The logical negation is used in symbolic logic to indicate that the truth value of the statement that follows is reversed. The symbol tilde (~) is used to indicate logical negation. A simple use of negation is as follows:
If p represents the statement "John is a boy", then ~p represents the statement "It is not the case that John is a boy" or "John is not a boy".
The Truth table given below shows the truth functionality of the propositions.
Here's the Truth table for negation:
p ~p
T F
F T

This table is easy to understand. If p is true, its negation ~p is false. If p is false, then ~p is true.
Example 1: p : John is a boy.
~p: It is not the case that John is a boy.
If p is true, i.e., "John is a boy" is true, then ~p is false, i.e., "It is not the case that John is a boy" is false. In the same way, if p is false, i.e., "John is a boy" is false, then ~p is true, i.e., "It is not the case that John is a boy" is true.
When we assign F (falsity) to the proposition p ("John is a boy"), we do not assert, however, that John is a girl. All that we do is stating that the proposition p is false. John being a girl is only one among many possibilities. It would also mean that John is a man or that John is a spiritual being who has no gender or that he is a transgender. All these are implied when we state that p, i.e., "John is a boy" is false. Nevertheless, all these possibilities are included in ~p. The set "John the non-boy"(~p) contains all the implications of the falsity of p. Therefore, no matter who John really is, if p(John is a boy) is false, then ~p(It is not the case that John is a boy) is necessarily true.
The falsity(F) of p necessarily implies the truth(T) of ~p. *
Let us now consider another proposition, Example 2:
p: John has stopped taking medicine.
~p: It is not the case that John has stopped taking medicine.
According to the truth table, if p is true, i.e., "John has stopped taking medicine" is true, then ~p is false, i.e., "It is not the case that John has stopped taking medicine" is false. In the same way, if p is false, i.e., "John has stopped taking medicine" is false, then ~p , i.e., "It is not the case that John has stopped taking medicine" must be true. But when we analyze it deeper we find that it is not so.
When we assign falsity(F) to the proposition p(John has stopped taking medicine), we do not assert that John continues to take medicine. All that we do is stating that the proposition p is false. John continuing to take medicine is only one among many possibilities. And if one holds on to that possibility alone, falsity(F) of p would imply the truth(T) of ~p. However, 'F' of p can also mean that "John has stopped taking medicine" is false because he never started taking it. The falsity of p can be understood in this way that John cannot stop an action which he had not even started doing. Surely, this is an assumption. But F of p interpreted as John continues to take medicine is as well an assumption. And only when we hold on to the latter assumption, ~p (It is not the case that John has stopped taking medicine) is true.
In this case, unlike the first example, p is false does not necessarily imply that ~p is true. In the first example, all the implications of F of p (John being a man, girl, spirit..) come under ~p(It is not the case that John is a boy). Whereas, in the second example, some (at least one) implication(s) of F of p lay outside the purview of ~p. The implication that John had not started taking medicine is beyond the range of ~p. Therefore, without a leap of assumption, one cannot conclude from the F of p that ~p has the truth function T.
Hence, the falsity(F) of p does not necessarily imply the truth(T) of ~p.( ~p can also have the truth function F). And so, * has an exception which proves it to be false.