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The traditional definition of 'God' (I don't necessarily here refer solely to the Judeo-Christian god by the capitalization, since this definition spreads to many other religious (and perhaps non-religious) claims of deities) is a perfect being that designed and created the universe. I won't go into an analysis of the nebulous word 'perfect' here, but in general it seems most people take this to include attributes such as omnipotence (unlimited power; or at most limited to that which is non-contradictory, to avoid the "can such a being create a stone so heavy it cannot be lifted?" paradox), omniscience (unlimited knowledge, say of the consequences of actions taken, of what goes on all over the universe, etc.) and benevolence (kindness, compassion, lacking ill-will or malevolence). These three attributes (along with that of being a creator, designer or first-cause-by-choice of the universe) are firmly ingrained into the meaning most religious believers give their god, and if one or more of these attributes should fall, I think it would require some rethinking on the part of the believer about the nature of the being they often heap so much praise upon.
The Problem of Suffering (I prefer this phrase to the more common Problem of Evil, because it better avoids the ambiguities and baggage of the term 'evil') is one that has been around a long time, and in my opinion one of the most powerful reasons why it seems extremely unlikely a deity, as described above, exists. It hinges on a sort of disproof by contradiction, so to speak. If it can be shown that the three characteristics listed above (along with that of being the creator/designer) are incompatible and cannot logically exist in the same being, then that demonstrates that, logically at least, such a being is impossible, cannot exist. I should add here, however, that I do not take this argument as an absolute proof, per se, but instead as at the very least a good reason not to give much consideration to the possibility of existence of such a deity (as I will explain further later on).
The idea behind the Problem of Suffering is fairly simple. Any one of us can look around and see the existence of pain and suffering in the world. We see people spending years slowly dying in hospital beds from terrible cancers; we see children abused, raped and killed by the adults who hold power over them; we see cities destroyed, families crippled, innocents dead from not only war, but unavoidable natural disasters as well; we see a world where age brings with it a terrible decay, mentally and physically, cured only by death itself. Suffering is all around us, that much is undeniable. And I think we all have some sort of natural aversion to it. If we can minimize our suffering, we tend to. That's not to say that suffering can't be a learning experience - but at some point in the extent to which people suffer, I think most people would draw a line. How much can a child of two learn from a torturous death by cancer? And so I think most of us do desire suffering to be, if not abolished or minimized, at least reduced from what we can so easily observe around us every day of our lives.
How then, can a world so full of unnecessary suffering exist, if it was created by a deity of not only unlimited power and knowledge, but benevolence as well? Even our limited human understanding can conceive of a logically possible world where there is less suffering. Why then, would not the god in question create such a world instead of the one it did? Perhaps it lacked the power to create a better world - in this case it is certainly not omnipotent, by the usual understanding of the word. Perhaps it lacked the foresight and knowledge to understand what it was creating, or to see how to create a better universe - in this case it is certainly not omniscient, by the usual understanding of the word. If it had both the power and the knowledge to create a better world, and did not do so, then its choice seems a malevolent one; not something a compassionate being would choose. So given the suffering that most certainly exists in our world, an omnipotent, omniscience and benevolent deity seems an absurdity. No such deity can exist - it would have to give up its claim to at least one of those attributes.
One common objection to the Problem of Suffering is the widespread claim that evil is necessary for good - that suffering is necessary for happiness. This is an opinion often shared by believers and non-believers alike, and it basically states that if we never experienced any suffering, we would have nothing against which to gauge our happiness - we would exist in a single static state which we could not truly enjoy without knowing what another state is like. The argument has some intuitive appeal to it, but that alone does not establish it. After all, contrary to intuition, cold is not the opposite of heat, but the lack of it - there is in fact no 'opposite' of heat, and no such opposite is needed for heat to be a meaningful concept. It seems at least somewhat plausible then that an analogy could be made between heat and happiness. If we lived in a world where there were merely degrees of happiness, but no suffering, happiness could still remain a meaningful thing. Of course, one might define suffering as the lack or relative lack of happiness. Even in this case, however, I think most of us can imagine happiness still existing meaningfully in a universe where there is less of even this type of suffering (in other words, where no one is so lacking in happiness as to have a terrible cancer or be killed by a natural disaster); thus again the universe could have been created with less suffering without destroying happiness. Happiness could still be balanced by suffering in a universe where innocent children (who lack even basic understanding of the situation) don't die slow deaths from disease.
There is another important point to be made regarding the claim that suffering is necessary for happiness. In the Euthyphro, Plato posed the question - are good things good (and evil things evil) because 'God' makes them so, or does 'God' make them so because they are that way (i.e. 'God' follows the rules already set)? In a similar fashion, one might ask - did the deity in question fully design 'the way things are' (from the laws of physics to the nature of consciousness) or did it work within constraints already set forth somehow. If the latter, the deity seems to, in some way, lose it's place as first-cause and creator/designer; also, its power becomes quite limited, and omnipotence is lost. If the former, then the deity in question designed the very 'rules' by which happiness requires suffering (if that is indeed the case). An omnipotent, omniscient and benevolent designer deity could, it seems plausible, have created a universe where happiness did not require suffering. The suggested relation between happiness and suffering, even were it true, is not a defense for the deity's choice, since it chose to design that very relation in the first place.
Another common objection to the Problem of Suffering is that evil or suffering is necessary for free will (assuming such a thing exists). That free will implies having a choice is generally accepted - that it implies having a choice between acting good and acting evil (or acting to cause happiness vs. suffering; I will use the terms interchangeably herein) is less clear, and that is why this objection is most common, in my experience at least, to Christians, whose notions of salvation include humans having the ability to choose between good and evil actions (thus evil actions must be possible). I don't personally see that that follows from free will, but that might be because I personally find the idea of free will far from clear.
At any rate, even if I were to grant that it is the case that we have free will, and that free will implies a choice between good and evil actions, there are two reasons why I don't think this solves the Problem of Suffering for believers. First, even if it were necessary for free will that humans be able to choose acts that cause suffering, natural disasters and disease would not be necessary. Thus, a benevolent deity with the power to do so would at least rid the world of killer earthquakes, tornados, plagues and diseases. The second reason why the free will argument doesn't solve the problem follows the same argument given above for the necessity of suffering to have happiness. If the deity in question did in fact design the way things are, it designed the necessity of possible evil actions under free will, whereas it could have (if omnipotent) designed free will to exist without the possibility of evil actions. Someone might object that the deity only created the possibility of evil actions, but that it was our choices that brought the actual evil about. This only puts off the problem however. An omniscient deity would know that in the universe it created (ours), people would indeed choose evil acts and suffering would take place. It would thus have chosen to create such a world (where suffering took place), instead of one where no suffering took place but free will still exists (in other words, the deity would design free will differently).
Given the ideas presented above, it seems to me not unreasonable to conclude that the omnipotent, omniscient and benevolent creator deity is a contradiction, and thus not logically possible. Because there is so much unnecessary suffering in the world, it seems that if any omnipotent and omniscient being created the world, that being could not have been benevolent as well. If benevolent, it seems it must have lacked one of the other attributes. There is one objection to the Problem of Suffering which I have not yet covered and that is that a being of such power and knowledge would be so far beyond our limited human understanding that we cannot come to any sure conclusions about its nature, its choices, the possibilities involved, and so forth. This is indeed a valid objection, and certainly one of the reasons I do not categorically claim that no such god exists (it could, for example, 'transcend' logic itself, and contradiction would thus present no problem to its existence). However, all the evidence we do have, all that is at our disposal as beings of limited understanding, points towards a universe where such a being is not possible, does not exist. While the possibility for error is certainly there, I see no reason to give it any more thought than other seemingly-contradictory possibilities which we pragmatically rule out every day - that the sun will simply disappear from existence tomorrow, or that the universe is actually the mixed droppings of a pair of transcendent invisible pink unicorns, or that the universe was simply created by an omnipotent, omniscient evil deity. They, and literally anything else (!), may in fact be the case, given the fallible understanding we humans have, but that doesn't give us any reason not to rule them out as far as living our lives goes. The existence of something that is so far beyond us is indistinguishable from that thing simply not existing.