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{{More citations needed|date=April 2018}}
'''Paraconsistent logic''' is a type of [[non-classical logic]] that allows for the coexistence of contradictory statements without leading to a logical explosion
Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of [[Aristotle]]);<ref>{{cite encyclopedia|url=http://plato.stanford.edu/entries/logic-paraconsistent/|title=Paraconsistent Logic|encyclopedia=[[Stanford Encyclopedia of Philosophy]]|access-date=1 December 2015|archive-url=https://web.archive.org/web/20151211014311/http://plato.stanford.edu/entries/logic-paraconsistent/|archive-date=2015-12-11|url-status=live}}</ref> however, the term ''paraconsistent'' ("beside the consistent") was first coined in 1976, by the [[Peru]]vian [[philosopher]] [[Francisco Miró Quesada Cantuarias]]
"[http://philsci-archive.pitt.edu/14115/1/letj.pdf An epistemic approach to paraconsistency: a logic of evidence and truth]"
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==Definition==
In [[classical logic]] (as well as [[intuitionistic logic]] and most other logics), contradictions [[Entailment|entail]] everything. This feature, known as the [[principle of explosion]] or ''ex contradictione sequitur quodlibet'' ([[Latin]], "from a contradiction, anything follows")<ref>{{cite journal|last1=Carnielli
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A primary motivation for paraconsistent logic is the conviction that it ought to be possible to reason with inconsistent [[information]] in a controlled and discriminating way. The principle of explosion precludes this, and so must be abandoned. In non-paraconsistent logics, there is only one inconsistent theory: the trivial theory that has every sentence as a theorem. Paraconsistent logic makes it possible to distinguish between inconsistent theories and to reason with them.
Research into paraconsistent logic has also led to the establishment of the philosophical school of [[dialetheism]] (most notably advocated by [[Graham Priest]]), which asserts that true contradictions exist in reality, for example groups of people holding opposing views on various moral issues.<ref name="Fisher2007">{{cite book|author=Jennifer Fisher|title=On the Philosophy of Logic|url=https://books.google.com/books?id=k8L_YW-lEEQC&pg=PT142|year=2007|publisher=Cengage Learning|isbn=978-0-495-00888-0|pages=132–134}}</ref> Being a dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise embracing [[trivialism]], i.e. accepting that all contradictions (and equivalently all statements) are true.<ref name="GabbayWoods2007">{{cite book|editor1=Dov M. Gabbay|editor2=John Woods|title=The Many Valued and Nonmonotonic Turn in Logic|chapter-url=https://books.google.com/books?id=3TNj1ZkP3qEC&pg=PA131|year=2007|publisher=Elsevier|isbn=978-0-444-51623-7|page=131|author=Graham Priest|chapter=Paraconsistency and Dialetheism}}</ref> However, the study of paraconsistent logics does not necessarily entail a dialetheist viewpoint. For example, one need not commit to either the existence of true theories or true contradictions, but would rather prefer a weaker standard like [[empirical adequacy]], as proposed by [[Bas van Fraassen]].<ref name="Allhoff2010">{{cite book |
==Philosophy==
In classical logic, Aristotle's three laws, namely, the excluded middle (''p'' or ¬''p''), non-contradiction ¬ (''p'' ∧ ¬''p'') and identity (''p'' iff ''p''), are regarded as the same, due to the inter-definition of the connectives. Moreover, traditionally contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory entails all possible consequences) are assumed inseparable, granted that negation is available. These views may be philosophically challenged, precisely on the grounds that they fail to distinguish between contradictoriness and other forms of inconsistency.
On the other hand, it is possible to derive triviality from the 'conflict' between consistency and contradictions, once these notions have been properly distinguished. The very notions of consistency and inconsistency may be furthermore internalized at the object language level.
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Both of these principles have been challenged.
One approach is to reject disjunction introduction but keep disjunctive [[syllogism]] and transitivity. In this approach, rules of [[natural deduction]] hold, except for [[disjunction introduction]] and [[excluded middle]]; moreover, inference A⊢B does not necessarily mean entailment A⇒B. Also, the following usual Boolean properties hold: [[double negation]] as well as [[associativity]], [[commutativity]], [[distributivity]], [[De Morgan's laws|De Morgan]], and [[idempotence]] inferences (for conjunction and disjunction). Furthermore, inconsistency-robust proof of negation holds for entailment: (A⇒(B∧¬B))⊢¬A.
Another approach is to reject disjunctive syllogism. From the perspective of [[dialetheism]], it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism is that, if ''¬ A'', then ''A'' is excluded and ''B'' can be inferred from ''A ∨ B''. However, if ''A'' may hold as well as ''¬A'', then the argument for the inference is weakened.
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== Criticism ==
Logic, as it is classically understood, rests on three main rules ([[Law of thought|Laws of Thought]]): The [[Law of identity|Law of Identity]] (''LOI''), the [[Law of noncontradiction|Law of Non-Contradiction]] (''LNC''), and the [[Law of excluded middle|Law of the Excluded Middle]] (''LEM''). Paraconsistent logic deviates from classical logic by refusing to accept ''LNC''. However, the ''LNC'' can be seen as closely interconnected with the ''LOI'' as well as the ''LEM'':
Paraconsistent logic is challenging to argue against. It is obvious that paraconsistent logic leads to contradictions. However, the paraconsistent logician embraces contradictions, including any contradictions that are a part or the result of paraconsistent logic. As a consequence, much of the critique has focused on the applicability and comparative effectiveness of paraconsistent logic. This is an important debate since embracing paraconsistent logic comes at the risk of losing a large amount of [[Theorem|theorems]] that form the basis of [[mathematics]] and [[physics]].▼
''LoI'' states that ''A'' is ''A'' (''A''≡''A''). This means that ''A'' is distinct from its opposite or negation (''not A'', or ¬''A''). In classical logic this distinction is supported by the fact that when ''A'' is true, its opposite is not. However, without the ''LNC'', both ''A'' and ''not A'' can be true (''A''∧¬''A''), which blurs their distinction. And without distinction, it becomes challenging to define identity. Dropping the ''LNC'' thus runs risk to also eliminate the ''LoI''.
''LEM'' states that either ''A'' or ''not A'' are true (''A''∨¬''A''). However, without the ''LNC'', both ''A'' and ''not A'' can be true (''A''∧¬''A''). Dropping the ''LNC'' thus runs risk to also eliminate the ''LEM''
Hence, dropping the ''LNC'' in a careless manner risks losing both the ''LOI'' and ''LEM'' as well. And dropping ''all'' three classical laws does not just change the ''kind'' of logic—it leaves us without any functional system of logic altogether. Loss of ''all'' logic eliminates the possibility of structured reasoning, A careless paraconsistent logic therefore might run risk of disapproving of any means of thinking other than chaos. Paraconsistent logic aims to evade this danger using careful and precise technical definitions. As a consequence, most criticism of paraconsistent logic also tends to be highly technical in nature (e.g. surrounding questions such as whether a paradox can be true).
▲
Logician [[Stewart Shapiro]] aimed to make a case for paraconsistent logic as part of his argument for a pluralistic view of logic (the view that different logics are equally appropriate, or equally correct). He found that a case could be made that either, [[Intuitionistic logic|intuitonistic logic]] as the "One True Logic", or a pluralism of [[Intuitionistic logic|intuitonistic logic]] and [[classical logic]] is interesting and fruitful. However, when it comes to paraconsistent logic, he found "no examples that are ... compelling (at least to me)".<ref>{{Cite book |last=Shapiro |first=Stewart |title=Varieties of Logic |publisher=Oxford University Press |year=2014 |isbn=978-0-19-882269-1 |___location=Oxford, UK |pages=82}}</ref>
In "Saving Truth from Paradox", [[Hartry Field]] examines the value of paraconsistent logic as a solution to [[Paradox|paradoxa]].<ref>{{Cite book |last=Field |first=Hartry |title=Saving Truth from Paradox |publisher=Oxford University Press |year=2008 |isbn=978-0-19-923074-7 |___location=New York}}</ref>
In his book "In Contradiction", which argues in favor of paraconsistent dialetheism, [[Graham Priest]] admits to metatheoretic difficulties: "Is there a metatheory for paraconsistent logics that is acceptable in paraconsistent terms? The answer to this question is not at all obvious."<ref>{{Cite book |last=Priest |first=Graham |title=In Contradiction. A Study of the Transconsistent |publisher=Oxford University Press |year=1987 |isbn=0-19-926330-2 |___location=New York |pages=258}}</ref>
Littmann and [[Keith Simmons (philosopher)|Keith Simmons]] argued that dialetheist theory is
Some philosophers have argued against dialetheism on the grounds that the counterintuitiveness of giving up any of the three principles above outweighs any counterintuitiveness that the principle of explosion might have.
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*[[Deviant logic]]
*[[Formal logic]]
*[[Fuzzy logic]]
*[[Probability logic]]
*[[Intuitionistic logic]]
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