# Principle of explosion

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Principle_of_explosion
> Markdown URL: https://mediated.wiki/source/Principle_of_explosion.md
> Source: https://en.wikipedia.org/wiki/Principle_of_explosion
> Source revision: 1355110239
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Theorem in formal logic

"EFQ" redirects here. For the literary baseball journal, see [Elysian Fields Quarterly](/source/Elysian_Fields_Quarterly).

"Ex falso quodlibet" redirects here. For the musical form, see [Quodlibet](/source/Quodlibet). For the audio player and library organizer, see [Quod Libet (software)](/source/Quod_Libet_(software)).

This article needs more citations. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Principle of explosion" – news · newspapers · books · scholar · JSTOR (August 2020) (Learn how and when to remove this message)

In [classical logic](/source/Classical_logic), [intuitionistic logic](/source/Intuitionistic_logic), and similar [logical systems](/source/Formal_system), the **principle of explosion**[a][b] is the [theorem](/source/Theorem) according to which any [statement](/source/Statement_(logic)) can be proven from a [contradiction](/source/Contradiction).[1][2][3] That is, from a contradiction, any [proposition](/source/Proposition) (including its [negation](/source/Negation)) can be inferred; this is known as **deductive explosion**.[4][5]

The proof of this principle was first given by 12th-century French philosopher [William of Soissons](/source/William_of_Soissons).[6] Due to the principle of explosion, the existence of a contradiction ([inconsistency](/source/Inconsistency)) in a [formal axiomatic system](/source/Formal_system) is disastrous; since any statement—true or not—can be proven, it trivializes the concepts of truth and falsity.[7] Around the turn of the 20th century, the discovery of contradictions such as [Russell's paradox](/source/Russell's_paradox) at the foundations of mathematics thus threatened the entire structure of mathematics. Mathematicians such as [Gottlob Frege](/source/Gottlob_Frege), [Ernst Zermelo](/source/Ernst_Zermelo), [Abraham Fraenkel](/source/Abraham_Fraenkel), and [Thoralf Skolem](/source/Thoralf_Skolem) put much effort into revising [set theory](/source/Set_theory) to eliminate these contradictions, resulting in the modern [Zermelo–Fraenkel set theory](/source/Zermelo%E2%80%93Fraenkel_set_theory).

As a demonstration of the principle, consider two contradictory statements—"All [lemons](/source/Lemon) 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 "[unicorns](/source/Unicorn) exist", by using the following argument:

1. We know that "Not all lemons are yellow", as it has been assumed to be true.

1. We know that "All lemons are yellow", as it has been assumed to be true.

1. 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.

1. 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](/source/Disjunctive_syllogism)).

1. 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](/source/Well-formed_formula). Thus, there is an *explosion* of provable statements.

In a different solution to the problems posed by the principle of explosion, some mathematicians have devised alternative theories of [logic](/source/Logic_(mathematics)) called [*paraconsistent logics*](/source/Paraconsistent_logic), which allow some contradictory statements to be proven without affecting the truth value of (all) other statements.[7]

## Symbolic representation

In [symbolic logic](/source/Symbolic_logic), the principle of explosion can be expressed schematically in the following way:[8][9]

        P
        ,
        ¬
        P
        ⊢
        Q

    {\displaystyle P,\lnot P\vdash Q}

For any statements *P* and *Q*, if *P* and not-*P* are both true, then it logically follows that *Q* is true.

## Proof

Below is the **Lewis argument**,[10] a formal proof of the principle of explosion using [symbolic logic](/source/Symbolic_logic).

Step Proposition Derivation 1 P ∧ ¬ P {\displaystyle P\land \neg P} Premise[c] 2 P {\displaystyle P} Conjunction elimination (1) 3 ¬ P {\displaystyle \neg P} Conjunction elimination (1) 4 P ∨ Q {\displaystyle P\lor Q} Disjunction introduction (2) 5 Q {\displaystyle Q} Disjunctive syllogism (4,3)

This proof was published by [C. I. Lewis](/source/C._I._Lewis) and is named after him, though versions of it were known to medieval logicians.[11][12][10]

This is just the symbolic version of the informal argument given in the introduction, with P {\displaystyle P} standing for "all lemons are yellow" and Q {\displaystyle Q} standing for "Unicorns exist". We start out by assuming that (1) all lemons are yellow and that (2) not all lemons are yellow. From the proposition that all lemons are yellow, we infer that (3) either all lemons are yellow or unicorns exist. But then from this and the fact that not all lemons are yellow, we infer that (4) unicorns exist by disjunctive syllogism.

### Semantic argument

An alternate argument for the principle stems from [model theory](/source/Model_theory). A [sentence](/source/Sentence) P {\displaystyle P} is a *[semantic consequence](/source/Semantic_consequence)* of a set of sentences Γ {\displaystyle \Gamma } only if every model of Γ {\displaystyle \Gamma } is a model of P {\displaystyle P} . However, there is no model of the contradictory set ( P ∧ ¬ P ) {\displaystyle (P\wedge \lnot P)} . *[A fortiori](/source/A_fortiori)*, there is no model of ( P ∧ ¬ P ) {\displaystyle (P\wedge \lnot P)} that is not a model of Q {\displaystyle Q} . Thus, vacuously, every model of ( P ∧ ¬ P ) {\displaystyle (P\wedge \lnot P)} is a model of Q {\displaystyle Q} . Thus Q {\displaystyle Q} is a semantic consequence of ( P ∧ ¬ P ) {\displaystyle (P\wedge \lnot P)} .

## Paraconsistent logic

[Paraconsistent logics](/source/Paraconsistent_logic) have been developed that allow for [subcontrary](/source/Subcontrary)-forming operators. [Model-theoretic](/source/Formal_semantics_(logic)) paraconsistent logicians often deny the assumption that there can be no model of { ϕ , ¬ ϕ } {\displaystyle \{\phi ,\lnot \phi \}} and devise semantical systems in which there are such models. Alternatively, they reject the idea that propositions can be classified as true or false. [Proof-theoretic](/source/Proof-theoretic_semantics) paraconsistent logics usually deny the validity of one of the steps necessary for deriving an explosion, typically including [disjunctive syllogism](/source/Disjunctive_syllogism), [disjunction introduction](/source/Disjunction_introduction), and *[reductio ad absurdum](/source/Reductio_ad_absurdum)*.

## Usage

The [metamathematical](/source/Metamathematics) value of the principle of explosion is that for any logical system where this principle holds, any derived [theory](/source/Mathematical_theory) which proves [⊥](/source/False_(logic)) (or an equivalent form, ϕ ∧ ¬ ϕ {\displaystyle \phi \land \lnot \phi } ) is worthless because *all* its [statements](/source/Statement_(logic)) would become [theorems](/source/Theorem), making it impossible to distinguish [truth](/source/Truth) from falsehood. That is to say, the principle of explosion is an argument for the [law of non-contradiction](/source/Law_of_non-contradiction) in classical logic, because without it all truth statements become meaningless.

Reduction in proof strength of logics without the principle of explosion is discussed in [minimal logic](/source/Minimal_logic).

## See also

- *[Consequentia mirabilis](/source/Consequentia_mirabilis)* – Clavius' Law

- [Dialetheism](/source/Dialetheism) – belief in the existence of true contradictions

- [Law of excluded middle](/source/Law_of_excluded_middle) – every proposition is true or false

- [Law of noncontradiction](/source/Law_of_noncontradiction) – no proposition can be both true and not true

- [Paraconsistent logic](/source/Paraconsistent_logic) – a family of logics used to address contradictions

- [Paradox of entailment](/source/Paradox_of_entailment) – a seeming paradox derived from the principle of explosion

- *[Reductio ad absurdum](/source/Reductio_ad_absurdum)* – concluding that a proposition is false because it produces a contradiction

- [Trivialism](/source/Trivialism) – the belief that all statements of the form "P and not-P" are true

## Notes

1. **[^](#cite_ref-1)** [Latin](/source/Latin_language): *ex falso [sequitur] quodlibet*, 'from falsehood, anything [follows]'; or *ex contradictione [sequitur] quodlibet*, 'from contradiction, anything [follows]'.

1. **[^](#cite_ref-2)** Also known as the **principle of Pseudo-Scotus** (falsely attributed to [Duns Scotus](/source/Pseudo-Scotus)).

1. **[^](#cite_ref-13)** Burgess2005 uses 2 and 3 as premises instead of this one

## References

1. **[^](#cite_ref-3)** [Carnielli, Walter](/source/Walter_Carnielli); Marcos, João (2001). ["Ex contradictione non sequitur quodlibet"](https://www.advancedreasoningforum.org/galeria/docs/2023/07/25/1690305935.pdf) (PDF). *Bulletin of Advanced Reasoning and Knowledge*. **1**: 89–109.

1. **[^](#cite_ref-4)** Smith, Peter (2020). [*An Introduction to Formal Logic*](https://www.logicmatters.net/resources/pdfs/IFL2_LM.pdf) (PDF) (2nd ed.). Cambridge University Press. Chapter 17.

1. **[^](#cite_ref-5)** MacFarlane, John (2021). *Philosophical Logic: A Contemporary Introduction*. Routledge. Chapter 7.

1. **[^](#cite_ref-6)** Başkent, Can (2013). "Some topological properties of paraconsistent models". *[Synthese](/source/Synthese)*. **190** (18): 4023. [doi](/source/Doi_(identifier)):[10.1007/s11229-013-0246-8](https://doi.org/10.1007%2Fs11229-013-0246-8). [S2CID](/source/S2CID_(identifier)) [9276566](https://api.semanticscholar.org/CorpusID:9276566).

1. **[^](#cite_ref-7)** Carnielli, Walter; Coniglio, Marcelo Esteban (2016). *Paraconsistent Logic: Consistency, Contradiction and Negation*. Logic, Epistemology, and the Unity of Science. Vol. 40. Springer. ix. [doi](/source/Doi_(identifier)):[10.1007/978-3-319-33205-5](https://doi.org/10.1007%2F978-3-319-33205-5). [ISBN](/source/ISBN_(identifier)) [978-3-319-33203-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-33203-1).

1. **[^](#cite_ref-8)** [Priest, Graham](/source/Graham_Priest). 2011. "What's so bad about contradictions?" In *The Law of Non-Contradiction*, edited by Priest, Beal, and Armour-Garb. Oxford: Clarendon Press. p. 25.

1. ^ [***a***](#cite_ref-McKubre-Jordens_9-0) [***b***](#cite_ref-McKubre-Jordens_9-1) McKubre-Jordens, Maarten (August 2011). ["This is not a carrot: Paraconsistent mathematics"](https://plus.maths.org/content/not-carrot). *Plus Magazine*. Millennium Mathematics Project. Retrieved January 14, 2017.

1. **[^](#cite_ref-10)** de Swart, Harrie (2018). *Philosophical and Mathematical Logic*. Springer. p. 47.

1. **[^](#cite_ref-11)** Gamut, L. T. F. (1991). *Logic, Language and Meaning, Volume 1. Introduction to Logic*. University of Chicago Press. p. 139.

1. ^ [***a***](#cite_ref-MacFarlane2021_12-0) [***b***](#cite_ref-MacFarlane2021_12-1) MacFarlane, John (2021). *Philosophical Logic: A Contemporary Introduction*. Routledge. p. 171. [ISBN](/source/ISBN_(identifier)) [978-1-315-18524-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-315-18524-8).

1. **[^](#cite_ref-14)** Lewis, C I; Langford, C H (1959). *Symbolic Logic* (2nd ed.). Dover. p. 250. [ISBN](/source/ISBN_(identifier)) [9780486601700](https://en.wikipedia.org/wiki/Special:BookSources/9780486601700). {{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date))

1. **[^](#cite_ref-Burgess2005_15-0)** Burgess, John P (2005). *The Oxford Handbook of Philosophy of Mathematics and Logic (ed Stewart Shapiro)*. Oxford University Press. p. 732. [ISBN](/source/ISBN_(identifier)) [9780195325928](https://en.wikipedia.org/wiki/Special:BookSources/9780195325928).

v t e Classical logic General Quantifiers Predicate Connective Tautology Truth tables Truth function Truth value Well-formed formula Logicism Problem of multiple generality Associativity Distribution Validity Soundness Classical logics Propositional First-order Second-order Higher-order Principles Commutativity of conjunction Excluded middle Bivalence Noncontradiction Monotonicity of entailment Explosion Rules De Morgan's laws Material implication Transposition modus ponens modus tollens modus ponendo tollens Constructive dilemma Destructive dilemma Disjunctive syllogism Hypothetical syllogism reductio ad absurdum Absorption Introduction Negation Double negation Existential Universal Biconditional Conjunction Disjunction Elimination Double negation Existential Universal Biconditional Conjunction Disjunction People Bernard Bolzano George Boole Georg Cantor Richard Dedekind Augustus De Morgan Gottlob Frege Kurt Gödel Hugh MacColl Giuseppe Peano Charles Sanders Peirce Bertrand Russell Ernst Schröder Henry M. Sheffer Alfred Tarski Willard Van Orman Quine Ludwig Wittgenstein Jan Łukasiewicz Works Begriffsschrift Function and Concept The Principles of Mathematics Principia Mathematica Tractatus Logico-Philosophicus

---
Adapted from the Wikipedia article [Principle of explosion](https://en.wikipedia.org/wiki/Principle_of_explosion) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Principle_of_explosion?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.