Understanding the negation of a statement Let's say
$p$: All dogs are fierce
$q$: No dogs are fierce
Then, why $q$ is not the negation of the $p$?
Edit: I just starting out with logic (and having a tough time understanding it). So, I am looking for an explanation in plain English.
 A: For $p$ not to be true, it's enough that at least one dog is not fierce. You don't need all dogs not to be fierce. If you think in terms of sets, $p$ means that the set $\mathit{dogs}$ is included in the set $\mathit{fierce}$. If $\mathit{dogs}$ is not included in $\mathit{fierce}$ it can still have some elements in common.
A: Formally, statement $p$ is $\forall d [dog(d)\Rightarrow fierce(d)]$. (For all d, if d is a dog, then d is fierce)
Statement $q$ is $\forall d [dog(d)\Rightarrow \neg fierce(d)]$. (For all d, if d is a dog, then d is not fierce)
The negation of $p$ is $\exists d[dog(g)\wedge \neg fierce(d)]$. (There exists d, such that d is a dog, and d is not fierce)
A: Rewording a bit,
$p$: All dogs are fierce
$q$: No dogs are fierce = All dogs are not fierce
$\neg p$: Not all dogs are fierce = Some dogs are fierce
$\neg q$: No dogs are not fierce = Not all dogs are not fierce = Some dogs are not fierce
Some of the quantifier negation laws are: $$\neg \exists x : P(x) = \forall x : \neg P(x)$$
$$\neg \forall x : P(x) = \exists x : \neg P(x)$$
As an example, take the statement where $A,B$ are sets : $$\forall x(x\in A \wedge x\in B)$$
The negation of this statement is : $$\neg \forall x(x\in A \wedge x\in B) =  \exists x \neg (x\in A \wedge x\in B) = \exists x (x\not\in A\, \vee \, x\not\in B)$$
So the first statement says that for all objects $x$,$x$ is a member of set $A$ and $B$, while the second statement, the negation of the former, say that there exists an object $x$ not in the set $A$ or $B$ or both, that is, there is some object $x$ not in both $A$ and $B$ at the same time.
Just for fun, let $x$ be the set of all hot dogs and $A$ be the set of hot dog buns and $B$ be the set of plates. The first statement says that all hot dogs are in hot dog buns and on plates while the second says that there are some hot dogs not in buns and on plates, or some that are in buns but not on plates, or some that are not in buns and not on plates.
A: It is intuitive to think of the negation of some statement as its 'opposite'.
And, it does seem to make to make sense that the 'opposite' of "All dogs are fierce" would be "No dogs are fierce"
So, I can understand why you would think that these are the negation of each other.
However, two statements are each other's negation if they have the opposite truth-value.  That is, when two statements are each other's negation, then no matter what, exactly one of them will be true, and thew other one will be false.
And note, that is clearly not the case with "All dogs are fierce" and "No dogs are fierce":  If only some dogs are fierce (and some are not), then both statement would be false. So, these are not the negation of each other.
So, to find the negation, don't necessarily ask yourself: "what is the opposite?", but instead ask: "what would it take for some statement not to be true?"
Let's do a somewhat more intuitive example. Suppose I say "All numbers are even". Of course, you say: "False!". Now, why do you say that?  Is it because no numbers are even?  No, clearly not, because some numbers are even.  Rather, "All numbers are even" is false because some numbers are not even.  And indeed, "Some numbers are not even" is the negation of "All numbers are even"
Likewise, "Some dogs are not fierce" is the negation of "All dogs are fierce"
