The mind has private experiences. You can watch someone wince in pain, and even see the brain activity on an MRI, but you can never experience their actual experience. This is qualia; the privacy of personal experience. Material things are public and can be observed. The mind is private and can only be observed from a first person perspective.
The mind cannot be divided into parts, like material things can. Multiple Personality Disorder is a case of another mind in the same body, not the mind splitting up into two. MPD might not even really exist, as many cases have been exaggerations or hoaxes. Split brain cases may be a case of mind splitting, but evidence shows that it might not be so much a case of one mind splitting into two as much as just an “absence of mind” type of phenomenon, that anybody experiences from time to time.
If two things are identical, then it is logically impossible to have the one without the other; it isn’t even coherent to imagine the one without the other. You can’t imagine water without H20; they are identical. But it seems that we can coherently imagine mind without brain: imagine yourself disembodied, for example. It is at least logically coherent, even if not physically so. This indicates that they are not identical.
This argument requires “rigid designators”, or words that refer to the same thing no matter what. Neil Armstrong is identical with the first person to walk on the moon, but there is a possible world where someone else walked on the moon first. Hence, the two are identical in our world, but “first person to walk on the moon” is not a rigid designator; it could have been anybody. H20 and water, on the other hand, are in fact rigid designators. So are “my mind” and “my brain.” Yet, it’s logically possible to have “my mind” without “my brain.”
Another possible problem with this argument is that perhaps we can’t really imagine our minds disembodied at all, like we think we can. When light bounces off an object, the photons go into your eyes and cause the sensation of sight; but if you are disembodied, what happens? The photons bounce off the object and go to nowhere. So perhaps you can’t really imagine yourself disembodied like you thought you could. On the other hand, take away the sight and you still have a thinking person, and hence the argument may still work.
The main argument against dualism is: how can something non-physical interact with something physical? No mechanism has been forthcoming. Also, the interaction problem implies that energy would have to come from somewhere else (because the mind is non-physical, but interacts with the physical), and this is against the laws of thermodynamics. This does not necessarily refute dualism (if the arguments work, then so much the worse for physics), but it is a huge problem. There are a few answers to this problem.
God sees that the phone rings, and simultaneously causes you to have the sensation of the phone ringing.
God set everything in motion at the beginning of time, like synced clocks. You receive the sensation of the phone ringing at the same time that the phone rings because the two have been timed to happen simultaneously from the beginning.
The mind is a byproduct of physical processes, like smoke is the byproduct of fire. There is only a one-way causal connection. Your physical body does what it does, and your mind is just along for the ride. The problem is you could never affirm your mental state, such as your belief that epiphenominalism is true, because your mind has no causal power over your lips.