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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Minimal Logics in the 2020's: A Meteoric Rise
Date: Fri, 5 Jul 2024 13:24:10 -0400
Organization: i2pn2 (i2pn.org)
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On 7/5/24 12:47 PM, olcott wrote:
> On 7/5/2024 11:23 AM, Richard Damon wrote:
>> On 7/4/24 11:19 PM, olcott wrote:
>>> On 7/4/2024 10:00 PM, Richard Damon wrote:
>>>> On 7/4/24 10:38 PM, olcott wrote:
>>>>> On 7/4/2024 8:58 PM, Mild Shock wrote:
>>>>>>
>>>>>> When red means blue, and yellow means
>>>>>> green, then black is white. Thanks for your hint!
>>>>>>
>>>>>> If my Grandmother had wheels she would have been a bike
>>>>>> https://www.youtube.com/watch?v=OplyHCIBmfE
>>>>>>
>>>>>
>>>>> *Here is the same thing more clearly*
>>>>> Every expression of language that is {true on the basis of
>>>>> its verbal meaning} is only made true by a sequence of truth
>>>>> preserving operations to this {verbal meaning}.
>>>>>
>>>>> The only way that we know that puppies are not fifteen
>>>>> story office buildings is that the accurate verbal model
>>>>> of the actual world tells use so.
>>>>
>>>> But, even if we can't find that sequence of truth perserving 
>>>> operations, but one exists (which might be infinite) makes the 
>>>> statement true, but not known.
>>>>
>>>> This is one of your confusions, You confuse a statment being True, 
>>>> with the statement being KNOWN to be True.
>>>>
>>>> There are a number of great problems and conjectures that seem to be 
>>>> true, but we can not prove them. They MUST be either True or False, 
>>>> as by their nature, there is no middle ground (something either 
>>>> exsits or it doesn't, or the count of something is either finite or 
>>>> infinite).
>>>>
>>>> The ACTUAL TRUTH  (or falsehood) of such a statement is thus firmly 
>>>> established by the system in which the conjeture is embedded, even 
>>>> if our knowledge of the value of the truth of the statement is not 
>>>> known, or possible even knowable.
>>>>
>>>> The concept of "incompleteness" for a logical system is a 
>>>> recognition that the system has grown powerful enough that there 
>>>> exist some truths in the system that no finite proof of those 
>>>> statements exist, and only infinite chains of inference in the 
>>>> system can establish it.
>>>>
>>>> Mathematics is one source for these sorts of truths, as the 
>>>> possiblity of problems having NO number that satisfy them, or an 
>>>> infinite number that satisfy them show paths that can use in 
>>>> infinite number of steps to prove them, and might only be provable 
>>>> if some "inductive" shortcut can be found.
>>>>
>>>
>>> Yet my system screens out pathological expressions that
>>> are incorrectly determined to be incompleteness of the
>>> formal system. When we do that then True(L,x) can be defined
>>> for every expression not requiring an infinite sequence
>>> of steps. True(L,x) or True(L,~x) or not a truth bearer in L.
>>
>> No, it dies in self-inconsistency.
>>
>> Note "Every expression BUT ..."  isn't "Every expresion ."
>>
> 
> Every expression such that neither X nor ~X is provable in L
> is simply not a truth bearer in L. This does correctly reject
> self-contradictory expressions that wold otherwise be interpreted
> as the incompleteness of L.

BUT THAT IS A WRONG STATEMENT.

There exists in many logic systems, including those of most mathematics, 
that are TRUE, based on an infinite chain of connections to the 
truth-makers of the system, but are not provable in that system because 
no finite chain exists.

Your definition rejefcts too much, not rejects too much, not only the 
self-contradictory statements, but every system of sufficient 
mathematics to create the truths that can only be established by an 
infinite series of steps.

Your logic is based on faulty reasoning, because you don't understand 
the greater nuances of logic.

> 
> This works correctly for every element of the accurate verbal
> model of the actual world. Since we can see that things like
> the Goldbach conjecture can be proven in an infinite sequence
> then an algorithm can see this too. For everything else it is
> an infallibly correct system of reasoning.

Nope.

No, we can NOT see that Goldbach conjecture can be proven in an infinite 
sequence, that is only true if the conjecture is actually true. There 
may exist an even Natural Number that is NOT the sum of two primes.

If we could "see" it, then we could construct the fimite proof from how 
we see it.

If the Goldbach conjecture is false, then its falsity can be proven.

There MAY be a finite proof of the Goldbach conjecture, but there might 
not be.

> 
>> So, your logic only works in systems small enough to be somewhat akin 
>> to toys. Those that are limited enough not to be able to cause the 
>> problems, which means it excludes most systems that support math.
>>
>>>
>>>>>
>>>>>> olcott schrieb:
>>>>>>> When provable means true and false means unprovable
>>>>>>> then (Γ ⊢ X) means X is true in Γ.
>>>>>>> then (Γ ⊢ ~X) means X is conventional false  in Γ.
>>>>>>> the (Γ ⊬ X) ∧ (Γ ⊬ ~X) X is not a truth bearer in Γ.
>>>>>
>>>>
>>>
>>
>