Possibility is the condition or fact of being possible. Latin origins of the word hint at ability. Possibility also refers to something that “could happen”, that is not precluded by the facts, but usually not probable.
In philosophy and modal logic, epistemic possibility relates a statement under consideration to the current state of our knowledge about the actual world: a statement is said to be:
epistemically possible if it may be true, for all we know
epistemically necessary if it is certain (or must be the case), given what we know
epistemically impossible if it cannot be true, given what we know
Epistemic possibility is often contrasted with subjunctive possibility (or alethic possibility), and although epistemic and subjunctive possibilities are often expressed using the same modal terms (such as possibly, could be, must be) or similar modal terms that are sometimes confused (such as may be and might be), statements that are qualified in terms of epistemic possibility and statements that are qualified in terms of subjunctive possibility have importantly different meanings.
The contrast is best explained by example. Consider the two statements:
Hitler might have been victorious in World War II
Hitler may have been victorious in World War II
Although these two statements are often confused with one another, they mean two different things: the first says something true about the vagaries of war; the second says something that is certainly false. The difference comes from the fact that the first statement—a statement of subjunctive possibility—says something about how things might have been under counterfactual conditions, whereas the second—a statement of epistemic possibility—says something about the relation between a particular outcome (a victory by Hitler) and our knowledge about the actual world (since, as it happens, we know perfectly well that that particular outcome did not actually obtain, we know that what it says is false).
The parallel distinction arises between types of conditionals (if-then statements). Consider the difference between the epistemic connection expressed by an indicative conditional and the causal or metaphysical relation expressed by a subjunctive conditional:
If Oswald didn’t shoot Kennedy, someone else did
If Oswald hadn’t shot Kennedy, someone else would have
The first statement says something that is certainly true, and will be accepted as such by anyone who is convinced that somebody shot Kennedy. It’s clearly true because it expresses this epistemic relation between its antecedent (the “if”-clause) and its consequent (the “then”-clause): The antecedent, if we came to know it was true, would provide us with excellent evidence that the consequent is true. The second statement, on the other hand, expresses a causal or metaphysical relation: It says that the world was set up so that the consequent would have been made true if the antecedent were true. One will accept that second statement to the extent that one thinks the world was set up in that way. (Conspiracy theorists who think there was a back-up shooter, for instance, may accept the second statement.)
Because of these differences, epistemic possibility bears on the actual world in ways that subjunctive possibility does not. Suppose, for example, that I want to know whether or not to take an umbrella before I leave. If you tell me “It’s possible that it is raining outside”—in the sense of epistemic possibility—then that would weigh on whether or not I take the umbrella. But if you just tell me that “It’s possible for it to rain outside”—in the sense of metaphysical possibility—then I am no better off for this bit of modal enlightenment.