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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: How a True(X) predicate can be defined for the set of analytic
 knowledge
Date: Sun, 30 Mar 2025 21:01:06 -0500
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On 3/30/2025 7:25 PM, Richard Damon wrote:
> On 3/30/25 7:34 PM, olcott wrote:
>> On 3/30/2025 5:47 PM, Richard Damon wrote:
>>> On 3/30/25 5:47 PM, olcott wrote:
>>>> On 3/30/2025 3:59 PM, Richard Damon wrote:
>>>>> 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:
>>>>>>>>>>>
>>>>>>>>>>>> On 3/29/2025 5:20 AM, Mikko wrote:
>>>>>>>>>>>>> On 2025-03-28 19:59:16 +0000, olcott said:
>>>>>>>>>>>>>
>>>>>>>>>>>>>> On 3/28/2025 7:12 AM, Mikko wrote:
>>>>>>>>>>>>>>> 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.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> 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
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