On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference rules
ensures that semantic problems don't affect the study of proofs
and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an
indirect proof is simpler than a direct one, and therefore more
convincing. But without the principle of explosion it might be
harder or even impossible to find one, depending on what there is
instead.
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference rules
ensures that semantic problems don't affect the study of proofs
and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an
indirect proof is simpler than a direct one, and therefore more
convincing. But without the principle of explosion it might be
harder or even impossible to find one, depending on what there is
instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
By combining the ideas from about seven papers together
we can derive: reCx (Provable(x) rcA True(x))
Makes "true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference rules
ensures that semantic problems don't affect the study of proofs
and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an
indirect proof is simpler than a direct one, and therefore more
convincing. But without the principle of explosion it might be
harder or even impossible to find one, depending on what there is
instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference rules
ensures that semantic problems don't affect the study of proofs
and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an
indirect proof is simpler than a direct one, and therefore more
convincing. But without the principle of explosion it might be
harder or even impossible to find one, depending on what there is
instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be
meaningful even when it is not known whether it is provable. For
example, the program fragment
-a if (x < 5) {
-a-a-a show(x);
-a }
is quite meaningful even when one cannot prove or even know whether
x at the time of execution is less than 5.
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference rules
ensures that semantic problems don't affect the study of proofs
and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an
indirect proof is simpler than a direct one, and therefore more
convincing. But without the principle of explosion it might be
harder or even impossible to find one, depending on what there is
instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be
meaningful even when it is not known whether it is provable. For
example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
On 2/10/26 8:37 AM, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference rules
ensures that semantic problems don't affect the study of proofs
and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an
indirect proof is simpler than a direct one, and therefore more
convincing. But without the principle of explosion it might be
harder or even impossible to find one, depending on what there is
instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
By combining the ideas from about seven papers together
we can derive: reCx (Provable(x) rcA True(x))
Makes "true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The problem is that trying to do that (the way you are trying to define
it) just removes the ability to define mathematics.
Since Mathematics is part of our current "Knowledge", it means you claim
of result is just a lie.
We completely replace the foundation of Truth Conditional Semantics with Proof Theoretic Semantics (PTS). Then expressions are "true on the basis
of meaning expressed in language" only to the extent that all their
meaning comes from inferential relations to other expressions of that language. This is the purely linguistic PTS notion of truth having no connections outside the inferential system.
"true on the basis of meaning expressed in language" are elements of the body of verbal knowledge. This can include basic facts of the actual
world as stipulated axioms of the verbal model of the actual world. This bridges the divide between the analytic/synthetic distinction.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
-a What is the appropriate notion of truth for sentences
-a whose meanings are understood in epistemic terms such
-a as proof or ground for an assertion? It seems that the
-a truth of such sentences has to be identified with the
-a existence of proofs or grounds...
-a Prawitz, D. (2012). Truth as an Epistemic Notion. Topoi, 31(1), 9rCo16
-a https://doi.org/10.1007/s11245-011-9107-6
-a 1.2 Inferentialism, intuitionism, anti-realism
-a Proof-theoretic semantics is inherently inferential,
-a as it is inferential activity which manifests itself
-a in proofs. It thus belongs to inferentialism (a term
-a coined by Brandom, see his 1994; 2000) according to
-a which inferences and the rules of inference establish
-a the meaning of expressions
-a Schroeder-Heister, Peter, 2024 "Proof-Theoretic Semantics"
https://plato.stanford.edu/entries/proof-theoretic-semantics/ #InfeIntuAntiReal
When we understand that linguistic truth (just like
an ordinary dictionary) expressions of language only
get their semantic meaning from other expressions of
language then we directly understand entirely based on
the meaning of words that when no such connection exists
then no semantic meaning is derived.
When we understand this then we can see that
"true on the basis of meaning expressed in language"
is reliably computable for the entire body of knowledge
by finite string transformations applied to finite strings.
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference rules >>>>>>> ensures that semantic problems don't affect the study of proofs
and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an
indirect proof is simpler than a direct one, and therefore more
convincing. But without the principle of explosion it might be
harder or even impossible to find one, depending on what there is
instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be
meaningful even when it is not known whether it is provable. For
example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
On 2/11/26 9:17 PM, olcott wrote:
We completely replace the foundation of Truth Conditional Semantics
with Proof Theoretic Semantics (PTS). Then expressions are "true on
the basis of meaning expressed in language" only to the extent that
all their meaning comes from inferential relations to other
expressions of that language. This is the purely linguistic PTS notion
of truth having no connections outside the inferential system.
Now, since you just changed the basic operation of ALL logic, you need
to re-prove what each system can do.
You also need to handle the axioms that don't really have meaning under Proof Theoretic Semantics, like induction, that validate that a
statement had "meaning" without proof, and provide a way to sometimes
prove it.
"true on the basis of meaning expressed in language" are elements of
the body of verbal knowledge. This can include basic facts of the
actual world as stipulated axioms of the verbal model of the actual
world. This bridges the divide between the analytic/synthetic
distinction.
But, "the basis of meaning" in some systems specifiically ALLOW for "unprovable" things to be true.
Note, "Facts" of the actual world can NOT be axioms, as they are categorically different type of things.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
-a-a What is the appropriate notion of truth for sentences
-a-a whose meanings are understood in epistemic terms such
-a-a as proof or ground for an assertion? It seems that the
-a-a truth of such sentences has to be identified with the
-a-a existence of proofs or grounds...
-a-a Prawitz, D. (2012). Truth as an Epistemic Notion. Topoi, 31(1), 9rCo16 >> -a-a https://doi.org/10.1007/s11245-011-9107-6
Note, "approximate".
You are quoting ideas about general Philosophy, which seek a way to try
to define what truth means, vs Formal Logic, which STARTS with a
definition of what Truth is, a definition you began by changing, and
thus you need to reexamine ALL the works of logic to see what changes,
As mentioned, it seems you lose mathematics, as you suddenly can't
finitely express it without an axiom which is based on Truth-Conditional logic, which you reject.
-a-a 1.2 Inferentialism, intuitionism, anti-realism
-a-a Proof-theoretic semantics is inherently inferential,
-a-a as it is inferential activity which manifests itself
-a-a in proofs. It thus belongs to inferentialism (a term
-a-a coined by Brandom, see his 1994; 2000) according to
-a-a which inferences and the rules of inference establish
-a-a the meaning of expressions
-a-a Schroeder-Heister, Peter, 2024 "Proof-Theoretic Semantics"
https://plato.stanford.edu/entries/proof-theoretic-semantics/
#InfeIntuAntiReal
When we understand that linguistic truth (just like
an ordinary dictionary) expressions of language only
get their semantic meaning from other expressions of
language then we directly understand entirely based on
the meaning of words that when no such connection exists
then no semantic meaning is derived.
And, you thus get a system with no "root" of meaning, and thus no actual ability to prove things.
When we understand this then we can see that
"true on the basis of meaning expressed in language"
is reliably computable for the entire body of knowledge
by finite string transformations applied to finite strings.
Nope, Just shows that you don't understand what you are talking about.
Either you definitions define that math (and related systems) is outside your logic, or it accepts that some things are not computable.
The problem is either you accept math with its infinite chains of
deduction, or you need an infinite number of "facts" to express "all knowledge".
TRY to express all that can be known about arithmatic, even simple
addition, without either rules that are allowed to be applied in
unbounded number, or an infinite number of base axioms.
Your problem is you mind just doesn't seem to understand the concept of
the infinite system, bevause it is just too small.
On 2026-02-10 21:59, olcott wrote:
We completely replace the foundation of truth conditional
semantics with proof theoretic semantics. Then expressions
are "true on the basis of meaning expressed in language"
only to the extent that all their meaning comes from
inferential relations to other expressions of that language.
This is a purely linguistic PTS notion of truth with no
connections outside the inferential system.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Proof-theoretic semantics makes no such claim.
That's your claim and you
should stop attributing it to others.
Andr|-
On 2/12/2026 6:29 AM, Richard Damon wrote:
On 2/11/26 9:17 PM, olcott wrote:
We completely replace the foundation of Truth Conditional Semantics
with Proof Theoretic Semantics (PTS). Then expressions are "true on
the basis of meaning expressed in language" only to the extent that
all their meaning comes from inferential relations to other
expressions of that language. This is the purely linguistic PTS
notion of truth having no connections outside the inferential system.
Now, since you just changed the basic operation of ALL logic, you need
to re-prove what each system can do.
You also need to handle the axioms that don't really have meaning
under Proof Theoretic Semantics, like induction, that validate that a
statement had "meaning" without proof, and provide a way to sometimes
prove it.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
"true on the basis of meaning expressed in language" are elements of
the body of verbal knowledge. This can include basic facts of the
actual world as stipulated axioms of the verbal model of the actual
world. This bridges the divide between the analytic/synthetic
distinction.
But, "the basis of meaning" in some systems specifiically ALLOW for
"unprovable" things to be true.
Only with a wrong-headed notion of:
"true on the basis of meaning expressed in language"
Note, "Facts" of the actual world can NOT be axioms, as they are
categorically different type of things.
Facts are expressions of language that are necessarily true.
Without language the world is merely a continuous stream
of physical sensations not even a "state of affairs" exists.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
-a-a What is the appropriate notion of truth for sentences
-a-a whose meanings are understood in epistemic terms such
-a-a as proof or ground for an assertion? It seems that the
-a-a truth of such sentences has to be identified with the
-a-a existence of proofs or grounds...
-a-a Prawitz, D. (2012). Truth as an Epistemic Notion. Topoi, 31(1), 9rCo16 >>> -a-a https://doi.org/10.1007/s11245-011-9107-6
Note, "approximate".
"appropriate" not "approximate"
You are quoting ideas about general Philosophy, which seek a way to
try to define what truth means, vs Formal Logic, which STARTS with a
incorrect
definition of what Truth is, a definition you began by changing, and
thus you need to reexamine ALL the works of logic to see what changes,
As mentioned, it seems you lose mathematics, as you suddenly can't
finitely express it without an axiom which is based on Truth-
Conditional logic, which you reject.
*This is ALL that changes*
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
-a-a 1.2 Inferentialism, intuitionism, anti-realism
-a-a Proof-theoretic semantics is inherently inferential,
-a-a as it is inferential activity which manifests itself
-a-a in proofs. It thus belongs to inferentialism (a term
-a-a coined by Brandom, see his 1994; 2000) according to
-a-a which inferences and the rules of inference establish
-a-a the meaning of expressions
-a-a Schroeder-Heister, Peter, 2024 "Proof-Theoretic Semantics"
https://plato.stanford.edu/entries/proof-theoretic-semantics/
#InfeIntuAntiReal
When we understand that linguistic truth (just like
an ordinary dictionary) expressions of language only
get their semantic meaning from other expressions of
language then we directly understand entirely based on
the meaning of words that when no such connection exists
then no semantic meaning is derived.
And, you thus get a system with no "root" of meaning, and thus no
actual ability to prove things.
No you get a system that knows how to reject the
Liar Paradox as meaningless nonsense instead of
the foundation of Tarski Undefinability.
Here are the Tarski Undefinability Theorem proof steps
(1) x ree Provable if and only if p
(2) x ree True if and only if p
(3) x ree Provable if and only if x ree True.
(4) either x ree True or x|a ree True;
(5) if x ree Provable, then x ree True;
(6) if x|a ree Provable, then x|a ree True;
(7) x ree True
(8) x ree Provable
(9) x|a ree Provable
These two pages are his actual complete proof https://liarparadox.org/Tarski_275_276.pdf
Within PTS Tarski's line (5) becomes an axiom
that rejects his line (3) thus causing his
whole proof to completely fail.
When we understand this then we can see that
"true on the basis of meaning expressed in language"
is reliably computable for the entire body of knowledge
by finite string transformations applied to finite strings.
Nope, Just shows that you don't understand what you are talking about.
No it shows that you don;t understand Proof Theoretic Semantics
deeply enough.
Understanding that this is true is all that you need.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
Either you definitions define that math (and related systems) is
outside your logic, or it accepts that some things are not computable.
The problem is either you accept math with its infinite chains of
deduction, or you need an infinite number of "facts" to express "all
knowledge".
TRY to express all that can be known about arithmatic, even simple
addition, without either rules that are allowed to be applied in
unbounded number, or an infinite number of base axioms.
Your problem is you mind just doesn't seem to understand the concept
of the infinite system, bevause it is just too small.
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference rules >>>>>>>> ensures that semantic problems don't affect the study of proofs >>>>>>>> and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an
indirect proof is simpler than a direct one, and therefore more
convincing. But without the principle of explosion it might be
harder or even impossible to find one, depending on what there is
instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be
meaningful even when it is not known whether it is provable. For
example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference rules >>>>>>>>> ensures that semantic problems don't affect the study of proofs >>>>>>>>> and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an >>>>>>> indirect proof is simpler than a direct one, and therefore more
convincing. But without the principle of explosion it might be
harder or even impossible to find one, depending on what there is >>>>>>> instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be
meaningful even when it is not known whether it is provable. For
example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of
a sentence and no way to find out the truth of that sentence is not
known and cannot be computed.
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference rules >>>>>>>>>> ensures that semantic problems don't affect the study of proofs >>>>>>>>>> and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an >>>>>>>> indirect proof is simpler than a direct one, and therefore more >>>>>>>> convincing. But without the principle of explosion it might be >>>>>>>> harder or even impossible to find one, depending on what there is >>>>>>>> instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be
meaningful even when it is not known whether it is provable. For
example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know whether >>>>>> x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of
a sentence and no way to find out the truth of that sentence is not
known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference >>>>>>>>>>> rules
ensures that semantic problems don't affect the study of proofs >>>>>>>>>>> and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an >>>>>>>>> indirect proof is simpler than a direct one, and therefore more >>>>>>>>> convincing. But without the principle of explosion it might be >>>>>>>>> harder or even impossible to find one, depending on what there is >>>>>>>>> instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know whether >>>>>>> x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024) >>>> reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of
a sentence and no way to find out the truth of that sentence is not
known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano arithmetic is infinite so it cannot be in a finite graph.
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False.
Logic is not paralyzed. Separating semantics from inference >>>>>>>>>>> rules
ensures that semantic problems don't affect the study of proofs >>>>>>>>>>> and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an >>>>>>>>> indirect proof is simpler than a direct one, and therefore more >>>>>>>>> convincing. But without the principle of explosion it might be >>>>>>>>> harder or even impossible to find one, depending on what there is >>>>>>>>> instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know whether >>>>>>> x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024) >>>> reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of
a sentence and no way to find out the truth of that sentence is not
known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano arithmetic is infinite so it cannot be in a finite graph.
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from inference >>>>>>>>>>>> rules
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False. >>>>>>>>>>>>
ensures that semantic problems don't affect the study of proofs >>>>>>>>>>>> and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an >>>>>>>>>> indirect proof is simpler than a direct one, and therefore more >>>>>>>>>> convincing. But without the principle of explosion it might be >>>>>>>>>> harder or even impossible to find one, depending on what there is >>>>>>>>>> instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know whether >>>>>>>> x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024) >>>>> reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of
a sentence and no way to find out the truth of that sentence is not
known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano
arithmetic is infinite so it cannot be in a finite graph.
So it looks like you are saying that no one can count
until after they first count to infinity?
I intend that the above axioms mean that every expression
in T is "true on the basis of meaning expressed in language"
in T is meaningless iff it is unprovable in T.
reCx (Provable(x) rco True(x)) is probably best as a bijection
because an expression that is not connected by back-chained
inference to the axioms of T is untrue in T. This is the
same as a word that is not defined in a dictionary has
no meaning in this dictionary.
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from inference >>>>>>>>>>>> rules
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False. >>>>>>>>>>>>
ensures that semantic problems don't affect the study of proofs >>>>>>>>>>>> and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an >>>>>>>>>> indirect proof is simpler than a direct one, and therefore more >>>>>>>>>> convincing. But without the principle of explosion it might be >>>>>>>>>> harder or even impossible to find one, depending on what there is >>>>>>>>>> instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know whether >>>>>>>> x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024) >>>>> reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of
a sentence and no way to find out the truth of that sentence is not
known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano
arithmetic is infinite so it cannot be in a finite graph.
The specialized nature of my work has exceeded the technical
knowledge of people here and most everywhere else.
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from inference >>>>>>>>>>>> rules
Conventional logic and math have been paralyzed for
many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False. >>>>>>>>>>>>
ensures that semantic problems don't affect the study of proofs >>>>>>>>>>>> and provability.
Then you end up with screwy stuff such as the psychotic
https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. Often an >>>>>>>>>> indirect proof is simpler than a direct one, and therefore more >>>>>>>>>> convincing. But without the principle of explosion it might be >>>>>>>>>> harder or even impossible to find one, depending on what there is >>>>>>>>>> instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression
is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless.
reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know whether >>>>>>>> x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024) >>>>> reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of
a sentence and no way to find out the truth of that sentence is not
known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano
arithmetic is infinite so it cannot be in a finite graph.
So it looks like you are saying that no one can count
until after they first count to infinity?
On 2/14/26 3:59 PM, polcott wrote:
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from inference >>>>>>>>>>>>> rules
Conventional logic and math have been paralyzed for >>>>>>>>>>>>>> many decades by trying to force-fit semantically
ill-formed expressions into the box of True or False. >>>>>>>>>>>>>
ensures that semantic problems don't affect the study of >>>>>>>>>>>>> proofs
and provability.
Then you end up with screwy stuff such as the psychotic >>>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. >>>>>>>>>>> Often an
indirect proof is simpler than a direct one, and therefore more >>>>>>>>>>> convincing. But without the principle of explosion it might be >>>>>>>>>>> harder or even impossible to find one, depending on what >>>>>>>>>>> there is
instead.
Completely replacing the foundation of truth conditional
semantics with proof theoretic semantics then an expression >>>>>>>>>> is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised
of its inferential relations to other expressions of that
language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions
lacking a "well-founded justification tree" as meaningless. >>>>>>>>>> reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know
whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024) >>>>>> reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of >>>>> a sentence and no way to find out the truth of that sentence is not
known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano
arithmetic is infinite so it cannot be in a finite graph.
So it looks like you are saying that no one can count
until after they first count to infinity?
No, he is pointing out that if you claim to encode ALL the knowledge
that is expressible in language, you can't stop until you finish, and
since there are an infinite number of those in Peano Arithmetic, you
can't stop at any finite number.
On 15/02/2026 01:41, Richard Damon wrote:
On 2/14/26 3:59 PM, polcott wrote:
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from inference >>>>>>>>>>>>>> rules
Conventional logic and math have been paralyzed for >>>>>>>>>>>>>>> many decades by trying to force-fit semantically >>>>>>>>>>>>>>> ill-formed expressions into the box of True or False. >>>>>>>>>>>>>>
ensures that semantic problems don't affect the study of >>>>>>>>>>>>>> proofs
and provability.
Then you end up with screwy stuff such as the psychotic >>>>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. >>>>>>>>>>>> Often an
indirect proof is simpler than a direct one, and therefore more >>>>>>>>>>>> convincing. But without the principle of explosion it might be >>>>>>>>>>>> harder or even impossible to find one, depending on what >>>>>>>>>>>> there is
instead.
Completely replacing the foundation of truth conditional >>>>>>>>>>> semantics with proof theoretic semantics then an expression >>>>>>>>>>> is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised >>>>>>>>>>> of its inferential relations to other expressions of that >>>>>>>>>>> language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions >>>>>>>>>>> lacking a "well-founded justification tree" as meaningless. >>>>>>>>>>> reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know >>>>>>>>>> whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/ >>>>>>>>>
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024) >>>>>>> reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of >>>>>> a sentence and no way to find out the truth of that sentence is not >>>>>> known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano >>>> arithmetic is infinite so it cannot be in a finite graph.
So it looks like you are saying that no one can count
until after they first count to infinity?
No, he is pointing out that if you claim to encode ALL the knowledge
that is expressible in language, you can't stop until you finish, and
since there are an infinite number of those in Peano Arithmetic, you
can't stop at any finite number.
You're talking about decompressing the encoding of knowledge. Stating
the axioms (and full detail of inference rules) symbolically is
sufficient to /encode/ the knowledge if you have lambda calculus or
illative combinatory logic.
On 15/02/2026 01:41, Richard Damon wrote:
On 2/14/26 3:59 PM, polcott wrote:
So it looks like you are saying that no one can count
until after they first count to infinity?
No, he is pointing out that if you claim to encode ALL the knowledge
that is expressible in language, you can't stop until you finish, and
since there are an infinite number of those in Peano Arithmetic, you
can't stop at any finite number.
You're talking about decompressing the encoding of knowledge. Stating
the axioms (and full detail of inference rules) symbolically is
sufficient to /encode/ the knowledge if you have lambda calculus or
illative combinatory logic.
----
Tristan Wibberley
[ Followup-To: set ]
In comp.theory Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 15/02/2026 01:41, Richard Damon wrote:
On 2/14/26 3:59 PM, polcott wrote:
[ .... ]
So it looks like you are saying that no one can count
until after they first count to infinity?
No, he is pointing out that if you claim to encode ALL the knowledge
that is expressible in language, you can't stop until you finish, and
since there are an infinite number of those in Peano Arithmetic, you
can't stop at any finite number.
You're talking about decompressing the encoding of knowledge. Stating
the axioms (and full detail of inference rules) symbolically is
sufficient to /encode/ the knowledge if you have lambda calculus or
illative combinatory logic.
Olcott's system, by his own admission, is insufficiently powerful to
express a true proposition it can't prove. Thus Peano arithmetic is
outside of its scope. You can't count in Olcott's system.
--
Tristan Wibberley
Olcott's system, by his own admission, is insufficiently powerful to
express a true proposition it can't prove. Thus Peano arithmetic is
outside of its scope. You can't count in Olcott's system.
On 15/02/2026 01:41, Richard Damon wrote:
On 2/14/26 3:59 PM, polcott wrote:
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from inference >>>>>>>>>>>>>> rules
Conventional logic and math have been paralyzed for >>>>>>>>>>>>>>> many decades by trying to force-fit semantically >>>>>>>>>>>>>>> ill-formed expressions into the box of True or False. >>>>>>>>>>>>>>
ensures that semantic problems don't affect the study of >>>>>>>>>>>>>> proofs
and provability.
Then you end up with screwy stuff such as the psychotic >>>>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. >>>>>>>>>>>> Often an
indirect proof is simpler than a direct one, and therefore more >>>>>>>>>>>> convincing. But without the principle of explosion it might be >>>>>>>>>>>> harder or even impossible to find one, depending on what >>>>>>>>>>>> there is
instead.
Completely replacing the foundation of truth conditional >>>>>>>>>>> semantics with proof theoretic semantics then an expression >>>>>>>>>>> is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised >>>>>>>>>>> of its inferential relations to other expressions of that >>>>>>>>>>> language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions >>>>>>>>>>> lacking a "well-founded justification tree" as meaningless. >>>>>>>>>>> reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know >>>>>>>>>> whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/ >>>>>>>>>
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024) >>>>>>> reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of >>>>>> a sentence and no way to find out the truth of that sentence is not >>>>>> known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano >>>> arithmetic is infinite so it cannot be in a finite graph.
So it looks like you are saying that no one can count
until after they first count to infinity?
No, he is pointing out that if you claim to encode ALL the knowledge
that is expressible in language, you can't stop until you finish, and
since there are an infinite number of those in Peano Arithmetic, you
can't stop at any finite number.
You're talking about decompressing the encoding of knowledge. Stating
the axioms (and full detail of inference rules) symbolically is
sufficient to /encode/ the knowledge if you have lambda calculus or
illative combinatory logic.
On 2/15/2026 1:27 PM, Alan Mackenzie wrote:
[ Followup-To: set ]
In comp.theory Tristan Wibberley
<tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
On 15/02/2026 01:41, Richard Damon wrote:
On 2/14/26 3:59 PM, polcott wrote:
[ .... ]
So it looks like you are saying that no one can count
until after they first count to infinity?
No, he is pointing out that if you claim to encode ALL the knowledge
that is expressible in language, you can't stop until you finish, and
since there are an infinite number of those in Peano Arithmetic, you
can't stop at any finite number.
You're talking about decompressing the encoding of knowledge. Stating
the axioms (and full detail of inference rules) symbolically is
sufficient to /encode/ the knowledge if you have lambda calculus or
illative combinatory logic.
Olcott's system, by his own admission, is insufficiently powerful to
express a true proposition it can't prove.-a Thus Peano arithmetic is
outside of its scope.-a You can't count in Olcott's system.
Unintentionally counter-factual.
You merely lack a sufficient grasp of Proof Theoretic Semantics.
PTS is the exact same ideas that-a have been saying for many
years. The only difference is that now you can look up and
see all of the details of exactly how I was right all along.
--
Tristan Wibberley
On 2/15/2026 3:18 AM, Mikko wrote:
On 14/02/2026 17:31, polcott wrote:
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from >>>>>>>>>>>>>> inference rules
Conventional logic and math have been paralyzed for >>>>>>>>>>>>>>> many decades by trying to force-fit semantically >>>>>>>>>>>>>>> ill-formed expressions into the box of True or False. >>>>>>>>>>>>>>
ensures that semantic problems don't affect the study of >>>>>>>>>>>>>> proofs
and provability.
Then you end up with screwy stuff such as the psychotic >>>>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
That you call it psychotic does not make it less useful. >>>>>>>>>>>> Often an
indirect proof is simpler than a direct one, and therefore more >>>>>>>>>>>> convincing. But without the principle of explosion it might be >>>>>>>>>>>> harder or even impossible to find one, depending on what >>>>>>>>>>>> there is
instead.
Completely replacing the foundation of truth conditional >>>>>>>>>>> semantics with proof theoretic semantics then an expression >>>>>>>>>>> is "true on the basis of meaning expressed in language"
only to the extent that its meaning is entirely comprised >>>>>>>>>>> of its inferential relations to other expressions of that >>>>>>>>>>> language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions >>>>>>>>>>> lacking a "well-founded justification tree" as meaningless. >>>>>>>>>>> reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know >>>>>>>>>> whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/ >>>>>>>>>
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from
"true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024) >>>>>>> reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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...
Which means that if it is not determined whether there is a proof of >>>>>> a sentence and no way to find out the truth of that sentence is
not known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano >>>> arithmetic is infinite so it cannot be in a finite graph.
The specialized nature of my work has exceeded the technical
knowledge of people here and most everywhere else.
That an inifinite sent cannot be in a finite graph may exceed your
technical knowledge but certainly doesn't everyone else's.
reCx((x > 10) rcA (x > 0))
Does not mean to test every x.
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
On 15/02/2026 15:02, polcott wrote:
On 2/15/2026 3:18 AM, Mikko wrote:
On 14/02/2026 17:31, polcott wrote:
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:That you call it psychotic does not make it less useful. >>>>>>>>>>>>> Often an
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from >>>>>>>>>>>>>>> inference rules
Conventional logic and math have been paralyzed for >>>>>>>>>>>>>>>> many decades by trying to force-fit semantically >>>>>>>>>>>>>>>> ill-formed expressions into the box of True or False. >>>>>>>>>>>>>>>
ensures that semantic problems don't affect the study of >>>>>>>>>>>>>>> proofs
and provability.
Then you end up with screwy stuff such as the psychotic >>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion >>>>>>>>>>>>>
indirect proof is simpler than a direct one, and therefore >>>>>>>>>>>>> more
convincing. But without the principle of explosion it might be >>>>>>>>>>>>> harder or even impossible to find one, depending on what >>>>>>>>>>>>> there is
instead.
Completely replacing the foundation of truth conditional >>>>>>>>>>>> semantics with proof theoretic semantics then an expression >>>>>>>>>>>> is "true on the basis of meaning expressed in language" >>>>>>>>>>>> only to the extent that its meaning is entirely comprised >>>>>>>>>>>> of its inferential relations to other expressions of that >>>>>>>>>>>> language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions >>>>>>>>>>>> lacking a "well-founded justification tree" as meaningless. >>>>>>>>>>>> reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know >>>>>>>>>>> whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/ >>>>>>>>>>
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from >>>>>>>>> "true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012) >>>>>>>>
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...
Which means that if it is not determined whether there is a proof of >>>>>>> a sentence and no way to find out the truth of that sentence is >>>>>>> not known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano >>>>> arithmetic is infinite so it cannot be in a finite graph.
The specialized nature of my work has exceeded the technical
knowledge of people here and most everywhere else.
That an inifinite sent cannot be in a finite graph may exceed your
technical knowledge but certainly doesn't everyone else's.
reCx((x > 10) rcA (x > 0))
Does not mean to test every x.
Irrelevant. That is only one sentence, not infinitely many.
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate
(which, if interpreted accordint to the usual meaning of either symbol,
is uncomputable).
But that is irrelevant, too. The set of provable sentences is infinite
so it cannot be in a finite graph.
On 15/02/2026 15:02, polcott wrote:
On 2/15/2026 3:18 AM, Mikko wrote:
On 14/02/2026 17:31, polcott wrote:
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:That you call it psychotic does not make it less useful. >>>>>>>>>>>>> Often an
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from >>>>>>>>>>>>>>> inference rules
Conventional logic and math have been paralyzed for >>>>>>>>>>>>>>>> many decades by trying to force-fit semantically >>>>>>>>>>>>>>>> ill-formed expressions into the box of True or False. >>>>>>>>>>>>>>>
ensures that semantic problems don't affect the study of >>>>>>>>>>>>>>> proofs
and provability.
Then you end up with screwy stuff such as the psychotic >>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion >>>>>>>>>>>>>
indirect proof is simpler than a direct one, and therefore >>>>>>>>>>>>> more
convincing. But without the principle of explosion it might be >>>>>>>>>>>>> harder or even impossible to find one, depending on what >>>>>>>>>>>>> there is
instead.
Completely replacing the foundation of truth conditional >>>>>>>>>>>> semantics with proof theoretic semantics then an expression >>>>>>>>>>>> is "true on the basis of meaning expressed in language" >>>>>>>>>>>> only to the extent that its meaning is entirely comprised >>>>>>>>>>>> of its inferential relations to other expressions of that >>>>>>>>>>>> language. AKA linguistic truth determined by semantic
entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions >>>>>>>>>>>> lacking a "well-founded justification tree" as meaningless. >>>>>>>>>>>> reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined to be >>>>>>>>>>> meaningful even when it is not known whether it is provable. For >>>>>>>>>>> example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know >>>>>>>>>>> whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/ >>>>>>>>>>
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from >>>>>>>>> "true on the basis of meaning expressed in language". There
is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012) >>>>>>>>
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...
Which means that if it is not determined whether there is a proof of >>>>>>> a sentence and no way to find out the truth of that sentence is >>>>>>> not known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order Peano >>>>> arithmetic is infinite so it cannot be in a finite graph.
The specialized nature of my work has exceeded the technical
knowledge of people here and most everywhere else.
That an inifinite sent cannot be in a finite graph may exceed your
technical knowledge but certainly doesn't everyone else's.
reCx((x > 10) rcA (x > 0))
Does not mean to test every x.
Irrelevant. That is only one sentence, not infinitely many.
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate
(which, if interpreted accordint to the usual meaning of either symbol,
is uncomputable).
But that is irrelevant, too. The set of provable sentences is infinite
so it cannot be in a finite graph.
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
On 2/15/2026 3:18 AM, Mikko wrote:
On 14/02/2026 17:31, polcott wrote:
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:That you call it psychotic does not make it less useful. >>>>>>>>>>>>>> Often an
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from >>>>>>>>>>>>>>>> inference rules
Conventional logic and math have been paralyzed for >>>>>>>>>>>>>>>>> many decades by trying to force-fit semantically >>>>>>>>>>>>>>>>> ill-formed expressions into the box of True or False. >>>>>>>>>>>>>>>>
ensures that semantic problems don't affect the study of >>>>>>>>>>>>>>>> proofs
and provability.
Then you end up with screwy stuff such as the psychotic >>>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion >>>>>>>>>>>>>>
indirect proof is simpler than a direct one, and therefore >>>>>>>>>>>>>> more
convincing. But without the principle of explosion it >>>>>>>>>>>>>> might be
harder or even impossible to find one, depending on what >>>>>>>>>>>>>> there is
instead.
Completely replacing the foundation of truth conditional >>>>>>>>>>>>> semantics with proof theoretic semantics then an expression >>>>>>>>>>>>> is "true on the basis of meaning expressed in language" >>>>>>>>>>>>> only to the extent that its meaning is entirely comprised >>>>>>>>>>>>> of its inferential relations to other expressions of that >>>>>>>>>>>>> language. AKA linguistic truth determined by semantic >>>>>>>>>>>>> entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions >>>>>>>>>>>>> lacking a "well-founded justification tree" as meaningless. >>>>>>>>>>>>> reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined >>>>>>>>>>>> to be
meaningful even when it is not known whether it is provable. >>>>>>>>>>>> For
example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know >>>>>>>>>>>> whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/ >>>>>>>>>>>
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from >>>>>>>>>> "true on the basis of meaning expressed in language". There >>>>>>>>>> is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister >>>>>>>>> 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012) >>>>>>>>>
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...
Which means that if it is not determined whether there is a
proof of
a sentence and no way to find out the truth of that sentence is >>>>>>>> not known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order
Peano
arithmetic is infinite so it cannot be in a finite graph.
The specialized nature of my work has exceeded the technical
knowledge of people here and most everywhere else.
That an inifinite sent cannot be in a finite graph may exceed your
technical knowledge but certainly doesn't everyone else's.
reCx((x > 10) rcA (x > 0))
Does not mean to test every x.
Irrelevant. That is only one sentence, not infinitely many.
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate
(which, if interpreted accordint to the usual meaning of either symbol,
is uncomputable).
What do you think that this means: PA reo x ?
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
On 2/15/2026 3:18 AM, Mikko wrote:
On 14/02/2026 17:31, polcott wrote:
On 2/14/2026 3:14 AM, Mikko wrote:
On 13/02/2026 15:32, olcott wrote:
On 2/13/2026 2:30 AM, Mikko wrote:
On 12/02/2026 17:48, olcott wrote:
On 2/12/2026 2:11 AM, Mikko wrote:
On 11/02/2026 14:38, olcott wrote:
On 2/11/2026 4:51 AM, Mikko wrote:
On 10/02/2026 15:37, olcott wrote:
On 2/10/2026 3:06 AM, Mikko wrote:
On 09/02/2026 17:36, olcott wrote:
On 2/9/2026 8:57 AM, Mikko wrote:That you call it psychotic does not make it less useful. >>>>>>>>>>>>>> Often an
On 07/02/2026 18:43, olcott wrote:
Logic is not paralyzed. Separating semantics from >>>>>>>>>>>>>>>> inference rules
Conventional logic and math have been paralyzed for >>>>>>>>>>>>>>>>> many decades by trying to force-fit semantically >>>>>>>>>>>>>>>>> ill-formed expressions into the box of True or False. >>>>>>>>>>>>>>>>
ensures that semantic problems don't affect the study of >>>>>>>>>>>>>>>> proofs
and provability.
Then you end up with screwy stuff such as the psychotic >>>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion >>>>>>>>>>>>>>
indirect proof is simpler than a direct one, and therefore >>>>>>>>>>>>>> more
convincing. But without the principle of explosion it >>>>>>>>>>>>>> might be
harder or even impossible to find one, depending on what >>>>>>>>>>>>>> there is
instead.
Completely replacing the foundation of truth conditional >>>>>>>>>>>>> semantics with proof theoretic semantics then an expression >>>>>>>>>>>>> is "true on the basis of meaning expressed in language" >>>>>>>>>>>>> only to the extent that its meaning is entirely comprised >>>>>>>>>>>>> of its inferential relations to other expressions of that >>>>>>>>>>>>> language. AKA linguistic truth determined by semantic >>>>>>>>>>>>> entailment specified syntactically.
Well-founded proof-theoretic semantics reject expressions >>>>>>>>>>>>> lacking a "well-founded justification tree" as meaningless. >>>>>>>>>>>>> reCx (~Provable(T, x) rco Meaningless(T, x))
Usually it is thought that an expression can be determined >>>>>>>>>>>> to be
meaningful even when it is not known whether it is provable. >>>>>>>>>>>> For
example, the program fragment
-a-a if (x < 5) {
-a-a-a-a show(x);
-a-a }
is quite meaningful even when one cannot prove or even know >>>>>>>>>>>> whether
x at the time of execution is less than 5.
Only Proof-Theoretic Semantics
https://plato.stanford.edu/entries/proof-theoretic-semantics/ >>>>>>>>>>>
Can make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
In order to achieve that all arithmetic must be excluded from >>>>>>>>>> "true on the basis of meaning expressed in language". There >>>>>>>>>> is no way to compute wheter a sentence of the first order
Peano arithmetic is provable.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister >>>>>>>>> 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012) >>>>>>>>>
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...
Which means that if it is not determined whether there is a
proof of
a sentence and no way to find out the truth of that sentence is >>>>>>>> not known and cannot be computed.
Its all in a finite directed acyclic graph of knowledge.
No, it is not. The set of provable statements of the first order
Peano
arithmetic is infinite so it cannot be in a finite graph.
The specialized nature of my work has exceeded the technical
knowledge of people here and most everywhere else.
That an inifinite sent cannot be in a finite graph may exceed your
technical knowledge but certainly doesn't everyone else's.
reCx((x > 10) rcA (x > 0))
Does not mean to test every x.
Irrelevant. That is only one sentence, not infinitely many.
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate
(which, if interpreted accordint to the usual meaning of either symbol,
is uncomputable).
But that is irrelevant, too. The set of provable sentences is infinite
so it cannot be in a finite graph.
What do you think that this means: PA reo x
for a specific (finite string) x ?
On 16/02/2026 15:47, olcott wrote:
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate
(which, if interpreted accordint to the usual meaning of either symbol,
is uncomputable).
What do you think that this means: PA reo x ?
The exact meaning depends on the context and the meanings of the types
of the left and right side expressions. The usual metalogical meaning
is that x is a theorem of some variant of PA. If something else is
meant that should be specified in the opus where the expression is used.
On 2/17/2026 3:03 AM, Mikko wrote:
On 16/02/2026 15:47, olcott wrote:
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate
(which, if interpreted accordint to the usual meaning of either symbol, >>>> is uncomputable).
What do you think that this means: PA reo x ?
The exact meaning depends on the context and the meanings of the types
of the left and right side expressions. The usual metalogical meaning
is that x is a theorem of some variant of PA. If something else is
meant that should be specified in the opus where the expression is used.
Yes that is correct. What does that mean?
On 2/17/2026 3:03 AM, Mikko wrote:
On 16/02/2026 15:47, olcott wrote:
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate
(which, if interpreted accordint to the usual meaning of either symbol, >>>> is uncomputable).
What do you think that this means: PA reo x ?
The exact meaning depends on the context and the meanings of the types
of the left and right side expressions. The usual metalogical meaning
is that x is a theorem of some variant of PA. If something else is
meant that should be specified in the opus where the expression is used
Yes that is correct. What does that mean?
On 17/02/2026 14:59, polcott wrote:
On 2/17/2026 3:03 AM, Mikko wrote:
On 16/02/2026 15:47, olcott wrote:
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate
(which, if interpreted accordint to the usual meaning of either
symbol,
is uncomputable).
What do you think that this means: PA reo x ?
The exact meaning depends on the context and the meanings of the
types of the left and right side expressions. The usual metalogical
meaning
is that x is a theorem of some variant of PA. If something else is
meant that should be specified in the opus where the expression is used
Yes that is correct. What does that mean?
It means that the author must define the symbols in the opus they are
used.
On 2/18/2026 3:10 AM, Mikko wrote:
On 17/02/2026 14:59, polcott wrote:
On 2/17/2026 3:03 AM, Mikko wrote:
On 16/02/2026 15:47, olcott wrote:Yes that is correct. What does that mean?
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate >>>>>> (which, if interpreted accordint to the usual meaning of either
symbol,
is uncomputable).
What do you think that this means: PA reo x ?
The exact meaning depends on the context and the meanings of the
types of the left and right side expressions. The usual metalogical
meaning
is that x is a theorem of some variant of PA. If something else is
meant that should be specified in the opus where the expression is used >>>
It means that the author must define the symbols in the opus they are
used.
Is this your best answer or are you trying to be evasive?
On 18/02/2026 21:48, polcott wrote:
On 2/18/2026 3:10 AM, Mikko wrote:
On 17/02/2026 14:59, polcott wrote:
On 2/17/2026 3:03 AM, Mikko wrote:
On 16/02/2026 15:47, olcott wrote:
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate >>>>>>> (which, if interpreted accordint to the usual meaning of either >>>>>>> symbol,
is uncomputable).
What do you think that this means: PA reo x ?
The exact meaning depends on the context and the meanings of the
types of the left and right side expressions. The usual metalogical >>>>> meaning
is that x is a theorem of some variant of PA. If something else is
meant that should be specified in the opus where the expression is
used
Yes that is correct. What does that mean?
It means that the author must define the symbols in the opus they are
used.
Is this your best answer or are you trying to be evasive?
Whether another answer would be better is a matter of taste, at least
to some extent.
On 2/19/2026 4:06 AM, Mikko wrote:
On 18/02/2026 21:48, polcott wrote:
On 2/18/2026 3:10 AM, Mikko wrote:
On 17/02/2026 14:59, polcott wrote:
On 2/17/2026 3:03 AM, Mikko wrote:
On 16/02/2026 15:47, olcott wrote:
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate >>>>>>>> (which, if interpreted accordint to the usual meaning of either >>>>>>>> symbol,
is uncomputable).
What do you think that this means: PA reo x ?
The exact meaning depends on the context and the meanings of the
types of the left and right side expressions. The usual
metalogical meaning
is that x is a theorem of some variant of PA. If something else is >>>>>> meant that should be specified in the opus where the expression is >>>>>> used
Yes that is correct. What does that mean?
It means that the author must define the symbols in the opus they are
used.
Is this your best answer or are you trying to be evasive?
Whether another answer would be better is a matter of taste, at least
to some extent.
PA reo x
The correct answer is
A back-chained inference from x to the axioms of PA exists
On 2/19/26 6:47 AM, polcott wrote:
On 2/19/2026 4:06 AM, Mikko wrote:
On 18/02/2026 21:48, polcott wrote:
On 2/18/2026 3:10 AM, Mikko wrote:
On 17/02/2026 14:59, polcott wrote:
On 2/17/2026 3:03 AM, Mikko wrote:
On 16/02/2026 15:47, olcott wrote:
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one
predicate
(which, if interpreted accordint to the usual meaning of either >>>>>>>>> symbol,
is uncomputable).
What do you think that this means: PA reo x ?
The exact meaning depends on the context and the meanings of the >>>>>>> types of the left and right side expressions. The usual
metalogical meaning
is that x is a theorem of some variant of PA. If something else is >>>>>>> meant that should be specified in the opus where the expression >>>>>>> is used
Yes that is correct. What does that mean?
It means that the author must define the symbols in the opus they are >>>>> used.
Is this your best answer or are you trying to be evasive?
Whether another answer would be better is a matter of taste, at least
to some extent.
PA reo x
The correct answer is
A back-chained inference from x to the axioms of PA exists
No, that is *ONE* definition of it, but in other contexts, it might mean something different.
All you are doing is proving you don't understand the importance of
context for definitions.
In fact, I rarely hear about it specifing that a "back chain" exists, instead it normally is described as there exists a "proof" of x in PA.
Proofs, are more normally talked about as something that moves in the FORWARD direction, from the axioms of the system to the conclusion, not about "back-chaining".
On 2/19/2026 8:44 PM, Richard Damon wrote:
On 2/19/26 6:47 AM, polcott wrote:
On 2/19/2026 4:06 AM, Mikko wrote:
On 18/02/2026 21:48, polcott wrote:
On 2/18/2026 3:10 AM, Mikko wrote:
On 17/02/2026 14:59, polcott wrote:
On 2/17/2026 3:03 AM, Mikko wrote:
On 16/02/2026 15:47, olcott wrote:
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one >>>>>>>>>> predicate
(which, if interpreted accordint to the usual meaning of
either symbol,
is uncomputable).
What do you think that this means: PA reo x ?
The exact meaning depends on the context and the meanings of the >>>>>>>> types of the left and right side expressions. The usual
metalogical meaning
is that x is a theorem of some variant of PA. If something else is >>>>>>>> meant that should be specified in the opus where the expression >>>>>>>> is used
Yes that is correct. What does that mean?
It means that the author must define the symbols in the opus they are >>>>>> used.
Is this your best answer or are you trying to be evasive?
Whether another answer would be better is a matter of taste, at least
to some extent.
PA reo x
The correct answer is
A back-chained inference from x to the axioms of PA exists
No, that is *ONE* definition of it, but in other contexts, it might
mean something different.
All you are doing is proving you don't understand the importance of
context for definitions.
In fact, I rarely hear about it specifing that a "back chain" exists,
instead it normally is described as there exists a "proof" of x in PA.
Proofs, are more normally talked about as something that moves in the
FORWARD direction, from the axioms of the system to the conclusion,
not about "back-chaining".
That is correct
On 2/19/2026 4:06 AM, Mikko wrote:
On 18/02/2026 21:48, polcott wrote:PA reo x
On 2/18/2026 3:10 AM, Mikko wrote:
On 17/02/2026 14:59, polcott wrote:
On 2/17/2026 3:03 AM, Mikko wrote:
On 16/02/2026 15:47, olcott wrote:
On 2/16/2026 5:25 AM, Mikko wrote:
On 15/02/2026 15:02, polcott wrote:
reCx ree PA (True(PA, x) rao PA reo x)
Does not mean to test every x in PA
No, it merely declares that there are two symbols for one predicate >>>>>>>> (which, if interpreted accordint to the usual meaning of either >>>>>>>> symbol,
is uncomputable).
What do you think that this means: PA reo x ?
The exact meaning depends on the context and the meanings of the
types of the left and right side expressions. The usual
metalogical meaning
is that x is a theorem of some variant of PA. If something else is >>>>>> meant that should be specified in the opus where the expression is >>>>>> used
Yes that is correct. What does that mean?
It means that the author must define the symbols in the opus they are
used.
Is this your best answer or are you trying to be evasive?
Whether another answer would be better is a matter of taste, at least
to some extent.
The correct answer is
A back-chained inference from x to the axioms of PA exists
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