Is the world arranged into the two separate but intersecting spheres of the spiritual and the natural, or just the natural?
The best argument, by my lights, against the existence of a supernatural sphere takes the form of what’s called an abductive argument: an argument to the best explanation. Take the issue of human psychology. Is the best explanation of moral decision-making purely natural, or is there a better supernatural explanation on hand?
Somewhat surprisingly, there’s a lot of agreement among philosophers (though of course, not complete agreement) about what constitutes a good explanation: falsifiability, consistency with data, simplicity, consistency with background knowledge, and so on.
So which kind of explanation–supernatural or natural–about human moral psychology does better, according to the standards just outlined? Well,
I cannot falsify the proposition that there’s a soul making moral decisions, for example. (How could I?) On the other hand, I can falsify the proposition that the prefrontal lobes are involved in moral decision making. One point for natural explanations, none for supernatural.
Then there is the issue of simplicity. The least “ontologically bloated” explanations are usually right. Again, one for natural explanations, none for supernatural.
One common objection is that it’s certainly possible this or that supernatural entity exists. But this misses its mark. While it may be possible that souls interfere with our brains, that does nothing to establish how probable the notion is. The relevant question centers around what good reason we have to think something is true; not merely what is possible. After all, it’s possible our behavior is constantly being manipulated by invisible pink unicorns. But of course, we have no good reason to think that’s true.
One final word in favor of natural over supernatural explanations. The history of science is the history of natural constantly overrunning supernatural explanations. On this line of thought, the naturalist’s objection to supernatural explanations might be pithily phrased “Why bet on the horse that’s lost every race it’s ran?”