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

On 2025-03-25 14:58:29 +0000, olcott said:

On 3/25/2025 4:54 AM, Mikko wrote:

On 2025-03-22 16:22:46 +0000, olcott said:
 

On 3/22/2025 8:37 AM, Richard Damon wrote:

On 3/21/25 11:03 PM, olcott wrote:

On 3/21/2025 9:31 PM, Richard Damon wrote:

On 3/21/25 9:24 PM, olcott wrote:

On 3/21/2025 7:50 PM, Richard Damon wrote:

On 3/21/25 8:40 PM, olcott wrote:

On 3/21/2025 6:49 PM, Richard Damon wrote:

On 3/21/25 8:43 AM, olcott wrote:

On 3/21/2025 3:41 AM, Mikko wrote:

On 2025-03-20 14:57:16 +0000, olcott said:
 

On 3/20/2025 6:00 AM, Richard Damon wrote:

On 3/19/25 10:42 PM, olcott wrote:

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.

 Which just means that you have stipulated yourself out of all classical logic, since Truth is different than Knowledge. In a good logic system, Knowledge will be a subset of Truth, but you have defined that in your system, Truth is a subset of Knowledge, so you have it backwards.
 

 True(X) always returns TRUE for every element in the set
of general knowledge that can be expressed using language.
It never gets confused by paradoxes.

 Not useful unless it returns TRUE for no X that contradicts anything
that can be inferred from the set of general knowledge.
 

 I can't parse that.
 > (a) Not useful unless
 > (b) it returns TRUE for
 > (c) no X that contradicts anything
 > (d) that can be inferred from the set of general knowledge.
 >
Because my system begins with basic facts and actual facts
can't contradict each other and no contradiction can be
formed by applying only truth preserving operations to these
basic facts there are no contradictions in the system.
 

 No, you system doesn't because you don't actually understand what you are trying to define.
 "Human Knowledge" is full of contradictions and incorrect statements.
 Adittedly, most of them can be resolved by properly putting the statements into context, but the problem is that for some statement, the context isn't precisely known or the statement is known to be an approximation of unknown accuracy, so doesn't actually specify a "fact".

 It is self evidence that for every element of the set of human
knowledge that can be expressed using language that undecidability
cannot possibly exist.
 

 SO, you admit you don't know what it means to prove something.
 

 When the proof is only syntactic then it isn't directly
connected to any meaning.

 But Formal Logic proofs ARE just "syntactic"
 

 When the body of human general knowledge has all of its
semantics encoded syntactically AKA Montague Grammar of
Semantics then a proof means validation of truth.

 Yes, proof is a validatation of truth, but truth does not need to be able to be validated.
 

 True(X) ONLY validates that X is true and does nothing else.
 

 But can't do that, as Tarski shows, as it creates contradictions when the system is able to generate unprovable truths.

 Unless we do what ZFC did to redefine the foundations
of set theory and redefine the notion of a formal system.

 The notion of a formal system is sufficiently generic that there is no
need to redefine it. If you want something else then call it something
else.

 ZFC got rid of the issues of pathological self-reference
from set theory. The same thing can be done for formal
systems.

Plain Z did that. But his theory is called "Zermelo's set theory" or
"Z set theory", which names are not used for any other theory. ZF
and ZFC are two other set theories that alse avoid pathological (and
other) self-reference. However, in all these theories it is possible
to construct a Gödel's set and use in the proof of incompleteness.
--
Mikko