The notion of logical possibility lies behind several different kinds of analysis, most notably supervenience and identity. For example, we might say that X is only identical to Y if X is Y in all possible worlds. But to use this in practice requires us to define what is and isn’t possible. Now when working with these ideas formally it isn’t a problem, we simply say that a collection of objects is possible if it satisfies certain constraints (such as: Pa & a = b -> Pb). However, when dealing with claims about real objects the formal method isn’t very useful. And some thus define it as what is conceivable, meaning that if a state of affairs is conceivable then it is possible. But this, I would maintain, is wrong.
There are three ways in which something conceivable may turn out to be in fact impossible. The first is cognitive error. Consider someone who knows that a bachelor is an unmarried man, and knows what it is to be married, but at the same time can conceive of a married bachelor (by some kind of cognitive defect). If we ask this individual if the person they are thinking of is unmarried they say yes, but they also agree that he is married. So this person is able to conceive of a married bachelor, but only because something has gone wrong, so that they cannot be made to see the inherent contradiction. But I will admit that perhaps this doesn’t count as “proper” conceivability.
The second way is through hidden contradictions. For example consider the following properties: A, B, C, D, E. Ax -> Bx, Ax -> ~Cx, Bx -> Dx, Cx -> Ex, ~Dx -> Cx, Ex -> ~Dx. Is it conceivable that Az & Ez? I think many of us would say that yes, it is. In fact right now I am thinking of just such an object. But such an object is impossible. Because if we follow the chain of inferences Az implies that Bz. And Bz implies that Dz. But Ez implies that ~Dz. And it is impossible for the same object to be both Dz and ~Dz. So, even though we thought we could conceive of it, it can’t be possible. This is because in thinking about it we don’t necessarily follow the chain of inferences to its conclusion and see the contradiction. And so it is quite possible that some of the things we can conceive of imply contradictions that we are unaware of, simply because our thinking about them doesn’t explore every detail. But perhaps this isn’t “proper” conceivability either, maybe what we meant by conceivability was a kind of ideal conceivability, in the sense that it is conceivable in any level of detail. This does seem like a bit of a questionable rejection, since many of the situations that are said to be conceivable, and thus possible by proponents of identifying conceivability with possibility, are not fully explored, and the potential for hidden contradictions remains. But let me assume that there is some valid reason to reject this complaint.
Even so there is a third way in which the conceivable can be impossible, namely when our conception of certain real kinds of things is too “loose”. For example, if we weren’t sure what heat was we might think it possible for heat to move through a vacuum (if we thought of heat as some kind of substance). But in fact it is impossible for heat, being molecular motion, to move through a region where there are no molecules. In general the problem can be framed as follows: assume there is some real thing X that we wish to think about it. To do so we have the concept X*. Assuming that we have only accurate information about X it certainly falls under the concept X*, but since we don’t have all the facts about X some things other than X fall under X* as well. For example our primitive concept of heat, heat*, may have been simply whatever made water boil and matches burn. Thus molecular motion fell under heat*, as it was one possible thing that could have those effects, but other possibilities, such as phlostigon fell under it as well. As we learn more about heat our concept of heat* becomes narrower, and certain things which once fell under the concept no longer do. And thus certain things which were once conceptual possibilities are now conceptual impossibilities. (Another way of understanding this point is to consider the example given in the previous paragraph, except that this time we assume that certain inferences, such as Bx -> Dx are unknown to us. Thus, even ideally, Az & Ez is conceptually possible, but when we learn that Bx -> Dx it becomes a conceptual impossibility.) Now I will grant that the possibility of whether the original concept, that of heat*, can move through a vacuum is unchanged by our new discoveries. And so in some sense a conceptual possibility or impossibility about heat* remains as it was. But we know now that heat* is not actually a description of real heat, and when we are talking about possibilities we want to know about the possibilities for real things, about the possibility of real heat moving through a vacuum, not about whether it was possible for our old concept, heat*, to.
But who would be foolish enough to rely on our conception of things to determine if something is possible or impossible? After all hasn’t the progress of science shown us that many of the things we may have once thought possible aren’t? (For example, that you can’t go faster than the speed of light.) Well some philosophers of mind, notably Chalmers, do. They argue that it is possible that facts about consciousness are independent of the physical facts, because they are conceptually independent, and thus that consciousness cannot be reduced to the physical. But this is obviously silly, or so I should hope, because for all we know further discoveries about consciousness may narrow our conception of it down, to the point where we may realize that it is impossible for consciousness to be independent of the physical facts as it was for heat to be independent of molecular motion. Of course we may not make these discoveries. But whether consciousness can or can’t be reduced to physical facts is something that must be discovered, not something we can know by thinking about the nature of our conception of consciousness.