Logic per se can prove anything and noting. It depends on the axioms you start with. For instance, take the argument for God's existence from design. It goes along the lines of "Complex things need an intelligence to design them. The universe is complex. Therefore there is a designer. We call that designer 'God.'"
I have yet to see an atheist who was convinced by that logic, even though it is perfectly sound. The problem is that the atheist rejects the axiom that complex things need an intelligent designer. Therefore the proof, while perfectly valid as a proof, is rejected. But let's continue to follow that chain of logic where it leads.
Ask the atheist why he rejects the notion that complex systems have a designer and he will ask you to show why he should accept that notion. You reply that everything we see that is complex has a designer. My iPhone, car, house all had an intelligent designer. "Ah" replies the atheist, "but you are overlooking the prime example of a complex thing that has no designer - the universe!" And of course they hold the truth of the statement that the universe has no designer based on the axiom that there isn't a God.
And so their logic is consistent as well, only it is circular. If you are trying to determine whether or not there is a God, you can't start with the axiom "there is no God" and see where logic takes you. You have to start with the axiom "there may or may no be a God" or you are being intellectually dishonest. It has nothing to do with logic, and everything to do with faith. The atheist has blind faith in his axiom.
N.B. As Deacon Will on Plurk pointed out, the first reading for the next day was Wisdom 13:1-9 (emphasis mine):
All men were by nature foolish who were in ignorance of God,
and who from the good things seen did not succeed in knowing him who is,
and from studying the works did not discern the artisan;
But either fire, or wind, or the swift air,
or the circuit of the stars, or the mighty water,
or the luminaries of heaven, the governors of the world, they considered gods.
Now if out of joy in their beauty they thought them gods,
let them know how far more excellent is the Lord than these;
for the original source of beauty fashioned them.
Or if they were struck by their might and energy,
let them from these things realize how much more powerful is he who made them.
For from the greatness and the beauty of created things
their original author, by analogy, is seen.
But yet, for these the blame is less;
For they indeed have gone astray perhaps,
though they seek God and wish to find him.
For they search busily among his works,
but are distracted by what they see, because the things seen are fair.
But again, not even these are pardonable.
For if they so far succeeded in knowledge
that they could speculate about the world,
how did they not more quickly find its Lord?