Michael Egnor discusses six aspects of our mental experience that make a dualist explanation of that experience more plausible than a materialist explanation. The six he describes are: Intentionality, Qualia, Persistence of Self-Identity, Restricted Access, Incorrigibility, and Free Will.
Egnor gives a good summary of each of these and concludes that materialism, the view that the material brain is the sole locus and cause of our mental life, can't explain any of them. It's worth the read especially for philosophy students. For example, here's what he says about free will:
If the mind is entirely caused by matter, it is difficult to understand how free will can exist. Matter is governed by fixed laws, and if our thoughts are entirely the product of brain chemistry, then our thoughts are determined by brain chemistry. But chemistry doesn't have "truth" or "falsehood," or any other values for that matter. It just is. Enzymatic catalysis isn't true or false, it just is. In fact, the view that "materialism is true" is meaningless...if materialism is true. If materialism is true, then the thought "materialism is true" is just a chemical reaction, neither true nor false. While there are some philosophers who assert that free will can exist in a deterministic materialistic world (they're called "compatibilists"), and some have argued that quantum indeterminacy may leave room for free will, the most parsimonious explanation for free will is that there is an immaterial component of the mind that is undetermined by matter.
Egnor is correct, of course. If a belief, say, is just a particular chemical reaction that occurs in my brain what sense does it make to talk of beliefs being true? Chemical reactions aren't the sorts of things that are true or false. Furthermore, how do mere chemical reactions produce something like an understanding or a doubt? To say that matter is the ultimate source of all our mental experience really does seem to fly in the face of our deepest intuitions about that experience.
Check out what Egnor says about the other five elements of our mental lives for which materialism seems ill-equipped to offer a plausible explanation.RLC