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A New Model for the Liar
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Luca Castaldo
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Abstract.AnewmodelforLt (thelanguageofarithmeticenhancedby F
pa
theunarytruthpredicateT)ispresented,whichextendsKripke’smini- i
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malfixedpoint. Thelatter,itwillbeargued,doesnotadequatelymodel o
thetruthpredicate,sincenodifferencebetweenLiarsandTruth-tellers s
o
canbemade.Thenewmodel,whichcontainsanextensionofKripke’sin-
fi
terpretationof alongwithanew4-valuedlogic,overcomesthisinade-
T a
quacy.Thegistofmyproposalisthat‘paradoxical’oughttobetreatedas
A
atruthvalue: Liarsentences,unlikeTruth-tellersentences,donotsim-
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plylackatruthvalue.Theydopossesone:theyareparadoxical. a
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Keywords.Formaltheoriesoftruth,Liarparadox,4-valuedlogic,Kripke.
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6
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*IwishtothankProfessorKarl-GeorgNiebergallforvaluablecommentsandsuggestions.
Copyright. (cid:13)CC (cid:13)BY: (cid:13)\$ (cid:13)C 2016LucaCastaldo.PublishedinItaly.Somerightsreserved.
Author.LucaCastaldo,[email protected]. 17
Received.7September2016.Accepted.29October2016.
LucaCastaldo ANewModelfortheLiar
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Introduction
i
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Itseemsthat
i
s
P Everysentenceiseither trueor untrue1. t
a
But,ifso,whatabouttheLiarsentencebelow? I
t
a
$ Thesentencemarkedwithadollarisuntrue.
l
i
$contradictsP,foritistrueif,andonlyif,itisuntrue. To(dis)solvetheproblem, a
n
Kripke(1975)proposestorejectPand, exploitingthe3-valuedlogiccalledStrong
a
Kleene,constructsapartialmodelforthetruthpredicate,wheresentenceslike$are
d
‘undefined’,i.e.theylackatruthvalue.
i
WithinKripke’smodel,however,alsotheso-calledTruth-teller F
i
e Thesentencemarkedwithaeuroistrue. l
o
lacksatruthvalue. Yet,$and eare,admittedly,verydifferent: thelattercancon- s
o
sistently bedeclaredtrueoruntrue;theformercannot. Anadequatemodelforthe
fi
truthpredicateoughttoaccountfortheirdiversity.
a
ThepurposeofthispaperistoputforwardanewresponsetotheLiarparadox,
A
whichextendsandimprovestheworkdonebySaulKripkeinhisseminalOutlineof n
aTheoryofTruth. a
The plan is as follows: after technical preliminaries in § 1 (including the con- l
i
structionoftheformalLiarsentence),Igoonin§2topresentanewmodelforthe t
i
truthpredicatealongwithanew4-valuedlogic,therebyproposingthenewresponse c
a
totheLiarparadox.Thefinalsection3examinesthepropertiesofthemodel,prov-
J
ingwhatIshallcall‘metalinguisticT-Schema’.
u
AlastremarkbeforeIbegin:InwhatfollowsIassumethereaderisfamiliarwith n
(i)Peanoarithmetic,(ii)thearithmetizationofsyntax,and(iii)Kripke’sOutlineofa i
o
TheoryofTruth2.
r
1Theeither...oristobereadhereasanexclusivedisjunction.
2ThereisanextensiveliteratureonKripke’sOutline. Amorephilosophicalandinformalintroduc- 7
tionisofferedbyBurgess,(2011). FormoreinformationonthemathematicalaspectsofKripke’scon-
:
structionsee,forexample,Fitting,(1986)andMcGee,(1991,§§4-5). Theaxiomatictheoryknownas 2
Kripke-Feferman(KF)wasfirstgivenbyReinhardt,(1986)andFeferman,(1991).Feferman,(1991)also
determinesitsproof-theoreticstrength. Cantini,(1989)givesamoredirectproof-theoreticanalysisof (
KFandsomeofitssubsystems.InKF,thepartialnotionoftruthadvancedbyKripkeisaxiomatisedin 2
classicallogic. Therefore,outerlogic(whatisprovable)andinnerlogic(whatisprovablytrue)ofthat
systemdifferssubstantially. HalbachandHorsten,(2006)(seealsoHorsten,2011,§9.5)haveproposed 0
aninterestingaxiomatisationinpartiallogic,creatingasystem,called“partialKripke-Feferman”(PKF), 1
withinwhichthetwologicscoincide. Inthatsystem,gapsbutnoglutsareadmitted. Halbach,(2014,
6
§16)proposesasystemthatadmitsboth.ForcriticaldiscussionsofKripke’spositionsee,amongothers,
Gupta,(1982,pp.30-37)andField,(2008,§3). )
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LucaCastaldo ANewModelfortheLiar
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1 The Formalised Liar
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1.1 TechnicalPreliminaries i
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t
TheobjectlanguageofthisworkwillbethelanguageofPeanoarithmetic(PA)ex-
a
tendedbytheunarytruthpredicateT.IshallcallthelanguageofPA,withoutT,Lpa; I
the extended language will be called Lt 3. As “official” logical vocabulary, I shall t
pa a
usetheexistentialquantifier∃,thenegationanddisjunctionsymbols¬,∨,andthe l
i
identitysymbol(cid:17). Asusual,however,abbreviationswillbeused. AstandardGödel a
numbering ofLt -expressionswillbeassumedthroughoutthework,withoutgo- n
pa
ingintodetails4.TheGödelnumber(orcode)ofaformulaϕisgn(ϕ),and(cid:112)ϕ(cid:113)isthe a
numeralofgn(ϕ). IshalldistinguishbetweennaturalnumbersandLt -numerals d
pa
i
exploitingboldfacedcharacters:thenaturalnumbersarewritten“0,1,2,...,n”(not
F
boldfaced)andtheLpta-numerals“0,1,2,...,n”(boldfaced),where“1,2,3...”ab- i
l
breviates“0(cid:48),0(cid:48)(cid:48),0(cid:48)(cid:48)(cid:48)...”.Formulaewithonefreevariableareindicatedbyϕ(vi);ϕ(t) o
denotesϕ[t/v ],i.e. theresultofsubstitutingt forv inϕ. Iwriteϕ ≡ ψtoindicate s
i i
o
thatϕandψarenamesofthesameformula.
fi
(cid:104)M,(E∞,A∞)(cid:105) is Kripke’sminimalfixedpoint(henceforth: MFP), and
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‘(cid:104)M,(E∞,A∞)(cid:105) |=sk ϕ’meansthat ϕistrueinMFP,accordingtotheStrongKleene.
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Furthermore,Ishallmakeuseofthefollowingmetalinguisticsymbols:
n
• ¬¬ for “non...”. a
l
• (cid:48) for “... or...”. i
t
• (cid:49) for “... and...”. i
c
• ⇒ for “if...,then...”.
a
• ⇔ for “... if,andonlyif,...”.
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• ∃∃ for “thereis...”. u
• \∀ for “forall...”. n
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1.2 λ ↔ ¬T(cid:112)λ(cid:113)
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TheDiagonalLemma5 is,asMcGee,(1991,p.24)putit,“acornerstoneofmodern 7
logic”. He even adds that “most of the results of [Truth, Vagueness, and Paradox] :
2
canberegardedascorollariestothisbasicresult”. InthissectionIshallexploitthe
(
typicaldiagonalconstruction,inordertoobtaintheformalisedliarantinomy.
2
3NoticethatwearejustextendingthelanguageofPA,notthetheory,i.e. wearenotaddingaxioms 0
forT,creatinganewtheory,sayPAT.Inaddition,wecanimposearestrictionontheinductionschema 1
toL -formulae,i.e.,aninstanceof
pa 6
(ϕ(0)∧∀vi(ϕ(vi)→ϕ(vi(cid:48))))→∀vi(ϕ(vi)) )
isanaxiom,onlyifT doesnotoccurinϕ.
4See,forinstance,Boolos,Burgess,andJeffrey,(2007)andSmith,(2013).
5OrFixedPointLemma,orSelf-ReferentialLemma.
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LucaCastaldo ANewModelfortheLiar
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BeforeIbegin,theconceptofdiagonalizationofaformulamustbeintroduced:
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Thediagonalizationofϕistheexpression∃v0(v0 (cid:17) (cid:112)ϕ(cid:113)∧ϕ).
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s
Evenifthisnotionmakessenseforarbitraryexpressions,itisofmostinterestinthe t
a
caseofaformulaϕ(v0)withjustonevariablev0free.Sinceanexpressionoftheform
I
ϕ(t) is equivalent to ∃v0(v0 (cid:17) t ∧ ϕ(v0)), the diagonalization of ϕ(v0) is equivalent t
a
to ϕ((cid:112)ϕ(cid:113)). That is: the diagonalization of a formula ϕ(v0) is true (in the standard
l
interpretation)if,andonlyif,itissatisfiedbyitsowncode. i
a
Thereisalsoarecursivefunction that,whenappliedtotheGödelnumber
diag n
ofaformula, yieldstheGödelnumberofitsdiagonalization. Thatistosay: ifthe
a
codeofaformulaϕisn andthecodeofitsdiagonalizationism,thendiag(n) = m. d
Amoreformaldefinitionis: i
F
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
diag(n)= gn ∃v0(v0 (cid:17) (cid:63)num n (cid:63)gn ∧ (cid:63)n(cid:63)gn ), il
o
s
where(cid:63)and num represent,respectively,theconcatenationandthenumeralfunc-
o
tions,bothrecursive6.
fi
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Lemma1.1. (THEFORMALISEDLIAR)ThereisaLt -sentenceλ,suchthat
pa
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PA(cid:96) λ ↔¬T(cid:112)λ(cid:113) n
a
Proof. SincePArepresentseveryprimitiverecursivefunction,diag isrepresentable li
t
in PA. Let Diag(v0,v1) be a formula representing diag, so that for any a and b, if i
c
( )= ,then
diag a b
a
PA(cid:96) ∀v1(Diag(a,v1)↔v1 (cid:17) b) (1)
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u
DiagisacomplexL -formula,not containingthenewpredicateT.
pa n
Letnow β(v0)betheformula i
o
∃v1(Diag(v0,v1)∧¬T(v1)) (β(v0)) r
7
Intuitively, β(v0)saysthatthediagonalizationofa formulaisnottrue,withoutyet :
2
sayingwhichformula.Let’snowconsiderthediagonalizationof β(v0),andlet’scall
itλ: (
2
∃v0(v0 (cid:17) (cid:112)β(cid:113) ∧ β(v0)) (λ) 0
6Theconcatenationfunction(cid:63)issuchthat,ifsandtarethecodesoftwoexpressions,thens(cid:63)tisthe 1
codeofthefirstexpressionfollowedbythesecond.Thenumeralfunctionnummapseachntothecode 6
ofthenumeraln.Thefunctiondiag couldhavebeendefinedmorepreciselybyfirstshowingthatalso
thelogicaloperationsofconjunctionandexistentialquantificationarerecursive.Formoreinformation )
seeBoolos,Burgess,andJeffrey,(2007,p.221,§15).
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Inothersymbols,λis
i
(cid:0) (cid:1) v
∃v0 v0 (cid:17) (cid:112)β(cid:113)∧∃v1(Diag(v0,v1)∧¬T(v1)) i
s
t
Thisislogicallyequivalentto β((cid:112)β(cid:113)),i.e.theresultofsubstituting(cid:112)β(cid:113)forv0in β(v0): a
I
t
∃v1(Diag((cid:112)β(cid:113),v1)∧¬T(v1)) (2) a
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Reading(2)inEnglish,wegetsomethinglike:“thereisanumberthathastwoprop- i
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erties:first,itisthecodeofthediagonalizationof β(v0);second,itisnotelementof n
theextensionofT”. Or,moreintuitively: “thediagonalizationof β isnottrue”. In- a
terestingenough,thediagonalizationof βispreciselyλ. Accordingly, λ islogically d
equivalenttoasentencethatsaysthatλisnottrue. i
F
Wehavethusfarconstructed,withintheformallanguageLt ,asentencesaying
pa i
ofitselfthatitisnottrue7. Thenextstepconsistsinproving,withinPA,something l
o
aboutthissentence.Sinceλislogicallyequivalentto(2),wehave: s
o
PA(cid:96) λ ↔∃v1(Diag((cid:112)β(cid:113),v1)∧¬T(v1)) (3) fi
a
Wedonotknow,whetherλ isatheoremof PA.Wedoknow,however,thatitisthe A
(cid:0) (cid:1)
diagonalizationof β,andhencediag gn(β) = gn(λ).Fromthis,by(1),follows n
a
PA(cid:96) ∀v1(Diag((cid:112)β(cid:113),v1)↔v1 (cid:17) (cid:112)λ(cid:113)) (4) li
t
i
Thatis,(cid:112)λ(cid:113)istheonlyclosedtermsatisfyingtheopenformulaDiag((cid:112)β(cid:113),v1)8.Sim- c
a
plelogicthengives,from(3)and(4):
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u
PA(cid:96) λ ↔∃v1(v1 (cid:17) (cid:112)λ(cid:113)∧¬T(v1)) (5) n
i
Since∃v1(v1 (cid:17) (cid:112)λ(cid:113)∧¬T(v1))isequivalentto¬T((cid:112)λ(cid:113)),wehave: o
r
PA(cid:96) λ ↔¬T(cid:112)λ(cid:113) (cid:3) 7
:
2
ThisistheformalcounterpartoftheparadoxicalLiarsentence: asentencethat
isprovablyequivalenttoasentencesayingthatitscodeisnotelementoftheexten- (
2
sionofthetruthpredicate. “Butnotethat[λ]isproducedbyasimplediagonaliza-
0
tionconstruction[...];andtheconstructionyieldsatheorem,notaparadox”(Smith,
1
7Whetherthissentence“saysofitselfthatitisnottrue”isnotasobviousasonemightthink.Foran 6
insightfuldiscussionaboutself-referenceinarithmetic,seeHalbachandVisser,(2014a,b). )
8Notethat(4)isequivalenttotheconjunctionofPA(cid:96) Diag((cid:112)β(cid:113),(cid:112)λ(cid:113))and
PA(cid:96) ∀v1(¬(v1(cid:17)(cid:112)λ(cid:113))→¬Diag((cid:112)β(cid:113),v1)).
21
LucaCastaldo ANewModelfortheLiar
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2013,p.198).The“formalLiarparadox”arisesifwewantourtheoryoftruthtoprove
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theT-Schemaϕ↔T(cid:112)ϕ(cid:113)forallsentencesϕ ∈Lt . v
pa
Yet,thisisbynomeansnecessary.Kripke(1975)proposestogiveupthebeloved i
s
T-Schema, constructing a partial model for the truth predicate, where both λ ↔ t
a
T(cid:112)λ(cid:113)andλ ↔ ¬T(cid:112)λ(cid:113)areneithertruenoruntrue,i.e. theyareundefined. Asindi-
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catedintheINTRODUCTION,IassumethereaderbeingfamiliarwithKripke’sOutline. t
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Iomitcompletelythepresentationofhiswork.HereIshalljuststatetwoimportant
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featuresof MFP,describedbyKripke(1975,p. 708)as“probablythemostnatural i
a
modelfortheintuitiveconceptoftruth”.
n
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Fact1.2. MFPverifiesthemetalinguisticT-Schema,i.e.:forallsentencesϕ ∈Lt ,
pa
d
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(cid:104)M,(E∞,A∞)(cid:105) |=sk ϕ ⇔ (cid:104)M,(E∞,A∞)(cid:105) |=skT(cid:112)ϕ(cid:113) F
(cid:104)M,(E∞,A∞)(cid:105) |=sk ¬ϕ ⇔ (cid:104)M,(E∞,A∞)(cid:105) |=sk ¬T(cid:112)ϕ(cid:113) i
l
o
Fact1.3. InMFPboththeLiarsentenceλandtheTruth-tellerτareundefined. s
o
fi
2 Towards a New Model a
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InthissectionIshallputforwardthenewresponsetotheLiarantinomy. Thegist
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ofmyproposalisthat‘paradoxical’oughttobetreatedasatruthvalue. Liarsen-
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tences,accordingtothepresentsuggestion,donotsimplylack atruthvalue. They l
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dopossessone:theyareparadoxical. AshasbeennotedintheINTRODUCTION,the t
i
triggerofmyconsiderationswillbethedifferencebetweentheLiarandtheTruth- c
a
teller.Themaingoalistoconstructamodelwithinwhich(i)thedifferencebetween
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paradoxicalandunparadoxicalstatementsisdetected,and(ii)everyLt -sentence
pa u
ϕhasthesametruthvalueasT(cid:112)ϕ(cid:113)(that’sthemetalinguisticT-Schema). n
Theplanisasfollows: thenextsubsectioncontainsphilosophicalarguments: I i
o
trytoexplainwhyKripke’sproposalisnotsufficientlysatisfactoryasresponsetothe
r
Liar,andwhy,moregenerally,hisMFPdoesnotadequatelymodelthetruthpredi-
7
cate.Inaddition,Ishallexplainwhy‘paradoxical’shouldbetreatedasatruthvalue.
:
Theremainingsubsectionscarryoutthisideaformally. 2
(
2
2.1 Why?
0
WithoutaimingtobecensorioustowardKripke’sproposal,butratherwiththeinten- 1
6
tionoffurtherdevelopinghiselegantideas,Ithinkthathisconstructionsuffersfrom
)
twoinadequacies,whichcan(Ihope)beremoved. Afirst,minorproblemhispro-
posalisconfrontedwithisthatusingthevalue‘undefined’forparadoxicalsentences
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LucaCastaldo ANewModelfortheLiar
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doesnotseementirelyadequate9–atleastifweadheretotheoriginalmeaningat-
i
tributedtoitbyKleene,(1971). Asecond,majorproblemisthatKripke’sMFPdoes
v
notmodelthetruthpredicateinasatisfactoryway. Letmeelaboratethesereasons i
s
inturn. t
a
InbothKleene’slogics(theStrong andtheWeak)10,thevalue‘undefined’(u)is
I
not treatedon a pairwith ‘true’ (1)and ‘false’ (0): u is nota third truth value11; it t
a
onlyrepresentsformallythelackoftruthvalues.Secondly,andmoreimportantfor
l
the present purposes, u is open to “arbitrariness for a classical value”: undefined i
a
sentencescanturnouttobetrueorfalse,orcanarbitrarilybedeclaredtrueorfalse.
n
Lesstersely: asiswellknown,Kleeneintroducedthenewlogicsinthestudyof
a
partialrecursivefunctions,speakingofwhichhewrites(Kleene,1971,p.334):
d
i
ifwhenQ(x)isu,Q(x)∨R(x)receivesthevalue1,thedecisionmust(inthe
F
generalcase)havebeenmadeinignoranceabout ( ),andinthefaceof
Q x i
l
thepossibilitythat,atsomestageinthepursuitofthealgorithmfor ( )
Q x o
laterthanthelastoneexamined,Q(x)mightbefoundtobe1ortobe0. s
o
Hegoeson(ibid.,p.335)toobservethat1,0,andu“mustbesusceptibleofanother fi
meaningbesides(i)‘true’,‘false’,‘undefined’,namely(ii)‘true’,‘false’,‘unknown(or a
valueimmaterial)’.Here‘unknown’isacategory,whosevalueweeitherdonotknow A
orchooseforthemomenttodisregard;anditdoesnotthenexcludetheothertwo n
possibilities‘true’or‘false’”12. a
Myquestionnowis:areparadoxicalsentencesliketheLiaropentothesamekind l
i
t
ofarbitrarinessforaclassicalvalue? Mightthesesentencesturnouttobetrue,or
i
c
false? Canwearbitrarilyassignthematruthvalue? Hardlyso. Thesesentencesare
a
paradoxicalpreciselybecausetheassumptionthattheyaretrue,orfalse,generates
J
inconsistencies. u
Asalreadyremarked,thisisaminorproblem.Onemightquiteeasilychangethe n
interpretationofuandadjustitaspleasedtoparadoxes13. Nonetheless,themajor i
o
problemcontinuestoflutter:MFPdoesnotmodelthetruthpredicateadequately,as
r
itdoesnotaccountforthedifferencebetweenLiarandTruth-teller–thisdifference
7
havingitsrootsinapeculiarityof .Letmemakethisclaimprecise,byfirstrepeating
T
:
2
thatthedifferencebetween
9Someauthorshavesuggestedthatparadoxesareoverdefined(bothtrueandfalse),andnotunder- (
defined(neithertruenorfalse).See,forexample,Dunn,(1969,1976)andPriest,(1979). 2
10SeeKleene,(1971,§64).
0
11Kripke(1975,fn18)stressesthesamepoint.
12Otherphilosophershavealsosuggested,asreportedbyvanFraassen,(1966,pp.482-483),thatsen- 1
tencesthatarenormallytakentobeneithertruenorfalse(forinstance“thekingofFranceiswise”)“are 6
‘don’tcares’forordinarypurposes,andthereisthereforenoreasonwhyweshouldnotarbitrarilyassign
themsometruthvalue”. )
13Forexample,Priest,(1979)introducedtheso-called‘LogicofParadox’(LP),whichhasthesametruth
tablesastheStrongKleene,buttheinterpretationofthethirdvalueis‘trueandfalse’,anditis,moreover,
adesignatedvalue.
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$ Thesentencemarkedwithadollarisuntrue.
i
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and
i
s
e Thesentencemarkedwithaeuroistrue.
t
a
isthatonecanmoreorlessarbitrarilydeclare etrue,oruntrue,withoutstumbling
I
onlogicalissues;onthecontrary,theonlywaytodeclare$true,oruntrue,requires t
a
theabandonmentofanimportantprincipleabouttruth, i.e. thatnothingisboth
l
trueanduntrue.Therefore,doingnothingmoreandnothinglessthandescribinga i
a
simplestateofaffairs,wecanstatethat n
a
(Fact) thetruthpredicateissuchthat,therearesentencesthatcanconsistentlybein
d
itsextensionorinitsanti-extension;therearesentencesthatcannot.
i
Everytheoryoftruthoughttotake(Fact)intoaccount14. F
i
As a matter of fact, in a substantial portion of the Outline, Kripke shows how l
o
tocategorisedifferentkindsofsentence. Asentenceisparadoxical, e.g., “ifithas
s
notruthvalueinanyfixedpoint”(Kripke,1975,p.708)15.Asentenceisungrounded o
andunparadoxical,ifithasatruthvalueinsomefixedpoint,differentfromthemin- fi
imalone–anexamplebeingtheTruth-teller.Heevenemphasisesthat“theassign- a
mentofatruthvalueto[theTruth-teller]isarbitrary”(ibid.,p.709)16. A
Thereadermightthereforeask,whatthepointofmyobjectionis–Kripkedoes n
offer a way to distinguish between paradoxical and simply undefined sentences; a
l
Kripke does account for the difference between Liars and Truth-tellers. He surely i
t
does. Butthepointisthatonlywithinthemetatheory onecanimplementthatdis- i
c
tinction. Onlywithinaninformal“metamodel”ofthevariousfixed-pointmodels
a
areweabletodifferentiatebetweenparadoxicalandunparadoxicalsentences.The
J
minimalfixedpoint,which(repetitaiuvant)isdescribedbyKripkeas“probablythe u
n
mostnaturalmodelfortheintuitiveconceptoftruth”(ibid.,p.708),doesn’tseethe
i
difference:inthismodeltheLiarandtheTruth-tellerarebothsimplyundefined.
o
IfIamright,andifthedifferencebetween$and e isduetothepeculiarityof r
T expressedby(Fact)17,thenIbelieveitisjustifiedtomaintaintheKripke’smodel
7
14AsimilarpointismadebyGuptaandBelnap,(1993,p.100):“TheessentialthingabouttheLiarap- :
pearstobeitsinstabilityundersemanticevaluation: Nomatterwhatwehypothesizeitsvaluetobe, 2
semanticevaluationrefutesourhypothesis. Atheoryoftruthoughttocapturethisintuition. Itshould
provideawayofdistinguishingsentencesthatexhibitthisbehaviourfromthosethatdonot,anditshould (
explainwhycertainsentencesbehavethisway”. 2
15Kripkeconsidersonlyconsistentfixedpoint,i.e.fixedpointwhereE∩A=∅.SodoI. 0
16Halbach,(2014,p.196)observesthat“Kripke’smaincontributionwasnotsomuchtheconstruction
ofthesmallestfixedpoint[...]butratherhisclassificationofthedifferentconsistentfixedpointsandthe 1
discussionoftheirusefordiscriminatingbetweenungroundedsentences,paradoxicalsentences,andso 6
on”.
17ArethereanyotherpredicateswhichareakintoT inthisrespect?Oneisthereforsure:thepredicate )
“isheterological”introducedbyKurtGrellingandLeonardNelson(seeGrellingandNelson,1907).Ina
parallelwork,IamtryingtoextendthesolutionpresentedheretohandletheGrelling-Nelsonparadox
too.
24
LucaCastaldo ANewModelfortheLiar
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isnotquiteaccurate. Ibelieveitisjustifiedtomaintainthatweshouldtrytofinda
i
waytoimproveit.
v
Some suggestions have already been made: it is what McGee, (1991, pp. 110- i
s
111)callsa‘liberalisationofKripke’sconstruction’,whichallowsextensionandanti- t
a
extensionof tooverlap. Thisrequiresareplacementofa3-valuedlogicwitha4-
T
I
valuedlogichavingbothtruthvaluegapsandtruthvaluegluts.Thelogical-mathe-
t
a
maticalpropertiesofsuchaliberalisationhavebeenstudiedbyWoodruff,(1984)18.
l
Suchsystemsareofgreatinterestfordialetheists19. Butforthosewhodonotbe- i
a
lievethatsomethingcanbeeverbothtrueandfalse,theyareoflittlehelp.Iamone
n
of those, and additionally I really do not believe that declaring the Liar both true
a
andfalsecanrepresentanykindofsolutiontotheparadox.Itseemstomethatthe
d
paradoxispreciselythatsomesentenceshouldbebothtrueandfalse.Ican’tdigress, i
F
however,todiscussdialetheism–intriguingthoughitmightbe.
i
l
o
2.2 How?
s
o
AlthoughIamnotanadvocateofdialetheism,IsubscribeVisser’swords,whenhe
fi
saysthat“[o]neattractivefeatureoffourvaluedlogicforthestudyoftheLiarParadox
a
isthepossibilityofmakingcertainintuitivedistinctions[thatis:thedistinctionbe-
A
tweenLiarsandTruth-tellers. L.C.]withinonesinglemodel”(Visser,1984,pp.181- n
182).AndthatiswhyIamabouttointroduceanew4-valuedlogic,whosevaluesare:
a
true,false,paradoxical,andundefined.“Why‘paradoxical’?”–thereadermightask. l
i
Toproperlyanswerthisquestion,Ifirstneedtointroducetheideaunderlyingthe t
i
newinterpretationof . c
T
a
We all agree (I venture) that an adequate interpretation of the truth predicate
J
oughttohaveanextension andananti-extension . Now,since(i)Idonotwant
E A u
LiarsentencestosimplylayoutsideE ∪AwithTruth-tellersentences,andsince(ii) n
I do not wantE and A to overlap, I propose to extend Kripke’s interpretation ofT i
o
by adding a third set to it, which will contain those (codes of) sentences that, as
r
statedin(Fact),cannotconsistentlybecontainedinE orinA. Ishallcallthisthird
7
set(duetolackofimagination)X. Inparticular: (E,A,X)willbetheinterpretation
:
ofT, theinterpretationofL remainingasbefore, i.e. weletM bethestandard 2
pa
interpretationofLpa.Consequently,(cid:104)M,(E,A,X)(cid:105)willbetheinterpretationofLpta (
2
with,informally:
0
(i) E = {gn(ϕ) | ϕistrue};A = {gn(ϕ) | ϕisuntrue};X = {gn(ϕ) | ϕisparadoxical}; 1
6
18SeealsoVisser,(1984).
19Dialetheism,roughly,istheviewthattherearetruecontradictions,andafullexpositionofitwould )
involveagreatdealoftechnicalmaterialthatwewillnotgointohere.SeePriestandBerto,(2013)foran
overview.
25
LucaCastaldo ANewModelfortheLiar
R
(ii) E ∩A =∅,E ∩X =∅,A∩X =∅;
i
v
(iii) E ∪A∪X (cid:44)(cid:78).
i
s
Andsonowthequestionarises,whattruthvaluesentenceslikeT(cid:112)ϕ(cid:113)shouldhave, t
a
whenever gn(ϕ) ∈ X. The answer suggested here, unsurprisingly, is that they are
I
paradoxical. Hence,thereasonwhyIamproposingtotake‘paradoxical’asatruth
t
a
valueisthatIthinkthebestwaytoformalise(Fact)ishavingathreefoldinterpre-
l
tationofT,withextension,anti-extension,andparadox-set.Accordingly,exactlyas i
a
thoughwewereallowing and tooverlap,afourthtruthvalueisneeded.Andno
E A n
valuebut‘paradoxical’seemstoproperlysuittheparadox-set .
X a
Now,tocarryoutthisprojectformally,thereareaboveallthreethingstobedone:
d
first, we need a new 4-valued logic to handle the value ‘paradoxical’; second, we i
needrulesdeterminingwhetherasentenceistrue,false,paradoxical,orundefined F
i
inthepartialmodel(cid:104)M,(E,A,X)(cid:105);third,weneedaformaldefinitionof(E,A,X). l
o
s
2.3 TheNewLogic o
fi
2.3.1 TruthValuesandtheirStructure
a
Let(cid:67)betheclassofconnectivesofclassicalpropositionallogic. Thenew4-valued A
logicisdefinedbythestructure: n
a
W = {1,0,p,u} li
t
D = {1} i
c
C = {f |c ∈(cid:67)} a
c
J
whereW isthesetoftruthvalues(true,false,paradoxical,undefined),Dthesetof u
n
thesoledesignatedvalue,Cthesetoftruthfunctions:foreveryconnectivec ∈(cid:67), f
c
i
isthecorrespondingtruthfunction.Thatis:ifc ∈ (cid:67)isann-placeconnective,f isa o
c
n-placefunctionwithinputsandoutputsinW. r
As usual, one might order the element of W by the relation ≤. Since u repre-
7
sentsthelackoftruthvalues,wewillhave: u ≤ 1;u ≤ 0;u ≤ p. Thedecisiontobe :
2
madeconcernsthenewvaluep.Therearethreepossibilities.Onemightarguethat
‘paradoxical’representssomesenseof‘overdefined’,inwhichcasewewouldhave (
2
1 ≤ p,0 ≤ p. Oronemightsaythat, likeu,p standsforanothercaseof‘underde- 0
fined’,inwhichcasewewouldhavep ≤ 1andp ≤ 0.Alternatively,onemightsay,as 1
Ishalldohere,thatitisneither‘overdefined’,nor‘underdefined’,whencewehave: 6
)
1,0,andparenotcomparable.
ThisyieldsastructureP =(cid:104)W, ≤(cid:105),whichcanbepicturedthus:
26