Things being possible — or impossible, or necessarily true — is a fundamental concept of thought that we use all the time, that we “understand” without an explanation, and that we take for granted. So naturally philosophers have spent ages thinking about it, presented strange-sounding theories about it, and made it seem like an unsolvable problem. (In spite of my ironic tone, I of course think that things taken for granted should be thought about.) But some such explanations are actually quite good and sensible, and I am here going to present one that resolves the whole question, provided an answer is not expected to have some strange, over-thought properties.
What does it mean to think about what it means for something to be possible? A state of affairs is possible if it could be true. But that’s just another way of putting the same thing. You could say that nothing prevents it that things could be so, but that’s not so simple either. If things are some particular way at the moment, they can’t be differently at the same time. They “could” be, but what does that mean?
It is clear, though, how possibility, impossibility and necessity relate to each other. That which is not possible is impossible. When something is necessarily true, that means it’s not possible that it could be otherwise. So as soon as we answer the question of what possibility is, we get these other two concepts on the side.
It’s customary in philosophy to say that what is possible is what is true in some possible world. This may sound strange, but it follows quite sensibly from what we usually think when we say that something is possible. When we say that something is or would be possible, we are thinking of the simple (unless it’s complicated) state of its being true. But we’re not denying that the rest of the world would still exist around the imagined situation. If we were to say that I could have black hair (which I don’t have), we are not usually thinking of a situation in which I have black hair and nothing else exists in the universe. (In that case I would say “I could have black hair and be the only thing in the universe.”) We assume that the world would be otherwise the same except that I would have black hair — and a few other minor details would have changed in order to enable my having black hair, such as that I would have dyed it, or I and my relatives having slightly different genes. So a possible world is a whole assumed universe of which it is stated how it differs from the known universe — called the actual world.
The unstated properties of the possible world can be relevant — if for example we were to say that it would be possible for there to exist a golden mountain in Finland, but then a physicist would point out that Finland is on Earth and Earth’s gravity is too great for a mountain there to be made of such a soft and heavy substance as gold without collapsing under its own weight. (I don’t know whether this is really the case. I just made it up.) In this case, features of the possible world that no-one had thought of would turn the possible into impossible. Of course, it would be possible then to readjust what was being described as possible — for example by saying that the mountain need not be made of pure gold, that it could be alloyed with a suitable touch of another metal to make it sturdier.
Because possibility is explained via possible worlds, next the question arises of what possible worlds are. This may be a misleading way of posing the question, though. It has led among others to David Lewis‘s famously absurd theory that all possible worlds physically exist; we just have access to only one, which we just call the world (or the actual world). At this point, I give official permission for even those without philosophical training to think the thought makes no sense. Its motivated by trying to explain possibility through something else. In that light, many counterarguments against it are actually rather poor, such as that “that isn’t what we mean by possibility,” because if thing A is explained through thing B, of course B is no longer the same as A, or nothing would have been explained. But Lewis (of course) cannot give any evidence of the sort that would be required to prove the existence of other physical universes. And since we can gain no information from these other worlds, they in no case explain what we think we know about possibilities. This being so, we need another explanation for how we know anything about the contents of other possible worlds — but, then, why do we need any other kind of possible worlds than those in our minds that we use to reach conclusions about possibility? Possible worlds have also have been explained as being linguistic constructions, and my own answer will be of this sort. But first more of something else.
Since people have started thinking about possible worlds as something that we as it were observe in order to know what is possible, questions have also been raised about how we can know which object in a possible world corresponds to which object in the actual world. (Of course, not all possible worlds have counterparts for all objects of other worlds.) If there is some person partly resembling you in a possible world, how can we tell whether by following this person we know what’s possible for you? What if there are two who both resemble you to an equal degree? How can we measure degrees of resemblance anyway? Lewis said that each person’s counterpart in a possible world is a person suitably resembling them, but that just raises the above questions. (It could still have been the right answer even if we did not believe possible worlds to be of the sort Lewis claimed otherwise.) Such thoughts have a seed of truth, but there’s also much sense in Saul Kripke‘s view that it makes no sense to act as if possible worlds were something that we are observing from the outside, and then we have to think about who there corresponds to whom. When we say that something would be possible to me, then we are talking about me in another possible world, not some counterpart; it is already assumed to be me, and that doesn’t need to be established separately. Kripke of course also thinks of possible worlds as constructions.
Naturally, there are also questions about what actually is possible and what isn’t, and we can speak of different kinds of possibility. No-one much thinks it’s possible for there to be logically contradictory things, such as a round square. The idea that the laws of nature would be different can be spoken of as impossible in one sense and possible in another. And apparently Kripke also does not take his above position all the way, and instead says that an object has some defining traits without which it could not exist, because then it would no longer be the same object. These might include a living creature’s DNA, or for a human person that they must always be a human rather than some other species.
In trying to explain possibility, there is yet another alternative: that it is primitive, not based on anything else, something we just have to take as given and on which other things can be based. At some point we always must have something that is not explained further through something else, lest we face an endless chain of explanations. And it is true that we usually use the concept of possibility without analysing it any further. But this is not a good answer at this point.
So much for the exposition. Now let me answer all the questions myself.
The answer is: Something’s being possible means that it doesn’t contradict any of our chosen assumptions.
What do we need to explain when we explain the “phenomenon” of possibility? That people speak of possibilities and use them as an aid for thinking. Here we can immediately distinguish between two uses of possibility.
In the first instance there is saying that something is possible. In this case, we’re not saying that it isn’t so but could be, but that it may or may not be so, or it may or may not happen in the future. This is called epistemic (meaning related to knowledge) possibility, and it does not appear to be what usually raises the questions. In any case, it’s easy to understand and explain. We know some things about the world, and based on those, we do not know whether something else holds or not. So it is possible in this sense. It’s easy to see why people need such a category of thought, since it doesn’t make sense to treat such things as either true or untrue yet.
On the other hand, there is the case in which we say something would be possible. In this case, insofar as it doesn’t just mean the same as above (they get a bit mixed up in the case of the future, and in general, the choice of words isn’t always a reliable indication as to which one is meant), we mean that things are not this way, but they… well, they what? They could be, it would be possible, but what does that mean? This class also encompasses cases in which we say that if things were so and so, something else would follow from that.
In speaking about the possibility of something being differently from how it is, we always assume some things to be different from how they are, but the world to be otherwise the same as now, except where it would contradict the things assumed — as I already stated above when explaining the concept of possible worlds. My claim is that there is no more to it, and no more is needed. We start with certain assumptions and state that something is possible in relation to them, meaning not in contradiction with itself or them. Round squares are impossible with any assumptions whatsoever if we really mean by “squares” and “roundness” what we usually do (involving an assumption of Euclidian geometry); the only assumption that could make such a round square “possible” would be that logic does not apply, but in that case everything else would also be possible and impossible and true and untrue at the same time, and nothing could be said of anything.
There are no other preset rules for what can be called possible or impossible. Anything else than the actual world is impossible if we take the whole actual world as a starting point. In this case it of course makes no more sense to even speak of possibility, so this is not usually done. But it makes sense in a way to say that it’s impossible for anything at all to be the least bit differently, such as my having one more hair on my head than now, because things are the way they are and they can’t simultaneously be some other way. If we were to say that I could have different DNA, that refers to a situation in which the being thought of as me would be similar in some other respect, such as looking exactly the same and having the same personality, but would have been born from different DNA. It’s another question how this would be possible, and for that one would have to look at other unsaid assumptions. If, in turn, we were to say that I could not have a different DNA, that might mean for example that the above assumed situation would be extremely unlikely due to the way genes function or to how complicated the world is — or just that I was being defined in the first place based on having a certain DNA code, in which case the contradiction would be akin to the square circle.
Perhaps I could be a dog, but then we would have to understand what that would mean — in what sense would this dog be “me” any more? Of course, we could develop Kripke’s first idea above further and say that we could just decide to speak of me as a dog. But then one would transmit no meaning in saying that I could be a dog, because by that system we could always say so, it would always be possible. If we really want to say something about something, we have to set the conditions that we start from. In this case they would be conditions for in what way the dog resembles me. (So counterpart theories make some sense, and in addition their problems disappear when we set the conditions separately in each instance, which in turn corresponds partly to Kripke’s idea.) We don’t usually talk this way, at least very seriously, because in imagining such great changes as a human being as a dog, not many identifying properties can be preserved, and the meaningfulness and content of the claims get thin. But you can make such counterfactual claims in normal speech. I could for example say, “If I were a dog, I wouldn’t behave that way.” This would almost certainly be at least partly joking speech, but it would work by the same logic — I could mean, for example, that the dog that I would be would have my temperament, or, at the more joking end, my good manners, and that would contradict the way in which I was saying it wouldn’t act.
Thus, the conditions delimiting possibility can be chosen arbitrarily. They need to be chosen for the current purpose in each situation. In the case of epistemic possibility, the choice is easy: choose as conditions everything we know about the actual world and call possible that which does not contradict it. We do the same thing with other kinds of possibility, but in that case the model of the world that we build contains deviations from the actual world, and vague assumptions that it is otherwise suitably similar to the current world, which may of course involve great changes as well. In this case, the choice of assumptions depends on the purpose of thought experiment.
Different kinds of possibilities look more or less possible or impossible than each other for different reasons. If something requires breaking the laws of nature, for example, it’s pretty clear in practice that it’s not going to happen (unless you happen to believe in miracles of a suitable sort), so it can be thought of as simply impossible. Other kinds of highly improbable events can be considered as impossible as well, even if it’s not impossible to imagine some unlikely way in which they could happen.
But there is nothing else impossible in principle than logical impossibility. Only the impossible by definition is impossible from all points of view. Even if some kind of possibility seemed much more likely or believable than some other, it should be kept in mind that things are only one way. (Maybe not counting something in quantum mechanics, but I’ll say no more of that here.) No other possibility exists in the same sense. They all exist only as thought and logical possibilities. That we can think of such logical possibilities certainly does not prove any independent existence (except in the completely different sense in which concepts “exist”). Each “me” conceived of with different properties from now, be it a dog or have it identical DNA to mine, is still just a thought-up object whose imagined properties coincide with mine only partially. Something can be defined as the defining property of an object, but that’s just part of setting the conditions too, not something that is already there.
From this it also follows that possibility as a general category is in a way uninteresting, and there’s more of interest in its delimited special cases, such as logical possibility (the basis of all the rest), possibility under the current laws of nature, possibility under some emergent “rules” such as natural selection, epistemic possibility, and so on, which all must be treated as different. There is no metaphysical truth about what is just “possible” and what isn’t, no possible worlds of some specific sort floating around somewhere.