Systems of modal logic are systems of logic concerned with modalities, primarily “neccessity” and “possibility”.

The term “modal logic” is often used more ambiguously and generally to refer to any system of logic that deals with operators which are not truth-functional. Systems concerned with temporal order or moral reasoning, for example, are often labeled as modal logics. However, for the sake of keeping things neatly categorized, this site refers to the broader category as contextual logics, and considers modal logic to be a type of contextual logic.

Modals

The term “modal” refers to modes of truth — different ways in which things are true. Propositions in a modal logic system are thought to have some kind of modal state for their truth. So, any proposition, P, may be considered true in one of several ways. P may be actually true, but it may also be necessarily true, meaning that it's impossible for P to be false. P may also be possibly true.

In most cases, modal logics consider actuality to be a modal state as well. In this case, the proposition is simply stated as P without any additional operator. Note that any propositional variable in a modal logic formula has no truth value until it is forced into some interpretation, so a statement "P". In other words, without a modal operator, the term is considered to be contingent. If some interpretation makes P true, then we say that P is actually true — it is true in the modal state of actuality. (Classical logic systems consider only one modal state, so all things that are true are actually true.)

Modal Operators

The two main operators that appear in modal logic systems are necessarily (□) and possibly (◊).

Some contextual logical systems, such as deontic logic, which deals with moral concepts such as permissibility and obligation, occasionally use the same symbols to represent different concepts, though keeping the same operational rules. Other times, those systems will employ different symbols or letters to represent operators.

Possible World Semantics

One method of providing a semantics to a modal logic is through the means of possible worlds, introduced in 1959. This method is also sometimes known as Kripke semantics after its creator Saul Kripke. This semantic approach envisions a model of possible worlds, or alternative states of affairs as Hintikka refers to them, in which atomic formulae may be true or false. Necessity is then defined semantically as ‘true in all possible worlds‘, while possibility is defined as ‘true in at least one accessible possible world’.

For a more detailed explanation, see possible world semantics.