I am looking for some alternative logic where a sentence p could not only be true or false but also absurd, 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 absurd 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 absurd.
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 P(G(P)) is false, 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.