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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 18:45:47 +0200
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On 2025-03-22 00:47:33 +0000, olcott said:

> On 3/21/2025 6:49 PM, Richard Damon wrote:
>> On 3/21/25 8:49 AM, olcott wrote:
>>> 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.
>> 
>> And thus your concept of truth breaks.
>> 
>> Truth, by its definition is an immutable thing, but you just defined it 
>> to be mutable.
>> 
>> How often do we need to re-verify our truths?
>> 
>>> 
>>>>> 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.
>>> 
>> 
>> But you aren't begining with basic facts, but with what has been 
>> assumed to be the basic facts.
> 
> That is not what I stipulated.
> When we begin with what actual are the set of basic
> facts and are only allowed to apply truth preserving
> operations to these basic facts then it is self-evident
> that True(X) must always be correct.
> 
>> We don't actually KNOW the basics principles for many things, but have 
>> been working to understand them.
> Then these are not included in the set of knowledge.

If nothing is inculded in the set of knowledge then nothing is provable
in your system.

-- 
Mikko