Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
This is best understood within the proof theoretic
notion of epistemic justification.
Truth as an Epistemic Notion --- Dag Prawitz (2011) https:// link.springer.com/article/10.1007/s11245-011-9107-6
On 9/22/2026 11:01 PM, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
This is best understood within the proof theoretic
notion of epistemic justification.
Truth as an Epistemic Notion --- Dag Prawitz (2011) https://
link.springer.com/article/10.1007/s11245-011-9107-6
Why do you restrict your proof to Pennsylvania? Does that mean if
you exit the state for any reason, your hypothesis becomes false?
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
On 9/22/2026 10:32 AM, Johann 'Myrkraverk' Oskarsson wrote:
On 9/22/2026 11:01 PM, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
This is best understood within the proof theoretic
notion of epistemic justification.
Truth as an Epistemic Notion --- Dag Prawitz (2011) https://
link.springer.com/article/10.1007/s11245-011-9107-6
Why do you restrict your proof to Pennsylvania? Does that mean if
you exit the state for any reason, your hypothesis becomes false?
The axioms of Peano Arithmetic (PA) https://mathworld.wolfram.com/PeanosAxioms.html
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and Gödel is wrong.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences
whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified
with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011) https://link.springer.com/article/10.1007/s11245-011-9107-6
Provability in PA requires a sequence of syntactic
inference steps to remain totally within PA and not
leap outside of PA into any model of arithmetic.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences
whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified
with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011) https://link.springer.com/article/10.1007/s11245-011-9107-6
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and Gödel is wrong.
That is clearly false. The meaning of the word "provability" (or amy
word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant
to my comment above.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of their
useful meanings. In order to say "proved in Russell's system" one can
say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make
sense. Gödel's most important results are not about numbers but about
formal theories. A map of Italy does not help if you have problems
with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences
whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified
with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually want.
Many useful things can be done with no semantics at all.
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and Gödel is wrong.
That is clearly false. The meaning of the word "provability" (or amy
word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant
to my comment above.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of their
useful meanings. In order to say "proved in Russell's system" one can
say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make
sense. Gödel's most important results are not about numbers but about
formal theories. A map of Italy does not help if you have problems
with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences
whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified
with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually want.
Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability
within such a system.
On 27/09/2026 05:59, olcott wrote:
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and Gödel is wrong.
That is clearly false. The meaning of the word "provability" (or amy
word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant
to my comment above.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of their
useful meanings. In order to say "proved in Russell's system" one can
say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make
sense. Gödel's most important results are not about numbers but about
formal theories. A map of Italy does not help if you have problems
with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences
whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified
with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually want.
Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability
within such a system.
Forst of all, that is a change of topic, apparently in order to make
people to forget what you said earlier. WHich is in fact a good idea,
there is no need to remember your errors. But it is useful to remember
that most of what you say is erroneous.
Whether "true on the basis of meaning expressed in language" is
computable depends on a language. For an uninterpreted formal
language nothing in the language is true, so the computation is
trivial.
Anyway, "true on the basis of meaning expressed in language" does
not cover the truths we usually need to know. An example is the
whether tomorrow: it is not computable. A prediction can be computed
but sometimes the truth is different.
There is no complete method to compute about a sentence is some lanugage
of natural number arithmetic whther it is true about natural numbers.
It is not quite clear whether such sentence can be said to be "true
on the basis of meaning expressed in language".
More often a truth is the basis of an expression than an expression
of a truth.
On 9/28/2026 4:31 AM, Mikko wrote:
On 27/09/2026 05:59, olcott wrote:
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means
for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and Gödel is wrong.
That is clearly false. The meaning of the word "provability" (or amy
word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant
to my comment above.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of their >>>> useful meanings. In order to say "proved in Russell's system" one can
say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make
sense. Gödel's most important results are not about numbers but about >>>> formal theories. A map of Italy does not help if you have problems
with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences
whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified
with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually want. >>>> Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability
within such a system.
Forst of all, that is a change of topic, apparently in order to make
people to forget what you said earlier. WHich is in fact a good idea,
there is no need to remember your errors. But it is useful to remember
that most of what you say is erroneous.
Whether "true on the basis of meaning expressed in language" is
computable depends on a language. For an uninterpreted formal
language nothing in the language is true, so the computation is
trivial.
The body of general knowledge expressed in formalized English
can be formalized in CycL. That "cats are animals" is one
example of "true on the basis of meaning expressed in language"
Anyway, "true on the basis of meaning expressed in language" does
not cover the truths we usually need to know. An example is the
whether tomorrow: it is not computable. A prediction can be computed
but sometimes the truth is different.
The scope of the body of general knowledge that is
"true on the basis of meaning expressed in language"
Tomorrow's weather is outside of this scope.
There is no complete method to compute about a sentence is some lanugage
of natural number arithmetic whther it is true about natural numbers.
It is not quite clear whether such sentence can be said to be "true
on the basis of meaning expressed in language".
Proof theoretic semantics is the way that "true on the
basis of meaning expressed in language" has always
worked in actual reality.
Mathematical incompleteness and undecidability has always been
artificially contrived by choosing model theory as a foundation.
When we switch> to proof theoretic semantics all that we lose is "undecidability" and "incompleteness", they were
never inherently actually there.
More often a truth is the basis of an expression than an expression
of a truth.
On 28/09/2026 15:10, olcott wrote:
On 9/28/2026 4:31 AM, Mikko wrote:
On 27/09/2026 05:59, olcott wrote:
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means >>>>>>> for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and Gödel is wrong.
That is clearly false. The meaning of the word "provability" (or amy >>>>> word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant
to my comment above.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of their >>>>> useful meanings. In order to say "proved in Russell's system" one can >>>>> say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make >>>>> sense. Gödel's most important results are not about numbers but about >>>>> formal theories. A map of Italy does not help if you have problems
with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences
whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified
with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually
want.
Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability
within such a system.
Forst of all, that is a change of topic, apparently in order to make
people to forget what you said earlier. WHich is in fact a good idea,
there is no need to remember your errors. But it is useful to remember
that most of what you say is erroneous.
Whether "true on the basis of meaning expressed in language" is
computable depends on a language. For an uninterpreted formal
language nothing in the language is true, so the computation is
trivial.
The body of general knowledge expressed in formalized English
can be formalized in CycL. That "cats are animals" is one
example of "true on the basis of meaning expressed in language"
An ordinary dictionary like https://en.wiktionary.org/wiki/cat can
tell that cats are animals. If you want to get anybody interested
you need more impressive examples.
Anyway, "true on the basis of meaning expressed in language" does
not cover the truths we usually need to know. An example is the
whether tomorrow: it is not computable. A prediction can be computed
but sometimes the truth is different.
The scope of the body of general knowledge that is
"true on the basis of meaning expressed in language"
Tomorrow's weather is outside of this scope.
And so are most of the things that people might want to find out.
There is no complete method to compute about a sentence is some lanugage >>> of natural number arithmetic whther it is true about natural numbers.
It is not quite clear whether such sentence can be said to be "true
on the basis of meaning expressed in language".
Proof theoretic semantics is the way that "true on the
basis of meaning expressed in language" has always
worked in actual reality.
It works in the sense of never giving a false answer
(as long as
applied correctly) but not in the sense of giving the answers
people most often need.
Mathematical incompleteness and undecidability has always been
artificially contrived by choosing model theory as a foundation.
That is false. A surprising discovery is not "artificially contrived".
Undecidability is not related to model theory. It is a conseqeunce of
the meaning of the words expressed in language.
The term "incomplete" has several meanings. One is that a theory is incomplete if there is an undecidable sentence. This meaning is unreated
to model theory. Another meaning is that there is a sentence that is
true is some model but cannot be proven. If no models are considered
this latter meaning is not applicable.
When we switch> to proof theoretic semantics all that we lose is
"undecidability" and "incompleteness", they were
never inherently actually there.
There still is undecidability as it is a purely proof theoretic
concept.
Undecidable sentences don't became decidable by avoiding all
consideration of decidability.
--More often a truth is the basis of an expression than an expression
of a truth.
-- Mikko
No, Pete. Calling an undecidable sentence "out-of-scope" does not make undecidability disappear.
For a formal theory T, "G is undecidable in T" simply means
T |-/- G and T |-/- ~G
That is a statement entirely about proofs. No model theory is involved.
And G is not "out-of-scope": it is a perfectly well-formed sentence of
the language of T. What is out-of-scope, apparently, is merely anything
for which your proposed semantics fails to supply an answer.
If you define your scope as--
{ A : T proves A or T proves ~A }
then of course everything "in scope" is decidable.
But that does not eliminate incompleteness. It eliminates the incomplete cases from your definition of "scope".
You can make a hospital 100% successful by defining "patient" to mean "person we successfully cured", too.
On 9/29/2026 4:25 AM, Mikko wrote:
On 28/09/2026 15:10, olcott wrote:
On 9/28/2026 4:31 AM, Mikko wrote:
On 27/09/2026 05:59, olcott wrote:
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means >>>>>>>> for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and Gödel is wrong.
That is clearly false. The meaning of the word "provability" (or amy >>>>>> word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant >>>>>> to my comment above.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of
their
useful meanings. In order to say "proved in Russell's system" one can >>>>>> say "proved in Russell's system". A statement proved in Russell's
system can also be called a "theorem of Russell's system".
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make >>>>>> sense. Gödel's most important results are not about numbers but about >>>>>> formal theories. A map of Italy does not help if you have problems >>>>>> with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences
whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified
with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually >>>>>> want.
Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability
within such a system.
Forst of all, that is a change of topic, apparently in order to make
people to forget what you said earlier. WHich is in fact a good idea,
there is no need to remember your errors. But it is useful to remember >>>> that most of what you say is erroneous.
Whether "true on the basis of meaning expressed in language" is
computable depends on a language. For an uninterpreted formal
language nothing in the language is true, so the computation is
trivial.
The body of general knowledge expressed in formalized English
can be formalized in CycL. That "cats are animals" is one
example of "true on the basis of meaning expressed in language"
An ordinary dictionary like https://en.wiktionary.org/wiki/cat can
tell that cats are animals. If you want to get anybody interested
you need more impressive examples.
Just that one example refutes Quine's
Two Dogmas of Empiricism
https://www.ditext.com/quine/quine.html
Anyway, "true on the basis of meaning expressed in language" does
not cover the truths we usually need to know. An example is the
whether tomorrow: it is not computable. A prediction can be computed
but sometimes the truth is different.
The scope of the body of general knowledge that is
"true on the basis of meaning expressed in language"
Tomorrow's weather is outside of this scope.
And so are most of the things that people might want to find out.
The best that any system of knowledge can possibly
provide is elements of the body of knowledge. Requiring
psychic ability is a bad requirement.
There is no complete method to compute about a sentence is some
lanugage
of natural number arithmetic whther it is true about natural numbers.
It is not quite clear whether such sentence can be said to be "true
on the basis of meaning expressed in language".
Proof theoretic semantics is the way that "true on the
basis of meaning expressed in language" has always
worked in actual reality.
It works in the sense of never giving a false answer
Unlike current LLM systems.
(as long as
applied correctly) but not in the sense of giving the answers
people most often need.
Mathematical incompleteness and undecidability has always been
artificially contrived by choosing model theory as a foundation.
That is false. A surprising discovery is not "artificially contrived".
Undecidability is not related to model theory. It is a conseqeunce of
the meaning of the words expressed in language.
It is not any consequence of
"the meaning of the words expressed in language"
within proof theoretic semantics.
The body of general> knowledge expressed in language is merely relations
between finite strings some of them are basic facts
that are stipulated to be true. It is infallible and
complete because it is a semantic tautology.
The term "incomplete" has several meanings. One is that a theory is
incomplete if there is an undecidable sentence. This meaning is unreated
to model theory. Another meaning is that there is a sentence that is
true is some model but cannot be proven. If no models are considered
this latter meaning is not applicable.
This cannot possibly occur within proof theoretic semantics
when we exclude Base-extension semantics (B-eS).
When we switch> to proof theoretic semantics all that we lose is
"undecidability" and "incompleteness", they were
never inherently actually there.
There still is undecidability as it is a purely proof theoretic
concept.
Try to explain the details of how that is true when we exclude
Base-extension semantics (B-eS).
Undecidable sentences don't became decidable by avoiding all
consideration of decidability.
Undecidable sentences in model theory becomes sentences that are
out-of-scope for the formal system in question within PTS.
--More often a truth is the basis of an expression than an expression
of a truth.
On 9/29/2026 11:01 AM, Python wrote:
No, Pete. Calling an undecidable sentence "out-of-scope" does not make
undecidability disappear.
For a formal theory T, "G is undecidable in T" simply means
T |-/- G and T |-/- ~G
That is a statement entirely about proofs. No model theory is involved.
And G is not "out-of-scope": it is a perfectly well-formed sentence of
the language of T. What is out-of-scope, apparently, is merely
anything for which your proposed semantics fails to supply an answer.
Within PTS any expression that is neither provable nor refutable
never acquires any semantic meaning.
On 29/09/2026 18:48, olcott wrote:
On 9/29/2026 4:25 AM, Mikko wrote:
On 28/09/2026 15:10, olcott wrote:
On 9/28/2026 4:31 AM, Mikko wrote:
On 27/09/2026 05:59, olcott wrote:
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means >>>>>>>>> for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and Gödel is wrong.
That is clearly false. The meaning of the word "provability" (or amy >>>>>>> word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant >>>>>>> to my comment above.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of >>>>>>> their
useful meanings. In order to say "proved in Russell's system" one >>>>>>> can
say "proved in Russell's system". A statement proved in Russell's >>>>>>> system can also be called a "theorem of Russell's system".
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results make >>>>>>> sense. Gödel's most important results are not about numbers but >>>>>>> about
formal theories. A map of Italy does not help if you have problems >>>>>>> with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences
whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified
with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually >>>>>>> want.
Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability
within such a system.
Forst of all, that is a change of topic, apparently in order to make >>>>> people to forget what you said earlier. WHich is in fact a good idea, >>>>> there is no need to remember your errors. But it is useful to remember >>>>> that most of what you say is erroneous.
Whether "true on the basis of meaning expressed in language" is
computable depends on a language. For an uninterpreted formal
language nothing in the language is true, so the computation is
trivial.
The body of general knowledge expressed in formalized English
can be formalized in CycL. That "cats are animals" is one
example of "true on the basis of meaning expressed in language"
An ordinary dictionary like https://en.wiktionary.org/wiki/cat can
tell that cats are animals. If you want to get anybody interested
you need more impressive examples.
Just that one example refutes Quine's
Two Dogmas of Empiricism
https://www.ditext.com/quine/quine.html
Quine's "Two Dogams of Empiricism" is an example of what most
people consider uninteresting.
Anyway, "true on the basis of meaning expressed in language" does
not cover the truths we usually need to know. An example is the
whether tomorrow: it is not computable. A prediction can be computed >>>>> but sometimes the truth is different.
The scope of the body of general knowledge that is
"true on the basis of meaning expressed in language"
Tomorrow's weather is outside of this scope.
And so are most of the things that people might want to find out.
The best that any system of knowledge can possibly
provide is elements of the body of knowledge. Requiring
psychic ability is a bad requirement.
Most people care about real world knowledge, which tends to be
uncertain but is often useful anyway.
finite strings some of them are basic factsThere is no complete method to compute about a sentence is some
lanugage
of natural number arithmetic whther it is true about natural numbers. >>>>> It is not quite clear whether such sentence can be said to be "true
on the basis of meaning expressed in language".
Proof theoretic semantics is the way that "true on the
basis of meaning expressed in language" has always
worked in actual reality.
It works in the sense of never giving a false answer
Unlike current LLM systems.
(as long as
applied correctly) but not in the sense of giving the answers
people most often need.
Mathematical incompleteness and undecidability has always been
artificially contrived by choosing model theory as a foundation.
That is false. A surprising discovery is not "artificially contrived".
Undecidability is not related to model theory. It is a conseqeunce of
the meaning of the words expressed in language.
It is not any consequence of
"the meaning of the words expressed in language"
within proof theoretic semantics.
That "proof theoretic semants" fails to find the connection does
not make the connection disappear.
The body of general
knowledge expressed in language is merely relations>> between
that are stipulated to be true. It is infallible and
complete because it is a semantic tautology.
A false claim does not become true by stipulation.
General knowledge expressed in language cannot be complete if it
covers natural number arithmetic.
That system can be complete and infallible about real world knowledge.
If real world knowledge is completely excluded it is not interesting.
The term "incomplete" has several meanings. One is that a theory is
incomplete if there is an undecidable sentence.
This meaning is unreated
to model theory. Another meaning is that there is a sentence that is
true is some model but cannot be proven. If no models are considered
this latter meaning is not applicable.
This cannot possibly occur within proof theoretic semantics
when we exclude Base-extension semantics (B-eS).
Doesn't matter. People don't care about proof theoretic semantics.
There care about getting answers they need or want.
When we switch> to proof theoretic semantics all that we lose is
"undecidability" and "incompleteness", they were
never inherently actually there.
There still is undecidability as it is a purely proof theoretic
concept.
Try to explain the details of how that is true when we exclude
Base-extension semantics (B-eS).
A simple example is ∀a ∀b (a * b = b * a) in the theory of groups. Neither it nor its negation can be proven.
Whether a sentence can
be proven does not depend on semantics.
Undecidable sentences don't became decidable by avoiding all
consideration of decidability.
Undecidable sentences in model theory becomes sentences that are
out-of-scope for the formal system in question within PTS.
If you declare too many qestions "out-of-scope" your system is
not useful for anybody.
On 10/01/2026 02:54 AM, Mikko wrote:
On 29/09/2026 18:48, olcott wrote:
On 9/29/2026 4:25 AM, Mikko wrote:
On 28/09/2026 15:10, olcott wrote:
On 9/28/2026 4:31 AM, Mikko wrote:
On 27/09/2026 05:59, olcott wrote:
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means >>>>>>>>>> for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and Gödel is wrong.
That is clearly false. The meaning of the word "provability" (or >>>>>>>> amy
word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant >>>>>>>> to my comment above.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of >>>>>>>> their
useful meanings. In order to say "proved in Russell's system" one >>>>>>>> can
say "proved in Russell's system". A statement proved in Russell's >>>>>>>> system can also be called a "theorem of Russell's system".
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results >>>>>>>> make
sense. Gödel's most important results are not about numbers but >>>>>>>> about
formal theories. A map of Italy does not help if you have problems >>>>>>>> with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences >>>>>>>>> whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified >>>>>>>>> with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we usually >>>>>>>> want.
Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability >>>>>>> within such a system.
Forst of all, that is a change of topic, apparently in order to make >>>>>> people to forget what you said earlier. WHich is in fact a good idea, >>>>>> there is no need to remember your errors. But it is useful to
remember
that most of what you say is erroneous.
Whether "true on the basis of meaning expressed in language" is
computable depends on a language. For an uninterpreted formal
language nothing in the language is true, so the computation is
trivial.
The body of general knowledge expressed in formalized English
can be formalized in CycL. That "cats are animals" is one
example of "true on the basis of meaning expressed in language"
An ordinary dictionary like https://en.wiktionary.org/wiki/cat can
tell that cats are animals. If you want to get anybody interested
you need more impressive examples.
Just that one example refutes Quine's
Two Dogmas of Empiricism
https://www.ditext.com/quine/quine.html
Quine's "Two Dogams of Empiricism" is an example of what most
people consider uninteresting.
Anyway, "true on the basis of meaning expressed in language" does
not cover the truths we usually need to know. An example is the
whether tomorrow: it is not computable. A prediction can be computed >>>>>> but sometimes the truth is different.
The scope of the body of general knowledge that is
"true on the basis of meaning expressed in language"
Tomorrow's weather is outside of this scope.
And so are most of the things that people might want to find out.
The best that any system of knowledge can possibly
provide is elements of the body of knowledge. Requiring
psychic ability is a bad requirement.
Most people care about real world knowledge, which tends to be
uncertain but is often useful anyway.
There is no complete method to compute about a sentence is some
lanugage
of natural number arithmetic whther it is true about natural numbers. >>>>>> It is not quite clear whether such sentence can be said to be "true >>>>>> on the basis of meaning expressed in language".
Proof theoretic semantics is the way that "true on the
basis of meaning expressed in language" has always
worked in actual reality.
It works in the sense of never giving a false answer
Unlike current LLM systems.
(as long as
applied correctly) but not in the sense of giving the answers
people most often need.
Mathematical incompleteness and undecidability has always been
artificially contrived by choosing model theory as a foundation.
That is false. A surprising discovery is not "artificially contrived". >>>>
Undecidability is not related to model theory. It is a conseqeunce of
the meaning of the words expressed in language.
It is not any consequence of
"the meaning of the words expressed in language"
within proof theoretic semantics.
That "proof theoretic semants" fails to find the connection does
not make the connection disappear.
The body of general> knowledge expressed in language is merelyrelations
between finite strings some of them are basic facts
that are stipulated to be true. It is infallible and
complete because it is a semantic tautology.
A false claim does not become true by stipulation.
General knowledge expressed in language cannot be complete if it
covers natural number arithmetic.
That system can be complete and infallible about real world knowledge.
If real world knowledge is completely excluded it is not interesting.
The term "incomplete" has several meanings. One is that a theory is
incomplete if there is an undecidable sentence. This meaning is
unreated
to model theory. Another meaning is that there is a sentence that is
true is some model but cannot be proven. If no models are considered
this latter meaning is not applicable.
This cannot possibly occur within proof theoretic semantics
when we exclude Base-extension semantics (B-eS).
Doesn't matter. People don't care about proof theoretic semantics.
There care about getting answers they need or want.
When we switch> to proof theoretic semantics all that we lose is
"undecidability" and "incompleteness", they were
never inherently actually there.
There still is undecidability as it is a purely proof theoretic
concept.
Try to explain the details of how that is true when we exclude
Base-extension semantics (B-eS).
A simple example is ∀a ∀b (a * b = b * a) in the theory of groups.
Neither it nor its negation can be proven. Whether a sentence can
be proven does not depend on semantics.
Undecidable sentences don't became decidable by avoiding all
consideration of decidability.
Undecidable sentences in model theory becomes sentences that are
out-of-scope for the formal system in question within PTS.
If you declare too many qestions "out-of-scope" your system is
not useful for anybody.
More often a truth is the basis of an expression than an expression >>>>>> of a truth.
Most people consider "formal logic" un-interesting, then as with
regards to Quine on empiricism, it might help if first you bring
Goodman's "Riddle of Induction" then as for "Calculus of Individuals",
since they're a refresher on the fact that it's well-known since
antiquity that empirical reasoning is un-sound in the surrounds.
Most people with "common-sense logic" think that "20'th century
'classical logic'" which is plainly Philo and Plotinus' "quasi-modal
logic" and not "Aristotle and Chrysippus' classical modal relevance
logic", is non-sensical, since false-antecedent/false-consequent.
Then the ex falso quodlibet and principle of explosion is its
own fault
since really it was merely erroneous, for "ex falso nihilum"
and "classical logic".
So, "caring what answers they need or want", sounds simply self-serving
and not conscientious about being a conscientious logician.
Modern accounts of the fragmented or synthetic are examples of--
the in-valid and un-sound and hypocritical, puffing itself expedient.
Most philosophers take Quine ideas to be an established
verified fact, thus shutting off making
"true on the basis of meaning expressed in language"
reliably computable before it has a chance to begin.
On 10/1/2026 10:17 PM, olcott wrote:
Most philosophers take Quine ideas to be an established
verified fact, thus shutting off making
"true on the basis of meaning expressed in language"
reliably computable before it has a chance to begin.
How do you define /most philosophers/?
Then the ex falso quodlibet and principle of explosion is its
own fault
Yes exactly. I have held this view for many years. People
that never bother to think these things through never see this.
On 10/1/2026 10:03 AM, Johann 'Myrkraverk' Oskarsson wrote:
On 10/1/2026 10:17 PM, olcott wrote:
Most philosophers take Quine ideas to be an established
verified fact, thus shutting off making
"true on the basis of meaning expressed in language"
reliably computable before it has a chance to begin.
How do you define /most philosophers/?
Many conversations on Philosophy specific groups over
many years.
[ Followup-To: set ]
In comp.theory Ross Finlayson <[email protected]> wrote:
On 10/01/2026 08:03 AM, Johann 'Myrkraverk' Oskarsson wrote:
On 10/1/2026 10:17 PM, olcott wrote:
Most philosophers take Quine ideas to be an established
verified fact, thus shutting off making
"true on the basis of meaning expressed in language"
reliably computable before it has a chance to begin.
How do you define /most philosophers/?
The "philocasuy" or "philocasuistry" is a stronger, more technical
account than "philosophistry", that the philocasuy is naturally logical,
then that some philosophers have most philosophers are
not philosophers.
It's not dissimilar the "un-scientific", for example like where
Dark Matter/Energy long ago saw falsified many "standard" accounts,
which are mostly cases of reductionism calling itself complete,
which is wrong.
"Great minds think alike" is a usual idea,
yet eventually there's "the great mind thinks".
Reading Quine, Quine acknowledges the "ultimate classes",
outside the ordinary theory, and "ur-elements" or "atoms",
outside the ordinary theory, and makes accounts of "modal
relativism", outside the ordinary theory, and as for a
"cosmic complement", outside the ordinary theory, and
deconstructive accounts of descriptive number theory,
outside the ordinary theory, then that as with regards
to "established verified fact": most philosophers
haven't actually read Quine, and have the perspective
of life in a wet, cardboard box, which some insist
can't be shred.
PA, Peano Arithmetic, is ordinary, yet, one can always
build the extra-ordinary in it, like Mirmanoff, or
Russell, before Russell's retro-thesis lied its way
to "an ordinary inductive set".
Yes, that's all very well, but how do you define "most philosophers",
like the previous poster challenged his predecessor to do?
On 10/1/2026 4:54 AM, Mikko wrote:
On 29/09/2026 18:48, olcott wrote:
On 9/29/2026 4:25 AM, Mikko wrote:
On 28/09/2026 15:10, olcott wrote:
On 9/28/2026 4:31 AM, Mikko wrote:
On 27/09/2026 05:59, olcott wrote:
On 9/24/2026 6:24 AM, Mikko wrote:
On 23/09/2026 16:39, olcott wrote:
On 9/23/2026 4:45 AM, Mikko wrote:
On 22/09/2026 18:01, olcott wrote:
Provability in PA requires a sequence of syntactic
inference steps that remain totally in PA and do not
leap outside of PA into any model of arithmetic.
That is not specific to PA. That is what formal provability means >>>>>>>>>> for every formal theory.
If that was true then everyone would understand that
Wittgenstein is correct and Gödel is wrong.
That is clearly false. The meaning of the word "provability" (or >>>>>>>> amy
word) has no consequences beyond meanings of words.
The rest of the quoted message and my response to it are irrelevant >>>>>>>> to my comment above.
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's
system' means: the opposite has been proved in
Russell's system.
https://www.liarparadox.org/Wittgenstein.pdf
With that defimition the words "true" and "false" lose a part of >>>>>>>> their
useful meanings. In order to say "proved in Russell's system" >>>>>>>> one can
say "proved in Russell's system". A statement proved in Russell's >>>>>>>> system can also be called a "theorem of Russell's system".
Gödel's result is only achieved by going outside of Peano
Arithmetic into the standard model of Arithmetic.
No, it is only necessary to go to some system where the results >>>>>>>> make
sense. Gödel's most important results are not about numbers but >>>>>>>> about
formal theories. A map of Italy does not help if you have problems >>>>>>>> with finding your way in Tokyo.
Proof Theoretic Semantics does this differently:
What is the appropriate notion of truth for sentences >>>>>>>>> whose meanings are understood in epistemic terms
such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified >>>>>>>>> with the existence of proofs or grounds.
Truth as an Epistemic Notion --- Dag Prawitz (2011)
https://link.springer.com/article/10.1007/s11245-011-9107-6
Truth theoretic semantics is not the kind of semantics we
usually want.
Many useful things can be done with no semantics at all.
"true on the basis of meaning expressed in language" is
reliably computable for the entire body of general knowledge.
There is no coherent sense of "incompleteness" or undecidability >>>>>>> within such a system.
Forst of all, that is a change of topic, apparently in order to make >>>>>> people to forget what you said earlier. WHich is in fact a good idea, >>>>>> there is no need to remember your errors. But it is useful to
remember
that most of what you say is erroneous.
Whether "true on the basis of meaning expressed in language" is
computable depends on a language. For an uninterpreted formal
language nothing in the language is true, so the computation is
trivial.
The body of general knowledge expressed in formalized English
can be formalized in CycL. That "cats are animals" is one
example of "true on the basis of meaning expressed in language"
An ordinary dictionary like https://en.wiktionary.org/wiki/cat can
tell that cats are animals. If you want to get anybody interested
you need more impressive examples.
Just that one example refutes Quine's
Two Dogmas of Empiricism
https://www.ditext.com/quine/quine.html
Quine's "Two Dogams of Empiricism" is an example of what most
people consider uninteresting.
Math and logic people don't care about this because they
take alternative foundations as inherently incorrect.
Most philosophers take Quine ideas to be an established
verified fact, thus shutting off making
"true on the basis of meaning expressed in language"
reliably computable before it has a chance to begin.
Anyway, "true on the basis of meaning expressed in language" does
not cover the truths we usually need to know. An example is the
whether tomorrow: it is not computable. A prediction can be computed >>>>>> but sometimes the truth is different.
The scope of the body of general knowledge that is
"true on the basis of meaning expressed in language"
Tomorrow's weather is outside of this scope.
And so are most of the things that people might want to find out.
The best that any system of knowledge can possibly
provide is elements of the body of knowledge. Requiring
psychic ability is a bad requirement.
Most people care about real world knowledge, which tends to be
uncertain but is often useful anyway.
I am not sure what you mean when we limit the
scope to the body of general knowledge. Unknowns
are definitely excluded.
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