Arguments against the existence of God
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Arguments against the existence of God attempt to establish that God's existence is unlikely or logically impossible. However, just showing that God's existence is unlikely doesn't really prove anything since improbable things happen.
The problem of evil
The omnipotence paradox
Omnipotence seems to have a logical contradiction built into it. An all-powerful being should be able to be more powerful than itself since It should be able to "be" anything. However, by definition, nothing can be more powerful than an all-powerful being.
Argument from poor design
The argument from poor design is based on the reasoning that an omnipotent, omniscient and omnibenevolent God would not create the Universe with the perceived suboptimal designs that can be seen in nature. Examples would be 99.999% of all species being extinct, millions of dead stars and galaxies, and bone cancer in children. Fun stuff like that. The argument can be used against the existence of God and/or his competence.
Free will argument for the nonexistence of God (FANG)
The FANG argument from Dan Barker says that since an omniscient God would know its own future, and therefore its own decisions, it does not have free will and thus cannot be a personal being. In fact, if the deity has never changed — it was always omniscient — then what is the origin of the decisions of God? So to explain the decisions of God; we need another agent. That would seem to make an omniscient deity impossible. For example, the Abarahamic deity (Who is largely believed to be omniscient by His followers) constrains His future action by giving information of the future in his Book of Revelation. Additionally, since (according to most Christians) many details of Jesus's life are foretold in the Old Testament, Jesus would have had no free will.
Albeit, this presupposes that an omnipotent being would somehow interact with time in the same manner that we non-omnipotent beings do or, at the least, that time and causal effect would somehow bind an omnipotent being at all. This is similar to Dr. Square trying to understand the third dimension. A truly omnipotent being would already be free from causality and logic as we know it. Within this existence, given our limitations, such a level of ability would seem to be a paradox, in and of itself.
Argument from incompatible attributes
The Argument from incompatible attributes attempts to prove that God is logically contradictory, much like a married bachelor, and therefore that specific God cannot exist. An example would be that God cannot be both all merciful and all just since mercy is a suspension of justice.
The omnipotence paradox, referenced above, might be considered a flavour of the "argument from incompatible attributes".
The problem of divine hiddenness attempts to show an inconsistency between the existence of God and a world where people fail to recognize him.
Argument from inconsistent revelations
The argument from inconsistent revelations attempts to show that an all loving God is incompatible with a Universe with so many mutually exclusive revelations, since an all loving God would not be the author of confusion and would attempt to clear it up.
An omniscient mind needs an extremely powerful information processing system. It has to make all the correct connections. How could such a mind come into existence? If the universe's complexity needs explanation — then the even more complex mind of God needs explanation. The monotheistic God is by definition uncreated — thus there is no explanation of its vast specified complexity. So, it is better to assume that the Universe exists without a creator, than that a God exists without a creator.
That being said, the same argument around the cause of God would seem to apply to the singularity (or near singularity) which resulted in the Big Bang. High gravitational wells mess the heck out of time, rendering the question "What existed before the Big Bang to cause it" almost inapplicable.