Why is the negation of $A \Rightarrow B$ not $A \Rightarrow \lnot B$? The book I am reading says that the negation of "$A$ implies $B$" is "$A$ does not necessarily imply $B$" and not "$A$ implies not $B$". I understand the distinction between the two cases but why is the first one considered true?

$ \newcommand{\T} {\text{True}}
 \newcommand{\F} {\text{False}}$
$$\begin{array} {cc|cc}
A & B & A \implies \lnot B & A \not \implies B \\ \hline
\T & \T & \F & \F \\
\T & \F & \T & \T \\
\F & \T & \T & \F \\
\F & \F & \T & \F \\
\end{array}$$
So in fact for material implication, $A \implies \lnot B$ and $A \not \implies B$ are the same except for when $A$ is false.  This makes it possible to construct a counter example:
True statement $A \implies B$:

If you are an alien, then you have not posted on math.se.

That is a true statement, because the qualifying phrase is false.
Incorrect negation: $A \implies \lnot B$

If you are an alien, then you have posted on math.se.

The qualifying phrase is still unchanged and so false, so the statement is true, so it is not a negation of the original statement.
Correct negation $A \not \implies B$:

You are an alien and you have posted on math.se.

That statement is (regrettably) false.  That is because it is the negation of the original statement.
