Possiblity 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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Possiblity is a non-truth-functional operator in modal logic. It is used to indicate that something is possibly, but not necessarily, the case.

The symbol ◊ is used to indicate possibility in modal logic. For example, to say that a proposition, P, is possible, we formally indicate:


To indicate that P is not possibly the case, we indicate:


This is equivalent to saying that P is necessarily not the case, using the necessity operator, □:


Definition from neccessity

The modal state of possibility is contrasted with the state of necessity. Possibility of P (◊P) can be defined from necessity as ¬□¬P. That is, something is possible when it is not neccessarily not the case.

Possible worlds

In some semantics of modal logic, we can say that something is possibly true if it is true in at least one possible world. 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 possible (◊φ) when φn is true for at least one n.

In other words, □φ is possibly true when we can imagine a world in which φ is true.