While "I think, therefore I am" is Descartes' most famous one-liner, substance dualism is his most famous philosophical position. According to substance dualism, our minds and our bodies are two distinct substances capable of existing apart.
The argument for substance dualism relies heavily on all the work that Descartes has done up until this point. The argument rests on the claim that the essence of mind is thought while the essence of body is extension, and this duality of essences implies a duality of corresponding substances. The full argument can be broken down into seven steps: (1) If I can clearly and distinctly perceive something, then God makes something that exists that corresponding to my clear and distinct perception, otherwise God would be a deceiver. (2) If I can clearly and distinctly perceive X and Y as complete things whose principal attributes exclude each other, then God can make X and Y exist apart from each other. (3) If X and Y can exist apart from each other (whether or not they really do) then they are really distinct. (4) I can clearly and distinctly perceive mind as a complete thing to which extension does not pertain. (5) I can clearly and distinctly perceive body as a complete thing to which thought does not pertain. (6) Therefore, God could bring it about that mind and body exist apart from each other. (7) Therefore, mind and body are really distinct.
What is Descartes' motivation in arguing for substance dualism? One practical upshot of this claim is the possibility of life after death. If mind and body can exist apart, then our soul can survive the death of our body. More important than that, though, (for Descartes, at least) is that substance dualism ensures that physics boils down to geometry. Physics is the study of physical substances. If things like sensations and thoughts belonged to physical substance then physics would have to account for them. This would be a problem for Descartes, because sensations and thoughts cannot possibly be expressed mathematically, or mechanistically for that matter. They would destroy the entire neat, mathematical, mechanistic expression of the physical world. Descartes, therefore, takes these items out of the realm of the physical (and, thus, of physics) by claiming that in addition to physical substance there is an entirely different substance in the world: mental substance. Substance dualism, then, is a necessary precursor to the rest of the text.
Since this claim is so significant for Descartes' project, it is important to get clear on every step of the argument. There is a lot going on in every premise, and so a lot of unpacking must be done in order to get a perfect grasp of how the whole thing is supposed to work.
Step (1) seems clear enough. This is just a corollary of Descartes' guarantee of clear and distinct perception. Step (2), though, confronts us with a question: why is it important that the principal attributes exclude each other? Why is step (2) not just, "If I can clearly conceive of X and Y as complete things then God could bring it about that they exist apart"? To understand why this is necessary consider the following example of bad reasoning: I know that Lilly is a woman. I, therefore, know that Lilly has no talent for football. This reasoning fails because there is nothing in the idea of being a girl that excludes the possibility of being good at football. One can be both a girl and talented at football. Imagine this other example: I can conceive of a square while only thinking of four equal sides. Therefore, I can conceive of a square without conceiving of shape. It is true that I can think of a square by thinking only of four equal sides. However, having four equal sides is just a way of being shaped. When you think about something having four equal sides, you are necessarily also thinking about shape.
Now consider what would happen if Descartes' second step lacked the proviso about exclusion. The argument would go like this: I know that thought is the essence of mind. I know that extension is the essence of body. I, therefore, know that mind and body have different essences. Can he draw this conclusion from the premises? In order to draw this conclusion, he has to show that extension and thought are not like femininity and football talent, or squareness and shape, but more like squareness and circularity. In other words, he must show that they exclude each other. What if thought was just another way of being extended (that is, a mode of extension), or extension was a kind of thought? Then the case would be exactly analogous to the square/shape example. By clearly and distinctly conceiving of mind as a thinking thing, you would also secretly be conceiving of body and vice versa. In other words, you would not be able to clearly and distinctly conceive of them apart at all. And if you cannot clearly and distinctly conceive of them apart, then the whole game is lost.
The next obvious question that arises is how Descartes actually proves that extension and thought exclude each other. Nowhere do we see him explicitly proving this. In fact, steps (4) and (5) are terribly compressed. Both contain two claims: P is the essence of S; P is not connected to Q.
First, how does Descartes know that thought is the essence of mind and extension is the essence of body? For this part of the claim he is resting on the suppressed premise we saw in the last section: P is the essence of S if and only if I can conceive of S attributing only P to it. He sees that he can conceive of mind by attributing only thought to it and of body by attributing only extension.
Now, though, the big question: How does he know that extension and thought are not like squareness and shape? How does he know that they exclude each other? Though he never explicitly states his reasoning, it is not that difficult to figure out. Extension and all of its modes can be captured by mathematical language. It seems intuitive to believe, however, that the modes of thought (experience or consciousness) cannot be expressed mathematically. After all, one would be hard pressed to imagine how to mathematically express the experience of seeing red. Since it is of the nature of extension to be expressible in mathematical terms, and of the nature of thought to lack this possibility, he is able to conclude that neither is a mode of the other. Extension and thought exclude one another.
Given all this, we can now see clearly how Descartes is able to conclude the argument. He knows that mind and body have different essences that are mutually exclusive. That means that when he clearly and distinctly conceives of one, he is not secretly also conceiving of the other. He also knows that God can bring about the separation of anything we can clearly and distinctly perceive. Since we can clearly and distinctly perceive of mind without body and of body without mind, God can bring it about that mind can exist without body and body without mind. In other words, they are really distinct.