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From FreeThoughtPedia

The Incompatible Properties argument is used to disprove the existence of various supernatural claims. It's particularly effective to illustrate the contradiction in biblical and other scripture.

Here are examples of Incompatible Property arguments (Note that most of these use Judeo-Christian scripture as as examples but similar arguments could be made for many other religions):

Absolute good

If the Judeo-Christian god is "all good" then why would he allow evil in the world? Why would an "all loving" god allow people to suffer? This concept is also known as The Problem of Evil. For a very good elaboration on this topic, I recommend visiting this site: [1]

The common Christian response to this invokes the parable of Free will. The problem with free will is that it introduces yet another incompatible property.

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Omnipotence

If god is all-knowing and all-seeing, then he surely knows what has happened and what will happen, so why would he insist on creating man knowing man will fall from grace? If god is "all good", "omniscient" and "omnipotent" then he knew he would be bringing suffering into the world. Why would an omnipotent/omniscient creator be surprised or disappointed in his creation if he knew what was to happen? Why create incentives or punishment?

Omnipotence / Omniscience paradox

It is not logically possible for god to be both omnipotent and omniscient. If god is omniscient, he would know the future with absolute certainty; but his 'omnipotence' would be rendered powerless if he ever tried to alter the future, due to his 'omniscience'.

Immorality of god

God claims to be the purveyor of absolute moral standards, but his scripture is not indicative of this. Many activities that are considered immoral by our standards, such as genocide, rape, slavery, etc., are promoted in scripture. For many examples of this, I recommend clicking on either "injustice" or "cruelty and violence" on this popular site: [2] or visit this site: [3]

Creation argument

If God is unchangeable, then he cannot go from not creating to creating or vice versa. He must have for the universe to be created. If every creation needs a creator, who created god?

Non-belief argument

God supposedly wants people to believe in him, yet there are people who do not, or people who believe in different gods. It makes no sense that god would allow this to happen if having people believe in him is important.

Confusion argument

If it is that important to believe in god, why are there so many people who have such diverse ideas of god and his nature? Surely if god wanted us to believe, he would have made it more clear. Books like the Bible and the Koran are anything but un-ambiguous.

Consciousness argument

Based on all available knowledge, consciousness requires a physical component (brain, etc.) So if god is conscious, where is his physical component?

Slavery argument

God wants us to worship him. He supposedly deserves this. But humans were created to make their own choices. If god truly deserves to be worshipped by all, then there is no freedom of choice, and in such a case, the significance of our commitment to him is nullified and we are basically slaves.