The symbol “ʌ” means “and also”.
So the question is: If a middle is allowed between A and not-A, then can one have both A and also not-A at the same time?
And the answer is that it is anyone’s guess because if logic doesn’t apply, then there are no consistent rules (because consistency in language is what logic is).
The new possible cases would be; that both A and also not-A exited, that neither A nor not-A existed, or that neither A nor not-A applied to the situation at all (such as the presence of an nonsensical statement; e.g. “this statement is false”). Take your pick