Possibility seems to be something that we often invoke, but which is rarely discussed by itself. And because possibility is used as a basis for certain distinctions confusion about possibility can lead to problems with any reasoning that depends on it. Here I will define three kinds of possibility, and hopefully these divisions will make some mistakes less likely (mistakes of appealing to one kind of possibility when really another was appropriate).
The first kind of possibility is meta-logical possibility. Specifically, it is the sense of possibility in which every internally consistent system of logic, and hence every set of rules for the preservation of truth, are considered as possibilities. This kind of possibility is probably the least useful, since it can’t even rule out what may seem like obvious contradictions as impossibilities, because there are some logical systems in which those apparent contradictions are in fact true. The only use I have ever found for this kind of possibility is when arguing that nothing can be known a priori, the argument goes that even contradictions can’t be ruled out a priori because it is impossible to know how truth works in the real world, as there are many possibilities for truth, in this sense.
The second kind of possibility is logical possibility, or possibility with respect to a logical system. Under this kind of possibility a statement is possible if it isn’t a contradiction within the logical system we are working with (usually classical logic). It is in this sense of possibility which contradictions are said to be impossible. Usually if possibility is defined it is defined in this way. But I think that this kind of possibility is the wrong kind to appeal to, at least in most cases. For example, even if we were in a perfectly Newtonian world, it would still be possible, in this sense, for my computer to float off my desk without any change in its current state, simply because to assert that it floats up without being changed in any way is not a logical contradiction. But there is a sense in which it isn’t possible, just like it isn’t possible to exceed the speed of light or to ride a unicorn.
This third kind of possibility is restricted possibility. Under this kind of possibility something is possible if it isn’t a contradiction and doesn’t violate the background conditions. What the background conditions are exactly can vary, but they can usually be deduced from the context. For example, if I say that it is impossible for heat to move through a vacuum, then my background conditions are the laws of physics. If I say that it is impossible for this post to complete itself then my background conditions are the laws of physics plus the construction and operation of my computer. In my opinion this restricted sense of possibility is the most useful, but it is the least often appealed to.
For example, consider a definition of supervenience in terms of possibility. Our definition goes as follows: a higher level fact, S, only supervenes on a collection of lower level facts, T, if and only if it is impossible for S to be absent when T is present. If we understand possibility here as the second kind of possibility then there are no supervenient facts. To see why consider heat. Let us define heat, our S, as the cause of boiling and burning. And let our T be the facts about molecular motion, which are the facts that heat really is supervenient on. But it is perfectly possible, in the sense of logical possibility, for molecular motion, of any sort, to fail to cause boiling or burning. And thus heat would not supervene on molecular motion, which means that we are working with a bad definition of supervenience. However, if we understand possibility here as the third kind, restricted possibility, then we can consider whether heat must be present whenever certain kinds of molecular are, if the laws of physics are held constant. And now it is true that whenever the right kinds of molecular motion are present around the right substances burning or boiling must follow, by the laws of physics, and thus that heat supervenes on the molecular motion, as it should.
Of course some won’t be happy with using the third kind of possibility, because by its very definition whether something is possible in this sense is not related to whether it is conceivable. However I doubt this shocks you, dear reader, since I have already detailed how conceivability and possibility come apart.