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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: How a True(X) predicate can be defined for the set of analytic
 knowledge
Date: Thu, 27 Mar 2025 08:03:21 -0500
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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. The moment
that new knowledge that can be expressed in language
comes into existence it is added to the set.

True(X) merely tests for membership in this set;
(a) Is X a Basic Fact? Then X is true.

(b) Can X be derived by applying truth preserving operations
     to Basic Facts? Then X is true.

(c) Otherwise X is not true, this does not always mean X is false.
Gibberish is not true. Self-contradictory expressions are not true.

-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer