In the philosophy of mind there are a number of powerful arguments that demonstrate consciousness cannot in principle be explained on a physicalistic ontology. In other words, presupposing that mindless particles organised in various ways by mindless forces is all that exists leaves us without the explanatory resources to account for our mental life. Most of these arguments examine some basic property of consciousness (qualia, intentionality, etc.) and give an a priori proof that each is insusceptible of psychophysical reduction. And if the mind cannot possibly be reduced to the brain then mind and brain are not identical. Some form of substance dualism is implicated. 
The Conceivability Argument for Substance Dualism is different: It demonstrates through natural reason that the mind and brain are nonidentical without reference to any particular property of consciousness. In what follows I will be summarising the discussion provided by Edward Feser in Philosophy of Mind: A Beginner’s Guide. The argument begins with a few preliminary precepts.
Physical Impossibility vs. Metaphysical Impossibility
Feser first introduces a distinction between two kinds of impossibility: physical impossibility and metaphysical impossibility. It is helpful here to think of this as a distinction between strong and weak forms of impossibility. A state of affairs is merely physically impossible if, though impossible in the actual world, we can give a description of it obtaining in some possible world without contradiction.  In this connection consider the proposition,
A man survived a headlong fall from the top of the Empire State Building.
This proposition is merely physically impossible because we can describe a possible world (say, one with very weak gravity) in which such a thing is possible. By contrast, a state of affairs is metaphysically impossible if it is impossible in the actual world and we cannot give a description of it obtaining in any possible world without contradiction. In this connection consider the proposition,
A married bachelor drafted a square circle.
This proposition is metaphysically impossible because we cannot coherently describe any possible world in which such a thing obtains.
From this distinction we can derive a terse precept,
Conceivability entails metaphysical possibility.
A Related Principle of Identity
Let us now use this distinction to articulate a principle of identity: A is identical to B if and only if it is metaphysically impossible for A to exist apart from B; that is, only if we cannot conceive of any possible world in which A exists apart from B. Consider the claim that water is identical to H2O. If you can conceive of a possible world in which you have water without H2O, or H2O without water, then, sensu stricto, water and H2O are not identical but different substances.
Applying this Principle of Identity to the Mind and Body
Let us finally apply this principle of identity to the mind and body. If one can conceive of a possible world in which you have a mind without a body then mind and body are not identical. And indeed one can conceive of such a possible world. W. D. Hart, for instance, invites us to imagine a man who wakes up one day and shuffles sleepily into the bathroom to wash his face. Looking in the mirror, he sees two empty sockets where his eyes should be. With a hacksaw, he then removes the top of his head and discovers that he has no brain. In a panic he removes his head, his neck, his torso. At last his body is completely disposed of and he sees nothing in the mirror but the wall behind him. Of course, all of this is physically impossible but it also conceivable and therefore metaphysically possible.
W. D. Hart’s example is appealingly ghoulish but there are many other ways to conceive of mind and body existing apart from one another. Solipsism is another example. Out of body experiences a third. All of them are eminently conceivable. And from each of them it follows, ex hypothesi, that mind and body are not identical.
An Objection from Opponents of Substance Dualism
Against this, some opponents of substance dualism have argued that it is possible to conceive of two identical substances existing separately. For example: Water is identical to H2O. But now let there be a substance having the properties of liquidity, quenching thirst, freezing at low temperatures, etc. whose chemical composition is XYZ. If this is conceivable, then it is metaphysically possible; and if it is metaphysically possible, then A and B can be identical and conceived to exist separately and so the operating precept is violated.
Kripke’s Objection to the Objection
However, Kripke, the American logician, fussily dispatches this objection. Let water be that substance which in every possible world has those properties which water has in the actual world; i.e., liquidity, quenching thirst, freezing at low temperatures, etc. Let H2O, meanwhile, be that substance which in every possible world has that chemical composition which H2O has in the actual world. Trivially, the substance in the actual world having the properties of water is the same substance in the actual world having the chemical composition H2O. But since “water” in any given possible world is the same substance having the properties of water in the actual world, and the substance having the properties of water in the actual world is H2O, so the substance having those properties in every possible world is H2O. And so water and H20 are identical in every possible world.
In other words, to conceive of a substance similar to water that is not H2O is not to conceive of water existing apart from H2O but simply to conceive of a substance similar to water that is not water. The case of water and H2O does not therefore offer a counterexample to the test for metaphysical identicality. And so, we may conclude, the Conceivability Argument for Substance Dualism obtains.
A Final Point
As a final point it is worth noting that nonconceivability does not necessarily entail metaphysical impossibility. In other words, it does not follow from the fact that we cannot conceive of A existing apart from B that A and B are metaphysically identical. Maybe we just aren’t creative enough or intelligent enough to conceive of how it is possible. But conceivability of separateness does entail metaphysical nonidenticality—which simple precept does all the work of the argument. And unless the physicalist can demonstrate that that precept is wrong, substance dualism intrudes upon and falsifies his physicalistic ontology and the shadow of theism begins to darken his door.
 See my discussion of the Argument from Consciousness here.
 It is important to understand that in this discussion “a possible world” is not another planet or a parallel universe. In modal logic a possible world is just a comprehensive description of a possible reality where “possible reality” is analogous to “hypothetical state of affairs” with the added condition that its description entails no logical contradictions. And just as there are infinitely many sets in set theory, so there are infinitely many possible worlds in modal logic.