Re: Minimal Logics in the 2020's: A Meteoric Rise

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.

 

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.