Bridges between Classical and Nonmonotonic Logic - cenecc經(jīng)典和非單調(diào)邏輯cenecc之間的橋梁

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BridgesbetweenClassicalandNonmonotonicLogic

DavidMakinson

Contents

1.Introduction

1.1.Purposeofthispaper

1.2.Recallingclassicalconsequence

1.3.Somemisunderstandings

1.4.Ahabittosuspend

1.5.Fourwaysofgettingmoreconclusions

2.Firstbridge-usingbackgroundassumptions

2.1.Fromclassicalconsequencetopivotalassumptions

2.2.Frompivotalassumptionstodefaultassumptions

2.3.Generalizationsandvariants

2.4Recapitulation

3.Secondbridge-restrictingthesetofvaluations

3.1.Fromclassicalconsequencetopivotalvaluations

3.2.Frompivotalvaluationstodefaultvaluations

3.3.Generalizationsandvariants

3.4.Recapitulation

4.Thirdbridge-usingadditionalrules

4.1.Fromclassicalconsequencetopivotalrules

4.2.Frompivotalrulestodefaultrules

4.3.Generalizationsandvariants

4.4.Recapitulation

5.Relationswithprobabilisticreasoning

5.1.Probabilisticinference

5.2.Na?veprobabilisticinferenceasabridgesystem

6.Conclusion

1.Introduction

1.1.PurposeofthisPaper

Thepurposeofthispaperistotakesomeofthemysteryoutofwhatisknownasnonmonotoniclogic,byshowingthatitisnotasunfamiliarasmayatfirstsightappear.Infact,itiseasilyaccessibletoanybodywithabackgroundinclassicalpropositionallogic,providedthatcertainmisunderstandingsareavoidedandatenacioushabitisputaside.

Ineffect,therearelogicsthatactasnaturalbridgesbetweenclassicalconsequenceandtheprincipalkindsofnonmonotoniclogictobefoundintheliterature.Likeclassicallogic,theyareperfectlymonotonic,buttheyalreadydisplaysomeofthedistinctivefeaturesofthenonmonotonicsystems.Aswellasprovidingeasyconceptualpassagetothenonmonotoniccasetheselogics,whichwecallparaclassical,haveaninterestoftheirown.

1.2.RecallingClassicalConsequence

Weassumefamiliaritywithclassicalpropositionalconsequence,butrefreshthememorywithsomepointsthatareessentialforwhatistofollow.

Classicallogicusesaformallanguagewhosepropositions(orformulae–wewillusethetwotermsinterchangeably)aremadeupfromaninfinitelistofelementarylettersbymeansofthetwo-placeconnectives,andtheone-placeconnective,understoodintermsoftheirusualtruth-tables,respectivelyforconjunction,disjunction,andnegation.ThesetofalltheseformulaeiscalledL.

Anassignmentisafunctiononthesetofallelementarylettersintothetwo-elementset{1,0}.Eachassignmentmaybeextendedinauniquewaytoavaluation,thatis,afunctionvonthesetofallformulaeintothetwo-elementset{1,0}thatagreeswiththeassignmentonelementarylettersandbehavesinaccordwiththestandardtruth-tablesforthecompoundformulaemadeupusing,,.WhenAisasetofformulae,onewritesv(A)=1asshorthandforv(a)=1forallaA.

LetAbeanysetofformulae,andletxbeanindividualformula.OnesaysthatxisaclassicalconsequenceofAiffthereisnovaluationvsuchthatv(A)=1whilstv(x)=0.ThestandaerdnotationisA|-x,andthesign|-iscalled‘gate’or‘turnstile’.Whendealingwithindividualformulaeontheleft,thenotationissimplifiedalittlebydroppingparentheses,writinga|-xinplaceof{a}|-xandxCn({a}).

Thusclassicalconsequenceisarelationbetweenpropositions,ormoregenerallybetweensetsAofsuchpropositionsandindividualpropositionsx.ItmayalsobeseenasanoperationactingonsetsAofpropositionstogivelargersetsCn(A).Thesetworepresentationsofclassicalconsequencearetriviallyinterchangeable.Givenarelation|-,wemaydefinetheoperationCnbysettingCn(A)={x:A|-x};andconverselywemaydefine|-fromCnbytheruleA|-xiffxCn(A).

Bothoftherepresentationsareuseful.Sometimesoneismoreconvenientthananother.Forexample,itisofteneasiertovisualizethingsintermsoftherelation,butmoreconcisetoformulateandprovethemusingtheoperation.Thesamewillbetruewhenwecometonon-classicalconsequence.Forthisreason,inthispaperwewillconstantlybehoppingfromonenotationtotheother,astwowaysofsayingthesamething,andweencouragethereadertodothesame.

Classicalconsequencehasanumberofusefulproperties.Tobeginwith,itisaclosurerelation,inthesensethatitsatisfiesthefollowingthreeconditionsforallformulaea,xandallsetsA,Bofformulae:

ReflexivityaliasInclusion

A|-awheneveraA

CumulativeTransitivity,CT

(aliasCut)

WheneverA|-bforallbBandAB|-xthenA|-x

Monotony

WheneverA|-xandABthenB|-x

ExpressedinthelanguageofCn,thismeansthatclassicalconsequenceisaclosureoperationinthesensethatitsatisfiesthefollowingconditionsforallsetsA,Bofformulae.

ReflexivityaliasInclusion

ACn(A)

CumulativeTransitivity,CT(aliasCut)

ABCn(A)impliesCn(B)Cn(A)

Monotony

ABimpliesCn(A)Cn(B)

Thosealreadyfamiliarwiththeconceptofaclosureoperationinabstractalgebraortopologymaybealittlesurprisedtoseetheconditionofcumulativetransitivitywheretheyareaccustomedtoseeingidempotence,i.e.theconditionthatCn(A)=Cn(Cn(A)).Thetwoareequivalent,notasindividualconditions,butascomponentsofthetrioofconditionsdefiningthenotionofaclosureoperation.Theleft-in-righthalfofidempotencealreadyfollowsfrominclusion,whileitsconversehalfiscoveredbyalimitinginstanceofCT,namelytheonewhereBischosentobeCn(A).Conversely,CTisimmediatefrommonotonyandidempotence.Whenconsideringrelationsthatfailmonotony,itturnsoutbesttofocusonCTratherthanidempotence.

ThethreeconditionsdefiningthenotionofaclosurerelationareexamplesofwhatareknownasHornrules.Roughlyspeaking,aHornruletellsusthatifsuch-and-suchandso-and-so(anynumberoftimes)areallelementsoftherelation,thensoissomethingelse.NoneofthesuppositionsofaHornrulemaybenegative–noneofthemcanrequirethatsomethingisnotanelementoftherelation.Noristheconclusionallowedtobedisjunctive–itcannotsaythatgiventhesuppositions,eitherthisorthatisintherelation.Hornruleshaveveryusefulproperties,mostnotablythatwheneveraHornruleissatisfiedbyeveryrelationinafamily,thentherelationformedbytakingtheintersectionoftheentirefamily,alsosatisfiestherule.

Finally,werecallthatclassicalconsequenceiscompact,inthesensethatwheneverA|-xthenthereisafinitesubsetAAwithA|-x.Inthelanguageofoperations,wheneverxCn(A)thenthereisafinitesubsetAAwithxCn(A).

Theseareabstractpropertiesofclassicalconsequence,inthesensethattheymakenoreferencetoanyoftheconnectives,,.Evidently,therelationalsohasanumberofpropertiesconcerningeachoftheseconnectivesarisingfromtheirrespectivetruth-tables,forexamplethepropertythatabCn(a).Weshallnotenumeratethese,butrecalljustonethatwillplayaveryimportantroleinwhatfollows:thepropertyofdisjunctioninthepremises,aliasOR.ItsaysthatwheneverA{a}|-xandA|-xthenA{ab}|-x.Inthelanguageofoperations,Cn(A{a})Cn(A)Cn(A{ab}).

1.3.SomeMisunderstandings

Forapersoncomingtononmonotonicreasoningforthefirsttime,itcanberatherdifficulttogetacleargriponwhatisgoingon.Thisispartlyduetosomemisunderstandingswhich,whileverynatural,distortunderstandingfromthebeginning.Wewarnthereaderinadvance,evenifsomeofthetechnicaldetailscanbecomefullyclearonlyasthestorydevelops.Theimportantthingistobeginwiththerightgestalt.

WeakerorStronger?

Thefirstthingthatonehearsaboutnonmonotoniclogicis,evidently,thatitisnotmonotonic.Inotherwords,itfailstheprinciplethatwheneverxfollowsfromasetAofpropositionsthenitalsofollowsfromeverysetBsuchthatBA.Bycontrast,classicallogicsatisfiesthisprinciple,asisimmediatefromitsdefinitionintheprecedingsection.

Giventhefailureofthisclassicalprinciple,itisnaturaltoimaginethatnonmonotoniclogicisweakerthanclassicallogic.Andindeed,inonesenseitis.Tobetechnical,foratypicalclassofnonmonotonicconsequencerelations,thesetofHornrulesholdingthroughouttheclassisapropersubsetofthoseholdingforclassicalconsequence.Forexample,takingtheclassofpreferentialinferencerelations(tobeexplainedbelow),therulesofreflexivityandcumulativetransitivityholdforallsuchrelationswhilemonotonydoesnot.

Butinanother,andmuchmorebasicsense,nonmonotoniclogicsarestrongerthantheirclassicalcounterpart.Recallthatclassicalconsequenceisarelation,i.e.undertheusualunderstandingofrelationsinset-theoreticalterms,asetoforderedpairs.Specifically,|-isasetoforderedpairs(A,x),whereAisasetofpropositionsandxisanindividualproposition.IfwrittenasanoperationCn,itisasetoforderedpairs(A,X),wherebothcomponentsaresetsofpropositions.Itisatthislevelthatthemostbasiccomparisonofclassicalwithnonmonotonicconsequencearises.

Supposewetakeanonmonotonicconsequencerelation|~(usuallycalled‘snake’).Ittooisasetoforderedpairs(A,x).Andaswewillsee,forthetypicalinstancesintheliterature,itisasupersetoftheclassicalconsequencerelation.Inotherwords,|-|~,whereissetinclusion.Likewise,supposethatwetakeanonmonotonicconsequenceoperation,usuallyreferredtobytheletterC.ThenwehaveCnC,whereisnotquitesetinclusionbetweenthetwooperations,butsetinclusionbetweentheirvalues,i.e.Cn(A)C(A)forallsetsAofpropositions.

Itisinthissensethatnonmonotonicconsequencesrelationsarestrongerthanclassicalconsequence,andforthisreasontheyarereferredtoassupraclassicalrelations.Althoughtheyarestronger,theyareratherlessregularintheirbehaviour.CertainHornrules,suchasmonotonyandtransitivity,whichholdforthe(smaller)classicalrelation,canfailforthe(larger)nonmonotonicones.Intechnicallanguage,wehave|-|~andCnC,evenwhenH(|~)H(|-)andH(C)H(Cn),whereH(.)isthesetofHornrulesthattherelationoroperationsatisfies.

Inwhatfollows,thisrelationshipshouldbeborneinmind.Therelationsthatwewillbeconsidering,bothmonotonicandnonmonotonic,areallsupraclassicalinthesenseofincludingclassicalconsequence,evenwhenfailingcertainoftheclassicalHornprinciples.

ClassicalorNon-Classical?

Insofarmonotonyfails,thelogicofnonmonotonicconsequencerelationscertainlydiffersfromclassicallogic.Butitwouldbequitemisleadingtorefertoitasakindofnon-classicallogicasthattermisusedwhenreferringto,say,intuitionisticlogic.Forincontrasttothatcase,wedonotrejectclassicalconsequenceasincorrect–asalreadyremarked,itisincludedinthenonmonotonicrelationsunderstudy.Nordowesaythatthereisanythingwrongwithmonotonyitself.Weareshowinghowthe‘goodoldrelationofclassicalconsequence’maybedeployedincertainways,todefinefurther,strongerrelationsthatareofpracticalvalue,butwhichhappentofailmonotony.

OneLogicorMany?

Thereisathirdcommonmisunderstandingofwhatnonmonotoniclogicisallabout.Fromtheclassicalcontext,wearefamiliarwiththeideathatthereisjustonecorelogic,uptonotationaldifferencesandmatterslikechoiceofprimitives.Thatcoreisclassicallogic,anditisalsothelogicthatweusewhenreasoningourselvesinthemetalanguage.

Evenintuitionistsandrelevantists,whodonotacceptallofclassicallogic,feelthesameway,butabouttheirownsystems,whicharesubsystemsoftheclassicalone.Theyhavesomedifficulties,onemightadd,inreconcilingthisviewwiththeirownpracticeinthemetalanguage,wheretheyusuallyuseclassicallogic.

Giventheunicityofclassicalinference,itisnaturalforthestudent,puzzledbyseeingseveraldifferentkindsofnonmonotonicconsequence,toask:whatisrealnonmonotonicinference?Whichisthecorrectnonmonotonicconsequencerelation?Whatistheonethatweuseinpractice,evenifwecanstudyothers?

Theansweristhatthereisnone.Thereisnouniquenonmonotonicconsequencerelation,butindefinitelymanyofthem.Wehaveallthoserelationsthatcanbegeneratedfromcertainkindsofstructure,whoseingredientsareallowedtovaryfreelywithintheboundariesofsuitableformalconditions.Likewise,thereareallthoserelationssatisfyingcertainsyntacticconditionssuchasconjunctionofconclusions,disjunctionofpremises,cumulativetransitivity,orcautiousmonotony.Moreover,ifonetriestogetawayfromnon-uniquenessbyintersectingallthemanyrelationsoroperations,theresultisjustclassicallogic.

Leavingasidetechnicaldetails,theessentialmessageisasfollows.Don’texpecttofindthenonmonotonicconsequencerelationthatwillalways,inallcontexts,betherightonetouse.Rather,expecttofindseveralfamiliesofsuchrelations,someinterestingconditionsthattheysometimessatisfy,sometimesfail,andsomeinterestingwaysofgeneratingthemmathematicallyfromunderlyingstructures.

Thisintrinsicnon-uniquenesswas,inthehistoricaldevelopmentoflogic,arathernewfeatureofthetheoryofnonmonotonicinferenceand,weshouldadd,oftheparalleldevelopmentofAGMbeliefrevision.Itappearstoberesponsibleforsomeoftheinitialdifficultiesoftheenterprisetobeassimilatedbythebroadercommunityoflogicians.

1.4.AHabittoSuspend

Wewillbeshowingthattherearesystemsthatactasnaturalbridgesbetweenclassicalconsequenceandnonmonotoniclogics.Thesebridgesystemsarealsosupraclassical,buttheyareperfectlymonotonic,indeedtheyareclosureoperations.

‘Butifthesebridgesystemsaresupraclassical’,puzzledstudentshaveaskedme,‘howaretheypossible?Ilearnedinclassthatclassicalconsequenceisalreadymaximal,inthesensethatthereisnostrongerclosureoperationinthesameunderlyinglanguage,otherthanthetrivialonebywhicheverypropositionofthelanguageimplieseveryotherone.Sohowcanyour‘bridgelogics’beclosureoperationsandatthesametimesupraclassical?’

Indeed,whatthestudentlearnedinclasshasbeenpartofthefolkloresincetheearlytwentiethcentury.Buttheformulationaboveomitsavitalelement,whichisnotalwaysmadeasexplicitasitshouldbe.Thatistheconditionofbeingclosedundersubstitution,whichwenowexplain.

Bysubstitutionwemeanwhatisoftencalled,atgreaterlength,uniformsubstitutionofarbitraryformulaefortheelementarylettersinaformula.Forexample,whenaistheformulap(qr),wherep,q,rarethreedistinctelementaryletters,thenonesubstitutionmayreplacealloccurrencesofpby,say,r,alloccurrencesofqbyp,and(simultaneously,notsubsequently)alloccurrencesofrby(ps).Thatwillgiveus(a)=r(p(ps)).Simplifications,suchastheeliminationofthedoublenegation,arenotpartofthesubstitution,butpossiblelateroperations.Uniformsubstitutionforelementarylettersisakindofconstruction,notaninference.Itshouldnotbeconfusedwithanotheroperationsubstitution,ratherconfusinglyalsosometimescalledsubstitution,whichreplacesoneofmoreoccurrencesofaformula(notjustanelementaryletter)byanotherformulatowhichitisclassicallyequivalent(notbyanarbitraryformula).

WhenAisasetofformulae,substitutiononitisunderstoodpointwise;thatis,(A)isunderstoodtobe{(a):aA}.

Classicalconsequenceisclosedundersubstitution,inthesensethatwheneverA|-xthen(A)|-(x).Inthelanguageofoperations,wheneverxCn(A)then(x)Cn((A)),ormoreconcisely(Cn(A))Cn((A)).

Themaximalityofclassicallogicmaynowbeexpressedasfollows:thereisnosupraclassicalclosurerelationinthesamelanguageasclassical|-,thatisclosedundersubstitution,exceptfor|-itselfandthetotalrelation.Likewiseforoperations.Here,thetotalrelationistheonethatrelateseverypremise(orsetofpremises)toeveryconclusion;asanoperationitsendsanysetofformulaetothesetofallformulae.Theproofoftheobservationisindeedasimpleone,andwerecallithere.

Proofofthemaximalityofclassicallogic.Let|-beanyclosurerelationthatisclosedundersubstitutionandalsoproperlysupraclassical,i.e.|-|-.Bythelasthypothesis,thereareA,xwithA|-xbutA|-/x.Fromthelatter,thereisaclassicalvaluationvwithv(A)=1,v(x)=0.Substitutingtautologiesforelementarylettersthataretrueunderv,andcontradictionsforlettersthatarefalseundervwesee,byaneasyinductionondepthofformulae,that(A)isasetoftautologiesand(x)isacontradiction.Sincebyhypothesis|-isclosedundersubstitution,A|-ximplies(A)|-(x).Butsince(A)isasetoftautologieswehavebyclassicallogicthatforarbitraryB,B|-(a)forall(a)(A),andlikewisesince(x)isacontradictionwehave(x)|-yforeveryformulay.Thussince|-|-wehaveB|-(a)forall(a)(A),and(A)|-(x),and(x)|-y.Puttingthesethreetogetherwithcumulativetransitivityandmonotonyof|-,wegetB|-yandtheproofiscomplete.

Themoralofthisstoryisthatthesupraclassicalclosurerelationsthatweshallbeofferingasbridgesbetweenclassicalconsequenceandnonmonotonicconsequencerelationsarenotclosedundersubstitution.Nor,forthatmatter,arethenonmonotonicrelationsthatissuefromthem.Thisrunsagainstingrainedhabit.Studentsoflogicarebroughtupwiththeideathatanydecentconsequencerelationshouldbepurelyformal,orstructural,andhencesatisfysubstitution.Indeed,thosetermsareoftenusedinthetextsassynonymsforclosureundersubstitution.Tounderstandnonmonotoniclogic,thisisahabittosuspend.

1.5.ThreeWaysofGettingMoreConclusions

Inthefollowingsections,wewillconsiderthreedistinctivewaysofgettingmoreoutofasetofpremisesthanisauthorizedbystraightforwardapplicationofclassicalconsequence.Roughlyspeaking,thefirstusesadditionalbackgroundassumptions.Thesecondrestrictsthesetofvaluationsthatareconsideredpossible.Andthethirdusesadditionalbackgroundrules.

Eachoftheseproceduresgivesrisetoacorrespondingkindofmonotonicconsequenceoperation.Theyarenotentirelyequivalenttoeachother.Buttheyallgiveussupraclassicalclosureoperations,i.e.operationssatisfyingreflexivity,cutandmonotonyandalsoincludeclassicalconsequence.Wecallsuchconsequencerelationsparaclassical.

Thethreekindsofparaclassicalconsequenceserveasconceptualbridgestocorrespondingfamiliesofnonmonotonicconsequence,formedessentiallybyallowingkeyelementsoftherespectiveconstructionstovarywiththepremisesunderconsideration.

Webeginbyexaminingthesimplestkindofparaclassicalconsequenceanditstransformationintoaformofnonmonotonicreasoning,namelyinferencewithadditionalbackgroundassumptions.

2.FirstBridge-UsingAdditionalBackgroundAssumptions

2.1.FromClassicalConsequencetoPivotalAssumptions

Indailylife,theassumptionsthatwemakewhenreasoningarenotallofthesamelevel.Generally,therewillbeafewthatwedisplayexplicitly,becausetheyarespecialtothesituationunderconsiderationorinsomeotherwaydeservingparticularattention.Therewillusuallybemanyothersthatwedonotbothereventomention,becausewetakethemtobepartofsharedcommonknowledge,orinsomeotherwaytrivial.ThisphenomenonwasalreadyknowntotheancientGreeks,whousedthetermenthymemetorefertoanargumentinwhichoneormorepremisesareleftimplicit.Thisistheideathatwedevelopinthissection.

Weworkwiththesamepropositionallanguageasinclassicallogic,withthesetofallitsformulaecalledL.LetKLbeasetofformulae.IntuitivelyKwillbeplayingtheroleofasetofbackgroundassumptionsor,astheyarecalledinG?rdenforsandMakinson(1994),‘expectations’.LetAbeanysetofformulae,andletxbeanindividualformula.

WesaythatxisaconsequenceofAmodulotheassumptionsetK,andwriteA|-KxaliasxCnK(A)iffthereisnovaluationvsuchthatv(KA)=1whilstv(x)=0.Equivalently,iffKA|-x.Assimpleasthat!Andwecallarelationoroperationapivotal-assumptionconsequenceiffitisidenticalwith|-K(resp.CnK)forsomesetKofformulae.Notethatthereisnotauniquepivotal-assumptionconsequencerelation,butmany–oneforeachvalueofK.

Sinceclassicalconsequenceismonotonic,pivotal-assumptionconsequencerelationsandoperationsaresupraclassicalinthesensedefinedearlier.Thatis,foreveryKwehave|-|-KandCnCnK.Theyalsoshareanumberofabstractpropertieswithclassicalconsequence.Inparticular,asisimmediatefromthedefinition,theysatisfyinclusion,cumulativetransitivityandmonotony,andthusareclosureoperations.Theyarealsocompact,andhavethepropertyofdisjunctioninthepremises.

Giventhattheyaresupraclassicalclosurerelations,weknowfromthegeneralobservationinsection1.4thatthe|-Karenot,ingeneral,closedundersubstitution.Itisinterestingtoseethroughanexamplewhythisisso.LetK={p}wherepisanelementaryletter.Chooseanyotherelementaryletterq,andputA={q}andx=pq.ThenA|-Kxsince{p,q}|-pq.Nowletbethesubstitutionthatreplaceseveryelementaryletterby,say,itsownnegation,sothat(p)=pand(q)=q.Then(A)|-/K(x)sinceK{q}={p,q}|-/pq.Analysingthisexample,weseewhatmakesitwork:thesubstitutionisappliedtotheexplicitpremisesAandtheconclusionx,butnottothebackgroundassumptionsetK,whichisheldconstant.

Astrikingfeatureofpivotal-assumptionconsequence(whichseparatesitfromthenextbridgesystemthatwewillbedescribing)isthattheabovepropertiesalsosufficetocharacterizeit.Inotherwords,wehavethefollowingrepresentationtheoremforpivotal-assumptionconsequence.Let|-(resp.Cn)beanysupraclassicalclosurerelation(resp.operation)thatiscompactandsatisfiestheconditionofdisjunctioninthepremises.ThenthereisasetKofformulaesuchthat|-=|-K(resp.Cn=CnK).

ThisrepresentationtheoremisformulatedinRott(2001)(section4.4observation5),butappearstohavebeenpartofthefolkloreformanydecades.Itsproofisabstractbutshort.Itismosteasilyexpressedusingtheoperationnotation.Wegiveithere,butsomereadersmayprefertoskipit,passingdirectlytosection2.2.

Proofofrepresentationtheoremforpivotal-assumptionconsequence.LetCnbeanysupraclassicalclosureoperationthatiscompactandsatisfiestheconditionofdisjunctioninthepremises.PutK=Cn(),whereistheemptysetofformulae.WeclaimthatCn=CnK.ItwillsufficetoshowbothCnKCnandCnCnK.

FortheinclusionCnKCnweneedtoshowthatforanyA,Cn(Cn()A)Cn(A).SincebysupraclassicalityCnCn,wehaveCn(Cn()A)Cn(Cn()A)=Cn(A)bywell-knownpropertiesofclosureoperations.

FortheconverseinclusionCnCnKweneedtoshowthatforanyA,Cn(A)Cn(Cn()A).Thisiswherewealsoneedcompactnessanddisjunctioninthepremises.SupposexCn(A).ThenbythecompactnessofCnthereisafiniteBAwithxCn(B).LetbbetheconjunctionofallthefinitelymanyelementsofB.SinceCnisasupraclassicalclosureoperation,wegetxCn(b)andthusinturnbxCn(b).ButagainbysupraclassicalityofCn,bxCn(b).Applyingdisjunctioninthepremisesasthecoupdegrace,wehavebxCn(bb)=Cn()withthehelpofsupraclassicalityandclosureagain.ToshowthatxCn(Cn()A)asdesired,itwillthussufficetoshowthatxCn({bx}A).Butbytheconstructionofb,wehavebypropertiesofclassicalconsequencethatbCn(A)andsoagainbypropertiesofclassicalconsequencexCn({bx}A),therebycompletingtheproof.

2.2.FromPivotalAssumptionstoDefaultAssumptions

Whathasallthistodowithnonmonotonicinference?Theconsequencerelations|-Kare,aswehaveseen,monotonic.ButnonmonotonicityisgeneratedifweallowthebackgroundassumptionsetKtovarywithA,inparticular,tobediminishedwhenAisinconsistentwithK.

Specifically,weobtainaninterestingnonmonotonicrelationifweworkwiththemaximalsubsetsKofKthatareconsistentwithA,andacceptasoutputonlywhatiscommontotheirseparateconsequences.Wecallthisrelationdefault-assumptionconsequence,tobringoutitscloserelationtotheprecedingpivotal-assumptionconsequence.

Togivethedefinitionmoreexplicitly,letKLbeasetofformulae,whichagainwillplaytheroleofasetofbackgroundassumptions.LetAbeanysetofformulae,andletxbeanindividualformula.

WesaythatasubsetKofKisconsistentwithA,iffKA|-/L,where|-isclassicalconsequenceandtheslashindicatesthatitdoesnothold.Equivalently,iffthereisaclassicalvaluationvwithv(KA)=1.

AsubsetKofKiscalledmaximallyconsistent(morebriefly,maxiconsistent)withAiffitisconsistentwithAbutisnotapropersubsetofanysubsetKKthatisconsistentwithA.

Finally,wedefinetherelation|~KofconsequencemodulothedefaultassumptionsKbyputtingA|~KxiffKA|-xforeverysubsetKKthatismaxiconsistentwithA.WritingCKforthecorrespondingoperation,thisputsCK(A)={Cn(KA):KKandKmaxiconsistentwithA}.

Wecallarelationoroperationadefault-assumptionconsequenceiffitisidenticalwith|~K(resp.CK)forsomesetKofformulae.Noteagainthatthereisnotauniquedefault-assumptionconsequencerelation,butmany–oneforeachvalueofK.

Default-assumptionconsequenceoperations/relationsarenonmonotonic.Thatis,wemayhaveA|~KxbutnotAB|~KxwhereA,Baresetsofpropositions.Likewise,wemayhavea|~Kxwithoutab|~Kxwherea,bareindividualpropositions.

Toillustratethefailureofmonotony,supposeK={pq,qr}wherep,q,raredistinctelementarylettersofthelanguageandisthetruth-functional(aliasmaterial)conditionalconnective.Thenp|~KrsincethepremisepisconsistentwiththewholeofKandclearly{p}K|-r.Butpq|~/Kr,forthepremisepqisnolongerconsistentwiththewholeofK.ThereisauniquemaximalsubsetKK