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While I read the SEP article and Eliezer's discussion, I don't understand much more than the basics of the theory. My biggest question is why Occam's razor cannot be used to eliminate the zombie theory?

The core of the zombie argument states that it can never be proved, even with perfect information. This is a perfect, stereotypical, textbook, etc. example of what Occam's razor is used against. From Wikipedia: "...eliminating those that make no difference in the observable predictions of the explanatory hypothesis or theory."

Occam's states that the central thesis of zombie theory should be eliminated, thus destroying the rest of it.

What is even more confusing to me is the fact that Occam's razor started as a key tenet of philosophy not science, yet it doesn't seem to apply here.