[ edited 11th april 2019]
Does the distinction between syntax and semantics imply that ( rigorously) the negation of P should not be read as " P is false"?
I'll try by the following comments to explain why the question is not off-topic from my point of view.
It can be argued that logical operators have a syntactic side and a semantic side.
From a syntactic point of view , the negation operator is simply ( if I am correct) a function from the set of wff to the set of wff. So the expression "~P" does not refer to the truth value of P, it only refers to the image of the proposition symbol P under the function " ~ " ( that is , the function : negation).
But from a semantic point of view, the negation operator is a function from {T,F} to {T,F}, namely the function { (T,F), (F,T) }.
So could one say that, at least from a semantic point of view, " ~ P " could be read as " P is false" ?
Or is this reading absolutely erroneous, as not taking into account the strict distinction between syntax and semantics?
Let me add an example.
Does the formula (P --> ~P) rigorouly mean that " P is false" ? I think it would be more correct to say that the formula only means " not-P". So there would be a difference between " not-P" and "P is false".
If $P$ is false, then $\lnot P$ is true; if $P$ is true, then $\lnot P$ is false. It is the negation of $P$, whatever truth value $P$ might have, so perhaps that's where the confusion lies.