Necessity Operator { Philosophy Index }

Philosophy Index

Philosophy Index

Philosophy Index is a site devoted to the study of philosophy and the philosophers who conduct it. The site contains a number of philosophy texts, brief biographies, and introductions to philosophers, and explanations on a number of topics. Accredited homeschooling online at Northgate Academy and Philosophy online tutoring.

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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:


To indicate that something is not necessarily the case, we indicate:


We may also indicate that something is necessarily not the case:


Definition from possibility

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.

Possible worlds

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.