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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 22:22:17 -0500
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On 3/30/2025 9:31 PM, Richard Damon wrote:
> On 3/30/25 10:01 PM, olcott wrote:
>> 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 
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