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Contextualizes the naming and credits the author behind much of the logic behind the term, with two sources to back up both the context and da Costa's importance in this field, one of them by Miró himself. |
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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 where anything can be proven true. Specifically, paraconsistent logic is the subfield of [[logic]] that is concerned with studying and developing "inconsistency-tolerant" systems of logic, purposefully excluding the [[principle of 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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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 unintelligible: "Once we realize that the theory includes not only the statement '(L) is both true and false' but also the statement '(L) isn't both true and false' we may feel at a loss."<ref>{{Cite book |
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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