A logical case for the existence of God:
Scenario #1: A homicide detective is investigating a murder. He begins his search for the murderer at the end result of the action (i.e. the crime scene). He uses blood splatter evidence to trace the trajectory of the bullet. He finds the bullet and analyzes the firing pattern to determine the gun it was fired from, etc., working his way back to the murderer (watch any episode of CSI to see this in action).
Scenario #1: In this scenario, a major assumption that the homicide detective makes is the belief that every action has a cause that precedes it, and thus the series of events leading to the murder can be reconstructed and traced back to the originator of the crime, i.e., the first cause of the murder. Whether that is an affair, or money, or abuse as a child, whatever it is, there is the assumption that something (or some-things) happened. And that something(s) set in motion a chain of causal events that led ultimately to the resulting action, i.e. the murder.
This basic premise can be taken even further, because everything that exists in the universe can be traced down the chain of causality all the way to the beginning; one event causes another, which causes another, which causes another, ad infinitum. In other words, the whole universe is a vast, interlocking chain of things that come into existence because other things cause them to be. Our murderer would not be here to murder anyone without billions of causes, from the marriages of his parents and their parents all the way back through the development of the first protein molecule to the cooling of the galaxies and the Big Bang.
Everything that comes into existence must either exist by itself (i.e. by its own essence or nature), called an Independent Being, or it must exist because of something else (it was brought into existence/caused), called a Dependent Being. If it is an Independent Being and exists by its own essence/nature, then it’s being-ness is sufficient to explain its own existence, and it cannot have been created because that would mean it was caused to exist by something else, and thus it exists eternally. It cannot not have these qualities and still exist as an Independent Being, just as a triangle cannot not have three sides and still be a triangle.
If, on the other hand, something is a Dependent Being and exists not by its own essence, then it needs a cause, a reason outside itself for its existence. Dependent Beings cannot cause themselves. They are dependent on their causes. But does the universe as a whole have a cause? Is there a First Cause, an uncaused cause of the whole chain of causality in the universe? If not, then there is an infinite regress of causes, with no first link in the great cosmic chain. If so, then there is an eternal, necessary, independent, self-explanatory being with nothing above it, before it, or supporting it. It would have to explain itself as well as everything else, for if it needed something else as its explanation, its reason, its cause, then it would not be the first and uncaused cause. Such a being would have to be God. If we can prove there is such a First Cause, we will have proved there is a God.
Why must there be a First Cause? If, as previously mentioned, the universe contains only Dependent Beings, then the whole universe is unexplained without a First Cause. If there is no First Cause, each particular thing in the universe is explained in the short run by some other thing, but nothing is explained in the long run, and the universe as a whole is not explained. If there is no First Cause, then the universe is like a train moving without an engine. Each car's motion is explained proximately by the motion of the car in front of it: the caboose moves because the boxcar pulls it, the boxcar moves because the cattle car pulls it, etc. But if there is no engine to pull the first car and the whole train, the train cars cannot move of their own accord. The universe as a whole existing wholly independent of some First Cause is like a train moving without an engine.
Therefore, the universe must have a First Cause, and that cause must be an Independent Being. If there is no Independent Being, then the whole chain of causality in the whole of the universe is dependent on nothing and could not exist. But it does exist. Therefore there is a First Cause, that First Cause is itself uncaused and must then be an Independent Being, and this Independent Being is necessarily eternal and explained and justified wholly by its own being-ness, i.e. God.