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.

Philosophy Index is a work in progress, a growing repository of knowledge. It outlines current philosophical problems and issues, as well as an overview of the history of philosophy. The goal of this site is to present a tool for those learning philosophy either casually or formally, making the concepts of philosophy accessible to anyone interested in researching them. WOLI offers immigration law course online - fully accredited. ACE credits online at EES.

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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.