On 16/01/2026 01:40, olcott wrote:
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:24 AM, olcott wrote:
On 1/14/2026 8:57 PM, Richard Damon wrote:
On 1/13/26 1:43 PM, olcott wrote:
On 1/13/2026 6:10 AM, Richard Damon wrote:
On 1/12/26 11:46 PM, olcott wrote:
On 1/12/2026 9:16 PM, Richard Damon wrote:
On 1/12/26 4:41 PM, olcott wrote:
How The Well-Founded Semantics for General Logic Programs
of (Van Gelder, Ross & Schlipf, 1991)
Journal of the Association for Computing Machinery,
volume 38, number 3, pp. 620{650 (1991).
https://users.soe.ucsc.edu/%7Eavg/Papers/wf.pdf
handle the Liar Paradox when we construe
non-well-founded / undefined as not a truth-bearer?
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
WFS assigns undefined to self-referential paradoxes
without external support.
When we interpret undefined as lack of truth-bearer
status the Liar sentence fails to be about anything
that can bear truth values
The paradox dissolves - there's no contradiction
because there's no genuine proposition
This is actually similar to how some philosophers
(like the "gap theorists") handle the Liar: sentences
that fail to achieve determinate truth conditions
simply aren't truth-bearers. WFS's undefined value
provides a formal mechanism for identifying exactly
these cases.
A Subtle Point The occurs-check failure in Prolog is
slightly different from WFS's undefined assignment -
it's a structural constraint on term formation. But
both point to the same insight: circular, unsupported
self-reference doesn't create genuine semantic content.
I thought you said that no one in the past handled the liar >>>>>>>>> paradox?
That is no one in the past handling the Liar Paradox.
That all happened today.
So, today is 1991?
The paper provides the basis for me to
handle the Liar Paradox today. The Paper
does not mention the Liar Paradox it
only shows how to implement Proof Theoretic
semantics in a logic programming system.
I guess you are just admitting you are just a liar.
Note, since Prolog's logic is not sufficient to handle PA,
I never said it was. A formal system anchored in
Proof Theoretic Semantics is powerful enough.
Nope. It can't handle PA.
It definitely can. I already showed you the details
of how.
Nope, you PRESUME that Godel is non-sense.
“When PA is interpreted within proof‑theoretic semantics, only
well‑founded inferential structures are admissible as meaningful
statements. Gödel’s diagonal construction produces an ungrounded,
self‑referential formula whose proof‑dependency graph contains a
cycle. Since such expressions are not truthbearers in this
framework, the classical incompleteness phenomenon does not arise.
PA itself remains sound and complete with respect to its grounded
proof rules.”
In other words, you are just admitting to be an idiot that deosn't
care what your words actually mean.
The term *proof‑theoretic semantics* has always
proved my point long before I ever heard of it.
A term does not prove anything. Only a proof proves.
On 1/18/2026 5:54 AM, Mikko wrote:
On 16/01/2026 01:40, olcott wrote:
On 1/15/2026 5:50 AM, Richard Damon wrote:
On 1/15/26 12:24 AM, olcott wrote:
On 1/14/2026 8:57 PM, Richard Damon wrote:
On 1/13/26 1:43 PM, olcott wrote:
On 1/13/2026 6:10 AM, Richard Damon wrote:
On 1/12/26 11:46 PM, olcott wrote:
On 1/12/2026 9:16 PM, Richard Damon wrote:
On 1/12/26 4:41 PM, olcott wrote:
How The Well-Founded Semantics for General Logic Programs >>>>>>>>>>>
of (Van Gelder, Ross & Schlipf, 1991)
Journal of the Association for Computing Machinery,
volume 38, number 3, pp. 620{650 (1991).
https://users.soe.ucsc.edu/%7Eavg/Papers/wf.pdf
handle the Liar Paradox when we construe
non-well-founded / undefined as not a truth-bearer?
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
WFS assigns undefined to self-referential paradoxes
without external support.
When we interpret undefined as lack of truth-bearer
status the Liar sentence fails to be about anything
that can bear truth values
The paradox dissolves - there's no contradiction
because there's no genuine proposition
This is actually similar to how some philosophers
(like the "gap theorists") handle the Liar: sentences
that fail to achieve determinate truth conditions
simply aren't truth-bearers. WFS's undefined value
provides a formal mechanism for identifying exactly
these cases.
A Subtle Point The occurs-check failure in Prolog is
slightly different from WFS's undefined assignment -
it's a structural constraint on term formation. But
both point to the same insight: circular, unsupported
self-reference doesn't create genuine semantic content.
I thought you said that no one in the past handled the liar >>>>>>>>>> paradox?
That is no one in the past handling the Liar Paradox.
That all happened today.
So, today is 1991?
The paper provides the basis for me to
handle the Liar Paradox today. The Paper
does not mention the Liar Paradox it
only shows how to implement Proof Theoretic
semantics in a logic programming system.
I guess you are just admitting you are just a liar.I never said it was. A formal system anchored in
Note, since Prolog's logic is not sufficient to handle PA, >>>>>>>>>
Proof Theoretic Semantics is powerful enough.
Nope. It can't handle PA.
It definitely can. I already showed you the details
of how.
Nope, you PRESUME that Godel is non-sense.
“When PA is interpreted within proof‑theoretic semantics, only
well‑founded inferential structures are admissible as meaningful
statements. Gödel’s diagonal construction produces an ungrounded, >>>>> self‑referential formula whose proof‑dependency graph contains a >>>>> cycle. Since such expressions are not truthbearers in this
framework, the classical incompleteness phenomenon does not arise.
PA itself remains sound and complete with respect to its grounded
proof rules.”
In other words, you are just admitting to be an idiot that deosn't
care what your words actually mean.
The term *proof‑theoretic semantics* has always
proved my point long before I ever heard of it.
A term does not prove anything. Only a proof proves.
Proof Theoretic Semantics with the notion of
non-well-founded expressions is the same thing
that I have been saying for years.
True and False in PA have always been x or ~x is
provable from the actual axioms of PA, otherwise
x is simply not a truth bearer in PA. The only
clarification that I make now explicitly adding a
truth predicate to PA.
∀x ∈ PA ((True(PA, x) ≡ (PA ⊢ x))
∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~x))
∀x ∈ PA ((~True(PA, x) ∧ (~False(PA, x) ≡ ~TruthBearer(PA, x))
On 1/18/2026 5:54 AM, Mikko wrote:
On 16/01/2026 01:40, olcott wrote:
The term *proof‑theoretic semantics* has always
proved my point long before I ever heard of it.
A term does not prove anything. Only a proof proves.
Proof Theoretic Semantics with the notion of--
non-well-founded expressions is the same thing
that I have been saying for years.
True and False in PA have always been x or ~x is
provable from the actual axioms of PA, otherwise
x is simply not a truth bearer in PA. The only
clarification that I make now explicitly adding a
truth predicate to PA.
∀x ∈ PA ((True(PA, x) ≡ (PA ⊢ x))
∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~x))
∀x ∈ PA ((~True(PA, x) ∧ (~False(PA, x) ≡ ~TruthBearer(PA, x))
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