On 4/15/2026 1:54 AM, Mikko wrote:No, but that measn that for some sentences X True(X) is unknown and thereOn 14/04/2026 16:48, olcott wrote:Unknown truths are not elements of the body ofOn 4/14/2026 12:55 AM, Mikko wrote:>On 13/04/2026 17:24, olcott wrote:>On 4/13/2026 2:03 AM, Mikko wrote:>On 12/04/2026 16:17, olcott wrote:>On 4/12/2026 4:26 AM, Mikko wrote:>On 11/04/2026 17:14, olcott wrote:>On 4/11/2026 2:30 AM, Mikko wrote:>On 10/04/2026 14:18, olcott wrote:>On 4/10/2026 2:30 AM, Mikko wrote:>On 09/04/2026 16:34, olcott wrote:>On 4/9/2026 4:17 AM, Mikko wrote:>On 08/04/2026 17:13, olcott wrote:>On 4/8/2026 6:52 AM, olcott wrote:>On 4/8/2026 2:08 AM, Mikko wrote:>On 07/04/2026 17:49, olcott wrote:>On 4/7/2026 3:00 AM, Mikko wrote:>On 06/04/2026 14:21, olcott wrote:>On 4/6/2026 3:27 AM, Mikko wrote:>On 05/04/2026 14:25, olcott wrote:>On 4/5/2026 2:05 AM, Mikko wrote:>On 04/04/2026 19:23, olcott wrote:>On 4/4/2026 2:53 AM, Mikko wrote:>On 03/04/2026 16:35, olcott wrote:>On 4/3/2026 2:13 AM, Mikko wrote:On 02/04/2026 23:58, olcott wrote:>To be able to properly ground this in existing foundational>
peer reviewed papers will take some time.
Do you think 100 years would be enough, or at least some finite time?
I have to carefully study at least a dozen papers
that may average 15 pages each. The basic notion
of a "well founded justification tree" essentially
means the Proof Theoretic notion of reduction to
a Canonical proof.
>
% This sentence is not true.
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
>>>>
The above Prolog determines that LP does not
have a "well founded justification tree".
If you want to illustrate with examples you should have two examples:
one with a negative result (as above) and one with a positive one.
So the above example should be paired with one that has someting
else in place of not(provable(F, G)) so that the result will not be
false.
>
THIS IS NOT A PROLOG SPECIFIC THING
That's mainly true. However, in como.lang.prolog the discussion should
be restricted to Prolog specific things, in this case to the Prolog
example above and the contrasting Prolog example not yet shown.
>
In order to elaborate the details of my system
I require some way to formalize natural language.
Montague Grammar, Rudolf Carnap Meaning Postulates,
the CycL language of the Cyc project and Prolog
are the options that I have been considering.
>
The notion of how a well-founded justification tree
eliminates undecidability is a key element of my system.
Prolog shows this best.
It is not Prolog computable to determine whether a sentence of Peano
arithmetic has a well-founded justification tree in Peano arithmetic.
A formal language similar to Prolog that can represent
all of the semantics of PA can be developed so that
it detects and rejects expressions that lack well- founded
justification trees.
A language does not detect. For detection you need an algorithm.
unify_with_occurs_check(LP, not(true(LP))).
is a function of the Prolog language that
implements the algorithm.
No, it is not. The question whether a sentence has a well- founded
justification tree is a question about one thing so it needs an
algrotim that takes only one input but uunify_with_occurs_check
takes two.
>
The number of inputs does not matter.
If BY ANY MEANS a cycle is detected in the
directed graph of the evaluation sequence of
the expression then the expression is rejected.
>
True(L, X) := ∃Γ ⊆ BaseFacts(L) (Γ ⊢ X) // copyright Olcott 2018
If for any reason a back chained inference does
not reach BaseFacts(L) then the expression is untrue.
>
CORRECTION:
...then the expression is untrue [within the body
of knowledge that can be expressed in language].
That is not useful unless there are methods to determine whether
∃Γ ⊆ BaseFacts(L) (Γ ⊢ X) for every X in some L.
>
You can't use nify_with_occurs_check/2 to deremine whether True(X, L).
If there is a finite back-chained inference path from X
to Γ then X is true.
That does not help if you don't know whether there is a finite
back-chained inference path from X to Γ.
You simply do the back-chained inference and it reaches
the subset of BaseFacts or it does not. If it reaches
a loop then it is rejected as semantically incoherent.
And if it does neither ?
It either finds a finite path or finds that no
finite path exists.
There is no such it.
This is the it:
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
A goal that is not and cannot be achieved.
All instances of undecidability have either been provably
semantically incoherent input:
>
Olcott's Minimal Type Theory
G ↔ ¬Prov[PA](⌜G⌝)
Directed Graph of evaluation sequence
00 ↔ 01 02
01 G
02 ¬ 03
03 Prov[PA] 04
04 Gödel_Number_of 01 // cycle
>
Or outside of the body of knowledge such as the
truth value of the Goldbach conjecture.
It is not known whether there is a finite back-chained inference path
from Goldbach conjecture to the body of knowledge. If there is then
the conjecture is both true and untrue according to your statements
above.
It is known that the truth value of the Goldbach
conjecture is unknown this is out-of-scope for
>
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
If there is a finite back-chained interference path from Goldbach
conjecture to the body of knowledge then how it is out of scope ?
>
knowledge is a semantic tautology. Did you think
that things that are unknown are known?
is no method to find out.
I don't know about philosophers but mathematicians and logicians don't
find it interesting if all you can say that all knowledge is knowable
and everything else is not.
--
Mikko
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