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.
reCx (Provable(T, x) rco Meaningful(T, x)) --- (Schroeder-Heister 2024)
reCx (Provable(x) rcA True(x)) --- Anchored in (Prawitz, 2012)
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 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.
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