Dualism, in its various forms, is the broad claim that there are two separate and irreducible types of things in the world; the physical and the mental. The intuitive appeal of this position is pretty apparent from its central role in many of the world's major religions. A common thread across the whole gamut of faiths from traditional Christianity to various forms of new-age transcendentalism is the claim that our bodies are in some sense mere finite temporal homes for our real selves; our 'souls' or 'spirits'. The classic philosophical statement of the dualist position can be found in Descartes. He observed that if we subject all of our beliefs to a program of rigorous skepticism we find that our perceptions of the physical world are always prone to the possibility of deception. The one thing that we cannot doubt is that we exist; that we are conscious thinking beings. As such, he argued, it follows logically that you, the real you, are something distinct from your body. Descartes identified this 'real self' with the Mind, describing it as a non-physical, non-spatial 'mental substance', characterised by its essential activity of thinking.
The greatest strength of such a theory is the strong resonance it possesses with our intuitive, common sense assumptions about ourselves and the world. When we consider our beliefs, our emotions, our thoughts and so on, we tend not to think of them as being the sorts of things that are in any sense 'physical'. If we take as an example my belief that, say, "Bobby Hazlehurst wanks dogs for coins", it seems nonsensical to ascribe physical properties such as length, breadth or colour to this belief, and seems just as absurd to ask where exactly the belief is located. Of course the simple fact that an argument has strong common-sense appeal is no guarantee of its validity. I am merely seeking to show something of the scope of the task facing the reductive physicalist in knocking such a theory down.
Fortunately for that reductive physicalist, there are numerous massive and intractable logical deficiencies in the Cartesian dualist theory of mind. Firstly and vitally, the notion of a 'mind' given to us by Descartes is woefully under-described. He sketches a very vague picture of the supposed 'mental substance' and, crucially, gives us no criteria of identity or difference for minds. There is no explanation of what constitutes one mind, what makes it distinct from other minds, or why we should justifiably suppose there to be one particular mind associated with one particular body, and so on. This might sound a little stupid, but think about it: how do you know you only have one mind? Is it the same one you've always had? And the same one you always will have? And how do you know your mind only has one body? When an entity is left so ill-defined as this, it loses all explanatory power and we might as well keep things tidy and be shot of it altogether.
Related to this problem of the vagueness of the posited minds is the mysterious question of mind-body interaction. Clearly minds and bodies do interact; mental event such as decisions lead to physical actions, and physical events such as bodily injury have mental effects such as the feeling of pain. But if minds and bodies are such fundamentally different substances as the dualist supposes, then how on earth is this constant two-way interaction possible? What could the causal mechanism between the physical and non-physical worlds be? This issue is left fatally unclear. And if it does somehow occur, it violates the principle of the causal closure of the physical universe, an assumption at the very heart of all scientific thought; of the way we know the world operates.
To conclude, then: a nice idea, but finding flaw with it is like shooting tuna in a tin.