Boolean algebras of conditionals, probability and logic

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Contents lists available atScienceDirect

Artificial

Intelligence

www.elsevier.com/locate/artint

Boolean

algebras

of

conditionals,

probability

and

logic

Tommaso Flaminio

a,

, Lluis Godo

a

,

Hykel Hosni

b aArtificialIntelligenceResearchInstitute(IIIA- CSIC),CampusUAB,Bellaterra08193,Spain bDepartmentofPhilosophy,UniversityofMilan,ViaFestadelPerdono7,20122Milano,Italy

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received17May2019

Receivedinrevisedform27January2020 Accepted8June2020

Availableonline10June2020

Keywords:

Conditionalprobability Conditionalevents Booleanalgebras

Preferentialconsequencerelations

Thispaper presents aninvestigation onthe structure of conditional events and onthe probability measures whicharisenaturallyinthatcontext.In particular weintroducea constructionwhichdefinesa(finite)Booleanalgebraofconditionalsfromany(finite)Boolean algebraofevents.Bydoingsowedistinguishthepropertiesofconditionaleventswhich depend onprobabilityand those whichare intrinsicto thelogico-algebraicstructure of conditionals. Ourmain result providesa way to regard standard two-placeconditional probabilities as one-place probability functions onconditional events. We alsoconsider a logical counterpart ofour Boolean algebras of conditionalswith links to preferential consequencerelationsfornon-monotonicreasoning.Theoverallframework ofthispaper providesanovel perspectiveonthe rich interplaybetweenlogicand probability inthe representationofconditionalknowledge.

©2020TheAuthors.PublishedbyElsevierB.V.Thisisanopenaccessarticleunderthe CCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introductionandmotivation

Conditionalexpressions are pivotalin representingknowledge andreasoning abilitiesofintelligent agents.Conditional reasoningfeaturesinawiderangeofareasspanningnon-monotonicreasoning,causalinference,learning,andmore gener-allyreasoningunderuncertainty.

Thispaperproposesanalgebraicstructureforconditionaleventswhichservesasalogicalbasistoanalysetheconcept ofconditionalprobability–afundamentaltoolinArtificialIntelligence.

Atleastsince theseminal workofGaifman[22], whointurn developstheinitialideas ofhis supervisorAlfredTarski [31],ithasbeenconsiderednaturaltoinvestigatetheconditionsunderwhichBooleanalgebras–i.e.classicallogic–played theroleofthelogicofeventsforprobability.Thepointisclearlymadein[23]:

Sinceeventsare alwaysdescribed insomelanguage they canbeidentified withthesentences thatdescribe themand theprobabilityfunctioncanberegardedasanassignmentofvaluestosentences.Theextensiveaccumulatedknowledge concerningformallanguagesmakessuchaprojectfeasible.

Weareinterestedinpursuingthesameidea,buttaking

conditional probability

asaprimitivenotionandobtain uncondi-tionalprobabilitybyspecialisation.Takingconditionalprobabilityasprimitivehasalongtraditionwhichdatesbackatleast to[12] andincludes[32,44,45,51].Thekeyjustificationfordoingthisliesinthemethodologicalviewthatnoassessmentof

*

Correspondingauthor.

E-mailaddresses:tommaso@iiia.csic.es(T. Flaminio),godo@iiia.csic.es(L. Godo),hykel.hosni@unimi.it(H. Hosni).

https://doi.org/10.1016/j.artint.2020.103347

0004-3702/©2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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probabilitytakesplaceinavacuum.Onthecontrary,eachprobabilisticevaluationmustbedoneinthelightofallandonly theavailableevidence.Inthissense,anyprobabilisticassessmentofuncertaintyisalwaysconditional.

The first step in achieving our goal is to clarify how conditional knowledge and information should be represented. To do thiswe putforward a structure forrepresenting conditional events, takenas the primitiveobjects of uncertainty quantification.Inother wordsweaimtocapturethelogic/algebrawhichplaystheroleofclassicallogicwhenthefocusof probabilitytheoryisshiftedonconditionalprobability.Inourpreliminaryinvestigations[20,21] onthesubjectwesuggested takingthemethodologicalapproachofaskingthefollowingquestions:

(i) which propertiesof conditionalprobabilities depend on propertiesofthe measureanddo not depend onthe logical propertiesofconditionalevents?

(ii) whichpropertiesdoinsteaddependonthe

logic

–whateveritis–ofconditionalevents?

Bruno deFinetti was thefirstnottotake thenotionofconditionaleventsforgranted andargued thatthey cannotbe described by truth-functionalclassicallogic.He expressedthisbyreferring toconditionaleventsas

trievents

[12,14],with the followingmotivation.Since, intuitively,conditional eventsoftheform“a givenb”expresssome formofhypothetical assertion–theassertionoftheconsequent

a

based

on

the suppositionthattheantecedent

b

issatisfied–thelogical evalua-tionofaconditionalamountstoatwo-stepprocedure.Wefirstchecktheantecedent.Ifthisisnotsatisfied,theconditional ceases tomean anythingatall. Otherwisewe moveon toevaluating theconsequent andtheconditional eventtakes the samevalueastheconsequent.

ThisinterpretationalloweddeFinettitousetheclassicalnotionofuncertaintyresolutionforconditionaleventsimplicitly assumedbyHausdorffandKolmogorov,exceptforthefactthatdeFinettiallowedtheevaluationofconditionaleventstobe a

partial

function.Thisisillustratedclearly byreferringtothe

betting interpretation

ofsubjectiveprobability,whichindeed can be extended to a number of coherence-based measures of uncertainty [18,19]. To illustrate this, fix an uncertainty resolving valuation v,orinotherwordsatwo-valuedclassicallogicvaluation.Then deFinettiinterprets conditionalevents “

θ

given

φ

”asfollows: a bet on “

θ

given

φ

” is

won ifv(φ)

=

v(θ )

=

1

;

lost ifv(φ)

=

1 andv(θ )

=

0

;

called-off ifv(φ)

=

0

.

Thisideahasbeendevelopedinuncertainreasoning,withlinkswithnonmonotonicreasoning,in[17,36,35,34].Inthe context ofprobability logic, thisapproach has been pursued in detail in [9]. Note that thislatter approach is measure-theoretically oriented, and yet the semantics of conditional events is three-valued. The algebra of conditional events developed in the present paper, on the contrary, will be a Boolean algebra. Hence, as we will point out in due time, thethree-valuedsemanticsofconditionaleventsisnotincompatiblewithrequiringthatconditionaleventsformaBoolean algebra.Whatmakesthispossibleisthatuncertainty-resolvingvaluationsnolongercorrespondtoclassicallogicvaluations, asin deFinetti’swork. Rather,asitwillbe clearfromouralgebraic analysis, they willcorrespondto finite total orders of valuationsofclassicallogics.Thiscruciallyallowsfortheformalrepresentationofthe“gaps”inuncertaintyresolutionwhich arisewhentheantecedentofaconditionalisevaluatedto0,forcingthebettobecalledoff.

Some readersmaybe familiarwiththecopiousandmultifariousliteraturespanningphilosophicallogic,linguisticsand psychologywhichseekstoidentify,sometimesprobabilistically,how“conditionals”departfromBoolean(akamaterial) im-plication.Withinthisliteratureemergedaviewaccordingtowhichconditionalprobabilitycanbeviewedastheprobability of asuitablydefinedconditional.A detailedcomparisonwithaproposal, duetoVan Fraassen,in thisspirit willbe done inSubsection8.2.Howeveritmaybepointedoutimmediatelythatakeycontributionofthisliteraturehasbeenthevery useful argument,dueto DavidLewis [39],according towhich the conditioningoperator “

|

” cannot be taken, on pain of trivialisingprobability functions, tobea Booleanconnective,andinparticularmaterialimplication.Thisclearly reinforces the view,heldsince de Finetti’searly contributions,that conditionalevents havetheirown algebraandlogic.A key con-tribution ofthispaperis toargue thatthisrole can beplayed by what wetermBoolean Algebras ofConditionals(BAC). Armed withthese algebraic structures, we can proceed toinvestigate the relation betweenconditional probabilities and (plain)probability measures on BooleanAlgebras ofConditionals.In particularwe construct, foreachpositive probability measureonafiniteBooleanalgebra,itscanonicalextensiontoaBooleanAlgebraofConditionalswhichcoincideswiththe conditional probability onthestarting algebra.Hence we provideaformal settinginwhichtheprobability ofconditional eventscanberegardedasconditionalprobability.Thiscontributestoalong-standingquestionwhichhasbeenputforward, re-elaborated anddiscussed bymanyauthors alongthe years,andwhosegeneralformcan be roughlystatedasfollows: conditional probability is the probability of conditionals[1,39,49,51,26,33].1

1 TosomeextentitcanberegardedasasimplifiedversionofAdams’sthesis[1,2],claimingthattheassertibilityofaconditionala

b correlateswith

theconditionalprobabilityP(b|a)oftheconsequentbgiventheantecedenta.AmoreconcretestatementofthisthesiswasputforwardbyStalnakerby equatingAdam’snotionofassertabilitywiththatofprobability:P(ab)=P(ab)/P(a),wheneverP(a)>0,knownintheliteratureasStalnaker’sthesis [49].

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Inthelate1960logic-basedArtificialIntelligencestartedtoencompassqualitativeuncertainty.Firstthroughthenotion ofnegation-as-failureinlogicprogramming,thenwiththeriseofnon-monotoniclogics,afieldwhichowessubstantiallyto the1980doublespecialissueof

Artificial Intelligence

editedby D.G.Bobrow.Muchofthefollowingdecadewasdevotedto identifying

general patterns

innonmonotonic reasoning,inthefelicitous turnofphraseduetoDavidMakinson [41].One prominentsuch patternemerged fromthesemantical approachputforward byShoham [48]. Accordingto it,a sentence

θ

isanon-monotonic consequenceofa sentence

φ

if

θ

is(classically)satisfiedbyall

preferred or most normal models

of

φ

. This equippedthe syntactic notionof defaults– i.e.conditionalswhich are takento be defeasiblytrue – witha natural semantics: defaultsareconditionalswhichare“normally” true,where normalityiscapturedby suitablyordering classical models.Orderedmodelshavethenbeenthekeytoprovidingremarkableunity[38] tonon-monotonicreasoning,whichby theearly 2000sencompassed notonly avariety ofdefaultlogics[42],butalsoAGM-style theoryrevision [40] andsocial choicetheory[46].

Inlightofallthis,itisnoteworthythattheBooleanAlgebrasofConditionalslendthemselvestoanaxiomatisationwhich turnsouttobesoundandcompletewithrespecttoaclassofpreferential structures.ThedetailsaredeferredtoSection 7, where inaddition we show that the logic ofboolean conditionalstherein definedsatisfies the propertiesof preferential non-monotonic consequence relations,in thesense pioneered by theseminal paper[37] and refined by [38]. Inspite of itstechnicalsimplicity,wethinkthisresultismethodologicallyverysignificantforitprovidesstrongreasonsinsupportof theverydefinitionoftheBooleanAlgebraofConditionals.Inotherwords,thefactthatitslogicalcounterpartleadstothe mostwidely investigatedframework fornonmonotonicreasoning,justifiesourinterpretationofthealgebra investigatedin Section3asthealgebraof

conditionals

.

Structure and summary of contributions of the paper The paperisstructured asfollows.After thisintroductionandrecalling some basic facts about Boolean algebras in Section 2, we present in Section 3 the main construction which allows us to define,starting fromanyBooleanalgebra A ofevents,acorresponding

Boolean algebra of conditional events

C

(

A

)

. These algebras

C

(

A

)

,whoseelementsareobjectsoftheform

(

a

|

b

)

for

a

,

b

A andtheirbooleancombinations,arefiniteifthe originalalgebrasAareso.

Section4isdedicatedtotheatomicstructureofBooleanalgebrasofconditionals.Themainresultisafull characteriza-tionoftheatomsofeachfinite

C

(

A

)

intermsoftheatomsofA.Thischaracterizationisafundamentalstepfortherestof thepaper.Furtherelaboratingontheatomicstructure,Section5presentstwo

tree-like

representationsforthesetofatoms ofanalgebraofconditionalswhichwillbedecisiveinestablishingthemainresultofthepaperinSection 6.

Infact,Section6introducesprobabilitymeasuresonBooleanalgebrasofconditionalsandpresentsourmainresulttothe effectthateverypositive probability P onafiniteBooleanalgebra Acanbecanonicallyextendedtoa positiveprobability

μ

P on

C

(

A

)

whichagrees withthe“conditionalised”version oftheformer. Thatis,weprovethat, forevery

a

,

b

A with b

= ⊥

,

μ

P

(

(a

|

b)

)

=

P(a

b)/P(b).

Asawelcomeconsequenceofourinvestigationweprovideanalternative,finitary,solutiontotheproblemknowninthe literatureas

the strong conditional event problem

,introducedandsolvedintheinfinitesettingoftheGoodmanandNguyen’s ConditionalEventAlgebrasof[26].

Although our Boolean algebras ofconditionals do not allow foran equational description, the characterizing proper-tiesofthesealgebrasareexpressibleinan expansion ofthelanguage ofclassicalpropositional logicandhencethey give rise naturally to a simplelogic of (non-nested)conditionals, that we nameLBC (for

Logic of Boolean Conditionals

). Thisis investigated inSection 7,where we axiomatizethe logic andprovesoundness andcompleteness withrespect toa class ofpreferentialstructures. Moreover,we showthat LBCsatisfiesthepropertiesofpreferential non-monotonicconsequence relations,inthesensepioneeredbytheseminalpaper[37] andrefinedby[38].Finally,Section8drawssomedetailed com-parisonsbetweenour contributionsandthe research on Measure-free conditionals(Subsection8.1) andwithConditional EventAlgebras(Subsection8.2).Section9outlinesasetofkeyissuesforfuturework.

Tofacilitatethereadingofthepaper,mostproofsarerelegatedtoanappendix.

2. Preliminaries:Booleanalgebrasinanutshell

ThealgebraicframeworkofthispaperisthatofBooleanalgebrasandhenceitslogicalsettingisthatofclassical propo-sitionallogic(CPL).Here,we willbrieflyrecaponsome needednotionsandbasicresultsaboutBooleanalgebrasandCPL, foramoreexhaustiveintroductionaboutthissubjectweinvitethereadertoconsult[7,§IV],and[10,25,29].

Givena countable(finite orinfinite) set V of

propositional variables

,theCPLlanguage L

(

V

)

(orsimply Lwhen V will beclearbythecontext)isthesmallestsetcontaining

V

andclosedundertheusual connectives

∧,

∨,

¬,

,and oftype

(

2

,

2

,

1

,

0

,

0

)

.Alongthispaperwewilluse thenotation

ϕ

,

ψ

,etc(withpossible subscript)forformulas.Further,weshall adoptthefollowingabbreviations:

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Weshalldenoteby

C P LtheprovabilityrelationofCPL,inparticularwewillwrite

C P L

ϕ

todenotethat

ϕ

isatheorem. A

logical valuation

(orsimplya

valuation

) ofLisamap fromv

:

V

→ {

0

,

1

}

,whichuniquelyextendstoafunction,that we denoteby thesamesymbol v,fromLto

{

0

,

1

}

inaccordancewiththe usualBoolean truthfunctions, i.e.v

(

ϕ

ψ)

=

min

{

v

(

ϕ

),

v

(ψ)}

, v

(⊥)

=

0,

v

ϕ

)

=

1

v

(

ϕ

)

,etc.Weshalldenoteby

thesetofallvaluationsofL.Foragivenformula

ϕ

andagivenvaluation v

,wewillwrite

v

|=

ϕ

whenever v

(

ϕ

)

=

1.

Wewillbroadlyadopt,analogouslytotheaboverecalledlogicalframe,thesignature

(

,

,

¬

,

,

)

oftype

(

2

,

2

,

1

,

0

,

0

)

for the algebraic language upon which Boolean algebrasare defined. Thus, the sameconventions andabbreviationsof L can beadoptedalsointhealgebraicsetting.Further,ineveryBooleanalgebraA

=

(

A

,

,

,

¬

,

,

)

weshallwrite

a

b, whenever

a

b

=

.Therelation

isindeedthelattice-orderinA.Thus,

a

b iff

a

b

=

aiff

a

b

=

b.

Along thispaper,inorderto distinguishan algebrafromits universe,we willdenotethe formerby A,Betc, andthe latterby A,

B

etc,respectively.

Recall that amap

h

:

A

BbetweenBooleanalgebrasisa

homomorphism

if

h

commuteswiththeoperationsoftheir language, that is, h

(

A

)

=

B, h

(

¬

Aa

)

= ¬

Bh

(

a

)

,

h

(

a

Ab

)

=

h

(

a

)

Bh

(

b

)

etc, (notice that we adoptsubscripts to

distin-guish the operationsof A fromthose of B). Bijective (or 1-1) homomorphisms are called

isomorphisms

and ifthere is a isomorphismbetweenAandB,theyaresaidtobe

isomorphic

(andwewriteA

=

B).

A

congruence

ofaBooleanalgebraAisanequivalencerelation

on Awhichiscompatiblewithitsoperations(see[25, §17]),thatis,forevery

a

,

a

,

b

,

b

A,if

a

aand

b

bthen

¬

a

≡ ¬

a

,

a

b

a

band

a

b

a

b.Thecompatibility property allowsustoequiptheset A

/

= {[

a

]

|

a

A

}

ofequivalenceclasseswithoperationsinheritedfromA,endowing A

/

≡withastructureofBooleanalgebra,writtenA

/

≡,andwhichiscalledthe

quotient

ofAmodulo

.Foralateruse,we furtherrecallthatforall

a

,

a

Asuchthat

a

a,theequality

[

a

]

= [

a

]

holdsinA

/

.Recallthatforanysubset X

A

×

A, the

congruence generated by X

isthesmallestcongruence

X whichcontains

X

.Thecongruence

X alwaysexists[7,§5].

Boolean algebrasformavariety,i.e.an equationalclass,inwhich,forany(countable)set V,the

free V -generated

alge-bra Free

(

V

)

(see [7, §II]) isisomorphic to theLindenbaumalgebra ofCPL over alanguage whose propositional variables belong to V (see for instance [6]). Since thesestructures will play a quite importantrole in the main construction we willintroduceinSection3,letusbrieflyrecaponthem.Givenanyset

V

ofpropositionalvariables,wedenotebyL

(

V

)/

the setofequivalenceclassesofformulasofthelanguage L

(

V

)

modulo thecongruencerelation

ofequi-provability,i.e., two formulas

ϕ

and

ψ

are equi-provableiff

C P L

ϕ

ψ

.Thealgebra L

(

V

)

=

(

L

(

V

)/

,

∧,

∨,

¬,

⊥,

)

isa Booleanalgebra calledtheLindenbaumalgebra ofCPLoverthelanguage L

(

V

)

.Therefore,a map v

:

L

(

V

)

→ {

0

,

1

}

isa valuationiffitis a homomorphismofL

(

V

)

into2(where2denotestheBooleanalgebraoftwoelements

{

0

,

1

}

).

Definition2.1.An element

a

ofaBoolean algebraA issaidto bean

atom

of Aif

a

>

andforanyother element

b

A suchthat

a

b

≥ ⊥

,either

a

=

b or

b

= ⊥

.

ForeveryalgebraA,weshallhenceforthdenoteby

at

(

A

)

thesetofitsatomsandwewilldenoteitselementsby

α

,

β,

γ

etc. If

at

(

A

)

= ∅

,A iscalled

atomic

,otherwise Ais saidto be

atomless

. IfAis finite,thenit isatomic. Inparticular,if V is finite,theLindenbaumalgebra L

(

V

)

isfiniteaswell andthusatomic[4].Infact, if

|

· |

denotesthecardinality map,if

|

V

|

=

nthen

|

at

(

L

(

V

))|

=

2n,and

|

L

(

V

)|

=

22n

.Thefollowingpropositioncollectswell-knownandneededfactsaboutatoms (see e.g.[7,§I] and[29,§16]).Itrecalls,amongotherthings,that

at

(

A

)

isapartitionofanatomicalgebraA.Recallthat a partitionofaBooleanalgebraisacollectionofpairwisedisjointelementsdifferentfrom

whosesupremumis .

Proposition2.2.

Every finite Boolean algebra is atomic. Further, for every finite Boolean algebra

Athe following hold: (i) for every

α

,

β

at

(

A

)

,

α

β

= ⊥

;

(ii) for every a

A, a

=

αa

α

. Thus, in particular,

αat(A)

α

=

;

(iii) for each

α

at

(

A

)

, the map hα

:

A

2such that hα

(

a

)

=

1if a

α

and hα

(

a

)

=

0otherwise, is a homomorphism. Furthermore, the map

λ

:

α

hαis a 1-1 correspondence between

at

(

A

)

and the set of homomorphisms of Ain 2;

(iv) if A

=

L

(

V

)

with V finite, the map

λ

as in(iii)

is a 1-1 correspondence between the atoms of

L

(

V

)

and the set

of valuations of V .

Moreover, a subset X

= {

x1

,

. . . ,

xm

}

A coincides with

at

(

A

)

iff the following two conditions are satisfied: (a) X is a partition of A(i.e. xi

xj

= ⊥

if i

=

j, and

mi=1xi

=

);

(b)every xi

X is such that

<

xiand there is no b

A such that

<

b

<

xi. 3. Booleanalgebrasofconditionals

In thissection we introducethe notion ofBoolean algebrasof conditionalsandprove some basic properties.Forany Boolean algebraA,theconstructionweare goingtopresentbuildsa Booleanalgebraofconditionalsthatwe shalldenote by

C

(

A

)

.Inthefollowing,givenaBooleanalgebraA,wewillwrite Afor A

\ {⊥}

.

Intuitively,inaBooleanalgebraofconditionalsoverAwewillallow

basic conditionals

,i.e.objectsoftheform

(

a

|

b

)

for a

Aand

b

A,tobefreelycombinedwiththeusualBooleanoperationsuptocertainextent.Recallfromtheintroduction

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thatourmaingoalistodistinguish,asfarasthisispossible,thepropertiesoftheuncertaintymeasurefromthealgebraic propertiesofconditionals.Thismeansthat wemustpin downpropertieswhichmakesense inthe contextofconditional reasoningunderuncertainty.Thosepropertiesaresummedupinthefollowingfourinformalrequirements,whichguideour construction.

R1 Forevery

b

A,theconditional

(

b

|

b

)

willbethetopelementof

C

(

A

)

,while

(

¬

b

|

b

)

willbethebottom;

R2 Given

b

A,thesetofconditionals A

|

b

= {(

a

|

b

)

:

a

A

}

willbethedomainofaBooleansubalgebraof

C

(

A

)

,andin particularwhen

b

=

,thissubalgebrawillbeisomorphictoA;

R3 Inaconditional

(

a

|

b

)

wecanreplacetheconsequent

a

by

a

b,thatis,werequiretheconditionals

(

a

|

b

)

and

(

a

b

|

b

)

torepresentthesameelementof

C

(

A

)

;

R4 Forall

a

Aandall

b

,

c

A,if

a

b

c,thentheresultofconjunctivelycombiningtheconditionals

(

a

|

b

)

and

(

b

|

c

)

mustyieldtheconditional

(

a

|

c

)

.

WhilstconditionsR1-R3donotrequiredelvingintoparticularjustifications,itisworthnotingthatR4encodesasortof restrictedchainingofconditionalsanditisinspiredbythechainruleofconditionalprobabilities:

P

(

a

|

b

)

·

P

(

b

|

c

)

=

P

(

a

|

c

)

whenever

a

b

c.

Giventhesefourrequirements,theformalconstructionofthealgebra

C

(

A

)

isdoneinthreestepsdescribednext.2 Thefirstoneistoconsiderthesetofobjects

A

|

A

= {

(

a

|

b

)

:

a

A

,

b

A

}

andthealgebra

Free

(A

|

A)

=

(

F ree(A

|

A),

,

,

,

,

).

RecallfromSection2thatFree

(

A

|

A

)

is(uptoisomorphism)theBooleanalgebrawhoseelementsareequivalenceclasses (moduloequi-provability)ofBoolean termsgeneratedby allpairs

(

a

|

b

)

A

|

A takenaspropositionalvariables. Inother words, in Free

(

A

|

A

)

two Boolean terms can be identified (i.e. they belong to the sameclass) only ifone term can be rewritten intothe other one by using only thelaws ofBoolean algebras. For instance

(

a

|

b

)

(

c

|

b

)

and

(

c

|

b

)

(

a

|

b

)

clearlybelongtothesameclassinFree

(

A

|

A

)

,but

(

a

c

|

b

)

doesnot,afactthatisnotinagreementwithrequirementR2. Therefore,inasecond step,inordertoaccommodate therequirementsR1-R4abovewe needtoidentifymore classes inFree

(

A

|

A

)

.Inparticular,we wouldlike

(

a

c

|

b

)

,

(

a

|

b

)

(

c

|

b

)

and

(

c

|

b

)

(

a

|

b

)

torepresentthesameelement in thealgebra

C

(

A

)

.Thus,toenforcethisandalltheotherdesiredidentificationsinFree

(

A

|

A

)

,weconsiderthecongruence relationonFree

(

A

|

A

)

generatedbythesubset

C

F ree

(

A

|

A

)

×

F ree

(

A

|

A

)

containingthefollowingpairsofterms: (C1)

((

b

|

b

),

)

,forall

b

A;

(C2)

((

a1

|

b

)

(

a2

|

b

),

(

a1

a2

|

b

))

,forall

a

1

,

a2

A,

b

A; (C3)

(∼(

a

|

b

),

a

|

b

))

,forall

a

A,

b

A;

(C4)

((

a

b

|

b

),

(

a

|

b

))

,forall

a

A,

b

A;

(C5)

((

a

|

b

)

(

b

|

c

),

(

a

|

c

))

,forall

a

A,

b

,

c

Asuchthat

a

b

c.

Note that (C1)-(C5) faithfully account for the requirements R1-R4where, in particular, (C2) and(C3) account for R2. In particular, observethat, continuingthediscussion above, nowthe elements

(

a

c

|

b

)

,

(

a

|

b

)

(

c

|

b

)

and

(

c

|

b

)

(

a

|

b

)

belongtothesameclassundertheequivalence

C. Then,wefinallyproposethefollowingdefinition.

Definition3.1.ForeveryBooleanalgebraA,wedefinethe

Boolean algebra of conditionals

ofAasthequotientstructure

C

(

A

)

=

Free

(A

|

A)/C

.

Notethat,byconstruction,ifAisfinite,sois

C

(

A

)

.Forthesakeofanunambiguousnotation,wewillhenceforth distin-guishtheoperationsofAfromthoseof

C

(

A

)

byadoptingthefollowingsignature:

C

(

A

)

=

(

C

(

A),

,

,

,

C

,

C

).

Remark3.2

(Notational convention).

Since

C

(

A

)

isa

quotient

ofFree

(

A

|

A

)

,itsgeneric elementisaclass

[

t

]

≡C,for

t

being a Boolean term, whose members are equivalent to t under

C.For the sake ofa clearnotation and without danger of confusion,we will henceforth identify

[

t

]

≡C withone ofits representative elements and, in particular,by

t

itself. Given twoelements

t

1

,

t2 of

C

(

A

)

,wewillwrite

t

1

=

t2 meaningthat

t

1 and

t

2 determinethesameequivalenceclassof

C

(

A

)

or, equivalently,that

t

1

Ct2.

Itisthenclearthat,usingtheabovenotationconvention,thefollowingequalities,whichcorrespondto(C1)–(C5)above, holdinanyBooleanalgebraofconditionals

C

(

A

)

.

(6)

Proposition3.3.

Any Boolean algebra of conditionals

C

(

A

)

satisfies the following properties for all a

,

a

A and b

,

c

A: (i)

(

b

|

b

)

=

C; (ii)

(

a

|

b

)

(

c

|

b

)

=

(

a

c

|

b

)

; (iii)

(

a

|

b

)

=

(

¬

a

|

b

)

; (iv)

(

a

b

|

b

)

=

(

a

|

b

)

; (v) if a

b

c, then

(

a

|

b

)

(

b

|

c

)

=

(

a

|

c

)

.

Straightforwardconsequencesof(iv)and(v)abovearethefollowing.

Corollary3.4.

(i)

(

b

a

|

b

)

=

(

a

|

b

)

; (ii)

(

a

b

|

)

=

(

a

|

b

)

(

b

|

)

; (iii)

(

a

b

|

c

)

=

(

a

|

b

c

)

(

b

|

c

)

.

Noticethat(iii)abovecorrespondstothequalitativeversionofaxiomCP3of[30,Definition3.2.3].

It isconvenienttodistinguishtheelementsof

C

(

A

)

in

basic

and

compound

conditionals.Theformerareexpressions of the form

(

a

|

b

)

, whilethe latterarethose terms

t

whichare (nontrivial)Boolean combinationofbasicconditionalsbut which are not equivalent modulo

C (and hencenot

equal

), to anyelement of

C

(

A

)

ofthe form

(

a

|

b

)

.For instance,if

b1

=

b2

Athereisnogeneralrule,among(C1)–(C5)above,whichallowsustoidentifyin

C

(

A

)

theterm

(

a

|

b1

)

(

a

|

b2

)

withabasicconditional oftheform

(

x

|

y

)

whilst,theterm

(

a1

|

b

)

(

a2

|

b

)

coincides in

C

(

A

)

withthebasicconditional

(

a1

a2

|

b

)

,asrequiredby(C2).

Example3.5.Letusconsider thefour elements Boolean algebra Awhose domain is

{

,

a

,

¬

a

,

⊥}

.Then, A

|

A

= {

(

,

)

,

(

,

a

)

,

(

,

¬

a

)

,

(

a

,

)

,

(

a

,

a

)

,

(

a

,

¬

a

)

,

(

¬

a

,

)

,

(

¬

a

,

a

)

,

(

¬

a

,

¬

a

)

,

(

,

)

,

(

,

a

)

,

(

,

¬

a

)

}

hascardinality12 andFree

(

A

|

A

)

is thefree Boolean algebraof 2212 elements, i.e.the finiteBooleanalgebra of 212 atoms.However, in

C

(

A

)

the following equationshold(andtheconditionalsbelowarehenceidentified):

1. C

=

(

|

)

=

(

a

|

)

(

¬

a

|

)

=

(

|

a

)

=

(

a

|

a

)

=

(

¬

a

| ¬

a

)

; 2.

( | )

(

a

| )

=

( ∧

a

| )

=

(

a

| )

= ∼(¬

a

| )

;

3.

( | )

a

| )

=

( ∧ ¬

a

| )

=

a

| )

= ∼

(

a

| )

;

4.

C

= ∼

(

|

)

=

(

⊥ |

)

=

(

a

|

)

(

¬

a

|

)

=

(

a

|

)

(

a

|

)

.

Thus, it iseasy to seethat

C

(

A

)

contains onlyfour elements that arenot redundantunder

C:

(

| ),

(

a

| ),

a

|

),

(⊥

| )

.AswewillshowinSection4(seeTheorem4.4)

C

(

A

)

has2 atomsanditisindeedisomorphictoA.

Next,we presentsomefurther basicpropertiesofBooleanalgebrasofconditionalswhicharenot immediatefromthe construction.However,sincetheirproofsareessentiallytrivial,wealsoomitthem.

Proposition3.6.

The following conditions hold in every Boolean algebra of conditionals

C

(

A

)

: (i) for all a

,

c

A,

(

a

| )

=

(

c

| )

iff a

=

c;

(ii) for all b

A,

(

¬

b

|

b

)

= ⊥

C;

(iii) for all a

,

c

A, and b

A,

(

a

|

b

)

(

c

|

b

)

=

(

a

c

|

b

)

;

Foreveryfixed

b

A,wecannowconsiderthesetA

|

b

= {(

a

|

b

)

|

a

A

}

ofallconditionalshaving

b

asantecedent.The followingisanimmediateconsequenceof(i-iii)ofProposition3.3andProposition3.6(iii)above.

Corollary3.7.

For every algebra

C

(

A

)

and for every b

Athe structure A

|

b

=

(

A

|

b

,

,

,

¬,

C

,

C

)

is a Boolean subalgebra of

C

(

A

)

. In particular, the algebra A

|

is isomorphic to A.

AsinanyBooleanalgebra,thelatticeorderrelationin

C

(

A

)

,denotedby

,isdefinedasfollows:forevery

t

1

,

t2

C

(

A

)

,

t1

t2ifft1

t2

=

t1ifft1

t2

=

t2

.

The following propositionscollect some general propertiesrelatedto the lattice order

definedabove. Nevertheless, somefurtherandstrongerpropertiesonthe

relationbetweenbasicconditionalswillbeprovidedattheendofSection4, oncetheatomicstructureofthealgebrasofconditionals

C

(

A

)

willbecharacterisedinthatsection.

(7)

Proposition3.8.

In every algebra

C

(

A

)

the following properties hold for every a

,

c

A and b

A: (i)

(

a

|

b

)

(

b

|

b

)

iff a

b;

(ii) if a

c, then

(

a

|

b

)

(

c

|

b

)

; in particular a

c iff

(

a

|

)

(

c

|

)

; (iii) if a

b

d, then

(

a

|

b

)

(

a

|

d

)

; in particular

(

a

|

b

)

(

a

|

a

b

)

; (iv) if

(

a

|

b

)

=

(

c

|

b

)

, then a

b

=

c

b;

(v)

(

a

b

|

)

(

a

|

b

)

(

b

a

|

)

;

(vi) if a

d

= ⊥

and

<

a

b, then

(

a

|

)

(

d

|

b

)

= ⊥

C; (vii)

(

b

|

)

(

a

|

b

)

(

a

|

)

;

Proof. SeeAppendix.

Somepropertiesinthepropositionabovehaveaclearlogicalreading.Forinstance,(v)tellsusthatinaBooleanalgebra ofconditionals,a basicconditional

(

a

|

b

)

isaweakerconstructthantheconjunction

a

b butstrongerthanthematerial implication

b

a,inaccordancetopreviousconsiderationsintheliterature,seee.g.[17].As aconsequence,thissuggests thata conditional

(

a

|

b

)

can beevaluatedto

true

when both

b

and

a

are so(i.e.when

a

b istrue), while

(

a

|

b

)

canbe evaluated as

false

when

a

is false andb istrue (i.e.when falsifying b

a). Furthermore,(vii) canbe read asa form of modus ponens withrespectto conditional expressions:fromb and

(

a

|

b

)

it follows

a

.We referthisdiscussion onlogical issuesofconditionalstoSection7inwhichwewillintroduceandstudyalogicofconditionalsandwherewewillpropose aformaldefinitionof

truth

forthem.

WenowendthissectionpresentingafewfurtherpropertiesofBooleanalgebrasofconditionalsregardingthedisjunction intheantecedents.

Proposition3.9.

In every algebra

C

(

A

)

the following properties hold for all a

,

a

A and b

,

b

A: (i)

(

a

|

b

)

(

a

|

b

)

(

a

|

b

b

)

; in particular,

(

a

|

b

)

(

a

| ¬

b

)

(

a

| )

;

(ii) if a

b

b, then

(

a

|

b

)

(

a

|

b

)

=

(

a

|

b

b

)

; (iii)

(

a

|

b

)

(

b

a

|

b

b

)

;

(iv)

(

a

|

b

)

(

a

|

b

)

((

b

a

)

(

b

a

)

|

b

b

)

. Proof. SeeAppendix.

Observe that the logical readingof property (i) above is the well-known OR-rule, typical of nonmonotonic reasoning (see [2,17]).Thisfact, althoughnot beingparticularlysurprising, willbe furtherstrengthened inSection 7wherewewill showthat,indeed,Booleanalgebrasofconditionalsprovideasortofalgebraicsemanticsforanonmonotoniclogicrelated to System P. Further, (iv)shows that the algebraic conjunction of two basic conditionalsis stronger than the operation of quasi-conjunction introduced in the settingof measure-free conditionals(see [2] and [17, Lemma2]) and recalledin Subsection8.1.Alsonoticethat,thepoint(iv)above,inthespecialcaseinwhich

a

=

b,

b

=

cand

a

b

c actuallygives

(

a

|

b

)

(

b

|

c

)

=

(

a

|

c

)

=

((

b

a

)

(

c

b

)

|

b

c

)

.Therefore,the requirementR4isinagreement withthe definitionof quasi-conjunction.

4. TheatomsofaBooleanalgebraofconditionals

Aswe alreadynoticed,ifAisfinite,

C

(

A

)

is finiteaswell andhenceatomic.Thissection isdevotedtoinvestigatethe atomicstructureoffiniteBooleanalgebrasofconditionals.Inparticular,inSubsection4.1weprovideacharacterizationof theatomsof

C

(

A

)

intermsoftheatomsofA.Thatcharacterizationwill beemployed inSubsections4.2and4.3togive, respectively,a fulldescription oftheatomswhichstandbelowa basicconditional

(

a

|

b

)

andto proveresultsconcerning equalitiesandinequalitiesamongconditionalswhichimprovethoseofSection3.

Inthissectionandinrestofthepaper,wewillonlydealwithfiniteBooleanalgebras. 4.1. The atomic structure of

C

(

A

)

Letusrecallthenotation introducedinSection2:foreveryBooleanalgebraA,wedenoteby

at

(

A

)

thesetofitsatoms, thatwillbedenotedbylower-casegreekletters,

α

,

β,

γ

etc.

Proposition4.1.

In a conditional algebra

C

(

A

)

, the following hold: (i) each element t of

C

(

A

)

is of the form t

=

i

(

j

(

aij

|

bij

))

; (ii) each basic conditional is of the form

(

a

|

b

)

=

αa

(

α

|

b

)

;

(8)

(iii) in particular, every element of

C

(

A

)

is a

-

combination of basic conditionals in the form

(

α

|

X

)

where

α

at

(

A

)

and X

at

(

A

)

.

Proof. (i).Itreadilyfollowsbyrecallingthat(1)everyelement

t

of

C

(

A

)

is,byconstruction,aBooleancombinationofbasic conditionals, (2)itcan beexpressedinconjunctive normalform,and(3)thenegationofa basicconditional

(

a

|

b

)

isthe basicconditional

(

¬

a

|

b

)

.

(ii).TheclaimdirectlyfollowsfromProposition3.6(iii)takingintoaccountthat

a

=

αa

α

(recallProposition2.2(ii)). (iii).Itisadirectconsequenceof(i)and(ii).

Now, letA be aBoolean algebrawith

n

atoms, i.e.

|

at

(

A

)

|

=

n.Foreach i

n

1,let usdefine Seqi

(

A

)

to bethe set ofsequences

α

1

,

α

2

,

. . . ,

α

i

of

i

pairwisedifferentelementsof

at

(

A

)

.Thus,forevery

α

=

α

1

,

α

2

,

. . . ,

α

i

Seqi

(

A

)

,letus considerthecompoundconditionalof

C

(

A

)

definedinthefollowingway:

ω

α

=

(

α

1

|

)

(

α

2

| ¬

α

1

)

. . .

(

α

i

| ¬

α

1

. . .

∧ ¬

α

i−1

).

(1)

Intuitively,suchaconjunctionofconditonalsencodesasortofchained‘defeasible’conditionalstatementsaboutasetof mutuallydisjointevents:inprinciple

α

1 holds,butif

α

1 turnsouttobefalsetheninprinciple

α

2 holds,butifbesides

α

2 turnsouttobefalseaswell,theninprinciple

α

3 holds,andsoon. . .

These conjunctions of basic conditionalswill play an important role in describing the atomic structure of

C

(

A

)

and enjoy suitable properties.To beginwith, letusconsider sets ofthose compoundconditionalsofa givenlength: foreach 1

i

n

1,let

P arti

(

C

(

A

))

= {

ω

α

|

α

Seqi

(

A

)

}

.

Example4.2.LetAbetheBooleanalgebrawith4 atoms,

at

(

A

)

= {

α

1

,

. . . ,

α

4

}

.For

i

=

1,theset P art1

(

C

(

A

))

iseasilybuilt byconsideringallsequencesoflength1ofatomsofA,

Seq

1

(

C

(

A

))

= {

α

1

,

α

2

,

α

3

,

α

4

}

,andhence:

P art1

(

C

(

A

))

= {

ω

α1

, . . . ,

ω

α4

} = {

(

α

1

|

), . . . , (

α

4

|

)

}

.

For

i

=

2,wehavetoconsidersequencesofatomsoflength2,i.e.

Seq2

(

C

(

A

))

= {

α

1

,

α

2

,

α

1

,

α

3

,

,

α

1

,

α

4

,

α

2

,

α

3

, . . .

}

,

andthenthecorrespondingsetP art2

(

C

(

A

))

iscomposedby12Booleantermslike

ω

α12

=

(

α

1

| )

(

α

2

| ¬

α

1

)

;

ω

α13

=

(

α

1

| )

(

α

3

| ¬

α

1

)

;

ω

α14

=

(

α

1

|

)

(

α

4

| ¬

α

1

)

;

ω

α23

=

(

α

2

|

)

(

α

3

| ¬

α

2

)

; . . .

Finally,for

i

=

3,considerthesetsequencesofatomsoflength3,

Seq3

(

C

(

A

))

= {

α

1

,

α

2

,

α

3

,

α

1

,

α

2

,

α

4

,

α

2

,

α

1

,

α

3

, . . .

}

.

Therefore,theset P art3

(

C

(

A

))

contains24Booleanterms:

ω

α123

=

(

α

1

| )

(

α

2

| ¬

α

1

)

(

α

3

| ¬

α

1

∧ ¬

α

2

)

;

ω

α124

=

(

α

1

|

)

(

α

2

| ¬

α

1

)

(

α

4

| ¬

α

1

∧ ¬

α

2

)

;

ω

α213

=

(

α

2

|

)

(

α

1

| ¬

α

2

)

(

α

3

| ¬

α

2

∧ ¬

α

1

)

; . . .

Thefollowingresultshowsthat,foreach

i

, P arti

(

C

(

A

))

isapartitionof

C

(

A

)

(recallSection2),andthehighertheindex i,themorerefinedthepartitionis.

Proposition4.3.

P art

i

(

C

(

A

))

is apartition

of

C

(

A

)

. Proof. SeeAppendix.

As we alreadysawin Example4.2,if

at

(

A

)

= {

α

1

,

. . . ,

α

n

}

,then P art1

(

C

(

A

))

= {(

α

1

| ),

. . . ,

(

α

n

| )}

andit givesthe coarsestpartitionamongthosethatwedenotedby P arti

(

C

(

A

))

.

(9)

Fig. 1.The Boolean algebra with 3 atoms and 8 elements.

In the following, if

|

at

(

A

)

|

=

n, for simplicity, we will write Seq

(

A

)

and P art

(

C

(

A

))

instead of Seqn−1

(

A

)

and P artn−1

(

C

(

A

))

respectively. Note that in thiscase, for every

α

=

α

1

,

α

2

,

. . . ,

α

n−1

Seq

(

A

)

, the compound conditional

ω

α definedasin(1),canbeequivalentlywrittenas

ω

α

=

(

α

1

|

)

(

α

2

|

α

2

∨ · · · ∨

α

n

)

(

α

3

|

α

3

. . .

α

n

)

. . .

(

α

n−1

|

α

n−1

α

n

).

Nexttheoremshowsthattheseconditionalsareinfacttheatomsof

C

(

A

)

.

Theorem4.4.

Let

Abe a Boolean algebra such that

|

at

(

A

)

|

=

n. Then,

at

(

C

(

A

))

= {

ω

α

|

α

Seq(A

)

} =

P art(

C

(

A

)).

As a consequence,

|

at

(

C

(

A

))

|

=

n

!

and

|

C

(

A

)

|

=

2n!.

Proof. Toshowthat P art

(

C

(

A

))

coincideswith

at

(

C

(

A

))

,byProposition2.2,wehavetoprovethefollowingtwoconditions: (a) P art

(

C

(

A

))

is a partition of

C

(

A

)

.

(b)

Any

ω

α

P art

(

C

(

A

))

is such that

C

<

ω

αand there is no t

C

(

A

)

such that

C

<

t

<

ω

α.

Itisclearthat(a)isjust thecase

i

=

n

1 inProposition4.3.As for(b),note firstthateach

ω

α

P art

(

C

(

A

))

isdifferent from

C.Indeed,itfollowsfrom(a)andasymmetryargumentontheelementsof P art

(

C

(

A

))

.Thus,itisenoughtoshow that,if

t

isanyelementof

C

(

A

)

whichisnot

C,theneither

t

ω

α

=

ω

α,or

t

ω

α

= ⊥

C.Weshowthisclaimbycaseson theformof

t

:

(i) Assume

t

isbasicconditionaloftheform

t

=

(

γ

|

b

)

with

γ

at

(

A

)

.Let

α

=

α

1

,

α

2

,

. . . ,

α

n−1

.Since

γ

at

(

A

)

,then

γ

=

α

i for some 1

i

n. Then we have two cases: eitherb

=

α

i

∨ · · · ∨

α

n,and in that case

ω

α

(

γ

|

b

)

=

ω

α, or otherwise b is of the form b

=

α

i

α

k

c, for some k

<

i and some c

A. If the latter is the case, we have

(

γ

|

b

)

(

α

k

|

α

k

. . .

α

n

)

=

(

α

i

|

α

i

α

k

c

)

(

α

k

|

α

k

. . .

α

n

)

(

α

i

|

α

i

α

k

)

(

α

k

|

α

k

α

i

)

= ⊥

C, andhence

(

γ

|

b

)

ω

α

= ⊥

Caswell.

(ii) Assume

t

isabasicconditional

t

=

(

a

|

b

)

.By(ii)ofProposition4.1,wecanexpress

(

a

|

b

)

=

αat(A):αa

(

α

|

b

)

.Hence,

ω

α

(

a

|

b

)

=

γ

ω

α

(

γ

|

b

)

,butby(i),foreach

γ

,

ω

α

(

γ

|

b

)

iseither

ω

α or

C.Sothisisalsothecasefor

ω

α

t. (iii) Finally,assume

t

isanarbitraryelementof

C

(

A

)

.ByProposition4.1above,

t

isa

-

combinationofbasicconditionals,

i.e.itcanbedisplayedas

t

=

i

(

j

(

aij

|

bij

))

.Thenwehave

ω

α

t

=

ω

α

i

j

(a

ij

|

bij

)

=

i

j

ω

α

(a

ij

|

bij

)

.

By(ii),each

ω

α

(

aij

|

bij

)

iseitherequalto

C orto

ω

α,andhencesois

ω

α

t. Therefore,wehaveprovedthat P art

(

C

(

A

))

=

at

(

C

(

A

))

.

Example4.5.LetAbe theBooleanalgebraof3atoms

α

1

,

α

2

,

α

3,and8elements,seeFig.1.Theorem4.4tellsusthatthe atomsoftheconditionalalgebra

C

(

A

)

areasfollows:

at

(

C

(

A

))

= {

(

α

i

|

)

(

α

j

| ¬

α

i

)

:

i,j

=

1

,

2

,

3 andi

=

j

}

.

Figure

Fig. 1. The Boolean algebra with 3 atoms and 8 elements.
Fig. 1. The Boolean algebra with 3 atoms and 8 elements. p.9
Fig. 2. The algebra of conditionals C( A ) of Example 4.5, where at( A ) = { α 1 , α 2 , α 3 } and at(C( A )) = { ω 1 , ω 2 , ω 3 , ω 4 , ω 5 , ω 6 }
Fig. 2. The algebra of conditionals C( A ) of Example 4.5, where at( A ) = { α 1 , α 2 , α 3 } and at(C( A )) = { ω 1 , ω 2 , ω 3 , ω 4 , ω 5 , ω 6 } p.10
Fig. 3. The tree T = T( 3 ) whose 24 paths describe the atoms of C( A ) with |at( A ) | = 4
Fig. 3. The tree T = T( 3 ) whose 24 paths describe the atoms of C( A ) with |at( A ) | = 4 p.13
Fig. 4. The tree B for S 5 when A has 5 atoms and b = α 1 ∨ α 2 .
Fig. 4. The tree B for S 5 when A has 5 atoms and b = α 1 ∨ α 2 . p.15