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

On 3/30/25 3:39 PM, olcott wrote:

On 3/30/2025 1:53 PM, Richard Damon wrote:

On 3/30/25 1:16 PM, olcott wrote:

On 3/30/2025 6:24 AM, Richard Damon wrote:

On 3/30/25 7:20 AM, olcott wrote:

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

On 2025-03-29 14:06:17 +0000, olcott said:
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On 3/29/2025 5:20 AM, Mikko wrote:

On 2025-03-28 19:59:16 +0000, olcott said:
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On 3/28/2025 7:12 AM, Mikko wrote:

On 2025-03-28 01:04:45 +0000, olcott said:
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On 3/27/2025 5:48 AM, Mikko wrote:

On 2025-03-26 17:58:10 +0000, olcott said:
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On 3/26/2025 3:39 AM, Mikko wrote:

On 2025-03-26 02:15:26 +0000, olcott said:
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On 3/25/2025 8:08 PM, Richard Damon wrote:

On 3/25/25 10:56 AM, olcott wrote:

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

On 2025-03-22 17:53:28 +0000, olcott said:
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On 3/22/2025 11:43 AM, Mikko wrote:

On 2025-03-21 12:49:06 +0000, olcott said:
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On 3/21/2025 3:57 AM, Mikko wrote:

On 2025-03-20 15:02:42 +0000, olcott said:
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On 3/20/2025 8:09 AM, Mikko wrote:

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

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A simple example is the first order group theory.
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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.

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

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However, it is possible that someone finds a proof of the conjecture
or its negation. Then the predicate True is no longer complete.
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The set of all human general knowledge that can
be expressed using language gets updated.
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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.

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

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The set of human knowledge that can be expressed using language
is not a tautology.
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tautology, in logic, a statement so framed that
it cannot be denied without inconsistency.

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And human knowledge is not.
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What is taken to be knowledge might possibly be false.
What actually <is> knowledge is impossibly false by
definition.
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How do you DEFINE what is actually knowledge?
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*This is a good first guess*
The set of expressions of language that have the
semantic property of true that are written down
somewhere.

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

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Only general knowledge

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What is "general" intended to mean here? In absense of any definition
it is too vague to really mean anything.
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Reverse-engineer how you could define a set of
knowledge that is finite rather than infinite.

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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.

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A list of sentences would not make for efficient processing.

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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.
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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.
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A knowledge ontology inheritance hierarchy is most efficient.
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However, there could be no uncertain sentences as they are not known
(sensu Olcotti).

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

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

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But not knowable.
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Most of this would be
specific knowledge Pete's dog was named Bella.
Some is general dogs are animals.
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Ae also know that many expressions of language that are written down
somewhere lack the semantic property of true.

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False statements do not count as knowledge.

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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.

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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.

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But we can't use it.

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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.

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Understanding that Tarski has been refuted hardly counts as understanding
as Tarstki has not been refuted.
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When Tarski said True(X) cannot be defined, he is proved wrong.

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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.

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Mere more verbose way of saying the same thing.

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The point is that an unimplmentable defintion doesn't define an existing predicate.
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If you reject
the idea that a sentence derived from true sentences with turth preserving
transformations is always true then you may disagree.

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

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

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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.
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But the problme is that you just rejected a sentence created by an (infinite) chain of truth preserving operations.
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 As not in the domain.

WHy not?
Your "domain" is the logic system created by your set of initial truths, plus your logical operations, plus EVERYTHING that can be developed by the (unlimted) application of those operations to those truths and other truths developed by that method.
You can't limit your domain to only finite numbers of operations without admitting that you logic system can't understand the properties of the Natural Numbers.

When we try to find the sum of an actual rabbit
with an actual pallet of bricks we fail because
these are not in the domain of sum().
 

But I didn't do that, so that is just red herring.

Thus you disagree with your own claims.
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The problem seems to be you don't understand the potential for a logic system to have things only shown by an infinite chain of operations, because you just don't understand infinity.
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 I did prove a whole huge very relevant domain where
True(X) always provides the correct return value.
Such a system wold also know that:
This sentence is not true: "This sentence is not true."
is true because the inner sentence is not a truth bearer.

Nope, It might be "huge" to you, but if it can't support the Natural Numbers, it can't support much of what is considered Human Knowledge (as you lost just about everything mathematical)

 

That, our you think you CAN see an infinite series of steps. IF that is the case, please present an actual infinite chain of steps that proves something (not an infinite chain that has a induction that reduces it to a finite chain, but an actual infinite chain).
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 Do you know what the domain of a function is, or is this
a brand new idea for you?

I do, but you don't seem to, as the domain of the Truth Predicate needs to be all the possible sentences in the logic systems grammar, which you just tried to say yours doesn't do.

 

You also can't just use ... because you need to show that each one DOES satisfy the requirements, and we assume an induction isn't available.
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 Starting with true expressions and deriving other expressions
only through semantic logical entailment necessarily derives
true expressions.

Yes, and some of those are established by an INFINITE series of operations, and thus are not necessarily provable. And part of the essence of Tarski's proof is that the existance of unprovable statements in the system breaks any definition of a Truth Predicate.

 

For example, show that no number satisfies that relationship, by showing it actually applied to EVERY possible number.
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This is what makes Godel's G true, and may be what establishes the Goldbach conjecture.
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 The some screwy systems can be defined with undecidability
merely proves that some screwy systems do exist.

But the proof shows that ALL system, with a certain level of complexity, a level not that high, has undecidable problems.

 

Sorry, you are just showing you don't understand what you are talking about.

 I am showing that YOU don't understand what I am talking about.
You have never pointed to any actual error.
 

No, I see that YOU don't understand what you are talking about, as you don't realize how little you need to include in your system to get you into the trouble you want to define out of it.
Sorry, you are just showing you don't understand what you are talking about, and thus many of your statements are just stupid lies, lies that come out of a reckless disregard for the truth, and thus are lies even if you ernestly beleive them, beause your "earnest belief" is based on a clearly illogical and irresponsible avoidance of the facts.