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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: How a True(X) predicate can be defined for the set of analytic knowledge
Date: Tue, 1 Apr 2025 09:36:01 +0300
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On 2025-03-31 18:29:32 +0000, olcott said:

> On 3/31/2025 4:04 AM, Mikko wrote:
>> On 2025-03-30 11:20:05 +0000, olcott said:
>> 
>>> 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
>>>>> that feline kittens have always been 15 story office buildings
>>>>> and we have been deluded into thinking differently.
>>>>> 
>>>>>>> A knowledge ontology inheritance hierarchy is most efficient.
>>>>>>> 
>>>>>>>> However, there could be no uncertain sentences as they are not known
>>>>>>>> (sensu Olcotti).
>>>>>>> 
>>>>>>> Scientific theories would be uncertain truth.
>>>>>>> It is a known fact that X evidence seems to make Y
>>>>>>> a reasonably plausible possibility.
>>>>>> 
>>>>>> 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?
>>>>>> 
>>>>>>>>> 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 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.
>>>>>> 
>>>>>> Understanding that Tarski has been refuted hardly counts as understanding
>>>>>> as Tarstki has not been refuted.
>>>>>> 
>>>>> 
>>>>> When Tarski said True(X) cannot be defined, he is proved wrong.
>>>> 
>>>> 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.
>>> 
>>> Mere more verbose way of saying the same thing.
>>> 
>>>> If you reject
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