What follows is one of the most challenging subjects you will ever hear about. It is the best way to get a head ache, but I think it proves the existence of God. The problem is it requires a very specialized background to understand it. First you have to understand modal logic.
Modal Logic is so called because it turns upon the use of so called "modal operators." It's called "modal" because it is the logic of modes of being. "modes" as in what type of existence something exits in, whether it is dependent upon other things, whether it can cease or fail to exist and so forth. The modal operators are "necessity," "contingency" "impossibly," "possibility."
Necessity and contingency lie at the base of our modern understanding of cause and effect. They come from scholastic notions of logic, but the distinction between the notion our modern notions of c/e and the scholastic ones in the middle ages is not that great. The scholastic had more levels of cause, efficient cause, final cause and several others. But one could everything we have done in modern science using the scholastic ideas of c/e.
Necessity doesn't mean has to exist. It doesn't mean God is necessary to the existence of the world (except in so far as if God exists then of closure God is necessary to the world as creator--without God there would be no world).The modal argument does not begin with the assumption that God has to exist. It begins with the assumption that there is a valid distinction between necessity and contingency, which there must be. It proceeds along the lines of hypothetical consequence that obtain from different scenarios of God's existence. It concludes that is necessary. But by "necessary" it means not contingent, or not dependent upon something else for its' existence.
This is often misconstrued by atheists and taken to mean the argument proceeds from God's existence as an assumed first premise. This is not the case, the first premise is either/or. Either God's existence is necessary or it is impossible. This allows for the possibility that there is no God. So the argument does not begin by "defining God into existence."
Necessity essentially not contingent, it also conveys the idea of he can't cease or fail to exist, stemming from his eternal nature.
Contingent means the opposite: that a thing is dependent upon a prior thing for existence, or that it could cease or fail to exist.
Impossible means logically impossible, something in the structure of the idea contradictions, such as square circles.
One of the sore spots that atheists get stuck on is the idea that God cannot be contingent. They will always leap to the conclusion that this is defining God into existence, because they don't understand the concept of God. God, by the nature of the concept, carries certain parameters just as the existence of any human assumes humanity, or the existence of any tree assumes that the tree in question is a plant. To have to define that God is not contingent should not even come into it. The idea of God is that of eternal creator of all things. Thus God cannot cease to exist and cannot be dependent upon anything (or he wouldn't be the creator of all things). Atheists usually assume that all knowledge has to be empirical. they will argue this is defining God into existence. maybe God is contingent.
Close to Hartshorne's version
1. God is either necessary or impossible.
2. God can be conceived without contradiction.
3. Whatever can be conceived without contradiction is not impossible.
4. God is not impossible.
5. God's existence is a necessity (from 1-4, not contingent or impossible means necessary)
6. If God is necessary, then God exists.
7. Belief in God's existence is warranted
Hartshorne Lived to be 103, at the time of his death in the Fall of 2000, he was known as "the greatest living Metaphysician." Hartshorne was one of the major forces in the "back to God" movement in Philosophy (a term coined by Christianity Today in a 1979 article. His first and greatest calim to fame is as the second most influential voice in process philosophy, along with Alfred North Whtiehead, but he is also credited as the man who brought the Ontological argument back from ignominious defeat by Kant almost two centuries earlier. Hartshorne was also a recognized authority on birdsong, and an authority on bycicles, having never driven a car a single time in his centogenerian lifespan. Hartshorne devoted the last years of life to waging a letter's to the editor campaign to advocate social issues such as medical care.