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I am looking for some alternative logic where a sentence p could not only be True or False but also Meaningless, which is different of false in such a logic.

I see that there is some content about three valued logic : https://en.wikipedia.org/wiki/Three-valued_logic. However mine would be different because Meaningless would be an absorbing element for any operation.

A formal system of this logic would be complete if for any true sentence there is a proof that the sentence is true.

A formal system of this logic would be consistent if you can't prove 0=1.

For sure there is no excluded middle in such a logic since a proposition can be neither true nor false, but meaningless.

The goal of this logic would be to get a formalization for arithmetic both complete and consistent. In classical logic Godel theorem forbid it but in such a logic the proof of Godel theorem is no longer possible because Godel assumes at some point that either $P(G(P))$ or $\neg P(G(P))$, but in this logic $P(G(P))$ could also be absurd.

Such a logic could also formalize the "not well defined sentence", like division by 0 and so on.

I wonder also if we could speak about the false only as a negation of the true and thus remove the false of this logic. It would have only 2 syntaxical values : true/meaningless and only one semantical value : true. The meaningless syntax would not have a semantical interpretation because meaningless only belong to syntaxical world and not to semantical world, which is single-valued in my idiosyncrasy.

François
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    See the post Three valued logic with the third value being "undecidable," applying to undecidable propositions – Mauro ALLEGRANZA Jul 19 '22 at 11:12
  • Yes this is exactly the kind of things I am looking for. My personnal view is that the distinction between 'true' and 'provable' has not been the "most important advances in mathematical logic" but an historical error and actually a bad way to handle absurdity. – François Jul 19 '22 at 11:25
  • It is all about to see mathematics as an open world and thus to formalize mathematics in a way you can extend it indefinitely. Or to see mathematics as a close, complete, finite thing, and then you have to handle what is outside of your formalization as absurd and non existing. It's mainly an esthetical point. – François Jul 19 '22 at 11:31
  • "to see mathematics as an open world and thus to formalize mathematics in a way you can extend it indefinitely." This is Brouwer's Intuitionism – Mauro ALLEGRANZA Jul 19 '22 at 11:44
  • In my opinion classical logic see mathematics as an open world that can be extended since there are sentences which are independent of axioms. May be Brouwer's Intuitionism is already the system I try to get but I am not sure because some people say that Godel theorem is still possible in Brouwer's Intuitionism, in my logic it is no longer possible, I think my logic would go in the same direction but further than Brouwer's Intuitionism. – François Jul 19 '22 at 12:13
  • Actually I think that Godel theorem has a transposition in my logic which would be : if F is consistent there exists a proposition in F which is meaningless. – François Jul 19 '22 at 12:49
  • Actually I guess what I am trying to do is to import classical logic into intuitionistic logic, just like Godel imported intuitionistic logic into classical logic. It does not mean the two logics are equivalent because the interpretation is very different. – François Jul 19 '22 at 12:54

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