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How does the atheist account for the acceptance and validity of the laws of logic?

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The same way I'm sure the sun isn't going to turn into a dragon's ass and shoot fire at me.

Self organising systems such as the brain necessarily develop certain internal rules that make them functional. We observe events and generalise them. We see effects and postulate causes. We draw links between related items. This is a basic component of our brain architecture - as well as that of animals. Logic is just a refinement of this process.