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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: Sat, 22 Mar 2025 18:43:45 +0200
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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.

-- 
Mikko