Re: How a True(X) predicate can be defined for the set of analytic knowledge

On 2025-04-01 17:51:29 +0000, olcott said:

On 4/1/2025 5:31 AM, Richard Damon wrote:

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On 3/30/2025 4:57 AM, Mikko wrote:

On 2025-03-29 14:06:17 +0000, olcott said:
 

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On 2025-03-28 19:59:16 +0000, olcott said:
 

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On 2025-03-26 17:58:10 +0000, olcott said:
 

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On 2025-03-26 02:15:26 +0000, olcott said:
 

On 3/25/2025 8:08 PM, Richard Damon wrote:

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On 3/25/2025 5:19 AM, Mikko wrote:

On 2025-03-22 17:53:28 +0000, olcott said:
 

On 3/22/2025 11:43 AM, Mikko wrote:

On 2025-03-21 12:49:06 +0000, olcott said:
 

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On 2025-03-20 15:02:42 +0000, olcott said:
 

On 3/20/2025 8:09 AM, Mikko wrote:

On 2025-03-20 02:42:53 +0000, olcott said:
 

It is stipulated that analytic knowledge is limited to the
set of knowledge that can be expressed using language or
derived by applying truth preserving operations to elements
of this set.

 A simple example is the first order group theory.
 

When we begin with a set of basic facts and all inference
is limited to applying truth preserving operations to
elements of this set then a True(X) predicate cannot possibly
be thwarted.

 There is no computable predicate that tells whether a sentence
of the first order group theory can be proven.
 

 Likewise there currently does not exist any finite
proof that the Goldbach Conjecture is true or false
thus True(GC) is a type mismatch error.

 However, it is possible that someone finds a proof of the conjecture
or its negation. Then the predicate True is no longer complete.
 

 The set of all human general knowledge that can
be expressed using language gets updated.
 

When we redefine logic systems such that they begin
with set of basic facts and are only allowed to
apply truth preserving operations to these basic
facts then every element of the system is provable
on the basis of these truth preserving operations.

 However, it is possible (and, for sufficiently powerful sysems, certain)
that the provability is not computable.
 

 When we begin with basic facts and only apply truth preserving
to the giant semantic tautology of the set of human knowledge
that can be expressed using language then every element in this
set is reachable by these same truth preserving operations.

 The set of human knowledge that can be expressed using language
is not a tautology.
 

 tautology, in logic, a statement so framed that
it cannot be denied without inconsistency.

 And human knowledge is not.
 

 What is taken to be knowledge might possibly be false.
What actually <is> knowledge is impossibly false by
definition.
 

 How do you DEFINE what is actually knowledge?
 

 *This is a good first guess*
The set of expressions of language that have the
semantic property of true that are written down
somewhere.

 We already know that many expressions of language that have the semantic
proerty of true are not written down anywhere.

 Only general knowledge

 What is "general" intended to mean here? In absense of any definition
it is too vague to really mean anything.
 

 Reverse-engineer how you could define a set of
knowledge that is finite rather than infinite.

 First one should define what the elements of that set could be.
If sentences, and there are not too many of them, a set of knowledge
could be presented as a book that contains those sentences and nothing
else.

 A list of sentences would not make for efficient processing.

 Unless you want to exclude uncertain facts the set of know facts is
small, probably empty. If you include many uncertain facts then
almost certainly your True(X) is true for some false X.
 

 Yes of course there are no known facts it might be the case
that feline kittens have always been 15 story office buildings
and we have been deluded into thinking differently.
 

A knowledge ontology inheritance hierarchy is most efficient.
 

However, there could be no uncertain sentences as they are not known
(sensu Olcotti).

 Scientific theories would be uncertain truth.
It is a known fact that X evidence seems to make Y
a reasonably plausible possibility.

 A good example is Newtonial mchanics, which is known to be wrong but is
useful and used for practical purposes. How should your True(X) handle
that?
 

The set of everything that anyone ever wrote
down would be finite.

 But not knowable.
 

Most of this would be
specific knowledge Pete's dog was named Bella.
Some is general dogs are animals.
 

Ae also know that many expressions of language that are written down
somewhere lack the semantic property of true.

 False statements do not count as knowledge.

 No, but your "the set of expressions of language that have the semantic
property of true that are written down somewhere" is not useful because
there is no way to know that set.

 We can know that the set of general knowledge that can
possibly be written down (formerly the analytic aspect
of the analytic/synthetic distinction) exists without
enumerating its elements.

 But we can't use it.

 We can use it right now to understand that Tarski
has been refuted and that True(X) does exist for
a specific and crucially relevant domain.

 Understanding that Tarski has been refuted hardly counts as understanding
as Tarstki has not been refuted.
 

 When Tarski said True(X) cannot be defined, he is proved wrong.

 He didn't say that True(X) cannot be defined. He proved that no definition
defines a predicate that tells whether a sentence is true.

 Mere more verbose way of saying the same thing.

 The point is that an unimplmentable defintion doesn't define an existing predicate.
 

 

If you reject
the idea that a sentence derived from true sentences with turth preserving
transformations is always true then you may disagree.

 Since this <is> my own design, I do not reject it.

 So, you think we can derive a non-true statement from truth preserving operations on true sentences?
 I think you just rejected your own logic by not understanding what you are talking about.

 No this is your ADD again.
That expressions of language derived only from
applying truth preserving operations to expressions
that are true are always true is necessarily true.
 

 But the problme is that you just rejected a sentence created by an (infinite) chain of truth preserving operations.
 

 As not in the domain.

 WHy not?
 

 You don't understand that unknown things are not in
the body of knowledge? The body of knowledge
Expressed in language INHERENTLY HAS A True(X) predicate.
 The body of knowledge that cannot be expressed in
language cannot have a True(X) predicate.
 

 But they ARE in the logic system based on the body of knowledge as its basic truths.
 

 Provide a citation that says this.
 

You just don't understand what a logic system is.
 IF you want to restrict your domain to just the body of knowledge, then you have no "logic system" as you can't allow the logic system to increase what it knows outside your initial knowledge, and your "truth predicate" is defined in the wrong type of domain, and is misnamed, it is just a knowledge predicate.
 

 The actual body of knowledge that can be expressed in
language (a) Inherently has a True(X) predicate
(b) Is continuously and immediately updated.

  No it doesn't, as a Truth Predicate needs to be over a domain of TRUTH, which means the full output of a LOGIC SYSTEM, which you just denied that you system has.
 

 Even the system of computing the sum of finite
strings of numeric digits is a complete logic
system over its domain.
 

 No, it isn't a *LOGIC* system, as it has no LOGIC, in that it has no INFERENCE operation.
 

 It applies truth preserving operations to pairs of
finite strings of numeric digits deriving their sum.
It computes the mapping from inputs to outputs
as Turing computable functions must do.
 

 Which isn't a "LOGIC", it is a COMPUTATION.
 

 https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence
 

 But your "Computation" system isn't good enought to get there.
 Your need a Turing Complete system, which yours isn't
 In particular, your definition has no conditional operation.
 

 Deterministic finite automatons have a lookup
table (like a JMP table) form of conditional
branches. I have an issued patent on a DFA.
This is the same kind of thing as a type hierarchy.
 

 But your system didn't have that.
 It just had a machine that adds two numbers together, and you used that limitiation to keep the creation of the unprovable truth,
 

 A machine that can correctly answer the question:
How do we know that anthropogenic climate change
is real? (The body of knowledge expressed in language)
can do more than sum two numbers.
 

 No, because we are talking about the machine you claimed:
 

It applies truth preserving operations to pairs of
finite strings of numeric digits deriving their sum.
It computes the mapping from inputs to outputs
as Turing computable functions must do.

 If you have a machine that can do actual logic to try to prove your statement, then it can do the full mathematics to allow Godel's proof, and we have a true statement that can not be proven.

 All we have to do is make a C program that does this
with pairs of finite strings then it becomes self-evidently
correct needing no proof.

There already are programs that check proofs. But you can make your own
if you think the logic used by the existing ones is not correct.
If the your logic system is sufficiently weak there may also be a way to
make a C program that can construct the proof or determine that there is
none.
--
Mikko