Normative conflicts in legal reasoning

Artificial Intelligence and Law 1: 209-235, 1992. 209 © 1992 Kluwer Academic Publishers. Printed in the Netherlands.

Normative Conflicts in Legal Reasoning

G I O V A N N I S A R T O R CIRFID, University of Bologna, Via Galliera 3/5, 40121, Bologna; IDG-CNR, Via Panciatichi 56/16, 50127, Florence

(Received 7 January 1992; accepted 2 November 1992)

Abstract. This article proposes a formal analysis of a fundamental aspect of legal reasoning: dealing with normative conflicts. Firstly, examples are illustrated concerning the dynamics of legal systems, the application of rules and exceptions, and the semantic indeterminacy of legal sources. Then two approaches to cope with conflicting information are presented: the preferred theories of Brewka, and the belief change functions of Alchourr6n, Gardenfors, and Makinson. The relations between those approaches are closely examined, and some aspects of a model of reasoning with normative conflicts are outlined. Since this model takes into account an ordering of the involved regulations, criteria to order legal norms are finally specified.

Key words: non-monotonic reasoning; belief revision; rules and exceptions; normative conflicts.

1. Introduction: normative conflicts and legal consequences

Lawyers frequently deal with normative conflicts when applying the law: (possibly) valid norms establish incompatible legal qualifications for the same concrete case. This fact has three main causes: (a) the dynamics of the legal system, (b) the legal protection of conflicting interests, and (c) the uncertainty concerning the content of legal sources.

(a) To cope with social and political change, law defines procedures through which norms can be produced or eliminated. Special legal rules, the authorizing or empowering rules [cf. Kelsen 1960, 57ff; Hart 1961, 77ff], attribute the power to produce legal norms and establish procedures for exercising this power. The freedom of the legislator is limited only by the need to follow those procedures, and by the hierarchical relations among the legal sources. So, new norms can be enacted that conflict with norms already in force. When introducing a new regulation, to prevent conflicts the legislator may explicitly eliminate (abrogate) or modify previous prescriptions. Nevertheless, not all normative conflicts can be (given the complexity of modern legal systems) or should be (given the positive functions of certain normative conflicts, such as those between rules and exceptions) prevented by the legislator. It would obviously be preferable if the legislator acted with full knowledge of the impact of the new norms on the preexisting legislative corpus, and were able to prevent unwanted normative conflicts, and various computer technologies (from information retrieval to expert systems) can contribute to this aim. However, technology must not be overrated: the automatic detection of possible conflicts presupposes an understanding (an interpretation) of the texts involved, that can be done automatically only to a very limited extent.

210 GIOVANNI SARTOR

(b) Law has to deal with complex social contexts, in which conflicting interests are

legally relevant. Two types of situations can be distinguished: (o0 the conflict concerns

scalable objectives (purposes or policies); (13) the conflict concerns well-specified alterna-

tive legal solutions. In (cx) situations the compromise maximizing the overall utility (in a

wide sense) must be pursued, and this normally requires that each of the conflicting

objectives is satisfied to a certain extent. In ([3) situations, instead, a choice is required in

favor of just one of the alternative solutions (a contract is valid or invalid, a person is

guilty or not guilty, etc.), although the choice criteria may be indeterminate. In these ([3)

situations the balancing of the conflicting interests is frequently implemented through a

combination of rules and exceptions. The solution established by the rule, i.e., corre-

sponding to the interest normally protected by the law in a certain type of situation, can

be derived (once such a situation has been ascertained in a concrete case) unless the rule

is contradicted (in that case) by an exception. The exception establishes that a certain pre-

vailing circumstances requires a different legal solution, even when the conditions for the

application of the rule are satisfied. In many cases the rule-exception hierarchy is estab-

lished by the legislator, who uses to this purpose a number of linguistic devices (for

example, norms with a negative consequent, unless clauses, and facts whose proof blocks

a legal conclusion are normally to be interpreted as exceptions, cf. Sartor 1991). In many

other cases, the interpreter has to individuate, evaluate, and balance the conflicting

profiles of legal relevance, to establish which is to be considered as prevailing, and has to

translate the result of this judgment in qualifying the corresponding norm as an

exception 1.

(c) Legal sources can be semantically indeterminate. In particular, documented

sources, such as statutory law, may be ambiguous (susceptible to being attributed alterna-

tive meanings) or vague (susceptible to being applied in different ways to borderline

cases). Legal science has developed methods to deal with semantic uncertainty, but it is

quite doubtful whether those methods - such as the traditional techniques of literal, sys-

tematic, and teleological interpretation - can really contribute to the certainty of law,

since they give only general directives that may lead to diverging results. In dealing with

semantically indeterminate legal sources two different aspects may be distinguished: (e 0

establishing the alternative possible interpretations of the legal sources; (13) reasoning in a

f ramework of alternative conflicting interpretations. Under the heading of semantic

undeterminateness or 'open texture ' , normally only the (a) aspect is considered, that

involves difficult problems, such as natural language understanding and modeling the

dynamics of conceptual structures. The ([3) aspect here considered is computationally

1 Every legal rule is defeasible in a set of not fully predetermined situations. It establishes only a primafacie legal consequence, that can be overridden by prevailing considerations to the contrary, and in many cases the only function of explicit exceptions is just to recall, without specifying them, all the possible situations that might overturn the qualification established by the rule (for example, let us think to the exceptions to the valid- ity of contracts grounded on the violation of boni mores or of public policy, explicitly stated by Italian law). Therefore, defeasibility cannot be the criterion for distinguishing two categories of legal norms, rules and principles, as Dworkin [1977, 24ff] seems to affirm. A distinction can instead be established between norms indi- cating scalable objectives and those prescribing behaviors, cf. Dworkin [1977, 26] and, more clearly, Peczenick [1991]. On comparative evaluations and defeasibility in moral thinking, of. W.D. Ross [1930; 1939].

NORMATIVE CONFLICTS IN LEGAL REASONING 211

more tractable [cf. Bench-Capon & Sergot 1985; Gardner 1987; Gordon, 1989] and is rel-

evant to purposes such as modeling the reasoning of authorities disagreeing on those

meanings, or anticipating the possible case outcomes in a f ramework of semantically

indeterminate norms. Besides the semantic indeterminacy of the law, factual uncertainty

may also be considered. When alternative reconstructions of the facts are possible on the

basis of the available evidence, it might be interesting to compare the legal consequences

deriving from each of them, so that the party involved can see which additional elements

of evidence could lead to a solution in his favor.

Dynamic normative systems, combinations of rules and exceptions (more in general, of

rules establishing conflicting profiles of legal relevance), and semantic indeterminacy

(and also factual uncertainty) share a common aspect that justifies their consideration in a

unifying framework: all of them can be modeled by an inconsistent premises set, from

which the classical notion of logical inference cannot be used to derive legal conse-

quences (since ex fa l so sequi tur quidlibet).

To deal with such situations the notion of logical consequence must be distinguished

from the notion of legal consequence: not every consequence logically derivable from a

normative set is also a justified legal consequence of that set. By justification here we

mean the so-called ' internal just if icat ion' of legal consequences, to wit their derivation

from the premises assumed as valid, to be distinguished from the so-called external

justification of those premises [cf. Wroblewski [1969] 1983; Alexi 1978, 273ff]. Legal

reasoning has developed principles for establishing a hierarchy (an ordering) over the

legal system, and techniques to reason with ordered conflicting legal norms.

Those principles and techniques allow three types of consequences of an inconsistent

normative set being distinguished: - - Grounded consequences of the normative set, i.e., logically derivable from a consistent subset of it. Those

statements are the ones in favor of which there are legal reasons, possibly overridden by reasons to the contrary.

- - Plausible or undefeated consequences of the normative set, i.e., those grounded consequences such that the reasons allowing them to be derived are not worse than the reasons to the contrary.

- - Certain or justified consequences of the normative set, i.e., those undefeated consequences such that the reasons allowing them to be derived them are better than the reasons to the contrary.

I f a normative set grounds some merely undefeated (undefeated and not justified) con-

sequences, we say that it is indeterminate. Merely undefeated consequences have a

special relevance for legal thinking. The legal decision-maker cannot arbitrarily accept a

merely undefeated legal consequence, since this would violate the principle of universal-

izability. This principle requires that equal (in all relevant aspects) cases are treated in the

same way, and it will be violated if equal cases were solved differently by choosing alter-

native undefeated consequences. The legal decis ion-maker cannot also refuse all those

consequences, since he is obliged to decide every case under the assumptions that he

knows the law, and that law is determinate. The 'postulate ' o f the determinateness of the

legal system (often, but inexactly, it said that 'consis tency ' is to be postulated) means

exactly that relevant indeterminacies have to be solved by means of interpretative

choices. Two different aspects of legal reasoning can therefore be distinguished. (a) reasoning inside a (possibly inconsistent but) determinate normative set: here formal methods of legal rea-

soning should allow a legal conclusion to be obtained;

212 GIOVANNI SARTOR

(b) reasoning inside an indeterminate set of legal premises: here interpretative choices may be necessary, and the analysis of merely undefeated legal consequences allows 'choice points', hidden in the premises set, to be detected.

We should not try to obtain consistency in a determinate normative set by eliminating

or modifying some of the conflicting norms (i.e., by ' interpreting' the p r i m e fac i e

conflicting norms into a consistent whole). All conflicting norms may be considered valid

by the lawyer, who uses an ordering over those norms to determine consistent legal solu-

tions in concrete cases: the weaker norms continue to be applied, but only by default, i.e.,

unless stronger conflicting norms are applicable in the same case.

In using an indeterminate legal system, instead, interpretative choices are not only

possible, but legally necessary. These choices, to be grounded in the framework of

legal argumentation (this is the so called external justification), may consist in discarding

some premises in the normative set (as when rejecting a certain interpretation of an

ambiguous legal text) or in extending the ordering relation established over it (as when a

certain rule is qualified by the interpreter as an exception). As a result, one of the unde-

feated legal conclusions becomes legally justified, and must be accepted as the solution to

the case. Nevertheless, before making those interpretative choices, it is necessary to

analyze the indeterminate normative system, to identify the alternatives plausible conse-

quences and the choices required to eliminate indeterminateness. This analysis may also

be useful to establish the possible outcomes of a case in a context of legal uncertainty (for

example, when legal authorities disagree on the meaning of legal sources or on their

ranking), or to identify plausible arguments leading to a desired legal conclusion [cf.

Gardner 1987].

We shall try to build a framework modeling the aspects of legal reasoning just sketched

by relating two conceptual frameworks: the preferred theories approach, proposed by

Brewka (1991b) for nonmonotonic reasoning, and the belief change functions defined by

Alchourr6n et al. [cf. Alchourr6n & Makinson 1981, 1982, 1985; Alchourrrn et al. 1985;

Makinson 1985; G~denfors 1988; Gardenfors & Makinson 1991].

Let us first introduce some terminology.

Let Cn be a logical consequence operation [cf. Makinson 1985, 348] including classi-

cal logic, and let us write Cn(A) for the deductive closure of A, and A F-p for p C Cn (A).

If A ~-p we say that p is a logical consequence of A, or that p is logically derivable from

A. Let us call theory a set of statement closed under Cn.

Let a norm set be a set of norms ordered by a preference relation < : we write nl -< n2 to

mean that n 2 is at least as strong as n 1, n 1 < n 2 for n I ~ n 2 ^ n 2 ~ n 1, and n 1 _= n 2 for

nl <-- n 2 ̂ n 2 ~ n 1. Let a case be a consistent set of factual statements, i.e., statements in

which predicates expressing legal qualifications do not occur. Let a normat ive set be a

norm set, or the union of a norm set and a case. In normative sets the < relation is

extended assuming that each fact is superior to any norm: for each norm n and fact f, n < f.

Let us call a norm set A inconsis tent iff A ~- F, where F denotes contradiction. Let us

call a norm set A potent ia l ly inconsis tent iff A can lead to contradiction in some case C,

i.e., iff there is an inconsistent normative system A u C. In other words, A is defined as

potent ia l ly inconsis tent when it contains norms n l , . . . , n n that are conflicting in a case C,

in the sense that n 1 . . . . . n n establish inconsistent legal qualifications for the case C.


We assign names to norms, and use those names to express cross references. The name

of a norm is a predicate fol lowed by the free variables in the norm. By replacing variable

names with constants, we have names for all ground instances of the same norm [cf.

Poole 1987; Poole 1988, 32ff]. So, a norm will be expressed as follows:

name (x): conclusion ~-- condition

where x is the tuple of the free variables included in conclusion and condition. In all our

examples, the conclusion is a literal and the condition is a conjunction of literals. Formally,

such a norm should be read as a couple of formulae: the schema name (x), representing all

its ground instances, and the material implicat ion Vx (name (x) ---> (conclusion +-- condi-

tion)). Each instance of the schema affirms that the corresponding concrete norm is applic-

able; the implicat ion states that, whenever the norm is applicable to individuals x, the

implication established by the norms holds for them. The priority relation < holds for both

norm schemata and corresponding material implications.

This naming device is especial ly useful for formulating exceptions. Two kinds of

exceptions are here distinguished: exceptions to norms and exceptions to effects [cf.

Sartor 1991, 156-157]. Exceptions to norms state that particular norms, unambiguously

identified, do not apply in a given situation; exceptions to effects lay down that a particu-

lar legal qualification does not occur, is excluded, in a given situation. Exceptions to

norms override the norms to which they refer; exceptions to effects override any norm

establishing the excluded effect. Here we represent exceptions to norms as conditional

clauses whose conclusion is the negation of the name of the norm not to be applied, and

exceptions to effects as clauses whose conclusion is the negation of the excluded effect.

2. Three examples

To make our discussion more concrete, we introduce three examples, concerning the

legal dynamics, the application of rules and exceptions, and semantic indeterminacy.

2.1. REASONING WITHIN A DYNAMIC LEGAL SYSTEM

In the Italian legal system there is a general principle stating contractual liberty: the

parties are free to determine the content of their contracts, i.e. to establish whatever

contractual relations they like 2. Let us represent this general rule, stated by Art. 1322 of

the Italian Civil Code, issued in 1945, as follows:

C o n t r a e t F o r e e (r, c):

it holds between the parties that r <---

a contract c is accomplished ^

c establishes r.

2 Art. 1322 Italian Civil Code states that 'the parties can freely determine the content of the contract... ' . The Italian legal system (like any other system) limits this liberty by exceptions, as the same paragraph anticipates, continuing with 'in the limits established by the law'.

214 GIOVANNI SARTOR

In the Seventies, strict limitations to contractual freedom were introduced in tenancy

law. In particular, Art. 12 State Act 329, of 1978, stated that rent prices for dwellings

could not be freely established by the parties, but were determined by fixed legal criteria.

The contractual determination of a higher rent price was to be considered void (not

holding).

FixedRent Prices(P1, h, P2): -fi t holds between the parties that (Pl is the rent price of h)~--

h is a dwelling ^

P2 is the legal rent price of h ^P2 < Pl.

As a result of the enactment of this norm, prevailing over the general principle of

contractual liberty by both criteria of lex posterior and lex special& (Cont rac tForce

< FixedRentPrices) , that principle could not be applied to rent prices for dwellings

(but it could be applied to other aspects of tenancy contracts, as to other types of con-

tracts).

In the Eighties, the political climate changed (deregulation became the new slogan),

and new norms were introduced, as exceptions excluding the constraints on rent prices, in

certain parts of the Italian territory. Here is the formalization of one of such exceptions.

FixedRentPr icesExcl(Pl , h, P2, m, n):

--,Fixed RentPrices(p 1, h, p2)3~ -

the house h is situated in the municipality m ^

n is the number of the inhabitants of m ^ n < 20 000.

The constraint on rent prices is blocked by this last exception (assuming that

FixedRentPrices < FixedRentPricesExcl), so that the general principle of contractual

liberty expands, again regulating the rent prices in the municipalities with less than

20 000 inhabitants.

Now, it seems that the Italian Parliament intends to abrogate Art. 12 State Act

329/1978 (FixedRentPrices). As a consequence, art. 1438 Civil Code (FixedRent-

PricesExcl) would recover its full applicability in the rent price domain.

The example shows that preserving conflicting norms in a dynamic legal system has

some important advantages: - - Completeness: the weaker norm (as art. 1322 Italian Civil Code) continues to be applicable in cases in

which stronger conflicting norms (as art. 12 Parliament Act 329/1978) are not applicable. So, normative gaps can be avoided.

- - Elasticity: if a stronger norm is abrogated or derogated, the weaker norm can recover its former application ambit.

This approach minimizes the costs of the modification of the legal system

(modification takes place simply adding axioms to the base of the system), relying on the

conflict solving mechanism to avoid inconsistent legal conclusions.

3 The negation of the name of a rule is to be read as the statement of the inapplicability of that rule. For example, the norm FixedRentPricesExcl, with consequent --FixedRentPrices(pl, h, p2), is to be read as: 'the norm FixedRentPrices is not applicable (to the rent price of a dwelling having a certain legal prices), if the dwelling is situated in a municipality having less than 20000 inhabitants'.


Normative hierarchies explain the dynamic function of norms establishing negative legal qualifications. When the legislator wants to eliminate a certain legal content p in circumstances q, he enacts a norm n: ~ p +-- q, stronger than any norm establishing p. In this way, he makes p no longer derivable in case that q, but he also obtains an additional result: any future norms establishing p, and weaker than n, will be blocked whenever q holds. Permissive norms (such as typically, constitutional norms establishing liberty rights) are typical instances of this technique. Therefore, we can consider strong permission P(p) as equivalent to ~O(-~p), i.e. as the negation of the obligation O(~p). But permission has no special role: what happens with obligations (negated by permissions) happens with any legal predicate (contradicted by a norm establishing its negation). For example the constitutional rule stating that the President of the Italian Republic is not liable for his acts is certainly not a permission, but blocks any rule establishing liabilities for the acts he has accomplished.

2.2 REASONING WITH RULES AND EXCEPTIONS

In the previous example, normative conflicts resulted from the temporal succession of normative acts determined by conflicting political purposes. But conflicting norms are frequently contained in the same normative act, structured into a combination of rules and exceptions [cf. Gordon 1988; Sartor 1991]. Let us consider, for example, the following set of rules and exceptions from Italian tort law.

ParentLiability(p, d,f, s): p is liable for the damage d caused by the f ac t f <----

p is a parent of s A S is liable for the damage d caused by the factf.

ParentLiabil i tyExcl(p, d,f , s): --,parentLiability(p, d, f, s) ~--

~(the parent p could prevent the fact f). 4

FauitLiability(x, d,f): x is liable for the damage d caused by the f ac t f +--

x accomplished the fact f A the factfcaused the wrongful damage d A x is at fault for the f a c t f 5

Selfdefense(x, d, fi: ~(x is liable for the damage d caused by the fact f) <--

4 The first two norms formalize a part of Art. 2048 Italian Civil Code(Parent liability . . . ): 'The father and the mother . . . are liable for the damages caused by torts committed by their under age children .. . [They] are freed from their liability only if they prove they could not prevent the fact.'

This norm corresponds to Art. 2043 Italian Civil Code (Tort compensation), establishing the general principle of fault liability: 'Any intentional or negligent act, causing unjust damage to other persons, places any one who performs that act under a duty to pay compensation for damages'.

216 GIOVANNI SARTOR

x did the f a c t f b y self defense. 6

Incapacity(x, d, f):

~ ( x is l iable for the damage d caused by the fact f ) ~--

x was incapable during the factf .

IncapacityExcl(x, d, f i :

~Incapaci ty(c , d , f )

x ' s incapacity during the f a c t f w a s due to his fault. 7

Let us assume the following ordering:

ParentLiability < ParentLiabilityExcl; FaultLiability < {Incapacity, SelfDefense}; Incapacity < IncapacityExc 1.

The norm set A = {ParentLiability, ParentLiabilityExcl, FaultLiability, Incapacity, SelfDefense, IncapacityExcl} is consistent, but potentially inconsistent. Let

us consider what happens by adding convenient facts. Let us assume, for example, that

Mary wants compensation from Mark, father of John, saying that John, who was invited

to a party in her house, has broken her precious Chinese vase. She has been able to prove

the following facts:

F21: John has accomplished the fact ChineseVasePush

F22: the fact ChineseVasePush has caused the wrongful damage ChineseVaseDe-

struction

F23: John is at fault for the fact ChineseVasePush

F24: Mark is a parent of John

The normative set A1 = A u {F21, F2z, F23, F24} is consistent, and it implies that

both Mark and John are l iable for the destruction of the vase (since the conditions of the

l iabili ty norms are satisfied and the condition of no exception is).

Mark can free himself by satisfying the condition of one of the exceptions, and hence

causing a contradiction with the rules establishing liability. So, for example, if he proves

that John was incapable at the moment of the accident, i.e., that

F25: John was incapable during the fact ChineseVasePush,

the normative set A 2 = A 1 u {F25} is obtained, where John 's l iabili ty is excluded by

the exception Incapacity, whose consequent contradicts the effect of the rule

FaultLiability. Mary can prove that John 's incapabil i ty derived from his fault, since it was caused by

his drunkenness:

F26: John's incapacity during the fact ChineseVasePush was due to his fault

6 Art. 2044 Italian Civil Code (Self Defense). He who causes the fact for the defense of himself or of others is not liable. 7 These two last norms formalize art. 2046 Italian Civil Code (Liability for the harmful fact): 'He who was incapable at the time when he committed the harmful fact, is not liable for its consequences, unless the state of incapability derives from his fault'.


In the normative set A 3 --- A 2 u {F2n} the exception Incapacity is denied (its applica-

bility is excluded) by the stronger exception IncapacityExcl, John appears liable again,

and therefore Mark. Then Mark could try to prove that he could not control his son, so as to prevent the

fact, that is

F27: ~ (the parent Mark could prevent ChineseVasePush),

and free himself again from the liability, through a new contradiction (between the rule ParentLiability and the exception ParentLiabilityExcl) arising in the normative system

A 4 = A 3 u {F27}. In every phase of the dispute, contradictions among rules and exceptions assume a peculiar relevance: if the condition of an exception denying a rule or its effect is proved, the derivation of the consequent of that rule is blocked.

The set A of rules and exceptions is inconsistent, but determinate: no uncertainty arises, since in every possible case grounding alternative consequences just one of these

is justified (all other grounded consequences are defeated). To prevent inconsistencies,

each norm should be rewritten so that its condition includes the complement of a literal occurring in the body of any exception to the norm itself or to its effect. In this way we would be sure that whenever the norm is to be applied, its exceptions are not. Therefore,

no ordering would be required to solve conflicts. For example, the reformulation of the

norm set A including all norms above gives the following result.

1. p is liable for the damage d caused by the fac t f +-- p is a parent of s ^

s is liable for the damage d caused by the fact f ^ ~(the parent p could prevent the fact f)

2. x is liable for the damage d caused by the fac t f

x accomplished the fact f ^ the factfcaused the wrongful damage d A

X is at fault for the fact f ^ --,(x did the fac t fby self defense) ^ -,(x was incapable during the facty)

3. x is liable for the damage-d caused by the f ac t f4 -

x accomplished the fact f ^ the factfcaused the wrongful damage d A

x is at fault for the fact f ^ ~(X did the fac t fby self defense) ^

x's incapacity during the factfwas due to his fault

4. --,(x is liable for the damage d caused by the fact f) 4-- x did the fac t fby self defense

5. -,(x is liable for the damage d caused by the factO) <-- x was incapable during the fact f ^ -,X'S incapacity during the factfwas due to his fault.

218 GIOVANNI SARTOR

This reformulation of A 8 corresponds to the (erroneous) opinion that legal norms are

only p r i m a f a c i e in conflict so that contradictions can be eliminated by means of interpre-

tation. Under this assumption it should be possible to translate any norm set X of ordered

conflicting norms into a set R x of non conflicting unordered statements, such that, in

every possible case, R x logical ly imply all and only the legal consequences derivable

from X.

Unfortunately, R A = {1, 2, 3, 4, 5} does not behave exactly as A. The main differences

are the following.

- - N o n m o n o t o n i c reasoning. Normative systems including rules and exceptions

allow nonmonotonic reasoning. The conclusions of a rule can be drawn whenever the

condition of the rule is satisfied, provided that the conclusion of no exception (to that

rule or to its conclusion) is derivable. If further information becomes available, al lowing

the condition of an exception to be satisfied, the conclusion of the rule must be

retracted. Therefore, the legal decision-maker (e.g., the judge) must derive the con-

clusion of the rule not only when he has posit ively ascertained that no exception is

satisfied, but also when he has no information about it, and he must retract this conclu-

sion if further evidence allows an exception to be satisfied. This aspect is lost in the

consistent ' interpretation' . For example, from A 1 = A u {F21, 1722, F23, F24} it is

possible to derive John 's responsibili ty. This result can be obtained by using the norm

17aultLiabili ty, since the condition of that norm is satisfied, while there is no information

concerning its exceptions SelfDefense and Incapac i ty . Instead from the premises set Rsl

= A s w {1721, F22, F23, F24} it is not derivable that John is responsible, since the con-

dition of none of clauses 1, 2, or 3 is satisfied. If further information is added to A1,

al lowing the condition of an exception to be posit ively satisfied (such as fact 172s), the

assertion of John 's responsibil i ty has to be retracted. Nothing is changed, instead by

adding F2 s to Rsl.

- - The dialectic o f rules and exceptions. Rules and exceptions have a dynamic aspect:

if a new rule is enacted 9 exceptions automatically apply to it, and if an exception is abro-

gated the rules expand themselves, i.e., are applicable in cases where they were previ-

ously blocked by the exception 1° Conversely, if an exception is added, the possibil i ty of

applying all rules conflicting with it is correspondingly reduced. We cannot obtain these

effects by adding or removing statements to a consistent set of unordered statements: for

this purpose a reformulation of those statements would be required.

s We have completely omitted exceptions to norms (whose only effect is to exclude the application of the norm to which they refer), and we can also eliminate statements 4 and 5, derived from exceptions to effects, if we assume that only positive legal effects are relevant (since the negative legal qualifications are only intended to block the derivation of the positive ones). 9 For example, a new form of liability, such as that for environmental damages, recently introduced in the Italian legal system. lo As in the example above, where the elimination of the constraint on rent prices determined the expansion of the principle of contractual freedom.


2.3 MODELING ALTERNATIVE INTERPRETATIONS

The Italian Constitution establishes that private property is protected (it cannot be expro-

priated). Nevertheless, the Constitution also states that property can be expropriated if the

expropriation is based on a statute, and a sufficient compensation is provided. The concept of sufficient compensation is at the center of an enduring legal dispute: must a sufficient compensation correspond, more or less, to the market price, or can it be established by the legislator at a lower level? The Italian legislator adopted the second inter-

pretation, while the Constitutional Court chose the first, and the problem is still unsettled.

Property(x, y):

~ (x may be expropriated of y) <--- x is the legitimate owner of y

Expropriation(x, y): x may be expropriated of y <---

y ' s expropriation is based on a statute A X is sufficiently compensated for y

Meaningl(x, y, Zl, z2): x is sufficiently compensated for y <---

x gets for y the amount of z I lire established by the law ^ the market price o f y is z 2 lire ^ z 1 ~-~ Z 2

Meaning2(x, y, z): x is sufficiently compensated ~--

x gets for y the amount of z life established by the law.

Meaning 1 and Meaning 2 are alternative, in the sense that both of them cannot be used to ground a legal conclusion. This must be specified by an additional axiom (as it necessary whenever dealing with alternative meanings):

MeaningChoice(x, y, zl, z2): ~(Meaningl(x, y, zl, z2) ^ Meaning2 (x, y, Zl) ).

Let us introduce now the facts:

F31: Mark gets for BlackAcre the amount of 100 000 000 lire established by the law F32: the market price of BlackAcre is 200 000 000 lire

F33: Mark is the legitimate owner of BlackAcre

Let the following ordering: P rope r ty < Expropriation < {Meaningl, Meaning2} < MeaningChoice < [F31, F32}. The normative set {Property, Expropriation, Meaning1, Meaning 2, MeaningChoiee, F31, F32} is indeterminate: by Meaningl BlackAcre is sufficiently compensated, and therefore, by Expropr ia t ion , it may be

expropriated; by Meaning 2 it is not possible to draw this conclusion. In such a situation the legal decision-maker must make a choice between the two alternative interpretations, and only after that choice he can take a (justified) decision.

2 2 0 GIOVANNI SARTOR

3. The preferred subsets framework

To give a formal account of the aspects of legal reasoning just outlined, we will consider first the preferred subsets 11 framework proposed by Brewka [1991b, 64ff], an approach

generalizing abduction [cf. Poole 1988], and inspired by Rescher [1964].

Brewka defines the notion of provability from a possibly inconsistent set of premises (a

base) A, by considering just certain maximal consistent subsets of A, i.e., its preferred subsets. So, a formula p is defined as weakly provable from a set of premises A iff there is

a preferred subset B ~ A such that B J-p. A formula p is defined as strongly provable from A, iff, for all preferred subsets B of A, B t-p [Brewka 1991, 65]. To build preferred

subsets Brewka [(1991, 75] assumes a strict partial ordering < 12 over a finite set of premises A, and considers all possible linearizations of the < relation.

Let < be a strict partial ordering on a finite set of premises A. B is a preferred subset of

A iff there exists a strict total ordering (P0, Pl . . . . . Pn) of A respecting < (i.e., ifpk < pj then j < k) such that B = B0 with

B 0 : = { } , a n d f o r 0 < i < n .

B i + 1 : = if t i + 1 consistent with B i , then B i u { t i + 1 }, else B i.

Here is an equivalent redefinition of Brewka's notion of preferred subset, that will be

used the following: a preferred subset B of a premises set A, over which a preordering _< is established, is obtained by inserting progressively into B the elements of A that are

maximal among those consistent with the elements already in B. In other words: (a) Let B0 be the empty set, and A0 be the premises set under consideration.

(b) For all i, while A i contains a non empty subset C i of elements that are consistent with B i , let B i + 1 be

B i k.) {p }, and A i + 1 be A i - {p }, where p is a _< maximal element of C i (for every q E Ci , p 4:. q).

(c) I r A i does not contain any element consistent with B i , then B i is a preferred subset of A.

In a declarative style, we could equivalently say that B = B n is preferred subset of A iff

there are sequences (A o . . . . . An) and (B o . . . . Bn) such that - - B0= { } , a n d A 0 = A ;

- - for all i < n, B i + 1 = Bi • {P}, and Ai + 1 = Ai - {P}' w h e r e p is < maximal in the set of elements o f A i consistent with Bi;

- - A n does not contain any element consistent with B n.

These last definitions manifest the intuition at the basis of Brewka's approach. A preferred subset B of a premises set A is a maximally consistent subset of A that is in

a situation of equilibrium: it is not possible that new statements are inserted in B without rejecting from it, in order to maintain consistency, statements not worse than those new statements. B can include statements incompatible with stronger statements in A, under the condition that those stronger statements are incompatible with other (non weaker) statements in B. Premises excluded from B, because of their incompatibility with (non weaker) premises already in B cannot block other premises from being inserted in it.

11 Brewka speaks o f p r e f e r r e d sub theor i e s , but, since the world t heory is here used only for deductive closures, we prefer to speak o f p r e f e r r e d subse t s .

12 To be consistent with the fol lowing discussion, the < relation assumed in Brewka [1991b, 75] has been inverted, and the definitions have been modif ied accordingly.


In this framework, to which we will refer when speaking of preferred subsets, the

notions of undefeated and justified legal consequences are definable as follows: - - Weakly provable statements, to wit statements derivable from a preferred subsets of A (the set of weakly

provable formulae is given by u {Cn(B): B is a preferred subset of A}), can be considered plausible (undefeated) legal consequences of A. Therefore, a legal system is determinate (has no merely undefeated consequences) when it has just one preferred subset.

-- Strongly provable statements, to wit statements derivable from all preferred statements of A (the set of strongly provable formulae is given by n {Cn(B): B is preferred subset of A}), can be considered certain (justified) legal consequences of A.

A third notion of provability might also be useful, along with those introduced by

Brewka, to wit the notion of strongly grounded statements. It might be argued that to

accept a certain statement p, it is not sufficient that p is implied by all preferred subsets of

the available premises set A: it is necessary that p can be derived by established premises,

to wit that p has a unique foundation in all preferred subsets. So, let us say that p is

strongly grounded in A iff p E Cn ({ r iB: B is a preferred subset of A}). Strongly

grounded statements are also strongly provable, but the converse is not always true.

The application of the framework just introduced to the examples above gives intu-

itively satisfying results. Let us consider a simplified version of example 2.2, to wit the

norm set A = {FaultLiabi l i ty , Incapaci ty , Incapaci tyExcl} , over which the ordering

Fau l tL iab i l i ty < Incapac i ty < Incapac i tyExc I is established. The normative set

A 1 = A u {F2 l, F22, F23, F24} is consistent. Therefore, it is trivially the only preferred

subset, and all its consequences are strongly provable statements. Among those conse-

quences, there is the following C2.

C2: John is liable for the damage ChineseVaseDestruction caused by the fact

ChineseVasePush

The s e t A 2 = A 1 u {F25} is inconsistent, but determinate, since its has the only pre-

ferred subset {F2 l, F22, F2 a, F24, F2 s, Incapaci ty , Incapaci tyExcl} from which C2 is

no longer deducible. 13 The normative set A 3 = A 2 u {F26} has the only preferred subset

{ F21, F22, F23, F24, F2 s, F26, Faul tLiabi l i ty , Incapaci tyExc 1 }, from which C2 is again

derivable.

Let us consider, instead example 2.3. Here the ordering is P roper ty < Expropriation < {Meaning1, Meaning2} < MeaningChoice < {F3 l, F32}. The normative set {F31,

F32, MeaningChoice, Meaning l, Meaning 2, Expropriation, Property} is inconsistent

and indeterminate, since it has two preferred subsets, to wit {F31, F32, MeaningChoice, Meaning 1, Expropriation, Property} and {F31, F32, MeaningChoice, Meaning 2, Expropriation, Property}. The first implies

C31: ~ M a r k may be expropriated of BlackAcre

13 In reality, not all instances of the rule FaultLiability are inconsistent with the other statements in the premises set, but only the instance FaultLiability (John, ChineseVaseDestruction, ChineseVasePush). So, the preferred subset is {F21, F22, F23, F2a, F25, FaultLiability - {FaultLiability (John, ChineseVaseDestruction, ChineseVasePush), Incapacity, Incapac!tyExcl}. We omit the instances FaultLiability - (FaultLiability (John, ChineseVaseDestruction, ChineseVasePush } since in the example here considered they are not relevant (in other situations this is not the case, cf. Sartor 1991, footnote 30).

222 GIOVANNI SARTOR

while the latter implies

C32: M a r k may be expropriated of BlackAcre

Both C31 and C32 are weak consequences of the considered premises set, confirming

our intuition that both are plausible but not certain legal consequences.

4. The AGM model of belief change

Alchourr6n, G~denfors , and Makinson (referred hereafter as AGM) have developed, a

general theory of bel ief change, for representing 'knowledge in flux'. 14 They consider

three fundamental operations on sets of sentences: - - E x p a n s i o n : a new proposition (axiom) p, hopefully consistent with a set A, is added to A, and this expanded

set is closed under logical consequence. The expansion of A by p is denoted A + p. - - C o n t r a c t i o n : a proposition p, derivable from A, is rejected from it, i . e . , a set that does not imply p is

obtained from A. This set, the contraction of A by p, is denoted A "- p. - - R e v i s i o n : a proposition p, inconsistent with A, is added to A with the requirement that the result be consis-

tent and closed under logical consequence, i . e . , a consistent theory containing p is obtained from A (for this purpose, obviously, some of the statements in A must be eliminated). This theory, the revision of A by p, is denoted A ) p.

The core of the A G M proposal resides in their analysis of contraction functions. In

fact, the definition of an expansion function does not present difficulties (expansion is

based on set union: A + p = Cn (A u {p})), while revision can be defined, through the

so-called Levi identity (cf. Levi 1977) in terms of contraction and expansion: A 4- p =

(A - ~ p ) + p.

The definition of a contraction function is problematic, since there can be different

alternative ways to obtain a set A "- p from which p is not derivable. Let us denote by

A ± p the family of all maximal subsets X ~ A such that X ~t p.

Alchourr6n & Makinson [1982] tested the idea to define A ' - p as a maxichoice con-

traction, i.e., as one of the X E A ± p. Unfortunately, they showed that maxichoice con-

traction functions give results contrasting with intuition when applied to theories: the

result of the contraction of a theory T is too big to correspond to reasonable expectations.

In particular, maxichoice revisions of theories, obtained by means of the Levi identity

T4- p = ( T ' - ~ p ) + p from maxichoice contractions have the property of creating com-

pleteness: any maxichoice revision T+ p is complete (for any proposit ion q, q E T4- p or

~ q E T 4- p) whenever p is inconsistent with T, even if T is not complete [Alchourr6n &

Makinson 1982, 19if].

Another candidate as a contraction function is the meet contraction, i.e., the intersec-

tion of all elements of A _ p. But this result is too small to correspond to intuition. In par-

ticular, Alchourr6n & Makinson [1982, 18-19] showed that the revision derived from the

meet contraction of a theory T collapse into the underlying consequence operation:

T ' - p = Cn(p).

14 Cf., for a systematic introduction, Makinson [1985], Gfirdenfors [1988]. The AGM model seems to have originated also from the study of normative systems (cf. Alchourrdn & Bulygin [1978,1981], Alchourrdn [1982], Alchourrdn & Makinson [1982].


Intuitively satisfying contraction functions can be characterized, instead, as partial meet contractions, i.e., by defning A "- p as the intersection of a certain subset of A ± p. This subset is specified by a selection function ~t, which singles out the most important ele-

ments of A ~ p. The contraction function A "- p is defined, so, as n~/(A ± p), i.e., A "- p contains exactly the proposition common to all X E ~/(A ± p).

Alchourrrn et al. [1985] prove some interesting properties of partial meet contraction, that will be briefly introduced in the following (the reader not interested in the technical

aspects of the AGM proposal can go directly to the end of this paragraph, where the

notion of safe contraction is introduced and applied to a legal example). In particular, they show that partial meet contraction functions, and the corresponding revision func-

tions (obtained through the Levi identity) are fully characterized by the following rationality postulates.

Contraction postulates

( ' - 1) A ' - p is a theory whenever A is a theory (closure). ( "- 2) A "- p ___ A (inclusion). ( ' - 3) I f p ~ Cn (A), thenA "- p = A (vacuity).

( ' - 4) I f p ~ (o), thenp ~ Cn (A'-- p) (success).

( ' - 5) If Cn (p) = Cn (q), then A "- p = A "- q (preservation).

( "- 6) A c_ (A "- p) + p whenever A is a theory (recovery).

Revision postulates

(+ 1) A+ p is always a theory. (q- 2 )p E A S p . (~- 3) If ~ p ~ Cn (A), then A ~ p -= A + p.

(~- 4) If ~ p ~ Cn (~), then A ~- p is consistent under Cn. (q- 5) If Cn (p) = Cn (q), then A ~- p --- A Jr q.

(q- 6) (A ~ p) n A = A "- ~ p whenever A is a theory.

Moreover, it is shown that additional rationality postulates are satisfied when the selec-

tion function "¢ is defined according to a relation < over the subsets of A, so that it selects the maximal elements (over < ) of A ± p. More exactly, ]t is said to be relational over A iff there is a relation < over 2A (the powerset of A) such that for all p ~ Cn (~), <

marks off 7( (A __ p), in the sense that the following identity holds, for all p:

7 ( A ± p ) = {B E ( A _ p ) : B ' < B foral l B ' E (A±p)}

Relationality of 7 implies that the corresponding contraction and revision functions satisfy the following additional postulates:

( ' - 7 ) ( ( A ' - p ) n ( A ' - q)) c A "--(p ̂ q) for any theory A. (Jr 7) (A $ (p ^ q )) c_ ((A + p) + q) for any theory A.

224 GIOVANNI SARTOR

Transitivity of the < relation has further consequences:

( "-- 8) (A-" (p ^ q) ___ A "- p whenever p ~ A "- (p ^ q) ( ~- 8) (A $ p) + q) c_ A ~- (p ^ q) for any theory A, whenever ~ p ~ A "- p

The idea behind postulates ( "- 7) and ( "-- 8) is that revising a theory by p ^ q should be equivalent to revising it by p and then consistently adding q.

The concept of ep i s t emic e n t r e n c h m e n t [cf. Gfirdenfors 1988; G~irdenfors & Makinson

1988; Gfirdenfors 1990] allows contraction and revision functions to be built according to an ordering over A: the statements in A are assumed to have differen~ degrees of impor-

tance (of value for planning or problem solving purposes), and the sentences that should be preserved when A is contracted by p (i.e., that should be contained in A "-- p) are those

which have a higher epistemic entrenchment, that is, more exactly, any sentence q such that p < (p v q).

The following five rationality postulates for epistemic entrenchment are introduced [Gfirdenfors 1988; G~denfors & Makinson 1988].

(EEl) I f p _< q and q _< r, thenp _< r (transitivity)

(EE2) I f p k- q, thenp _< q (dominance)

(EE3) For any p and q, p _< p ^ q (conjunctivenss)

(EE4) I fA is consistent, p ~ A i f fp _< q for all q (minimality) (EE5) q _< p for all q, only i fp is logically valid (maximality)

It is shown that, if the ordering of the sentences of a theory respects those postulates, then it is possible to construct contraction (and revision) functions satisfying the rationality postulates above.

Other interesting results are in Alchourr6n & Makinson [1985, 1986], where the

concept of safe con t rac t ion is proposed. A non-c i r cu lar relation < is assumed over A (by

non-circularity it is meant that ' for no a l, a 2 . . . . a n E A (n >- 1), a 1 < a 2 < . . . < a n <

a l ' [Alchourr6n & Makinson 1985, 407]: and the safe contraction function ' - determined

by < is defined by putting A ' - p = A n Cn (A/p), where A/p is the set of all elements a E A that are safe with respect top . 'An element a of A is said to be safe with respect t o p

. . . iff a is not a minimal element (under < ) of any minimal subset (under inclusion)B of A such that p E Cn(B)' [Alchourr6n & Makinson 1985, 407-8] . The idea is to discard

from A the elements that can be 'blamed' for the implication o f p (these elements are the weakest ones necessary for a subset of A to imply p). Given certain restrictions on the < relation it is shown that any safe contraction function satisfies the rationality postulates for contractions, and so, is a partial meet contraction function.

Let us apply the belief revision approach, in the safe contraction version, to a simplification of example 2.2. Let us start with the set A = {FanltLiability, Incapaci ty,

Incapac i tyExc 1, F21, F2 z, F23, F24}, implying John's liability, and let us revise A by F2 s. The minimal subset of A from which ~ F 2 s is (contrapositively) derivable is [FauitLiability, Incapaci ty , F21, F2z, F23, F24} whose minimal element, under < , is FanltLiabil i ty, that has to be discarded from A '-- p. Therefore, John's liability is not

N O R M A T I V E CONFLICTS IN LEGAL REASONING 225

derivable. Let us revise, instead, A by F2s ^ F26. The minimal subsets from which ~(F25

A F26) is derivable are {FaultLiability, Incapacity, F21, F22, F2 a, F24}, and

{Incapactiy, IncapacityExcl, F21, F22, F2 a, F24}. Therefore, we should reject both

FaultLiability and Incapacity, and we obtain the set {IncapacityExc l, F21, F22, F23, F24} from which John's responsibility is always not derivable. The preferred subsets approach gave a different result in this case. The reason seems to be a different ground

intuition concerning the role of information having an intermediate position in the _< hierarchy: in safe revisions a statement p (such as FaultLiability) conflicting with a higher statement q becomes unusable, even q is in conflict with a higher statement r; in

the preferred subset approach, instead, the conflict of q with r allows p being recovered.

5. Relations of the two approaches

The motivations behind the AGM's and Brewka's approaches seem quite different: in the first approach, the fundamental problem is maintaining consistency through change, in

the latter, it is reasoning from inconsistent premises. Nevertheless, Brewka has tried to apply his model to belief change, while Alchourr6n and Makinson have treated the problem of reasoning from inconsistent premises.15 in the following pages, we shall con-

sider first how to model contraction and revision inside the preferred subsets approach, and then how to reason from inconsistent premises sets in the framework of belief revi-

sion. Let us first introduce a new definition of Brewka's preferred subsets, that can easily be

shown to be equivalent to the ones introduced above.

Let us say that X is strictly more exposed than Y [cf. Alchourr6n & Makinson 1981, 127], iff there is ap E X such that for all q E Y ,p < q.

Let us denote by A 3_ F the family of the maximal sets of A from which inconsistency

is not derivable, to wit the set of the maximal consistent subsets of A. For X, Y E A ± F, we say that Y is protected against X, and write X _~ Y iff for every p E X such that p ~ Y there is a Y' c y such that {p} u Y' is inconsistent and Y' is not strictly more

exposed than {p} (for every q E Y', q g: p) .16 In other words, it would not be possible to insert in Y any element p E X, without having to reject statements q at least as

valuable as p (under the _< relation). The '~ maximal elements in A _L F, that we denote

as [l(A 3- F ) - - to wit 13(A ± F) = {B E (A ± F): B' <1B, for all B' E (A ± F)} - - are exactly the preferred subsets of A, 17 as can be checked by using the definitions proposed

under 3.

15 The relation between nonmonotonic reasoning and bel ief change is considered also in Gfirdenfors & Makinson [ 1991 ]. 16 The <1 relat ion can also be extended to non maximal subsets. We can say that X <1 y iff for every X' c_ X is X' u Y is inconsistent, then there is a subset Y' of Y such that X' u Y' is inconsistent and Y' is not strictly more exposed than X'. 17 We can also say, equivalent ly that a maximal ly consistent B c_ A is a best subset o f A iff for every p E A such that p ~ B there is a B' ~ B such that {p } u B' is inconsistent and B' is not strictly more exposed than {p } (no element of B' is inferior to p).

226 GIOVANNI SARTOR

5.1. CONTRACTION

Let us consider now the contraction of a consistent premises set A. By contraction of A by

p we mean, in general, the construction of a new set of statements, from which p is not derivable. We will distinguish contraction in the AGM and in the preferred subsets approach, by denoting the first as A~Mcontraction and the second as PScontraction.

The PScontraction of A by p can be defined as A u {=/2}, where for all q in A,

q < _-112, and _-1t2 is a constraint, to wit a formula that can be used only in consistency

checking [cf. Brewka 1991a; Brewka 1991b, 78ff]. So, any preferred subset of the r'Scontraction of A by p contains _-1t2, and, therefore, does not include any X such that X ~-

p. The statements weakly provable from A u {_-1/2} are given by u {Cn (X): X E ~ ((A

u {7/2} J- F)}. The strongly provable ones are n {Cn (X): X E ~ ((A u 7/2}) ,L F)}. The strongly grounded ones are given by Cn (n ~(A u {=/z}) ,LF)).

Let us now define a selection function y(A ~ p), that selects, in the maximal subsets of A not implying p, exactly the most protected ones (but considering their extension with the 7t2 constraints):

]'(A A_ F ) = {B E (A_l_p): ( B ' u ]=/2} < l (Bu {~/2}) for a l lB ' E (A_Lp)}.

It is easy to see that for every X E y(A _Lp) there is an Y E [~((A u {=~}) _LF) such that

Y = X u {_~/2}, and conversely. Therefore n [3 ( (Au {~/2}) _L F) = ny(A A_ p) u

{=/2}, and since constraints can be used only in consistency checking, Cn(n [3((A u {7/2}) ± F)) = Cn(ny(A _L p)). The consequences of the A~M contraction function A "- p based on

the selection function ~/are exactly the consequences of the statements strongly grounded in the PScontraction: Cn(A ' - p) = Cn (n (y(A ± p))) = Cn (n(~((A u {_~/2 }) _L F))).

5.2. REVISION

Let us consider now revision of a consistent premises set A. By revision of A by p, we mean, in general, the construction of a consistent set of statements, from which p is deriv-

able. Obviously, we do not want the output of the revision to be a gratuitous distortion or impoverishment of A. As with contractions, we distinguish PSrevision and A~Mrevision.

The PSrevision of A by p can be defined simply as the addition of p to A, i.e., as

A u {p}, p being greater than any element of A over < (cf. Brewka 1991a; 1991b 78ff). A formula q is weakly provable from the PSrevision of A by p iff it is derivable from a set Y E 13(A u {p } _1_ F). The set of the weakly provable formulae is therefore given by u {Cn(Y): E 13(Au {p} ± F)}, while strongly provable and strongly grounded formulae are n{Cn (Y): Y E ~ ( A u {p} _L F)} andCn (n{Y: Y E I](A u {p} A_ F)}).

A correspondence can be established between strongly grounded consequences of the PSrevision and partial meet ACMrevision based on the selection function y defined above: the set of the strongly grounded statements coincides with the partial meet aaMrevision A + p. In fact, it would be easy to show that

Cn(n{Y:YE ~(Au {p} _LF)}) = C n ( n { X u {p}: XEqt(A J_ ~ p ) } ) = Cn(n{X: X E ~{(A _l.~p)} u p ) = A~- p.

NORMATIVE CONFLICTS IN LEGAL REASONING 2 2 7

6. Reasoning with inconsistencies

We have observed above that the lawyer, faced with an inconsistent normative system,

normally prefers to use conflict handling procedures to derive consistent legal conclu-

sions, rather than transform the system into a set of compatible norms. Although in the

AGM model of belief change consistency is assumed and maintainedJ 8 Alchourr6n & Makinson [1981] proposed a formal analysis of reasoning from inconsistent normative sets, further developed in Alchourr6n [1986].

Alchourr6n & Makinson [1981] affirm that it is possible to deliver a justified verdict on

the basis of an inconsistent normative set A by using (and eventually extending) an ordering _< over A. They introduce the following notions: - - Indicat ing . B c_ A indicates a consequence p iff p E Cn(B), and moreover for all C c_ A, if --~p E Cn(C),

then B is not strictly more exposed than C.

- - De te rmin ing . B d e t e r m i n e s p i f fp E Cn(B), and moreover, for all C ~ A, if -~p E Cn(C), then C is strictly more exposed than B.

- - Del iver ing . A premises set A de l i vers a proposition p, iff some subset B ___ A determines p.

It is shown that the delivered propositions are those derivable from the set of the

normal elements of A. An element a E A is normal iff for every inconsistent C CA, there is a c E C with c < a.

Indicated statements are candidate plausible (undefeated) consequences of the premises

set, and should correspond, therefore, to Brewka's weakly provable statements, Nevertheless, it is easy to see that weakly provable statements and indicated statements do

not coincide: statements indicated from a premises set A may be strictly included in the weakly provable statements. For example, given the premises set {~p, p ^ q, ~q} and the

ordering ~p < p ^ q < ~q, {~p, ~q} is the only preferred subset and, therefore, ~p is weakly derivable (and also strongly derivable). Nevertheless, it is easy to see that ~p is not indicated. The problem is the same already seen in safe contraction: in Alchourrrn &

Makinson's intuition intermediate statements have the power of inhibiting the use of the

incompatible lower ones independently of higher statements; in Brewka's approach, instead, lower statements may be recovered when intermediate statements conflict with

higher ones. Similar considerations distinguish strongly provable and strongly grounded statements from the delivered ones.

Alchourr6n & Makinson's model is developed by Alchourr6n [1986], where condi-

tions are defined for the truth of propositions such as O(q/p), to be read as the meta-theo- retic assertion 'it is obligatory to do q (referring to a system A of norms - formulated in the language) in 'p ' conditions' [Alchourrrn 1986, 182]. A is said to be a 'critical set for p if and only if A k ~p and furthermore, there is no typical subset of A that implies -~p'

[Alchourr6n 1986, 183]. Then, the notion of 'healthy norms for p ' , denoted by A/p, is introduced (healthy norms are strictly related to the safe elements of Alchourrrn & Makinson [ 1985], and to the normal elements of Alchourrrn & Makinson [1981 ]:

A norm a belongs toA/p if and only if (1) a belongs to A

18 AGM contraction and revision functions give consistent results even when they are applied to inconsistent sets.

228

and

GIOVANNI SARTOR

(2) there is no B c_ A for which

(i) B is a cri t ical set for p.

(ii) a is < - m i n i m a l in B [Alchourr6n 1986, 184].

Final ly , 'O(q/p) i f and only i f A / p ~-(p~" Oq)' (ibid.).

In our f r amework the assertions that O(q/p), in relat ion to the normat ive sys tem A,

wou ld be translated in the assert ion that O(q) is a just i f ied consequence o f the premises

set A u p. Those assertions do not co inc ide with strongly provable statements, nor with

s t rongly g rounded ones.

Fo r example , let A be {nt: O(a A b) <--- p, n2: O ~ a <--- q, n3:Oa <-- r}, and let the fol-

lowing relat ions hold: n t < n 2 < n 3. The no rm nl is not in the heal thy set for the case

p ^ q ^ r, since it is the min ima l e l emen t in the cri t ical set {n 1, n2}. 19 Therefore n 1 is not

in A/(p A q ^ r). So, A/(p A q A r) b L O(a ^ b) and not O(a ^ b/p ^ q ^ r).

In such a case, instead, the preferred subtheories approach wou ld have g iven to us the

only preferred subset {nl: O(a A b) +-- p, n3: Oa +-- r, p ^ q A r} imply ing O(a ^ b), as

s trongly provable (and strongly grounded) statements.

As we have already seen for Alchour r6n & Makinson [1981], Alchour r6n [1986] also

expresses an intuit ion genuine ly different f rom that o f B r e w k a ' s preferred subsets: - - In Brewka's model statements conflicting with higher ones can be recovered, when these last statements are

contradicted by stronger assumptions; - - In Alchourr6n's model any statement inconsistent with stronger ones is definitively inapplicable (when the

conflict arises).

Both intuit ions can be legal ly relevant . In fact, jurists seem to dis t inguish two different

situations: - - Partial incompatibility. A norm nl is incompatible with stronger norms contained in the legal system, but

there are possible cases in which nl is applicable. The consequence of partial incompatibility is derogation sensu stricto: all incompatible norms are valid, and their conflicts are solved by means of the hierarchical ordering over the legal system. If a norm n2 higher than nl and conflicting with it, is contradicted, the applicability of n l may expand correspondingly.

- - Total incompatibility. After the enactment of a new norm n2, there is no possible case in which a preexisting norm nl can be applied. The consequence of total incompatibility is tacit abrogation. The norm n 1 is considered deleted from the system, and can no longer be recovered (the norm could be subtracted from the system without changing its legal consequences, at least as far as future cases are concerned).

The first si tuation can poss ibly be mode l ed in B r e w k a ' s f ramework , the latter in

Alchour r6n & Makinson ' s . The not ion o f total incompat ib i l i ty also establishes a new way

to m o d e l expl ic i t abrogat ions, to wit prescript ions expl ic i t ly stating that a certain no rm is

no longer ' va l id ' , Norma l ly the abrogat ion o f a no rm nl is interpreted as its subtraction

f rom the legal system: the expl ic i t abrogat ion o f nl is interpreted as an act (a funct ion)

t ransforming the base A into (A-{ n t }). It seems better to represent expl ic i t abrogat ion o f a

no rm nl s imply as a no rm n2 stating that nl is uncondi t iona l ly inapplicable, to wi t as n2:

n l. In this way abrogated norms wou ld remain part o f the k n o w l e d g e base and could still

be used for cases which happened before their abrogat ion, a l though being comple te ly

i r re levant for subsequent cases (this presupposes a representat ion o f time).

19 This set implies O(a ^ b) ^ O(~a) ~ p A r, and, therefore, contrapositively, since its consequent is contra- dictory in standard deontic logic, ~(p n r), from which ~(p ^ r A q).


7. The hierarchy of the legal norms

In both Brewka ' s and Alchourr6n & Makinson ' s approaches, reasoning with inconsisten-

cies in the law relies on the hierarchical ordering _< over the legal system. Let us

examine how such an ordering is defined.

For this purpose, the traditional principles regulating the relation between norms must

be considered: the source criterion ( lex superior) , the chronological criterion ( lex pos te-

rior), and the speciali ty criterion ( lex specialis) . 2° Two additional criteria will be here

also examined, the inferential and the hermeneutic ones.

7.1. THE SOURCE CRITERION (LEX SUPERIOR DEROGAT LEGI INFERIOR1)

Each normative act has a certain degree of 'normative strength' , depending on the power

(the competence) in whose exercise it has been accomplished, that is on the 'source of

law' to which the act belongs. For example, in the Italian legal system, constitutional acts

(the acts of production of constitutional law) are superior to ordinary Parliament acts,

these being superior to Government regulations, the latter being superior to acts of subor-

dinate authorities, and so on.

This source-ordering is transmitted from the types of normative acts (the sources), to

the acts themselves, and to the norms contained in (expressed by) the latter. We indicate

the source ordering by -< SR and write nl ~< SR n2 meaning that n 2 is at least as strong by

source as n I. We define correspondingly < SR and --=-SR-

So, for example, if n 2 is contained in a Parl iament Act, and nl is contained in a

Government Regulations then, in the Italian legal systems, n I < SR n2"

7.2. THE CHRONOLOGICAL CRITERION (LEX SUPERIOR DEROGAT LEGI INFERIORI)

Normative acts are also ordered according to the time when they were accomplished: the

later normative acts prevail over the preceding ones, are 'chronological ly superior ' to

them. Also the chronological ordering, denoted here by _< x (and by the corresponding

< T and -=T), can be extended from the acts to the norms produced by them. So, n 2 is

chronological ly superior to nl (n 1 <T n 2) iff n 2 has been produced by an act a 2 later that

the act a l through which nl was produced.

7.3. THE SPECIALITY CRITERION (LEX SPEC1AL1S DEROGAT LEGI GENERA1_2)

The source and the chronological criteria refer to the type of act through which a

certain norm has been created, so that all the norms contained in one act must be

equivalent in this respect. The speciali ty principle applies, instead, directly to the single

norms, so that the speciali ty ordering, that we indicate by < sP (and by the corresponding

20 On these principles cf., for all, Peczenick [1982, 1983] who qualifies them as derogation-norms, that is as metanorms that 'help to set aside both incompatibilities of rules and collision of principles' [Peczenick 1983, 66].

230 GIOVANNI SARTOR

<sP and --sP), can assign different degrees of normative strength to norms included in the

same act. We can distinguish two notions of speciality: special i ty sensu stricto (or specificity)

and special i ty sensu largo (or predominance).

Special i ty sensu stricto (specificity)

In a strict sense, a norm nl: ql ~-- Pl is more special than a norm n2: qz ~ P2, iff, given a set of assumption B, B u {Pl} ~-P2, while B u {P2} ~ P I "21 The principle of speciality sensu stricto coordinates specific and general provisions, by giving precedence to the first

over the latter.

Special i ty sensu largo (predominance)

Lawyers speak of speciality, not only in the hypothesis just considered, but also when a

certain legal consequence c is established for situations presenting a certain feature f, and

f is considered legally predominant over other possible features fl . . . . . fn of those situations. So, the norm n: c ~- - f i s held as stronger then any norm n i" c i ~"' f i establishing an

incompatible legal consequence c i for a fact f/. Therefore, in case of conflict (when b o t h f and f /a re satisfied), the precedence is given to n. In such contexts, we can say that n is an

exception. Normally, exceptions have a negative consequent establishing that, given determined conditions, a certain legal effect does not hold or a certain norm does not

apply. Norms with a negative consequent have to be interpreted as exceptions, since oth- erwise it would be difficult to attribute to them any legal relevance. This is typically the

case for norms establishing justification causes. These norms are not more special in a strict sense to any particular criminal rule, since they intend to block all of them. The

justification cause (e.g. self defense) cannot imply the condition of any particular criminal

norm, because in this case it would concern just that norm. More complex normative connections are possible: there may be different levels of

exceptions, positive and negative, or two norms may establish incompatible legal effect, none of those being the negation of the other. In such contexts, grading the competing norms involves the assessment of the relative importance of the aspects they consider (we

must establish which norm contemplates circumstances so important to override the legal relevance of those in the antecedents of conflicting norms). This assessment involves, obviously, a noteworthy ambit of discretion, as often in legal interpretation.

7.4. THE JOINT APPLICATION OF THE THREE CRITERIA

The three criteria just mentioned can be in conflict. It has been authoritatively said that there is no criterion of the criteria, allowing every conflict to be solved [cf. Bobbio 1958,

2l On speciality, cf. Poole [1985], who defines rigorously the notion of speciality, and Delgrande [1988] and Simari & Loui [1991], whose nonmonotonic logics are based on this notion. The concept of speciality has been widely treated in the framework of inheritance networks, cf. Tourezky [1986], Tourezky & Thomason [1988], Tourezky et al. [1988]. On speciality in law, cf. Prakken [1991a].

N O R M A T I V E CONFLICTS IN LEGAL REASONING 231

113 ff]. Nevertheless it is generally accepted that, as a rule, the source criterion prevails

over both the speciality and the chronological ones, while the speciality criterion prevails

over the chronological one.

The three relations _< SR, -< T, and -< sP are reflexive and transitive. Therefore, given

the priorities just described among the corresponding criteria, it is not difficult to combine

them into an ordering <__ over a legal system. For this purpose, it is easier to start from

the relations < SR, < T, < SP, and define first < .

A norm n 2 prevails over a norms n 1 (n 1 < n2) iff 1. n 1 '<SR n2; or 2. n 1 <SP n2, provided that n2 ~SR nl; or

3. n 1 < T n2, provided that n2 ,~SRnl a n d n 1 g:SRnl .

The source and chronological relations are connected (all sources and time instants

must be comparable); the same cannot be assumed for speciality. 22 So, we can define two

norms as equivalent iff they are equivalent by source, time, and speciality:

H1 --~/'/2 i f f / ' /1 -'~SP n2, a n d H 1 ~--sP n2, and n 1 _= T n 2.

Finally, we can say that

n 1 _.< n 2 iff n 1 < n 2 or n 1 -_- n 2.

The relation < is transitive and reflexive, b u t is not connected, since _< sp is not con-

nected.

7.5. INFERENTIAL ORDERING

In some cases the application of the ordering criteria just indicated gives counterintuitive

results. Let us consider the following example:

Property(x, y):

~(x may be expropriated of y) <---

x is the legitimate owner of y

Expropriat ionz(x, y):

x may be expropriated of y <---

x is useful for public purposes

Ownership(x, y):

x is the legitimate owner of y if

x has bought y.

Let us assume that P roper ty is a Constitutional rule, while Expropr i a t ion 2 and

Ownersh ip are statutory prescriptions, the first issued after the second: Ownersh ip < T

Expropr ia t ion2 < SR Proper ty . The facts are the following

F31: Mark has bought BlackAcre

22 Connectivi ty of the -----sP relation would amount to the (erroneous) assumption that the special reasons making it so that n 2 --<sP nl , also al low nl, to prevail over norms for which such reasons do not exist, but which are considered equivalent to n 2 on the basis of the other criteria.

232 GIOVANNI SARTOR

F 3 4 : BlackAcre is useful for public purposes

In such a case the intuitive consequence should be that BlackAcre is not to be expropri-

ated. Instead, this is not the result obtained, given the ordering above, in both Brewka ' s

and Alchourr6n & Makinson ' s approaches, where we can derive that BlackAcre can be

expropriated. This is because in both approaches the weaker rule, to wit Ownersh ip ,

would be rejected.

The problem is the following: when a certain regulation nl: p 4 - ql ^ - . . ^ qn, is

issued, stating the legal consequence p for the legal situation ql ^ . . . ^ qn, usually the

normative framework establishing ql ^ . . . ^ qn is implici t ly accepted by this regulation.

Therefore, when p is derived by using nl and p is inconsistent only with regulations n 2

lower than n 1, we do not want the derivation o f p to be blocked by putting into question

the regulations establishing ql A . . . A qn, even when those regulations are weaker than n 2

(for a formal development of this intuition, cf. Prakken [1991b]). When this assumption

is correct (as it is not always the case) we might add a further ordering principle, that we

might call inferential criterion: norms n2: qi ~-- r and n3: --~qi 4- s are inferentially supe-

rior to norms nl: p 4-- ql ^ . - . ^ qn, ( n l < I n2 and n 1 < I n3).23 The inferential principle

intends to block contrapositive reasoning: without such principle from the normative

conflict involving the rule p 4 - q, to wit from ~ p and p <--- q, ~ q would be contraposi-

t ively de r ived , so that the rule q ~-- r would be put into question.

In the above discussed case, if the inferential criterion is given the highest-priority, the

ordering E x p r o p r i a t i o n 2 < P r o p e r t y < O w n e r s h i p < {F31, F34} is obtained.

Therefore, the justified legal conclusion is the intuitive one: BlackAcre is not to be expro-

priated.

7.6. HERMENEUTIC ORDERING

In example 2.3 above we have considered a case in which the choice between two alter-

native meanings was required. This aspect of legal reasoning can be generalized to the

situations in which many alternative interpretations of a legal text are possible, that can

be ordered on the basis of their plausibility.

Let P l , . . . , Pn be n alternative interpretations of the legal provision p (we can think o f p

as any statement included in a normative text), Pl < n . . . < H Pn be their 'hermeneut ic '

ordering ( < n expresses the grading of their plausibil i ty as interpretations of p, estab-

lished by the interpreter according to his own hermeneutic criteria), and let q be a new

statement, asserting that no more than one interpretation can be accepted. If _< n is placed

just after _< SR in the priority grading of the criteria, we obtain a partial formalization of

the principle of systematic interpretation, prescribing to choose the interpretation that best

fits in the system of legal norms. We also have some aspects of the principle of evolution-

ary interpretation: since the choice of Pi depends on the content of A, i f A is modified (for

example, if a higher level norm contradicting the previously chosen interpretation is

enacted) a different interpretation can be selected.

23 This is a simple formulation of this criterion, sufficient for the type of norms and facts here considered.

N O R M A T I V E C O N F L I C T S IN L E G A L R E A S O N I N G 2 3 3

8. Conc lu s ion

Let us briefly summarize the results obtained in formalizing the notions of plausible (undefeated) legal consequence, and of certain (justified) legal consequence of an inconsistent but ordered normative sets.

The analysis of Brewka's approach allowed the notion of undefeated legal consequence to be translated into the concept of weakly provable statements (statements provable from a preferred subsets), and the notion of justified legal consequence to be translated into the concepts of strongly provable statements (statements derivable from all preferred subsets) and strongly grounded statements (statements derivable from the inter-

section of the preferred subsets). Then AGM's notions of contraction and revision were considered-- in particular

partial meet revisions and contractions, and safe contractions and revis ions--and the results of revision functions were evaluated as candidate justified legal conclusions.

A relation was established with Brewka's model, on the basis of an equivalence between strongly grounded statements from premises sets A u 5/2 and A u p and the consequences of contractions and revisions of A based on a certain selection function.

The comparison of Brewka's preferred subsets model on the one hand and of the approach developed in Alchourr6n & Makinson [1981, 1985] and Alchourr6n [1986] on the other, disclosed two alternative intuitions to deal with inconsistent normative set, both legally relevant: - - In B r e w k a ' s mode l no rms confl ic t ing with h igher ones can be recovered when the latter prescr ipt ions are

rejected; - - In A lchour r6n & M a k i n s o n ' s approach statements inconsis tent wi th s t ronger ones are definit ively inappl ica-

ble.

The criteria to order normative set were then considered: besides the traditional principles of lex superior, lex specialis and lex posterior, two additional criteria, the inferential criterion and the hermeneutic one, were proposed.

Many problems have yet to be solved, but nevertheless the methods for reasoning from inconsistent premises seem to offer a significant possibility of extending formal reasoning beyond the domain of logical deduction in strict sense.

These methods have a noteworthy importance for legal theory. In fact, if normative conflicts can be solved in applying the law, by considering an ordering established over legal norms, then a legal system containing incompatible norms can be acceptable. There is no need to 'postulate' the consistency of the legal system, that is to imagine a consistent system as the mysterious reality (to be recognized by the faculty of 'legal interpretation') hiding behind the illusory phenomenon of inconsistency. Interpretation is required, instead to obtain determinateness.

The capability of reasoning from inconsistent premises has a fundamental significance also for AI and law: it allows more intuitive and isomorphic representations of legal knowledge, since law is naturally represented as a set of inconsistent ordered norms. Moreover, the methods to reason with inconsistent legal premises can be related to the techniques for nonmonotonic reasoning developed in AI, so disclosing an important inter- disciplinary area of research. For this purpose two directions must be considered:

234 GIOVANNISARTOR

- - Approaches explicitly intended to deal with conflicting premises sets, such as, for example, abduction [Poole 1988], rules and exceptions (Kowalski & Sadri 1991), defeasible arguments [Simari & Loin 1991], the various techniques for reasoning with defeasible inheritance networks, etc.

- - The possibility to map conflicting norms into sets of non conflicting statements (or inference rules) expressed in a nonmonotonic formalism (as we have seen a translation in classical logic would not be sufficient), such as default logic or autoepistemic logic, negation by failure (cf. Kowalski & Sadri 1991), cir- cumscription, etc. (some aspects of such a translation are sketched in Sartor 1991).

R e f e r e n c e s

Alchourr6n, C.E. 1982. Normative Order and Derogation. In Deontic Logic, Computational Linguistics and Legal Information Systems, ed. A.A. Martino, 51-63. Amsterdam: North Holland.

Alchourr6n C.E. 1986. Conditionality and the Representation of Legal Norms. In Automated Analysis of Legal Texts, eds. A.A. Martino & F. Socci-Natali, 175-186. Amsterdam: North Holland.

Alchourr6n C.E. & Bulygin, E. 1977. 'Unvollst[indigkeit, Widersptichlichkeit und Unbestimmheit der Normenordnung. In Deontische Logik und Semantik. Wiesbaden: Athenaion.

Alchourr6n C.E. & Bulygin, E. 1981. The Expressive Conception of Norms. In New Studies in Deontic Logic, ed. R. Hilpinen, 95 -124. Dordrecht: Reidel.

AlchotLrr6n C.E. & Makinson, D. 1982. On the Logic of Theory Change: Contraction Functions and their Associated Revision Functions. Theoria 48 (1): 14-37.

Alchourr6n C.E. & Makinson, D. 1985. On the Logic of Theory Change: Safe Contractions. Studia Logica: 44: 405-422.

Alchourr6n C.E., Gardenfors, P. & Makinson, D. 1985. On the Logic of Theory Change: Partial Meet Functions for Contractions and Revisions. Journal of Symbolic Logic 50: 510-530.

Alexy, R. 1978. Theorie derjuristischen Argumentation. Frankfurt: Suhrkamp. Bench-Capon, T.J.M & Sergot M.J., 1985. Towards a Rule-based Representation of Open Texture in Law.

London: Imperial College, Department of Computing. Bobbin, N. 1958. Teoria dell' ordinamento giuridico. Torino: Giappichelli. Brewka, G. 1991a. Belief Revision in a Framework for Default Reasoning. In The Logic of Theory Change.

Workshop, Konstanz, FRG, October 13-15, 1898. Proceedings, eds. A. Fuhrmann & M. Morreau, 206-222. Berlin: Springer.

Brewka, G. 1991b. Nonmonotonic Reasoning. Logical Foundations of Commonsense. Cambridge: Cambridge University Press.

Delgrande, J.P. 1988. An Approach to Default Reasoning Based on a First-Order Conditional Logic: Revised Report. Artificial Intelligence 26:63-90.

G~denfors, P. 1988. Knowledge in Flux. Cambridge (Massachusetts): MIT Press. G~denfors, P. 1989. The Dynamics of Normative Systems. In Preproceedings of the III International

Conference on Logica, Informatica, Diritto, edited by A.A. Martino, 293-299. Firenze: Istituto per la docu- mentazione giuridica.

G~denfors, P. 1990. The Dynamics of Belief Systems: Foundations vs. Coherence Theories. Revue interna- tionale de Philosophic. 172 (1): 24-46.

Gardenfors, P. & Makinson, D. 1988. Revision of Knowledge Systems Using Epistemic Entrenchment. In Proceedings of the Second Conference on Theoretical Aspects of Reasoning about Knowledge, ed. I.Y. Vardi. Los Altos (CA): Morgan Kaufmann.

Gardenfors, P. & Makinson, D. 1991. Relations between the Logic of Theory Change and Nonmonotonic Logic. In The Logic of Theory Change. Workshop, Konstanz, FRG, October 13-15, 1989. Proceedings, eds. A. Fnhrmann & M. Morrean, 185-205. Berlin: Springer.

Gardner, A.v.d.L. 1987. An Artificial Intelligence Approach to Legal Reasoning. Cambridge (Massachusetts): MIT Press.

Gordon, T.F. 1988. The Importance of Nonmontonicity for Legal Reasoning. In Expert Systems in Law: Impacts on Legal Theory and Computer Law, eds. H. Fiedler, F. Haft & R. Traunmiiller, 111-126. Ttibingen: Attempto.

Gordon, T.F. 1989. Issue Spotting in a System for Searching Interpretation Spaces. In The Second International Conference on Artificial Intelligence and Law: Proceedings of the Conference, 157-164. New York: ACM Press.


Hart, H.L.A. 1961. The Concept of Law. London: Oxford University Press. Kowalski, R.A. & Sadri, F. 1990. Logic Programming with Exceptions. In Proceedings of the Seventh

International Logic Programming Conference, 598-613. New York: MIT Press. Levi, L 1977. Subjunctives, Dispositions, and Change. Synthdse 34: 423-455. Makinson, D. 1985. How to Give up: A Survey of Some Formal Aspects of the Logic of Theory Change.

Synthdse 62: 347-363. Peczenick, A. 1982. Legal Errata. In Deontic Logic, Computational Linguistics and Legal Information Systems,

ed. A.A. Martino, 103-125. Amsterdam: North Holland. Peczenick, A. 1983. The Basis of Legal Justification. Lund: Peczenik. Peczenick, A. 1987. Legal Rules and Moral Principles. In Technischer lmperativ und Legitimationskrise des

Rechts, eds. W. Krawietz, A.A. Martino & K.L. Winston, 151-167. Rechtstheorie, Beiheft 11. Berlin: Dunckner und Humblot.

Poole, D.L. 1985. One the Comparison of Theories: Preferring the Most Specific Explanation. In Proceedings IJCA11985, 144-147.

Poole, D.L. 1987. Variables in Hypotheses. In Proceedings IJCA11987, Milano. Poole, D.L. 1988. A Logical Framework for Default Reasoning. Artificial Intelligence 36: 27-47. Poole, D.L., Goebel, R.G. & Aleliunas, R. 1987. Theorist: A Logical Reasoning System for Defaults and

Diagnosis. In The Knowledge Frontier: Essays in the Representation of Knowledge, eds. N. Cercone, & G. McCalla, 331-352. New York: Springer.

Prakken, H. 1991a. A Tool to Model Disagreement in Law: Preferring the Most Specific Argument. In The Third International Conference on Artificial Intelligence and Law. Proceedings of the Conference, 165 - 174. New York: ACM Press.

Prakken, H. 1991b. Reasoning with Normative Hierarchies. In Deon'91. First International Workshop on Deontic Logic in Computer Science. Amsterdam, The Netherlands, December 11-13. Proceedings, eds. Ch. Meyer & R.J. Weiringa. Amsterdam: Vrije Universiteit Amsterdam.

Rescher, N. 1964. Hypothetical Reasoning. Amsterdam: North Holland. Ross. W.D. 1930. The Right and the Good. Oxford: Claredon Press. Ross, W.D. 1939. Foundations of Ethics. Oxford: Claredon Press. Sartor, G. 1991. The Structure of Legal Norms and Nonmonotonic Reasoning in Law. In The Third

International Conference on Artificial Intelligence and Law. Proceedings of the Conference, 155-164. New York: ACM Press.

Simari, G. & Loni, R.P. 1991. A Mathematical Treatment of Defeasible Reasoning and its Implementation. Artificial Intelligence 53: 125-157.

Tourezky, D.S. 1986. The Mathematics of Inheritance. London: Pitnam Research Notes in Artificial Intelligence.

Tourezky, D.S, Horty J.F., & Thomason, R.H. 1988. A Clash of Intuitions: The Current State of Nonmonotonic Multiple Inheritance Systems. Proceedings AAAI 87.

Tourezky, D.S. & Thomason, R.H. 1988. Nonmonotonic Inheritance and Generic Reflexives. Proceedings AAAI 88. Wroblewski, J. [1969] 1983. Justification of Legal Decisions. In Meaning and Truth in Judicial Decision,

49-70. Helsinki: A-Thieto Oy. First published in Logique et Analyse 3.

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