Necessity is a non-truth-functional operator in modal logic. It is used to indicate that something is necessarily the case.
The symbol □ is used to indicate necessity in modal logic. For example, to say that a proposition, P, is necessary, we formally indicate:
□P
To indicate that something is not necessarily the case, we indicate:
¬□P
We may also indicate that something is necessarily not the case:
□¬P
Necessity is compared to the modal state of possibility. The neccessity of P (□P) can be defined from possiblity as ¬◊¬P. By this definition, something is neccessary when it is not possible for it to not be the case.
In some semantics of modal logic, the way to define the modal state of neccessity is to say that something is necessarily true if it is true in all possible worlds. 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 necessarily true (□φ) when φn is true for every n.
In other words, □φ is necessarily true when we cannot logically conceive of a world in which φ is false.