Basic question about logic If P implies Q, what can one say about the truth of not P implies not Q?
Note: I've revised the statement above. Sorry. 
I'll also provide some context. I am considering the following statement on English Language and Usage Stack Exchange:

Violation of rights triggers war. Prevention of violation of rights
  prevents war.

Please correct me if I'm mistaken, but this statement seems to have the form:

P implies Q. Not P implies not Q. 

What can one say in general about the truth of not P implies not Q? I assume nothing, but I would appreciate an explanation/example. 
Also, is "not P implies not Q" perhaps an example of a tautology in the sense that all wars involve the violation of rights, so if one can prevent the violation of rights, one can prevent all wars? The second part of this question seems like the contrapositive of the first part.
 A: Given a statement $P \rightarrow Q$, you are asking about what we can deduce of $\lnot P  \rightarrow \lnot Q$.
Your intuition is correct in that one can say nothing about $\lnot P  \rightarrow \lnot Q$. Here are a few examples:
Example 1:
If it is raining, then it is cloudy outside .
From this, you cannot conclude that if it is not raining, then it is not cloudy (it might be cloudy outside but not raining for ex.)
Example 2:
If you are a human, then you have two legs.
From this, you can't conclude that if you're not a human, then you don't have two legs. What about monkeys and kangaroos etc.
Example 3:
If you are not alive, then you are dead
From this you actually can conclude that if you are not not alive (i.e. alive), then you are not dead. In this case, the statement $P \rightarrow Q$ should actually be $P \iff Q$.

A lot of English statements get fudged up when converted to logic. The statement:

Violation of rights triggers war. Prevention of violation of rights prevents war.

is more like saying $P \iff Q$ which is kind of dodgy because it is saying a war is caused by violation of rights and nothing else. That is, if rights are not violated, then a war will always be prevented. You can take that as you will, but as the statement stands, a biconditional is probably closer to what it means and so yes, the statement is of the form $P \rightarrow Q$ and $\lnot P \rightarrow \lnot Q$ (which is equivalent to $P \iff Q$.
A: 

Violation of rights triggers war. Prevention of violation of rights prevents war.

Please correct me if I'm mistaken, but this statement seems to have the form:

P implies Q. Not P implies not Q. 


You are correct.   However, the argument is not sound.    We can make the statements equivalent by correcting the direction of the conditional connectives.


Violation of rights are sufficient to trigger war.   Prevention of violations of rights is necessary to prevent war.
$$(P\to Q) ~~\iff~~ (\neg P\gets \neg Q)$$ 


There are, after all, other things that trigger wars, so preventing violation of rights alone will not guaranteed war is prevented.   But if you wish to prevent war you will need to prevent violations among other things.
A: In colloquial English
Violation of rights trigger wars
Ergo
Maintaining rights prevents wars
does have the form P implies Q therefore not P implies Q.
And this is simply not valid logic as everyone else has pointed out.
However colloquially translating 
Violation of rights trigger wars
as P implies Q is not quite accurate.  It isn't "if P then Q" (always; no exception).  It's more "higher incidence of P => higher probability of Q".
In which case "higher incidence of not P => higher probability of not Q" is a valid inference.  
But "high prob P => high prob Q" which can be an English interpretation of "implies", is most certainly not the mathematical interpretation of "implies".
A: $Q$ might be a tautology - that is, always true. Then $P$ implies $Q$ and $\neg P$ implies $Q$ - indeed, everything implies $Q$! So you can't conclude anything like what you want.
What we can conclude from $P\implies Q$ is $\neg Q\implies \neg P$ - this is the contrapositive. Think about it this way: supposing $P\implies Q$, if $Q$ is false then $P$ can't be true, since if $P$ were true then $Q$ would be true. (This goes the other way, too - $P\implies Q$ is equivalent to $Q\implies P$.)
A: This no necesary is true. Results equivalent are a example of this. 
If P imply Q, is true that not Q imply not P.
If "k even then 2k is even" and  "if k is odd then 2k is even".
