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Importing Wikidata short description: "Type of formal logic without explosion principle" |
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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{{Short description|Type of formal logic without explosion principle}}
{{more citations needed|date=April 2018}}▼
A '''paraconsistent logic''' is an attempt at a [[logical system]] to deal with [[contradiction]]s in a discriminating{{What|date=December 2022}} way. Alternatively, paraconsistent logic is the subfield of [[logic]] that is concerned with studying and developing "inconsistency-tolerant" systems of logic which reject 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]].<ref>Priest (2002), p. 288 and §3.3.</ref> The study of paraconsistent logic has been dubbed '''paraconsistency,'''<ref>Carnielli, W.; Rodrigues, A. ▼
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▲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/
"[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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Which means: if ''P'' and its negation ¬''P'' are both assumed to be true, then of the two claims ''P'' and (some arbitrary) ''A'', at least one is true. Therefore, ''P'' or ''A'' is true. However, if we know that either ''P'' or ''A'' is true, and also that ''P'' is false (that ¬''P'' is true) we can conclude that ''A'', which could be anything, is true. Thus if a [[theory (logic)|theory]] contains a single inconsistency,
The characteristic or defining feature of a paraconsistent logic is that it rejects the principle of explosion. As a result, paraconsistent logics, unlike classical and other logics, can be used to formalize inconsistent but non-trivial theories.
==Comparison with classical logic==
== Motivation ==
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.
==Tradeoffs==
Paraconsistency involves tradeoffs. In particular, abandoning the principle of explosion requires one to abandon at least one of the following two principles:<ref>See the article on the [[principle of explosion]] for more on this.</ref>
{| class="wikitable" style="margin: auto;"
![[Disjunction introduction]]
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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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![[Negation#
|If <math> A \vdash B \land \neg B</math>, then <math> \vdash \neg A</math>
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== An ideal three-valued paraconsistent logic ==
Here is an example of a [[three-valued logic]] which is paraconsistent and ''ideal'' as defined in "Ideal Paraconsistent Logics" by O. Arieli, A. Avron, and A. Zamansky, especially pages 22–23.<ref>{{Cite web |url=https://www.cs.tau.ac.il/~aa/articles/ideal.pdf
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* [[Set theory]] and the [[foundations of mathematics]]
* [[Epistemology]] and [[belief revision]]: Paraconsistent logic has been proposed as a means of reasoning with and revising inconsistent theories and belief systems.
* [[Knowledge management]] and [[artificial intelligence]]: Some [[computer scientist]]s have utilized paraconsistent logic as a means of coping gracefully with inconsistent<ref>See, for example, [[truth maintenance systems]] or the articles in Bertossi et al. (2004).</ref> or contradictory<ref>Gershenson, C. (1999). Modelling emotions with multidimensional logic. In Proceedings of the 18th International Conference of the North American Fuzzy Information Processing Society (NAFIPS ’99), pp. 42–46, New York City, NY. IEEE Press. http://cogprints.org/1479/</ref> information. Mathematical framework and rules of paraconsistent logic have been proposed as the [[activation function]] of an [[artificial neuron]] in order to build a [[neural network]] for [[function approximation]], [[model identification]], and [[Control (management)|control]] with success.<ref>{{Cite journal|last1=de Carvalho Junior|first1=A.|last2=Justo|first2=J. F.|last3=Angelico|first3=B. A.|last4=de Oliveira|first4=A. M.|last5=da Silva Filho|first5=J. I.|date=2021|title=Rotary Inverted Pendulum Identification for Control by Paraconsistent Neural Network|journal=IEEE Access|volume=9|pages=74155–74167|doi=10.1109/ACCESS.2021.3080176|bibcode=2021IEEEA...974155D |issn=2169-3536|doi-access=free}}</ref>
* [[Deontic logic]] and [[metaethics]]: Paraconsistent logic has been proposed as a means of dealing with ethical and other normative conflicts.
* [[Software engineering]]: Paraconsistent logic has been proposed as a means for dealing with the pervasive inconsistencies among the [[documentation]], [[use cases]], and [[Source code|code]] of large [[software systems]].<ref name="Hewitt 2008b">Hewitt (2008b)</ref><ref name="Hewitt 2008a">Hewitt (2008a)</ref><ref>Carl Hewitt. "Formalizing common sense reasoning for scalable inconsistency-robust information coordination using Direct Logic Reasoning and the Actor Model". in Vol. 52 of ''Studies in Logic''. College Publications. {{isbn|1848901593}}. 2015.</ref>
* [[Expert system]]. The Para-analyzer algorithm based on paraconsistent annotated logic by 2-value annotations (PAL2v), also called paraconsistent annotated evidential logic (PAL ''E''t), derived from paraconsistent logic, has been used in decision-making systems, such as to support medical diagnosis.<ref>{{cite journal |last1=de Carvalho Junior |first1=Arnaldo |last2=Justo |first2=João Francisco |last3=de Oliveira |first3=Alexandre Maniçoba |last4=da Silva Filho |first4=João Inacio |title=A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives |journal=Engineering Applications of Artificial Intelligence |date=1 January 2024 |volume=127 |issue=B |pages=107342 |doi=10.1016/j.engappai.2023.107342|s2cid=264898768 }}</ref>
* [[Electronics]] design routinely uses a [[four-valued logic]], with "hi-impedance (z)" and "don't care (x)" playing similar roles to "don't know" and "both true and false" respectively, in addition to true and false. This logic was developed independently of philosophical logics.
* [[Control system
*[[Digital filter]]: PAL2v Filter Algorithm, using a paraconsistent artificial neural cell of learning by contradiction extraction (PANLctx) in the composition of a paraconsistent analysis network (PANnet), based on the PAL2V rules and equations, can be used as an estimator, average extractor, filtering and in signal treatment for industrial automation and robotics.<ref>{{cite journal |last1=de Carvalho Junior |first1=Arnaldo |last2=Justo |first2=João Francisco |last3=de Oliveira |first3=Alexandre Maniçoba |last4=da Silva Filho |first4=João Inacio |title=A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives |journal=Engineering Applications of Artificial Intelligence |date=1 January 2024 |volume=127 |issue=B |pages=107342 |doi=10.1016/j.engappai.2023.107342|s2cid=264898768 }}</ref><ref>{{cite book |last1=de Carvalho Jr. |first1=Arnaldo |last2=Da Silva Filho |first2=João Inácio |last3=de Freitas Minicz |first3=Márcio |last4=Matuck |first4=Gustavo R. |last5=Côrtes |first5=Hyghor Miranda |last6=Garcia |first6=Dorotéa Vilanova |last7=Tasinaffo |first7=Paulo Marcelo |last8=Abe |first8=Jair Minoro |title=Advances in Applied Logics |chapter=A Paraconsistent Artificial Neural Cell of Learning by Contradiction Extraction (PANCLCTX) with Application Examples |series=Intelligent Systems Reference Library |date=2023 |volume=243 |pages=63–79 |doi=10.1007/978-3-031-35759-6_5|isbn=978-3-031-35758-9 }}</ref><ref>{{cite journal |last1=Carvalho |first1=Arnaldo |last2=Justo |first2=João F. |last3=Angélico |first3=Bruno A. |last4=de Oliveira |first4=Alexandre M. |last5=da Silva Filho |first5=João Inacio |title=Paraconsistent State Estimator for a Furuta Pendulum Control |journal=SN Computer Science |date=22 October 2022 |volume=4 |issue=1 |doi=10.1007/s42979-022-01427-z|s2cid=253064746 }}</ref>
*[[Contradiction]] Extractor. A recurrent algorithm based on the PAL2v rules and equations has been used to extract contradictions in a set of statistical data.<ref>{{cite journal |last1=de Carvalho Junior |first1=Arnaldo |last2=Justo |first2=João Francisco |last3=de Oliveira |first3=Alexandre Maniçoba |last4=da Silva Filho |first4=João Inacio |title=A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives |journal=Engineering Applications of Artificial Intelligence |date=1 January 2024 |volume=127 |issue=B |pages=107342 |doi=10.1016/j.engappai.2023.107342|s2cid=264898768 }}</ref>
* [[Quantum physics]]
* [[Black hole]] physics
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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'':
''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).
However, even on a highly technical level, paraconsistent logic can be 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]].
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> Field argues for a view that avoids both truth gluts (where a statement can be both true and false) and truth gaps (where a statement is neither true nor false). One of Field's concerns is the problem of a paraconsistent [[metatheory]]: If the logic itself allows contradictions to be true, then the metatheory that describes or governs the logic might also have to be paraconsistent. If the metatheory is paraconsistent, then the justification of the logic (why we should accept it) might be suspect, because any argument made within a paraconsistent framework could potentially be both valid and invalid. This creates a challenge for proponents of paraconsistent logic to explain how their logic can be justified without falling into paradox or losing explanatory power. [[Stewart Shapiro]] expressed similar concerns: "there are certain notions and concepts that the dialetheist invokes (informally), but which she cannot adequately express, unless the meta-theory is (completely) consistent. The insistence on a consistent meta-theory would undermine the key aspect of dialetheism"<ref>{{Cite book |last=Shapiro |first=Stewart |title=Simple Truth, Contradiction, Conistency |publisher=Oxford University Press |isbn=978-0-19-920419-9 |editor-last=Priest |editor-first=Graham |___location=New York |publication-date=2004 |pages=338 |editor-last2=Beall |editor-first2=JC |editor-last3=Armour-Garb |editor-first3=Bradley}}</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 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 |last1=Littmann |first1=Greg |title=A Critique of Dialetheism |last2=Simmons |first2=Keith |publisher=Oxford University Press |year=2004 |isbn=978-0-19-920419-9 |editor-last=Priest |editor-first=Graham |___location=New York |pages=314–335 |editor-last2=Beall |editor-first2=JC |editor-last3=Armour-Garb |editor-first3=Bradley}}</ref>
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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== Alternatives ==
Approaches exist that allow for resolution of inconsistent beliefs without violating any of the intuitive logical principles. Most such systems use [[multi-valued logic]] with [[Bayesian inference]] and the [[Dempster-Shafer theory]], allowing that no non-tautological belief is completely (100%) irrefutable because it must be based upon incomplete, abstracted, interpreted, likely unconfirmed, potentially uninformed, and possibly incorrect knowledge (of course, this very assumption, if non-tautological, entails its own refutability, if by "refutable" we mean "not completely [100%] irrefutable")
== Notable figures ==
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* [[Bryson Brown]] (Canada)
* [[Walter Carnielli]] ([[Brazil]]). The developer of the ''possible-translations semantics'', a new semantics which makes paraconsistent logics applicable and philosophically understood.
* [[Newton da Costa]] ([[Brazil]],
* [[Itala M. L. D'Ottaviano]] ([[Brazil]])
* [[J. Michael Dunn]] (United States). An important figure in relevance logic.
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*[[Deviant logic]]
*[[Formal logic]]
*[[Fuzzy logic]]
*[[Probability logic]]
*[[Intuitionistic logic]]
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* {{cite journal | author=Slater, B. H. | title= Paraconsistent Logics? | journal=Journal of Philosophical Logic | year=1995 | volume=24 | pages= 451–454 | doi=10.1007/BF01048355 | issue=4| s2cid= 12125719 }}
* {{cite book | last=Woods | first=John | title=Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences | year=2003 | publisher=[[Cambridge University Press]] | ___location=Cambridge | isbn=0-521-00934-0}}
*{{cite journal|author1=De Carvalho, A.|author2=Justo, J. F.|author3=De Oliveira, A. M.|author4=Da Silva Filho, J. I.|title=A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives|journal=Engineering Applications of Artificial Intelligence|year=2024|volume=127B|pages=107342|issn=0952-1976|doi=10.1016/j.engappai.2023.107342}}
==External links==
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{{Non-classical logic}}
{{Authority control}}
[[Category:Paraconsistent logic| ]]▼
[[Category:Belief revision]]
[[Category:Non-classical logic]]
▲[[Category:Paraconsistent logic| ]]
[[Category:Philosophical logic]]
[[Category:Systems of formal logic]]
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