Validity of an argument Can anyone help me solve this question?
Determine whether the following argument is valid. Explain why:
"If Batman were able and willing to prevent corruption, then he would do so. If Batman were unable to prevent
corruption, he would be ineffective; if he were unwilling to prevent corruption, then he would be cruel. Batman
does not prevent corruption. If Batman exists, he is neither ineffective nor cruel. Therefore, Batman does not
exist."
 A: What do you know that you can bring to bear on this question???
It could be that you are being asked to informally assess the argument (in something of the manner suggested in @Lord_Farin's answer). But (since you are asking in a maths forum!), here's how I imagine you are intended to tackle the question:

The inference here turns on the propositional connectives "if ..then ...", "and", "not" and "or" -- so obvious first step: render the argument into the language of the propositional calculus. 

Do you know how to do that?

Second step: use the truth-table test to determine whether the formalized argument is valid. 

Do you know how to do that?
If you in principle know how to do both steps, this example is straightforward  and should present no problems (this site isn't for doing all your homework for you! -- but ask again if you got stuck at a particular stage)  If you don't know how to do both steps then you'd better do some background reading and get up to speed! 
A: This argument reminds me of some "proofs" of the non-existence of god. First, you assume that god is good and omnipotent and then notice that despite god being all that there is still evil in the world. Ergo, god doesn´t exist.
The argument is valid, and to see that all you need to know is that P => Q is equivalent to not Q => not P. And that if any proposition is implying a contradiction, then that proposition must be false.
Batman exist. => Batman is not cruel. Batman exists also implies that Batman is not ineffective. Bataman is neither ineffective nor cruel => he prevents corruption. And so Batman exists implies both corruption is prevented AND corruption is not prevented. Therefore that statement must be false.
A: Let us recall the definition of valid argument:

An argument is valid iff the truth of all its premises implies the truth of its conclusion.

So let us assume that the following are true:


*

*If Batman were able and willing to prevent corruption, then he would do so.

*If Batman were unable to prevent corruption, he would be ineffective.

*If Batman were unwilling to prevent corruption, then he would be cruel. 

*Batman does not prevent corruption.

*If Batman exists, he is neither ineffective nor cruel.


The key argument that we will need is the equivalence of an implication with its contrapositive. That is:

$p \implies q$ is equivalent to $\neg q \implies \neg p$

In words:

If $p$ implies $q$, and $q$ is false, then $p$ is also false. Moreover, if the falsehood of $q$ implies the falsehood of $p$, then the truth of $p$ implies the truth of $q$.

We obtain the following contrapositives:


*

*If Batman would not prevent corruption, he would be either unwilling or unable to do so.

*If Batman would not be ineffective, he would be able to prevent corruption.

*If Batman would not be cruel, he would be willing to prevent corruption.

*N/A

*If Batman is not neither ineffective nor cruel (i.e., if Batman is ineffective or cruel), he does not exist.



So let us get started with these nine statements.
We see that it suffices to verify the antecedent of 5. from the second list: that Batman is ineffective or cruel.
By 2. and 3. of the first list, it is sufficient that Batman be unable or unwilling to prevent corruption.
By 1. of the second list, it is sufficient that Batman not prevent corruption.
This is precisely the statement of 4.
We conclude that the truth of the premises implies the truth of the conclusion.
That is, the argument is valid.

As a side remark, I think it is sad that mathematical writers consider it necessary or pleasing to let their view on religion (a distinctly non-scientific thing) permeate into their work, thinly veiled or not. Just sad. You'd expect them to know better.
