I think law of excluded middle makes sense to mean that a statement should be either logical or illogical but in this case I don't assume "not logical" = "illogical" since the author didn't say "illogical", the author said "not logical."
Can we formalize logic about logic in a way so that a statement such as "That's not logical" is formal when it's now used in a non-formal way while I interpret it as "metalogic" - logic about logic and just a negation. Logic doesn't formalize itself, logic formalizes statements, so what does logic mean? Is logic itself also just a statement similarly to how an axiom is also just a statement and a rule is also just a statement so what we could agree on is what type of statement is meant with just a negation so that "nothing exists" i.e.
- "It's not technical." this statement doesn't even say that there is anything technical
- "It's not accounting" ..and therefore not economics which might as well be the empty set
- "It's not a detail" ..ergo what is meant is "the big picture" and only if I understand correctly
- "No prisoner escaped during the night" - this statement doesn't even say that there exists any prisoners or prisons when "All prisoners were prevented from escaping" at least tells us that there was something instead of nothing
- "It's not logical/It's not rational" - What is meant is not even "illogical" or "irrational" since I can't assume that "not logical" = "illogical" so again it's the empty set and even a statement about the system itself which a mathematical system like theoretical philosophical logic is so a meaning with meta level still not ruling out that it's the empty set. And there are mathematical truths that are not logical for instance the so-called Gabriel's horn that has infinite area and finite volume which is not logical and still mathematically correct
- "I don't understand." - The statement neither says what you do when you don't understand nor does it define what it means to understand and again the statement is on the negation form "There is nothing..."
So how can I rule out that all these statements that are just a negation indeed are about something instead of nothing / the empty set or undefined?
So that the statement A is neither logical nor illogical it is simply not logical as how it is perceived as "not logical" or "not a logical sequence" disctinct from being an illogical sequence where a logical sequence typically would be "A causes B" like cause and effect while a statement that is about logic and a negation also can be true, false, a negation or provable.