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 Gödel-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 Gödel-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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded when its justification cannot be grounded in a finite, well‑structured chain of inferential steps. It lacks a terminating, well‑ordered proof tree that would normally establish its truth or falsity. This often happens with self‑referential or circular statements whose “proofs” loop back on themselves rather than bottoming out in basic axioms or introduction
rules. // Copilot
In proof-theoretic semantics, saying that something is “not well- founded” 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 — 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded when its >> justification cannot be grounded in a finite, well‑structured chain of
inferential steps. It lacks a terminating, well‑ordered proof tree
that would normally establish its truth or falsity. This often happens
with self‑referential or circular statements whose “proofs” loop back >> on themselves rather than bottoming out in basic axioms or
introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not well-
founded” 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 — 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded when
its justification cannot be grounded in a finite, well‑structured
chain of inferential steps. It lacks a terminating, well‑ordered
proof tree that would normally establish its truth or falsity. This
often happens with self‑referential or circular statements whose
“proofs” loop back on themselves rather than bottoming out in basic >>> axioms or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not well-
founded” 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 — 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded when >>>> its justification cannot be grounded in a finite, well‑structured
chain of inferential steps. It lacks a terminating, well‑ordered
proof tree that would normally establish its truth or falsity. This
often happens with self‑referential or circular statements whose
“proofs” loop back on themselves rather than bottoming out in basic >>>> axioms or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not well-
founded” 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 — 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded when >>>>> its justification cannot be grounded in a finite, well‑structured >>>>> chain of inferential steps. It lacks a terminating, well‑ordered
proof tree that would normally establish its truth or falsity. This >>>>> often happens with self‑referential or circular statements whose
“proofs” loop back on themselves rather than bottoming out in basic >>>>> axioms or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not well- >>>>> founded” 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 — 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded when >>>>>> its justification cannot be grounded in a finite, well‑structured >>>>>> chain of inferential steps. It lacks a terminating, well‑ordered >>>>>> proof tree that would normally establish its truth or falsity.
This often happens with self‑referential or circular statements >>>>>> whose “proofs” loop back on themselves rather than bottoming out >>>>>> in basic axioms or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not well- >>>>>> founded” 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 — 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded >>>>>>> when its justification cannot be grounded in a finite,
well‑structured chain of inferential steps. It lacks a
terminating, well‑ordered proof tree that would normally
establish its truth or falsity. This often happens with
self‑referential or circular statements whose “proofs” loop back >>>>>>> on themselves rather than bottoming out in basic axioms or
introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not well- >>>>>>> founded” 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 — 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded >>>>>>>> when its justification cannot be grounded in a finite,
well‑structured chain of inferential steps. It lacks a
terminating, well‑ordered proof tree that would normally
establish its truth or falsity. This often happens with
self‑referential or circular statements whose “proofs” loop back
on themselves rather than bottoming out in basic axioms or
introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not >>>>>>>> well- founded” 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 — 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded >>>>>>>> when its justification cannot be grounded in a finite,
well‑structured chain of inferential steps. It lacks a
terminating, well‑ordered proof tree that would normally
establish its truth or falsity. This often happens with
self‑referential or circular statements whose “proofs” loop back
on themselves rather than bottoming out in basic axioms or
introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not >>>>>>>> well- founded” 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 — 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded >>>>>>>>> when its justification cannot be grounded in a finite,
well‑structured chain of inferential steps. It lacks a
terminating, well‑ordered proof tree that would normally
establish its truth or falsity. This often happens with
self‑referential or circular statements whose “proofs” loop >>>>>>>>> back on themselves rather than bottoming out in basic axioms or >>>>>>>>> introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not >>>>>>>>> well- founded” 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 — 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded >>>>>>>>> when its justification cannot be grounded in a finite,
well‑structured chain of inferential steps. It lacks a
terminating, well‑ordered proof tree that would normally
establish its truth or falsity. This often happens with
self‑referential or circular statements whose “proofs” loop >>>>>>>>> back on themselves rather than bottoming out in basic axioms or >>>>>>>>> introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not >>>>>>>>> well- founded” 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 — 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 well‑founded. A well‑founded graph always has a terminating evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
non‑well‑founded has no terminating evaluation, so it is
not meaningful and not truth‑apt. 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded >>>>>>>>>> when its justification cannot be grounded in a finite,
well‑structured chain of inferential steps. It lacks a
terminating, well‑ordered proof tree that would normally >>>>>>>>>> establish its truth or falsity. This often happens with
self‑referential or circular statements whose “proofs” loop >>>>>>>>>> back on themselves rather than bottoming out in basic axioms >>>>>>>>>> or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not >>>>>>>>>> well- founded” 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 — 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 well‑founded. A well‑founded graph always has a terminating
evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
non‑well‑founded has no terminating evaluation, so it is
not meaningful and not truth‑apt. 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 Gödel-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 proof‑theoretic semantics, a statement is not well‑founded >>>>>>>>>> when its justification cannot be grounded in a finite,
well‑structured chain of inferential steps. It lacks a
terminating, well‑ordered proof tree that would normally >>>>>>>>>> establish its truth or falsity. This often happens with
self‑referential or circular statements whose “proofs” loop >>>>>>>>>> back on themselves rather than bottoming out in basic axioms >>>>>>>>>> or introduction rules. // Copilot
In proof-theoretic semantics, saying that something is “not >>>>>>>>>> well- founded” 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 — 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 well‑founded. A well‑founded graph always has a terminating
evaluation. Truth is defined as the result of that terminating
evaluation. Any sentence whose justification graph is
non‑well‑founded has no terminating evaluation, so it is
not meaningful and not truth‑apt. Therefore truth is total
and computable over the meaningful fragment.
And thus your criteria for well-foundedness isn't itself well founded.
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