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Logic-based models of analogical reasoning: A fundamental basis for the design of legal expert systems
ExpertSystems WithApplications,Vol.4, pp. 369-378, 1992 Printedin the USA.
0957-4174/92 $5.00+ .00 © 1992PergamonPressLtd.
Logic-Based Models of Analogical Reasoning: A Fundamental Basis for the Design of Legal Expert Systems IVIATTHIAS BAAZ InstitutflirAlgebraund DiskreteMathematik,TcchniscbeUniversit/ItWien, Wien, Austria
GERALD QUIRCHMAYR InstitutfiirInformatik,JohannesKeplerUniversit/R,Linz,Austria
Abstract--Traditional approaches in the domain of legal expert systems do largely neglect the importance of analogy and the weight of decisions. That is why this paper is focussing on a formal model of Reasoning by Analogy based on Gentzen's sequent calculus LK, which is then used to develop a formal representation of the evolutionary development of law. 1. INTRODUCTION: EXPERT SYSTEMS IN LAW AND FORMS OF LEGAL REASONING
IN HIS 1987 overview of expert systems applications in the legal domain, Susskind clearly shows that advanced forms of knowledge representation are scarce (Susskind, 1987 ). Apart from the approach developed by McCarty (1983) and Gordon (1986), existing systems confine themselves to traditional rule-based inference. So elementary principles of legal decision making such as Reasoning by Analogy are not yet supported. Systems based on concepts developed in the field of machine learning seem to be a solution, but an indepth analysis reveals that these systems should rather be called classification algorithms (Hall, 1989) and if Reasoning by Analogy is supported, it is only a support of building analogies of facts. From a theoretical point of view this is merely equivalence of sets (Alschwee, 1986). What has so far been ignored is the structure of cases, the relationship between the elements (facts) of a case. The worst deficiency of these models is perhaps that none of them is able to define the meaning and the weight of a legal decision because all of them reduce legal reasoning to mere subsumption, thereby neglecting the importance of the interpretation and evaluation done by the judge. That is why they can serve for deal-
ing with Analogy in a limited way, but are not suitable for modelling the evolutionary development of law. To overcome this shortage it is necessary to develop a formal model of Reasouing by Analogy which is based on the analysis of decisions. Reasoning by Analogy does extend the available knowledge, but as continuous growth would lead to an uncontrollable mess there must also be mechanisms for reducing the number of rules. These are mainly overwriting strategies for canceiling obsolete and contradictory rules and generalizations for classifying cases. The best example of this development is perhaps the evolution of rules concerning the tort of negligence (ef. Marsh & Soulby, 1987, pp. 124 if). The goal of this paper is to develop a formal model of these processes which is based on Gentzen's LK-calculus. 2. ABSTRACT FORMS OF DECISIONS
We base our formalization on a quantifier-free variant LK* of L K , the famous calculus ofGentzen (cf. Gentzen, 1934; Takeuti, 1975), that includes extended resolution. Definition 1. The formal language of L K * is constructed from the usual terms and predicate symbols together with the logical connectives -1, A, V, --~. Sequents 1-1 •, r consist of (possibly empty) sequences of formulas H, r. (The intended meaning is AAEw4 --~ VA~rA. Sequents consisting of predicates only correspond to clauses by H •. r _ VA~nTA V VA~rA.) Logical axioms: A =* A Inference rules: Logical rules:
Requests for reprints shouldbe sent to MatthiasBaaz,Institutfiir Algebra und Diskrete Mathematik,TeehniseheUniversit~t Wien, WiednerHauptstr. 8-10, A-1040Wien,Austria. 369
370
L1
M. Baaz and G. Quirchmayr
F~A,D
(q-left):
qD, r = , A '
D, F = * A F =* A, -qD '
(-q-right): L2
C,D,F~A CAD, F~A'
(A-left):
( A -right ): L3
C,F~AD,
(V-left):
I" ~ A, C r ==~ A, D
r~A, F~A
CVD, F~A r ~ A, C, D
(V-right): IA
CAD
F=.A, CVD'
r = * A, C D, r = A
(-*-left):
C--~D,r~A
(--~-right):
C, P ~ A, D F ~ A, C " ~ D "
Structural rules: Sl weakening (w-left):
r~A
•
A, r==. A '
(w-right):
r~A r=.
.
A,A'
S2 exchange F ~ A r' is rearrangement of r, r' ~ A' ' A' is rearrangement of A;
(exchange):
$3 factorization (f-left):
A,A',F=*
A
Ao, F * =~ A~ '
(f-right):
F = * A, A, A' r a =* A~, Act '
o is most general unifyer of A and A'; $4 resolution (res):
r ==*
A, A A ' , II ~ A
(r, n ) ~
(& A)~
o is most general unifier of A and A'. (The numbers ( i , j ) of the formulas in I" and A are called the index of the inference) Definition 2. An LK*-deduction of H out of Ht, . . . . H . is a tree of sequents such that: (a) the top nodes are axioms or variants of Hi . . . . . H . by variable
renaming; (b) each other node is the direct consequence of its ancestor nodes; (c) the bottom node is a sequent H ' such that H = H ' o for some o. Let at . . . . . a. be all substitutions subsequently generated by the rule applications. The total substitution of the derivation is ~o. • • • at. Hi . . . . . H . I- H iffthere is any derivation of H out of Ht . . . . . H~. Definition 3. Let T(II --~ r ) = AAenA -~ VAerA, AAenA is true for II empty, V a e r A is false for r empty• T ( ~ ) = { T ( H ) IH ~ ~/}. ~ is a valid consequence of f ( f H-- Yf) iff the universal closures of formulas in T ( . £ ) imply all universal closures of formulas in T(~/). Theorem 1. S1 . . . . .
S , F S *"-, St . . . . .
S , : ~, S.
Proof. Calculate the total substitution of the derivation and use the usual completeness theorem (cf. Takeuti, 1975, Theorem 8.2) in connection with lifting lemma (of. Chang & Lee, 1973, Lemma 5.1 ) extended to L K * . Definition 4. The skeleton of a deduction is: (a) a number ( = an initial skeleton); (b) of the form ( S , D ) where Sis a skeleton and D ~ {(-q-left), (-q-right), (A-left), (V-right), (-~.-right), (w-left), (w-right), (fleft), (f-right) } ; (c) of the form ($1, $2, D ) where St, $2 are skeletons and D = {(A-right), (V-left), (--~left)}; (d) of the form ( S t , $2, (res), ( i , j ) ) ; and (e) of the form ( S , (exchange), it . . . . . i., jt . . . . , i s ) , where {it . . . . . ir} = {1 . . . . . r}, {j, . . . . . j,} = {1, ...,S}.
All initial skeletons within a given skeleton have to be different. Explanation. The skeleton of a deduction is an abstract description of the deduction without specifying assumptions and goal. For the general notion of skeleton see Parikh (1973). Definition 5. A decision path is a quadruple (A, B, C, D ) such that: (a) A is a skeleton of a deduction; (b) B is a finite sequence of assignments i : Hi of sequents to initial skeletons in A; (c) C is a sequence of assignments j : I'j of finite sets of constants I'j to initial skeletons which occur in A but not in B; and (d) D is a variable free non-empty sequent. We use the convention for (b) and (c) that in presence of several decision paths containing the initial skeleton k either always the same sequent Hk is assigned to k or always the same set of constants rk is assigned to k. Explanation. The skeleton (A) represents the figure of a deduction together with the indices of the rule applications, but gives no information on initial and final clauses• The sequence of assignments (B) fixes some of the initial clauses of the skeleton. In interesting cases some of them remain unknown• (C) represents objects from which the "unknown" hypotheses are assumed to be independent. The final sequent (D) is
Logic-Based Models of Analogical Reasoning
371
fixed. Decision paths are formal objects and can therefore be treated in a formal manner. The information given by facts and statements and the additional information given by the decision itself are used to derive the weakest preconditions of the decision. These weakest preconditions have to be accepted, because otherwise the decision itselfwould have to be rejected as false. The initial sexluents of the decision are usually at least known to be independent from certain elements, such as time, location, person, and other elements not occurring in the general form of the statements in the decision (in the decision path these elements are represented by C). Therefore, the weakest preconditions are also considered as independent from these elements. For the modelling of the evolutionary development of law we have to consider sets of decision paths: Definition 6. An interpretation/(a) of a set of decision paths $ = { . • . , ( A i , Bi, Ci, Di ), • • • } is a set of deductions in accordance with Ai and Bi with final sequents D;. (Initial skeletons have to be realized by the same formulas throughout/($).) An interpretation of a decision path E is defined b y / ( {E} ). (We write/(E) instead o f / ( { E } ) . ) Definition 7. The propositional ( k, /)-realization of a skeleton S is constructed as follows (reducing S from outside to inside): ( l ) Write Xt . . . . . Xk =* Y~ . . . . . Yt as bottom node. (2) Assume Ui . . . . . fir =, Vl . . . . . Vs is constructed and that the subskeleton S' has to be reduced. (a) S' is initial skeleton k: Ul . . . . . U~ k VI, . . . . V~is top node. (b) S' = ( S ~, D ) , D = (~-left) (resp. (a-right), (A-left), (V-right), --*-right), (w-left), (w-fight), (fleft), (f-fight)): for the logical rules anywhere apply the following corresponding substitutions: {-aU/Ul } (r~sp. { T U ] V s } , { V ]~ V / V l } , { U V V]V$}, { V - . ~ V / V , } ) . Write U2. . . . . U,=* V ~ , . . . , V~, U(resp. U, UI . . . . . Or ==0 V I , . . . , Vs-l, U, V, U 2 , . . . , Ur===lb Vl, . . . . V$, U i , . . . , Ur==~ V I s . . . , V$_I, U, V, U, UI, ....
U,~
Vs, U I , . . .
Vi . . . . .
V~-I, V, U2 . . . . .
, Ur==~ V I , . . .
U,~
, V$_I, UI, U 1. . . .
VI . . . . .
, Urm=~ V1 . . . . . V~, Ui . . . . . U, =* Vi . . . . . Vs, Vs) above the transformation of Ui . . . . . U, ~ V~. . . . , V~, reduce the skeleton S' to S". (c) S' = (S", S", D ) , D = (A-right) (resp. (Vleft), (-'*-left)): for the logical rules anywhere apply the following corresponding substitutions: { U A V~ V, } (resp. { U V V~ U, }, { V --~ V~ V~ } ). Write UI, . . . . U,=* VI . . . . . Vs-m, U a n d Ul . . . . . Ur ===0Vl, . . . . V~_~, V(resp. U, U2. . . . . /Jr =* V~. . . . . Vs and V, U2 . . . . . Ur=~ VI . . . . . Vs, U2 . . . . . Ur==o Vi . . . . . V~, U and V, /.12. . . . . U, =* Vl, . . . , Vs, above the transformation of Ul . . . . . U, ~ Vl . . . . . Vs, reduce the skeleton S' to S" and S ' .
(d) S' = (S", S ' , (res), (i, j ) ) : write Ut . . . . . vt . . . . . v/, u and U, Ui+~ . . . . . U, ~ Vj+,, . . . . Vs above UI . . . . . U, ~ VI . . . . . Vs. (e) S' = ( S", (exchange), i l , . . . , it, j r , . . . ,is): write Ui~. . . . . Ui, =* V/~. . . . . Vj, above Um. . . . . U, Vt . . . . . Vs and reduce the skeleton S' to S". Explanation. The propositional (k, /)-realization of a skeleton represents the propositional structure of any adequate derivation with this skeleton. Proposition 1. If the construction of Definition 7 is impossible for some skeleton S and some (k, l), no set of decision paths containing E = (S, B, C, D ) with D = Ai . . . . . Ak =~ B I , • • . , B l has an interpretation. Let any deduction that is an interpretation of the decision path E = (S, B, C, D ) be given. The deduction is transformed into a propositional realization in a unique way if the total substitution (cf. Definition 2) is applied and atomic formulas are 1-1 replaced by propositional variables. This motivates the following definition. Definition 8. Let E = (A, B, C, D ) , D = Ai . . . . . Ak =* B~ . . . . . Bt, be a decision path. E determines a propositional variable X in the ( k, /)-realization of A iffX occurs in the top nodes fixed by B or in the bottom node. Definition 9. Let a set of decision paths ~ = { • • . , ( A i , Bi, Ci, Di, • • • } be given and let il . . . . . i, be the sequence of numbers representing those sequents in ~ to which no sequents of B; are assigned. Variable-free sequents Hi, . . . . . Hi, are called precondifi0ns of a itf (a) there exists an interpretation/($') of ~' = { . . . , ( A i , B~, Ci, Di), . . . } where B~ is the extension of Bi by ik : Hik for ik occurring as initial skeleton in Ai and Ci is correspondingly restricted. (b) the groups of propositional variables not determined by E in $ are realized by a previously fixed list of zero-placed predicates. Explanation. The preconditions complete the incomplete information of ~. In case of not determined variables no information is given. So it is useful (cf. condition (b)) to replace them by previously fixed zeroplaced predicates. Definition 10. Let f * be the extension o f . £ by the set of new constants { ci } and let ~ be a set of decision paths. Wm. . . . . Wn are weakest preconditions for and .£ iff (a) WR. . . . . W, are preconditions for E in the language .£*; (b) For any sequence HI . . . . . Hn of preconditions for ~, any H a n d any F in .£, the following holds: ui ~
Wl . . . . .
W,,,
F I- H=* HI . . . . .
H,,,
F I- H
using the same derivation with the exception that W~ is replaced by Hi.
372
M. Baaz and G. Quirchmayr
Explanation. I f a decision path E is to be completed to a proof, the assumptions have to be made at least as strong as the weakest preconditions. The consequence is: if a decision path E is accepted as corresponding to a real conclusion, the weakest preconditions have to be accepted as true statements. (The definition of the weakest preconditions corresponds to the interrelation { ( 3 x ) A ( x ) } O r k B = , {A(t)} O I~ ~-B for any term t in usual first order predicate logic. If existential quantitiers have to be avoided (like in LK* ), new constants have to be introduced. Theorem 2. 1. It is decidable whether any interpretation for a set of decision paths ~ exists. 2. If an interpretation exists, the weakest preconditions in the extension .£* of .£ by { ci } can be constructed. Proof. Let ¢~ = { . . . , ( Ai , B,, Ci , Di ) , " ' " },let it . . . . . in be the sequence of numbers representing those initial sequents in A to which no sequents of B are assigned. Weakest preconditions are constructed as follows: ( 1 ) Construct the propositional realizations of Ai relative to D~. (2) Identify the final sequents of the propositional realizations with Di and identify the corresponding initial sequents with the sequents fixed by Bi using unification. (3) Replace remaining propositional variables by the previously fixed sequence of zero-placed predicates (these are the variables not determined by the skeleton, cf. Definition 8). (4) Let W * ( g ) , . . . , W * i,(x) be the resulting initial clauses not determined by B~. Replace g by new constants 3. The result is a sequence of weakest preconditions w,, . . . . .
w~.
To prove 1 simply note that no interpretation is possible if this construction is not successful. Apply the usual lifting lemma to go back to the clauses fixed by Bi. To prove 2 note that any sequence of preconditions Hh . . . . . Hi. has the property
w ~ ( £ ) ~ =H,, . . . . .
w ~ ( £ ) ~ =Hi..
Assume Wh . . . . . IV/,, r t- H, where r and H do not contain the new constants. Assume that the new constants 5 have replaced ~ in the construction of weakest preconditions. Now replace 5anywhere in the derivation by ~ . Thus H h . . . . . /-/in, F t- H using the same derivation with the exception that W~ is replaced by Hi. Corollary. The number of new constants needed to construct the weakest preconditions for the decision path E is bounded by the structure of E .
3. REASONING BY ANALOGY Using the notion of weakest precondition it is possible to reconstruct the "minimal" assumptions from a decision. Definition 11. Simultaneous inversion of a set of conclusions: Let ¢ = { . . . , ( Ai , Bi , Ci , Di >, ' ' ' } be a set of decision paths, let W~, . . . , W, be the weakest preconditions corresponding to initial skeletons il . . . . . i, with assigned sets of constants I't . . . . . I'~ in . . . Di ' ' ' . I N V ( g ) = {W'l . . . . . W~}, where W'~ . . . . , W" result from Wt . . . . . W, by replacing all constants in r l , . . . , I'~ by new variables. (Note that INV (~) may extend the formal language by new constants.) We use the notion of inversion of sets of conclusions given by decision paths to define analogical reasoning. Definition 12. Let I' be a set of accepted sequents; let ~ be a set of decision paths in .£ of accepted decisions. A is derived from F and ~ by Reasoning by Analogy iff r U INV(~)FA in the language extended by the adequate constants. This concept of Reasoning by analogy reasoning depends on the following principle: A successful conclusion based on nonjustified assumptions gives some kind of justification to these assumptions. Example. In Brown v. Ziirich Insurance (c.o. 1954) a car with bald front tires was hold not to be in "roadworthy condition" and therefore the insured was not covered (cf. Marsh & Soulby, 1987, p. 270). In analogy to this decision, cars with damaged breaks are not considered to be insured. We use the following constants and relations to formalize the decision: the cars of the case under consideration bald front tires damaged breaks I ( x ) . • car x is insured H D ( x , y ) . • car x is heavily damaged by y Re(x)- • car x is in roadworthy condition C,C?.
•
BFT.. DB..
The decision takes the form: 1
2
3 assumption a R C ( x ) ~ 7I(x) H D ( C , BFT ) H D ( C , BFT ) =0 7I(C) 7I(C) 1 is the assumption of the court that a car with bald front tires is not in roadworthy condition, 2 is a reg-
Logic-Based Models of Analogical Reasoning
3
(1, 2, (res), (1, 0 ) )
model we need the notion of stepwise inversion of a sequence of decision paths: Definition 13. Let M be a set of statements, ~ be a finite set of decision paths, ,~ an ordering principle for d. INVSEQ (d~, ~, ~ ), the stepwise inversion of ~ relative to • and 4, is defined as follows: Let E0 ,~ • • • ~ E, be the elements o f ~ , Ei =
==o.Xi
XI ==o.X3 =~,X3
xlto = ~ ,
ulation, 3 is a fact. The assumption is considered to be independent from the concrete car and the concrete damage. The decision path takes the form: E = ( ( 3, (1, 2, (res), (1, 0 ) ) , (res), (0, 0 ) ) , 2 : 7 n C ( x ) =* 7I(x), 3 :=* H D ( C , BFF), 1 : {C, BFT},=* 7 I ( C ) ) . We construce the propositional (0, 1 )-realization of the skeleton according to Definition 7 step by step as
follows: (a) (3, ( I, 2, (res),( I, 0)>, (res),(0, 0)> ==~X3
(b)
1 2 3 XI==*X2 X2=~ X3 ==*X!
(c)
373
x~=, x3
==*'X3
Bj' is the extension of B~.by assignments
i : Hi ~ ggj for i in Aj, D~ is the corresponding restriction of D~, E)'
= (aj, B;, C;,/gj), Unify 2 with aRC(x) =* 7I(x), unify 3 with =~, HD(C,
BFT).
~'j+l
= ¢/jU { i : H i l H i 6?. INV(E~'), H; assigned to i}. INV(E) = { H D ( x , y) =* aRC(x) } Then reconstructs the assumption of the court. For another car C' with damaged breaks (DB) we conclude that the car is not insured (-1I(C')) using the fact HD(C', DC) (the damage of breaks is heavy damage) and Reasoning by Analogy. (For an example of Reasoning by Analogy modelling a construction of MARPOL international treaty on maritime pollution regulation concerning the dumping of waste from oil tankers in a logic programming framework see Baaz & Quirehmayr (1990).) 4. A M O D E L OF THE EVOLUTIONARY DEVELOPMENT OF LAW For the construction of a formal model of the evolutionary development of law a timeline is introduced together with the following assumptions: (a) There are different priority levels of statements and decisions represented by decision paths. These priority levels represent the legal hierarchy. (b) At each stage of the timeline either one statement is added to one statement level or one decision is added to one decision level. The added statement (decision) has to be consistent with the statements and decisions at higher levels and cancels the statements and decisions at the same or at lower levels that are contradictory (using Reasoning by Analogy) to the added statement (decision). For the formal construction of the evolutionary
INVSEQ (d~, •, ~ ) = {Hi Ii : Hi ~ ~tr+l }. Explanation. The ordering principle ,~ is necessary to model a chain of decisions. This corresponds to the reality of legal decision making, where information gained from prior cases is used in subsequent ones. So the set of weakest preconditions of cases dependent on preceding ones is a continuously growing one. Initial skeletons appearing in decision path E and the subsequent decision path E', E ,~ E', therefore represent the information gained in E and used in E'. That is why INVSEQ represents an argumentation chain of Reasoning by Analogy. Definition 14. An evolutionary model is a quintuple (SL, DE,
M. Baaz and G. Quirchmayr
374 A "
iff for some A ~ X and A ~ Y and for all B such that A "< B, B E X i f f B ~ Y. Obviously there is a < * - m a x i m a l subset Z _ A,. Let Z' = Z tO {extend(n) }. Define
= extend(l) and k < / . ga,,+l = {S[(S, a ) E Z', a E SL} extend(n) has to fulfill the following additional conditions in accordance with the inductive definition of the actual sets of statements • .... a ~ SL, and the actual sets of decision paths ~0.... a ~ DL: (a') #~,0 = ~ and £),,0 = ~ . (b') If extend(n) = (E, a), a ~ DL, E = (A, B, C, D ) , then B may only contain assignments i : H~ with Hi ~ t-J~SLg~,,. If extend(n) = (S, a ) , a ~ SL, then
U '$o,n U {S} to INVSEQ( U #,,,n, to ~0t~,,,, "
a<~
aESL
a<~
is consistent. If extend(n) = (E, a ) , a ~ DL, then U #t~,,, U INVSEQ( U ¢~,,n, U ~0t~,nU {E}, -
a
aESL
a<~8
is consistent. (c') Assume extend(n) = (S, a), a ~ SL, and
U d'¢,,, U INVSEQ( U g .... U .Ot~,n, ~(t) ~
a~fl~
aESL
a~p3
S
or extend(n) = (E, a), a ~ DL, and U d't~.nU INVSEQ( O ~,,,n, U Z)t~,n,-<,)
a~t~8
ot~SL
II
a~
INVSEQ( t0 ~ .... to ~0a,n tO {E}, -
a~r,O
Then #,,,,,+l = #,,,,,
and
~,,.n+l = ~),,,n.
Otherwise use the ordering "
~0,,,n}.
Now consider the subsets X of A, that have the following properties: (a") Let Yt = { S ' I ( S ' , a ' ) E X , a ' ~ S L } , Y2 = {E'I(E', a') E X , a ' ~ DL}. If extend(n) = (S, a ) , a SL, then Yi to {S} to INVSEQ(Y~, Y2, "
X'<* Y,
X, Y~_ An,
and
~O~,n+~ = { E l ( E , a ) E Z', a ~ DL}.
Explanation. A formal model of the evolutionary development of law must comprise ordered levels of statements and decisions (represented by
Logic-Based Models of Analogical Reasoning --This maximal subset determines the statement levels and decision levels at the following time stage. As the maximal consistent subset of decisions and statements determines the new status of the knowledge base and all other statements and decisions are cancelled, the oscillation of legal knowledge can be modelled, which is a main difference to usual nonmonotonic approaches (cf. Reiter, 1981 ).
375 1 is assumption of the court, 2 is a regulation ("If a contract is completed in a country then the courts of this country decide legal actions related to the contract."), 3 is a fact ("The completion of the acceptance of the contract between E and MFE was done in England by Telex."). The assumption is assumed to be independent from the parties, the local circumstances and the way of posting (E,MFE,GB,TELEX). The decision of the House of Lords takes the related form:
1 4 assumption ACCEPTANCE(x,y,z) •. COURT(x.y,z) 3 COMP(B, SS,GB,TELEX) •* COURT(B, SS,GB) •*COMP(B, SS, GB,TELEX) COURT(B, SS,GB) 5. EXAMPLES OF THE DEVELOPMENT OF LEGAL KNOWLEDGE
The following facts and regulations are used: F1: ~=~C O M P ( E , M F E , G B , T E L E X )
5.1. Acceptance of Contracts by Telex Transmission
Entores Ltd. v. Miles Far East Corporation ( 1955 ) (cf. Padfield, 1987). Plaintiffs in London made an offer by Telex to defendants in Amsterdam. The defendants accepted by Telex message transmitted to London. Later defendants were in breach of contract, and the plaintiffs wished to establish that the contract by Telex was made in London where the acceptance took place, in which case the legal action could be decided in English courts. Held: that the contract had been made in London, since the defendant's acceptance of plaintiff's offer was not complete until actually received by plaintiff. This Court of Appeal decision was upheld by the House of Lords in Brinkibon Ltd. v. Stahag Stahl (1982). A possible model of the relevant elements of the court hierarchy for formalization is: FACI~
I LAWS
I HOUSE OF LORDS DECISIONS (HOFLORDS)
I
F2: =* COMP(B,SS,GB,TELEX ) L: ACCEPTANCE (x,y,z) =~ COURT(x,y,z) The corresponding decision paths take the form E l =
((3,(1,2,(res),(1,0)),(res),(0,0)), 2 : L, 3 : Fi, 1 : { E,MFE,GB,TELEX }, COURT(E,MFE,GB) ),
//:2 = ( ( 5 , ( 1,4,(res),(1,0)),(res),(0,0)), 4 : L, 5 : F2, 1: {B,SS,GB,TELEX }, ==~COURT(B,SS,GB)). The inversion of El yields INV(EI) = {COMP(x,y,u,v) =* ACCEPTANCE(x,y,u) } ("A contract whose offer is accepted by Telex is accepted in the country where the Telex is sent to.") The evolutionary model (up to stage 5 ) is given by ( { FACTS,LAWS },{ CAPP,HOFLORDS }, <,,
COURT OF APPEAL DECISIONS (CAPP) To formalize the decisions we need the following constants and relations: GB, TELEX, E . . . B... ACCEPTANCE(x,y,z)... COURT(x,y,z)... COMP(x,y,u,v)...
with
Entores Ltd., M F E . . . Miles Far East Corporation, Brinkibon Ltd., S S . . . Stahag Stahl. the acceptance of a contract between x and y takes place in z courts in z decide the legal action related to the contract between x and y the contract between x and y was completed in u using v.
The decision of the Court of Appeal takes the form: 1 2 3 assumption ACCEPTANCE(x,y,z) =, COURT(x,¥,z) ~COMP(E,MFE,GB,TELEX) COMP(E,MFE,GB,TELEX) ~ COURT(E, MFE,GB) COURT(E, MFE,GB)
376
M. Baaz and G. Quirchmayr extend(0) = (L,LAWS), extend(l) = (FI,FACTS), extend(2) = (EI,CAPP), extend(3) = (F2,FACTS), extend(4) = (E2,HOFLORDS), ~0,FACrS ~--"~ , ~3,FACrS ~--"{FI},
~0,taws = Z~, ~3,LAWS = {L}, ~)0,CAPP ---~~ , ~)3,CAPP = {El}, ~0,HOFLORDS = ~ , ~)3,HOFLORDS = ~ ,
e~I,FACrS ~~4,FACrS = ~,,taws = ~Y4.LAWS= ~)I,CAPP '~4,CAPP
~, {F,,F2}, (L}, {L}, ---- ~ , ~"
{El},
¢~2,FACTS-~" {FI}, ~5.FACrS = {F, ,F2}, ~2.~ws = {L}, "PS,LAWS= {L}, ~2,CAPP ---- ~ , *~5,CAPP
=
{E,},
*~I,HOFLORDS ~- ~ ,
~)2,HOFLORDS----D ,
~94,HO~S
~gS,HOFtOROS= {E2}.
(Note that by INVSEQ no assumption of E2 is open but that the House of Lords decision transfers the assumption of El to a higher priority level.)
5.2. The Tort of Negligence A representative example of the evolutionary development of law are the cases dealing with the tort of negligence. It arises when damage is caused to the person or property of another by failure to take such care as the law requires in the circumstances of the case. For succeeding in an action, it is necessary to prove that: (a) the defendant owed a legal duty of care to the plaintiff; (b) the duty was broken; and (c) damage was suffered in consequence. As there are many different situations, where duty of care is owed, Lord Atldns' generalization by creating an abstract principle, the so-called "neighbour principle" (cf. Donoghue v. Stevenson, 1932), was the natural consequence. The perhaps best example of extensions o f this principle by law courts is the House o f Lords" decision in Hedley Byrne v. Heller & Partners (1963). In this case a firm of advertising agents relied upon a banker's reference and gave credit to a client.
= ~,
When the client became insolvent, the firm suffered financial loss. The reference had been given carelessly but, since the bank had expressly disclaimed liability when giving it, the action failed. (In addition it was noted that financial losses were not considered as damage in the sense of law.) Nevertheless, the House of Lords stated that, contrary to what had previously been believed, liability for negligence may extend to careless words as well as to careless deeds and that damages may be awarded for financial loss as well as for physical injury to persons and property. A possible model of the relevant elements of the court hierarchy for formalization is: FACTS
I LAWS
I HOUSE OF LORDS DECISIONS ( H O F L O R D S )
I H I G H C O U R T DECISIONS ( H C O U R T ) To formalize the decisions we need the following constants and relations:
Heller & Partners H B . . . Hedley Byrne A . • . the advice in question H . . . DAMAGE(x,y). • x suffers a damage (in the sense of law) in case y FINLOSS(x,y). • x suffers financial loss only in case y DUTYTOCARE(x,y,z). • x has the duty to care for y in case z CARELESS(x,y,z) . • concerning z x sets a careless action relative to y INFORMAL(x,y,z). • a careless advice z given to y by x is merely informal LIABLE(x,y,z) . • x is liable to y for consequences of z 1. The High Court decision rejecting the liability of Heller & Partners for two independent reasons is expressed by two formal decisions. 1.1.
2 INFORMAL(H~HB~A)
0 1 assumption -aCARELESS(x,y,z) =* ~LIABLE(x,y~z) INFORMAL(H~HB,A) =~ -aLIABLE(H~HB,A) -1LIABLE(H,HB,A)
0 is the assumption of the court, 1 is a law, 2 is a fact. The assumption is considered to be independent from {H,I-IB,A}, the concrete persons and advices.
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1.2.
5 =* FINLOSS(HB,A)
3 4 assumption 7DAMAGE(x,y,z) =, 7LIABLE(x,y,z) FINLOSS(HB,A) =. 7LIABLE(H~HB~A) =* -7LIABLE(H, HB,A)
3 is the assumption of the court, 4 is a law, 5 is a fact. The assumption is considered to be independent from {H,HB,A}, the concrete persons and advices. 2. The House of Lords' decision rejecting the High Court decision can be expressed as follows: 10 INFORMAL(H, HB~A)
6 assumption 9 =* DUTYTOCARE(H,x,A)
8
DUTYTOCARE(u,v,w), CARELESS(u,v,w),
DAMAGE(v,w) =*LIABLE(u,v,w) 7 ~ CARELESS(H,HB~A) CARELESS(H,v,A), DAMAGE(v,A) =* LIABLE(H,v,A) 11 assumption DAMAGE(HB~A) =* LIABLE(H~HB,A) ==, FINLOSS(HB,A) FINLOSS(HB,A) =* LIABLE(H,HB,A) =* LIABLE(H,HB,A) 6, 7 are the assumptions of the House of Lords, 8, 9, are laws, 10, 11 are facts. The assumptions are considered to be independent from { H , H B ,A }, the concrete persons and advices. To put together, the following facts and laws are used: Facts Laws and regulations
{ H , H B , A }, 7 : { H , H B , A }, =* L I A B L E ( H , H B , A ) ) INVSEQ( {El, F2, L~, L2, L3, L4 }, {Ell, El2 }, ~) { INFORMAL(x,y,z) ==, 7CARELESS(x,y,z),
F,: •* INFORMAL(H,HB,A) F2: =* FINLOSS(HB,A) Ll: DUTYTOCARE(u,v,w), CARELESS(u,v,w),DAMAGE(v,w) =, LIABLE(u,v,w) L2: CARELESS (x,y,z) =* 7LIABLE(x,y,z) L3: 7DAMAGE(x,y,z) =* 7LIABLE(x,y,z) (these sequents represent the regulations 1-3) L4: =* DUTYTOCARE(H,x,A)
The decision paths take the form E~t = ((2,(0,1,(res),(l,0)),(res),(0,0)),
FINLOSS(y,z) =. 7DAMAGE(x,y,z) } are the reconstructed assumptions of the High Court (Ell ¢ E12). ("An explicitly informally advice is not careless; .... A financial loss is no damage in the sense of law")
1 : L2, 2 : Fl, 0 : { H , H B , A }, =* -aLIABLE ( H , H B , A ) ) El2 = ((5,(3,4,(res),(l,0)),(res),(0,0)),
INV(E2) = { INFORMAL(x,y,z) •* CARELESS(x,y,z), FINLOSS(y,z)
==}DAMAGE(x,y,z) } 4 : L3, 5 : F2, 3 : { H , H B , A }, =* ~LIABLE (H, HB, A ) )
are the reconstructed opposite assumptions of the House of Lords. The evolutionary model (up to stage 9) is given by
E 2 -- ( ( 11,(7,((10,6,(res),(0,0)), < { FACTS,LAWS }, { H C O U R T , H O F L O R D S }, (9,8,(res),(0,0)),(res),(1,0)),(res),(0,0)),
(res),(0,0)), 8 : LI, 9 : L4, 10 : El, 1 1 : F2, 6 : with
extend(I) = (L2,LAWS), extend(3) = (L,LAWS),
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M. Baaz and G. Quirchmayr
extend(5) = (F2,FACTS), extend(4) = (F,,FACTS), extend(6) = (E,,,HCOURT), extend(7) = (El2, HCOURT), extend(8) = (E2,HOFLORDS), ~'~0.FACTS= ~ , e~3.FACI'S = ~ , ~O6,FA~ = {F,,F2},
e~I.FACI'S : ~ ' ~'~4.FACTS- ~ ,
a_~2.FACTS = ~ , ~5.FACTS = {F,},
87,FACrS = {F,,F2}, g9,FACrS----{F,,F2},
gS,FACrS = {FbF2},
g0.LAWS = ~ , e~°3,LAIS m {L,,L2,L3} ~'°6,LAWS = {L,,L2,L3,L4}
~'~I,LAWS= {L,}, e~4,LAIs = {LI,L2,L3,L,,}, ~7,LAWS= {L,,L2,L3,L4}, ~9,LAWS = {LI,L2,L3,L4}, ~LHCOU~T = Z~, '~4,HCOURT = ~ , ~7.HCOURT = {Ell}, '~9,HCOURT = ~ , ~I,aO~I~ORDS = ~ ,
~00,.couaT = Z~, '~3,HCOURT : J~, ~6.HCOURT = ~ , ~0,HOFLORDS = ~ , '~3,HOFLORDS = ~ , '~6,HOFLORDS = ~ ,
~)2,HCOURT = ~ , ~5,HCOURT = ~ , ~D8.HCOUXT= {Ell,El2}, '~2,HOFLORDS = ~ , ~)5,HOFLORDS = ~ ,
~)7.HOFLORDS = ~ , ~9,HOFLORDS ~- {E2}.
'~8,HOFLORDS = ~ ,
6. IMPLEMENTATION As the approach inherits the problem ofundecidability of first order logic, some restrictions have to be made to make the model computable. Therefore, occurring contradictions can only be solved until a certain depth. This seems to be a nasty restriction but it corresponds to the reality of law court hierarchy, which is characterized by limited time (= limited depth of search). The higher the level of the law court is, the higher the complexity of the search can be. Also if the undecidability of first order logic is neglected, these restrictions have to be made, because in legal decision making time and space state are limited, too. 7. CONCLUSION As Reasoning by Analogy is the main aspect of legal reasoning, an approach which is trying to formalize the dynamic aspect of law has to be built around a formal model of Reasoning by Analogy. In addition, a model of the evolutionary development of law has to consider the continuous growth and subsequent reduction of legal knowledge which can be described as a sort of permanent oscillation of valid statements.
Brown v. Zurich Insurance, 1954. Entores Ltd. v. Miles Far East Corporation, 1955.
#8.LAWS "~" {LI,L2,L3,L4},
~)4,HOFLOm~S = ~ ,
(Note for the calculation that INVSEQ (g, {E },,~) is INV(E) and that E2 "refutes" E11 and El2 .) At stages 7 and 8 liability of Heller & Partners is denied, at stage 9 liability of Heller & Partners is confirmed.
CASES CITED
e~2.LAWS-----{L1,L2}, "-~5,LAWS= {L1,L2,L3,L4},
Brinkibon Ltd. v. Stahag Stahl, 1982. Donoghue v. Stevenson, 1932. Hedley Byrne v. Heller & Partners, 1963.
REFERENCES Alschwee, B. (1986). Analyse natiidicher Sprache in einem juristischen Expertensystem. In Fiedler, Haft, Erdmann, & Traunmoiler (Eds.), Neue Methoden im Recht (Bd. 1). Ttibingen: Attempto Verlag. Baaz, M., & Quirchmayr, G. (1990). A logic based model of legal decision making. Proc. of the lOth Int. Workshop on Expert Systems and their Applications. Avignon. Chang, C.-L., & Lee, R.C.-T. (1973). Symbolic logic and mechanical theorem proving. New York: Academic Press. Gentzen, G. (1934). Untersuchungen fiber das logische Schliel3en. Mathematische Zeitschrifi, 39, 176-210, 405--431. Gordon, T. (1986). The role of exceptions in models of law. In J. Schweitzer (Ed.), Formalisierung im Recht und Ansdtze juristischer Expertensysteme. Miinchen: Verlag. Hall, R.P. (1989). Computational approaches to analogical reasoning: A comparative analysis. Artificial Intelligence, 39, 39-120. Marsh, S.B., & Soulby, J. ( 1987 ). Outlines of English Law (4th ed.). London: McGraw-Hill. McCarty, L.T. ( 1983). Permissions and obligations. Proc. of the 8th Int. Joint Conf. on Artificial Intelligence. Los Angeles: Kaufmann. Padfield, C.F. (1987). LA 14/ made simple. London: Heinemann. Parikh, R. (1973). Some results on the length of proofs. TAMS, 177, 29-36. Reiter, R. ( 1981). A logic for default reasoning. Artificial Intelligence, 13, 81-132, Susskind, R. (1987). Expert systems in law. Oxford: Oxford University Press. Takeuti, G. (1975). Proof theory. Amsterdam: North-Holland.
0957-4174/92 $5.00+ .00 © 1992PergamonPressLtd.
Logic-Based Models of Analogical Reasoning: A Fundamental Basis for the Design of Legal Expert Systems IVIATTHIAS BAAZ InstitutflirAlgebraund DiskreteMathematik,TcchniscbeUniversit/ItWien, Wien, Austria
GERALD QUIRCHMAYR InstitutfiirInformatik,JohannesKeplerUniversit/R,Linz,Austria
Abstract--Traditional approaches in the domain of legal expert systems do largely neglect the importance of analogy and the weight of decisions. That is why this paper is focussing on a formal model of Reasoning by Analogy based on Gentzen's sequent calculus LK, which is then used to develop a formal representation of the evolutionary development of law. 1. INTRODUCTION: EXPERT SYSTEMS IN LAW AND FORMS OF LEGAL REASONING
IN HIS 1987 overview of expert systems applications in the legal domain, Susskind clearly shows that advanced forms of knowledge representation are scarce (Susskind, 1987 ). Apart from the approach developed by McCarty (1983) and Gordon (1986), existing systems confine themselves to traditional rule-based inference. So elementary principles of legal decision making such as Reasoning by Analogy are not yet supported. Systems based on concepts developed in the field of machine learning seem to be a solution, but an indepth analysis reveals that these systems should rather be called classification algorithms (Hall, 1989) and if Reasoning by Analogy is supported, it is only a support of building analogies of facts. From a theoretical point of view this is merely equivalence of sets (Alschwee, 1986). What has so far been ignored is the structure of cases, the relationship between the elements (facts) of a case. The worst deficiency of these models is perhaps that none of them is able to define the meaning and the weight of a legal decision because all of them reduce legal reasoning to mere subsumption, thereby neglecting the importance of the interpretation and evaluation done by the judge. That is why they can serve for deal-
ing with Analogy in a limited way, but are not suitable for modelling the evolutionary development of law. To overcome this shortage it is necessary to develop a formal model of Reasouing by Analogy which is based on the analysis of decisions. Reasoning by Analogy does extend the available knowledge, but as continuous growth would lead to an uncontrollable mess there must also be mechanisms for reducing the number of rules. These are mainly overwriting strategies for canceiling obsolete and contradictory rules and generalizations for classifying cases. The best example of this development is perhaps the evolution of rules concerning the tort of negligence (ef. Marsh & Soulby, 1987, pp. 124 if). The goal of this paper is to develop a formal model of these processes which is based on Gentzen's LK-calculus. 2. ABSTRACT FORMS OF DECISIONS
We base our formalization on a quantifier-free variant LK* of L K , the famous calculus ofGentzen (cf. Gentzen, 1934; Takeuti, 1975), that includes extended resolution. Definition 1. The formal language of L K * is constructed from the usual terms and predicate symbols together with the logical connectives -1, A, V, --~. Sequents 1-1 •, r consist of (possibly empty) sequences of formulas H, r. (The intended meaning is AAEw4 --~ VA~rA. Sequents consisting of predicates only correspond to clauses by H •. r _ VA~nTA V VA~rA.) Logical axioms: A =* A Inference rules: Logical rules:
Requests for reprints shouldbe sent to MatthiasBaaz,Institutfiir Algebra und Diskrete Mathematik,TeehniseheUniversit~t Wien, WiednerHauptstr. 8-10, A-1040Wien,Austria. 369
370
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M. Baaz and G. Quirchmayr
F~A,D
(q-left):
qD, r = , A '
D, F = * A F =* A, -qD '
(-q-right): L2
C,D,F~A CAD, F~A'
(A-left):
( A -right ): L3
C,F~AD,
(V-left):
I" ~ A, C r ==~ A, D
r~A, F~A
CVD, F~A r ~ A, C, D
(V-right): IA
CAD
F=.A, CVD'
r = * A, C D, r = A
(-*-left):
C--~D,r~A
(--~-right):
C, P ~ A, D F ~ A, C " ~ D "
Structural rules: Sl weakening (w-left):
r~A
•
A, r==. A '
(w-right):
r~A r=.
.
A,A'
S2 exchange F ~ A r' is rearrangement of r, r' ~ A' ' A' is rearrangement of A;
(exchange):
$3 factorization (f-left):
A,A',F=*
A
Ao, F * =~ A~ '
(f-right):
F = * A, A, A' r a =* A~, Act '
o is most general unifyer of A and A'; $4 resolution (res):
r ==*
A, A A ' , II ~ A
(r, n ) ~
(& A)~
o is most general unifier of A and A'. (The numbers ( i , j ) of the formulas in I" and A are called the index of the inference) Definition 2. An LK*-deduction of H out of Ht, . . . . H . is a tree of sequents such that: (a) the top nodes are axioms or variants of Hi . . . . . H . by variable
renaming; (b) each other node is the direct consequence of its ancestor nodes; (c) the bottom node is a sequent H ' such that H = H ' o for some o. Let at . . . . . a. be all substitutions subsequently generated by the rule applications. The total substitution of the derivation is ~o. • • • at. Hi . . . . . H . I- H iffthere is any derivation of H out of Ht . . . . . H~. Definition 3. Let T(II --~ r ) = AAenA -~ VAerA, AAenA is true for II empty, V a e r A is false for r empty• T ( ~ ) = { T ( H ) IH ~ ~/}. ~ is a valid consequence of f ( f H-- Yf) iff the universal closures of formulas in T ( . £ ) imply all universal closures of formulas in T(~/). Theorem 1. S1 . . . . .
S , F S *"-, St . . . . .
S , : ~, S.
Proof. Calculate the total substitution of the derivation and use the usual completeness theorem (cf. Takeuti, 1975, Theorem 8.2) in connection with lifting lemma (of. Chang & Lee, 1973, Lemma 5.1 ) extended to L K * . Definition 4. The skeleton of a deduction is: (a) a number ( = an initial skeleton); (b) of the form ( S , D ) where Sis a skeleton and D ~ {(-q-left), (-q-right), (A-left), (V-right), (-~.-right), (w-left), (w-right), (fleft), (f-right) } ; (c) of the form ($1, $2, D ) where St, $2 are skeletons and D = {(A-right), (V-left), (--~left)}; (d) of the form ( S t , $2, (res), ( i , j ) ) ; and (e) of the form ( S , (exchange), it . . . . . i., jt . . . . , i s ) , where {it . . . . . ir} = {1 . . . . . r}, {j, . . . . . j,} = {1, ...,S}.
All initial skeletons within a given skeleton have to be different. Explanation. The skeleton of a deduction is an abstract description of the deduction without specifying assumptions and goal. For the general notion of skeleton see Parikh (1973). Definition 5. A decision path is a quadruple (A, B, C, D ) such that: (a) A is a skeleton of a deduction; (b) B is a finite sequence of assignments i : Hi of sequents to initial skeletons in A; (c) C is a sequence of assignments j : I'j of finite sets of constants I'j to initial skeletons which occur in A but not in B; and (d) D is a variable free non-empty sequent. We use the convention for (b) and (c) that in presence of several decision paths containing the initial skeleton k either always the same sequent Hk is assigned to k or always the same set of constants rk is assigned to k. Explanation. The skeleton (A) represents the figure of a deduction together with the indices of the rule applications, but gives no information on initial and final clauses• The sequence of assignments (B) fixes some of the initial clauses of the skeleton. In interesting cases some of them remain unknown• (C) represents objects from which the "unknown" hypotheses are assumed to be independent. The final sequent (D) is
Logic-Based Models of Analogical Reasoning
371
fixed. Decision paths are formal objects and can therefore be treated in a formal manner. The information given by facts and statements and the additional information given by the decision itself are used to derive the weakest preconditions of the decision. These weakest preconditions have to be accepted, because otherwise the decision itselfwould have to be rejected as false. The initial sexluents of the decision are usually at least known to be independent from certain elements, such as time, location, person, and other elements not occurring in the general form of the statements in the decision (in the decision path these elements are represented by C). Therefore, the weakest preconditions are also considered as independent from these elements. For the modelling of the evolutionary development of law we have to consider sets of decision paths: Definition 6. An interpretation/(a) of a set of decision paths $ = { . • . , ( A i , Bi, Ci, Di ), • • • } is a set of deductions in accordance with Ai and Bi with final sequents D;. (Initial skeletons have to be realized by the same formulas throughout/($).) An interpretation of a decision path E is defined b y / ( {E} ). (We write/(E) instead o f / ( { E } ) . ) Definition 7. The propositional ( k, /)-realization of a skeleton S is constructed as follows (reducing S from outside to inside): ( l ) Write Xt . . . . . Xk =* Y~ . . . . . Yt as bottom node. (2) Assume Ui . . . . . fir =, Vl . . . . . Vs is constructed and that the subskeleton S' has to be reduced. (a) S' is initial skeleton k: Ul . . . . . U~ k VI, . . . . V~is top node. (b) S' = ( S ~, D ) , D = (~-left) (resp. (a-right), (A-left), (V-right), --*-right), (w-left), (w-fight), (fleft), (f-fight)): for the logical rules anywhere apply the following corresponding substitutions: {-aU/Ul } (r~sp. { T U ] V s } , { V ]~ V / V l } , { U V V]V$}, { V - . ~ V / V , } ) . Write U2. . . . . U,=* V ~ , . . . , V~, U(resp. U, UI . . . . . Or ==0 V I , . . . , Vs-l, U, V, U 2 , . . . , Ur===lb Vl, . . . . V$, U i , . . . , Ur==~ V I s . . . , V$_I, U, V, U, UI, ....
U,~
Vs, U I , . . .
Vi . . . . .
V~-I, V, U2 . . . . .
, Ur==~ V I , . . .
U,~
, V$_I, UI, U 1. . . .
VI . . . . .
, Urm=~ V1 . . . . . V~, Ui . . . . . U, =* Vi . . . . . Vs, Vs) above the transformation of Ui . . . . . U, ~ V~. . . . , V~, reduce the skeleton S' to S". (c) S' = (S", S", D ) , D = (A-right) (resp. (Vleft), (-'*-left)): for the logical rules anywhere apply the following corresponding substitutions: { U A V~ V, } (resp. { U V V~ U, }, { V --~ V~ V~ } ). Write UI, . . . . U,=* VI . . . . . Vs-m, U a n d Ul . . . . . Ur ===0Vl, . . . . V~_~, V(resp. U, U2. . . . . /Jr =* V~. . . . . Vs and V, U2 . . . . . Ur=~ VI . . . . . Vs, U2 . . . . . Ur==o Vi . . . . . V~, U and V, /.12. . . . . U, =* Vl, . . . , Vs, above the transformation of Ul . . . . . U, ~ Vl . . . . . Vs, reduce the skeleton S' to S" and S ' .
(d) S' = (S", S ' , (res), (i, j ) ) : write Ut . . . . . vt . . . . . v/, u and U, Ui+~ . . . . . U, ~ Vj+,, . . . . Vs above UI . . . . . U, ~ VI . . . . . Vs. (e) S' = ( S", (exchange), i l , . . . , it, j r , . . . ,is): write Ui~. . . . . Ui, =* V/~. . . . . Vj, above Um. . . . . U, Vt . . . . . Vs and reduce the skeleton S' to S". Explanation. The propositional (k, /)-realization of a skeleton represents the propositional structure of any adequate derivation with this skeleton. Proposition 1. If the construction of Definition 7 is impossible for some skeleton S and some (k, l), no set of decision paths containing E = (S, B, C, D ) with D = Ai . . . . . Ak =~ B I , • • . , B l has an interpretation. Let any deduction that is an interpretation of the decision path E = (S, B, C, D ) be given. The deduction is transformed into a propositional realization in a unique way if the total substitution (cf. Definition 2) is applied and atomic formulas are 1-1 replaced by propositional variables. This motivates the following definition. Definition 8. Let E = (A, B, C, D ) , D = Ai . . . . . Ak =* B~ . . . . . Bt, be a decision path. E determines a propositional variable X in the ( k, /)-realization of A iffX occurs in the top nodes fixed by B or in the bottom node. Definition 9. Let a set of decision paths ~ = { • • . , ( A i , Bi, Ci, Di, • • • } be given and let il . . . . . i, be the sequence of numbers representing those sequents in ~ to which no sequents of B; are assigned. Variable-free sequents Hi, . . . . . Hi, are called precondifi0ns of a itf (a) there exists an interpretation/($') of ~' = { . . . , ( A i , B~, Ci, Di), . . . } where B~ is the extension of Bi by ik : Hik for ik occurring as initial skeleton in Ai and Ci is correspondingly restricted. (b) the groups of propositional variables not determined by E in $ are realized by a previously fixed list of zero-placed predicates. Explanation. The preconditions complete the incomplete information of ~. In case of not determined variables no information is given. So it is useful (cf. condition (b)) to replace them by previously fixed zeroplaced predicates. Definition 10. Let f * be the extension o f . £ by the set of new constants { ci } and let ~ be a set of decision paths. Wm. . . . . Wn are weakest preconditions for and .£ iff (a) WR. . . . . W, are preconditions for E in the language .£*; (b) For any sequence HI . . . . . Hn of preconditions for ~, any H a n d any F in .£, the following holds: ui ~
Wl . . . . .
W,,,
F I- H=* HI . . . . .
H,,,
F I- H
using the same derivation with the exception that W~ is replaced by Hi.
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M. Baaz and G. Quirchmayr
Explanation. I f a decision path E is to be completed to a proof, the assumptions have to be made at least as strong as the weakest preconditions. The consequence is: if a decision path E is accepted as corresponding to a real conclusion, the weakest preconditions have to be accepted as true statements. (The definition of the weakest preconditions corresponds to the interrelation { ( 3 x ) A ( x ) } O r k B = , {A(t)} O I~ ~-B for any term t in usual first order predicate logic. If existential quantitiers have to be avoided (like in LK* ), new constants have to be introduced. Theorem 2. 1. It is decidable whether any interpretation for a set of decision paths ~ exists. 2. If an interpretation exists, the weakest preconditions in the extension .£* of .£ by { ci } can be constructed. Proof. Let ¢~ = { . . . , ( Ai , B,, Ci , Di ) , " ' " },let it . . . . . in be the sequence of numbers representing those initial sequents in A to which no sequents of B are assigned. Weakest preconditions are constructed as follows: ( 1 ) Construct the propositional realizations of Ai relative to D~. (2) Identify the final sequents of the propositional realizations with Di and identify the corresponding initial sequents with the sequents fixed by Bi using unification. (3) Replace remaining propositional variables by the previously fixed sequence of zero-placed predicates (these are the variables not determined by the skeleton, cf. Definition 8). (4) Let W * ( g ) , . . . , W * i,(x) be the resulting initial clauses not determined by B~. Replace g by new constants 3. The result is a sequence of weakest preconditions w,, . . . . .
w~.
To prove 1 simply note that no interpretation is possible if this construction is not successful. Apply the usual lifting lemma to go back to the clauses fixed by Bi. To prove 2 note that any sequence of preconditions Hh . . . . . Hi. has the property
w ~ ( £ ) ~ =H,, . . . . .
w ~ ( £ ) ~ =Hi..
Assume Wh . . . . . IV/,, r t- H, where r and H do not contain the new constants. Assume that the new constants 5 have replaced ~ in the construction of weakest preconditions. Now replace 5anywhere in the derivation by ~ . Thus H h . . . . . /-/in, F t- H using the same derivation with the exception that W~ is replaced by Hi. Corollary. The number of new constants needed to construct the weakest preconditions for the decision path E is bounded by the structure of E .
3. REASONING BY ANALOGY Using the notion of weakest precondition it is possible to reconstruct the "minimal" assumptions from a decision. Definition 11. Simultaneous inversion of a set of conclusions: Let ¢ = { . . . , ( Ai , Bi , Ci , Di >, ' ' ' } be a set of decision paths, let W~, . . . , W, be the weakest preconditions corresponding to initial skeletons il . . . . . i, with assigned sets of constants I't . . . . . I'~ in . . . Di ' ' ' . I N V ( g ) = {W'l . . . . . W~}, where W'~ . . . . , W" result from Wt . . . . . W, by replacing all constants in r l , . . . , I'~ by new variables. (Note that INV (~) may extend the formal language by new constants.) We use the notion of inversion of sets of conclusions given by decision paths to define analogical reasoning. Definition 12. Let I' be a set of accepted sequents; let ~ be a set of decision paths in .£ of accepted decisions. A is derived from F and ~ by Reasoning by Analogy iff r U INV(~)FA in the language extended by the adequate constants. This concept of Reasoning by analogy reasoning depends on the following principle: A successful conclusion based on nonjustified assumptions gives some kind of justification to these assumptions. Example. In Brown v. Ziirich Insurance (c.o. 1954) a car with bald front tires was hold not to be in "roadworthy condition" and therefore the insured was not covered (cf. Marsh & Soulby, 1987, p. 270). In analogy to this decision, cars with damaged breaks are not considered to be insured. We use the following constants and relations to formalize the decision: the cars of the case under consideration bald front tires damaged breaks I ( x ) . • car x is insured H D ( x , y ) . • car x is heavily damaged by y Re(x)- • car x is in roadworthy condition C,C?.
•
BFT.. DB..
The decision takes the form: 1
2
3 assumption a R C ( x ) ~ 7I(x) H D ( C , BFT ) H D ( C , BFT ) =0 7I(C) 7I(C) 1 is the assumption of the court that a car with bald front tires is not in roadworthy condition, 2 is a reg-
Logic-Based Models of Analogical Reasoning
3
(1, 2, (res), (1, 0 ) )
model we need the notion of stepwise inversion of a sequence of decision paths: Definition 13. Let M be a set of statements, ~ be a finite set of decision paths, ,~ an ordering principle for d. INVSEQ (d~, ~, ~ ), the stepwise inversion of ~ relative to • and 4, is defined as follows: Let E0 ,~ • • • ~ E, be the elements o f ~ , Ei =
==o.Xi
XI ==o.X3 =~,X3
xlto = ~ ,
ulation, 3 is a fact. The assumption is considered to be independent from the concrete car and the concrete damage. The decision path takes the form: E = ( ( 3, (1, 2, (res), (1, 0 ) ) , (res), (0, 0 ) ) , 2 : 7 n C ( x ) =* 7I(x), 3 :=* H D ( C , BFF), 1 : {C, BFT},=* 7 I ( C ) ) . We construce the propositional (0, 1 )-realization of the skeleton according to Definition 7 step by step as
follows: (a) (3, ( I, 2, (res),( I, 0)>, (res),(0, 0)> ==~X3
(b)
1 2 3 XI==*X2 X2=~ X3 ==*X!
(c)
373
x~=, x3
==*'X3
Bj' is the extension of B~.by assignments
i : Hi ~ ggj for i in Aj, D~ is the corresponding restriction of D~, E)'
= (aj, B;, C;,/gj), Unify 2 with aRC(x) =* 7I(x), unify 3 with =~, HD(C,
BFT).
~'j+l
= ¢/jU { i : H i l H i 6?. INV(E~'), H; assigned to i}. INV(E) = { H D ( x , y) =* aRC(x) } Then reconstructs the assumption of the court. For another car C' with damaged breaks (DB) we conclude that the car is not insured (-1I(C')) using the fact HD(C', DC) (the damage of breaks is heavy damage) and Reasoning by Analogy. (For an example of Reasoning by Analogy modelling a construction of MARPOL international treaty on maritime pollution regulation concerning the dumping of waste from oil tankers in a logic programming framework see Baaz & Quirehmayr (1990).) 4. A M O D E L OF THE EVOLUTIONARY DEVELOPMENT OF LAW For the construction of a formal model of the evolutionary development of law a timeline is introduced together with the following assumptions: (a) There are different priority levels of statements and decisions represented by decision paths. These priority levels represent the legal hierarchy. (b) At each stage of the timeline either one statement is added to one statement level or one decision is added to one decision level. The added statement (decision) has to be consistent with the statements and decisions at higher levels and cancels the statements and decisions at the same or at lower levels that are contradictory (using Reasoning by Analogy) to the added statement (decision). For the formal construction of the evolutionary
INVSEQ (d~, •, ~ ) = {Hi Ii : Hi ~ ~tr+l }. Explanation. The ordering principle ,~ is necessary to model a chain of decisions. This corresponds to the reality of legal decision making, where information gained from prior cases is used in subsequent ones. So the set of weakest preconditions of cases dependent on preceding ones is a continuously growing one. Initial skeletons appearing in decision path E and the subsequent decision path E', E ,~ E', therefore represent the information gained in E and used in E'. That is why INVSEQ represents an argumentation chain of Reasoning by Analogy. Definition 14. An evolutionary model is a quintuple (SL, DE,
M. Baaz and G. Quirchmayr
374 A "
iff for some A ~ X and A ~ Y and for all B such that A "< B, B E X i f f B ~ Y. Obviously there is a < * - m a x i m a l subset Z _ A,. Let Z' = Z tO {extend(n) }. Define
= extend(l) and k < / . ga,,+l = {S[(S, a ) E Z', a E SL} extend(n) has to fulfill the following additional conditions in accordance with the inductive definition of the actual sets of statements • .... a ~ SL, and the actual sets of decision paths ~0.... a ~ DL: (a') #~,0 = ~ and £),,0 = ~ . (b') If extend(n) = (E, a), a ~ DL, E = (A, B, C, D ) , then B may only contain assignments i : H~ with Hi ~ t-J~SLg~,,. If extend(n) = (S, a ) , a ~ SL, then
U '$o,n U {S} to INVSEQ( U #,,,n, to ~0t~,,,, "
a<~
aESL
a<~
is consistent. If extend(n) = (E, a ) , a ~ DL, then U #t~,,, U INVSEQ( U ¢~,,n, U ~0t~,nU {E}, -
a
aESL
a<~8
is consistent. (c') Assume extend(n) = (S, a), a ~ SL, and
U d'¢,,, U INVSEQ( U g .... U .Ot~,n, ~(t) ~
a~fl~
aESL
a~p3
S
or extend(n) = (E, a), a ~ DL, and U d't~.nU INVSEQ( O ~,,,n, U Z)t~,n,-<,)
a~t~8
ot~SL
II
a~
INVSEQ( t0 ~ .... to ~0a,n tO {E}, -
a~r,O
Then #,,,,,+l = #,,,,,
and
~,,.n+l = ~),,,n.
Otherwise use the ordering "
~0,,,n}.
Now consider the subsets X of A, that have the following properties: (a") Let Yt = { S ' I ( S ' , a ' ) E X , a ' ~ S L } , Y2 = {E'I(E', a') E X , a ' ~ DL}. If extend(n) = (S, a ) , a SL, then Yi to {S} to INVSEQ(Y~, Y2, "
X'<* Y,
X, Y~_ An,
and
~O~,n+~ = { E l ( E , a ) E Z', a ~ DL}.
Explanation. A formal model of the evolutionary development of law must comprise ordered levels of statements and decisions (represented by
Logic-Based Models of Analogical Reasoning --This maximal subset determines the statement levels and decision levels at the following time stage. As the maximal consistent subset of decisions and statements determines the new status of the knowledge base and all other statements and decisions are cancelled, the oscillation of legal knowledge can be modelled, which is a main difference to usual nonmonotonic approaches (cf. Reiter, 1981 ).
375 1 is assumption of the court, 2 is a regulation ("If a contract is completed in a country then the courts of this country decide legal actions related to the contract."), 3 is a fact ("The completion of the acceptance of the contract between E and MFE was done in England by Telex."). The assumption is assumed to be independent from the parties, the local circumstances and the way of posting (E,MFE,GB,TELEX). The decision of the House of Lords takes the related form:
1 4 assumption ACCEPTANCE(x,y,z) •. COURT(x.y,z) 3 COMP(B, SS,GB,TELEX) •* COURT(B, SS,GB) •*COMP(B, SS, GB,TELEX) COURT(B, SS,GB) 5. EXAMPLES OF THE DEVELOPMENT OF LEGAL KNOWLEDGE
The following facts and regulations are used: F1: ~=~C O M P ( E , M F E , G B , T E L E X )
5.1. Acceptance of Contracts by Telex Transmission
Entores Ltd. v. Miles Far East Corporation ( 1955 ) (cf. Padfield, 1987). Plaintiffs in London made an offer by Telex to defendants in Amsterdam. The defendants accepted by Telex message transmitted to London. Later defendants were in breach of contract, and the plaintiffs wished to establish that the contract by Telex was made in London where the acceptance took place, in which case the legal action could be decided in English courts. Held: that the contract had been made in London, since the defendant's acceptance of plaintiff's offer was not complete until actually received by plaintiff. This Court of Appeal decision was upheld by the House of Lords in Brinkibon Ltd. v. Stahag Stahl (1982). A possible model of the relevant elements of the court hierarchy for formalization is: FACI~
I LAWS
I HOUSE OF LORDS DECISIONS (HOFLORDS)
I
F2: =* COMP(B,SS,GB,TELEX ) L: ACCEPTANCE (x,y,z) =~ COURT(x,y,z) The corresponding decision paths take the form E l =
((3,(1,2,(res),(1,0)),(res),(0,0)), 2 : L, 3 : Fi, 1 : { E,MFE,GB,TELEX }, COURT(E,MFE,GB) ),
//:2 = ( ( 5 , ( 1,4,(res),(1,0)),(res),(0,0)), 4 : L, 5 : F2, 1: {B,SS,GB,TELEX }, ==~COURT(B,SS,GB)). The inversion of El yields INV(EI) = {COMP(x,y,u,v) =* ACCEPTANCE(x,y,u) } ("A contract whose offer is accepted by Telex is accepted in the country where the Telex is sent to.") The evolutionary model (up to stage 5 ) is given by ( { FACTS,LAWS },{ CAPP,HOFLORDS }, <,,
COURT OF APPEAL DECISIONS (CAPP) To formalize the decisions we need the following constants and relations: GB, TELEX, E . . . B... ACCEPTANCE(x,y,z)... COURT(x,y,z)... COMP(x,y,u,v)...
with
Entores Ltd., M F E . . . Miles Far East Corporation, Brinkibon Ltd., S S . . . Stahag Stahl. the acceptance of a contract between x and y takes place in z courts in z decide the legal action related to the contract between x and y the contract between x and y was completed in u using v.
The decision of the Court of Appeal takes the form: 1 2 3 assumption ACCEPTANCE(x,y,z) =, COURT(x,¥,z) ~COMP(E,MFE,GB,TELEX) COMP(E,MFE,GB,TELEX) ~ COURT(E, MFE,GB) COURT(E, MFE,GB)
376
M. Baaz and G. Quirchmayr extend(0) = (L,LAWS), extend(l) = (FI,FACTS), extend(2) = (EI,CAPP), extend(3) = (F2,FACTS), extend(4) = (E2,HOFLORDS), ~0,FACrS ~--"~ , ~3,FACrS ~--"{FI},
~0,taws = Z~, ~3,LAWS = {L}, ~)0,CAPP ---~~ , ~)3,CAPP = {El}, ~0,HOFLORDS = ~ , ~)3,HOFLORDS = ~ ,
e~I,FACrS ~~4,FACrS = ~,,taws = ~Y4.LAWS= ~)I,CAPP '~4,CAPP
~, {F,,F2}, (L}, {L}, ---- ~ , ~"
{El},
¢~2,FACTS-~" {FI}, ~5.FACrS = {F, ,F2}, ~2.~ws = {L}, "PS,LAWS= {L}, ~2,CAPP ---- ~ , *~5,CAPP
=
{E,},
*~I,HOFLORDS ~- ~ ,
~)2,HOFLORDS----D ,
~94,HO~S
~gS,HOFtOROS= {E2}.
(Note that by INVSEQ no assumption of E2 is open but that the House of Lords decision transfers the assumption of El to a higher priority level.)
5.2. The Tort of Negligence A representative example of the evolutionary development of law are the cases dealing with the tort of negligence. It arises when damage is caused to the person or property of another by failure to take such care as the law requires in the circumstances of the case. For succeeding in an action, it is necessary to prove that: (a) the defendant owed a legal duty of care to the plaintiff; (b) the duty was broken; and (c) damage was suffered in consequence. As there are many different situations, where duty of care is owed, Lord Atldns' generalization by creating an abstract principle, the so-called "neighbour principle" (cf. Donoghue v. Stevenson, 1932), was the natural consequence. The perhaps best example of extensions o f this principle by law courts is the House o f Lords" decision in Hedley Byrne v. Heller & Partners (1963). In this case a firm of advertising agents relied upon a banker's reference and gave credit to a client.
= ~,
When the client became insolvent, the firm suffered financial loss. The reference had been given carelessly but, since the bank had expressly disclaimed liability when giving it, the action failed. (In addition it was noted that financial losses were not considered as damage in the sense of law.) Nevertheless, the House of Lords stated that, contrary to what had previously been believed, liability for negligence may extend to careless words as well as to careless deeds and that damages may be awarded for financial loss as well as for physical injury to persons and property. A possible model of the relevant elements of the court hierarchy for formalization is: FACTS
I LAWS
I HOUSE OF LORDS DECISIONS ( H O F L O R D S )
I H I G H C O U R T DECISIONS ( H C O U R T ) To formalize the decisions we need the following constants and relations:
Heller & Partners H B . . . Hedley Byrne A . • . the advice in question H . . . DAMAGE(x,y). • x suffers a damage (in the sense of law) in case y FINLOSS(x,y). • x suffers financial loss only in case y DUTYTOCARE(x,y,z). • x has the duty to care for y in case z CARELESS(x,y,z) . • concerning z x sets a careless action relative to y INFORMAL(x,y,z). • a careless advice z given to y by x is merely informal LIABLE(x,y,z) . • x is liable to y for consequences of z 1. The High Court decision rejecting the liability of Heller & Partners for two independent reasons is expressed by two formal decisions. 1.1.
2 INFORMAL(H~HB~A)
0 1 assumption -aCARELESS(x,y,z) =* ~LIABLE(x,y~z) INFORMAL(H~HB,A) =~ -aLIABLE(H~HB,A) -1LIABLE(H,HB,A)
0 is the assumption of the court, 1 is a law, 2 is a fact. The assumption is considered to be independent from {H,I-IB,A}, the concrete persons and advices.
Logic-Based Models of Analogical Reasoning
377
1.2.
5 =* FINLOSS(HB,A)
3 4 assumption 7DAMAGE(x,y,z) =, 7LIABLE(x,y,z) FINLOSS(HB,A) =. 7LIABLE(H~HB~A) =* -7LIABLE(H, HB,A)
3 is the assumption of the court, 4 is a law, 5 is a fact. The assumption is considered to be independent from {H,HB,A}, the concrete persons and advices. 2. The House of Lords' decision rejecting the High Court decision can be expressed as follows: 10 INFORMAL(H, HB~A)
6 assumption 9 =* DUTYTOCARE(H,x,A)
8
DUTYTOCARE(u,v,w), CARELESS(u,v,w),
DAMAGE(v,w) =*LIABLE(u,v,w) 7 ~ CARELESS(H,HB~A) CARELESS(H,v,A), DAMAGE(v,A) =* LIABLE(H,v,A) 11 assumption DAMAGE(HB~A) =* LIABLE(H~HB,A) ==, FINLOSS(HB,A) FINLOSS(HB,A) =* LIABLE(H,HB,A) =* LIABLE(H,HB,A) 6, 7 are the assumptions of the House of Lords, 8, 9, are laws, 10, 11 are facts. The assumptions are considered to be independent from { H , H B ,A }, the concrete persons and advices. To put together, the following facts and laws are used: Facts Laws and regulations
{ H , H B , A }, 7 : { H , H B , A }, =* L I A B L E ( H , H B , A ) ) INVSEQ( {El, F2, L~, L2, L3, L4 }, {Ell, El2 }, ~) { INFORMAL(x,y,z) ==, 7CARELESS(x,y,z),
F,: •* INFORMAL(H,HB,A) F2: =* FINLOSS(HB,A) Ll: DUTYTOCARE(u,v,w), CARELESS(u,v,w),DAMAGE(v,w) =, LIABLE(u,v,w) L2: CARELESS (x,y,z) =* 7LIABLE(x,y,z) L3: 7DAMAGE(x,y,z) =* 7LIABLE(x,y,z) (these sequents represent the regulations 1-3) L4: =* DUTYTOCARE(H,x,A)
The decision paths take the form E~t = ((2,(0,1,(res),(l,0)),(res),(0,0)),
FINLOSS(y,z) =. 7DAMAGE(x,y,z) } are the reconstructed assumptions of the High Court (Ell ¢ E12). ("An explicitly informally advice is not careless; .... A financial loss is no damage in the sense of law")
1 : L2, 2 : Fl, 0 : { H , H B , A }, =* -aLIABLE ( H , H B , A ) ) El2 = ((5,(3,4,(res),(l,0)),(res),(0,0)),
INV(E2) = { INFORMAL(x,y,z) •* CARELESS(x,y,z), FINLOSS(y,z)
==}DAMAGE(x,y,z) } 4 : L3, 5 : F2, 3 : { H , H B , A }, =* ~LIABLE (H, HB, A ) )
are the reconstructed opposite assumptions of the House of Lords. The evolutionary model (up to stage 9) is given by
E 2 -- ( ( 11,(7,((10,6,(res),(0,0)), < { FACTS,LAWS }, { H C O U R T , H O F L O R D S }, (9,8,(res),(0,0)),(res),(1,0)),(res),(0,0)),
(res),(0,0)), 8 : LI, 9 : L4, 10 : El, 1 1 : F2, 6 : with
extend(I) = (L2,LAWS), extend(3) = (L,LAWS),
378
M. Baaz and G. Quirchmayr
extend(5) = (F2,FACTS), extend(4) = (F,,FACTS), extend(6) = (E,,,HCOURT), extend(7) = (El2, HCOURT), extend(8) = (E2,HOFLORDS), ~'~0.FACTS= ~ , e~3.FACI'S = ~ , ~O6,FA~ = {F,,F2},
e~I.FACI'S : ~ ' ~'~4.FACTS- ~ ,
a_~2.FACTS = ~ , ~5.FACTS = {F,},
87,FACrS = {F,,F2}, g9,FACrS----{F,,F2},
gS,FACrS = {FbF2},
g0.LAWS = ~ , e~°3,LAIS m {L,,L2,L3} ~'°6,LAWS = {L,,L2,L3,L4}
~'~I,LAWS= {L,}, e~4,LAIs = {LI,L2,L3,L,,}, ~7,LAWS= {L,,L2,L3,L4}, ~9,LAWS = {LI,L2,L3,L4}, ~LHCOU~T = Z~, '~4,HCOURT = ~ , ~7.HCOURT = {Ell}, '~9,HCOURT = ~ , ~I,aO~I~ORDS = ~ ,
~00,.couaT = Z~, '~3,HCOURT : J~, ~6.HCOURT = ~ , ~0,HOFLORDS = ~ , '~3,HOFLORDS = ~ , '~6,HOFLORDS = ~ ,
~)2,HCOURT = ~ , ~5,HCOURT = ~ , ~D8.HCOUXT= {Ell,El2}, '~2,HOFLORDS = ~ , ~)5,HOFLORDS = ~ ,
~)7.HOFLORDS = ~ , ~9,HOFLORDS ~- {E2}.
'~8,HOFLORDS = ~ ,
6. IMPLEMENTATION As the approach inherits the problem ofundecidability of first order logic, some restrictions have to be made to make the model computable. Therefore, occurring contradictions can only be solved until a certain depth. This seems to be a nasty restriction but it corresponds to the reality of law court hierarchy, which is characterized by limited time (= limited depth of search). The higher the level of the law court is, the higher the complexity of the search can be. Also if the undecidability of first order logic is neglected, these restrictions have to be made, because in legal decision making time and space state are limited, too. 7. CONCLUSION As Reasoning by Analogy is the main aspect of legal reasoning, an approach which is trying to formalize the dynamic aspect of law has to be built around a formal model of Reasoning by Analogy. In addition, a model of the evolutionary development of law has to consider the continuous growth and subsequent reduction of legal knowledge which can be described as a sort of permanent oscillation of valid statements.
Brown v. Zurich Insurance, 1954. Entores Ltd. v. Miles Far East Corporation, 1955.
#8.LAWS "~" {LI,L2,L3,L4},
~)4,HOFLOm~S = ~ ,
(Note for the calculation that INVSEQ (g, {E },,~) is INV(E) and that E2 "refutes" E11 and El2 .) At stages 7 and 8 liability of Heller & Partners is denied, at stage 9 liability of Heller & Partners is confirmed.
CASES CITED
e~2.LAWS-----{L1,L2}, "-~5,LAWS= {L1,L2,L3,L4},
Brinkibon Ltd. v. Stahag Stahl, 1982. Donoghue v. Stevenson, 1932. Hedley Byrne v. Heller & Partners, 1963.
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