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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 ("Impossibly, p", "It is impossible that p").〔"Formal Logic", by A. N. Prior, Oxford Univ. Press, 1962, p. 185〕 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"),〔"Temporal Logic", by Rescher and Urquhart, Springer-Verlag, 1971, p. 52〕〔"Past, Present and Future", by A. N. Prior, Oxford Univ. Press, 1967〕 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")〔"Knowledge and Belief", by Jaakko Hinntikka, Cornell Univ. Press, 1962〕 and doxastic modalities, or modalities of belief ("It is believed that p").〔"Topics in Philosophical Logic", by N. Rescher, Humanities Press, 1968, p. 41〕 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.〔So in the standard work ''A New Introduction to Modal Logic'', by G. E. Hughes and M. J. Cresswell, Routledge, 1996, ''passim''.〕 ==Development of modal logic== In addition to his non-modal syllogistic, Aristotle also developed a modal syllogistic in Book I of his ''Prior Analytics'' (chs 8–22), which Theophrastus attempted to improve. There are also passages in Aristotle's work, such as the famous sea-battle argument in ''De Interpretatione'' §9, that are now seen as anticipations of the connection of modal logic with potentiality and time. In the Hellenistic period, the logicians Diodorus Cronus, Philo the Dialectician and the Stoic Chrysippus each developed a modal system that accounted for the interdefinability of possibility and necessity, accepted axiom T and combined elements of modal logic and temporal logic in attempts to solve the notorious Master Argument.〔Bobzien, S. (1993). "Chrysippus' Modal Logic and its Relation to Philo and Diodorus", in K. Doering & Th. Ebert (eds), ''Dialektiker und Stoiker'', Stuttgart 1993, pp. 63–84.〕 The earliest formal system of modal logic was developed by Avicenna, who ultimately developed a theory of "temporally modal" syllogistic.〔(History of logic: Arabic logic ), ''Encyclopædia Britannica''.〕 Modal logic as a self-aware subject owes much to the writings of the Scholastics, in particular William of Ockham and John Duns Scotus, who reasoned informally in a modal manner, mainly to analyze statements about essence and accident. C. I. Lewis founded modern modal logic in his 1910 Harvard thesis and in a series of scholarly articles beginning in 1912. This work culminated in his 1932 book ''Symbolic Logic'' (with C. H. Langford), which introduced the five systems ''S1'' through ''S5''. Ruth C. Barcan (later Ruth Barcan Marcus) developed the first axiomatic systems of quantified modal logic — first and second order extensions of Lewis's "S2", "S4", and "S5". The contemporary era in modal semantics began in 1959, when Saul Kripke (then only a 19-year-old Harvard University undergraduate) introduced the now-standard Kripke semantics for modal logics. These are commonly referred to as "possible worlds" semantics. Kripke and A. N. Prior had previously corresponded at some length. Kripke semantics is basically simple, but proofs are eased using semantic-tableaux or analytic tableaux, as explained by E. W. Beth. A. N. Prior created modern temporal logic, closely related to modal logic, in 1957 by adding modal operators () and () meaning "eventually" and "previously". Vaughan Pratt introduced dynamic logic in 1976. In 1977, Amir Pnueli proposed using temporal logic to formalise the behaviour of continually operating concurrent programs. Flavors of temporal logic include propositional dynamic logic (PDL), propositional linear temporal logic (PLTL), linear temporal logic (LTL), computational tree logic (CTL), Hennessy–Milner logic, and ''T''. The mathematical structure of modal logic, namely Boolean algebras augmented with unary operations (often called modal algebras), began to emerge with J. C. C. McKinsey's 1941 proof that ''S2'' and ''S4'' are decidable, and reached full flower in the work of Alfred Tarski and his student Bjarni Jónsson (Jónsson and Tarski 1951–52). This work revealed that ''S4'' and ''S5'' are models of interior algebra, a proper extension of Boolean algebra originally designed to capture the properties of the interior and closure operators of topology. Texts on modal logic typically do little more than mention its connections with the study of Boolean algebras and topology. For a thorough survey of the history of formal modal logic and of the associated mathematics, see Robert Goldblatt (2006).〔()〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「modal logic」の詳細全文を読む スポンサード リンク
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