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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: Sun, 30 Mar 2025 12:57:43 +0300
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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. If you reject
the idea that a sentence derived from true sentences with turth preserving
transformations is always true then you may disagree.

-- 
Mikko