# Necessity

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 neccessity 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 possibility, ◊, which states that something may or may not be the case. Possibility of P (◊P) can be defined from necessity as ¬□¬P.

## Possible Worlds

One way to define the modal state of neccessity is to say that something is neccessarily 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 neccessarily true (□φ) when φn is true for every n.

In other words, □φ is neccessarily true when we cannot logically conceive of a world in which φ is false.