Having spent roughly 25 years or so of my life contemplating the question of God's existence, I've surveyed the depth and breadth of arguments both for and against the existence of a being we refer to as 'God'. In some cases, not much effort is made to distinguish between 'gods' and God. In at least classical theistic traditions, God refers to a being that is maximally great; possessing every great making property to the maximal degree possible. In other words, the greatest conceivable being; that than which it is not possible for something to be greater than.
This is important because, when people speak of 'gods' they often lump in beings that, whilst perhaps powerful and possessed of some kind of 'supernatural' quality, are nevertheless finite beings. Aside from many if not all being created beings and thus contingent, many suffer from a complete lack of anything resembling divine characteristics. The 'gods' of polytheism are all too finite. God, on the other hand, is metaphysically necessary, omnipotent, omniscient, morally perfect, and the ontological source of being for everything else that exists.
As such, when we speak of the existence of God, we can immediately and easily dispense with the various deities of paganism. Now, in many ways, the existence of God can said to be known immediately and a priori. Indeed, belief in God can be deemed simply as being properly basic. We do not form such a belief on the basis of underlying arguments, evidence, or reason, but form the belief as naturally as we do the belief in our own existence.
Of course, whilst this means we can be rational in our belief, it isn't really enough to show that God exists. The first argument for God's existence is based on modal logic and dubbed the Ontological Argument. The argument is fairly simple: if the concept of a maximally great being is coherent, it is therefore possible and thereby instantiated in some possible world.
The key premise is that if a being that exists in every possible world is possible, it exists in some possible world. However, it becomes immediately apparent that it follows logically and inescapably that this being must therefore exist. The reason for this is because the set of all logically possible worlds includes the actual world in which we live.
If the concept of God is coherent, and therefore logically possible, God exists in some possible world. But if God, who exists in every possible world if He exists at all, exists in some possible world, then that means He must exist in every logically possible world. Given that the set of all logically possible worlds contains the actual world, it follows logically and inescapably that God exists.
There are three responses to this argument made by atheists. The first is to attack the coherency of the concept of God. The second is to try and utilise a parody version of the argument in an attempted reductio ad absurdum counterargument. The third is to accuse those of putting forward this argument of begging the question.
There are various arguments against the existence of God alleging to show that various divine attributes are either themselves impossible, or at the least cannot be instantiated in the being of God in conjunction with other properties simultaneously. Whilst there are various arguments along these lines, this general approach has been dubbed the argument from divine incompatibilities.
Some versions of this approach are weaker than others. For instance, some atheists argue that God is an incoherent concept because there is no such thing as a being worthy of worship. However, it doesn't take a dyed-in-the-wool religious fundamentalist to recognise that this begs the question.
If God is defined as a being worthy of worship, and your argument for why we should believe He does not exist includes the premise that 'beings worthy of worship do not exist', then, aside from not being a convincing argument at all, you beg the question since your only reason for accepting that premise is because you already believe the conclusion.
Some argue that God cannot be perfectly just and perfectly merciful at the same time. Whilst this particular argument does not beg the question, it is nevertheless not a very convincing argument because it assumes definitions of justice and mercy that aren't immediately obvious. Moreover, there exist definitions of both that are mutually compatible.
So, you would need to go through the laborious task of making a case that your definitions of mercy and justice are at least more plausible than other versions before making such an argument and hope that there are no defeaters. So, we shall focus on stronger versions of this argument. Probably the best version of this argument is the claim that omnipotence is an incoherent concept.
It is claimed that, if God has the ability to do anything, then He can do things that are logically impossible, such as making square circles, or nonsensical things like making purple smells. The response to this is to simply point out that logical impossibilities aren't 'things'. As such, the ability to do anything hardly applied to non-things.
The classic example given by atheists is God being tasked with creating a rock too big for Him to lift. Now, God, being immaterial, isn't reliant on physical strength, so a better reconstruction might be to ask instead: could God create something more powerful than Himself? Now, the atheist alleges that this is a paradox that shows that omnipotence is incoherent, but the reality is that all the atheist has done is conjoined random words that have no overall meaning when taken together.
Take the example of a square circle. A shape cannot be fully square and fully circle at the same time because a square is defined as being a shape with 4 equal sides and 4 equal right-angles, and a circle is a one-sided shape whose center is equidistant from every point of that one side. Oh, but someone could argue that you could have a square with smooth corners.
However, note how, in this case, you would no longer have a shape that is fully square or fully circle. It would be neither, but it could not be both. By saying a squircle is a square circle, you are relying on an alternative definition of both square and circle. It's an 'if by whiskey' style argument that obfuscates by playing with words.
In the same way, the atheist that alleges that God must be able to create something more powerful than Himself in order to be omnipotent is likewise playing around with words in order to obfuscate. If God is the most powerful being possible, then the theist can confidently answer that the answer to the question is no, because a being more powerful than God is an incoherent concept.
In other words, the question 'can God make something more powerful than Himself' is just a nonsense question, akin to asking someone if the statement true or false is true or false. So, really, this approach is just a non-starter. You might as well think that asking 'can God absquatulate enamel from the singularity' is a meaningful question.
The second approach is, I think, more promising. Parody versions that seek to show that the argument would entail absurd results generally has more truck, in my experience. For instance, some might argue that someone could construct similar arguments about a maximally excellent lion, or a maximally great island.
These versions of such an approach are, in my opinion, very weak, since neither a maximally excellent lion nor a maximally great island are coherent concepts. The reason for this is because both lions and islands have features that are incompatible with maximal excellence. A lion cannot exist in every logically possible world, and neither can an island.
Another version of this approach would be to argue that maximal excellence or greatness would entail absurd properties, such as maximal smelliness. However, the obvious rejoinder to such arguments is to simply point out that these are not, in fact, great-making properties. Some have therefore chosen to argue, in the same vein as Immanuel Kant, that existence is not a property that can be added to the essence of something.
However, the obvious response to such a challenge is that we are not treating mere existence as a predicate. Rather, we are treating a specific mode of existence as a property; that of metaphysically necessary existence. This neatly sidesteps this objection entirely without committing us to a particular ontology.
The third approach alleges that theists who put forward the ontological argument are somehow begging the question; that we only believe that the existence of God is possible because we already believe that God exists. However, there are two things that can be said in reply to this. First, one can easily construct arguments for the possibility of a metaphysically necessary being. Second, however, is that, in the absence of any obvious logical contradictions and defeaters, we are rationally justified in believing that God is a coherent concept.
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