Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. Modals—words that express modalities—qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is usually happy, in which case the term "usually" is functioning as a modal. The traditional alethic modalities, or modalities of truth, include possibility ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("It is impossible that p"). Other modalities that have been formalized in modal logic include temporal modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p"), deontic modalities (notably, "It is obligatory that p", and "It is permissible that p"), epistemic modalities, or modalities of knowledge ("It is known that p") and doxastic modalities, or modalities of belief ("It is believed that p").
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 with negation:
Thus it is possible that it will rain today if and only if it is not necessary that it will not rain today; and it is necessary that it will rain today if and only if it is not possible that it will not rain today. Alternative symbols used for the modal operators are "L" for Necessarily and "M" for Possibly.
Read more about Modal Logic: Development of Modal Logic, Alethic Logic, Epistemic Logic, Temporal Logic, Deontic Logic, Doxastic Logic, Other Modal Logics, The Ontology of Possibility, Applications, Further Applications, Controversies
Famous quotes containing the word logic:
“Our argument ... will result, not upon logic by itself—though without logic we should never have got to this point—but upon the fortunate contingent fact that people who would take this logically possible view, after they had really imagined themselves in the other man’s position, are extremely rare.”
—Richard M. Hare (b. 1919)