Just thought I'd write a brief essay explaining in simple terms why God does not exist. For the sake of this essay, I will be assuming that no argument proves that God necessarily exists or has an extremely high probability of existing. I don't believe any argument has been successful in providing anything more than very weak support for the existence of God, nevertheless, if someone reading this believed that moderate support for God's existence was proven by a few arguments, he should still agree with me since I believe that my arguments would outweigh any case for God that did not prove it to 100% certainty (or to 99.99999999% certainty before taking my arguments into account, and I mean that literally).
In philosophy every hypothesis is judged on two levels: a priori and a posteriori. A priori reasoning determines whether something is possible/impossible, likely or unlikely through abstract logic and backround knowledge (what we know from cases we have observed other than the one at hand). A posteriori reasoning deals with the facts that are specific to the case at hand.
Here's a quick example of a priori reasoning: If Bob's wife Sue turns up missing, and we know from statistical studies that 99% of married women who go missing are murdered by their husbands, then we can infer that the a priori probability that Bob murdered Sue is 99%.
Here's an example of a posteriori reasoning: Let's say that during the investigation of the missing woman Sue, we find evidence that Sue bought a plane ticket and went to Russia. Investigators go to Russia and find Sue there. We conclude that Bob did not kill Sue because of the a posteriori evidence (the evidence specific to this case).
I think I can show that the existence of God is incredibly unlikely on both a priori and a posteriori grounds, therefore making the case against God as airtight as possible.
I argue that a being like God has an extremely low a priori probability on the following grounds:
First, God is usually concieved of as a super-intelligent, conscious, creative mind. I argue that such a mind existing uncaused is highly improbable. Furthermore, the usual attempts to get away from this (like saying that God is necessary and therefore not improbable) fail because they do not take into account the fact that the hypothesis of being like God being necessary is just as unlikely as the original problem. Think about it: How likely is that, out of all the possible minds that exist, God's mind, and only God's mind, is necessary (if any mind exists necessarily)? A few philosophers have arguments to back up their assertion that God is necessary, but I believe that they fail-- miserably. See my book Atheism and Naturalism.
The classic argument from design points to living things, and suggests that design will explain the fact that so many living things have organs and organ systems made of many parts, all specified and tailored to perform some function. In Darwin's Dangerous Idea philosopher Daniel Dennett uses a metaphor for thought called "The Library of Mendel". This imaginary library contains all possible genomes (genetic codes). I suppose someone trying to make the design argument might ask us to think about this library, and would then ask us how many of these genomes could produce a viable animal. After all, even tiny changes in the genomes of living things will leave them unviable. If you think about it, very few genomes in Mendel's library would code for an organism of apparent design. But we see apparent design all around us. So the codes could not have been created by randomness, that would be terribly unlikely. William Paley, had he known about the genetic code, would have undoubtedly thought that the codes were determined by a master designer. We know better; we know that they were determined by natural selection.
I say that to introduce another metaphor: The library of Broca. The library of Broca is the set of all possible minds. Now, theists postulate that there is a necessary mind (God's mind). The hypothesis that there is one and only one necessary mind (rather than none or two-infinity) and that it is God's mind is an extremely specific and obviously improbable hypothesis. Why couldn't the necessary mind be an idiot mind, or the mind of an Asian Tiger, or any other mind from the practically infinite set contained in Broca's library? Unless some argument can be given to support the hypothesis that the mind of God is necessary, we must apply the principle of indifference and assume that all possible hypotheses are equally probable (I have defended this practice here). And there are billions and billions of hypotheses. There might not be any necessary minds. There might be ten necessary minds (which would be polytheism, not monotheism!). And so on.
Second, I'd like to follow a line of argument suggested by Geoffrey Berg in The Six Ways of Atheism : God is a being who is omnipotent, the creator of the universe, supremely good, and so on. But it must be statistically improbable for the creator of the universe to also possess attributes like supreme goodness which must occur in very few (if any) beings. To see this argument with more clarity, try thinking about it this way: suppose that tomorrow one person will win the lottery and one person will be struck by lightning. What are the odds that the lottery winner will be one and the same as the person who gets struck by lightning? Obviously, the odds are incredibly low that two improbable occurances will happen to the same person. Likewise, the odds that a rare (or perhaps even non-existent) attribute like supreme goodness will happen to exist in the unique being who created the universe is also improbable.
Now for the a posteriori argument: The hypothesis that there is a completely good, all knowing, and all powerful being predicts that the world will contain little or no evil. A good being always removes as much evil as it can (that it knows about). An all powerful being could remove all evil. An all knowing being would know about all evils. Therefore, an all good, all powerful and all knowing being would remove all evil if it existed. But evil exists. Therefore, the being does not exist.
Various responses have been made to this argument: God does not remove all evil because he must allow some evil which we humans do with our free will (and free will is supposedly such a great good that God is willing to allow some evil in order for us to have it).
Leaving aside the fact that this is a mere possiblity (we don't know with any precision whether or not free will is more valuable than the abuse that the gift of free will will lead to), the defense is a poor one. The actions of human beings are determined by what they can do, their knowledge of the options available to them, and by their desires (humans choose the option that they believe will best fulfill their desires). If that view is correct, then God could have created humans with pure desires, thereby determining that their choices would always be good and evil would not exist. I've written more on this here.
Addendum: On the possibility of arguments for God.
Gregory Dawes' book Theism and Explanation, which I have reviewed here, shows that any inference to the best explanation to God must be weak, at best.