The members of this group share the view that there is a sharp separation between the mind and the body. Not all of them go as far as Descartes, who claims that the mind and body are two distinct kinds of substances that can exist entirely independently of one another. But they all believe in a basic split between an inner, subjective realm and an outer, objective one.
Now, we'll admit that not all of them fought in a duel. Okay, so Descartes was the only one who actually did. But they all at least liked to argue, and that's sort of like a duel, isn't it? Kinda, sorta?
Malebranche was one of the most famous early Cartesian dualists. He holds that mind and body are completely distinct, but he doesn't accept Descartes's way of explaining their interaction. (When it comes to mind-body interaction, you'll note that none of the dualists agree with Descartes... or with each other.) In fact, Malebranche says that the mind and the body do not have any causal impact on each other at all.
No causal impact? Then why does your finger move when you tell it to, you ask? Well, Malebranche hauls out the 17th century's favorite explanation for all mysteries: God does it. According to his theory of "occasionalism," when you decide to move your finger, it's God who actually causes that finger to move (source). So Malebranche's theory avoids Descartes's hypothesis that the pineal gland (hehe) mediates between the mind and body... but it sure does make God an awfully busy guy.
All organizations want to show their best face to the world. And who would be a more impressive person to welcome you to this organization than the Father of Western Philosophy himself? After all, Plato is typically known as the original mind/body dualist, especially for the position he presents in his dialogue the Phaedo.
Unfortunately, if you start looking at what Plato actually says, you start to realize that his dualism is fundamentally different from Descartes's. Plato's view in the Phaedo is not that the mind and body are totally distinct from each other, and we have to figure out how they interact; Plato says that, ordinarily, mind and body are all mixed together, and the goal of philosophy is to try and separate them. Almost the opposite of Descartes's view.
So don't ask too many questions, okay? Just say hi to Plato when you see him at the door, and then come right on in.
Chalmers is proof that Cartesian dualism is not dead in the 21st century. Of course, he wouldn't describe himself as a Cartesian (no one does these days), but in the most important ways, Chalmers is carrying on René's noble tradition.
Chalmers describes himself as a "property dualist." That means that he is not like Descartes in claiming that there are mental substances. Instead, he claims that the mind is only a property of certain physical substances and cannot exist apart from them. At the same time, though, he thinks that the mind cannot be reduced to, or explained in terms of, physical things (source).
Nagel is further proof that Cartesian dualism is still alive in the 21st century. Like Chalmers, Nagel doesn't buy Descartes's idea that the mind is a thing, but he certainly believes there is an inner world of consciousness that cannot be explained in the terms of the physical sciences (source).