At first glance, the question in the title seems like a no-brainer. When we say an event is possible, we’re saying that it could happen. It is possible for me to meet an old friend tomorrow, because it could happen (note that it is possible from my point of view regardless of whether it will actually happen tomorrow). It is not possible that the Earth will fall into the Sun tomorrow, because that can’t happen according to the laws of physics.
So what are we talking about when we talk about the existence of something being possible or not? Is it possible for God to exist? Is it possible for there to be a teacup on the dark side of the Moon? Is it possible for ghosts to exist? In order to make such pronouncements, we must have evidence of what can exist. We know what can or cannot exist based on what we know of nature, history, the laws of nature, and so on. So for example, minds are predicated upon working brains, therefore it is not possible for ghosts to exist.
People who do believe in ghosts may very well reply that it is possible for ghosts to exist because some people have seen them. No one is denying that people see something that they identify as ghosts, but this does not prove that disembodied minds or spirits can exist. The identification is done with the concept of ghosts already in our minds, and all we’re doing is associating assorted phenomena with the concept of ghosts. We know that identification changes depending on prevalent beliefs: we used to believe that sleep paralysis was the result of demons, and now we associate them with aliens. So this is a cultural construct, not a fact.
What I am talking about is actual evidence. We have no actual evidence that ghosts could exist, because we have no evidence that there can be such a thing as a disembodied mind (let alone a disembodied mind that can still do bodily things like talk or make a room colder). Can you conceive of a ghost? Sure. In fact, it’s quite an ordinary thing for us to conceive of ghosts. But that doesn’t mean ghosts could actually exist.
And this is the big problem that I see with Christian apologetics: they can’t understand that being able to conceive of something does not necessarily mean it is possible for that thing to exist. Christian belief is all imagination-based: no one’s ever actually seen a god or a soul, and in order to make sense of those things we have to imagine them. This leads believers to blur the line between imagination and reality. If God is real (according to their worldview) and can only be apprehended through the imagination, then the imagination becomes, at some level, evidence of the reality of something.
We construct our beliefs, in a large part, from narratives. Older religions exploit this quite heavily, by presenting the believer with all sorts of stories about the creation of the universe, the early history of humankind, gods and demons traipsing around and manipulating humans, human heroes or demigods, and so on. These stories all have attributes of myths and fables, and lack any sort of realism, but most believers accept them as being at least partially real.
This is in contrast with other areas which are also populated with narratives. Take politics, for example. While the domain of politics is full of false beliefs and logical fallacies, there are still measurable aspects to the things that we have beliefs about, and therefore those beliefs can be verified (whether the believers care or not is another matter). By and large, people agree with what is being observed, they disagree on how to interpret it, and I would say that’s in a large part due to religion and prejudice, not for any rational reason. We can observe people and institutions, we can’t observe gods and demons.
We see this confusion in many areas of Christian apologetics. For example, they are always very insistent to point out that “Creationism is a theory, just like evolution.” The main issue with that sort of pronouncement is that they don’t understand the word “theory.” A theory is an explanation for the observed data. Evolution is a theory because it explains the data, and Creationism does not.
But even if we interpret the statement to mean that Creationism is a hypothesis (which seems to be what they really mean), well, that implies that Creationism is possible. Of course this is predicated on the belief that God is possible. But it’s also predicated on the belief that God creating all the life on Earth ex nihilo is possible, and we know of no mechanism that can explain such a process. The possibility of Creation has never been demonstrated, it has only been assumed, and we have no reason to assume that it might be true.
The most important thing we must have in order to ground a belief in reality is material evidence. Supernatural entities either do not interact with reality, in which case no evidence could ever be found and we must reject their possibility out of hand, or they do interact with reality, in which case we need to demand to know how those interactions are possible. Belief in the soul, for example, begs the question of how a soul, which is supernatural, could interact with a material body. If we cannot even begin to explain how such communication could take place, then we cannot deem souls possible, no matter how much we conceive of them.