54:
1615:
624:
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
265:
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
1046:
979:
205:
is disastrous; since any statement can be proven, it trivializes the concepts of truth and falsity. Around the turn of the 20th century, the discovery of contradictions such as
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850:
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346:
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981:
and devise semantical systems in which there are such models. Alternatively, they reject the idea that propositions can be classified as true or false.
1511:
2000:
1711:
240:
are yellow" and "Not all lemons are yellow"âand suppose that both are true. If that is the case, anything can be proven, e.g., the assertion that "
71:
1427:
1321:
258:
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 "
289:
In a different solution to the problems posed by the principle of explosion, some mathematicians have devised alternative theories of
118:
1477:
1452:
137:
1918:
90:
230:
97:
1979:
1584:
1539:
985:
paraconsistent logics usually deny the validity of one of the steps necessary for deriving an explosion, typically including
167:
75:
1969:
1504:
1777:
1665:
104:
299:, which allow some contradictory statements to be proven without affecting the truth value of (all) other statements.
1818:
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1782:
1685:
86:
64:
1833:
1823:
1574:
982:
2005:
1802:
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943:
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209:
at the foundations of mathematics thus threatened the entire structure of mathematics. Mathematicians such as
31:
1933:
1843:
1838:
1828:
1680:
1102:
1022:
498:
463:
1011:
value of the principle of explosion is that for any logical system where this principle holds, any derived
949:
277:
exist (hence proving an additional contradiction where unicorns do and do not exist), as well as any other
1908:
1751:
1675:
1670:
1634:
1096:
1084:
1974:
1772:
1746:
1731:
1716:
1594:
1114:
986:
568:
267:
42:
895:
820:
765:
726:
316:
262:" means that if even one part of the statement is true, the statement as a whole must be true as well.
1964:
1736:
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1120:
1108:
995:
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659:
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155:
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2010:
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406:
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1943:
1639:
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171:
686:
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511:
1923:
1913:
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584:
This is just the symbolic version of the informal argument given in the introduction, with
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from falsehood. That is to say, the principle of explosion is an argument for the
311:, the principle of explosion can be expressed schematically in the following way:
1068:
in classical logic, because without it all truth statements become meaningless.
946:
paraconsistent logicians often deny the assumption that there can be no model of
581:
and is named after him, though versions of it were known to medieval logicians.
1564:
1554:
1090:
939:
578:
236:
As a demonstration of the principle, consider two contradictory statementsâ"All
179:
53:
1470:
The Oxford
Handbook of Philosophy of Mathematics and Logic (ed Stewart Shapiro)
193:
The proof of this principle was first given by 12th-century French philosopher
1629:
1313:
1282:
1126:
759:
226:
1308:. Logic, Epistemology, and the Unity of Science. Vol. 40. Springer. ix.
1267:
BaĆkent, Can (2013). "Some topological properties of paraconsistent models".
1604:
1357:
1123:â concluding that a proposition is false because it produces a contradiction
248:
We know that "Not all lemons are yellow", as it has been assumed to be true.
38:
1343:, edited by Priest, Beal, and Armour-Garb. Oxford: Clarendon Press. p. 25.
27:
Theorem which states that any statement can be proven from a contradiction
1579:
1269:
183:
1071:
Reduction in proof strength of logics without ex falso are discussed in
251:
We know that "All lemons are yellow", as it has been assumed to be true.
17:
1057:
241:
197:. Due to the principle of explosion, the existence of a contradiction (
1129:â the belief that all statements of the form "P and not-P" are true
625:
yellow, we infer that (4) unicorns exist by disjunctive syllogism.
1489:
1061:
237:
1614:
1493:
37:"Ex falso quodlibet" redirects here. For the musical form, see
47:
1306:
Paraconsistent Logic: Consistency, Contradiction and
Negation
30:"EFQ" redirects here. For the literary baseball journal, see
1401:
Logic, Language and
Meaning, Volume 1. Introduction to Logic
1117:â a seeming paradox derived from the principle of explosion
229:
to eliminate these contradictions, resulting in the modern
254:
Therefore, the two-part statement "All lemons are yellow
1185:
Burgess2005 uses 2 and 3 as premises instead of this one
273:
The procedure may be repeated to prove that unicorns do
723:. However, there is no model of the contradictory set
1304:
Carnielli, Walter; Coniglio, Marcelo
Esteban (2016).
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380:, a formal proof of the principle of explosion using
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1952:
1856:
1811:
1765:
1699:
1658:
1622:
1527:
1111:â a family of logics used to address contradictions
633:An alternate argument for the principle stems from
78:. Unsourced material may be challenged and removed.
1358:"This is not a carrot: Paraconsistent mathematics"
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340:
41:. For the audio player and library organizer, see
1413:
1411:
1351:
1349:
1339:. 2011. "What's so bad about contradictions?" In
1420:Philosophical Logic: A Contemporary Introduction
1253:Philosophical Logic: A Contemporary Introduction
1093:â belief in the existence of true contradictions
1105:â no proposition can be both true and not true
364:are both true, then it logically follows that
1505:
8:
1218:Bulletin of Advanced Reasoning and Knowledge
968:
953:
1239:(2nd ed.). Cambridge University Press.
1154:
1762:
1512:
1498:
1490:
1211:"Ex contradictione non sequitur quodlibet"
1159:, 'from contradiction, anything '.
1024:
951:
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822:
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604:standing for "all lemons are yellow" and
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318:
244:exist", by using the following argument:
138:Learn how and when to remove this message
1472:. Oxford University Press. p. 732.
1356:McKubre-Jordens, Maarten (August 2011).
386:
1198:
1139:
1060:, making it impossible to distinguish
1041:{\displaystyle \phi \land \lnot \phi }
974:{\displaystyle \{\phi ,\lnot \phi \}}
178:. That is, from a contradiction, any
7:
1447:(2nd ed.). Dover. p. 250.
1385:Philosophical and Mathematical Logic
1099:â every proposition is true or false
186:) can be inferred; this is known as
76:adding citations to reliable sources
938:have been developed that allow for
1443:Lewis, C I; Langford, C H (1959).
1153:, 'from falsehood, anything '; or
1032:
962:
908:
833:
817:. Thus, vacuously, every model of
778:
739:
690:
670:
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326:
25:
920:{\displaystyle (P\wedge \lnot P)}
845:{\displaystyle (P\wedge \lnot P)}
790:{\displaystyle (P\wedge \lnot P)}
751:{\displaystyle (P\wedge \lnot P)}
341:{\displaystyle P,\lnot P\vdash Q}
1613:
1364:. Millennium Mathematics Project
52:
2001:Theorems in propositional logic
1237:An Introduction to Formal Logic
63:needs additional citations for
1980:Tractatus Logico-Philosophicus
1585:Problem of multiple generality
1403:. University of Chicago Press.
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839:
824:
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745:
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225:put much effort into revising
1:
1970:The Principles of Mathematics
892:is a semantic consequence of
425:{\displaystyle P\land \neg P}
1666:Commutativity of conjunction
1156:ex contradictione quodlibet
577:This proof was published by
1341:The Law of Non-Contradicton
231:ZermeloâFraenkel set theory
2027:
1686:Monotonicity of entailment
1422:. Routledge. p. 171.
1170:principle of Pseudo-Scotus
36:
29:
1611:
1575:Idempotency of entailment
1418:MacFarlane, John (2021).
1383:de Swart, Harrie (2018).
1314:10.1007/978-3-319-33205-5
1283:10.1007/s11229-013-0246-8
1251:MacFarlane, John (2021).
1468:Burgess, John P (2005).
1399:Gamut, L. T. F. (1991).
1066:law of non-contradiction
1019:(or an equivalent form,
991:disjunction introduction
537:Disjunction introduction
87:"Principle of explosion"
32:Elysian Fields Quarterly
1934:Willard Van Orman Quine
1209:; Marcos, JoĂŁo (2001).
1172:(falsely attributed to
1103:Law of noncontradiction
1048:) is worthless because
797:that is not a model of
762:, there is no model of
696:{\displaystyle \Gamma }
683:only if every model of
676:{\displaystyle \Gamma }
527:{\displaystyle P\lor Q}
499:Conjunction elimination
464:Conjunction elimination
303:Symbolic representation
203:formal axiomatic system
170:according to which any
1909:Charles Sanders Peirce
1752:Hypothetical syllogism
1155:
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1097:Law of excluded middle
1085:Consequentia mirabilis
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663:of a set of sentences
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489:{\displaystyle \neg P}
455:
426:
342:
164:principle of explosion
1975:Principia Mathematica
1747:Disjunctive syllogism
1732:modus ponendo tollens
1235:Smith, Peter (2020).
1115:Paradox of entailment
1043:
987:disjunctive syllogism
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936:Paraconsistent logics
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569:Disjunctive syllogism
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343:
296:paraconsistent logics
268:Disjunctive syllogism
174:can be proven from a
43:Quod Libet (software)
1965:Function and Concept
1737:Constructive dilemma
1712:Material implication
1121:Reductio ad absurdum
1109:Paraconsistent logic
1023:
996:reductio ad absurdum
950:
942:-forming operators.
931:Paraconsistent logic
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707:
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660:semantic consequence
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550:
512:
477:
445:
407:
317:
281:. Thus, there is an
156:intuitionistic logic
72:improve this article
1939:Ludwig Wittgenstein
1742:Destructive dilemma
1570:Well-formed formula
1151:ex falso quodlibet
348:For any statements
285:of true statements.
279:well-formed formula
195:William of Soissons
188:deductive explosion
1884:Augustus De Morgan
1168:Also known as the
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1429:978-1-315-18524-8
1323:978-3-319-33203-1
1207:Carnielli, Walter
885:{\displaystyle Q}
865:{\displaystyle Q}
810:{\displaystyle Q}
716:{\displaystyle P}
650:{\displaystyle P}
629:Semantic argument
617:{\displaystyle Q}
597:{\displaystyle P}
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574:
559:{\displaystyle Q}
454:{\displaystyle P}
207:Russell's paradox
148:
147:
140:
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16:(Redirected from
2018:
1924:Henry M. Sheffer
1914:Bertrand Russell
1879:Richard Dedekind
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1707:De Morgan's laws
1681:Noncontradiction
1623:Classical logics
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2006:Classical logic
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1960:Begriffsschrift
1948:
1944:Jan Ćukasiewicz
1864:Bernard Bolzano
1848:
1819:Double negation
1807:
1778:Double negation
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1695:
1671:Excluded middle
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1521:Classical logic
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1005:
983:Proof-theoretic
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944:Model-theoretic
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182:(including its
160:logical systems
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35:
28:
23:
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15:
12:
11:
5:
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1919:Ernst Schröder
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1560:Truth function
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1453:
1445:Symbolic Logic
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1407:
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1375:
1345:
1337:Priest, Graham
1329:
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1259:
1243:
1227:
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1135:
1132:
1131:
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1118:
1112:
1106:
1100:
1094:
1088:
1087:â Clavius' Law
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1077:
1037:
1034:
1031:
1028:
1004:
1001:
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967:
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913:
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904:
901:
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861:
852:is a model of
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838:
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832:
829:
826:
806:
786:
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747:
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741:
738:
735:
732:
712:
703:is a model of
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672:
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572:
566:
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545:
541:
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382:symbolic logic
378:Lewis argument
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309:symbolic logic
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223:Thoralf Skolem
158:, and similar
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145:
60:
58:
51:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2023:
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1929:Alfred Tarski
1927:
1925:
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1917:
1915:
1912:
1910:
1907:
1905:
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1900:
1897:
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1889:Gottlob Frege
1887:
1885:
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1877:
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1867:
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1834:Biconditional
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1822:
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1801:
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1793:Biconditional
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1727:modus tollens
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1717:Transposition
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1708:
1705:
1704:
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1590:Associativity
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215:Ernst Zermelo
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211:Gottlob Frege
208:
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199:inconsistency
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176:contradiction
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83:Find sources:
77:
73:
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61:This article
59:
55:
50:
49:
44:
40:
33:
19:
1899:Hugh MacColl
1874:Georg Cantor
1869:George Boole
1766:Introduction
1722:modus ponens
1690:
1650:Higher-order
1645:Second-order
1595:Distribution
1555:Truth tables
1469:
1463:
1444:
1438:
1419:
1400:
1394:
1384:
1378:
1366:. Retrieved
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1277:(18): 4023.
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1255:. Routledge.
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635:model theory
632:
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393:Proposition
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70:Please help
65:verification
62:
1844:Disjunction
1839:Conjunction
1824:Existential
1812:Elimination
1803:Disjunction
1798:Conjunction
1783:Existential
1640:First-order
1565:Truth value
1535:Quantifiers
1387:. Springer.
1368:January 14,
1241:Chapter 17.
1174:Duns Scotus
1091:Dialetheism
940:subcontrary
579:C. I. Lewis
396:Derivation
180:proposition
128:August 2020
2011:Principles
1995:Categories
1894:Kurt Gödel
1757:Absorption
1659:Principles
1545:Connective
1257:Chapter 7.
1193:References
1127:Trivialism
1054:statements
760:A fortiori
227:set theory
98:newspapers
1829:Universal
1788:Universal
1691:Explosion
1676:Bivalence
1605:Soundness
1550:Tautology
1540:Predicate
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671:Γ
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417:¬
414:∧
333:⊢
327:¬
283:explosion
172:statement
39:Quodlibet
1773:Negation
1600:Validity
1580:Logicism
1270:Synthese
1079:See also
1058:theorems
434:Premise
368:is true.
360:and not-
242:unicorns
184:negation
18:Ex Falso
1528:General
1405:p. 139.
1291:9276566
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293:called
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166:is the
112:scholar
1857:People
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1389:p. 47.
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1013:theory
993:, and
571:(4,3)
238:lemons
221:, and
162:, the
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1953:Works
1700:Rules
1287:S2CID
1214:(PDF)
1147:Latin
1134:Notes
1062:truth
1003:Usage
657:is a
390:Step
372:Proof
356:, if
291:logic
119:JSTOR
105:books
1630:Term
1474:ISBN
1449:ISBN
1424:ISBN
1370:2017
1318:ISBN
1052:its
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539:(2)
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352:and
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168:law
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