Modal logic

Modal logic, or (less commonly) intensional logic is the branch of logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, and necessarily, and others. Any logical system making use of modal operators, such as possibly, or necessarily is thus also called a modal logic. Modal logics are characterized by semantic intensionality: non-modal logics all have the feature that the truth value of a complex sentence is determined by the truth values of its sub-sentences. They are thus extensional. In modal logics, by contrast, this does not hold: both "George W. Bush is President of the United States" and "2 + 2 = 4" are true, yet "Necessarily, George W. Bush is President of the United States" is false, while "Necessarily, 2 + 2 = 4" is true.

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Modal logic - Logic - George W. Bush

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A formal modal logic represents modalities using modal sentential operators. The basic set of modal operators are usually given to be Box and Diamond. In alethic modal logic (i.e. the logic of necessity and possibility) the Box represents necessity and the Diamond possibility. A sentence is said to be

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• possible if it might be true (whether it is actually true or actually false);
• necessary if it could not possibly be false;
• contingent if it is not necessarily true, i.e., is possibly true, and possibly false. A contingent truth is one which is actually true, but which could have been otherwise.
 
Modal logic: Metaphysical and other modalities ►