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A world with identical micro-physical structure, but with no qualia may be conceived of.
I can conceive of another world in which there is a person with identical micro-physical states, but lacks all qualia whatsoever, has no subjective experience, possesses no phenomenal states of consciousness. This is called the "philosophical zombie" thought experiment of Chalmers.
Conceivability entails possibility.
If it is possible that my mental properties are independent of physical properties, then it follows that mental properties and physical properties are independent.
Conceivability does not entail metaphysical possibility. I can imagine that I can break the laws of nature, but this does not make it possible evidently, therefore the conclusion does not follow.
I can know only of nomological possibilities, not metaphysical possibilities that cannot be supported by any evidence. Thus, the conclusion does not follow.
if my mind has non-physical properties, how can I know anything about them in the physical world?
Many mental properties are embodied in the physical world, such as neurotransmitters.