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Modal logic
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==History== The basic ideas of modal logic date back to antiquity. [[Aristotle]] developed a modal syllogistic in Book I of his ''[[Prior Analytics]]'' (ch. 8–22), which [[Theophrastus]] attempted to improve.<ref>{{cite SEP |url-id=logic-ancient |title=Ancient Logic |last=Bobzien |first=Susanne}}</ref> There are also passages in Aristotle's work, such as the famous [[problem of future contingents|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''' (see [[#Axiomatic systems|below]]), and combined elements of modal logic and [[temporal logic]] in attempts to solve the notorious [[Diodorus Cronus#Master argument|Master Argument]].<ref>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.</ref> The earliest formal system of modal logic was developed by [[Avicenna]], who ultimately developed a theory of "[[Temporal logic|temporally]] modal" syllogistic.<ref name=Britannica>[https://www.britannica.com/ebc/article-65928 History of logic: Arabic logic], ''[[Encyclopædia Britannica]]''.</ref> 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 (philosophy)|accident]]. In the 19th century, [[Hugh MacColl]] made innovative contributions to modal logic, but did not find much acknowledgment.<ref>{{cite journal | author=Lukas M. Verburgt | title=The Venn-MacColl Dispute in ''Nature'' | journal=History and Philosophy of Logic | volume=41 | number=3 | pages=244–251 | year=2020 | doi=10.1080/01445340.2020.1758387 | s2cid=219928989 | doi-access=free }} Here: p.244.</ref> [[C. I. Lewis]] founded modern modal logic in a series of scholarly articles beginning in 1912 with "Implication and the Algebra of Logic".<ref>Lewis, C. I. (1912). "Implication and the Algebra of Logic." ''[[Mind (journal)|Mind]]'', '''21'''(84):522–531.</ref><ref>{{cite web|last1=Ballarin|first1=Roberta|title=Modern Origins of Modal Logic|url=https://plato.stanford.edu/entries/logic-modal-origins/|website=The Stanford Encyclopedia of Philosophy|access-date=30 August 2020}}</ref> Lewis was led to invent modal logic, and specifically [[strict implication]], on the grounds that classical logic grants [[paradoxes of material implication]] such as the principle that [[Vacuous truth|a falsehood implies any proposition]].<ref>Lewis, C. I. (1917). "The issues concerning material implication." ''Journal of Philosophy, Psychology, and Scientific Methods'', '''14''':350–356.</ref> This work culminated in his 1932 book ''Symbolic Logic'' (with [[Cooper Harold Langford|C. H. Langford]]),<ref>{{cite book | author=Clarence Irving Lewis and Cooper Harold Langford | title=Symbolic Logic | publisher=Dover Publications | edition=1st | year=1932 }}</ref> which introduced the five systems ''S1'' through ''S5''. After Lewis, modal logic received little attention for several decades. [[Nicholas Rescher]] has argued that this was because [[Bertrand Russell]] rejected it.<ref>{{cite book|last=Rescher|first=Nicholas|title=Bertrand Russell Memorial Volume|year=1979|publisher=George Allen and Unwin|location=London|pages=146|editor=George W. Roberts|chapter=Russell and Modal Logic}}</ref> 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 modality to "come from confusing propositions with [[propositional function]]s", as he wrote in ''The Analysis of Matter''.<ref>{{cite journal| last=Dejnozka |first=Jan|title=Ontological Foundations of Russell's Theory of Modality|journal=Erkenntnis| year=1990| volume=32|issue=3| pages=383–418 |url=http://www.members.tripod.com/~Jan_Dejnozka/onto_found_russell_modality.pdf|access-date=2012-10-22|doi=10.1007/bf00216469|s2cid=121002878}}; quote is cited from {{cite book|last=Russell|first=Bertrand|title=The Analysis of Matter|url=https://archive.org/details/in.ernet.dli.2015.221533|year=1927|pages=[https://archive.org/details/in.ernet.dli.2015.221533/page/n183 173]}}</ref> Ruth C. Barcan (later [[Ruth Barcan Marcus]]) developed the first axiomatic systems of quantified modal logic — first and second order extensions of Lewis' ''S2'', ''S4'', and ''S5''.<ref>{{cite journal | author=Ruth C. Barcan | title=A Functional Calculus of First Order Based on Strict Implication | journal=Journal of Symbolic Logic | volume=11 | number=1 | pages=1–16 | date=Mar 1946 | doi=10.2307/2269159| jstor=2269159 | s2cid=250349611 }}</ref><ref>{{cite journal | author=Ruth C. Barcan | title=The Deduction Theorem in a Functional Calculus of First Order Based on Strict Implication | journal=Journal of Symbolic Logic | volume=11 | number=4 | pages=115–118 | date=Dec 1946 | doi=10.2307/2268309| jstor=2268309 | s2cid=31880455 }}</ref><ref>{{cite journal | author=Ruth C. Barcan | title=The Identity of Individuals in a Strict Functional Calculus of Second Order | journal=Journal of Symbolic Logic | volume=12 | number=1 | pages=12–15 | date=Mar 1947 | doi=10.2307/2267171| jstor=2267171 | s2cid=43450340 }}</ref> [[Arthur Norman Prior]] warned her to prepare well in the debates concerning quantified modal logic with [[Willard Van Orman Quine]], because of bias against modal logic.<ref>[[Ruth Barcan Marcus]], ''Modalities: Philosophical Essays'', Oxford University Press, 1993, p. x.</ref> The contemporary era in modal semantics began in <!--March-->1959, when [[Saul Kripke]] (then only a 18-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 [[Evert Willem Beth|E. W. Beth]]. [[A. N. Prior]] created modern [[temporal logic]], closely related to modal logic, in 1957 by adding modal operators [F] and [P] meaning "eventually" and "previously". [[Vaughan Pratt]] introduced [[dynamic logic (modal logic)|dynamic logic]] in 1976. In 1977, [[Amir Pnueli]] proposed using temporal logic to formalise the behaviour of continually operating [[concurrent program]]s. Flavors of temporal logic include [[propositional dynamic logic]] (PDL), (propositional) [[linear temporal logic]] (LTL), [[computation tree logic]] (CTL), [[Hennessy–Milner logic]], and ''T''.{{clarify|reason=Add a wikilink, give a longer name, or give a reference for the 'T' logic.|date=November 2016}} The mathematical structure of modal logic, namely [[Boolean algebra (structure)|Boolean algebra]]s augmented with [[unary operation]]s (often called [[modal algebra]]s), began to emerge with [[J.C.C. McKinsey|J. C. C. McKinsey]]'s 1941 proof that ''S2'' and ''S4'' are decidable,<ref>{{cite journal|author=McKinsey, J. C. C.|title=A Solution of the Decision Problem for the Lewis Systems S2 and S4, with an Application to Topology|journal=J. Symb. Log.|year=1941|volume=6|issue=4|pages=117–134|jstor=2267105|doi=10.2307/2267105|s2cid=3241516 }}</ref> 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 operator|interior]] and [[closure operator]]s of [[topology]]. Texts on modal logic typically do little more than mention its connections with the study of [[Boolean algebra (structure)|Boolean algebra]]s and [[topology]]. For a thorough survey of the history of formal modal logic and of the associated mathematics, see [[Robert Goldblatt]] (2006).<ref>Robert Goldbaltt, [http://www.mcs.vuw.ac.nz/~rob/papers/modalhist.pdf Mathematical Modal Logic: A view of its evolution]</ref>
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