Rivista Italiana di Filosofia Analitica Junior Vol. 7, n. 2

46 Pages·2017·1.68 MB·Italian

Patrocinatadal2011dalla ISSN2037-4445 (cid:13)CC
SocietàItalianadiFilosofiaAnalitica http://www.rifanalitica.it
R
i
v
i
s
t
a
I
t
a
l
Rivista Italiana di Filosofia Analitica Junior i
a
Vol. 7, n. 2
n
a
d
Indice i
F
i
l
o
Editoriale
s
StefanoCanali,PietroAngeloCasati
o
Lavoriincorso............................................................1 fi
a
Articoli A
DavideEmilioQuadrellaro n
EpistemicLogicandtheProblemofEpistemicClosure .................... 3 a
l
LucaCastaldo i
t
ANewModelfortheLiar ................................................17 i
c
a
J
u
n
i
o
r
7
:
2
(
2
0
1
6
)
Copyright. (cid:13)CC (cid:13)BY: (cid:13)\$ (cid:13)C 2016PubblicatoinItalia.Alcunidirittiriservati.
i
Patrocinatadal2011dalla ISSN2037-4445 (cid:13)CC
SocietàItalianadiFilosofiaAnalitica http://www.rifanalitica.it
R
i
v
i
s
t
a
I
t
Lavori in corso a
l
i
a
Stefano Canali, Pietro Angelo Casati
n
a
d
Cari lettori, è con piacere che apriamo la pubblicazione di RIFAJ Vol.7 N.2. Si
i
trattadiun’uscitaunpo’particolare,legataadalcunenovitàriguardoalfuturo F
i
delprogettoRIFAJ.
l
o
InseiannidiattivitàlaRivistahamantenutounapresenzacostante,pubbli-
s
candopuntualmenteduenumeriall’anno, ampliandoilraggiodelleiniziative
o
correlateecoinvolgendounmaggiornumerodicollaboratorisianellaRedazio- fi
nesianelComitatoScientifico.Hadatoatantilapossibilitàdimettersiingioco a
inprimapersonaeperlaprimavolta.Moltidinoisonopassatidalaureatrien- A
naleamagistraleedottorato,spostandosinelfrattempoingiroper(quasi)tutto n
ilmondo. a
l
Datalastabilitàraggiunta,pensiamosiagiuntoilmomentoperdedicarcial-
i
t
losviluppodialcunepotenzialitàinespresse,cosìdaportareilprogettoaduno
i
c
stadiosuccessivo.NeiprossimimesiRIFAJandràincontroadunaprofondarivi-
a
sitazione,voltaadunacrescitainterminidiqualitàscientifica,maturitàeattività
J
correlate.Inquest’otticaabbiamodecisodipubblicareilpresenteeilprossimo u
fascicoloconunnumeroridottodicontributi,cosìdainvestiretempoedener- n
i
gienellaristrutturazionedelprogetto.Saràperciòassenteilconsuetocontorno
o
direcensioni, intervisteereport, nonchéFirmad’Autore edEx-Cathedra, soli- r
tamentepresentineinostrinumeritematici. Inparticolare,ilvolumecorrente
7
presentaduearticoli,accomunatidalfattodiproporrerevisionidiprospettive
:
2
inauguratedaSaulKripke.
In “Epistemic Logic and the Problem of Epistemic Closure”, Davide Emilio (
2
Quadrellaropresentaunalogicadellaconoscenzaalternativaallelogichepropo-
0
sizionalimodaliditipokripkeano. L’intentodell’autoreèdiservirsidei“mondi 1
impossibili”perevitareilcompromessoconilcontroversoprincipiodichiusura 6
)
epistemica.
In“ANewModelfortheLiar”,LucaCastaldopresentaunnuovomodelloperil
linguaggiodell’aritmeticadiPeanoconl’aggiuntadelpredicatounariodiverità.
EstendendoilpuntofissominimodiKripkeeimpiegandounaparticolarelogi-
caaquattrovalori,l’autoreintendesopperireadun’inadeguatezzadelmodello
Copyright. (cid:13)CC (cid:13)BY: (cid:13)\$ (cid:13)C 2016StefanoCanali,PietroAngeloCasati.Pubblicatoin
Italia.Alcunidirittiriservati. 1
Autori.StefanoCanali,[email protected],
[email protected].
StefanoCanali,PietroAngeloCasati Lavoriincorso
R
kripkeano,chenonconsentedidistingueretraicosiddettimentitoriequelliche
i
potremmochiamareassertori. v
EntrambigliarticolivertonodunquesullaLogica,inlineaconquellocheera i
s
statopresentatocomeunoSpecialIssuediLogica. Tuttavia,datol’altonumero t
a
dipotenzialicontributi,laricchezzadeldibattoelaquantitàequalitàdieventie
I
pubblicazionisultema,crediamosianecessariocostruireunnumerotematico t
a
piùcorposo. Cifaquindipiacereanticiparefindasubitochestiamolavoran-
l
do alla costruzione di un nuovo Special Issue di Logica, con uscita prevista a i
a
novembre2017, percuiabrevediffonderemounCallforPapersandReviews. n
Inattesadellegrandinovità,invitiamoadinviarecontributi,sottoporrenuove a
proposteeleggereiprossiminumeri. d
Cogliamoinfinel’occasioneperringraziaretutticolorochecihannoscritto, i
F
lettoesupportatoinquestiseianni.
i
Restatesintonizzati1. l
o
s
o
fi
a
A
n
a
l
i
t
i
c
a
J
u
n
i
o
r
7
:
2
(
2
0
1
6
)
1Apropositodi“sintonizzazione”,viinvitiamoaseguircisullapaginaFacebookdiRIFAJ(face-
book.com/RifanaliticaJun),chegrazieall’amministrazionediDarioMortinièdecisamentepiùatti-
vacheinprecedenzaediventasemprepiùriccadisegnalazioni,linknotevoli,intervistevariegate,
spuntiinteressanti,nonchédilettevolimeme.
2
Sponsoredsince2011bythe ISSN2037-4445 (cid:13)
CC
ItalianSocietyforAnalyticPhilosophy http://www.rifanalitica.it
R
i
v
i
s
t
a
I
t
Epistemic Logic and the Problem of a
l
Epistemic Closure i
a
n
*
Davide Emilio Quadrellaro
a
d
i
F
Abstract.Thispaperarguesthatpropositionalmodallogicsbasedon i
l
Kripke-structurescannotbeacceptedbyepistemologistsasamini- o
malframeworktodescribepropositionalknowledge. Infact,many s
o
authorshaveraiseddoubtsoverthevalidityoftheso-calledprinci-
fi
pleofepistemicclosure,whichisalwaysvalidinnormalmodallogics.
a
Thispaperexamineshowthisprinciplemightbecriticizedanddis-
A
cusses one possible way to obtain a modal logic where it does not
n
hold,namelythroughtheintroductionofimpossibleworlds..
a
l
i
t
i
c
a
J
u
n
i
o
r
7
:
2
(
2
0
Keywords.Epistemology,EpistemicLogic,EpistemicClosure,Rantala 1
Semantics,LogicofKnowledge,ImpossibleWorlds. 6
)
*IwishtothankGregSax,GianmarcoDefendiandthreeanonymousrefereesforcommentsand
suggestions.
Copyright. (cid:13)CC (cid:13)BY: (cid:13)\$ (cid:13)C 2016DavideEmilioQuadrellaro.PublishedinItaly.Some
rightsreserved.Author.DavideEmilioQuadrellaro, [email protected]. 3
Received.18August2016.Accepted.19October2016.
D.E.Quadrellaro EpistemicLogicandtheProblemofEpistemicClosure
1 Introduction
R
i
Thepurposeofthisarticleistodescribeaminimallogicofknowledgewhichcan v
beusedbyepistemologistswithdifferentphilosophicalorientations.Afirstway i
s
toproceedisdescribingamodallogicbasedonaKripke-semantics,specifying t
a
howtheaccessibilityrelationshouldberestrictedinordertorepresentknowl-
I
edge.However,itisnotdifficulttoprovethatthisstandardformalepistemolog-
t
a
icalanalysisimpliesthevalidityoftheprincipleofepistemicclosure,namelyof
l
thefactthat,ifonebothknowsthatpandthatifpthenq,thenhe/shealsoknows i
a
that . Thisprinciple,however,hasbeenobjectofcriticismandobjectionsby
q n
someepistemologists. Therefore, ifwearelookingforageneralmodallogical
a
frameworkthatcanbeusedbyphilosopherswithdifferentorientations,wehave
d
toconstructaformalsystemwheretheclosureprincipledoesnothold.Aninter-
i
estingwaytoproceedisworkingwiththesemanticswhichhasbeendeveloped F
bylogicianstoaccountfortheparadoxofthelogicalomniscience.Infact,ifwe i
l
introducethe“impossibleworlds”andweconstructaRantala-semanticsbased o
s
onthem,weobtainaweakerlogicwheretheclosureprincipledoesnothold.
o
InthefirstpartofthisarticleIpresentthemodallogicT,whichisgenerally fi
consideredtheminimalformalsystemforthelogicofknowledge.FirstlyIintro- a
ducethesyntaxandthesemanticsofmodallogic,secondlyIcharacterizehow
A
the accessibility relationRa has to be restricted in order to obtain the logic T. n
InthesecondpartIprovethattheprincipleofepistemicclosurefollowsfrom
a
TandItrytounderlinesomecriticalaspectsofit. InthethirdpartIintroduce l
i
an alternative logic for knowledge where the closure principle does not hold, t
i
namelyamodallogicwithimpossibleworldsandaRantala-semantics.Finally, c
a
inthefourthpart,Ievaluatethisproposal,tryingtounderlinebothupsidesand
J
downsidesofit.
u
n
i
2 The standard logic of knowledge o
r
Afirstwaytogiveaformalaccounttoepistemologicalconceptssuchasbelief
andknowledgeistoadoptthelanguageofmodallogic.Evenifthemodaloper- 7
:
ators(cid:94)and(cid:3)areusuallyreadaspossibilityandnecessity,wecanalsoadoptan 2
epistemicinterpretationofthem.Onthisalternativereadingwewilltranslatea
(
logicalformulalike(cid:3)p notas“itisnecessarythatp”butratheras“itisknown 2
that ”, “it is believed that ” or “it is certain that ”. Following each of these 0
p p p
1
interpretationswecanformulateadifferentmodallogic,inordertoformalize
6
thespecificfeaturesoftheconsideredepistemicoperator.InwhatfollowsIwill
)
beinterestedexclusivelyintheformerofthesealternativesandIwillfocusmy
attentiononthelogicofknowledge.
Working with an epistemological interpretation of modal logic, it is worth
4
D.E.Quadrellaro EpistemicLogicandtheProblemofEpistemicClosure
specifyingwhoisthesubjectoftheknowledgewearespeakingabout.Ifweread
R
(cid:3) simplyas“itisknownthat ”,themeaningofthisoperatorremainsnotclear
p p
i
enough. What does it mean, in fact, that something is known? Does it mean v
i
thatsomeoneknowsit? Ordoesitmeanthateveryoneknowsit? Therefore,in s
t
ordertobeasclearaspossible,weshouldadoptamoreintuitiveterminology
a
and make explicit the fact that we are working with a propositional notion of
I
knowledgeandwithinalogicofindividualagents.Theboxoperatorwillbesub- t
a
stitutedbya (for“knowledge”),followedbyaletterthatindicateswhoisthe
K l
i
agent that knows the considered proposition. Modal formulas will look, thus,
a
likeK p andK p andtheywillbereadas“theagenta knowsthat p”and“the n
a b
agentbknowsthatp”. Inwhatfollows,wewillbeinterestedinformalsystems a
withonlyoneagent,butitisimportanttokeepinmindthatwecanintroduce d
many -operators,inordertomaptheknowledgeofmorethanonesubject1. i
K
F
Letusnowmove,aftertheseintroductoryremarks,togiveaprecisedefini-
i
tionofthesyntax ofthepropositionalmodallogicforknowledge. Weproceed l
o
extendingthealphabetofclassicalpropositionallogicwithaknowledgeopera-
s
torK . o
a
fi
Definition2.1(AlphabetofPropositionalModalLogicforKnowledge). Anal-
a
phabetforpropositionalmodallogicforknowledgeisdefinedastheunionof
A
thefollowingdisjointedsets:
n
• AdenumerablesetofatomicpropositionalvariablesP = {p0,p1,...}. a
l
i
• ThesetofthelogicalconnectivesC = {¬,∧,→}. t
i
c
• Thesetoftheknowledgeoperator O = {K }.
a a
• ThesetofauxiliarysymbolsA = {(,)}. J
u
Giventhealphabet,itispossibletodefineinductivelythesetoftheformulasof
n
thelogicofknowledge. i
o
Definition2.2(FormulasofPropositionalModalLogicofKnowledge). Thefor- r
mulasofthemodallogicofknowledgearegivenbythefollowingdefinitionby
7
induction:
:
2
1. Ifϕisanatomicpropositionalvariable,thenϕisaformula.
(
2
2. Ifϕisaformula,thenalsoitsnegation¬ϕisaformula.
0
3. Ifϕand χareformulas,thenalsotheirconjunction(ϕ∧ χ)isaformula. 1
6
4. Ifϕand χareformulas,thenalsotheconditional(ϕ→ χ)isaformula. )
5. Ifϕisaformula,thenalsoK ϕisaformula.
a
1FortheintroductionofmultipleagentsseebothHendricksandSymons,(2015,pp.9-11)and
Holliday,(forthcoming,pp.5-7).
5
D.E.Quadrellaro EpistemicLogicandtheProblemofEpistemicClosure
6. Nothingelseisaformula.
R
The semantics of the logic of knowledge is provided by a Kripke-structure, i
v
whichisthestandardwaytointerpretmodallanguages. i
s
t
Definition2.3(Kripke-structures). Givenapropositionalmodallogicofknowl-
a
edge,aKripke-structureMisatriple(cid:104)W,R ,V(cid:105),where:
a I
t
1. W isanon-emptyset.Intuitively,W isasetof“possibleworlds”or“possible a
scenarios”. l
i
a
2. R isabinaryrelationoverW,i.e. asubsetofW ×W. Intuitively,weread n
a
vR was“thepossibleworldwisepistemicallyaccessiblefromthepossible a
a
worldv bytheagenta”. d
i
3. isafunctionthatassignstoeveryatomicpropositionalformulaasubset
V
F
of .Intuitively, specifiesinwhichpossibleworldseachatomicformula
W V i
l
istrue. o
s
GiventheKripke-structures,wecandefinethenotionoftruthinaworld:
o
fi
Definition2.4(Truthinaworld). Givenapropositionalmodallogicforknowl-
a
edge,aKripke-structureMandaworldw,thenotionM (cid:15) ϕofbeingtrueina
w
A
worldisdefinedasfollows:
n
1. whenϕisatomic,thenM (cid:15) ϕiffw ∈V(ϕ); a
w
l
2. whenϕhastheform¬χ,thenM (cid:15) ϕiffM (cid:50) χ; i
w w t
i
3. whenϕhastheform(χ∧ψ),thenM (cid:15) ϕiffM (cid:15) χandM (cid:15) ψ; c
w w w
a
4. whenϕhastheform(χ →ψ),thenM (cid:15)w ϕiffM (cid:50)w χorM (cid:15)w ψ; J
u
5. whenϕhastheformKaχ,thenM (cid:15)w ϕiffforeverypossibleworldv such n
thatwR v,M (cid:15) χ. i
a v
o
Thedefinitionoftruthinaworldallowsustodefinetwofurtherimportantno- r
tions. We say that a formula ϕ is true in a model M if and only if it is true in
7
everyworldw ∈W oftheKripke-structureM.Wesaythataformulaϕisavalid
:
2
formula ifandonlyifitistrueineveryworldw ∈ W ofeveryKripke-structure
M. (
2
WhatwehavedescribedsofaristheminimalsystemKofmodallogic,with
0
theonlypeculiaritythattheinformalreadingthatwehaveassumedforthemodal
1
operator is “the agent a knows that...”. Nevertheless, it is clear that to obtain 6
a logic of knowledge this is not enough. What one needs, rather, is to specify )
theformalpropertiesthataretypicalofknowledgeandtorepresenttheminthe
logic.Puttingspecificrestrictionsovertheaccessibilityrelation ,itispossible
R
a
toobtainmanymodallogicsstrongerthanK,wheremoreprinciplesarevalid
6
D.E.Quadrellaro EpistemicLogicandtheProblemofEpistemicClosure
formula. The problem is that it is not sufficiently clear which modal system
R
photographsinthecorrectwaytheformalpropertiesofknowledge. Sincethe
i
purposeofthisarticleistoexaminewhichlogiccanbeacceptedbyepistemolo- v
i
gistswithdifferentphilosophicalorientations,wewillextendKonlywiththose s
t
principleswhicharegenerallytakenforgrantedintheepistemologicaldebate.
a
Therefore,theonlyrestrictionthatwewantimposetoourlogicalsystemisthat
I
ithastosatisfythefollowingprinciple: t
a
(T) K ϕ→ ϕ l
a i
a
What(T)saysisthat,ifoneknowsaproposition,thenthisverysameproposition
n
mustbetrue. Thisdoesnotonlyfollowfromanyanalysisofknowledgeastrue
a
beliefplussomething,butitalsoseemstobeavalidminimaldescriptionofthe
d
meaningofknowledge.Indeed,ifonesaysthathe/sheknowsthat butitisnot
p i
thecasethatp,itseemsreasonabletoconcludethathe/shedoesnot knowthat F
p,butratheronlybelievesthatp2. i
l
Ifwewantthattheprinciple(T)holdsinthelogicalframeworkthatweare o
s
considering,wehavetoputarestrictionontheaccessibilityrelation . More
R
a o
precisely,asweprovewiththefollowingtheorem,wehavetorestrictouratten-
fi
tiontothoseKripke-structureswheretheaccessibilityrelationisreflexive. The
a
modallogicthatweobtainwhenweworkonlywithreflexiveaccessibilityrela-
A
tionsiscalledT. n
a
Theorem2.1. GiventhelanguageofpropositionalmodallogicanditsKripke-
l
structureM = (cid:104)W,R ,V(cid:105),theformula(T)K ϕ → ϕisavalidformulaifandonly i
a a
t
iftheaccessibilityrelationRa isreflexive. i
c
Proof: AssumingthattheaccessibilityrelationsR inM isreflexive,thengiven a
a
anypossibleworldw ∈W wehavethatwRaw.Therefore,sinceM (cid:15)w Kaϕholds, J
u
then M (cid:15) ϕ holds in every worldv such thatv is accessible fromw. But for
v
n
reflexivitywehavethatw isaccessiblefromitselfand,therefore,thatM (cid:15) ϕ.
w
i
Vice versa, assuming that K ϕ → ϕ is a valid formula then, for every Kripke- o
a
structure M and every worldw in it M (cid:15) K ϕ → ϕ. Given the semantics of r
w a
theconditional, thisamountstosaythatitisnotthecasethatM (cid:15) K ϕand
w a 7
M (cid:50)w ϕ.But,ifRa wasnotreflexive,wecouldconstructaKripke-structuresuch :
2
as N = (cid:104)W,R ,V(cid:105), withW = {v,w} and R = {(cid:104)w,v(cid:105)}. In N we have that, if
a a
v ∈V(ϕ)butw (cid:60)V(ϕ),thenN (cid:15) K ϕbutN (cid:50) ϕ,contradictingourclaimthat (
w a w
2
K ϕ→ ϕisavalidformula.Therefore,R mustbereflexive.(cid:4)
a a 0
1
6
3 Theprincipleofepistemicclosureanditsproblems
)
InthepreviouspartofthisarticleIhaveintroducedthemodallogicT,inorderto
representsomeminimalformalpropertiesofknowledge.Movingastepfurther,
2ThisaspectisfamouslystressedbyWittgenstein,(1969).
7
D.E.Quadrellaro EpistemicLogicandtheProblemofEpistemicClosure
itisnowpossibletoproveaninterestingresult,whichsaysthattheprincipleof
R
epistemicclosureisavalidformulainT.Firstly,letusclarifywhatwemeanwith
i
thenameof“principleofepistemicclosure”. v
i
s
(CP) Ifanagentknowsthat ϕandhe/sheknowsthatif ϕthen χ,thenhe/she t
a
alsoknowsthat χ.
I
Itisstraightforwardtotranslatethisthesisintothelanguageofthelogicofknowl- t
a
edge.Wethusobtainthefollowingformalversionoftheclosureprinciple:
l
i
a
(FCP) (K ϕ∧K (ϕ→ χ))→K χ
a a a n
a
Wecannowprovethefollowingtheorem:
d
Theorem3.1. GiventhelogicofknowledgeT,theformalclosureprinciple(FCP)
i
isavalidformula. F
i
Proof: Wereasonforabsurd. If(FCP)wasnotavalidformula,therewouldbea l
o
worldw ofaKripke-structureM =(cid:104)W,Ra,V(cid:105),where(FCP)doesnothold.Given s
thesemanticsoftheconditional, thismeansthatM (cid:15) K ϕ∧K (ϕ → χ)but o
w a a
M (cid:50) K χ. Given M (cid:15) K ϕ, we have that in every world accessible fromw, fi
w a w a
a
ϕholds. GivenM (cid:15) K (ϕ → χ), wehavethatineveryworldaccessiblefrom
w a
w, ϕ → χholds. Moreover,sinceM (cid:50) K χ,thereisatleastoneworldv such A
w a
thatwR v whereM (cid:50) χ. Butinthissameworldv wehavethatM (cid:15) ϕ and n
a v v
a
M (cid:15) ϕ → χholdtoo,fromwhichitfollowsthatM (cid:15) χ. Therefore,weobtain
v v
l
thecontradictionthatM (cid:15) χandM (cid:50) χ.(cid:4) i
v v t
Ifourconcernsaremainlyepistemologicalthisresulthasaparticularrele- i
c
vance. Infact,whatwehaveprovedisthatevenifweworkwithaweakmodal
a
system,theprincipleofepistemicclosurewillholdinit3. Therefore,ifwehave
J
somereasontorefusetheprincipleofepistemicclosure,thenwecannotadopt u
theformallogicTanymore,foritdescribesknowledgeinawaywhichisincon- n
i
sistentwithourtheory. InparticularDretske(1970)offersatleasttwopossible
o
reasonstorefusetheclosureprinciple4.IntherestofthispartIwillpresentboth r
ofthem,butIwillnottrytosetthequestionabouttheirvalidity. Indeed,Ionly
7
wanttoshowthatitmightbereasonableforanepistemologisttorejecttheclo-
:
sureprinciple.Infact,giventhepossibilitythat(FCP)isnotacceptable,wehave 2
tolookforamodallogicforknowledgeweakerthanthestandardonedescribed
(
2
3Notice,moreover,thatintheproofofthetheorem3.1.wedidnotmakeanyuseofthefactthat
0
theaccessibilityrelationbetweenworldsisreflexive.Therefore,ourproofisvalidalsoforthebasic
1
modallogicK.
4Luper,(2016)synthesizesawiderangeofargumentsagainsttheclosureprinciple,oftenorigi- 6
nallyraisedbyDretskeandNozick.However,evenifLuper’sreconstructionisclear,Idonotagree )
withhispresentationoftheargumentsfromthe“analysisofknowledge”. Infact,thetheoriesof
knowledgesupportedbyDretskeandNozickareexplanationsofwhytheclosureprinciplefailsand
notreasonstorefuseit.Lupercommits,therefore,asortofinversionoftherightorderofexplana-
tion.
8