[PDF] L. Royakkers and F. Dignum. Defeasible reasoning with legal rules

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L. Royakkers and F. Dignum. Defeasible reasoning with legal rules. In M.A. Brown and J. Carmo, editors, Deontic Logic, Agency and Normative Systems (Workshops in Computing), pages 174--193. Springer-Verlag, 1996. Defeasible Reasoning with Legal Rules in aDeontic Logic based on Preferences

Lamber RoyakkersDept of Legal InformaticsTilburg UniversityPOb ox LE TilburgLMMRoyakkerskubnlFrank DignumDept of Mathematics and Computer ScienceEindhoven UniversityofTechnologyPOb ox

MB EindhovendignumwintuenlAbstract

Over the last few years several defeasible deontic reasoning formalismshave b een develop ed to solve the problem of deontic inconsistencyHowever these formalisms are unable to deal with some very common formsof deontic reasoning since eg their expressiveness is restricted In thispap er which is based on Royakkers and Dignum

we will establisha priority hierarchy of legal rules to solve the problem of deontic conictsand we will present a mechanism to reason ab out nonmonotonicityoflegal rules over the priority hierarchy The theory presented here based ondefault logic and b eing a mo dication and extension of Prakkens argumentation framework adequately deals with some shortcomings of otherdefeasible deontic reasoning approachesIntro ductionLogical analysis of reasoning with inconsistent rules is a very relevan

t area forAIandLaw research since rules used in the legal domain are often conictingPrioritised rules have received attention in research on the formalisationofnonmonotonic reasoning particularly as a way of mo delling the choice crite

This researchwas sp onsored by the foundation for Law and Public Administration Reobwhich is part of the Netherlands Organization for Scientic ResearchNWO

rion in dealing with exceptions cf Brewka Po ole Prakken

Shoham

To deal with the inconsistencies various sorts of consistencybased approaches have b een develop ed such as the nonmonotonic logic of McDermott andDoyle

and the default logic of Reiter

But these approaches failto reason ab out conicting norms since they are all based on nonmo dal logics As a way of solving the problems of deontic conicts forms of defeasiblereasoning cf Pollo ck

are adopted whichprovide a mechanism to establish preference hierarchies of norms and to select a more applicable normamong conicting ones in a sp ecic situation cf Alchourron and Makinson

Royakkers and Dignum The existing formalisations of defeasibledeontic reasoning approaches Horty Ryu Tan and Van der Torre are unable to deal with several highly common forms of deontic logic cfPrakken A rst problem isthe lack of notion of permissionin certain approachesHorty Tan and Van der Torre

In these approachesOpq pisobligatory in case ofq is treated as a normal default and can b e read as a Reiterdefaultqpp a nondeontic Reiter default Inherent in this treatmentis theabsence of a reasonable Reiter default for thenegated obligationp ermissionAnother problem isthe defeasibility of only a single opposing statementinsome approaches Horty

Prakken Tan and Van der Torre In these approaches only couples of statements are considered to chec

kinconsistencies For instance take the three statementsOaOabandObNo single statement is in conict with the other single statements Howeverthe group of statementsOaandOab impliesOb in standard deonticlogic which is in conict with the statementObThe third problem is that most approaches Horty

Ryu Tan andVan der Torre

can only deal with defeasible conditionals that are deonticBut deontic defaults are not the only defaults in legal reasoning Consider thedeontic defaultaObOb With this default it is very often the case thatais derived by another default rule egcaawhich is called a classicationrule or an interpretation rule In the legal domain it is accepted that theserules are also defeasible cfHart

Prakken shows this by extendingHarts standard example on a park regulation that forbids vehicles to enter theparknot only this rule itself may turn out to b e defeasible for exampleif the v

ehicle is an ambulance but also rules on when somethingcounts as a vehicle may b e defeasible imagine that a court says thatob jects on wheels that are meant for normal transp ort are vehiclesthen roller skates used by p eople on their way to the oce mightberecognised as an exception Prakken

In this pap er we will develop a theory of defeasible deontic reasoning whichadequately deals with the ab ovementioned problems The theory is an exten

sion and mo dication of the argumentation framework in default logic develop edbyPrakken Further our theory is an extension of the Dungs theory

which only considers argumentation frameworks with one kind ofconict b etween argumentsThe structure of this pap er is as follows In section we will give the representation of legal rules whichwe sub divide in rules and conditional norms Insection

we will discuss the argumentation framework for rules The argumentation framework for norms dep ending on rules selected from the argumentationframework for rules will b e discussed in section We will concentrate here ondefeasibility and violation We will end this pap er with some conclusionsLegal Rules Rules and Conditional NormsThe fundamental logical structure of legal knowledge gives rise to the nonmonotonicity of legal reasoning Reiter

Delgrande

the consequencesthat mayfollow from a set of legal and factual premises can b e invalidated byfurther information This means that rules can b e defeated by other or newrules and facts The principal idea of this pap er whichgoes backtoRescher

is to allow the rules to b e ordered and to use this ordering in suchaway that conicts can b e solved in a logical argumentation framework usingnonmonotonic logic Such an ordering can often b e discerned when consideringthe rules in a legal co de The Lex Sup erior principle for instance is basedon the general hierarchy of a legal system the rules are divided along the linesof the hierarchical structure of the normative system Rules with a lower rankof priorityhave to resp ect the consequences that follow from a higher rankedrule see pap er To describ e the ordering b etween the formulas we use thefollowing notation Letxandyb e legal rules then xy means thatyis

preferred toxxy is an abbreviation for xyandyx and xyisan abbreviation for xyandyx The ordering relationis reexive andtransitiveLegal rules or at least most of them sub ordinate a legal eect to a legalconditionBy legal eect we mean every qualication generated by a legalnorm the ascription of deontic or normative mo dalities status professionaltitles other legal qualities of p ersons and things By legal condition wemeanevery condition to which a legal eect is sub ordinatedThe legal rules arerepresented as conditional statements of the typ ea

a an whereis the legal eect anda a an are the elements of the antecedentthe conjunction of literals representing the legal condition Ifis a norm an A literal is any atomic prop ositionalformula and any negation of an atomic prop ositionalformula

obligation O or a p ermission P witha formula of the prop ositionallogic then the conditional statementiscalledaconditional normIfis aliteral then the statementiscalledaruleThe statementABhas to b e interpreted as anormal defaultaccording toReiters theory

ABBIfA and it can b e consistently assumedB then we can inferB This means thatisnotinterpreted as the materialimplication but as an inference rule that can b e defeated FromAandABwe can inferBunlessBcan b e proven This representation corresp onds tothe formalisations usually prop osed by legal theory and legal logic cf Sartor

In our theorywe distinguish b etween rules and norms for the followingreasonsRules cannot b e violatedThe defeasibility of rules is dierent from the defeasibility of norms cfdenition

and denition which is the most imp ortant dierenceThe most imp ortant thing ab out the dierence b etween rules and norms is notwhatdierences there are but simplythatthere are dierences This is whywediscuss our theory on dieren

t levels rst on the level of rules section

andsecond on the level of norms based on a given set of rules section The set of rules will b e denoted byWand the set of conditional normsbyFurthermore wehave a factual sentenceFrepresenting the factualsituation which consists of background knowledge and contingent facts Thebackground knowledge consists of necessary conditions for example a humanb eing is mortal A set of conditional norms a set of rules and a factual sentencewill b e called a deontic contextDenition Adeontic contextTWFconsists of a setof conditional norms a setWof rules a factual propositional sentenceFtheconjunction of background know ledgeFb

and contingent factsFc

and an orderingover rules and conditional normsRulesFacts formalised by the sentenceFcancontain material implications Ruleshowever are represented by normal defaults written as a conditional statementof the typ ea

a an

withthe legal eect formalised by a literalThe theory of defeasible reasoning for rules here is based on four notionsthe notion of anargumentdenition

the notion ofdefeatingdenition the notion of adefeasibility chaindenition the notion ofjustied defensible and overruledargumentsdenition

At the end of this section we will denemaximal coherent argument setsof rulesthat we will use for the notion of the applicability of norms and the violation ofobligations in section Before we discuss the notion of argument we will give three denitions whichwe will use in the sequelDenition LetFbe the factual propositional sentenceVbe a set of rulesandra literal thenVexplainsrV

fFgjrifFgrora a an rV fV fFgjai ji

fnggIntuitivelyexplainingis the same as logical consequence except that wenowdeal with defaults and not with implications

Denition LetVbe a set of rules then the consequences ofV fFgC onsV fFgis denedasConsV fFgfrjris a literal andV

fFgjrgThus theConsrelation is a transitive closure of the explain It gives the set ofall literals that can b e consistently derived fromVandfFgDenition LetVbe a set of rules ThenV

fFgis coherent i risaliteral r ConsV fFgr ConsV fFgThe notion of argumentcan no wbedened asfollowsDenition LetMWaliteral andM fFgcoherentThenMexplainsminimal ly ifFg Mjand M fFg Mnf

gjWe cal lMa minimal ly explaining set or an argument The set of al l argumentswil l be denotedasM Therelevant set ofW denotedbyM is the set ofal l arguments inMthat explainminimal lyM

is a subargument ofMiM MandM is an argument If thereisanargument for thus

Mthenis cal ledanoutcomeDenition LetM

MandM

M ThenM

is defeatedbyMM M i M M f gf g f g fFgis inconsistent

Thus an argumentM

defeats an argumentM iM andM havecontradictory conclusions and with resp ect to the factual sentenceFandtherule M resp onsible for the conict do es not havealower prioritythanthe rule M Note thatf g fFgandf g fFgare consistentwhich directly follows from denition

Relation

is not transitive and not asymmetric It is p ossible thatM M andM M b oth hold The following example illustrates this p ointExample abca da bcecFffdewith LetM fec cagandM fda ab bcgThenM M sinceecbcandM M sincedacaDenition A defeasibility chain is a sequenceofarguments inMM M M nwith the fol lowing conditions klnkl Mk Ml M n M fM Mn Mn Mn M n

gWe deneChMas the set of al l defeasibility chains of arguments inMThe rst condition ensures that cycles in defeasibilitychains are avoided Supp ose thatM

M andM M withM M

Wewould thus end upwith the endless chainM

M M M M

This would also b eaccomplished by instead of The reason whywedoneed b ecomesclear in prop osition

The second condition provides that a chain stops if there is no strongerargument than the last argumentinthe chainTake the example ab ove thenChMffda abg

fec cagfda ab bcg fec cagfecg fda ab bcggfdag fec cagfec cag fda ab bcg

Denition ChMis the set of al l defeasibility chains inChMstartingwithMThe defeasibilitychains inChM take the set of all p ossible arguments andtheir mutual relations of defeat as input They pro duce a distinction b etweenarguments in three classes as output

justiedargumentsoverruledarguments

defensibleargumentsA justied argument is a winning argument Such an argument can b e defeated by another argument but that argument will b e overruled An overruledargument is a losing argument A defensible argument is an argument that isneither justied nor overruled In other words an undeciding argumentDenition LetM

MThenMis a justiedargument i for al l chainsM

M M n

ChMit holds thatnis even

M M Mn M kis even M k is a justiedargumentMis an overruledargument i there is a chainM M M n

ChMnis oddM

M Mn M Mis a defensible argument iMis neither a justiedargument nor anoverruledargumentNote thatMn in the chain of denition is a justied argument sinceChMn fMn g whichisequivalenttoM M Mn M LetM M M n be a chain inChM then we call the argumentsMi withiis o ddodd arguments and the argumentsMandMi withiiseveneven argumentsIn a defeasibilitychainM M M n

withnis even o dd westipulate that the o dd even arguments are theattackedarguments and theeven o dd arguments thenonattackedarguments If the chain ends withMn

thenMn is an attacked argument b ecause it is defeated by a nonattackedargumentMn is not attacked b ecause it is defeated byanattack ed argumentMn and so on For exampleMis defeated and overruled ifM is a nonattacked argument and this follows ifnis o dd The terms justied overruled and defensible argumentwere intro duced by Prakken andSartor

Prop osition

LetMbe a justiedargument Then al l odd arguments in the chains ofChMare overruledarguments LetMbe an overruledargument Then there is a chain inChMwithal l even arguments overruled or defensiblePro of

LetMb e a justied argument Then for all chainsM M M n inChM it holds thatnis even andM M Mn M

LetMkwithkis o dd b e an argumentof a chainM

M M n inChMThen this chain without the rstkarguments th usMk M k M n isachain inChMk which satises the conditionsof an overruled argumentMk since the chain contains an even numberof arguments andChMn Mn Thus all o dd arguments are overruledarguments LetMbe an overruled argument Then there is a chainM M M n withnis o dd andM M Mn M

Supp ose an evenargumentMk

is justied then the chainM M M n withoutthe rstkarguments ieMk M k M n is a chain inChMk and do es not satisfy the conditions for a justied argumentMk

since thechain contains an even numb er of arguments Thus the even argumentsin suchchains inChM are overruled or defensibleThe condition that all even arguments in the chains ofChM withMbeing ajustied argumenthave to b e justied arguments is necessary since otherwisewe obtain some undesirable resultsExample

abcabd ebgfdhfdi fhj kh jg ljFceiklwith LetM fab cagM fbd ebgM fgf jg ljgM fdh fd ifgM fhj khgChM fM M M M M gandChM fM gthuswithout the condition that all even arguments hav e to b e justied argumentsthenM would b e a justied argument HoweverM is a defensible argumentsinceM is not a justied argument The set of defeasibilitychains startingwithM isChM fM M M M M M g SinceM is anoverruled argument the chainM M M M satises the conditionsfor the overruled argumentM the chainM M M do es not satisfythe conditions for a justied argumentM Thus M is not a justied argumentNote thatM is a defensible argument since neither of the twochains inChM satises the conditions for an overruled argumentM

Corollary

A justiedargument can only be defeated by an overruledargument If there is no justiedargument then there is no overruledargument There is no justiedargument i al l arguments are defensible

There is a justiedargument i there is a defeasibility chain with oneargumentPro of LetMb e a justied argument and defeated byM

Wehaveto provethatM

is an overruled argument For all chainsM M M ninChM it holds thatnis even andChMn fMn gFor all thesechains without the rst argumentM ieM M n it holds thatthey are elements ofChM and satisfy the conditions of an overruledargument ThusM is an overruled argument Supp ose that there is no justied argument Then there is no chainM M M n inChM withChMn fMn

g Hence there is nooverruled argumentThe converse do es not hold For example letWfabgandaa factThen the only argumentisfabg and this argument is justied

Supp ose that there is no justied argument Then there is no overruledargument th

us all arguments are defensibleEvidently if all arguments are defensible then there are no justied arguments

Supp ose thatMis a justied argument Then for all chainsM M M n inChM it holds thatChMn fMn

gThus there is adefeasibilitychain with one argumentIf there is a chain with one argument sayMthenMis a justied argument Hence there is a justied argumentThe converse of

do es not hold an overruled argument need not necessarilyb e defeated by a justied argumentExample abcdbe fbedgehdFacfghwith LetM fabgM fcdgM fbe fbgM fge edgM fhdgThenChM fM M M M M M M gM isan overruled argument since there is a chainM M M M withChM fM gFurtherM is only defeated by argumentM whichisanoverruled argument sinceM M ChM andChM fM

gThusan overruled argument can b e defeated byan overruled argumentProp osition Al l subarguments of a justiedargument are justiedargumentsPro ofSupp ose thatMis a justied argumentandM

is a subargumentofMWehavetoprove thatM is a justied argument and without loss of generalityw e assume thatM is a largest subargumentofM ieM M M M MFor if this argument is justied we can rep eat this pro cess forM to provethat all subarguments ofM are justied Supp ose thatM is not a justiedargument thenM is an overruled or a defensible argumentSupp ose thatM is an overruled argument Then there is a chainM M M n withChMn fMn g andnis o dd However thenthe chainM M M n isachain ofChM whic hisincontradiction with the assumption thatMis a justied argument

Supp ose nowthatM

is a defensible argument Then there is a chainM M M n andChMn fMn gNow the chainM M M n is not an elementofChM but part of a chain inChMMn can only b e followed byM thusM M M n M M M n withChM n fM n gHowever now it followsthatM is an overruled argument if it is an o dd argumentinthe chainprop osition or a justied argument if it is an even argumentdenition and this is in contradiction with the assumption thatM is defensibleThusM is a justied argumentExample abcbde efbehbfdifjdFachijwith Let M fa bgM fcbgM fde jdM fef be hbgM ffd ifgChM fM gthusM is a justied argumentChM fM M gthusM is an overruled argument ChM fM M M M M gM

is not overruled sinceotherwise the chain contains an even numb er of arguments It dep ends onargumentsM

andMquotesdbs_dbs29.pdfusesText_35

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