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What would constitute evidence of the supernatural?
There is a lot of evidence for the supernatural, in my view, and some of it is intrasubjectively verifiable. For example, that the physical constants of the universe had to be finely tuned in order for a life-permitting universe to come into existence is evidence of a causal influence that extends beyond that of particles and laws of nature. The wide-spread belief in the supernatural itself constitutes some evidence for the supernatural. The evidence we have in the occurrence of miracles is evidence in the existence of the supernatural.
Other evidence in the existence of the supernatural is not intrasubjectively verifiable. For example, I often have strong experiences as of God loving me, or forgiving me, or otherwise being present. Those experiences themselves are evidence (available to me, but not to you) of the existence of the supernatural.
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In other words, if we had evidence of something supernatural, that something would not be considered supernatural.
That makes sense. That's just what you should think, given your commitment to naturalism.