Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
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: Fri, 28 Mar 2025 14:16:14 +0200
Organization: -
Lines: 98
Message-ID: <vs63ue$2ngoo$1@dont-email.me>
References: <vrfvbd$256og$2@dont-email.me> <vrh432$39r47$1@dont-email.me> <vrhami$3fbja$2@dont-email.me> <vrj9lu$1791p$1@dont-email.me> <vrjn82$1ilbe$2@dont-email.me> <vrmpc1$bnp3$1@dont-email.me> <vrmteo$cvat$6@dont-email.me> <vru000$33rof$1@dont-email.me> <vrug71$3gia2$6@dont-email.me> <vs0e9v$1cg8n$1@dont-email.me> <vs1fda$296sp$3@dont-email.me> <vs3b1d$3aoq$1@dont-email.me> <vs3iap$9lob$1@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Fri, 28 Mar 2025 13:16:14 +0100 (CET)
Injection-Info: dont-email.me; posting-host="6b7417c2cac36ebd3726a4ec4446c661";
	logging-data="2867992"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX18dsdod7bxYNt2VXAa68u5Q"
User-Agent: Unison/2.2
Cancel-Lock: sha1:AVvn1VN9DG2CTsTiA2NIhWrwi64=

On 2025-03-27 13:03:21 +0000, olcott said:

> On 3/27/2025 5:58 AM, Mikko wrote:
>> On 2025-03-26 18:01:14 +0000, olcott said:
>> 
>>> On 3/26/2025 3:36 AM, Mikko wrote:
>>>> On 2025-03-25 14:56:33 +0000, olcott said:
>>>> 
>>>>> 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.
>>>> 
>>>> What is presented as the body of human knowledge either is a very small
>>>> part of actual knowledge or contains false claims.
>>>> 
>>> 
>>> I am NOT referring to what is merely presented as the body
>>> of general knowledge, I am referring to the actual body of
>>> general knowledge. Within this hypothesis it is easy to see
>>> that True(X) would be infallible.
>> 
>> In that case your True(X) is uncomputable and any theory that contains
>> it is incomplete.
>> 
> 
> The body of general knowledge that can be expressed
> using language is defined to be complete.

That doesn't prevent us from presenting general knowledge that is not
in that "complete" body.

-- 
Mikko