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Can someone tell me why, in a debate, one side would attempt to present evidence that, if accurate, would shoot down both sides of the debate?
It doesn't shoot down both sides. The proper question should be "Why owuld the creationist present evidence that shoots down his side?" Sediment doesn't stay on the bottom of the ocean, so this isn't really a problem for an old earth. Sediment can get scraped off of the oceanic crust as that crust is subducted under another plate, or lifted up as two plates crunch together, or the sediment might just get pulled into the mantle with the rest of the crust it rides upon.
I'm not sure what is going on with the short period comet argument. I have heard an argument which runs something like "If the earth were 4.5 billion years old there shouldn't be any short period comets as they would have all been swallowed by the sun by now." Maybe there is a connection. The problem with this argument is that there are mechanisms which can pull long period comets or other bodies into short period orbits thereby replenishing the stock of short period comets.
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