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

On 2025-03-28 01:04:45 +0000, olcott said:

On 3/27/2025 5:48 AM, Mikko wrote:

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

On 3/26/2025 3:39 AM, Mikko wrote:

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

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:
 

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

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

On 3/21/2025 3:57 AM, Mikko wrote:

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

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 also know that a True(X) predicate can be directly
defined for this set.

But it is not useful if the value of True(X) cannot be evaluated.
--
Mikko