Possiblity is a non-truth-functional operator in modal logic. It is used to indicate that something is possibly, but not necessarily, the case.
The symbol ◊ is used to indicate possibility in modal logic. For example, to say that a proposition, P, is possible, we formally indicate:
To indicate that P is not possibly the case, we indicate:
This is equivalent to saying that P is necessarily not the case, using the necessity operator, □:
The modal state of possibility is contrasted with the state of necessity. Possibility of P (◊P) can be defined from necessity as ¬□¬P. That is, something is possible when it is not neccessarily not the case.
In some semantics of modal logic, we can say that something is possibly true if it is true in at least one possible world. So if we have some formula, φn, where n is a rational number indicating some possible world (so φ0 is φ in World 0, φ1 is φ in World 1, and so on…), then φ is possible (◊φ) when φn is true for at least one n.
In other words, □φ is possibly true when we can imagine a world in which φ is true.