic semantics.
Generic PTS states that ~Provable(x) rco Meaningless(x).
Model theory and truth conditional semantics are rejected.
On 2/6/26 10:30 AM, olcott wrote:
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really define.
It is an axiom: reCx (Provable(x) rcA True(x))
But the axiom uses rcA which goes in just one direction, while you statements used rco which attempts to go both ways.
There are dozens of papers needed to verify this.
It will take me quite a while to form proper citations
of these papers. It is anchored in proof theoretic semantics.
Generic PTS states that ~Provable(x) rco Meaningless(x).
Model theory and truth conditional semantics are rejected.
And, I think your problem is you don't actually understand what you are reading. This shows in that you have been making the claim for years,
but you are now admitting you can't ACTUALLY show why it is (yet).
Your problem is it seems you fundamentally don't understand how
semantics work, and why it is important to put things into context.
This shows in part because you keep on trying to apply principles for general Philosophy to Formal Logic, where they do not apply.
Sorry, you are just showing your fundamental ignorance of what you are talking about.--
On 2/6/2026 12:15 PM, Richard Damon wrote:
On 2/6/26 10:30 AM, olcott wrote:
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really define.
It is an axiom: reCx (Provable(x) rcA True(x))
But the axiom uses rcA which goes in just one direction, while you
statements used rco which attempts to go both ways.
This was corrected by an expert that seems
to really know these things.
This same expert agrees that with within PTS:
"if x is provable, then it is true."
There are dozens of papers needed to verify this.
It will take me quite a while to form proper citations
of these papers. It is anchored in proof theoretic semantics.
Generic PTS states that ~Provable(x) rco Meaningless(x).
Model theory and truth conditional semantics are rejected.
And, I think your problem is you don't actually understand what you
are reading. This shows in that you have been making the claim for
years, but you are now admitting you can't ACTUALLY show why it is (yet).
reCx-a (~Provable(x) rco Meaningless(x))
Seems to be exactly and precisely what Proof Theoretic
Semantics actually says. Since the SEP article was
written by the guy that coined the term:
"Proof Theoretic Semantics"
It should be pretty definitive.
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Your problem is it seems you fundamentally don't understand how
semantics work, and why it is important to put things into context.
Not at all. It all in "Proof Theoretic Semantics"
This shows in part because you keep on trying to apply principles for
general Philosophy to Formal Logic, where they do not apply.
Try saying that after you spend three hours carefully studying
the linked article. That article is not the end-all be-all
of "Proof Theoretic Semantics", yet it does seem to be the
most definitive single source.
Sorry, you are just showing your fundamental ignorance of what you are
talking about.
On 2/6/26 3:00 PM, olcott wrote:
On 2/6/2026 12:15 PM, Richard Damon wrote:
On 2/6/26 10:30 AM, olcott wrote:
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really define. >>>>>
It is an axiom: reCx (Provable(x) rcA True(x))
But the axiom uses rcA which goes in just one direction, while you
statements used rco which attempts to go both ways.
This was corrected by an expert that seems
to really know these things.
This same expert agrees that with within PTS:
"if x is provable, then it is true."
Right, Provable leads to Truth. But Not Provable does not mean not true,
or Truth require provability by the axiom.
I gues you are just admitting that you are just a pathetic liar.
There are dozens of papers needed to verify this.
It will take me quite a while to form proper citations
of these papers. It is anchored in proof theoretic semantics.
Generic PTS states that ~Provable(x) rco Meaningless(x).
Model theory and truth conditional semantics are rejected.
And, I think your problem is you don't actually understand what you
are reading. This shows in that you have been making the claim for
years, but you are now admitting you can't ACTUALLY show why it is
(yet).
reCx-a (~Provable(x) rco Meaningless(x))
Seems to be exactly and precisely what Proof Theoretic
Semantics actually says. Since the SEP article was
written by the guy that coined the term:
"Proof Theoretic Semantics"
It should be pretty definitive.
No, which is part of your problem. Proof-Theoretic Semantics say we
can't talk about the truth of a statement we can not prove, NOT that the statement can't be true without the proof, just we can't talk about it.
Proof-Theoretic Semantics limits our way of looking at things to what
can be proven, and things outside of what can be proven are just outside
the domain of discussion.
The problem of using this Philosophical view in Formal Logic systems
that have the power to create the Natural Number system is that we
suddenly find we can't know if we can talk about a given statement until
we solve it.
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Your problem is it seems you fundamentally don't understand how
semantics work, and why it is important to put things into context.
Not at all. It all in "Proof Theoretic Semantics"
Which you don't understand, as that is all discussion in PHILOSOPHY, not FORMAL LOGIC, particularly those systems that can create infinite
domains of reguard.
This shows in part because you keep on trying to apply principles for
general Philosophy to Formal Logic, where they do not apply.
Try saying that after you spend three hours carefully studying
the linked article. That article is not the end-all be-all
of "Proof Theoretic Semantics", yet it does seem to be the
most definitive single source.
Maybe you should notice how many times they talk about removing things
like in standard logic. Since Formal Logic system include in there definitions, the mode of interpreation of the logic, you aren't allowed
to change that and keep the system being "the same".
In other words, if you want to change to your "Proof-Theoretic
Semantics", you FIRST need to show how much of the system services the change of rules.
Since the definition of arithmatic of Natural Numbers falls apart if you
try to force this on it, all you are doing is saying that you logic
can't handle mathematics.
On 2/6/2026 5:18 PM, Richard Damon wrote:
On 2/6/26 3:00 PM, olcott wrote:
On 2/6/2026 12:15 PM, Richard Damon wrote:
On 2/6/26 10:30 AM, olcott wrote:
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really define. >>>>>>
It is an axiom: reCx (Provable(x) rcA True(x))
But the axiom uses rcA which goes in just one direction, while you
statements used rco which attempts to go both ways.
This was corrected by an expert that seems
to really know these things.
This same expert agrees that with within PTS:
"if x is provable, then it is true."
Right, Provable leads to Truth. But Not Provable does not mean not
true, or Truth require provability by the axiom.
I gues you are just admitting that you are just a pathetic liar.
There are dozens of papers needed to verify this.
It will take me quite a while to form proper citations
of these papers. It is anchored in proof theoretic semantics.
Generic PTS states that ~Provable(x) rco Meaningless(x).
Model theory and truth conditional semantics are rejected.
And, I think your problem is you don't actually understand what you
are reading. This shows in that you have been making the claim for
years, but you are now admitting you can't ACTUALLY show why it is
(yet).
reCx-a (~Provable(x) rco Meaningless(x))
Seems to be exactly and precisely what Proof Theoretic
Semantics actually says. Since the SEP article was
written by the guy that coined the term:
"Proof Theoretic Semantics"
It should be pretty definitive.
No, which is part of your problem. Proof-Theoretic Semantics say we
can't talk about the truth of a statement we can not prove, NOT that
the statement can't be true without the proof, just we can't talk
about it.
Lets try to say this exactly accurately.
In PTS expressions that are unprovable are
ungrounded in semantic meaning.
When I refer to a formal system I am referring to
Russell's atomic facts written down and placed in
a simple Type Hierarchy.
-a atomic facts, which consist either of a simple
-a particular exhibiting a quality, or multiple
-a simple particulars standing in a relation. https://plato.stanford.edu/entries/logical-atomism/
Proof-Theoretic Semantics limits our way of looking at things to what
can be proven, and things outside of what can be proven are just
outside the domain of discussion.
"true on the basis of meaning expressed in language"
necessarily includes the entire body of knowledge
expressed in language.
The problem of using this Philosophical view in Formal Logic systems
that have the power to create the Natural Number system is that we
suddenly find we can't know if we can talk about a given statement
until we solve it.
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Your problem is it seems you fundamentally don't understand how
semantics work, and why it is important to put things into context.
Not at all. It all in "Proof Theoretic Semantics"
Which you don't understand, as that is all discussion in PHILOSOPHY,
not FORMAL LOGIC, particularly those systems that can create infinite
domains of reguard.
This shows in part because you keep on trying to apply principles
for general Philosophy to Formal Logic, where they do not apply.
Try saying that after you spend three hours carefully studying
the linked article. That article is not the end-all be-all
of "Proof Theoretic Semantics", yet it does seem to be the
most definitive single source.
Maybe you should notice how many times they talk about removing things
like in standard logic. Since Formal Logic system include in there
definitions, the mode of interpreation of the logic, you aren't
allowed to change that and keep the system being "the same".
In other words, if you want to change to your "Proof-Theoretic
Semantics", you FIRST need to show how much of the system services the
change of rules.
Since the definition of arithmatic of Natural Numbers falls apart if
you try to force this on it, all you are doing is saying that you
logic can't handle mathematics.
reCx (Provable(PA, x)-a rcA True(PA, x))
reCx (Provable(PA, ~x) rcA False(PA, x))
reCx (~True(PA, x) reo ~False(PA, x) rco ~Truth_Apt(PA, x))
On 2/6/26 7:10 PM, olcott wrote:
On 2/6/2026 5:18 PM, Richard Damon wrote:
On 2/6/26 3:00 PM, olcott wrote:
On 2/6/2026 12:15 PM, Richard Damon wrote:
On 2/6/26 10:30 AM, olcott wrote:
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really define. >>>>>>>
It is an axiom: reCx (Provable(x) rcA True(x))
But the axiom uses rcA which goes in just one direction, while you
statements used rco which attempts to go both ways.
This was corrected by an expert that seems
to really know these things.
This same expert agrees that with within PTS:
"if x is provable, then it is true."
Right, Provable leads to Truth. But Not Provable does not mean not
true, or Truth require provability by the axiom.
I gues you are just admitting that you are just a pathetic liar.
There are dozens of papers needed to verify this.
It will take me quite a while to form proper citations
of these papers. It is anchored in proof theoretic semantics.
Generic PTS states that ~Provable(x) rco Meaningless(x).
Model theory and truth conditional semantics are rejected.
And, I think your problem is you don't actually understand what you >>>>> are reading. This shows in that you have been making the claim for
years, but you are now admitting you can't ACTUALLY show why it is
(yet).
reCx-a (~Provable(x) rco Meaningless(x))
Seems to be exactly and precisely what Proof Theoretic
Semantics actually says. Since the SEP article was
written by the guy that coined the term:
"Proof Theoretic Semantics"
It should be pretty definitive.
No, which is part of your problem. Proof-Theoretic Semantics say we
can't talk about the truth of a statement we can not prove, NOT that
the statement can't be true without the proof, just we can't talk
about it.
Lets try to say this exactly accurately.
In PTS expressions that are unprovable are
ungrounded in semantic meaning.
Right, which means you can't talk about them.
When I refer to a formal system I am referring to
Russell's atomic facts written down and placed in
a simple Type Hierarchy.
Then you aren't talking about a real Formal System.
This is your problem, You don't understand what a Formal system actually
is.
You keep on thinking it is just a form of Philosophy, which it really
isn't.
-a-a atomic facts, which consist either of a simple
-a-a particular exhibiting a quality, or multiple
-a-a simple particulars standing in a relation.
https://plato.stanford.edu/entries/logical-atomism/
So, how do you fit Peano Arithmatic into that system?
Proof-Theoretic Semantics limits our way of looking at things to what
can be proven, and things outside of what can be proven are just
outside the domain of discussion.
"true on the basis of meaning expressed in language"
necessarily includes the entire body of knowledge
expressed in language.
Try to do it.
Since that body of knowledge expresses facts about mathematics and such system, it include things like the Pythagorean Theorem, which is *NOT*
true just on the meaning of its words,
Since the definition of arithmatic of Natural Numbers falls apart if
you try to force this on it, all you are doing is saying that you
logic can't handle mathematics.
reCx (Provable(PA, x)-a rcA True(PA, x))
reCx (Provable(PA, ~x) rcA False(PA, x))
reCx (~True(PA, x) reo ~False(PA, x) rco ~Truth_Apt(PA, x))
Which isn't what Proof Theoretic says,
As it doesn't introduce the concept of the predicate "True".
It says if you CAN prove the statement in the system, then you can say
the statement is true.
And, if you CAN prove the converse of the statement in the system, then
you can say the statement if false,
And, if you CAN prove that the you can never do either of the above, you
can say the statement is non-well-founded.
You might not be able to do any of the above, in which case you can't
talk about the statement and it truth.
Since in mathematics, there ARE statements for which you can't do any of
the above, Proof-Theoretic Semantic fall apart for it, as you start to
run into the issue of not knowing if you can talk about the statements.
It works better in simpler systems where there are many statements for
which you can reduce it to one of the three cases you can talk about.
On 2/7/2026 8:47 PM, Tristan Wibberley wrote:
On 07/02/2026 00:10, olcott wrote:
When I refer to a formal system I am referring to
Russell's atomic facts written down and placed in
a simple Type Hierarchy.
Is that, in effect, the conventional meaning of "formal system"? It is
not normally expressed so, see Curry and Feys.
It is the axiomatic foundation of this:
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
On 2/7/26 11:59 PM, olcott wrote:
On 2/7/2026 8:47 PM, Tristan Wibberley wrote:
On 07/02/2026 00:10, olcott wrote:
When I refer to a formal system I am referring to
Russell's atomic facts written down and placed in
a simple Type Hierarchy.
Is that, in effect, the conventional meaning of "formal system"? It is
not normally expressed so, see Curry and Feys.
It is the axiomatic foundation of this:
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
No, because that doesn't exist.
The problem is "True" can't be computable in a system that handles all knowledge, as you have been shown.
Your problem is you just don't understand what you are talking about, as truth is a concept you, as a pathological liar, can't understand.
On 2/9/2026 6:51 AM, Richard Damon wrote:
On 2/7/26 11:59 PM, olcott wrote:
On 2/7/2026 8:47 PM, Tristan Wibberley wrote:
On 07/02/2026 00:10, olcott wrote:
When I refer to a formal system I am referring to
Russell's atomic facts written down and placed in
a simple Type Hierarchy.
Is that, in effect, the conventional meaning of "formal system"? It is >>>> not normally expressed so, see Curry and Feys.
It is the axiomatic foundation of this:
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
No, because that doesn't exist.
The problem is "True" can't be computable in a system that handles all
knowledge, as you have been shown.
Your problem is you just don't understand what you are talking about,
as truth is a concept you, as a pathological liar, can't understand.
I warned you about disrespect
On 2/6/2026 6:23 PM, Richard Damon wrote:
On 2/6/26 7:10 PM, olcott wrote:
On 2/6/2026 5:18 PM, Richard Damon wrote:
On 2/6/26 3:00 PM, olcott wrote:
On 2/6/2026 12:15 PM, Richard Damon wrote:
On 2/6/26 10:30 AM, olcott wrote:
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics >>>>>>>>> Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really
define.
It is an axiom: reCx (Provable(x) rcA True(x))
But the axiom uses rcA which goes in just one direction, while you >>>>>> statements used rco which attempts to go both ways.
This was corrected by an expert that seems
to really know these things.
This same expert agrees that with within PTS:
"if x is provable, then it is true."
Right, Provable leads to Truth. But Not Provable does not mean not
true, or Truth require provability by the axiom.
I gues you are just admitting that you are just a pathetic liar.
There are dozens of papers needed to verify this.
It will take me quite a while to form proper citations
of these papers. It is anchored in proof theoretic semantics.
Generic PTS states that ~Provable(x) rco Meaningless(x).
Model theory and truth conditional semantics are rejected.
And, I think your problem is you don't actually understand what
you are reading. This shows in that you have been making the claim >>>>>> for years, but you are now admitting you can't ACTUALLY show why
it is (yet).
reCx-a (~Provable(x) rco Meaningless(x))
Seems to be exactly and precisely what Proof Theoretic
Semantics actually says. Since the SEP article was
written by the guy that coined the term:
"Proof Theoretic Semantics"
It should be pretty definitive.
No, which is part of your problem. Proof-Theoretic Semantics say we
can't talk about the truth of a statement we can not prove, NOT that
the statement can't be true without the proof, just we can't talk
about it.
Lets try to say this exactly accurately.
In PTS expressions that are unprovable are
ungrounded in semantic meaning.
Right, which means you can't talk about them.
Your way of saying it is way too weak.
Is gibberish nonsense is more accurate.
When I refer to a formal system I am referring to
Russell's atomic facts written down and placed in
a simple Type Hierarchy.
Then you aren't talking about a real Formal System.
This is your problem, You don't understand what a Formal system
actually is.
You keep on thinking it is just a form of Philosophy, which it really
isn't.
-a-a atomic facts, which consist either of a simple
-a-a particular exhibiting a quality, or multiple
-a-a simple particulars standing in a relation.
https://plato.stanford.edu/entries/logical-atomism/
So, how do you fit Peano Arithmatic into that system?
Proof-Theoretic Semantics limits our way of looking at things to
what can be proven, and things outside of what can be proven are
just outside the domain of discussion.
"true on the basis of meaning expressed in language"
necessarily includes the entire body of knowledge
expressed in language.
Try to do it.
Since that body of knowledge expresses facts about mathematics and
such system, it include things like the Pythagorean Theorem, which is
*NOT* true just on the meaning of its words,
"true on the basis of meaning expressed in language"
NOT THE MEANING OF WORDS
NOT THE MEANING OF WORDS
NOT THE MEANING OF WORDS
NOT THE MEANING OF WORDS
NOT THE MEANING OF WORDS
NOT THE MEANING OF WORDS
*THE MEANING IN ALL FORMAL MATHEMATICAL AND NATURAL LANGUAGES*
*THE MEANING IN ALL FORMAL MATHEMATICAL AND NATURAL LANGUAGES*
*THE MEANING IN ALL FORMAL MATHEMATICAL AND NATURAL LANGUAGES*
I spent 25 years coming up with that you could
take two minutes to pay COMPLETE ATTENTION.
<snip>
Since the definition of arithmatic of Natural Numbers falls apart if
you try to force this on it, all you are doing is saying that you
logic can't handle mathematics.
reCx (Provable(PA, x)-a rcA True(PA, x))
reCx (Provable(PA, ~x) rcA False(PA, x))
reCx (~True(PA, x) reo ~False(PA, x) rco ~Truth_Apt(PA, x))
Which isn't what Proof Theoretic says,
As it doesn't introduce the concept of the predicate "True".
It says if you CAN prove the statement in the system, then you can say
the statement is true.
And, if you CAN prove the converse of the statement in the system,
then you can say the statement if false,
And, if you CAN prove that the you can never do either of the above,
you can say the statement is non-well-founded.
*Good job, you got the most important point exactly correctly* Non-well-founded means not truth-apt.
You might not be able to do any of the above, in which case you can't
talk about the statement and it truth.
The key thing is that PTS rejects cases of pathological
self-reference as lacking a well-founded justification
tree thus semantically ill-formed within PTS.
Since in mathematics, there ARE statements for which you can't do any
of the above, Proof-Theoretic Semantic fall apart for it, as you start
to run into the issue of not knowing if you can talk about the
statements. It works better in simpler systems where there are many
statements for which you can reduce it to one of the three cases you
can talk about.
You are conflating mathematics within the foundation of
Truth Conditional Semantics with mathematics itself.
Mathematics within Proof Theoretic Semantics cannot
be incomplete.
On 2/6/26 8:13 PM, olcott wrote:
On 2/6/2026 6:23 PM, Richard Damon wrote:
On 2/6/26 7:10 PM, olcott wrote:
On 2/6/2026 5:18 PM, Richard Damon wrote:
On 2/6/26 3:00 PM, olcott wrote:
On 2/6/2026 12:15 PM, Richard Damon wrote:
On 2/6/26 10:30 AM, olcott wrote:
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics >>>>>>>>>> Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really >>>>>>>>> define.
It is an axiom: reCx (Provable(x) rcA True(x))
But the axiom uses rcA which goes in just one direction, while you >>>>>>> statements used rco which attempts to go both ways.
This was corrected by an expert that seems
to really know these things.
This same expert agrees that with within PTS:
"if x is provable, then it is true."
Right, Provable leads to Truth. But Not Provable does not mean not
true, or Truth require provability by the axiom.
I gues you are just admitting that you are just a pathetic liar.
There are dozens of papers needed to verify this.
It will take me quite a while to form proper citations
of these papers. It is anchored in proof theoretic semantics.
Generic PTS states that ~Provable(x) rco Meaningless(x).
Model theory and truth conditional semantics are rejected.
And, I think your problem is you don't actually understand what >>>>>>> you are reading. This shows in that you have been making the
claim for years, but you are now admitting you can't ACTUALLY
show why it is (yet).
reCx-a (~Provable(x) rco Meaningless(x))
Seems to be exactly and precisely what Proof Theoretic
Semantics actually says. Since the SEP article was
written by the guy that coined the term:
"Proof Theoretic Semantics"
It should be pretty definitive.
No, which is part of your problem. Proof-Theoretic Semantics say we >>>>> can't talk about the truth of a statement we can not prove, NOT
that the statement can't be true without the proof, just we can't
talk about it.
Lets try to say this exactly accurately.
In PTS expressions that are unprovable are
ungrounded in semantic meaning.
Right, which means you can't talk about them.
Your way of saying it is way too weak.
Is gibberish nonsense is more accurate.
Nope, just because it doesn't know the answer, doesn't mean the question
is "gibberish".
That is the flaw in your thinking.
Unless the system can KNOW that the statement can not be proven, it
doesn't know that the statement is actually gibberish, or just to
difficult to understand.
Just like Tarski's proof that a Truth Primative can't exist, or Godel's proof that formal logic system that can handle the mathematcis is
incomplete SEEM LIKE GIBBERISH TO YOU, but are actually full of meaning
to those that actually understand their meaning.
Proof-Theoretic Semantics understands that just because we haven't found
a proof, doesn't make it non-well-founded, and accepts it.
Godel's proof shows the limitiation of Proof-Theoretic Semantics in a mathematical system.
He creates a perfectly semantic statement
as a relationship that can be
finitely tested for any number. He then ask a complete semantically reasonable question about that relationship, does a number exist that satisfies it, making an assertion that it does not.
This SHOULD have meaning in a system that understands the mathematics.
But, by the rules of proof-theoretic semantics, The statement can only
be considered to be true if we could prove it, and false if we could
prove its converse, or declared non-well-founded if we could prove that neither of those can be shown,
but it turns out that there is no proof
in the system for ANY of those options, and thus a statement that HAS semantic meaning by the structure of the system can't be given a PT semantics.
It turns out that Proof-Theoretic Semantics just fail for system that
handle mathematics as there exists statements like this that can not be proven true, can not be proven false, and can't be proven to not be able
for make either of those proofs.
But then, this comes out as a somewhat natural result of the axiom of induction, which defines a way to answer SOME truth-conditional
statements, but not all, and thus admits them into its semantics.
*THE MEANING IN ALL FORMAL MATHEMATICAL AND NATURAL LANGUAGES*
Then you accept the Formal Logic definition of "Truth" that means what
can be demonstrated by an infinite sequence of steps, and thus reject
you proposition that something is only true if it can be proven.
*Good job, you got the most important point exactly correctly*
Non-well-founded means not truth-apt.
So, you accept that your "Proof-Theoretic" system uses Truth-Conditional meaning, as you can't actually PROVE that something is Truth_Apt.
Sorry, you are just showing your world is just an imaginary fantasy.
You might not be able to do any of the above, in which case you can't
talk about the statement and it truth.
The key thing is that PTS rejects cases of pathological
self-reference as lacking a well-founded justification
tree thus semantically ill-formed within PTS.
But, the problem is that the things you want to reject don't HAVE a self-reference in the system you are apply the Proof-Theoretic semantics
to, and thus CAN'T reject the statements.
Either you accept the meaning in the meta-system, which proves the
statement to be true and unprovable, or you can't access that to show it
is not-well-founded.
On 2/6/2026 7:54 PM, Richard Damon wrote:
On 2/6/26 8:13 PM, olcott wrote:
On 2/6/2026 6:23 PM, Richard Damon wrote:
On 2/6/26 7:10 PM, olcott wrote:
On 2/6/2026 5:18 PM, Richard Damon wrote:
On 2/6/26 3:00 PM, olcott wrote:
On 2/6/2026 12:15 PM, Richard Damon wrote:
On 2/6/26 10:30 AM, olcott wrote:
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics >>>>>>>>>>> Tarski Undefinability is overcomeA definition in terms of an undefined symbol does not really >>>>>>>>>> define.
x ree Provable rco x ree True // proof theoretic semantics >>>>>>>>>>
It is an axiom: reCx (Provable(x) rcA True(x))
But the axiom uses rcA which goes in just one direction, while you >>>>>>>> statements used rco which attempts to go both ways.
This was corrected by an expert that seems
to really know these things.
This same expert agrees that with within PTS:
"if x is provable, then it is true."
Right, Provable leads to Truth. But Not Provable does not mean not >>>>>> true, or Truth require provability by the axiom.
I gues you are just admitting that you are just a pathetic liar.
There are dozens of papers needed to verify this.
It will take me quite a while to form proper citations
of these papers. It is anchored in proof theoretic semantics. >>>>>>>>> Generic PTS states that ~Provable(x) rco Meaningless(x).
Model theory and truth conditional semantics are rejected.
And, I think your problem is you don't actually understand what >>>>>>>> you are reading. This shows in that you have been making the
claim for years, but you are now admitting you can't ACTUALLY >>>>>>>> show why it is (yet).
reCx-a (~Provable(x) rco Meaningless(x))
Seems to be exactly and precisely what Proof Theoretic
Semantics actually says. Since the SEP article was
written by the guy that coined the term:
"Proof Theoretic Semantics"
It should be pretty definitive.
No, which is part of your problem. Proof-Theoretic Semantics say
we can't talk about the truth of a statement we can not prove, NOT >>>>>> that the statement can't be true without the proof, just we can't >>>>>> talk about it.
Lets try to say this exactly accurately.
In PTS expressions that are unprovable are
ungrounded in semantic meaning.
Right, which means you can't talk about them.
Your way of saying it is way too weak.
Is gibberish nonsense is more accurate.
Nope, just because it doesn't know the answer, doesn't mean the
question is "gibberish".
That is the flaw in your thinking.
What is an expression of language that has no meaning?
Unless the system can KNOW that the statement can not be proven, it
doesn't know that the statement is actually gibberish, or just to
difficult to understand.
Just like Tarski's proof that a Truth Primative can't exist, or
Godel's proof that formal logic system that can handle the mathematcis
is incomplete SEEM LIKE GIBBERISH TO YOU, but are actually full of
meaning to those that actually understand their meaning.
Proof-Theoretic Semantics understands that just because we haven't
found a proof, doesn't make it non-well-founded, and accepts it.
Godel's proof shows the limitiation of Proof-Theoretic Semantics in a
mathematical system.
Not in the least little bit.
He creates a perfectly semantic statement
In the wrong semantic system.
as a relationship that can be finitely tested for any number. He then
ask a complete semantically reasonable question about that
relationship, does a number exist that satisfies it, making an
assertion that it does not.
This SHOULD have meaning in a system that understands the mathematics.
But, by the rules of proof-theoretic semantics, The statement can only
be considered to be true if we could prove it, and false if we could
prove its converse, or declared non-well-founded if we could prove
that neither of those can be shown,
Yes.
but it turns out that there is no proof in the system for ANY of those
options, and thus a statement that HAS semantic meaning by the
structure of the system can't be given a PT semantics.
My system involves a type hierarchy with unlimited finite levels of
indirect reference.
It turns out that Proof-Theoretic Semantics just fail for system that
handle mathematics as there exists statements like this that can not
be proven true, can not be proven false, and can't be proven to not be
able for make either of those proofs.
But then, this comes out as a somewhat natural result of the axiom of
induction, which defines a way to answer SOME truth-conditional
statements, but not all, and thus admits them into its semantics.
<snip>
*THE MEANING IN ALL FORMAL MATHEMATICAL AND NATURAL LANGUAGES*
Then you accept the Formal Logic definition of "Truth" that means what
can be demonstrated by an infinite sequence of steps, and thus reject
you proposition that something is only true if it can be proven.
We have been over this quite a few time now.
Anything requiring an infinite number of steps
is outside the domain of knowledge.
<snip>
*Good job, you got the most important point exactly correctly*
Non-well-founded means not truth-apt.
So, you accept that your "Proof-Theoretic" system uses Truth-
Conditional meaning, as you can't actually PROVE that something is
Truth_Apt.
ABSOLUTELY NOT EVER. WHERE THE HELL DID YOU GET THAT ???
ABSOLUTELY NOT EVER. WHERE THE HELL DID YOU GET THAT ???
ABSOLUTELY NOT EVER. WHERE THE HELL DID YOU GET THAT ???
Truth conditional semantics has always been TOTALLY WRONG HEADED !!
Sorry, you are just showing your world is just an imaginary fantasy.
You might not be able to do any of the above, in which case you
can't talk about the statement and it truth.
The key thing is that PTS rejects cases of pathological
self-reference as lacking a well-founded justification
tree thus semantically ill-formed within PTS.
But, the problem is that the things you want to reject don't HAVE a
self-reference in the system you are apply the Proof-Theoretic
semantics to, and thus CAN'T reject the statements.
Either you accept the meaning in the meta-system, which proves the
statement to be true and unprovable, or you can't access that to show
it is not-well-founded.
"true on the basis of meaning expressed in language"
Including formal mathematical languages and natural language
Where the meaning of expressions is only defined in terms
of other expressions makes the entire body of knowledge
that can be expressed in language nothing more than
computable relations between finite strings.
On 2/6/26 9:16 PM, olcott wrote:
On 2/6/2026 7:54 PM, Richard Damon wrote:
On 2/6/26 8:13 PM, olcott wrote:
On 2/6/2026 6:23 PM, Richard Damon wrote:
What is an expression of language that has no meaning?
That is your problem, you can't figure out how to make it have the
meaning you want.
The language of mathematics FULLY understand Godel's statment, butyou > need it to have no meaning.
Either you accept the language of methematics, which assigns meaning
based on infinite sequences of operation, and thus is NOT compatible
with your proof-theoretic semantic, or you admit that your system can't actually handle the meaning of the langugae that you claim to handle.
Godel's proof shows the limitiation of Proof-Theoretic Semantics in a
mathematical system.
Not in the least little bit.
Sure it does, you are just to ignorant to understand the problem
He creates a perfectly semantic statement
In the wrong semantic system.
In other words, you ADMIT that your system can't handle the semantics of basic mathematics,
After all, the final statement is just expressed using the language of
basic mathematics.
as a relationship that can be finitely tested for any number. He then
ask a complete semantically reasonable question about that
relationship, does a number exist that satisfies it, making an
assertion that it does not.
This SHOULD have meaning in a system that understands the mathematics.
But, by the rules of proof-theoretic semantics, The statement can
only be considered to be true if we could prove it, and false if we
could prove its converse, or declared non-well-founded if we could
prove that neither of those can be shown,
Yes.
So, you ADMIT that you system is self-contradictory, by using Truth- Condition semantics but also rejects them.
*THE MEANING IN ALL FORMAL MATHEMATICAL AND NATURAL LANGUAGES*
Then you accept the Formal Logic definition of "Truth" that means
what can be demonstrated by an infinite sequence of steps, and thus
reject you proposition that something is only true if it can be proven.
We have been over this quite a few time now.
Anything requiring an infinite number of steps
is outside the domain of knowledge.
But not out of the domain of TRUTH.
And our domain of knowledge accepts that domain of truth.
Your problem is you don't understand the difference, because you just
don't understand truth.
<snip>
*Good job, you got the most important point exactly correctly*
Non-well-founded means not truth-apt.
So, you accept that your "Proof-Theoretic" system uses Truth-
Conditional meaning, as you can't actually PROVE that something is
Truth_Apt.
ABSOLUTELY NOT EVER. WHERE THE HELL DID YOU GET THAT ???
ABSOLUTELY NOT EVER. WHERE THE HELL DID YOU GET THAT ???
ABSOLUTELY NOT EVER. WHERE THE HELL DID YOU GET THAT ???
How does it determine that something is not well founded if you can't
prove it.
Your "Truth-Apt" needs Truth-Conditional interpretation or your last statement can't be correct.
The problem is that while you can determine SOME conditions that make a statement True, you can't determine if something is ~True, unless you
can can actually prove that converse.
"true on the basis of meaning expressed in language"
Including formal mathematical languages and natural language
Where the meaning of expressions is only defined in terms
of other expressions makes the entire body of knowledge
that can be expressed in language nothing more than
computable relations between finite strings.
Then it accept Truth by infinite sequence of operations.
And thus it accepts that Godel's G statement is true in PA, as by
testing ALL Natural Numbers (an infinite number of numbers to test) we
see that none of them satisfies the statement.
Note, the Relationship being tested is just defined by normal
mathematical operations, fully defined in PA. (It was BUILT in the meta- math, but to only use the operations defined in basic PA).
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really define.
It is an axiom: reCx (Provable(x) rcA True(x))
On 2/6/2026 3:01 AM, Mikko wrote:
On 05/02/2026 18:55, olcott wrote:
Changing the foundational basis to Proof Theoretic Semantics
Tarski Undefinability is overcome
x ree Provable rco x ree True // proof theoretic semantics
A definition in terms of an undefined symbol does not really define.
It is an axiom: reCx (Provable(x) rcA True(x))
On 2/6/2026 8:33 PM, Richard Damon wrote:
On 2/6/26 9:16 PM, olcott wrote:
On 2/6/2026 7:54 PM, Richard Damon wrote:
On 2/6/26 8:13 PM, olcott wrote:
On 2/6/2026 6:23 PM, Richard Damon wrote:
<snip>
What is an expression of language that has no meaning?
That is your problem, you can't figure out how to make it have the
meaning you want.
{Meaningless} did not occur to you?
you > need it to have no meaning.
The language of mathematics FULLY understand Godel's statment, but
Truth Conditional Semantics fails.
Proof Theoretic Semantics succeeds.
Either you accept the language of methematics, which assigns meaning
based on infinite sequences of operation, and thus is NOT compatible
with your proof-theoretic semantic, or you admit that your system
can't actually handle the meaning of the langugae that you claim to
handle.
If these are not algorithmically compressed
that remain outside of the body of knowledge.
<snip>
Godel's proof shows the limitiation of Proof-Theoretic Semantics in
a mathematical system.
Not in the least little bit.
Sure it does, you are just to ignorant to understand the problem
He creates a perfectly semantic statement
In the wrong semantic system.
In other words, you ADMIT that your system can't handle the semantics
of basic mathematics,
It utterly replaces Truth conditional semantics with
proof theoretic semantics.
After all, the final statement is just expressed using the language of
basic mathematics.
Not at all. The truth predicates have entirely
different basis.
as a relationship that can be finitely tested for any number. He
then ask a complete semantically reasonable question about that
relationship, does a number exist that satisfies it, making an
assertion that it does not.
This SHOULD have meaning in a system that understands the mathematics. >>>>
But, by the rules of proof-theoretic semantics, The statement can
only be considered to be true if we could prove it, and false if we
could prove its converse, or declared non-well-founded if we could
prove that neither of those can be shown,
Yes.
So, you ADMIT that you system is self-contradictory, by using Truth-
Condition semantics but also rejects them.
My system has always used "proof theoretic semantics"
and this is many years before I even knew that term.
<snip>
*THE MEANING IN ALL FORMAL MATHEMATICAL AND NATURAL LANGUAGES*
Then you accept the Formal Logic definition of "Truth" that means
what can be demonstrated by an infinite sequence of steps, and thus
reject you proposition that something is only true if it can be proven. >>>>
We have been over this quite a few time now.
Anything requiring an infinite number of steps
is outside the domain of knowledge.
But not out of the domain of TRUTH.
And our domain of knowledge accepts that domain of truth.
Your problem is you don't understand the difference, because you just
don't understand truth.
We have two entirely different foundations of truth
Truth conditional semantics
proof theoretic semantics
There truth predicates have an entirely different basis.
<snip>
*Good job, you got the most important point exactly correctly*
Non-well-founded means not truth-apt.
So, you accept that your "Proof-Theoretic" system uses Truth-
Conditional meaning, as you can't actually PROVE that something is
Truth_Apt.
ABSOLUTELY NOT EVER. WHERE THE HELL DID YOU GET THAT ???
ABSOLUTELY NOT EVER. WHERE THE HELL DID YOU GET THAT ???
ABSOLUTELY NOT EVER. WHERE THE HELL DID YOU GET THAT ???
How does it determine that something is not well founded if you can't
prove it.
Your "Truth-Apt" needs Truth-Conditional interpretation or your last
statement can't be correct.
It does not need Truth-Conditional interpretation
Truth-Conditional interpretation has always been
totally wrong-headed.
The problem is that while you can determine SOME conditions that make
a statement True, you can't determine if something is ~True, unless
you can can actually prove that converse.
~Provable(x) rcA ~True(x)
<snip>
"true on the basis of meaning expressed in language"
Including formal mathematical languages and natural language
Where the meaning of expressions is only defined in terms
of other expressions makes the entire body of knowledge
that can be expressed in language nothing more than
computable relations between finite strings.
Then it accept Truth by infinite sequence of operations.
It accepts that they exist and rejects that their result is
in the domain of knowledge.
And thus it accepts that Godel's G statement is true in PA, as by
testing ALL Natural Numbers (an infinite number of numbers to test) we
see that none of them satisfies the statement.
reCx (Provable(PA, x) rcA True(PA, x))
Note, the Relationship being tested is just defined by normal
mathematical operations, fully defined in PA. (It was BUILT in the
meta- math, but to only use the operations defined in basic PA).
Truth conditional semantics is model theory that has
been rejected as erroneous. G||del Incompleteness
cannot exist in Proof Theoretic Semantics.
When I refer to a formal system I am referring to
Russell's atomic facts written down and placed in
a simple Type Hierarchy.
You keep on thinking it is just a form of Philosophy, which it really
isn't.
You don't understand what Languages is as you think words are pliable
and can be changed.
On 07/02/2026 00:10, olcott wrote:
When I refer to a formal system I am referring to
Russell's atomic facts written down and placed in
a simple Type Hierarchy.
Is that, in effect, the conventional meaning of "formal system"? It is
not normally expressed so, see Curry and Feys.
On 07/02/2026 00:23, Richard Damon wrote:
You keep on thinking it is just a form of Philosophy, which it really isn't.
It's at least a form of a subset of philosophy because it's semantical.
The message body is Copyright (C) 2026 Tristan Wibberley exceptcitations and quotations noted. All Rights Reserved except that you may,
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