A modal logic is any system of formal logic that attempts to deal with modalities. Modals qualify the truth of a judgment. For example, if it is true that "John is happy," we might qualify this statement by saying that "John is very happy," in which case the term "very" would be a modality. Traditionally, there are three "modes" or "moods" or "modalities" represented by modal logic, namely, possibility, probability, and necessity.
A formal modal logic represents modalities using modal operators. For example, "It might rain today" and "It is possible that rain will fall today" both contain the notion of possibility. In a modal logic this is represented as an operator, Possibly, attached to the sentence It will rain today.
The basic unary (1-place) modal operators are usually written for Necessarily and for Possibly. In a classical modal logic, each can be expressed by the other and negation:
Thus it is possible that it will rain today if and only if it is not necessary that it will not rain today. For the standard formal semantics of the basic modal language, see Kripke semantics.
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Pace Plantinga's sleight of hand with modal logic (the premises of which I tend to think are meaningless in a way that eviscerates the coherence of the ...
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