Path: ...!eternal-september.org!feeder3.eternal-september.org!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: Thu, 27 Mar 2025 12:58:53 +0200
Organization: -
Lines: 88
Message-ID: <vs3b1d$3aoq$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>
MIME-Version: 1.0
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Thu, 27 Mar 2025 11:58:53 +0100 (CET)
Injection-Info: dont-email.me; posting-host="464c8a9049e6fcd42f4be7e97e4cff46";
	logging-data="109338"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1/mpRNlSQU7Exrj5e1s1jpj"
User-Agent: Unison/2.2
Cancel-Lock: sha1:em66xZbFVVnAslCCOqlSavMP+yQ=
Bytes: 4655

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.

-- 
Mikko