The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
*Proof Theoretic Semantics Blocks Pathological Self-Reference* https://philpapers.org/archive/OLCPTS.pdf
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
*Proof Theoretic Semantics Blocks Pathological Self-Reference*
https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies.
Your problem is that you system is based on a criteria that matches your
own definition of non-well-founded.
It seems that for many of the system you want to talk about, it is non- well-founded if statements are in fact non-well-founded because you--
can't KNOW if a proof exists (but isn't known yet) of the statement or
its negation.
This collapse your whole system into a ball of meaningless unless you restrict it to "toy" level where you can prove if a proof can exist.
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
*Proof Theoretic Semantics Blocks Pathological Self-Reference*
https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies.
You have to actually read the paper.
Your problem is that you system is based on a criteria that matches
your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics?
In proofrCatheoretic semantics, a statement is not wellrCafounded when its justification cannot be grounded in a finite, wellrCastructured chain of inferential steps. It lacks a terminating, wellrCaordered proof tree that would normally establish its truth or falsity. This often happens with selfrCareferential or circular statements whose rCLproofsrCY loop back on themselves rather than bottoming out in basic axioms or introduction
rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot well- foundedrCY means that the structure used to define or justify meanings
does not rest on a base case that is independent of itself. Instead, it involves circular or infinitely descending dependencies among rules or proofs. // ChatGPT
In proof-theoretic semantics, not well-founded typically refers to derivations or proof structures that contain infinite descending chains
or circular dependencies, violating the well-foundedness property.
In classical proof theory, well-founded derivations have a clear hierarchical structure where every inference rule application depends
only on "smaller" or "simpler" premises, eventually bottoming out in
axioms or basic rules. This ensures that proofs are finitely
constructible and verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom P is
called well-founded if every chain of successive
"definitions" (unfoldings) eventually terminates rCo i.e., there is no infinite descending chain of definitional dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in logical structure after finitely many unfoldings. // Grok
It seems that for many of the system you want to talk about, it is
non- well-founded if statements are in fact non-well-founded because
you can't KNOW if a proof exists (but isn't known yet) of the
statement or its negation.
This collapse your whole system into a ball of meaningless unless you
restrict it to "toy" level where you can prove if a proof can exist.
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
*Proof Theoretic Semantics Blocks Pathological Self-Reference*
https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies.
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your syste,/
Your problem is that you system is based on a criteria that matches
your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics?
So. how is your definition of the criteria to be non-well-founded not non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your system.
In proofrCatheoretic semantics, a statement is not wellrCafounded when its >> justification cannot be grounded in a finite, wellrCastructured chain of
inferential steps. It lacks a terminating, wellrCaordered proof tree
that would normally establish its truth or falsity. This often happens
with selfrCareferential or circular statements whose rCLproofsrCY loop back >> on themselves rather than bottoming out in basic axioms or
introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot well-
foundedrCY means that the structure used to define or justify meanings
does not rest on a base case that is independent of itself. Instead,
it involves circular or infinitely descending dependencies among rules
or proofs. // ChatGPT
In proof-theoretic semantics, not well-founded typically refers to
derivations or proof structures that contain infinite descending
chains or circular dependencies, violating the well-foundedness property.
In classical proof theory, well-founded derivations have a clear
hierarchical structure where every inference rule application depends
only on "smaller" or "simpler" premises, eventually bottoming out in
axioms or basic rules. This ensures that proofs are finitely
constructible and verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom P is
called well-founded if every chain of successive "definitions"
(unfoldings) eventually terminates rCo i.e., there is no infinite
descending chain of definitional dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in logical
structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
*Proof Theoretic Semantics Blocks Pathological Self-Reference*
https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies.
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your syste,/
Your problem is that you system is based on a criteria that matches
your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics?
So. how is your definition of the criteria to be non-well-founded not
non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your system.
In proofrCatheoretic semantics, a statement is not wellrCafounded when
its justification cannot be grounded in a finite, wellrCastructured
chain of inferential steps. It lacks a terminating, wellrCaordered
proof tree that would normally establish its truth or falsity. This
often happens with selfrCareferential or circular statements whose
rCLproofsrCY loop back on themselves rather than bottoming out in basic >>> axioms or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot well-
foundedrCY means that the structure used to define or justify meanings
does not rest on a base case that is independent of itself. Instead,
it involves circular or infinitely descending dependencies among
rules or proofs. // ChatGPT
In proof-theoretic semantics, not well-founded typically refers to
derivations or proof structures that contain infinite descending
chains or circular dependencies, violating the well-foundedness
property.
In classical proof theory, well-founded derivations have a clear
hierarchical structure where every inference rule application depends
only on "smaller" or "simpler" premises, eventually bottoming out in
axioms or basic rules. This ensures that proofs are finitely
constructible and verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom P is
called well-founded if every chain of successive
"definitions" (unfoldings) eventually terminates rCo i.e., there is no
infinite descending chain of definitional dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in logical
structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
*Proof Theoretic Semantics Blocks Pathological Self-Reference*
https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies.
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your syste,/
Your problem is that you system is based on a criteria that matches >>>>> your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics?
So. how is your definition of the criteria to be non-well-founded not
non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your system.
In proofrCatheoretic semantics, a statement is not wellrCafounded when >>>> its justification cannot be grounded in a finite, wellrCastructured
chain of inferential steps. It lacks a terminating, wellrCaordered
proof tree that would normally establish its truth or falsity. This
often happens with selfrCareferential or circular statements whose
rCLproofsrCY loop back on themselves rather than bottoming out in basic >>>> axioms or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot well-
foundedrCY means that the structure used to define or justify meanings >>>> does not rest on a base case that is independent of itself. Instead,
it involves circular or infinitely descending dependencies among
rules or proofs. // ChatGPT
In proof-theoretic semantics, not well-founded typically refers to
derivations or proof structures that contain infinite descending
chains or circular dependencies, violating the well-foundedness
property.
In classical proof theory, well-founded derivations have a clear
hierarchical structure where every inference rule application
depends only on "smaller" or "simpler" premises, eventually
bottoming out in axioms or basic rules. This ensures that proofs are
finitely constructible and verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom P is
called well-founded if every chain of successive "definitions"
(unfoldings) eventually terminates rCo i.e., there is no infinite
descending chain of definitional dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in logical
structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body of knowledge" is itself not built on those semantics,
It is a problem trying to process "knowledge" based on a different logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for instance
will assert that the Goldbach Conjecture is one of the great puzzles of mathematics, and must either be true or false, but that FACT is
incompatible with proof-theoretic semantics, as mathematics can show
that some true statements do not have proofs in the system.
Thus, your system colapses in a contradiction that the statement might
be not-well-founded, but that classification might be not-well-founded,
and that determination may be not-well-founded, and so on, so your
attempt to define you system runs into a possibly infinite loop of
asking if we can even talk about the statement.
If you disagree, it falls upon YOU to figure out how to handle that
issue, you can't just assume it can be done.
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
*Proof Theoretic Semantics Blocks Pathological Self-Reference*
https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies.
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your syste,/ >>>>
Your problem is that you system is based on a criteria that
matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics?
So. how is your definition of the criteria to be non-well-founded
not non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your system.
In proofrCatheoretic semantics, a statement is not wellrCafounded when >>>>> its justification cannot be grounded in a finite, wellrCastructured >>>>> chain of inferential steps. It lacks a terminating, wellrCaordered
proof tree that would normally establish its truth or falsity. This >>>>> often happens with selfrCareferential or circular statements whose
rCLproofsrCY loop back on themselves rather than bottoming out in basic >>>>> axioms or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot well- >>>>> foundedrCY means that the structure used to define or justify
meanings does not rest on a base case that is independent of
itself. Instead, it involves circular or infinitely descending
dependencies among rules or proofs. // ChatGPT
In proof-theoretic semantics, not well-founded typically refers to
derivations or proof structures that contain infinite descending
chains or circular dependencies, violating the well-foundedness
property.
In classical proof theory, well-founded derivations have a clear
hierarchical structure where every inference rule application
depends only on "smaller" or "simpler" premises, eventually
bottoming out in axioms or basic rules. This ensures that proofs
are finitely constructible and verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom P is >>>>> called well-founded if every chain of successive
"definitions" (unfoldings) eventually terminates rCo i.e., there is >>>>> no infinite descending chain of definitional dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in
logical structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body of
knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a different
logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for instance
will assert that the Goldbach Conjecture is one of the great puzzles
of mathematics, and must either be true or false, but that FACT is
incompatible with proof-theoretic semantics, as mathematics can show
that some true statements do not have proofs in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Thus, your system colapses in a contradiction that the statement might
be not-well-founded, but that classification might be not-well-
founded, and that determination may be not-well-founded, and so on, so
your attempt to define you system runs into a possibly infinite loop
of asking if we can even talk about the statement.
My paper already explains all of the details of that.
Proof Theoretic Semantics Blocks Pathological Self-Reference https://philpapers.org/archive/OLCPTS.pdf
If you disagree, it falls upon YOU to figure out how to handle that
issue, you can't just assume it can be done.
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
*Proof Theoretic Semantics Blocks Pathological Self-Reference* >>>>>>>> https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies.
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your syste,/ >>>>>
Your problem is that you system is based on a criteria that
matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics?
So. how is your definition of the criteria to be non-well-founded
not non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your system. >>>>>
In proofrCatheoretic semantics, a statement is not wellrCafounded when >>>>>> its justification cannot be grounded in a finite, wellrCastructured >>>>>> chain of inferential steps. It lacks a terminating, wellrCaordered >>>>>> proof tree that would normally establish its truth or falsity.
This often happens with selfrCareferential or circular statements >>>>>> whose rCLproofsrCY loop back on themselves rather than bottoming out >>>>>> in basic axioms or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot well- >>>>>> foundedrCY means that the structure used to define or justify
meanings does not rest on a base case that is independent of
itself. Instead, it involves circular or infinitely descending
dependencies among rules or proofs. // ChatGPT
In proof-theoretic semantics, not well-founded typically refers to >>>>>> derivations or proof structures that contain infinite descending
chains or circular dependencies, violating the well-foundedness
property.
In classical proof theory, well-founded derivations have a clear
hierarchical structure where every inference rule application
depends only on "smaller" or "simpler" premises, eventually
bottoming out in axioms or basic rules. This ensures that proofs
are finitely constructible and verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom P
is called well-founded if every chain of successive "definitions" >>>>>> (unfoldings) eventually terminates rCo i.e., there is no infinite >>>>>> descending chain of definitional dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in
logical structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body of
knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a different
logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for instance
will assert that the Goldbach Conjecture is one of the great puzzles
of mathematics, and must either be true or false, but that FACT is
incompatible with proof-theoretic semantics, as mathematics can show
that some true statements do not have proofs in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own classification
of things.
So, it can't talk about things like Global Warming, or f the Earth is
Round.
Thus, your system colapses in a contradiction that the statement
might be not-well-founded, but that classification might be not-well-
founded, and that determination may be not-well-founded, and so on,
so your attempt to define you system runs into a possibly infinite
loop of asking if we can even talk about the statement.
My paper already explains all of the details of that.
Proof Theoretic Semantics Blocks Pathological Self-Reference
https://philpapers.org/archive/OLCPTS.pdf
WHERE???
You have a less than one page prompt that defines what you are thinking of.
Everything after that is LLM garbage making comments of what you said.
I guess you are building a theory of nothing.
You are trying to define what is "true", but not a system that it works
in, which means you haven't actually shown it can do anything.
You are talking Philosophy, not Formal Logic.
If you disagree, it falls upon YOU to figure out how to handle that
issue, you can't just assume it can be done.
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences. >>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self-Reference* >>>>>>>>> https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies.
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your
syste,/
Your problem is that you system is based on a criteria that
matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics?
So. how is your definition of the criteria to be non-well-founded >>>>>> not non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your system. >>>>>>
In proofrCatheoretic semantics, a statement is not wellrCafounded >>>>>>> when its justification cannot be grounded in a finite,
wellrCastructured chain of inferential steps. It lacks a
terminating, wellrCaordered proof tree that would normally
establish its truth or falsity. This often happens with
selfrCareferential or circular statements whose rCLproofsrCY loop back >>>>>>> on themselves rather than bottoming out in basic axioms or
introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot well- >>>>>>> foundedrCY means that the structure used to define or justify
meanings does not rest on a base case that is independent of
itself. Instead, it involves circular or infinitely descending
dependencies among rules or proofs. // ChatGPT
In proof-theoretic semantics, not well-founded typically refers >>>>>>> to derivations or proof structures that contain infinite
descending chains or circular dependencies, violating the well- >>>>>>> foundedness property.
In classical proof theory, well-founded derivations have a clear >>>>>>> hierarchical structure where every inference rule application
depends only on "smaller" or "simpler" premises, eventually
bottoming out in axioms or basic rules. This ensures that proofs >>>>>>> are finitely constructible and verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom P >>>>>>> is called well-founded if every chain of successive
"definitions" (unfoldings) eventually terminates rCo i.e., there is >>>>>>> no infinite descending chain of definitional dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in
logical structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body of
knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a different
logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for
instance will assert that the Goldbach Conjecture is one of the
great puzzles of mathematics, and must either be true or false, but
that FACT is incompatible with proof-theoretic semantics, as
mathematics can show that some true statements do not have proofs in
the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
We do not get the psychotic nonsense that
(A & ~A) Proves that Donald Trump is Jesus the Christ.
the principle of explosion is the law according to
which any statement can be proven from a contradiction.
https://en.wikipedia.org/wiki/Principle_of_explosion
*Proof Theoretic Semantics Blocks Pathological Self-Reference* https://philpapers.org/archive/OLCPTS.pdf
Furthermore all undecidability and incompleteness is blocked.
So, it can't talk about things like Global Warming, or f the Earth is
Round.
Thus, your system colapses in a contradiction that the statement
might be not-well-founded, but that classification might be not-
well- founded, and that determination may be not-well-founded, and
so on, so your attempt to define you system runs into a possibly
infinite loop of asking if we can even talk about the statement.
My paper already explains all of the details of that.
Proof Theoretic Semantics Blocks Pathological Self-Reference
https://philpapers.org/archive/OLCPTS.pdf
WHERE???
You have a less than one page prompt that defines what you are
thinking of.
Everything after that is LLM garbage making comments of what you said.
I guess you are building a theory of nothing.
You are trying to define what is "true", but not a system that it
works in, which means you haven't actually shown it can do anything.
You are talking Philosophy, not Formal Logic.
If you disagree, it falls upon YOU to figure out how to handle that
issue, you can't just assume it can be done.
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences. >>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self-Reference* >>>>>>>>>> https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your
syste,/
Your problem is that you system is based on a criteria that >>>>>>>>> matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics?
So. how is your definition of the criteria to be non-well-founded >>>>>>> not non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your system. >>>>>>>
In proofrCatheoretic semantics, a statement is not wellrCafounded >>>>>>>> when its justification cannot be grounded in a finite,
wellrCastructured chain of inferential steps. It lacks a
terminating, wellrCaordered proof tree that would normally
establish its truth or falsity. This often happens with
selfrCareferential or circular statements whose rCLproofsrCY loop back
on themselves rather than bottoming out in basic axioms or
introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>> well- foundedrCY means that the structure used to define or
justify meanings does not rest on a base case that is
independent of itself. Instead, it involves circular or
infinitely descending dependencies among rules or proofs. //
ChatGPT
In proof-theoretic semantics, not well-founded typically refers >>>>>>>> to derivations or proof structures that contain infinite
descending chains or circular dependencies, violating the well- >>>>>>>> foundedness property.
In classical proof theory, well-founded derivations have a clear >>>>>>>> hierarchical structure where every inference rule application >>>>>>>> depends only on "smaller" or "simpler" premises, eventually
bottoming out in axioms or basic rules. This ensures that proofs >>>>>>>> are finitely constructible and verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom P >>>>>>>> is called well-founded if every chain of successive
"definitions" (unfoldings) eventually terminates rCo i.e., there >>>>>>>> is no infinite descending chain of definitional dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in
logical structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body of
knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a different
logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for
instance will assert that the Goldbach Conjecture is one of the
great puzzles of mathematics, and must either be true or false, but >>>>> that FACT is incompatible with proof-theoretic semantics, as
mathematics can show that some true statements do not have proofs
in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just defective.
We do not get the psychotic nonsense that
(A & ~A) Proves that Donald Trump is Jesus the Christ.
Which only happens in incoherent systems like yours.
the principle of explosion is the law according to
which any statement can be proven from a contradiction.
No, it says that if a systems says that a contradiction can be proven
true, then you can prove anything you want in the system.
Remember, a PROOF must be based on true statements. Thus to prove
something from a contradiction means the contradiction must have first
been proven to be true (in the system).
https://en.wikipedia.org/wiki/Principle_of_explosion
*Proof Theoretic Semantics Blocks Pathological Self-Reference*
https://philpapers.org/archive/OLCPTS.pdf
Furthermore all undecidability and incompleteness is blocked.
Nope, A Proof Theoretic Semantic system will still explode if it can
prove a contradiction.
The proof of the law of the principle of explosion works in Proof-
Theoretic Semantics.
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences. >>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self-Reference* >>>>>>>>>> https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your
syste,/
Your problem is that you system is based on a criteria that >>>>>>>>> matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics?
So. how is your definition of the criteria to be non-well-founded >>>>>>> not non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your system. >>>>>>>
In proofrCatheoretic semantics, a statement is not wellrCafounded >>>>>>>> when its justification cannot be grounded in a finite,
wellrCastructured chain of inferential steps. It lacks a
terminating, wellrCaordered proof tree that would normally
establish its truth or falsity. This often happens with
selfrCareferential or circular statements whose rCLproofsrCY loop back
on themselves rather than bottoming out in basic axioms or
introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>> well- foundedrCY means that the structure used to define or
justify meanings does not rest on a base case that is
independent of itself. Instead, it involves circular or
infinitely descending dependencies among rules or proofs. //
ChatGPT
In proof-theoretic semantics, not well-founded typically refers >>>>>>>> to derivations or proof structures that contain infinite
descending chains or circular dependencies, violating the well- >>>>>>>> foundedness property.
In classical proof theory, well-founded derivations have a clear >>>>>>>> hierarchical structure where every inference rule application >>>>>>>> depends only on "smaller" or "simpler" premises, eventually
bottoming out in axioms or basic rules. This ensures that proofs >>>>>>>> are finitely constructible and verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom P >>>>>>>> is called well-founded if every chain of successive
"definitions" (unfoldings) eventually terminates rCo i.e., there >>>>>>>> is no infinite descending chain of definitional dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in
logical structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body of
knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a different
logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for
instance will assert that the Goldbach Conjecture is one of the
great puzzles of mathematics, and must either be true or false, but >>>>> that FACT is incompatible with proof-theoretic semantics, as
mathematics can show that some true statements do not have proofs
in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just defective.
On 1/16/2026 5:21 PM, Richard Damon wrote:
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes >>>>>>>>>>> self-referential constructions such as G||del-type sentences. >>>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self-Reference* >>>>>>>>>>> https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your >>>>>>>> syste,/
So. how is your definition of the criteria to be non-well-
Your problem is that you system is based on a criteria that >>>>>>>>>> matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics? >>>>>>>>
founded not non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your
system.
In proofrCatheoretic semantics, a statement is not wellrCafounded >>>>>>>>> when its justification cannot be grounded in a finite,
wellrCastructured chain of inferential steps. It lacks a
terminating, wellrCaordered proof tree that would normally
establish its truth or falsity. This often happens with
selfrCareferential or circular statements whose rCLproofsrCY loop >>>>>>>>> back on themselves rather than bottoming out in basic axioms or >>>>>>>>> introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>>> well- foundedrCY means that the structure used to define or >>>>>>>>> justify meanings does not rest on a base case that is
independent of itself. Instead, it involves circular or
infinitely descending dependencies among rules or proofs. // >>>>>>>>> ChatGPT
In proof-theoretic semantics, not well-founded typically refers >>>>>>>>> to derivations or proof structures that contain infinite
descending chains or circular dependencies, violating the well- >>>>>>>>> foundedness property.
In classical proof theory, well-founded derivations have a
clear hierarchical structure where every inference rule
application depends only on "smaller" or "simpler" premises, >>>>>>>>> eventually bottoming out in axioms or basic rules. This ensures >>>>>>>>> that proofs are finitely constructible and verifiable. //
Claude AI
A set of introduction rules (definitional clauses) for an atom >>>>>>>>> P is called well-founded if every chain of successive
"definitions" (unfoldings) eventually terminates rCo i.e., there >>>>>>>>> is no infinite descending chain of definitional dependencies. >>>>>>>>> Intuitively:
The meaning of P is ultimately grounded in basic facts or in >>>>>>>>> logical structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body of >>>>>> knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a different >>>>>> logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for
instance will assert that the Goldbach Conjecture is one of the
great puzzles of mathematics, and must either be true or false,
but that FACT is incompatible with proof-theoretic semantics, as
mathematics can show that some true statements do not have proofs >>>>>> in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just defective.
We do not get the psychotic nonsense that
(A & ~A) Proves that Donald Trump is Jesus the Christ.
Which only happens in incoherent systems like yours.
the principle of explosion is the law according to
which any statement can be proven from a contradiction.
No, it says that if a systems says that a contradiction can be proven
true, then you can prove anything you want in the system.
I quoted the words that it said sheep dip !!!
Remember, a PROOF must be based on true statements. Thus to prove
something from a contradiction means the contradiction must have first
been proven to be true (in the system).
https://en.wikipedia.org/wiki/Principle_of_explosion
*Proof Theoretic Semantics Blocks Pathological Self-Reference*
https://philpapers.org/archive/OLCPTS.pdf
Furthermore all undecidability and incompleteness is blocked.
Nope, A Proof Theoretic Semantic system will still explode if it can
prove a contradiction.
The proof of the law of the principle of explosion works in Proof-
Theoretic Semantics.
No sheep dip it does not.
When we merely assume the axioms of a proof-theoretic
formal system are PA then incompleteness goes away
for PA.
On 1/16/2026 5:21 PM, Richard Damon wrote:
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes >>>>>>>>>>> self-referential constructions such as G||del-type sentences. >>>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self-Reference* >>>>>>>>>>> https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your >>>>>>>> syste,/
So. how is your definition of the criteria to be non-well-
Your problem is that you system is based on a criteria that >>>>>>>>>> matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics? >>>>>>>>
founded not non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your
system.
In proofrCatheoretic semantics, a statement is not wellrCafounded >>>>>>>>> when its justification cannot be grounded in a finite,
wellrCastructured chain of inferential steps. It lacks a
terminating, wellrCaordered proof tree that would normally
establish its truth or falsity. This often happens with
selfrCareferential or circular statements whose rCLproofsrCY loop >>>>>>>>> back on themselves rather than bottoming out in basic axioms or >>>>>>>>> introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>>> well- foundedrCY means that the structure used to define or >>>>>>>>> justify meanings does not rest on a base case that is
independent of itself. Instead, it involves circular or
infinitely descending dependencies among rules or proofs. // >>>>>>>>> ChatGPT
In proof-theoretic semantics, not well-founded typically refers >>>>>>>>> to derivations or proof structures that contain infinite
descending chains or circular dependencies, violating the well- >>>>>>>>> foundedness property.
In classical proof theory, well-founded derivations have a
clear hierarchical structure where every inference rule
application depends only on "smaller" or "simpler" premises, >>>>>>>>> eventually bottoming out in axioms or basic rules. This ensures >>>>>>>>> that proofs are finitely constructible and verifiable. //
Claude AI
A set of introduction rules (definitional clauses) for an atom >>>>>>>>> P is called well-founded if every chain of successive
"definitions" (unfoldings) eventually terminates rCo i.e., there >>>>>>>>> is no infinite descending chain of definitional dependencies. >>>>>>>>> Intuitively:
The meaning of P is ultimately grounded in basic facts or in >>>>>>>>> logical structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body of >>>>>> knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a different >>>>>> logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for
instance will assert that the Goldbach Conjecture is one of the
great puzzles of mathematics, and must either be true or false,
but that FACT is incompatible with proof-theoretic semantics, as
mathematics can show that some true statements do not have proofs >>>>>> in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just defective.
A sentence is meaningful only if its justification graph
is wellrCafounded. A wellrCafounded graph always has a terminating evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
nonrCawellrCafounded has no terminating evaluation, so it is
not meaningful and not truthrCaapt. Therefore truth is total
and computable over the meaningful fragment.
On 1/16/26 8:27 PM, olcott wrote:
On 1/16/2026 5:21 PM, Richard Damon wrote:
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes >>>>>>>>>>>> self-referential constructions such as G||del-type sentences. >>>>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self-Reference* >>>>>>>>>>>> https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your >>>>>>>>> syste,/
So. how is your definition of the criteria to be non-well-
Your problem is that you system is based on a criteria that >>>>>>>>>>> matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics? >>>>>>>>>
founded not non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your >>>>>>>>> system.
In proofrCatheoretic semantics, a statement is not wellrCafounded >>>>>>>>>> when its justification cannot be grounded in a finite,
wellrCastructured chain of inferential steps. It lacks a
terminating, wellrCaordered proof tree that would normally >>>>>>>>>> establish its truth or falsity. This often happens with
selfrCareferential or circular statements whose rCLproofsrCY loop >>>>>>>>>> back on themselves rather than bottoming out in basic axioms >>>>>>>>>> or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>>>> well- foundedrCY means that the structure used to define or >>>>>>>>>> justify meanings does not rest on a base case that is
independent of itself. Instead, it involves circular or
infinitely descending dependencies among rules or proofs. // >>>>>>>>>> ChatGPT
In proof-theoretic semantics, not well-founded typically
refers to derivations or proof structures that contain
infinite descending chains or circular dependencies, violating >>>>>>>>>> the well- foundedness property.
In classical proof theory, well-founded derivations have a >>>>>>>>>> clear hierarchical structure where every inference rule
application depends only on "smaller" or "simpler" premises, >>>>>>>>>> eventually bottoming out in axioms or basic rules. This
ensures that proofs are finitely constructible and
verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom >>>>>>>>>> P is called well-founded if every chain of successive
"definitions" (unfoldings) eventually terminates rCo i.e., there >>>>>>>>>> is no infinite descending chain of definitional dependencies. >>>>>>>>>> Intuitively:
The meaning of P is ultimately grounded in basic facts or in >>>>>>>>>> logical structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making >>>>>>>> "true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body >>>>>>> of knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a
different logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for
instance will assert that the Goldbach Conjecture is one of the >>>>>>> great puzzles of mathematics, and must either be true or false, >>>>>>> but that FACT is incompatible with proof-theoretic semantics, as >>>>>>> mathematics can show that some true statements do not have proofs >>>>>>> in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just defective.
A sentence is meaningful only if its justification graph
is wellrCafounded. A wellrCafounded graph always has a terminating
evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
nonrCawellrCafounded has no terminating evaluation, so it is
not meaningful and not truthrCaapt. Therefore truth is total
and computable over the meaningful fragment.
And thus your criteria for well-foundedness isn't itself well founded.
This is the problem of trying to redefine "truth" to be something other
than what it is.
The problem is, there are statements you can't show that they ARE not- well-founded, and thus you can't talk about them.
We can't tell of the Golfbach conjecture is well-founded or not, so your system ends up having many unkownable holes in it.
And because when you first want to pose the question, you likely don't
know if the answer will be available, or if it is in the realm of unprovable. This means your "logic" is mostly restrictred to talking
about what is already known, and is worthless for producing new knowledge.
It even has problem with much of the existing knowledge, as that is--
based on truth-conditional logic, so isn't even true anymore in your
system.
On 1/16/26 8:27 PM, olcott wrote:
On 1/16/2026 5:21 PM, Richard Damon wrote:
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes >>>>>>>>>>>> self-referential constructions such as G||del-type sentences. >>>>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self-Reference* >>>>>>>>>>>> https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your >>>>>>>>> syste,/
So. how is your definition of the criteria to be non-well-
Your problem is that you system is based on a criteria that >>>>>>>>>>> matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics? >>>>>>>>>
founded not non-well-founded for some questions?
Note, asking LLMs for a definition doesn't define it in your >>>>>>>>> system.
In proofrCatheoretic semantics, a statement is not wellrCafounded >>>>>>>>>> when its justification cannot be grounded in a finite,
wellrCastructured chain of inferential steps. It lacks a
terminating, wellrCaordered proof tree that would normally >>>>>>>>>> establish its truth or falsity. This often happens with
selfrCareferential or circular statements whose rCLproofsrCY loop >>>>>>>>>> back on themselves rather than bottoming out in basic axioms >>>>>>>>>> or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>>>> well- foundedrCY means that the structure used to define or >>>>>>>>>> justify meanings does not rest on a base case that is
independent of itself. Instead, it involves circular or
infinitely descending dependencies among rules or proofs. // >>>>>>>>>> ChatGPT
In proof-theoretic semantics, not well-founded typically
refers to derivations or proof structures that contain
infinite descending chains or circular dependencies, violating >>>>>>>>>> the well- foundedness property.
In classical proof theory, well-founded derivations have a >>>>>>>>>> clear hierarchical structure where every inference rule
application depends only on "smaller" or "simpler" premises, >>>>>>>>>> eventually bottoming out in axioms or basic rules. This
ensures that proofs are finitely constructible and
verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an atom >>>>>>>>>> P is called well-founded if every chain of successive
"definitions" (unfoldings) eventually terminates rCo i.e., there >>>>>>>>>> is no infinite descending chain of definitional dependencies. >>>>>>>>>> Intuitively:
The meaning of P is ultimately grounded in basic facts or in >>>>>>>>>> logical structure after finitely many unfoldings. // Grok
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making >>>>>>>> "true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body >>>>>>> of knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a
different logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for
instance will assert that the Goldbach Conjecture is one of the >>>>>>> great puzzles of mathematics, and must either be true or false, >>>>>>> but that FACT is incompatible with proof-theoretic semantics, as >>>>>>> mathematics can show that some true statements do not have proofs >>>>>>> in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just defective.
A sentence is meaningful only if its justification graph
is wellrCafounded. A wellrCafounded graph always has a terminating
evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
nonrCawellrCafounded has no terminating evaluation, so it is
not meaningful and not truthrCaapt. Therefore truth is total
and computable over the meaningful fragment.
And thus your criteria for well-foundedness isn't itself well founded.
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it.
A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T.
These are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
On 1/16/2026 9:24 PM, Richard Damon wrote:
On 1/16/26 8:27 PM, olcott wrote:
On 1/16/2026 5:21 PM, Richard Damon wrote:
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the >>>>>>>>>>>>> meaning of a statement is determined entirely by its >>>>>>>>>>>>> inferential role within a theory. A theory T consists >>>>>>>>>>>>> of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes >>>>>>>>>>>>> self-referential constructions such as G||del-type sentences. >>>>>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self-Reference* >>>>>>>>>>>>> https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your >>>>>>>>>> syste,/
So. how is your definition of the criteria to be non-well- >>>>>>>>>> founded not non-well-founded for some questions?
Your problem is that you system is based on a criteria that >>>>>>>>>>>> matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics? >>>>>>>>>>
Note, asking LLMs for a definition doesn't define it in your >>>>>>>>>> system.
In proofrCatheoretic semantics, a statement is not wellrCafounded >>>>>>>>>>> when its justification cannot be grounded in a finite,
wellrCastructured chain of inferential steps. It lacks a >>>>>>>>>>> terminating, wellrCaordered proof tree that would normally >>>>>>>>>>> establish its truth or falsity. This often happens with >>>>>>>>>>> selfrCareferential or circular statements whose rCLproofsrCY loop >>>>>>>>>>> back on themselves rather than bottoming out in basic axioms >>>>>>>>>>> or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>>>>> well- foundedrCY means that the structure used to define or >>>>>>>>>>> justify meanings does not rest on a base case that is
independent of itself. Instead, it involves circular or >>>>>>>>>>> infinitely descending dependencies among rules or proofs. // >>>>>>>>>>> ChatGPT
In proof-theoretic semantics, not well-founded typically >>>>>>>>>>> refers to derivations or proof structures that contain
infinite descending chains or circular dependencies,
violating the well- foundedness property.
In classical proof theory, well-founded derivations have a >>>>>>>>>>> clear hierarchical structure where every inference rule >>>>>>>>>>> application depends only on "smaller" or "simpler" premises, >>>>>>>>>>> eventually bottoming out in axioms or basic rules. This >>>>>>>>>>> ensures that proofs are finitely constructible and
verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an >>>>>>>>>>> atom P is called well-founded if every chain of successive >>>>>>>>>>> "definitions" (unfoldings) eventually terminates rCo i.e., >>>>>>>>>>> there is no infinite descending chain of definitional
dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in >>>>>>>>>>> logical structure after finitely many unfoldings. // Grok >>>>>>>>>>>
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making >>>>>>>>> "true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body >>>>>>>> of knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a
different logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for
instance will assert that the Goldbach Conjecture is one of the >>>>>>>> great puzzles of mathematics, and must either be true or false, >>>>>>>> but that FACT is incompatible with proof-theoretic semantics, as >>>>>>>> mathematics can show that some true statements do not have
proofs in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just defective. >>>>
A sentence is meaningful only if its justification graph
is wellrCafounded. A wellrCafounded graph always has a terminating
evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
nonrCawellrCafounded has no terminating evaluation, so it is
not meaningful and not truthrCaapt. Therefore truth is total
and computable over the meaningful fragment.
And thus your criteria for well-foundedness isn't itself well founded.
This is the problem of trying to redefine "truth" to be something
other than what it is.
The problem is, there are statements you can't show that they ARE not-
well-founded, and thus you can't talk about them.
We can't tell of the Golfbach conjecture is well-founded or not, so
your system ends up having many unkownable holes in it.
Within the set of knowledge that is
"true on the basis of meaning expressed in language"
the truth value of Goldbach is outside of the domain.
And because when you first want to pose the question, you likely don't
know if the answer will be available, or if it is in the realm of
unprovable. This means your "logic" is mostly restrictred to talking
about what is already known, and is worthless for producing new
knowledge.
It can catch all dangerous liars 100 million times a day.
It even has problem with much of the existing knowledge, as that is
based on truth-conditional logic, so isn't even true anymore in your
system.
On 1/16/2026 9:24 PM, Richard Damon wrote:
On 1/16/26 8:27 PM, olcott wrote:
On 1/16/2026 5:21 PM, Richard Damon wrote:
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the >>>>>>>>>>>>> meaning of a statement is determined entirely by its >>>>>>>>>>>>> inferential role within a theory. A theory T consists >>>>>>>>>>>>> of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes >>>>>>>>>>>>> self-referential constructions such as G||del-type sentences. >>>>>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self-Reference* >>>>>>>>>>>>> https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of your >>>>>>>>>> syste,/
So. how is your definition of the criteria to be non-well- >>>>>>>>>> founded not non-well-founded for some questions?
Your problem is that you system is based on a criteria that >>>>>>>>>>>> matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics? >>>>>>>>>>
Note, asking LLMs for a definition doesn't define it in your >>>>>>>>>> system.
In proofrCatheoretic semantics, a statement is not wellrCafounded >>>>>>>>>>> when its justification cannot be grounded in a finite,
wellrCastructured chain of inferential steps. It lacks a >>>>>>>>>>> terminating, wellrCaordered proof tree that would normally >>>>>>>>>>> establish its truth or falsity. This often happens with >>>>>>>>>>> selfrCareferential or circular statements whose rCLproofsrCY loop >>>>>>>>>>> back on themselves rather than bottoming out in basic axioms >>>>>>>>>>> or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>>>>> well- foundedrCY means that the structure used to define or >>>>>>>>>>> justify meanings does not rest on a base case that is
independent of itself. Instead, it involves circular or >>>>>>>>>>> infinitely descending dependencies among rules or proofs. // >>>>>>>>>>> ChatGPT
In proof-theoretic semantics, not well-founded typically >>>>>>>>>>> refers to derivations or proof structures that contain
infinite descending chains or circular dependencies,
violating the well- foundedness property.
In classical proof theory, well-founded derivations have a >>>>>>>>>>> clear hierarchical structure where every inference rule >>>>>>>>>>> application depends only on "smaller" or "simpler" premises, >>>>>>>>>>> eventually bottoming out in axioms or basic rules. This >>>>>>>>>>> ensures that proofs are finitely constructible and
verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an >>>>>>>>>>> atom P is called well-founded if every chain of successive >>>>>>>>>>> "definitions" (unfoldings) eventually terminates rCo i.e., >>>>>>>>>>> there is no infinite descending chain of definitional
dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in >>>>>>>>>>> logical structure after finitely many unfoldings. // Grok >>>>>>>>>>>
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making >>>>>>>>> "true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body >>>>>>>> of knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a
different logic than the logic you are trying to process it.
Also, part of our knowledge is about mathematics, which, for
instance will assert that the Goldbach Conjecture is one of the >>>>>>>> great puzzles of mathematics, and must either be true or false, >>>>>>>> but that FACT is incompatible with proof-theoretic semantics, as >>>>>>>> mathematics can show that some true statements do not have
proofs in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just defective. >>>>
A sentence is meaningful only if its justification graph
is wellrCafounded. A wellrCafounded graph always has a terminating
evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
nonrCawellrCafounded has no terminating evaluation, so it is
not meaningful and not truthrCaapt. Therefore truth is total
and computable over the meaningful fragment.
And thus your criteria for well-foundedness isn't itself well founded.
You cannot possibly show that.
It is actually the same semantic tautology
that I have been saying for years.
On 16/01/2026 19:47, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it.
Usually the expression "is a theorem of T" is used instead of "is true
in T". THe words "true" and "false" are usually reserved for truth in
a particular interpretation.
A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T.
Usually the expression "undecidable in T" is used instead of
"neither true nor false in T".
A theory where some statements that are neither provable or disprovable
is said to be incomplere.
These are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
G||del's sentence is a sentence of Peano arithmetic. When inpterpreted according to the arithmentic semantics it is not self-referential.
Peano's postulates just are insufficient for the proof of G||del's
sentence. The simlest fis is to simply add G||del's sentence to the postulates. This addition does not create any inconsistency. However,
the new theory is still incomlete.
One should also note that for some theories there is no way to determine whether a claim is provable. A simple example is the group theory.
On 1/16/26 10:42 PM, olcott wrote:
On 1/16/2026 9:24 PM, Richard Damon wrote:
On 1/16/26 8:27 PM, olcott wrote:
On 1/16/2026 5:21 PM, Richard Damon wrote:
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the >>>>>>>>>>>>>> meaning of a statement is determined entirely by its >>>>>>>>>>>>>> inferential role within a theory. A theory T consists >>>>>>>>>>>>>> of a finite set of basic statements together with
everything that can be derived from them using the >>>>>>>>>>>>>> inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded >>>>>>>>>>>>>> in a finite, well-founded proof structure. This includes >>>>>>>>>>>>>> self-referential constructions such as G||del-type sentences. >>>>>>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self- >>>>>>>>>>>>>> Reference*
https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of >>>>>>>>>>> your syste,/
So. how is your definition of the criteria to be non-well- >>>>>>>>>>> founded not non-well-founded for some questions?
Your problem is that you system is based on a criteria that >>>>>>>>>>>>> matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics? >>>>>>>>>>>
Note, asking LLMs for a definition doesn't define it in your >>>>>>>>>>> system.
In proofrCatheoretic semantics, a statement is not
wellrCafounded when its justification cannot be grounded in a >>>>>>>>>>>> finite, wellrCastructured chain of inferential steps. It lacks >>>>>>>>>>>> a terminating, wellrCaordered proof tree that would normally >>>>>>>>>>>> establish its truth or falsity. This often happens with >>>>>>>>>>>> selfrCareferential or circular statements whose rCLproofsrCY loop >>>>>>>>>>>> back on themselves rather than bottoming out in basic axioms >>>>>>>>>>>> or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>>>>>> well- foundedrCY means that the structure used to define or >>>>>>>>>>>> justify meanings does not rest on a base case that is >>>>>>>>>>>> independent of itself. Instead, it involves circular or >>>>>>>>>>>> infinitely descending dependencies among rules or proofs. // >>>>>>>>>>>> ChatGPT
In proof-theoretic semantics, not well-founded typically >>>>>>>>>>>> refers to derivations or proof structures that contain >>>>>>>>>>>> infinite descending chains or circular dependencies,
violating the well- foundedness property.
In classical proof theory, well-founded derivations have a >>>>>>>>>>>> clear hierarchical structure where every inference rule >>>>>>>>>>>> application depends only on "smaller" or "simpler" premises, >>>>>>>>>>>> eventually bottoming out in axioms or basic rules. This >>>>>>>>>>>> ensures that proofs are finitely constructible and
verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an >>>>>>>>>>>> atom P is called well-founded if every chain of successive >>>>>>>>>>>> "definitions" (unfoldings) eventually terminates rCo i.e., >>>>>>>>>>>> there is no infinite descending chain of definitional >>>>>>>>>>>> dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in >>>>>>>>>>>> logical structure after finitely many unfoldings. // Grok >>>>>>>>>>>>
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making >>>>>>>>>> "true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body >>>>>>>>> of knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a
different logic than the logic you are trying to process it. >>>>>>>>>
Also, part of our knowledge is about mathematics, which, for >>>>>>>>> instance will assert that the Goldbach Conjecture is one of the >>>>>>>>> great puzzles of mathematics, and must either be true or false, >>>>>>>>> but that FACT is incompatible with proof-theoretic semantics, >>>>>>>>> as mathematics can show that some true statements do not have >>>>>>>>> proofs in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just
defective.
A sentence is meaningful only if its justification graph
is wellrCafounded. A wellrCafounded graph always has a terminating
evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
nonrCawellrCafounded has no terminating evaluation, so it is
not meaningful and not truthrCaapt. Therefore truth is total
and computable over the meaningful fragment.
And thus your criteria for well-foundedness isn't itself well founded.
This is the problem of trying to redefine "truth" to be something
other than what it is.
The problem is, there are statements you can't show that they ARE
not- well-founded, and thus you can't talk about them.
We can't tell of the Golfbach conjecture is well-founded or not, so
your system ends up having many unkownable holes in it.
Within the set of knowledge that is
"true on the basis of meaning expressed in language"
the truth value of Goldbach is outside of the domain.
Then the truth value of most of mathematics is outside the domain.
Can you prove even the Pathagorean Theorem (the sum of the squares of
the length of the two sizes of a right triange is equal to the square of
the length of the hypotenuse)?
Can you prove based on the meaning expressed in langugage that the earth
is round?
The problem is your criteria doesn't handle most knowledge.
And because when you first want to pose the question, you likely
don't know if the answer will be available, or if it is in the realm
of unprovable. This means your "logic" is mostly restrictred to
talking about what is already known, and is worthless for producing
new knowledge.
It can catch all dangerous liars 100 million times a day.
Nope, as most lies are not lies ONLY by the meaning of words, but needs
to have other facts brought into the picture.
Your problem is you want to exclude the things you don't like but keep
what you want, but your definition doesn't do that.
It even has problem with much of the existing knowledge, as that is
based on truth-conditional logic, so isn't even true anymore in your
system.
On 1/16/26 10:44 PM, olcott wrote:
On 1/16/2026 9:24 PM, Richard Damon wrote:
On 1/16/26 8:27 PM, olcott wrote:
On 1/16/2026 5:21 PM, Richard Damon wrote:
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the >>>>>>>>>>>>>> meaning of a statement is determined entirely by its >>>>>>>>>>>>>> inferential role within a theory. A theory T consists >>>>>>>>>>>>>> of a finite set of basic statements together with
everything that can be derived from them using the >>>>>>>>>>>>>> inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it. A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T. These
are the non-well-founded statements: statements
whose inferential justification cannot be grounded >>>>>>>>>>>>>> in a finite, well-founded proof structure. This includes >>>>>>>>>>>>>> self-referential constructions such as G||del-type sentences. >>>>>>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self- >>>>>>>>>>>>>> Reference*
https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of >>>>>>>>>>> your syste,/
So. how is your definition of the criteria to be non-well- >>>>>>>>>>> founded not non-well-founded for some questions?
Your problem is that you system is based on a criteria that >>>>>>>>>>>>> matches your own definition of non-well-founded.
What does not well-founded mean in proof-theoretic semantics? >>>>>>>>>>>
Note, asking LLMs for a definition doesn't define it in your >>>>>>>>>>> system.
In proofrCatheoretic semantics, a statement is not
wellrCafounded when its justification cannot be grounded in a >>>>>>>>>>>> finite, wellrCastructured chain of inferential steps. It lacks >>>>>>>>>>>> a terminating, wellrCaordered proof tree that would normally >>>>>>>>>>>> establish its truth or falsity. This often happens with >>>>>>>>>>>> selfrCareferential or circular statements whose rCLproofsrCY loop >>>>>>>>>>>> back on themselves rather than bottoming out in basic axioms >>>>>>>>>>>> or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>>>>>> well- foundedrCY means that the structure used to define or >>>>>>>>>>>> justify meanings does not rest on a base case that is >>>>>>>>>>>> independent of itself. Instead, it involves circular or >>>>>>>>>>>> infinitely descending dependencies among rules or proofs. // >>>>>>>>>>>> ChatGPT
In proof-theoretic semantics, not well-founded typically >>>>>>>>>>>> refers to derivations or proof structures that contain >>>>>>>>>>>> infinite descending chains or circular dependencies,
violating the well- foundedness property.
In classical proof theory, well-founded derivations have a >>>>>>>>>>>> clear hierarchical structure where every inference rule >>>>>>>>>>>> application depends only on "smaller" or "simpler" premises, >>>>>>>>>>>> eventually bottoming out in axioms or basic rules. This >>>>>>>>>>>> ensures that proofs are finitely constructible and
verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an >>>>>>>>>>>> atom P is called well-founded if every chain of successive >>>>>>>>>>>> "definitions" (unfoldings) eventually terminates rCo i.e., >>>>>>>>>>>> there is no infinite descending chain of definitional >>>>>>>>>>>> dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or in >>>>>>>>>>>> logical structure after finitely many unfoldings. // Grok >>>>>>>>>>>>
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making >>>>>>>>>> "true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system.
I think not.
One problem you are going to run into is that this "entire body >>>>>>>>> of knowledge" is itself not built on those semantics,
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a
different logic than the logic you are trying to process it. >>>>>>>>>
Also, part of our knowledge is about mathematics, which, for >>>>>>>>> instance will assert that the Goldbach Conjecture is one of the >>>>>>>>> great puzzles of mathematics, and must either be true or false, >>>>>>>>> but that FACT is incompatible with proof-theoretic semantics, >>>>>>>>> as mathematics can show that some true statements do not have >>>>>>>>> proofs in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just
defective.
A sentence is meaningful only if its justification graph
is wellrCafounded. A wellrCafounded graph always has a terminating
evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
nonrCawellrCafounded has no terminating evaluation, so it is
not meaningful and not truthrCaapt. Therefore truth is total
and computable over the meaningful fragment.
And thus your criteria for well-foundedness isn't itself well founded.
You cannot possibly show that.
It is actually the same semantic tautology
that I have been saying for years.
I did.
How can you PROVE that a statement is not well founded in proof-
theoretic logic, when you can not search the entire body of possible
proofs to KNOW that one doesn't exist.
Yes, there are SOME statements you can do this for, but your system is--
built on a criteria that isn't always well-founded.
Goldbach's Conjecture is a good example. We don't KNOW if a proof
exists, or doesn't exist, or it a counter example exists.
Thus, any claim about its well-foundedness is just a lie, and is just not-well-founded.
The Conjecture seems true, as we have tested it to a very large value.
What is know is that we can not prove that no proof r refutation can
exist, so we can not prove it not to be well-founded, as such a proof
would mean that no counter example number can exist (as that is all that
is needed to refute it) and thus it must be true.
On 1/17/2026 6:03 AM, Richard Damon wrote:
On 1/16/26 10:44 PM, olcott wrote:
On 1/16/2026 9:24 PM, Richard Damon wrote:
On 1/16/26 8:27 PM, olcott wrote:
On 1/16/2026 5:21 PM, Richard Damon wrote:
On 1/16/26 5:09 PM, olcott wrote:
On 1/16/2026 3:54 PM, Richard Damon wrote:
On 1/16/26 3:51 PM, olcott wrote:
On 1/16/2026 2:34 PM, Richard Damon wrote:
On 1/16/26 3:24 PM, olcott wrote:
On 1/16/2026 1:34 PM, Richard Damon wrote:
On 1/16/26 2:16 PM, olcott wrote:
On 1/16/2026 12:52 PM, Richard Damon wrote:
On 1/16/26 12:47 PM, olcott wrote:
The system uses proof-theoretic semantics, where the >>>>>>>>>>>>>>> meaning of a statement is determined entirely by its >>>>>>>>>>>>>>> inferential role within a theory. A theory T consists >>>>>>>>>>>>>>> of a finite set of basic statements together with >>>>>>>>>>>>>>> everything that can be derived from them using the >>>>>>>>>>>>>>> inference rules. The statements derivable in this >>>>>>>>>>>>>>> way are the theorems of T. A statement is true in >>>>>>>>>>>>>>> T exactly when T proves it. A statement is false >>>>>>>>>>>>>>> in T exactly when T proves its negation. Some
statements are neither true nor false in T. These >>>>>>>>>>>>>>> are the non-well-founded statements: statements
whose inferential justification cannot be grounded >>>>>>>>>>>>>>> in a finite, well-founded proof structure. This includes >>>>>>>>>>>>>>> self-referential constructions such as G||del-type sentences. >>>>>>>>>>>>>>>
*Proof Theoretic Semantics Blocks Pathological Self- >>>>>>>>>>>>>>> Reference*
https://philpapers.org/archive/OLCPTS.pdf
WHAT system?
WHAT can you do in it?
Can you actually prove that, or is it just more of your lies. >>>>>>>>>>>>>>
You have to actually read the paper.
I did. Where do you actually define the initial axioms of >>>>>>>>>>>> your syste,/
So. how is your definition of the criteria to be non-well- >>>>>>>>>>>> founded not non-well-founded for some questions?
Your problem is that you system is based on a criteria >>>>>>>>>>>>>> that matches your own definition of non-well-founded. >>>>>>>>>>>>>>
What does not well-founded mean in proof-theoretic semantics? >>>>>>>>>>>>
Note, asking LLMs for a definition doesn't define it in your >>>>>>>>>>>> system.
In proofrCatheoretic semantics, a statement is not
wellrCafounded when its justification cannot be grounded in a >>>>>>>>>>>>> finite, wellrCastructured chain of inferential steps. It >>>>>>>>>>>>> lacks a terminating, wellrCaordered proof tree that would >>>>>>>>>>>>> normally establish its truth or falsity. This often happens >>>>>>>>>>>>> with selfrCareferential or circular statements whose rCLproofsrCY
loop back on themselves rather than bottoming out in basic >>>>>>>>>>>>> axioms or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is rCLnot >>>>>>>>>>>>> well- foundedrCY means that the structure used to define or >>>>>>>>>>>>> justify meanings does not rest on a base case that is >>>>>>>>>>>>> independent of itself. Instead, it involves circular or >>>>>>>>>>>>> infinitely descending dependencies among rules or
proofs. // ChatGPT
In proof-theoretic semantics, not well-founded typically >>>>>>>>>>>>> refers to derivations or proof structures that contain >>>>>>>>>>>>> infinite descending chains or circular dependencies, >>>>>>>>>>>>> violating the well- foundedness property.
In classical proof theory, well-founded derivations have a >>>>>>>>>>>>> clear hierarchical structure where every inference rule >>>>>>>>>>>>> application depends only on "smaller" or "simpler"
premises, eventually bottoming out in axioms or basic >>>>>>>>>>>>> rules. This ensures that proofs are finitely constructible >>>>>>>>>>>>> and verifiable. // Claude AI
A set of introduction rules (definitional clauses) for an >>>>>>>>>>>>> atom P is called well-founded if every chain of successive >>>>>>>>>>>>> "definitions" (unfoldings) eventually terminates rCo i.e., >>>>>>>>>>>>> there is no infinite descending chain of definitional >>>>>>>>>>>>> dependencies.
Intuitively:
The meaning of P is ultimately grounded in basic facts or >>>>>>>>>>>>> in logical structure after finitely many unfoldings. // Grok >>>>>>>>>>>>>
And, thus, your "definition" of non-well-founded
Is the standard definition in truth theoretic semantics making >>>>>>>>>>> "true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
This includes expressing all of PA in a complete system. >>>>>>>>>>>
I think not.
One problem you are going to run into is that this "entire >>>>>>>>>> body of knowledge" is itself not built on those semantics, >>>>>>>>>>
I knew that this would be philosophically too deep
for you so I am using PA to build a bridge.
It is a problem trying to process "knowledge" based on a
different logic than the logic you are trying to process it. >>>>>>>>>>
Also, part of our knowledge is about mathematics, which, for >>>>>>>>>> instance will assert that the Goldbach Conjecture is one of >>>>>>>>>> the great puzzles of mathematics, and must either be true or >>>>>>>>>> false, but that FACT is incompatible with proof-theoretic >>>>>>>>>> semantics, as mathematics can show that some true statements >>>>>>>>>> do not have proofs in the system.
You seem to keep forgetting the specified domain
is the body of knowledge that is
"true on the basis of meaning expressed in language"
Which means NOTHING about the real world, only man's own
classification of things.
When viewed within proof theoretic semantics it
specifies a precisely defined and coherent set
that shows all of the details of exactly how
conventional logic diverges from correct reasoning.
No, it shows how your concept of "correct reasoning" is just
defective.
A sentence is meaningful only if its justification graph
is wellrCafounded. A wellrCafounded graph always has a terminating
evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
nonrCawellrCafounded has no terminating evaluation, so it is
not meaningful and not truthrCaapt. Therefore truth is total
and computable over the meaningful fragment.
And thus your criteria for well-foundedness isn't itself well founded. >>>>
You cannot possibly show that.
It is actually the same semantic tautology
that I have been saying for years.
I did.
How can you PROVE that a statement is not well founded in proof-
theoretic logic, when you can not search the entire body of possible
proofs to KNOW that one doesn't exist.
If no proof exists in the entire body of general knowledge
that is "true on the basis of meaning expressed in language"
and no rebuttal exists in this body then the expression is
non-well-founded in this body.
In a proofrCatheoretic framework, an expression is wellrCafounded
only if the body of knowledge contains a proof or a rebuttal
grounded in the meaningrCagiving rules of the language. If
neither exists, the expression lacks inferential grounding
and is therefore nonrCawellrCafounded within that body.
Yes, there are SOME statements you can do this for, but your system is
built on a criteria that isn't always well-founded.
Goldbach's Conjecture is a good example. We don't KNOW if a proof
exists, or doesn't exist, or it a counter example exists.
Thus, any claim about its well-foundedness is just a lie, and is just
not-well-founded.
The Conjecture seems true, as we have tested it to a very large value.
What is know is that we can not prove that no proof r refutation can
exist, so we can not prove it not to be well-founded, as such a proof
would mean that no counter example number can exist (as that is all
that is needed to refute it) and thus it must be true.
On 1/17/2026 6:03 AM, Richard Damon wrote:
On 1/16/26 10:42 PM, olcott wrote:
Within the set of knowledge that is
"true on the basis of meaning expressed in language"
the truth value of Goldbach is outside of the domain.
Then the truth value of most of mathematics is outside the domain.
Everything that is
"true on the basis of meaning expressed in language"
in entire body knowledge is in the domain. Only a
subset of this is encoded in any physically existing system.
Can you prove even the Pathagorean Theorem (the sum of the squares of
the length of the two sizes of a right triange is equal to the square
of the length of the hypotenuse)?
Can you prove based on the meaning expressed in langugage that the
earth is round?
The problem is your criteria doesn't handle most knowledge.
And because when you first want to pose the question, you likely
don't know if the answer will be available, or if it is in the realm
of unprovable. This means your "logic" is mostly restrictred to
talking about what is already known, and is worthless for producing
new knowledge.
It can catch all dangerous liars 100 million times a day.
Nope, as most lies are not lies ONLY by the meaning of words, but
needs to have other facts brought into the picture.
Your problem is you want to exclude the things you don't like but keep
what you want, but your definition doesn't do that.
It even has problem with much of the existing knowledge, as that is
based on truth-conditional logic, so isn't even true anymore in your
system.
On 1/17/2026 3:22 AM, Mikko wrote:
On 16/01/2026 19:47, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it.
Usually the expression "is a theorem of T" is used instead of "is true
in T". THe words "true" and "false" are usually reserved for truth in
a particular interpretation.
That is what I changed. That is how Incompleteness arises.
I switched truth conditional semantics that uses model
theory for proof theoretic semantics. Then I had to add
Haskell Curry's notion of True in {T}.
-a Then the elementary statements which belong to {T} we
-a shall call the elementary theorems of {T} we also say
-a that these elementary statements are true for {T}
-a https://www.liarparadox.org/Haskell_Curry_45.pdf
This is the same thing that I have been saying for years,
the only difference now is the I can anchor my own ideas
in standard terms of the art.
A statement is false
in T exactly when T proves its negation. Some
statements are neither true nor false in T.
Usually the expression "undecidable in T" is used instead of
"neither true nor false in T".
What was undecidable in truth conditional semantics
becomes non-well-founded thus semantically illrCaformed
in proof theoretic semantics. Non-well-founded a
conventional term of the art from proof theoretic semantics.
A theory where some statements that are neither provable or disprovable
is said to be incomplere.
These are the non-well-founded statements: statements
whose inferential justification cannot be grounded
in a finite, well-founded proof structure. This includes
self-referential constructions such as G||del-type sentences.
G||del's sentence is a sentence of Peano arithmetic. When inpterpreted
according to the arithmentic semantics it is not self-referential.
G_F rao -4Prov_F (riLG_FriY) is self-referential https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
Peano's postulates just are insufficient for the proof of G||del's
sentence. The simlest fis is to simply add G||del's sentence to the
postulates. This addition does not create any inconsistency. However,
the new theory is still incomlete.
One should also note that for some theories there is no way to determine
whether a claim is provable. A simple example is the group theory.
Plain PA has no internal notion of truth; any truth
talk is metarCatheoretic. To work proofrCatheoretically,
we must add a rulerCaanchored truth predicate in the
sense of Curry, governed by elementary theorems of T.
If we then impose an objectrCalevel wellrCafoundedness
constraint on truthrCorejecting any cyclic truth
dependenciesrCoG||delrCOs fixedrCapoint sentence G becomes
syntactically nonrCawellrCafounded and is blocked before
any truth value is assigned. In such a system,
G||delrCOs G is not a deep undecidable truth, but
an illrCaformed attempt at selfrCareference.
On 1/17/2026 3:22 AM, Mikko wrote:
On 16/01/2026 19:47, olcott wrote:
The system uses proof-theoretic semantics, where the
meaning of a statement is determined entirely by its
inferential role within a theory. A theory T consists
of a finite set of basic statements together with
everything that can be derived from them using the
inference rules. The statements derivable in this
way are the theorems of T. A statement is true in
T exactly when T proves it.
Usually the expression "is a theorem of T" is used instead of "is true
in T". THe words "true" and "false" are usually reserved for truth in
a particular interpretation.
That is what I changed. That is how Incompleteness arises.
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