edit: looking at Wikipedia does seem to indicate that there might be systems of logic where "A and not A" might not be valid (paraconsistent logic), but I haven't really looked into that, and I don't know how much such systems are used.
Literally copying from Wikipedia: The system of classical logic is obtained by adding any one of the following axioms: ϕ ∨ ¬ϕ (Law of the excluded middle), ...
where ∨ means OR, not AND... right?
Nobody was saying p OR !p is necessarily true. The argument was about p AND !p. It really seems to me like you're confusing AND and OR...
The intuitionist will accept that "A and not A" cannot be true https://en.wikipedia.org/wiki/Intuitionism#Truth_and_proof
Unknown questions are A and not A.
There’s interesting alternative logic systems that try to tackle this.
We get into interesting territory when using English and then “is” as a statement. Most English is assumed to have a certain value because of to be.
English is almost inherently flawed with logic bugs and assumptions.
E-Prime may help with this.
1. https://en.wikipedia.org/wiki/Catu%E1%B9%A3ko%E1%B9%ADi
The point was that you can reject the law of the excluded middle without getting into the absurdity that would result from accepting "a and not a".