A capacity-based framework encompassing Belnap-Dunn logic for reasoning about multisource information

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To cite this version: Ciucci, Davide and Dubois, Didier A

capacity-based framework encompassing Belnap-Dunn logic for reasoning

about multisource information. (2019) International Journal of

Approximate Reasoning, 106. 107-127. ISSN 0888-613X

Official URL

DOI :

https://doi.org/10.1016/j.ijar.2018.12.014

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A

capacity-based

framework

encompassing

Belnap–Dunn

logic

for

reasoning

about

multisource

information

✩

,

✩✩

Davide Ciucci

a

,

Didier Dubois

b

,

∗

a_DISCo_–_Università_di_Milano_–_Bicocca,_Viale_Sarca₃₃₆_–_U14,₂₀₁₂₆_Milano,_Italy b_IRIT,_CNRS_&_Université_de_Toulouse,₁₁₈_route_de_Narbonne,₃₁₀₆₂_Toulouse_cedex_9,_France

a

b

s

t

r

a

c

t

Keywords:

Belnap–Dunnlogic Inconsistencyhandling KleeneandPriestlogics Capacities

Possibilitytheory Modallogic

Belnap–Dunn four-valued logicis oneofthe bestknownlogics forhandlingelementary information items coming from several sources. More recently, a conceptually simple framework, namely a two-tiered propositional logic augmented with classical modal axioms (here called BC logic), was suggested by the secondauthor and colleagues, for the handlingofmultisourceinformation.Itisafragmentofthenon-normalmodallogic EMN,whosesemanticsisexpressedintermsoftwo-valuedmonotonicsetfunctionscalled Boolean capacities. We show BC logic is more expressive than Belnap–Dunn logic by proposing aconsequence-preservingtranslationofBelnap–Dunn logicinthissetting.As special cases, wecanrecoveralreadystudiedtranslationsofthree-valuedlogicssuchas KleeneandPriestlogics.Moreover,BClogiciscomparedwiththesource-processorlogicof Avron,BenNaimandKonikowska.OurtranslationbridgesthegapbetweenBelnap–Dunn logic,epistemiclogic,andtheoriesofuncertaintylikepossibilitytheoryorbelieffunctions, and pavestheway toaunifiedapproachtovariousmethods forhandlinginconsistency duetoseveralconflictingsourcesofinformation.

1. Introduction

Reasoningwithinconsistentinformationisatopicthathasemergedinthelastthirtyyearsasakeyissueinlogic-based Artificial Intelligence,becausetheusualapproach inmathematicsaccordingtowhichfromacontradictioneverything fol-lowsisnotusefulinpractice.Duetothelargeamountofdataavailabletoday,inconsistenciesareunavoidable,andpeople oftencopewithinconsistentinformationindailylife.Manyworkshavebeenpublishedproposingapproachestodealwith inconsistentknowledgebasesinsuchawayastoextractusefulinformationfromitinanon-explosiveway.See[52,12,15] forsurveys.

Belnap–Dunnfour-valuedlogic[32,10,9] isoneoftheearliestapproachestothisproblem.Itisbasedonaverynatural set-up whereby several sources ofinformation tentativelydeclare some elementary propositions tobe trueor false.The set oftruth-valuesthuscollected foreach propositionfromallsources issummarizedby aso-calledepistemictruth-value referring towhethersourcesareinconflictornot,informedornotaboutthiselementaryproposition.Therearefoursuch

✩

ThispaperispartoftheVirtualspecialissueonthe14thEuropeanConferenceonSymbolicandQuantitativeApproachestoReasoningwithUncertainty (ECSQARU2017),EditedbyAlessandroAntonucci,LaurenceCholvyandOdilePapini.

✩✩

Thispaperisarevisedandexpandedversionof[22].

*

Correspondingauthor.

E-mailaddresses:ciucci@disco.unimib.it(D. Ciucci),dubois@irit.fr(D. Dubois).

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epistemictruth-values,twoofwhichrefertoignoranceandconflict.Truthtablesforconjunction,disjunction,andnegation areused tocompute theepistemic statusofother morecomplex formulas.Thislogicunderlies both Kleenethree-valued logic[40] (recovered whenno conflictbetweensources isobserved) andthethree-valuedPriest LogicofParadox[46,47] (whensourcesare neverignorantandalways assigntruth-valuestoeachelementaryproposition).Intheformerlogic,the truth-valuereferringtotheideaof“contradiction”iseliminated,whileinthelatter,thetruth-valuereferringtotheideaof “ignorance”iseliminated.Belnap–Dunnlogicusesallfourvaluesformingabilatticestructure,inwhichonepartialordering expressesrelative strength ofinformation(fromno informationto toomuchinformation), andtheother partial ordering representsrelativetruth(fromfalsetotrue).

Inrecentworks[20,21],wehavefocusedattentiononthree-valuedlogicsofuncertaintyorinconsistency(paraconsistent logics).We were able to expressthem in a two-tiered propositional logic calledMEL [8], which adopts the syntax of a fragment ofepistemic logic, andborrows axiomsof theKD logic. MELcan be viewed asthe simplestlogicalframework forreasoningaboutincompleteinformationandatthesemanticlevelitexpresses theall-or-nothingversionofpossibility theory[28].Thiskindoftwo-tieredlogicdatesbackstoworksbyHájekandcolleagues[38] whotriedtocaptureprobability andotheruncertaintylogics,usingatwo-tieredlogic,withpropositionallogicatthelowerlevelandamultivaluedlogicat thehigherlevel.Theideaisthatthedegreeoftruthofanatomicstatement p isprobable isthedegreeofprobabilityof p. Thisschemehasbeensystematizedin [34,18]. ThelogicMEL isinthesamevein, albeitusingpropositional logicatboth levels,andamodallanguage.

IntheMELlogic,itispossibletoexpressthat anatomicproposition, oritscomplement,isunknown(when translating Kleenelogic,forinstance),orthatitiscontradictory(whentranslatingPriestLogicofParadox,forinstance).Themainmerit ofsuchtranslationsin[20,21] istolaybarethemeaningofthevariousthree-valuedconnectivesusuallydefinedbymeans oftruthtables, andespecially toexhibit thelimitedexpressivepowerofsuch three-valuedlogics.Namely, modalitiescan onlyappearinfrontofliteralsinthetranslations.Theseresultsfacilitatethepracticaluseofthree-valuedlogicsandtheMEL logicseems tooffera unifiedlogicalframeworkforall ofthem,whereonlyusual binary truth-valuesareneeded. Inthis paperwefocusonthecaseofBelnap–Dunnlogic,forwhichweproposeasimilartranslationintoatwo-tieredpropositional settingthatborrowsaxiomsfromanonnormalmodallogicandsupportsbothideasofconflictandignorance.Indeed,the settingofKDlogicsisnotgeneralenoughtocaptureBelnap–Dunnapproach.

A more generalmodal framework for multiple source information, herecalled BC logic, first outlined by the second author[24],andproved soundandcompletein[30],isrecalled. Itisbasedonafragment ofthenon-normalmodallogic EMN[17]. Atthesemantic levelthislogicaccountsforthe notionofBooleancapacity, i.e.,set functionsvaluedon

{

0

,

1

}

that are monotonic under inclusion. This logic brings us closer to uncertainty theories and can encompass variants of probabilisticandbelieffunctionlogics,forinstancethelogicofriskyknowledge[43],wheretheadjunctionruleisnotvalid. Ourframeworkseemstobetailoredforreasoningfrommultiplesourceinformation.

Then,weshowthatourtwo-tieredpropositional settingrelatedtotheEMNlogiccanencodeBelnap–Dunnfour-valued logic,namelythatthefourtruth-valuesinthislogicarenaturallyrepresentedbymeansofcapacitiestakingvalueson

{

0

,

1

}

. We providea consequence-preserving translation ofthis four-valuedlogic into BC. Thus, we constructa bridge between Belnap–Dunn logic and uncertainty theories. As special cases, we recover our previous translations of Kleene logic for incompleteinformation[20] andPriestLogicofParadox[21].

Thistranslationisnot agratuitous exercise. BClogic haspotential tosupport anumberofinconsistencyhandling ap-proachesandtoactasaunifyingframeworkforseverallogicsofuncertaintyandconflictinginformation;so,itisimportant toshowthat thebestknownlogicforinconsistencyhandlingisencompassed.Theframework ofBC logicismore expres-sivethantheoneofBelnap–Dunnlogic,becauseitdoesawaywiththeassumptionthatsourcesprovideinformationabout propositional variables only,and itdoesnot require truth-functionalityassumptions. Thesource-processor logic ofAvron etal.[6] alsodoesaway withthe firstrestriction butitstill requiresa weak formoftruth functionality.We discussthe connectionsbetweenthat settingandours. TheBC logichasthus potentialtoencompass variousapproachesto inconsis-tency-tolerantformalisms,andiseasierto relatetoinformationfusionmethods thatareproposed inuncertaintytheories [26],duetotheuseofmonotonicsetfunctions. Forinstance,a possibleextensionofBelnapset-upswithbelieffunctions wasalreadysuggestedin[23].

Thispaper is organized as follows.Section 2 presents the propositional logic of Booleancapacities BC andshows its capability to capturethe notionof information coming from severalsources. Section 3 recalls Belnap–Dunnfour-valued logicfromthepointofviewofitsmotivation,itssyntacticalinferenceanditssemantics,aswellassomeofitsthree-valued extensions,i.e.,Kleene andPriest logics.Section 4containsthemain resultspertainingtothetranslationofBelnap–Dunn logicintoBC.Results inthissectionwere moresuccinctlyannouncedinaformerconference paper[22].Section 5shows how Kleene andPriest logic translations in [20,21] canbe retrieved by translating additional axioms orinference rules intoBC. Section 6discussessome relatedapproachesthat deal withconflictinginformation,includingthe generalsource processorlogicof[6].WeprovidesomeproofsaboutBelnap–DunnlogicinAppendixA,forthesakeofself-containedness.

2. ThelogicofBooleancapacitiesandmultisourceinformationmanagement

Inthissection,weconsiderageneralapproachtothehandlingofpiecesofincompleteandconflictinginformation com-ingfromseveralsources.Weshow,followinganintuitionalreadysuggestedin[5,24,30],thatifwerepresenteachbodyof informationitemssuppliedbyonesourcebymeansofasetofpossiblestatesofaffairs,thecollectiveinformationsupplied

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by thesourcescanbemodeled inalosslesswaybyamonotonicsetfunction calledacapacity,thattakesvalue on

{

0

,

1

}

. Thesesetfunctionscanserveasnaturalmulti-sourcesemanticsforasimplenon-normalmodallogicinwhichweshall cap-tureBelnap–Dunnfourtruth-valuesasalreadysuggestedin[24].Thislogic,formally studiedin[30] isdrivenbywhatwe believeistheminimalsetofaxiomstoexpressmultisourcereasoning.Itcannaturallyserveasatargetlanguagetoexpress Belnap–Dunn,KleeneandPriestlogics,aswellasthesourceprocessorlogicofAvronetal.[6],andotherapproachesaswell. 2.1. Booleancapacitiesandmultisourceinformation

Considerasetofstatesofaffairs

Ä

whichmaybethesetofinterpretationsofapropositionallanguage.Thecomplement ofasubset A

⊆ Ä

isdenotedby Ac.

Definition1.Acapacity(orfuzzymeasure)isamapping

γ

:

2Ä

_{→ [}

₀

_,

₁

_]

_such_that

γ

_(∅)

₌

_0;

_γ

_(Ä)

₌

_{1; and}_if _A

_⊆

_{B then}

γ

(A)

≤

γ

(B)

.

Thiscapacityissupposedtoinformusaboutwhatthestate oftheworld, says,is.Thevalue

γ

(A)

canbeinterpreted asthe degree ofsupport ofa propositionofthe forms

∈

A.Fora textbookon (numerical)capacities,see Grabisch[37]. A Boolean capacity(B-capacity,forshort) isacapacitywithvaluesin

{

0

,

1

}

[30].It canbe definedfromausual capacity andanythreshold

λ

>

0 as

β(A)

=

1 if

γ

(

A)

≥ λ

and0otherwise.

TheusefulinformationinaB-capacityconsistsofitsfocalsets.AfocalsetE issuchthat

β(E

)

=

1 (hencenotempty)and

β(E

\ {w})

=

0

,

∀w

∈

E. Let

F

_β be thesetoffocalsetsof

β

.

F

_β collectsallminimalsets forinclusionsuchthat

β(E)

=

1.

F

_β isthus an antichain(no inclusion betweensets).We cancheck that

β(

A)

=

1 ifandonlyifthere isasubset E of A in

F

_β with

β(E

)

=

1.So,the focalsets ofa Booleancapacityare thenecessaryandsufficient informationtorecover the capacity.

We can interpret a Boolean capacityas the information provided by a set of sources. Consider n sources providing information intheform ofnon-empty sets Ei

⊆ Ä

that are supposedto contain theactual world s aftereach source. In

other words,Ei canbeviewedastheepistemicstateofthesourcei: itisonlyknownfromsource i thattherealstateof

affairs s shouldlie in Ei.Acapacity

β

can bebuiltfromthesepieces ofinformationletting

β(A)

=

1 when

∃i such

that

Ei

⊆

A and0otherwise.Then

β(A)

=

1 reallymeansthatthereisatleastonesourcei thatclaimsthat s

∈

A istrue(that

is,s

∈

A istrueinallstatesofaffairsinEi).Thiswayofsynthesizinginformationisnotdestructive,sinceitpreservesevery

non-redundant initial piece of information: theset of focal sets

F

_β isclearly includedin

{E1,

E2,

. . . ,

En

}

asit contains

minimalsets Ei forinclusion.

Givenapropositions

∈

A,therearefourepistemicstatusesbasedontheinformationfromsources,thatthecapacitycan express:

•

Support:

β(A)

=

1 and

β(A

c

₎

₌

_0._Then_s

_∈

_{A is}_asserted_by_at_least_one_source_and_negated_by_no_other_one.

•

Rejection:

β(

Ac

)

=

1 and

β(

A)

=

0.Thens

∈

A isnegatedbyatleastonesourceandassertedbynootherone.

•

Ignorance:

β(

A)

= β(

Ac

)

=

0.Nosourceassertsnornegatess

∈

A.

•

Conflict:

β(

A)

= β(A

c

₎

₌

_1._Some_sources_assert_s

_∈

_A,_some_negate_it.

ImportantspecialcasesofBooleancapacitiesare[30]:

•

whenthereisasinglefocalset:

F

_β

= {E}

.Thisisequivalenttohaving

β

minitive,i.e.,

β(

A

∩

B)

=

min

(β(

A),

β(B))

.Itis thenanecessitymeasure [27].Thereisonlyonesourceanditsinformationisincomplete,butthereisnoconflict.

•

when the focal sets are singletons

{w

1},

{w

2},

. . . ,

{w

n

}

. This is equivalent to having

β

is maxitive, i.e.,

β(A

∪

B)

=

max

(β(

A),

β(B))

. All sources have complete information, so there are conflicts, but no ignorance. Letting E

=

{w1

,

w2,

. . . ,

wn

}

, then

β(

A)

=

1 if and onlyif A

∩

E

6= ∅

; so,formally

β

is apossibilitymeasure [27]. But here E is

aconjunctionofnon-mutuallyexclusiveelements(fullyinformedsources),notapossibilitydistribution.

• β

isbothmaxitiveandminitiveifandonlyifits uniquefocalsetisasingleton.ThisisusuallycalledaDiracfunction, representingcomplete(deterministic)information.

2.2. Themultisourceset-upinthepropositionalsetting

Thisapproachcanbedescribedinmoresyntacticterms.Considerastandardpropositionallanguage

L

withvariablesV

=

{a,

b,c,

. . . }

andconnectives

∧,

∨,

¬

, forconjunction, disjunction andnegation,respectively. We denote thepropositional formulas of

L

by letters p,q,r. The truth set is

{

0

,

1

}

andwe denote by v

:

V

→ {

0

,

1

}

a truth-assignment. The set of interpretationsof

L

is

Ä

.Thesetofmodelsofp isdenotedby

[p]

⊆ Ä

.Theinformationprovidedbyeachsourcecanthen equivalentlytake theformofaconsistentset Ki ofpropositionalformulasclaimedtobetrue.Ifweassumeeachsourceis

logicallysophisticated,1 then Ki isequivalently modeledby theset Ei ofits models.Moreover,foranyformula p, source

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i declares that p is true(1) whenever Ki

⊢

p, false (0)whenever Ki

⊢ ¬p,

and is silent otherwise. Then, we can again

compute thestatusofall propositionswithrespecttothe sources,usingpropositional inference,i.e., decidewhetherany propositionp is

•

supported ifforsomesourcei,Ki

⊢

p and Ki

⊢ ¬p for

noothersource;

•

rejected ifforsomesourcei, Ki

⊢ ¬p and

Ki

⊢

p fornoothersource;

•

unknown ifp isneithersupportednorrejected,i.e.,forallsources j,Kj

0

p andKj

0 ¬

p;

•

conflicting ifp isbothsupportedandrejected,i.e.,forsomesourcei,Ki

⊢

p andforanotherone j,Kj

⊢ ¬p.

This setting is somewhat similar to the one of Konieczny and Pino-Pérez [41,42] who consider the problem of syntax-independentmergingof setsofpropositional formulasthat maycontradicteach other.Theiraimis tocompute aunique consistentclosed setofformulas astheresultofthe mergingprocess. Theyproviderationality axiomsto doso,and ex-amplesofmergingoperationsthatfollowtheseprinciples.Howeverthusdoing,themethodisdestructive,namely,itdoes awaywithsome piecesofinformationprovidedbysources, inordertorestoreconsistency.Incontrast,thecapacity-based informationprocessingset-upproposedhere,whileitisalsosyntax-independent,isnon-destructive.Indeed,allinformation itemsprovidedby thesources remainunaltered(unless redundant): wedonot solveinconsistency, weonlyconfineit to someconclusions.

Example1.ThisexampleisinspiredbyoneproposedbyRevesz[50] intheliteratureonknowledgebasemerging.Thereare threevariables

{a,

b,c}.Therearethreesourceswhichdeclarethefollowingformulasandtheirconsequencesasbeingtrue:

•

K1

= {a

∨

c,

¬b}

withmodels E1

= {

100

,

001

,

101

}

•

K2

= {¬a,

(b

∨ ¬c)

∧ (c

∨ ¬b)}

withmodelsE2

= {

000

,

011

}

•

K3

= {a,

b}withmodels E3

= {

110

,

111

}

where models are denoted by triples of truth-values for a,b,c in this order. The reader can check that, following our approach,c isunknown(impliedbynoneofthebases),a

∧

b isconflicting(impliedby K3 but

¬a

∨ ¬b is

impliedby K2), a

∧

b

∧

c isrejected(impliedbynoneofthebasesbutitsnegationisimpliedby K1 andK2)aswellasa

∧ ¬b (implied

by noneofthebasesbutitsnegationisimpliedby K2 andK3),a

∨

b anda

∨

c aresupported(impliedby K3,butrejectedby none).

2.3.ThelogicBC

Inthis section, we recall the logical framework formultiple source information,proposed in [24,30], whichaccounts forthe abovecapacity-basedframework.Consider atwo-tieredpropositional language

L✷

basedonanotherpropositional language

L

.Theatomicpropositionsof

L✷

formtheset

V

_¤

= {✷p

:

p

∈ L}

.Thelanguage

L✷

,whoseformulasaredenoted byGreekletters

φ,

ψ,

. . .

,isthenoftheform:

•

Ifp

∈ L

then

✷p

∈ L✷

;

•

if

φ,

ψ ∈ L✷

then

¬φ ∈ L✷

,

φ ∧ ψ ∈ L✷

.

Notethatthelanguage

L

isembeddedinside (butdisjointfrom)

L✷

.Asusual

✸p stands

for

¬✷¬p.

Itdefinesavery elementaryfragment,proposedbyBanerjeeandDubois[7,8],ofamodallogiclanguage.

Aminimalepistemiclogicwithconflictsusingthelanguage

L

_¤definedabovehasbeenproposed[24].Itisatwo-tiered propositionallogicaugmentedwithsomemodalaxioms:

(PL) Allaxiomsofpropositionallogicfor

L✷

-formulas; (RM)

¤p

→ ¤q if

⊢

p

→

q inPL;

(N)

¤p,

wheneverp isapropositionaltautology; (P)

♦p,

wheneverp isapropositionaltautology.

Theonlyinferenceruleismodusponens:If

ψ

and

ψ → φ

then

φ

.

Thisisa fragmentofthe non-normallogic EMN[17].In particulartheaxiom (RE):

✷p

≡ ✷q if

andonlyif

⊢

p

≡

q is valid.Notethatthetwodualmodalities

✷

and

✸

playthesamerole.Namelytheaboveaxiomsremainvalidifweexchange

✷

and

✸

.Sothesemodalitiesarenotdistinguishable.

Semanticsof non-normallogics are usually expressed interms ofneighborhoods, which attacha familyof subsetsof interpretationstoeach possibleworld[17].Herewe donotneedthiscomplexsemanticssince modalitiesarenotnested. Capacitiesonthesetofinterpretations

Ä

ofthelanguage

L

encodeasetofsubsetsA suchthat

β(A)

=

1,whichisaspecial caseofworld-independentneighborhoodcontaining

Ä

andclosedunderinclusion(afteraxiomRM).

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WecallthisformalismBClogic,whereBCstandsforBooleancapacities.2

ModelsinthislogicareBooleancapacities.ABC-modelofanatomicformula

✷

p forthismodallogicisthenaBoolean capacity

β

suchthat

β([p])

=

1.ThesatisfactionofBC-formulas

φ ∈ L✷

isthendefinedrecursivelyasusual:

• β |= ✷p,

ifandonlyif

β([p])

=

1;

• β |= ¬φ

,

β |= φ ∧ ψ

aredefinedinthestandardway.

Satisfiabilitycanbeequivalentlyexpressedintermsofthefocalsetsof

β

,asfollows:

β |= ✷

p iff

∃

E

∈

F

_β

,

E

⊆ [

p

].

Soonemightaswelldefineamodelinthislogicasanyantichainofsubsetsthatformafamilyoffocalsetsofacapacity, thuslayingbarethemultiple-sourcenatureofthislogic.Theantichainconditioncanberelaxedtoanyn-tupleofnon-empty subsets

(E

1,

. . . ,

En

),

∀n

>

0.However,severalsuchtuplesmaygeneratethesameBooleancapacity.

Semanticentailmentisdefinedclassically,andsyntacticentailmentisclassicalpropositionalentailmenttakingRM,N,P asaxiomschemata:

Ŵ

⊢

BC

φ

ifandonlyif

Ŵ

∪ {

allinstancesofR M,N,P

}

⊢ φ

(classicallydefined).Ithasbeenprovedthat

BC logicissoundandcompletewrtBooleancapacitymodels[30].Infact,axiomRMclearlyexpressesthemonotonicityof capacities,anditiseasytorealizethataclassicalpropositionalinterpretationof

L✷

thatrespectstheaxiomsofBC canbe preciselyviewedasaBooleancapacity.BC isaspecialcaseofmodallogicofuncertaintywithatwo-layeredsyntaxinthe senseof[34,18].

2.4. Specialcase:incompleteinformation

A firstimportantspecial caseofBC iswhen thereis onlyone sourcewithepistemic state E. Then thecorresponding capacityN,callednecessitymeasureinpossibilitytheory[27],isminitiveanddefinedbyN(A)

=

1 ifE

⊆

A and0otherwise. ThelogicBCparticularizedtonecessitymeasuresisthelogicMEL[8].ThefollowingKDaxiomsandinferencerulearevalid inMELunderthisrestrictiontoasinglesource:

•

Allaxiomsofpropositionallogicfor

L✷

-formulas.

• (K

)

:

✷(p

→

q)

→ (✷p

→ ✷q)

.

• (N)

:

✷

p,wheneverp isapropositionaltautology.

• (D)

:

✷p

→ ✸p.

•

ModusPonens:If

φ,

φ → ψ

then

ψ

.

Axiom(C):

✷(p

∧

q)

≡ (✷p

∧ ✷q)

,whichistheadjunctionaxiom(theBooleanformoftheminitivityaxiom)isvalidinthis system. Satisfiabilityreducesto E

|= ✷p if

andonlyif E

⊆ [p]

.Thislogicis soundandcomplete forthisstandard system of modalaxiomsofthe logic KD.Note that theconflicting situationcannot be expressed usingnecessitymeasures since it never holdsthat N

([p])

=

N(¬[p])

=

1. Inprevious publications, we haveshownthat Kleene logic canbe encoded in MEL[20].ClassicallogiccanbeencodedinMELaswell,ifwerestrictthelanguagetoconjunctionsofatomicpropositions

✷p

∈ L✷

:ithasbeenshownthat

{✷p1,

. . . ,

✷p

k

}

⊢ ✷q in

MELifandonlyif

{p1,

. . . ,

pk

}

⊢

q inclassicallogic[8].

Givena Boolean capacity

β

, let Ni denotetheBoolean capacityinduced by thesinglefocal set Ei.Then, itis easy to

check thattheBooleancapacityinducedby thetuple

(E

1,

. . . ,

En

)

isoftheform

β(

A)

=

maxni=1Ni

(A)

[30].Denoteby

✷

i

the KD modality associated to the necessity measure Ni. As a consequence, forany capacityof BC-model,there are KD

models

{E1,

. . . ,

En

}

andKDmodalities

✷

1,

. . . ✷

n suchthat

β |=

BC

✷

p ifandonlyif

∃i

∈ {

1

,

. . . ,

n}suchthatEi

|=

K D

✷

ip.

WecanalsoconsiderdroppingaxiomKinMEL.Inthatcase,onlyAxiomDremains,whichstillentailsthat

✷p

∧ ✷¬p

isalwaysacontradiction,while

✸p

∨ ✸¬p is

atautology.IntermsofBooleancapacities,axiomDcorrespondstorequiring that aBooleancapacity

β

isdominatedbyitsconjugate

β

c

(A)

=

1

− β(

Ac

)

,i.e.,

β(

A)

≤ β

c

(A)

,forall A

⊆ Ä

.Wecallsuch a Boolean capacity pessimistic while its conjugate

β

c isthen said to be optimistic [29]. For pessimistic (resp. optimistic) capacities,itisequivalenttohavemin

(β(A),

β(

Ac

₎₎

₌

_{0 (resp.}_max

_(β(

_A),

_β(

_Ac

₎₎

₌

₁₎_for_all _A

_{⊆ Ä}

_.

PessimisticBooleancapacitieswerecharacterizedbymeansoftheirfocalsetsin[29]:

Proposition1.ABooleancapacityobeysmin

(β(

A),

β(A

c

))

=

0 forallA

⊆ Ä

ifandonlyifitsfocalsetsintersect.

Viewing the focal sets asinformation comingfrom varioussources, it is clearthat the sources leading to pessimistic Boolean capacitiesare globally coherentsince theinformationitemsdonot contradicteach other.Ifthere weretwo non

2 _In_[₃₀_],_it_is_called_QC_logic,_where_QC_stands_for_qualitative_capacities,_i.e.,_mappings_γ_:₂Ä_→_L,_where_{L is}_a_finite_bounded_chain_with_bottom

0 and top 1. A logic for qualitative capacities would extend the one for Boolean ones using atomic formulas of the form✷λp standing forγ([p]) ≥ λ,

forλ ∈L\ {0}. QC logic is a natural name for this multimodal setting, while here we use✷_λp with a fixedλ, which comes down to using a Boolean capacity. Such a QC logic would be to BC logic what generalized possibilistic logic GPL [31] is to the logic of necessity measures MEL [8], mentioned in the introduction.

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overlappingfocalsetsE and F therewouldbesomesetA containingE anddisjointfromF ,andthenmin

(β(

A),

β(A

c

₎₎

₌

_1.

So,noconflictsituation(Ei

⊆

A,Ei

⊆

Ac forsomei

6=

j)can beexpressedbypessimistic capacities.However,information

itemsfromthesourcescanbeincomplete.

Remark1. InBC, we can define a newmodality

¥

such that

¥p

≡ ✷p

∧ ✸p whose

dual is

¨p

≡ ✷p

∨ ✸p.

It is clear that

(¥,

¨)

satisfiestheaxiomsofBC andalsoaxiom D. Itreflects thefact that givenanyB-capacity

β

, thesetfunction

β∗(A)

=

min

(β(

A),

β

c

(A)),

∀A

⊂ Ä

isapessimisticB-capacity[29].

2.5.Specialcase:conflictinginformation

Anoppositeinterestingspecialcaseiswhenthefocalsetsaresingletons

{w1},

. . .

,

{w

n

}

.Thissituationcorrespondston

totallyinformedbutconflictingsources,eachofwhichiscapableofassertingthatanyproposition p istrueorfalse,while theyareintotaldisagreement.Then,asseenearlier,thecorrespondingBooleancapacity

5

isapossibilitymeasuredefined by

5(A)

=

1 ifandonlyif

∃w

i

∈

A,whichisequivalentto

5(A)

=

1 ifandonlyif A

∩

E

6= ∅

,withE

= {w1,

. . . ,

wn

}

.

Inpreviouspublications,wehaveshownthatPriest logiccanbe encodedinMEL[19–21], providedthat

5([p])

=

1 is understoodas“atleastonesourcein E supports p”andisexpressedas

✸p,

contrarytotheconventionusedhereforBC logic.The fact that in Priest logic p

∨ ¬p is

a tautology is then expressedby the fact that

✸

p

∨ ✸¬p is

a tautology in MEL.Asweusepossibilitymeasuresformodelingconflict,thesituationofignorancecannotbeexpressedusingpossibility measuresunderthisconventionsinceitneverholdsthat

5([p])

= 5(¬[p])

=

0.

ConsiderdroppingaxiomK,andaddingonlyDtoBCaxiomsfromtheMELsettingdevotedtoPriestLogic.Then,keeping theconvention of

β([p])

=

1 as

✷

p when

β

is pessimistic,and

✸

p if itisoptimistic, wenote thataxiom Dnowreflects thepropertymax

(β(A),

β(A

c

₎₎

₌

_{1 as}_{characteristic}_of_optimistic_capacities._We _can_specify_this_notion_using_focal_sets_(a

resultnothighlightedin[29]):

Proposition2.ABooleancapacityobeysmax

(β(

A),

β(

Ac

₎₎

₌

_{1 for}_all_A

_{⊆ Ä}

_if_and_only_if_it_possesses_at_least_one_focal_set_that_is_a

singletondisjointfromtheotherfocalsets.

Proof. If no focal set Ei is a singleton,then one can choose A such that neither A nor its complementcontain any Ei

(splittingeachfocalsetintotwonon-emptyparts).Then

β(

A)

= β(A

c

)

=

0.Suppose

β

hasonesingletonfocalset.Then,it canonlybedisjointfromtheotheronessince

F

_β isanantichain.Thus,foranyset A,either A contains w1 and

β(A)

=

1 orA doesnot,henceitcontainsanotherfocalset,and

β(A

c

₎

₌

_1.

_✷

In terms of information supplied by sources, Proposition 2 tells that an optimistic capacity corresponds to a set of sources, atleastone of whichis fullyinformed andcontradicts all other sources, which ensures that optimistic Boolean capacitiesaccountforconflictingsourcesandforbidignorance(atleastonesourcealwaysinformsaboutanypropositionp). WecannoticethattheroleofaxiomKisnotmoreessentialinthemodelingofconflictinginformationthanforincomplete information.

Inthe followingwe try to showthat the framework ofBoolean capacities andits logicare capable ofaccountingfor somepreviouslogicalapproachestothehandlingofinconsistentandincompleteinformationduetoconflictingsources.We startwithoneoftheoldestapproaches,thefour-valuedlogicofDunnandBelnap.

3. Belnap–Dunnfour-valuedlogic

Belnap–Dunnlogic[32,10,9] handlesbothincompleteandinconsistent informationcomingfromseveralsources.Forty yearslater,itremains oneofthemostinfluentialapproachestothisproblem. Thislogicisalsomathematicallyimportant asithasfocused attentionon aspecific algebraic structurecalleda bilattice,that hasbeenextensivelystudied andused fornon-monotonic reasoning and logic programming[36,33,13]. This section recalls the intuition behind thislogic, and presentsbothits four-valuedsemanticsandits syntax,aswellasits connectionswitha three-valuedlogic ofincomplete information (Kleenelogic) andanother three valued paraconsistent logic (Priest Logic ofParadox). Indeed, Belnap–Dunn four-valuedlogiccan be viewed,fromthepoint ofview ofits truth tables, asanaugmented Kleene logic: ontop ofthe truth-valuesupposedtorepresentignorance,oneaddsanothertruth-valuerepresentingconflict.However,thisfour-valued logiccan alsobeconsidered asmorebasicthanKleene logicasithaslessinference rulesfromasyntactic pointofview [51]. Likewise,adding theexcluded middlelawto Belnap–Dunnlogic yields PriestLogicof Paradox[48]. Thematerial in thissectionisscatteredinvariouspublications[32,10,9,48,35],anditisusefultopresentasynthesisinasingleplace. 3.1. TheBelnapmultisourceset-up

Belnap[10,9] considersanartificialinformationprocessor,fedfromavarietyofsources,andcapableofanswering queries onpropositionsofinterest.Inthiscontext,inconsistencythreatens,allthemoresoastheinformationprocessorissupposed nevertosubtractinformation.Thebasicassumptionisthatthecomputerreceivesinformationaboutatomicpropositions in

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Table 1

Belnap–Dunnnegation,disjunctionandconjunction. ¬ F T U U C C T F ∨ F U C T F F U C T U U U T T C C T C T T T T T T ∧ F U C T F F F F F U F U F U C F F C C T F U C T

acumulativewayfromoutsidesources,eachassertingforeachatomicpropositionwhetheritistrue,false,orbeingsilent (hence ignorant) aboutit. The notionof epistemicset-up is definedas an assignment, of one offour so-called epistemic truth-values,heredenotedbyT

,

F

,

C

,

U,toeachatomicpropositiona,b,

. . .

:

1. AssigningT toa meansthecomputerhasonlybeentoldthata istrue(1)byatleastonesource,andfalse(0)bynone. 2. AssigningF toa meansthecomputerhasonlybeentoldthata isfalsebyatleastonesource,andtruebynone. 3. AssigningC toa meansthecomputerhasbeentoldatleastthata istruebyonesourceandfalsebyanother. 4. AssigningU toa meansthecomputerhasbeentoldnothingabouta.

InBelnap’sepistemicsetting,theroleofthecomputerisnot tointerprettheinformationprovided bythesources,but justtostoreit.Inparticular,anypieceofinformationsupplied isviewedasalogicalatomtowhichanepistemicqualifier isattachedaspertheaboverules.Havingattachedanepistemictruth-valuetoallatomsofthelanguage

L

,thisassignment canbeextendedtoallformulasin

L

usingthetruthtablesinTable1.

If

{

0

,

1

}

isthesetofusualtruth-values(asassignedbytheinformationsourcestoatoms),thentheset

V4

= {

T

,

F

,

C

,

U

}

ofepistemictruth-valuescoincideswiththepowersetof

{

0

,

1

}

,lettingT

= {

1

}

,F

= {

0

}

,C

= {

0

,

1

}

,U

= ∅

,accordingtothe conventioninitiatedbyDunn[32].Then,theemptysetcorrespondstonoinformationreceived,while

{

0

,

1

}

representsthe presenceofconflictingsources,someclaimingthetruthofa someitsfalsity.C expressesanexcessoftruth-values,theset

{

0

,

1

}

beingviewedasexpressing0 and1 atthesametime.

Belnap’sapproachreliesontwoorderingsin

V4

= {

T

,

F

,

C

,

U

}

,equippingitwithtwolatticestructures:

•

Theinformationordering,

❁

whosemeaning is “less informativethan”, such that U

❁

T

❁

C

;

U

❁

F

❁

C. Thisordering reflectstheinclusionrelationofthesets

∅

,

{

0

},

{

1

}

,and

{

0

,

1

}

.Itisinfactalsoinagreementwiththespecificityordering ofpossibilitytheory[24].Itintendstoreflecttheamountof(possiblyconflicting)dataprovidedbythesources.U isat thebottom because(to quote)“itgivesnoinformationatall”. C isatthetopbecause(following Belnap)it givestoo much information.Ityields theinformationlattice, a Scottapproximation lattice withjoinandmeet definedby union andintersectionofsetsoftruth-values(inthislattice,themaximumofT andF is C).

•

Thetruthordering,

<

t,representing“moretruethan”accordingtowhichF

<

tC

<

tT andF

<

tU

<

tT,eachchain

reflect-ing thetruth-setofKleene’slogic.In otherwords,ignorance andconflictplaythe samerolewithrespect to F and T accordingto thisordering.Ityields thelogicallattice, basedonthe truthordering, andthe intervalextension of

∧

,

∨

and

¬

from

{

0

,

1

}

to2{0,1}

_{\ {∅}}

_,_as_appears_in_Table₁_._In_this_lattice,_the_maximum_of_{U and}_{C is T and}_their_minimum isF.

Remark2.Theconventionforrepresentingepistemicstancesintheprevioussection,inagreementwithpossibilitytheory [27],isoppositetoDunn’s,astheset

{

0

,

1

}

representsthehesitationbetweentrue andfalse andmeans0or1,justlikean epistemicstate E isadisjunctionofpossibleworlds.Then,conflictisexpressedbytheemptysetoftruth-values(reflecting thefactthataconflictingformulahasnomodel).Underthisconvention,subsetsof

{

0

,

1

}

representconstraintsonmutually exclusivetruth-values.Denotingby

π

aBooleanpossibilitydistribution,U canbe encodedby

π

(

0

)

=

π

(

1

)

=

1 (represent-ing ignorance), C by

π

(

0

)

=

π

(

1

)

=

0 representing the contradiction(correspondingto no possible truth-valueleft).The informationorderingbetweenepistemictruth-valuesinducedbythisconventionisthesameasintheBelnap–Dunnsetting, usingtheoppositeinclusionorderingbetweensubsetsof

{

0

,

1

}

.

3.2. Thesyntax

Consider thepreviously introduced propositional language

L

withvariables V

= {a,

b,c,

. . . }

withconnectives

∧,

∨,

¬

, for conjunction, disjunction and negation, respectively. Formulas p in

L

are generated as usual. However, as explained above,connectivesofnegation,conjunctionanddisjunctionaredefinedtruth-functionallyinBelnap–Dunnfour-valuedlogic, followingtherulesofthebilattice(seeTable1).

Belnap–Dunnfour-valuedlogichasnotautologies,hencenoaxioms.Itcanbedefinedonlyviaasetofinferencerules, asthosethatcanbefoundin[48,35]:

Definition2.Leta,b,c

∈

V .TheBelnap–Dunnfour-valuedlogic isdefinedbynoaxiomandthefollowingsetofrules

(

R1

) :

a

∧

b a

(

R2

) :

a

∧

b b

(

R3

) :

a b a

∧

b

(

R4

) :

a a

∨

b

(9)

(

R5

) :

a

∨

b b

∨

a

(

R6

) :

a

∨

a a

(

R7

) :

a

∨ (

b

∨

c

)

(

a

∨

b

) ∨

c

(

R8

) :

a

∨ (

b

∧

c

)

(

a

∨

b

) ∧ (

a

∨

c

)

(

R9

) :

(

a

∨

b

) ∧ (

a

∨

c

)

a

∨ (

b

∧

c

)

(

R10

) :

a

∨

c

¬¬

a

∨

c

(

R11

) :

¬(

a

∨

b

) ∨

c

(¬

a

∧ ¬

b

) ∨

c

(

R12

) :

¬(

a

∧

b

) ∨

c

(¬

a

∨ ¬

b

) ∨

c

(

R13

) :

¬¬

a

∨

c a

∨

c

(

R14

) :

(¬

a

∧ ¬

b

) ∨

c

¬(

a

∨

b

) ∨

c

(

R15

) :

(¬

a

∨ ¬

b

) ∨

c

¬(

a

∧

b

) ∨

c

Thesyntactic inference

Ŵ

⊢

B p, where

Ŵ

⊆ L

is aset offormulas,means that p can bederived from

Ŵ

usingthe above

inferencerules.

Theserulesexpressthatconjunctionisidempotent,andityieldsamorespecificpropositionthaneachoftheconjuncts (R1,R2,R3);disjunctionisidempotentandityieldsalessspecificpropositionthanitsdisjuncts(R4,R5,R6).Disjunctionis associative(R7),conjunctionisdistributiveon disjunction(R8)andconversely(R9).Negationisinvolutive(R10,R13)and DeMorganLawsaresatisfied(R11,R12,R14,R15).ItmakesitclearthattheunderlyingalgebraisaDeMorganalgebra.

Formulasinthislogiccanbeputinnormalform,byapplyingtheaboverules.Namely, distributivity,idempotenceand DeMorganlawsimplythata formulacan beexpressedasaconjunctionofclauses: p

=

p1

∧ . . . ∧

pn,wherethe pi’sare

disjunctionsofliteralsli j

=

a or

¬a,

fora

∈

V .

3.3.Thesemantics

Consideragainthefourepistemictruth-values

{

F

,

U

,

C

,

T

}

formingthebilattice

V4

.ABelnap–Dunnvaluation isa map-ping v4

: L

7→

V4

constructed from an assignment of epistemic truth-values V

→

V4

to atoms and the truth tables of Table1. Let

Ŵ

⊆ L

and p

∈ L

, thenwe define thesemantic consequence relationby means of thetruth ordering

≤

t as

follows:

Ŵ ²

Bp iff

∃

p1

, . . . ,

pn

∈ Ŵ, ∀

v4

,

v4

(

p1

) ∧ . . . ∧

v4

(

pn

) ≤

t v4

(

p

)

Usually,semantic inference isdefinedvia thepreservation of truth,andmore generallyofdesignated truth-values.In Belnap–Dunnsettingnaturaldesignatedtruth-valuesare

{

C

,

T

}

,sinceintheirset-theoreticencodingunderDunn’s conven-tionsbothinclude1.Analternativeconsequencerelationcanthusbedefinedby:

Ŵ ²

C p iff

∀

v4

,

if

∀

pi

∈ Ŵ,

v4

(

pi

) ∈ {

C

,

T

}

then v4

(

p

) ∈ {

C

,

T

}

Duetothefact thatU andC playthesamerole inthebilattice,Font[35] provesthattheconsequencerelation

Ŵ

²

C p

isequivalentto

Ŵ

²

U p,definedby:

Ŵ ²

U p iff

∀

v4

,

if

∀

pi

∈ Ŵ,

v4

(

pi

) ∈ {

U

,

T

},

then v4

(

p

) ∈ {

U

,

T

}

using

{

U

,

T

}

asdesignatedtruth-values.Font[35] alsoindependentlyprovesthat

Ŵ

²

B p iff both

Ŵ

²

U p and

Ŵ

²

C p.Thus,

eachoftheconsequencerelations

²

C and

²

U isequivalentto

²

B.WegiveexplicitproofsinAppendixA,inthecaseoftwo

premises,forthesakeofself-containedness.

Itshouldbeclearthat Belnap–Dunnlogichasneithertautologiesnorcontradictions,e.g.,a

∨ ¬a is

notatautology,nor isa

∧ ¬a a

contradiction.

Example2.a

∨

b

2

Ba

∨ ¬a since

forinstanceitfailsifv4

(b)

=

T and v4(a)

=

v4(¬a)

=

U.Alsoa

∧ ¬a

2

Bb

∨ ¬b,

sincefor

instanceitfailsifv4(a)

=

C

,

v4(b)

=

U.

The agreementbetween theHilbert-style axiomatizationof Belnap–Dunnlogic andthe above semantics isproved by Pynko[48] andFont[35]:

Theorem1.Belnap–Dunnlogicissoundandcompletewithrespecttothebilatticesemanticswithdesignatedtruth-values

{

C

,

T

},

that is

Ŵ

⊢

Bp iff

Ŵ

²

Bp.

3.4.Specialcases

TwospecialcasesofBelnap–Dunnlogic,namelyPriestLogicofParadox[46,47] andKleenelogic[40],thatusethesame language,canbeobtainedbyaddinganaxiomandaninferencerulerespectively.

In Priest Logic ofParadox [46,47], the truth-set is restricted to the three linearly ordered truth-values

P3

= {

F

,

C

,

T

}

, andthe designated set of truth-values is

{

C

,

T

}

; it comes down to merging U and F inBelnap

V4

. The truth tablesfor

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conjunction, disjunctionandnegationareobtainedfromtherestrictionofBelnap–Dunntruth tablesto

P3

.Itisthenclear that, letting v3 denotea three-valuedtruth assignment ranging on

P3

, we have

∀a

∈

V

,

v3(a

∨ ¬a)

∈ {

C

,

T

}

,which gives intuitionforthefollowingresultbyPynko[48]:

Proposition3.AHilbertsystemfor PriestLogicofParadoxisobtainedbyBelnap–Dunnlogicinferencerulesplusaxioma

∨ ¬a.

SemanticinferenceintheLogicofParadoxisthendefinedby:

Ŵ ²

P p iff

∀

v3

,

if v3

(

pi

) ∈ {

C

,

T

}, ∀

pi

∈ Ŵ

then v3

(

p

) ∈ {

C

,

T

}

Anysentenceoftheformp

∨ ¬

p isthenclearlyatautology,since v3(p

∨ ¬p)

∈ {

C

,

T

}

foranypropositionp intheLogicof Paradox.Infactalltautologiesofpropositionalcalculuscanberecovered,butmodusponensdoesnotholdinthislogic.

Onthe other hand,thestrong Kleenelogic isanother extensionof Belnap–Dunnlogicthat uses astruth-seta Kleene lattice

K3

= {

F

,

U

,

T

}

(mergingC andT).ThislogichasthesametruthtablesastheLogicofParadox,ifweidentifyU and C. Howeverthereisonedesignatedtruth-valueT,3sothatsemanticinference

²

K isoftheform

Ŵ ²

K p iff

∀

v3

,

if

∀

pi

∈ Ŵ,

v3

(

pi

) = {

T

},

then v3

(

p

) = {

T

}

ByaddingoneruletoBelnap–Dunnlogic,wegetaHilbertsystemforKleenelogic,asrecentlyexplainedbyAlbuquerque etal.[1],correctingaclaimin[35].

Proposition4.AHilbertsystemforstrong KleenelogicisobtainedbyBelnap–Dunnlogicinferencerulesplustheruleofsuppression ofcontradictions:

SC:a

∨ (

b

∧ ¬

b

)

a

.

Togetanintuitionofwhythisisso,we borrowsomeremarksfrom[1].First,itiseasy torealizethedualitybetween PriestandKleenelogicasp

⊢

Kq ifandonlyif

¬q

⊢

P

¬p,

since

v3

(

p

) =

T implies v3

(

q

) =

T

⇐⇒

v3

(¬

q

) ∈ {

F

,

U

}

implies v3

(¬

p

) ∈ {

F

,

U

}.

ThisisincontrastwithBelnap–Dunnlogicwhere p

²

Bq ifandonlyif

¬q

²

B

¬p.

Itiseasytoseethatrule(SC)is

semanti-callyvalidinstrongKleenelogic.Conversely,supposep

⊢

Kq,thatis,

¬q

⊢

P

¬p,

usingcompletenessoftheLogicofParadox.

It meansthatthere aresome atomicformulasai

,

i

=

1

,

. . . ,

n suchthat

¬q

∧

V

ni=1

(a

i

∨ ¬a

i

)

⊢

B

¬p in

Belnap–Dunnlogic.

Italsoreads p

⊢

Bq

∨

W

in=1

(a

i

∧ ¬a

i

)

,and,using(SC)q

∨

W

ni=1

(a

i

∧ ¬a

i

)

impliesq.So,inferencerelation

⊢

K iscapturedby

Belnap–Dunnlogicrules(R1–15)plus(SC).

Itisinterestingtonotice,asdonein[1],thatrule(SC)isequivalenttotheresolutionrule

RR:a

∨

b

,

c

∨ ¬

b

a

∨

c

.

Thisisbecauseontheonehand

(a

∨b)

∧ (c

∨ ¬b)

⊢

B

(a

∧

c)

∨ (a

∧ ¬b)

∨ (b

∧c)

∨ (b

∧ ¬b)

(distributivity)using(R1–15),which

implies

(a

∧

c)

∨ (a

∧ ¬b)

∨ (b

∧

c),using(SC),whichinturnimpliesa

∨

c using(R1–15).So(R1–15)

+

(SC) implies(RR). Conversely, thepremisseof(SC)alsoreads

(a

∨

b)

∧ (a

∨ ¬b)

using(R1–15).Then (SC)is (RR)wherea

=

c.Theinference systemmade ofthe15 rulesof Belnap–Dunnlogic plusresolution isthus soundandcomplete withrespectto semantic entailmentinstrongKleenelogic.

Remark3 (Nothingbutthetruth). As a consequence, modus ponens is validin Kleene logic, noticing that from

{b,

¬b

∨

c}, one can deduce

{b

∨

c,

¬b

∨

c} from (R4), and then c using (RR). However, adding modus ponens to the 15 rules of Belnap–Dunnlogic does not yieldKleene logic.It corresponds tousing T as the singledesignated value for semantic inferenceinBelnap–Dunnsetting,whilekeepingthefourepistemictruth-values.Thisfour-valuedlogic,preserving“nothing butthetruth”(andcalledNBT),hasbeenstudiedin[51,45].

Remark4(Kleenelogicoforder).ThefollowinginferenceruleholdsinstrongKleenelogic:

R16:a

∧ ¬

a

b

∨ ¬

b

since a

∧ ¬a

⊢

B

(a

∨

b) ∧ (¬a

∨ ¬b)

which entailsb

∨ ¬b,

using (RR). It also holds inNBT logic. AlthoughR16 reflects a

known property of Kleene algebras, it is insufficient to recover Kleene logic when added to Belnap–Dunn logic rules as

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explainedin[1],contrarytowhatisclaimedin[35]. Infact,(R16)isvalidinbothPriest andKleene logicsaswellasthe rule:

R17:

(

a

∧ ¬

a

) ∨

c

(

b

∨ ¬

b

) ∨

c

AddingR17toBelnap–Dunnlogicrules (R1–15)providesaHilbertstylesystemfortheso-calledKleenelogicoforder

K

≤ whichconsidersconsequencescommontoboththeLogicofParadoxandthestrongKleenelogic(seeTh.3.4in[1]).Ithas thesame three-valuedtruth set asstrong Kleene andPriest logics (say

V3

, asC and U areequal in

K

≤_), _and_semantic entailmentisoftheform

Ŵ ²

≤p iff v3

(∧

pi∈Ŵpi

) ≤

v3

(

p

)

iff

∀

v3

,

if v3

(

pi

) ∈ {

C

,

T

} ∀

pi

∈ Ŵ,

then v3

(

p

) ∈ {

C

,

T

}

and

∀

v

,

if v3

(

pi

) =

T

∀

pi

∈ Ŵ,

then v3

(

p

) =

T

.

OtherextensionsofBelnap–Dunnlogicarestudiedin[51,45,1].

4. ATranslationofBelnap–DunnlogicintoBC

Inprevious papers [20,21], we haveproposed a consequence-preserving translationofKleene and Priestlogics inthe two-tieredlogicMEL[8] withsemanticsintermsofnecessityandpossibilitymeasures.Theseresultsstronglysuggestthat Belnap–DunnlogiccanbeinturnexpressedinBC.Indeed,Belnap–Dunntruth-valuesdescribedintheprevioussectioncan beexpressedbymeansofpairs

(β(A),

β(

Ac

₎₎

_introduced_in_Section₂_,_restricting _{A to}_sets_of_classical_models

_[a]

_of_atomic

formulasa,namely:

•

v4(a)

=

T isinterpretedas

β([a])

=

1 and

β([¬a])

=

0

•

v4(a)

=

F isinterpretedas

β([a])

=

0 and

β([¬a])

=

1

•

v4(a)

=

U isinterpretedas

β([a])

=

0 and

β([¬a])

=

0

•

v4(a)

=

C isinterpretedas

β([a])

=

1 and

β([¬a])

=

1

Basedonthisintuition,thetranslationofBelnap–DunnlogicintoBCnaturally follows.Resultsofthissectionwerealready announcedintheconferencepaper[22].

4.1. Principleofthetranslation

As mentioned inSection 2,

✷p is

encoding

β([p])

=

1 for a Boolean capacity, whichclearly meansthat atleast one source(correspondingtoafocalsetin

F

_β)assertsp.Thecasewhere

β([¬p])

=

0 thuscorrespondsto

✸p

= ¬✷¬

p,which clearlymeansthatnosourceisasserting

¬p.

Hence,formulasinBCcanberelatedtoBelnap–Dunntruth-values[24]:

• ✷p

∧ ♦p holds

when

β([p])

=

1 and

β([¬p])

=

0 (relatedtotheepistemictruth-valueT).

• ✷¬p

∧ ♦¬p holds

when

β([¬p])

=

1 and

β([p])

=

0 (relatedtotheepistemictruth-valueF).

• ♦p

∧ ♦¬p holds

when

β([p])

=

0 and

β([¬p])

=

0 (relatedtotheepistemictruth-valueU).

• ✷p

∧ ✷¬

p holdswhen

β([p])

=

1 and

β([¬

p])

=

1 (relatedtotheepistemictruth-valueC).

Belnap–Dunnapproachfirstassignsepistemictruth-valuestoatomicpropositionsonly.Wecanencodethesetruth-qualified atomsintothemodallanguage ofBCasfollows.Let

T

denotethetranslationoperationthat takesapartialBelnap–Dunn truth-valueassignmentoftheformv4(a)

∈ 2

⊆

V4

toatomicpropositionalformulas,(indicatingtheirepistemicstatusw.r.t asetofsources),andturnsitintoamodalformulainthelogicBC.Forinstance:

T

(

v4

(

a

) =

T

) = ✷

a

∧ ♦

a

T

(

v4

(

a

) =

F

) = ✷¬

a

∧ ♦¬

a (1a)

T

(

v4

(

a

) =

U

) = ♦

a

∧ ♦¬

a

T

(

v4

(

a

) =

C

) = ✷

a

∧ ✷¬

a (1b)

T

(

v4

(

a

) ≥

tC

) = ✷

a

T

(

v4

(

a

) ≤

tC

) = ✷¬

a (1c)

T

(

v4

(

a

) ≥

tU

) = ♦

a

T

(

v4

(

a

) ≤

tU

) = ♦¬

a (1d)

(where

≥

t referstothetruthordering).InBelnap–Dunnlogic,sincevaluationsofpropositionsotherthanelementaryones

areobtainedviatruthtables,thetranslationof

V4

-truth-qualifiedformulaswillbecarriedoutbyrespectingthetruthtables ofthelogic.However,notallformulasof

L✷

canbereachedviathetranslation:asweshallsee,onlyliteralscanappearin thescopeofmodalities.

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Table 2

Translationtablefornegation,conjunctionanddisjunction.

T v4(¬a) v4(a∧b) v4(a∨b) =T ✷¬a∧ ✸¬a ✷a∧ ♦a∧ ✷b∧ ♦b ✷a∨ ✷b∨ (✸a∧U b) ∨ (✸b∧U a) =F ✷a∧ ✸a ✷¬a∨ ✷¬b∨ (✸¬a∧U b) ∨ (✸¬b∧U a) ✷¬a∧ ♦¬a∧ ✷¬b∧ ♦¬b ≥tU ✸a ✸a∧ ✸b ✸a∨ ✸b ≥tC ✷a ✷a∧ ✷b ✷a∨ ✷b ≤tU ✸¬a ✸¬a∨ ✸¬a ✸¬a∧ ✸¬a ≤tC ✷¬a ✷¬a∨ ✷¬a ✷¬a∧ ✷¬a

4.2. TranslatingformulasfromBelnap–DunnlogictoBC

WecanproceedtothetranslationofBelnap–DunntruthtablesintoBC.Firstconsidernegation.Itiseasytocheckthat

T

(

v4

(¬

a

) =

T

) =

T

(

v4

(

a

) =

F

)

T

(

v4

(¬

a

) =

F

) =

T

(

v4

(

a

) =

T

)

T

(

v4

(¬

a

) =

x

) =

T

(

v4

(

a

) =

x

),

x

∈ {

U

,

C

}

T

(

v4

(¬

a

) ≥

tx

) =

T

(

v4

(

a

) ≤

tx

),

x

∈ {

U

,

C

}

Forcompoundformulasbuiltwithconjunctionanddisjunction,weget:

T

(

v4

(

p

∧

q

) =

T

) =

T

(

v4

(

p

) =

T

) ∧

T

(

v4

(

q

) =

T

)

T

(

v4

(

p

∨

q

) =

T

) =

T

(

v4

(

p

) =

T

) ∨

T

(

v4

(

q

) =

T

)

∨(

T

(

v4

(

p

) =

U

) ∧

T

(

v4

(

q

) =

C

)) ∨ (

T

(

v4

(

p

) =

C

) ∧

T

(

v4

(

q

) =

U

))

T

(

v4

(

p

∧

q

) ≥

tx

) =

T

(

v4

(

p

) ≥

tx

) ∧

T

(

v4

(

q

) ≥

tx

)

T

(

v4

(

p

∨

q

) ≥

tx

) =

T

(

v4

(

p

) ≥

tx

) ∨

T

(

v4

(

q

) ≥

tx

)

T

(

v4

(

p

∧

q

) ≤

tx

) =

T

(

v4

(

p

) ≤

tx

) ∨

T

(

v4

(

q

) ≤

tx

)

T

(

v4

(

p

∨

q

) ≤

tx

) =

T

(

v4

(

p

) ≤

tx

) ∧

T

(

v4

(

q

) ≤

tx

)

withx

∈ {

U

,

C

}

.Forelementaryformulas

¬a,

a

∨

b,a

∧

b,wegiveexplicittranslationsusingthetruthtablesofBelnap–Dunn logicinTable2.Forsimplicityweshorten

♦a

∧ ♦¬a as

U a (a isunknown).Then,sometranslationscanbesimplifiedasfor theexpressionsof

T (v4(a

∨

b)

=

T

)

and

T (v4(a

∧

b)

=

F

)

.Thefirstoneisoftheform(aspertheaboveresults):

(✷

a

∧ ♦

a

) ∨ (✷

b

∧ ♦

b

) ∨ (✷

a

∧ ✷¬

a

∧

U b

) ∨ (

U a

∧ ✷

b

∧ ✷¬

b

).

Developing

(✷a

∧ ♦a)

∨ (✷a

∧ ✷¬a

∧

U b)simplifiesinto

✷a

∨ (✸a

∧

U b)andlikewiseexchanginga andb.Theresultappears onTable2line2,lastcolumn.Theresultforv4(a

∧

b)obtainssimilarlyonline3,column3ofthistable.

Beyondtheirtechnicalities,thesetranslationslaybarethepricepaidbyassumingatruth-functionalapproachtomultiple sourcereasoning,namelywecannotreallyassignanepistemicstatustogenuinecompoundBooleanformulas.Forinstance, consider the translation

T (v4(a

∧

b)

=

T

)

= ✷a

∧ ♦a

∧ ✷b

∧ ♦b.

It says that attaching the epistemic truth-value T toa conjunctionofatomsinBelnap–DunnlogicjustcomesdowntohavingtheBooleanatoma confirmedbyonesource(

✷a)

andnotdisconfirmedby otherones(

♦a),

andlikewisefortheBooleanatomb,whichisunsurprising.Butittellsnothing aboutthe epistemicstatusofthe Boolean formulaa

∧

b.Insome sense, thetruth-functionalconjunctionin Belnap–Dunn logicbecomestrivialinBC,sinceinBC,thetruth-functionalitystatementv4(a

∧b)

=

v4(a)

∧

v4(b)(Table1)justtellsusthat theconjunctionof

✷a

∧ ♦a and

✷b

∧ ♦b is

✷a

∧ ♦a

∧ ✷b

∧ ♦b;

itdoesnotinformaboutthetruthof

✷(a

∧

b)

∧ ♦(a

∧

b).In fact,inBC,wedonothavethat

✷a

∧ ♦a

∧ ✷b

∧ ♦b

⊢ ✷(a

∧

b)

∧ ♦(a

∧

b)(bylackofadjunctionaxiom).Thesamecomment appliestotheothertranslations.

4.3. Belnap–DunnlogicinferenceinBC

We havetwo designated values: T and C. So,for inference purposes,we only haveto consider thetranslation of ex-pressionsoftheform

T (v4(p)

≥

tC

)

.Since

T (v4(a)

≥

tC

)

= ✷a,

and

T (v4(¬a)

≥

tC

)

= ✷¬a for

literals, theother formulas

beingtranslatedviaBelnap–Dunntruthtables,thistranslationsendsBelnap–Dunnlogicintothefollowingfragmentofthe languageofBC:

L

B_✷

= ✷

a

|✷¬

a

|φ ∧ ψ |φ ∨ ψ

wherenonegationappearsinfrontof

✷

.

Conversely,fromthefragment

L

✷B wecangobacktoBelnap–Dunnlogic.Namelyanyformula

φ

in

L

✷B canbetranslated intoaformula

θ (φ)

ofthepropositionallogiclanguagebymeansofthefollowingtranslationrules:

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• θ (✷a)

=

a

• θ (✷¬a)

= ¬a

• θ (ψ ∧ φ)

= θ (ψ)

∧ θ (φ)

• θ (ψ ∨ φ)

= θ (ψ)

∨ θ (φ)

Weremark,onlyusingthepropertiesofclassicallogic,that

✷a

∨ ✷¬a is

notatautologyinBC.Moregenerally,notautologies canbeexpressedintheabovefragment.ThisiscoherentwiththefactthatBelnap–Dunnlogichasnotheorems.Likewise,

✷

a

∧ ✷¬a is

notacontradictioninBC.NowweshowthattheinferencerulesofBelnap–DunnlogicarevalidinBCviathe proposedtranslation.

Theorem2.Let

φ

ψ

beanyoftherulesofBelnap–Dunnlogic.Then,thefollowinginferencerule

T

(

v4

(ψ ) ≥

tC

)

T

(

v4

(φ) ≥

tC

)

holdsinBC.

Proof. The result easily follows by the definition of the translation. In order to give an example,let us show why the translationofrule(R11)isvalidinBC.Atfirst,wetranslate

T (v4(¬(a

∨

b)

∨

c)

≥

tC

)

.Bythetranslationrulesfordisjunction

andnegation givenpreviously,we get

T (v4(¬(a

∨

b))

≥

tC

)

∨ T (v4(c)

≥

tC

)

andthen

[T (v

4(a)

≤

tC

)

∧ T (v4(b)

≤

tC

)]

∨

T (v

4(c)

≥

tC

)

.Then,bytherulesonatomicpropositionsgiveninTable2,weget

(✷¬a

∧ ✷¬b)

∨ ✷c.

Now,theconsequent

T (v4((¬a

∧ ¬b)

∨

c)

≥

tC

)

givesexactlythe sameexpression,that is,oncetranslated, antecedentandconsequentare the

same.

✷

AsaconsequencewecanmimicksyntacticinferenceofBelnap–DunnlogicinBC, providedthat werestricttoformulas in

L

B✷.

4.4.CapacitysemanticsoftheBelnap–DunnfragmentofBC

Therestriction ofthe scopeof modalitiestoliterals requiredby thetranslationofBelnap–Dunn logicinto BC also af-fectsthesetofB-capacities thatcan act asa semanticcounterpartof thelogic.Wecan check thatsemanticinference in Belnap–Dunnlogiccanbe expressedinthemodalsettingofBCbyconstraining thecapacitiesthatcanbeusedasmodels of

L

B✷formulas.Namely,consideraBelnapset-upwhereeachsourcei providesasetTi ofatomsconsideredtrue,asetFi

ofatomsconsideredfalse,whereTi

∩

Fi

= ∅

.Shouldthesourcebehavelikeapropositionalreasoner,thisinformationwould

correspondtoapartialmodel,whichisaspecialkindofepistemicstateofrectangularshape,namely:

Ei

= [(

^

a∈Ti a

) ∧ (

^

b∈Fi

¬

b

)].

As there are n sources, we could consider n-tuples of partial models

(E1,

. . . ,

En

)

, and restrict to capacities with such

rectangularfocalsets.

In fact,Belnap sources are not propositional reasoners,and pieces of atomicinformation they provide cannot be ex-pressedbytheirpropositionalconjunctions.Thisisasiftherewere,behindeachsourcei,asmanyprimitivesourcesasthe numberofatomsinTi

∪

Fi.Thisisexpressedbytheuseofatomsin

L

B✷,oftheform

✷ℓ

where

ℓ

isaliteral.Aswecannot put

✷

infront ofconjunctionsnordisjunctions,we shouldusecapacitieswhosefocal setsare oftheform

[a],

a

∈ ∪

n_i₌₁Ti

and

[¬b],

b

∈ ∪

n_i₌₁Fi tointerpretformulasin

L

✷B.Wecallsuchcapacitiesatomic.

ConsideringtheBelnap–Dunn valuation v4 associated tothe informationsupplied by n sources,there isa one-to-one correspondencebetweenBelnap–DunnvaluationsandatomicB-capacities

α

inducedbythisinformation:

Proposition5.ForeachBelnap–Dunnvaluationv4,thereexistsauniqueatomicB-capacity

α

v4 suchthatv4

|=

p ifandonlyif

α

v4

|= T (v4(p)

≥

tC

).

Proof. GivenBelnap–Dunnvaluation v4,defineT

= {a

:

v4(a)

=

T orC

}

,F

= {a

:

v4(a)

=

F orC

}

,andlet

α

([a])

=

1 ifa

∈

T ,

α

([¬a])

=

1 ifa

∈

F .Clearly,ifa

∈

T ,thenv4

|=

a and

α

|= T (v

4(a)

≥

tC

)

= ✷a.

Ifa

∈

F ,thenv4

|= ¬a and

α

|= T (v4(¬a)

≥

t

C

) = ✷¬a.

TheremainderfollowsusingtruthtablesofBelnap–DunnlogictranslatedintoBC.

✷

Intheotherwayaround,

Proposition6.ForanyB-capacity

β,

thereisasingleBelnap–Dunnvaluationvβ₄ suchthat

β |= φ ∈ L

✷B ifandonlyifv β 4

(θ (φ))

∈

{

C

,

T

}.