Modal Logic

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 LogicDevelopment of Modal Logic, Alethic Logic, Epistemic Logic, Temporal Logic, Deontic Logic, Doxastic Logic, Other Modal Logics, The Ontology of Possibility, Applications, Further Applications, Controversies

Other articles related to "modal logic, modal, logic, modal logics":

Actualities - Post-Aristotelian Usage - Influence On Modal Logic
... by some to be an anticipation of modal logic and its treatment of potentiality and time ... Contemporary philosophy regards possibility, as studied by modal metaphysics, to be an aspect of modal logic ... Modal logic as a named subject owes much to the writings of the Scholastics, in particular William of Ockham and John Duns Scotus, who reasoned informally ...
Modal Logic - Controversies
... Modal logic has been rejected by many philosophers ... with Aristotle seem to have had priority of interest in modal logic over mathematicians.) Nicholas Rescher has argued that Bertrand Russell rejected Model Logic, and that this rejection led to the theory ... However, Jan Dejnozka has argued against this view, stating that a modal system which Dejnozka calls MDL is described in Russell's works, although Russell did believe the concept of ...
Logicus - Topics in Logic - Modal Logic
... sub-parts of a sentence may have their semantics modified by special verbs or modal particles ... of modality treated to include deontic logic and epistemic logic ... Arthur Prior applied the same formal language to treat temporal logic and paved the way for the marriage of the two subjects ...
Admissible Rule - Examples
... The same argument applies to any multi-valued logic L complete with respect to a logical matrix whose all elements have a name in the language of L.) The Kreisel–Putnam rule (aka Harrop's rule, or ... In fact, it is admissible in every superintuitionistic logic ... The rule is admissible in many modal logics, such as K, D, K4, S4, GL (see this table for names of modal logics) ...
S5 (modal Logic)
... In logic and philosophy, S5 is one of five systems of modal logic proposed by Clarence Irving Lewis and Cooper Harold Langford in their 1932 book Symbolic Logic ... It is a normal modal logic, and one of the oldest systems of modal logic of any kind ...

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