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

On 7/5/24 1:38 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:

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When red means blue, and yellow means
green, then black is white. Thanks for your hint!
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If my Grandmother had wheels she would have been a bike
https://www.youtube.com/watch?v=OplyHCIBmfE
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*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}.
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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.

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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.
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This is one of your confusions, You confuse a statment being True, with the statement being KNOWN to be True.
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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).
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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.
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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.
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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.
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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.

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No, it dies in self-inconsistency.
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Note "Every expression BUT ..."  isn't "Every expresion ."
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 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.

FALSE STATEMENT.
Some statements are true due to an infinite number of steps to ther truth-makers of the system.
You will lead your logic system into contradictions by your definition (or you just need to treat it as a worthless phrase that doesn't actually tell you anything, particually what you call non-truth-bearers, which might actuall be statement that are true or false).

 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 *OR REFUTED* in an infinite
sequence then an algorithm can see this too. For everything
else it is an infallibly correct system of reasoning.
 

So, you ADMIT that you definition doesn't work for some statements, and thus is not correct.
Note, the algorithm can not tell wether the statement like to Goldback conjecture is true or not, or even if it takes an infinite number of steps to come to that answer. Thus, you statement is just a FALSEHOOD.
You just don't understand logic well enough to understand that can't have definitions that just don't work as the basis of a system.
By your definition, the Goldbach conjecture must currently be consider a non-truth-bearer, but we KNOW that it must be either true or false, we just don't know which, so you definition of a truth-bearer is just a lie.
What you are defining are KNOWLEDGE bearers, statements that there truth can be known. But we can't even know if the Goldbach conjecture is a knowledge-bearer or not. If it turns out to be false, then that fact is knowable, but not yet known (since showing the number, as a simple finite proof that no pair of primes below it sum to it make it prove false), but if it is true, there might be a proof, or there might not be.
So even Knowledge-Bearers as a concept is has limited use. Knowledge, that which we currently know, is a valid concept, and one that admits things can be added to it.
And Truth-Bearers, with the allowance of infinite chains to establish the truth (or falseness) of the statement can be useful, though we do need to admit we don't know, and perhaps CAN'T know that truth value, and need to allow for some statements that we don't yet have the ability to know if they are truth-bearers or not.
But your definition of truth-bearers is just worthless for most logic systems, claiming to be about truth but actually being about knowledge isn't a good definition, and just shows your fundamental misunderstanding about what is actually truth and how it differs from knowledge.

 

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.
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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.

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