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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Why Tarski is wrong --- Montague, Davidson and Knowledge Ontology
 providing situational context.
Date: Fri, 21 Mar 2025 19:48:56 -0400
Organization: i2pn2 (i2pn.org)
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On 3/21/25 4:20 PM, Richard Heathfield wrote:
> On 21/03/2025 19:30, Keith Thompson wrote:
>> My understanding is that Gödel proved that there are statements
>> that are true but not provable.
> 
> Yes. Incompleteness.
> 
>> It's still not possible for both
>> X and not-X to be provable.  If proofs exist for both, at least
>> one of the proofs must be flawed.
> 
> I'd be interested to see a proof of that conjecture.
> 

It comes from the definition of consistency.

If something is provable, then it must be true, as the proof shows the 
path from the fundamental truths the the stateemnt, establishing its truth.

If both x and not-x are probable then both x and not-x are true, which 
is the definition of a contradiction.

Most of these proofs start with the requirement that the system we are 
starting with in non-contradictory, i.e, there are no contradictions 
probable in it.

There are some more outlying logic systems, that due to built in 
weakness of their logic, do not support the principle of explosion, and 
thus can be defined to allow limited contradictions to exist in them, 
but these are on the fringe and not commonly used.