contradiction

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Fall2010

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TheParadoxesofEubulidesofMiletus(4thcenturyB.C.)

r

“WhenIlieandadmitthatIlie,doIlieorspeakthetruth?”1

adox

Astatementthatdirectlyorindirectlypredicatesfalhoodofitlfislogically

pathologicalintheworst,ntencesarecalled„liarntences.‟

Version1a:Thisntenceisfal.

ProofofParadoxicality

-Thatntence,likeeveryother,mustbeeithertrueorfal.

-Ifthentenceistrue,thatitisfal,and,

sincewhatitsaysistheca,,ifthentenceistrue,thenitfollowsthat

diction.

-Therefore(byreductioadabsurdum/~I),thentence„Thisntenceisfal‟cannotbetrue

and,so,mustbefal.

-Butifthepropositionisfal,thatitis

fal,and,aswhatitsaysisnottheca,itfollowsthatitisnotthecathatitisfal.

Therefore,,ifthentenceisfal,thenitfollowsthatitistrue.

Contradiction.

-Therefore(byreductioadabsurdum),thentence„Thisntenceisfal‟cannotbefal

and,so,mustbetrue,which(asweproved)itcannotbe…???!!!

Astheargumentshows,apropositionthatdirectlyorindirectlysaysofitlfthatitis

falmustbeeithertrueoffal(likeanyotherproposition),and,pathologically,cannotbe

either.

Version1b:Iamlying.

Version2a:Thefollowingntenceisfal.

Theprecedingntenceistrue.

1AulusGellius,AtticNights,vol.3,,.

Version2b:

Thentencein

BoxBistrue.

BoxA

Thentencein

BoxAisfal.

BoxB

fullandprecidetail(onthemodelofmypreviousdemonstration)

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thatthethreesituationsaregenuinelyparadoxical,i.e.,somestatement

mustbefalifitistrue,mustbetrueifitisfal,andnothirdoptionis

possible.

Inallfourvariations,thetruthofantencedirectlyorindirectlyimpliesitsown

falhood(whichisacontradiction),andthentence‟sfalsitydirectlyorindirectlyimplies

itsowntruth(whichisacontradiction).Byitlf,oneofthecontradictoryresultsmerely

wouldconstituteavalidreductioadabsurdum/indirectproofforthenon-contradictorydenial

oftheassumption(ofeithertruthorfalhood).But,bothcontradictoryresultstogether

generateineliminablecontradictoriness:wemustholdastruewhatwecannotholdas

true…andthemindboggles.(Suchparadoxesarealsocalled„antinomies.‟)

ewTestament

InhisEpistletoTitus,theApostlePaulcomplainsoftheCretansadding:2

Oneofthemlves,evenaprophetoftheirown,said,„TheCretansarealwaysliars,

evilbeasts,slowbellies.‟Thiswitnessistrue.

,then,thereisaCretanprophetwhoasrtsthattheCretansalwayslie,

whichimplies,ofcour,thattheCretanprophethimlfislying.3Now,that‟sfine;butin

addingthattheCretanprophetistellingthetruth,Paulbrutelycontradictedhimlf.

Proof

idesisaCretan.

idessaid,“TheCretansarealwaysliars.”

*tnessistrue,i.e.,whatEpimenidessaidistrue.

ore,“TheCretansarealwaysliars”istrue.[by(2)&(3)]

ore,theCretansarealwaysliars.[by(4)]

ore,Epimenidesalwayslies.[by(1)&(5)]

ore,Epimenidesliedwhenhesaid,“TheCretansarealwaysliars.”[by(6)]

ore,whatEpimenidessaidisfal,i.e.,thiswitnessisfal.[by(7)]

Contradiction:(3)&(8)!‟sasrtionthatEpimenidestoldthetruth

impliesthatEpimenideslied.

ThoughtheApostleemsnottohaverealizedit,weknowthathislf-contradiction

constitutesavalidreductioadabsurdum(orisitaconquentiamirabilis?)that(3)—his

asrtionthatthewitnessistrue—isfal.

HavewejustdisproventhethesisofBiblicalliteralism—everystatementintheBibleis

literallytrue?

2Titus1:tionedCretanprophetwasprobablyEpimenides,andtheLiarParadoxisoftencalled

theEpimenideanParadox.

3Bytheway,‟reboundtoinsultsomebodyifyou

mixthetwowordsup,sogetitstraight.(Originally,acretinwassomeonewhosufferedthecripplingmental

d„cretin‟comesfromtheSwissFrench„crestin’meaning

„Christian.‟Thatis,theunfortunates,thoughretardedanddeformed,arestillhumanbeings.)

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Itthenfollowsthatthewitnessisfal,i.e.,r,

hat,let‟sfindoutwhatthat

conclusiondoentail.

Argument

nessisfal,i.e.,whatEpimenidessaidisfal.[byProof]

idessaid,“TheCretansarealwaysliars.”[by(2)]

ore,„TheCretansarealwaysliars‟isfal.[by(a)&(b)]

ore,„SomeCretansarenotalwaysliars‟istrue.

Proof:,in

thelanguageofpredicatelogic,~(x)(…x…)iquivalentto(x)~(…x…)(bythe

quantifiernegationrule).

ore,thereisatleastoneCretanwhodoesnotalwayslie,i.e.,whotellsthetruth

onatleastoneoccasion.

[by(d)]

rethreewaysfor(e)tobetrue:

idestoldthetruthonthisoccasion,

thetruthonatleastoneotheroccasion,or

satleastoneotherCretanwhosometimestellsthetruth.

Argument:Ican‟tthinkofanyother.

g.(i)isfal.[by(a)]

nogroundsfordenyingeither(ii)or(iii).

Argument:ThefactthatEpimenidesliedontherelevantoccasionaloneimpliesnothing

aboutthetruthfulnessofhisotherasrtionsorotherCretans‟asrtions.

ore,either(ii)or(iii)(orboth)istrue.

So,itfollowsfromEpimenides‟havingspokenfallywhenhesaidthattheCretans

arealwaysliarsthatatleastoneCretanspoke,orwillspeak,trulyonatleastoneoccasion.

,I‟msureit‟strue(andyou

aretoo).

IftheonlythinganyCretaneverhadsaid,oreverwouldsay,were,“TheCretansare

alwaysliars,”thenwe‟dbeintrouble;butsincethatisn‟ttheca,we‟renot.

Thus,Paul‟sblunderisn‟,it‟snotreallyaversionoftheLiar,but

Iincludeitforitshistoricalandreligiousinterest.4

ioadAbsurdum?

4Ican‟tresistgiving,forthe

sakeofargumentlet‟sacceptthemedievalclaimthat„Godexists‟erthefollowing

argument:ore,rgumentvalid?(Anynecessarilytrue

propositionmayreplace„Godexists.‟)

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It‟s,

whatunrecognized,implicitassumptionhavewemadesuchthatwecantaketheLiar‟s

double-contradictoryconclusiontoconstituteareductioadabsurdumproofforitsfalhood?

Well,tobegin,we‟,perhapstheLiarprovesthere

,thisisridiculous!Wedidn‟

ntenceisrightthereonpage1!(Youdon‟tneedtoturnback;therumored-to-be-non-

existentntencewilloccurinmynextntence.)

We‟vealsoimplicitlyassumedthatthentence,„Thisntenceisfal‟actuallysays

something,i.e.,thatthentenceismeaningful,heLiar

provesthatthatisaperfectlygrammaticalntencethatexpressnoproposition,is

meaningless.5Thissuggestionfailsfortworeasons.

entence„Thisntenceisfal‟expressnoproposition,thenit

ore,thatntenceisneithertruenorfal,and,so,we

,considerthentence:

Thisntenceisnottrue.

Thisnewntenceisalsoaparadoxicalliarntence.

t.

Now,accordingtothepropodsolution,thenewntenceisalsomeaningless,and,

therefore,neithertruenorfal,and,thus,,thenewntencejustsaysthatitis

ore,itistrue(forwhatitsaysistheca)!Butifthentenceistrue,thenit

uently,theproposaldoesnotresolvetheparadox.

posalonlyeverhadachanceofresolvingversions1a&1bin

ions2a&2b,wherea

ntenceindirectlypredicatesfalhoodtoitlfbyascribingatruth-valuetoanother

ntence,it‟sclearthatthentenceisperfectlymeaningfulbecauyoueffortlessly

ll,itwasonlyasaresultofreadingitthatyouwentontoconsider

precilythentencetowhichitreferred.

Perhapsanotherexampleshowsthepointevenmoreclearly:

Thentenceprintedintheboxonp.160ofHowardDeLong’sAProfileof

MathematicalLogic(Addison-Wesley,1970)istrue.

Goahead;lookitup.(I‟llwait.)

Finally,we‟veimplicitlyassumedtheprincipleofbivalence:(i)thereareonlytwo

truth-values,viz.,thetrueandthefal,and(ii)everyntencemusthaveone,andonlyone,

ore,we‟veimplicitlyassumedthatourliarntenceshavetobeeithertrueor

a

5TheStoiclogicianChrysippussuggestedthisresolution.

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verycomplexandabstractreply,toocomplexandabstractforustogiveitmuch

considerationnow.6However,wecanatleastconsiderhowdifficultitistomakenof

thepossibilitythatantenceexpresssomethingcoherentaboutthewaythingsareand,yet,

itisneithertruenorfal,i.e.,ay,then

arethingssuppodtobe?

Items,then,thattheLiar,unliketheBarberortheLawCourt,isagenuineparadox.

eLiarParadoxShows

LikequestionsaboutRusll‟sbarbershavinghimlf,theLiarrevealsthesurprising

,lf-referenceclearlyisnot

paradoxicalingeneral,ersome

examples.

tementissimplyfal,butitisn‟t

paradoxical.(Indeed,astatementmustbecoherenteventobefal.)

2.(i)-(iv)ofthefollowinglf-referentialntencesarejusttrue;(v)-(vi)arejustfal:7

ntencecontainsfivewords.

ntencecontainsthirty-sixletters.

refourteenvowelsinthisntence.

ntenceiswritteninEnglish.

ntencecontainsprecilyfiftycharacters.

tkeinDeutscherSatz.[Translation:ThisntenceisnotinGerman.]

idescanlf-referentiallyasrtthelf-referentialntence,“ThentenceIamnow

utteringcontainsninewords,”andhisremarkwillbeunproblematicallytrue.

Still,theLiarshowsthatthoinstancesoflf-referenceinwhichastatement

directlyorindirectlypredicatesfalhoodofitlfarelogicallypathologicalintheworst,

mostmind-bogglingway:theygenerateineliminable,,the

Liaralsoshowsthataliarntencecanbeconstructedinanylanguagewiththeresourcesto:

oitsownntences,and

atetruthorfalsityofthontences.

Unhappily,allnaturallanguages(likeGujarati,Magyar,Frisian,etc.)satisfy(A)and

(B).Thatisverybad,for,aswe‟veen,liarntencesgenerateunpreventablecontradiction,

and,aswe‟velearned,everystatementvalidlyfollowsandisderivablefromacontradiction.

Insuchalanguage,therefore,theentirepracticeofdeductivereasoningaspartofthearch

,deductiverationalitycarriedoninEnglish(Basque,

Korean,etc.)isafarceandadelusion…unlesswecanescapefromtheLiar.

6But,ethehandout“DeviantLogic,No.3:Bočvar‟s3-ValuedLogic.”

7Antencelikethethatdescribesitsowncontentorconstructioniscalledan„autogram.‟

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dSolution

Onewaytoescapewouldbetosomehowguaranteethatliarntencescouldnever

ari.8In1931,thelogicianAlfredTarskiarguedthatwecouldblockthepossibilityofliar

ntencesifwe:9

lydistinguishobjectlanguagesfrommetalanguages,and

ctlanguagesothatitisonlyinthemetalanguagethatonecanconstructntences

aboutthentencesofanobjectlanguage,and,so,predicatetruthorfalsityoftho

object-languagentences.

Satisfyingtheconditionswouldpreventalanguagefromdoing(A)and(B).

Forexample,theEnglishlanguagewouldbecomeaninfinitequenceofdistinct

languages:English

1

,English

2

,English

3

,…whereeachsucceedinglanguagewouldbethe

meta-languageforthepreviousone.

English

1

wouldcontainalmostallofthevocabularyoftheEnglishwecurrentlyspeak,butit

wouldlackallmanticvocabulary:wordslike„ntence,‟„proposition,‟„asrtion,‟

„meaning,‟„lies,‟„true,‟,wecouldtalkaboutsnow(andshoes,ships,andaling

wax,etc.)inEnglish

1

byasrtingtheEnglish

1

ntence,„Snowiswhite.‟However,

English

1

wouldlacktheresourcesforustobeabletotalkaboutanyEnglish

1

ntences(or

anyotherntences)inEnglish

1

.

So,ifwewouldwanttotalkabouttheEnglish

1

ntence„Snowiswhite‟inordertosay,for

example,thatitistrue,thenwewouldhavetotreatEnglish

1

astheobjectlanguageandspeak

initsmetalanguageEnglish

2

.Amongitsvocabulary,English

2

hasthemantictermsneeded

totalkaboutntencesinEnglish

1

.OnlyinEnglish

2

,then,couldwesaythat„Snowiswhite‟

istrue,whichwewoulddobyasrtingtheEnglish

2

ntence,“TheEnglish

1

ntence„Snow

iswhite‟istrue.”

Now,English

2

onlyhasthemanticvocabularyneededtotalkaboutntencesinEnglish

1

,

notitsownntences(orthoofanyotherlanguageinthequence).Similarly,then,it

wouldbeonlyinEnglish

3

—themetalanguageofEnglish

2

—thatwecouldconstructa

ntencetoasrtthetruthofthatEnglish

2

ntence,viz.“TheEnglish

2

ntence,„The

English

1

ntence,“Snowiswhite”istrue‟istrue.”AndsoonthroughtheriesofEnglish

metalanguages.

Asnolanguageintheinfiniteriescouldsatisfyeither(A)or(B),Tarski‟s

stratificationbanishesallsame-levelntencereferenceand,thus,allntencelf-reference,

8Ofcour,,thatprogramwouldberidiculouslyextreme.

Almostallinstancesoflf-referenceareharmlessand,indeed,veryufulifnotpracticallyirreplaceable.

9In“TheConceptofTruthinFormalizedLanguages,”butforaveryreadableversionIrecommendTarski‟s

“TruthandProof”[ScientificAmerican,vol.194,no.6,pp.63-77].

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asungrammaticaland,therefore,uently,nontenceinanyofthe

languagescoulddirectlysayanythingatallaboutitlf.

hatversion1aoftheLiarcannotbeconstructedinTarski‟shierarchy.

Ofcour,onecouldutterorwritethewords„Thisntenceisfal‟or„Iamlying,‟but

theputativelf-referentialconstructionswouldbeutterlyungrammatical—likethe

Englishconstruction„Shoespointedlymysleepeat‟—and,hence,notntences;therefore,

notmeaningful;and,so,withouttruth-values.

Equally,indirectlylf-referringntenceswouldbeimpossibletoconstruct.

hatversion2aoftheLiarcannotbeconstructedinTarski‟s

hierarchy.

Therefore,nontenceinanylanguageintheriescould(directlyorindirectly)predicate

uently,noliarntencecouldbeconstructedinanylanguageinthe

ries,andtheparadoxwouldberesolved.

DespitethefactthatTarski‟sproposalwould,ifcarriedout,quashtheliarparadox,

theproposalhasfewadherents.(Yet,it‟simpossibletooverstatethehistoricalimportanceof

Tarski‟swork.)TheproblemisthatTarski‟sproposalistoocrude,foritbanishesnotonly

liarntencesbutall(directlyorindirectly)uently,itistoo

extreme,for,again,thevastmajorityoflf-referringntencesareperfectlybanal,andsome

arepracticallyirreplaceable—considertheindirectlylf-referringntence,„AllEnglish

ntencesbeginwithacapitalletter.‟Thereisnogoodreasontoproscribeallofthe

ntencesandverygoodreasonsnottoproscribesomeofthem.

ItisalsointerestingtonotethatsuccessfullycarryingoutTarski‟sproposalmakesthat

ki‟sstratificationwerecarriedout,thedescriptionofthe

hierarchyoflanguageswouldhavetobeimpossible,forthentencesofthatdescription

wouldhavetobelongtosomefinitelevelofthehierarchyinordertobemeaningful,butin

ordertodescribethewholehierarchytheycouldn‟tbelongtoanyofitslevels.