You or me, we can't prove that God/Gods doesn't exist using logic, because for definition God is beyond the territory of logic, it can't be reached by our mind, even because if we were able to state, using some kind of algorithm, that God is non-existent, the game would be over, and we all would not be here, waiting for the next Islamic massacre, perpetrated in name of their God. So, sadly, up to now, we have not yet discovered the final equation of the non-existence of God. At this point, in my opinion, we have two main roads to choose:
1 - One road is saving our ass saying -
Who cares? I'm agnostic! - Congrats. Game over. You're a smart guy. Bravo.
2 - The other one instead is more complex, painful and dividing, but also more interesting: it consists in taking position, even though we can't prove it, we can believe in God or we can say -
God doesn't exist. - both these choices are in explicit contradiction, one excludes the other.
- That is, choosing God, you give a new meaning to your life, a new meaning which explains every aspect of our life experience: death, pain, love, passion, war, hate, suffering, time, space, universe, everything now finds a new meaning, every single part of our lives can be, if not fully understood, at least accepted as God's will. You're here, on this planet, because you're part of a big supernatural design, any action and decision taken, plays a role in the big fresco of the divine will. Everything is finalized to some kind of meaning, so everything finds a meaning in the end, even though our minds are too small to fully understand it, as part of a whole. Key words are: faith and hope.
- Instead if you choose the matter, then you exclude any kind of external explaination for the world in which you live, there is no meaning, there is no finality, there is just matter, colliding and competing to survive, we are just slow chemical combustions, we are on this planet without any reason, there is no finality in our existence, we have nothing to prove, there's no tribunal for us and for our actions, we are the judges and the tribunal of ourselves; there is not even any hope to save our ass if we fail, there is only one life lasting more or less 80 or 90 years then .. nothing, nothing at all, we have just one shot, then we will vanish as thinking subjects. In atheism the nothing is the most peaceful and damn hard concept to accept, accepting the real end of ourselves as thinking subjects is probably the hardest part of atheism, but, all in all, it's also very consolatory.
I've made my choice, I can't rationally prove that God doesn't exist, being God an irrational concept/presence. I firmly refuse to explain the world using God, for I believe in the non-existence of God, this is why
I'm theistically atheist, the non-existence of God has for me the same value of faith. This is called materialism, and believe me, not always it's an easy trail on which taking a walk.
Side note: Of course, when I write the word "God" I'm not talking of the Christain deity, I'm talking of any form of metaphysical presence/s beyond the physic space we can touch, measure, and understand (in some way).