Definition
Necessity is the status of what must be true — what could not have been otherwise. A necessary truth holds not merely in the actual world but in every possible world. That seven plus five equals twelve is necessary; that it is raining is, at best, merely actual.
In modal logic, necessity is written □P, read 'it is necessary that P'. It is the strong modal status: it rules out every alternative. Its weaker partner, possibility, only requires that some alternative succeed.
Why it matters
How it works
On the possible-worlds account, □P is true exactly when P is true in every world accessible from the world of evaluation. Necessity is universal quantification over worlds. This explains why necessary truths feel unshakeable: there is no way things could have gone that would make them fail.
Necessity also stands in tight logical relations with possibility. To say P is necessary is to say not-P is impossible; to say P is possible is to say not-P is not necessary. The two operators trade places under negation, so a logic of one is automatically a logic of the other. Which exact principles govern necessity — for instance, whether □P implies □□P — depends on the properties assumed for the accessibility relation.