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| '''Non-classical logics''' (and sometimes '''alternative logics''') is the name given to [[formal system]]s which differ in a significant way from [[Classical logic|standard logical systems]] such as [[Propositional logic|propositional]] and [[Predicate logic|predicate]] logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of [[logical consequence]] and [[logical truth]].<ref>''Logic for philosophy'', [[Theodore Sider]]</ref>
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| [[Philosophical logic]], especially in [[theoretical computer science]], is understood to encompass and focus on non-classical logics, although the term has other meanings as well.<ref name="Burgess2009i">{{cite book|author=[[John P. Burgess]]|title=Philosophical logic|url=http://books.google.com/books?id=k32w3_wjBoYC&pg=PR7|year=2009|publisher=Princeton University Press|isbn=978-0-691-13789-6|pages=vii-viii}}</ref>
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| == Examples of non-classical logics ==
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| *[[Fuzzy logic]] rejects the law of the excluded middle and allows as a [[truth value]] any real number between 0 and 1.
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| *[[Intuitionistic logic]] rejects the law of the excluded middle, [[double negative elimination]], and the De Morgan's laws;
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| *[[Linear logic]] rejects idempotency of [[entailment]] as well;
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| *[[Modal logic]] extends classical logic with [[Truth function|non-truth-functional]] ("modal") operators.
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| *[[Paraconsistent logic]] (e.g., [[dialetheism]] and [[relevance logic]]) rejects the law of noncontradiction;
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| *[[Relevance logic]], [[linear logic]], and [[non-monotonic logic]] reject monotonicity of entailment;
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| *[[Computability logic]] is a semantically constructed formal theory of computability, as opposed to classical logic, which is a formal theory of truth; integrates and extends classical, linear and intuitionistic logics.
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| ==Classification of non-classical logics==
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| In ''Deviant Logic'' (1974) [[Susan Haack]] divided non-classical logics into [[Deviant logic|deviant]], quasi-deviant, and extended logics.<ref name="Haack1974">{{cite book|author=Susan Haack|title=Deviant logic: some philosophical issues|url=http://books.google.com/books?id=ANg8AAAAIAAJ&pg=PA4|year=1974|publisher=CUP Archive|isbn=978-0-521-20500-9|page=4}}</ref> The proposed classification is non-exclusive; a logic may be both a deviation and an extension of classical logic.<ref name="Haack1978">{{cite book|author=Susan Haack|title=Philosophy of logics|url=http://books.google.com/books?id=0GsZ8SBQrUcC&pg=PA204|year=1978|publisher=Cambridge University Press|isbn=978-0-521-29329-7|pages=204}}</ref> A few other authors have adopted the main distinction between deviation and extension in non-classical logics.<ref name="Gamut1991">{{cite book|author=[[L. T. F. Gamut]]|title=Logic, language, and meaning, Volume 1: Introduction to Logic|url=http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156|year=1991|publisher=University of Chicago Press|isbn=978-0-226-28085-1|pages=156–157}}</ref><ref name="Akama1997">{{cite book|author=Seiki Akama|title=Logic, language, and computation|url=http://books.google.com/books?id=QyksEA5i-1QC&pg=PA3|year=1997|publisher=Springer|isbn=978-0-7923-4376-9|page=3}}</ref><ref name="Hanna2006">{{cite book|author=Robert Hanna|title=Rationality and logic|url=http://books.google.com/books?id=ka9BhOL1ev8C&pg=PA40|year=2006|publisher=MIT Press|isbn=978-0-262-08349-2|pages=40–41}}</ref> [[John P. Burgess]] uses a similar classification but calls the two main classes anti-classical and extra-classical.<ref name="Burgess2009">{{cite book|author=John P. Burgess|title=Philosophical logic|url=http://books.google.com/books?id=k32w3_wjBoYC&pg=PA1|year=2009|publisher=Princeton University Press|isbn=978-0-691-13789-6|pages=1–2}}</ref>
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| In an ''extension'', new and different [[logical constant]]s are added, for instance the "<math>\Box</math>" in [[modal logic]] which stands for "necessarily."<ref name="Gamut1991"/> In extensions of a logic,
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| * the set of [[well-formed formula]]s generated is a [[proper superset]] of the set of well-formed formulas generated by [[classical logic]].
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| * the set of [[theorem]]s generated is a proper superset of the set of theorems generated by classical logic, but only in that the novel theorems generated by the extended logic are only a result of novel well-formed formulas.
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| (See also [[Conservative extension]].)
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| In a ''deviation'', the usual logical constants are used, but are given a different meaning than usual. Only a subset of the theorems from the classical logic hold. A typical example is intuitionistic logic, where the [[law of excluded middle]] does not hold.<ref name="Burgess2009"/><ref name="Hanna2006"/>
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| Additionally, one can identify a ''variations'' (or ''variants''), where the content of the system remains the same, while the notation may change substantially. For instance [[many-sorted logic|many-sorted]] predicate logic is considered a just variation of predicate logic.<ref name="Gamut1991"/>
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| This classification ignores however semantic equivalences. For instance, [[Gödel]] showed that all theorems from intuitionistic logic have an equivalent theorem in the classical modal logic S4. The result has been generalized to [[superintuitionistic logic]]s and extensions of S4.<ref name="GabbayMaksimova2005">{{cite book|author1=Dov M. Gabbay|author2=Larisa Maksimova|title=Interpolation and definability: modal and intuitionistic logics|url=http://books.google.com/books?id=v6sDNSaW5wAC&pg=PA61|year=2005|publisher=Clarendon Press|isbn=978-0-19-851174-8|page=61}}</ref>
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| The theory of [[abstract algebraic logic]] has also provided means to classify logics, with most results having been obtained for propositional logics. The current algebraic hierarchy of propositional logics has five levels, defined in terms of properties of their [[Leibniz operator]]: [[protoalgebraic]], (finitely) [[equivalential]], and (finitely) [[algebraizable]].<ref>{{cite book|editor=M. Hazewinkel|title=Encyclopaedia of mathematics: Supplement Volume III|year=2001|publisher=Springer|isbn=1-4020-0198-3|chapter=Abstract algebraic logic|author=D. Pigozzi|pages=2–13}} Also online: {{SpringerEOM| title=Abstract algebraic logic | id=Abstract_algebraic_logic | oldid=21268 }}</ref>
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| == References ==
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| {{reflist}}
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| == Further reading ==
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| * {{cite book|author=[[Graham Priest]]|title=An introduction to non-classical logic: from if to is|year=2008|publisher=Cambridge University Press|isbn=978-0-521-85433-7|edition=2nd}}
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| * {{cite book|author=[[Dov M. Gabbay]]|title=Elementary logics: a procedural perspective|year=1998|publisher=Prentice Hall Europe|isbn=978-0-13-726365-3}} A revised version was published as {{cite book|author=D. M. Gabbay|title=Logic for Artificial Intelligence and Information Technology|year=2007|publisher=[[College Publications]]|isbn=978-1-904987-39-0}}
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| * {{cite book|author=[[John P. Burgess]]|title=Philosophical logic|year=2009|publisher=Princeton University Press|isbn=978-0-691-13789-6}} Brief introduction to non-classical logics, with a primer on the classical one.
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| * {{cite book|editor=Lou Goble|title=The Blackwell guide to philosophical logic|year=2001|publisher=Wiley-Blackwell|isbn=978-0-631-20693-4}} Chapters 7-16 cover the main non-classical logics of broad interest today.
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| * {{cite book|author=Lloyd Humberstone|title=The Connectives|year=2011|publisher=MIT Press|isbn=978-0-262-01654-4}} Probably covers more logics than any of the other titles in this section; a large part of this 1500-page monograph is cross-sectional, comparing—as its title implies—the [[logical connective]]s in various logics; decidability and complexity aspects are generally omitted though.
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| == External links ==
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| *[http://philosophy.commons.gc.cuny.edu/video-graham-priest-on-devient-logic/ Video of Graham Priest & Maureen Eckert on Deviant Logic]
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| {{logic}}
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| [[Category:Non-classical logic]]
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| [[Category:Philosophy of logic]]
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