Of all the various arguments that have been proffered over the centuries for why God exists, I find some more interesting and compelling than others. I am not such a big fan, in general, of arguments like, "Look at the hippopotamus. Such a beautiful thing could only have been created by God", although Tertullian did use such an argument to refute Marcion's claim that the devil had created the world. Such 'arguments from design' leave me cold, not least because we now know just how hippopotamuses and other living things came to be: they evolved from the first living organisms, and it was by the long, bloody, and brutal process known as natural selection, hardly something one associates with a loving God.
The arguments I find most convicing are the argument from causality (the cosmological), the argument from mystical visions and miracles (the experiential) and finally, the argument from the concept of perfection itself (the ontological). The ontological proof is, I think, the most interesting and the most powerful, since it purports to establish, not just the existence of a First Cause or a Prime Mover, but the existence of a Perfect Being (i.e. God). Also because it relies on logic alone. It was first developed by St. Anselm in the twelfth century, a great stalwart of the English Church, and in the twentieth century was revived by the philosopher Charles Hartshorne. Hartshorne was a theologically liberal Unitarian, and so of course I disagree with most of his theological viewpoints, but I think he made a powerful modification of the ontological argument, updating it for the twentieth century, and turned it into an argument that, it seems to me, is very hard to refute. I recently looked it up- though it's couched in the language of highly technical philosophy, here's the gist of the proof as I understand it.
Let's start by defining God as the most perfect being that can be conceived. Now, either He exists or He doesn't. But the important thing to realize about the nature of a perfect being is that he doesn't, and cannot, exist _contingently_. That is to say, God does not _owe_ his existence to anything outside Himself. If a perfect being were to be brought into existence, or depend for his existence, on some outside agent, He wouldn't be perfect. Because clearly it is better to be sufficient on one's own, and able to exist and spread one's goodness in any conceivable set of circumstances, then to only be able to exist if the right circumstances are met.
This is to say that God is a _necessarily existing_ being. Nothing- no agent, force, or set of circumstances- could allow God to come into existence if he doesn't right now. Conversely, no agent, force, or set of circumstances could result in God's _nonexistence_ if in fact he existed. Because again, a conceivable perfect being that exists self-sufficiently and eternally is more perfect, and more flawless, then a being who is perfect in every other way but has the flaw of not being able to exist except under the right conditions. The first God is greater with respect to power, and I'd argue also with respect to goodness, then the second, thus the second being is not the most perfect being conceivable.
We've established, then, that either A) a perfect being exists and nothing could stop him from existing- that is, he _necessarily_ exists, or B) a perfect being doesn't exist, and nothing could make him exist, that is it is impossible for him to exist. What cannot be the case is that God just 'happens' to exist, that He exists contingently on other circumstances, that if things were different He could exist or He could not exist. If the concept of a perfect being means anything, it means a being who is self-sufficient, eternal, and independent of anything else. Self-0sufficiency is a tricky thing here, as the Trinitarian conception of God involves three persons who in some sense 'depend on' each other, but let that pass for the moment. God cannot depend, for His existence, on anything outside Himself (as the existence of a tree depends on adequate rain, nutrients, and seeds in that place, or as the existence of an animal depends on its parents, or as the existence of the moon depends on the physical laws and processes that formed it).
So summing up, it is either NECESSARY that God exists, or it is IMPOSSIBLE that God exists. If it is possible that He exists, then He necessarily must exist= He cannot exist contingently.
The full power of Anselm's argument, as developed by Hartshorne, is evident here. We are suddenly brought to the brink, where one must fall off the fence to the left or right. There is no room left to say "Maybe God exists, or maybe He doesn't". Of course one can take that line, and many people do, but I think if we take the Anselm/Hartshorne argument seriously, it's logically insupportable to do so. We are required to make a leap of faith and decide whether we think it's fair to say that it is _impossible_ for God to exist. Naturally, I don't think so.
To say that it is _impossible_ for God to exist is to say that the concept of a perfect being is logically incoherent. People have tried to do that, of course, but I think such arguments fail. If perfect power and perfect knowledge are sufficiently qualified and rightly understood, such that God cannot do anything that detracts from His perfection, then there are no logical inconsistencies in the concept of God. Is it more likely that God doesn't exist, or that God does exist? One test of any theory over its rivals, say Theory A against Theory B, is that Theory A explains everything that Theory B does, and other things as well. In this light, a universe just like ours but including a God would account for everything we observe (if we assume that God allowed the universe to develop, for the most part, according to physical laws and life to evolve according to natural selection), but it would also explain some things that couldn't be explained otherwise (i.e. direct visionary experiences of God).
So there is at least some prima facie possibility that God exists. We have the concept in our minds, after all, and it is hard to find anything incoherent in the _concept_. But the moment we grant that the existence of God is a possibility, then we must also grant that it is a certainty. For God cannot exist contingently, as that would be inconcistent with the logical requirements of perfection.
Which leads me to conclude, as Anselm and Hartshorne did, that since it is logically _possible_ to conceive of the existence of a perfect being, such a perfect being, the most perfect being that the mind can conceive, must exist. "And this all men call God."
Indeed. Rest in peace, Charles Hartshorne: into Paradise may the angels lead you, may a choir of angels great you at your coming, and with Lazarus who once was poor, may you find eternal rest.