Editing Principle of explosion (section)

{{Short description|Theorem in formal logic}}
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{{redirect|Ex falso quodlibet|the musical form|Quodlibet|the audio player and library organizer|Quod Libet (software)||}}

{{More citations needed|date=August 2020}}

In [[classical logic]], [[intuitionistic logic]], and similar [[Formal system|logical systems]], the '''principle of explosion'''{{efn|{{Langx|la|ex falso [sequitur] quodlibet}}, 'from falsehood, anything [follows]'; or {{Langx|la|ex contradictione [sequitur] quodlibet|lit=from contradiction, anything [follows]|label=none}}.}}{{efn|Also known as the '''principle of Pseudo-Scotus''' (falsely attributed to [[Pseudo-Scotus|Duns Scotus]]).}} is the [[Laws of logic (disambiguation)|law]] according to which any [[Statement (logic)|statement]] can be proven from a [[contradiction]].<ref>{{cite journal |author1-link=Walter Carnielli |last1=Carnielli |first=Walter |first2=João |last2=Marcos |title=Ex contradictione non sequitur quodlibet |journal=Bulletin of Advanced Reasoning and Knowledge |volume=1 |issue= |pages=89–109 |date=2001 |doi= |url=https://www.advancedreasoningforum.org/galeria/docs/2023/07/25/1690305935.pdf }}</ref><ref>{{cite book |last1=Smith |first1=Peter |title=An Introduction to Formal Logic |date=2020 |publisher=Cambridge University Press |edition=2nd |url=https://www.logicmatters.net/resources/pdfs/IFL2_LM.pdf }} Chapter 17.</ref><ref>{{cite book |last1=MacFarlane |first1=John |title=Philosophical Logic: A Contemporary Introduction |date=2021 |publisher=Routledge}}
Chapter 7.</ref> That is, from a contradiction, any [[proposition]] (including its [[negation]]) can be inferred; this is known as '''deductive explosion'''.<ref>{{cite journal
| journal=[[Synthese]]
| title=Some topological properties of paraconsistent models
| last1=Başkent
| first1=Can
| date=2013 
| doi=10.1007/s11229-013-0246-8
| volume=190
| issue=18
| page=4023
| s2cid=9276566
}}</ref><ref>{{cite book
| series=Logic, Epistemology, and the Unity of Science
| publisher=Springer 
| title=Paraconsistent Logic: Consistency, Contradiction and Negation
| last1=Carnielli
| first1=Walter
| last2=Coniglio
| first2=Marcelo Esteban
| volume=40
| year=2016
| doi=10.1007/978-3-319-33205-5
| isbn=978-3-319-33203-1
| at=ix}}</ref>

The proof of this principle was first given by 12th-century French philosopher [[William of Soissons]].<ref> [[Graham Priest|Priest, Graham]]. 2011. "What's so bad about contradictions?" In ''The Law of Non-Contradicton'', edited by Priest, Beal, and Armour-Garb. Oxford: Clarendon Press. p. 25.</ref> Due to the principle of explosion, the existence of a contradiction ([[inconsistency]]) in a [[formal system|formal axiomatic system]] is disastrous; since any statement can be proven, it trivializes the concepts of truth and falsity.<ref name="McKubre-Jordens">{{cite web|last=McKubre-Jordens|first=Maarten|date=August 2011|title=This is not a carrot: Paraconsistent mathematics|url=https://plus.maths.org/content/not-carrot|access-date=January 14, 2017|work=Plus Magazine|publisher=Millennium Mathematics Project}}</ref> Around the turn of the 20th century, the discovery of contradictions such as [[Russell's paradox]] at the foundations of mathematics thus threatened the entire structure of mathematics.  Mathematicians such as [[Gottlob Frege]], [[Ernst Zermelo]], [[Abraham Fraenkel]], and [[Thoralf Skolem]] put much effort into revising [[set theory]] to eliminate these contradictions, resulting in the modern [[Zermelo–Fraenkel set theory]].

As a demonstration of the principle, consider two contradictory statements—"All [[lemon]]s are yellow" and "Not all lemons are yellow"—and suppose that both are true. If that is the case, anything can be proven, e.g., the assertion that "[[unicorn]]s exist", by using the following argument:
# We know that "Not all lemons are yellow", as it has been assumed to be true.
# We know that "All lemons are yellow", as it has been assumed to be true.
# Therefore, the two-part statement "All lemons are yellow ''or'' unicorns exist" must also be true, since the first part of the statement ("All lemons are yellow") has already been assumed, and the use of "''or''" means that if even one part of the statement is true, the statement as a whole must be true as well.
# However, since we also know that "Not all lemons are yellow" (as this has been assumed), the first part is false, and hence the second part must be true to ensure the two-part statement to be true, i.e., unicorns exist (this inference is known as the [[disjunctive syllogism]]).
# The procedure may be repeated to prove that unicorns do ''not'' exist (hence proving an additional contradiction where unicorns do and do not exist), as well as any other [[well-formed formula]]. Thus, there is an ''explosion'' of true statements. 

In a different solution to the problems posed by the principle of explosion, some mathematicians have devised alternative theories of [[logic (mathematics)|logic]] called [[paraconsistent logic|''paraconsistent logics'']], which allow some contradictory statements to be proven without affecting the truth value of (all) other statements.<ref name="McKubre-Jordens" />