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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic,comp.theory
Subject: Re: Minimal Logics in the 2020's: A Meteoric Rise --- Richard
 contradicts himself
Date: Mon, 8 Jul 2024 21:24:40 -0400
Organization: i2pn2 (i2pn.org)
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On 7/8/24 8:56 PM, olcott wrote:
> On 7/8/2024 7:37 PM, Richard Damon wrote:
>> On 7/8/24 8:28 PM, olcott wrote:
>>> On 7/8/2024 7:07 PM, Richard Damon wrote:
>>>> On 7/8/24 8:00 PM, olcott wrote:
>>>>> On 7/7/2024 10:09 PM, olcott wrote:
>>>>>> On 7/7/2024 10:02 PM, olcott wrote:
>>>>>>>
>>>>>>> Formal logic is a subset of this.
>>>>>>> Not-a-logic-sentence(PA,g) ≡ (~True(PA,g) ∧ ~True(PA,~g))
>>>>>>> There are no truth preserving operations in PA to g or to ~g
>>>>>>>
>>>>>>
>>>>>> https://liarparadox.org/Tarski_275_276.pdf
>>>>>>
>>>>>> Within my analytical framework this Tarski sentence is merely
>>>>>> self-contradictory
>>>>>>
>>>>>> (3) x ∉ Provable if and only if x ∈ True. // (1) and (2) combined
>>>>>>
>>>>>> There are no truth preserving operations in Tarski's
>>>>>> theory to x if and only if There are truth preserving
>>>>>> operations in Tarski's theory to x
>>>>>>
>>>>>
>>>>> There cannot possibly be an infinite proof that proves
>>>>> that there is no finite proof of Tarski x in Tarski's theory
>>>>
>>>> Who says there needs to be a infinite proof, since there is no such 
>>>> thing.
>>>>
>>>> As I said, one example of such an x is Godel's G.
>>>>
>>>>>
>>>>> The infinite proof of the Goldbach conjecture
>>>>> (if it is true) continues to find more true
>>>>> cases than it had before, thus makes progress
>>>>> towards its never ending goal (if its true).
>>>>
>>>> or, it continue to show that there is no counter examples.
>>>>
>>>> "Progress" on an infinite path isn't really measurable.
>>>>
>>>>>
>>>>> The cycles in the following two cases never make any progress
>>>>> towards any goal they are merely stuck in infinite loops.
>>>>
>>>> Which just means you are on the wrong path. One wrong path doesn't 
>>>> me that there is no path.
>>>>
>>>>>
>>>>> The Prolog unify_with_occurs_check test means that
>>>>> LP is stuck in an infinite loop that makes no progress
>>>>> towards resolution. I invented Minimal Type Theory to
>>>>> see this, then I noticed that Prolog does the same thing.
>>>>
>>>> Which is irrelevent, since Prolog can't handle the basics of the 
>>>> field that Traski assumes.
>>>>
>>>>>
>>>>> ?- LP = not(true(LP)).
>>>>> LP = not(true(LP)).
>>>>> ?- unify_with_occurs_check(LP, not(true(LP))).
>>>>> false.
>>>>>
>>>>> LP := ~(L ⊢ LP)
>>>>> 00 ~ 01
>>>>> 01 ⊢ 01, 00
>>>>> 02 L
>>>>>
>>>>> The cycle in the direct graph of LP is
>>>>> an infinite loop that make no progress
>>>>> towards the goal of evaluating LP as
>>>>> true or false.
>>>>>
>>>>>
>>>>
>>>> So?
>>>>
>>>> Failure to prove by example doesn't show something isn't true.
>>>>
>>>> You are just proving you are stupid and don't know what you are 
>>>> talking about.
>>>
>>> Every expression of language that cannot be proven
>>> or refuted by any finite or infinite sequence of
>>> truth preserving operations connecting it to its
>>> meaning specified as a finite expression of language
>>> is rejected.
>>>
>>
>> So?
>>
>> Tarski's x like Godel's G are know to be true by an infinite sequence 
>> of truth preserving operations.
>>
> 
> *You already agreed that such things can never be known*
> 
> The Goldbach conjecture is known to be true or false
> yet not which one. Anything known to be true by an infinite
> sequence of truth preserving operations contradicts the
> fact that nothing can be known to be true by an infinite
> sequence of truth preserving operations.
> 

Yes, if ALL we have is a statement that can only be shown by an infinite 
series of steps, then we can not know that.

But many things that take an infinite number of steps in one system, 
might have a finite proof in another system that can relate back to the 
original one. KNOWLEDGE can cross system boundaries under the right 
conditons, even if the proof doesn't transfer.

Just like Godel's G, that in F, needs an infinite number of tests to 
prove in F, but by knowing from the meta-F of the implication of that 
relationship, we can find the "shortcut" to proving it in a finite 
number of steps.
Goldbach's conjecture might be false, in which case that is provable by 
just showing the even number that can't be the sum of two primes.

There might be a finite proof of it, either in the normal mathematics, 
or in a meta-mathematics that allows us to transfer that knowledge back 
to ordinary arithmetic.

Or, it might be that no such proof exists in any meta-mathematics, and 
if so, it will just be unknown if it is true.

Such a statement can not be some how proven to be unprovable, as that 
would end up being a proof that it is true, as if it IS false, that 
falsehood is easily proven with a sample number, which must exist if it 
is false.