=Paper=
{{Paper
|id=Vol-2891/XAILA-2020_paper_3
|storemode=property
|title=Precedent Comparison in the Precedent Model Formalism: Theory and Application to Legal
Cases
|pdfUrl=https://ceur-ws.org/Vol-2891/XAILA-2020_paper_3.pdf
|volume=Vol-2891
|authors=Heng Zheng,Davide Grossi,Bart Verheij
|dblpUrl=https://dblp.org/rec/conf/jurix/ZhengGV20a
}}
==Precedent Comparison in the Precedent Model Formalism: Theory and Application to Legal
Cases==
https://ceur-ws.org/Vol-2891/XAILA-2020_paper_3.pdf
Precedent Comparison in
the Precedent Model Formalism:
Theory and Application to Legal Cases?
Heng Zheng1 , Davide Grossi1,2 , and Bart Verheij1
1
Bernoulli Institute, University of Groningen, The Netherlands
2
ILLC/ACLE, University of Amsterdam, The Netherlands
Abstract. Comparison between precedents and case facts is a core issue
in case-based reasoning, which has been discussed in a lot of research.
In this paper, we use a recently developed precedent model formalism
to discuss precedent comparison in case-based reasoning. With this for-
malism and a case study in a real legal domain, we show a new gener-
alization and a new refinement of precedent comparison with respect to
case-based reasoning approaches based on factors, such as HYPO and
CATO. 1) Generalization: precedents and case facts can now be com-
pared with general propositional formulas, and not only with factors. 2)
Refinement: a distinction can be made between current analogies and
distinctions in precedent models, and so-called relevances, i.e., unshared
formulas between two precedents that are relevant for possible additional
analogies and distinctions that can arise in a discussion. With these con-
tributions the role of factors in case-based reasoning can be refined and
compound formulas based on factors can be taken into account in case-
based reasoning.
Keywords: case-based reasoning· precedents· precedent comparison
1 Introduction
Case-based reasoning, one of the main legal reasoning types, has been discussed
in the Artificial Intelligence and Law community for years. It allows for a form
of analogical reasoning [1], and the core issue is how to make decisions for a
current case by comparing precedents, namely the doctrine of stare decisis. As
concluded by [1], there are three kinds of approaches for modeling case-based
reasoning: prototype and deformation [2]; dimension and legal factor [3–7]; and
exemplar-based explanation [8]. This paper follows the direction of factor-based
approaches.
Case-based reasoning with factors has been formalized using many different
approaches. For instance, abductive logic programming[9, 10], formal dialogue
?
Corresponding Author: Heng Zheng, University of Groningen, Nijenborgh 9, 9747
AG Groningen, The Netherlands; E-mail: h.zheng@rug.nl.
Copyright 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
2 H. Zheng et al.
games [11], context-related frameworks [12], dialectical arguments [13], ontolo-
gies in OWL [14], the ASPIC+ framework [15], reason models [16], abstract
argumentation [17], abstract dialectical frameworks [18] and case-based argu-
mentation frameworks [19]. These works often discuss precedent comparison in
terms of factors, following ideas developed in HYPO [3, 4].
In [20], a new formalism on modeling case-based reasoning has been dis-
cussed in a formal logical language. It can be used for evaluating the validity of
arguments in legal reasoning. These models are based on the case model formal-
ism [21], which has been implemented in a Prolog program [22]. The precedent
models we present in that paper represent precedents as conjunctions of factors
and outcomes. In our approach, factors are meant to represent generalized case
facts relevant to the outcome of the case decision. However, unlike in CATO
[6], our use of factors does not assume that factors favor a side of the decision,
either pro-plaintiff or pro-defendant, as such an assumption is not needed for
our logical definitions of precedent comparison. Unlike HYPO [4], our factors do
not come with a dimension that can express a magnitude.
In [20], we show that precedents can be compared through the preference re-
lation in precedent models, however, that paper only briefly mentions precedent
comparison in the form of case-based reasoning, which is the focus of the present
paper. In Section 2, we show the technical aspect of comparing precedents with
our formalism which can compare precedents not only in terms of shared factors,
but also in terms of other, more general shared propositional formulas, as also
presented in [23]. The present paper is an extension of [23], we continue our dis-
cussion by applying our approach to HYPO-style comparison in terms of factors
(Section 3.1), and discussing the approach in the context of a real legal domain
(Section 3.2). We generalize case-based reasoning by comparing precedents with
general formulas and refine case-based reasoning by introducing the new notion
of relevances (Section 2.2). In this way, we show that comparing precedents with
respect to general properties, represented by general propositional formulas, of-
fers a novel angle on case-based reasoning.
2 The precedent model formalism
In this section, we present the precedent model formalism and apply it to prece-
dent comparison in case-based reasoning (also shown in [23]). The precedent
models defined in Section 2.1 are based on the case models formalism addressed
by Verheij [21]. Precedent comparison is based on the analogies and distinctions
defined in the case models [21].
2.1 Precedents and precedent models
The formalism introduced in this paper uses a propositional logic language L
generated from a set of propositional constants. We fix language L. We write ¬
for negation, ∧ for conjunction, ∨ for disjunction, ↔ for equivalence, > for a tau-
tology, and ⊥ for a contradiction. The associated classical, deductive, monotonic
consequence relation is denoted .
Precedent Comparison in the Precedent Model Formalism 3
Precedents consist of factors and outcomes. As explained in the introduc-
tion, our use of factors is related to but differs from other uses of factors in
the literature, and we do not assume that factors favor a side since that is not
necessary for our focus of logical precedent comparison. We consider both fac-
tors and outcomes are literals. A literal is either a propositional constant or its
negation. We use F ⊆ L to represent a set of factors, O ⊆ L to represent a
set of outcomes. The sets F and O are disjoint and consist only of literals. If
a propositional constant p is in F (or O), then ¬p is also in F (respectively in
O). A factor represents an element of a case, namely a factual circumstance. Its
negation describes the opposite fact. For instance, if a factor ϕ is “A kills B”,
then its negation ¬ϕ is “A does not kill B”. An outcome always favors a side in
the precedent, its negation favors the opposite side. For instance, an outcome ω
is “A is guilty”, its negation ¬ω is “A is not guilty”.
Following existing work in case-based reasoning, a precedent is a logical con-
sistent conjunction of factors and outcomes. If a precedent contains an outcome,
then we say it is a proper precedent. If a precedent doesn’t have any outcome,
then it is a situation that describes a current case. The outcomes of these situ-
ations need to be decided upon.
Definition 1 (Precedents). A precedent is a logically consistent conjunction
of distinct factors and outcomes π = ϕ0 ∧ ϕ1 ∧ . . . ∧ ϕm ∧ ω0 ∧ ω1 ∧ . . . ∧ ωn−1 ,
where m and n are non-negative integers. We say that ϕ0 , ϕ1 , ..., ϕm are the
factors of π, ω0 , ω1 , ..., ωn−1 are the outcomes of π. If n = 0, then we say that
π is a situation with no outcomes, otherwise π is a proper precedent.
Notice that both m and n can be equal to 0. When m = 0, there is one single
factor. When n = 0, the precedent has no outcome and the empty conjunction
ω0 ∧ . . . ∧ ωn−1 is equivalent to >. We do not assume precedents are complete
descriptions. That is, factors may exist which do not occur in the precedent.
Furthermore, we do not assume that the negation of a factor holds when the
factor does not occur in the precedent.
Example 1. Assume sentences π0 , π1 ∈ L are two precedents. π0 = f1 ∧f2 ∧f3 ∧o,
π1 = f1 ∧ ¬f2 ∧ ¬o. f1 , f2 , ¬f2 and f3 are factors, o and ¬o are outcomes.
Precedents can be compared through the preference relation between precedents
in precedent models, which has been discussed in [20]. A precedent model is a
set of logically incompatible precedents forming a total preorder representing a
preference relation among the precedents.
Definition 2 (Precedent models). A precedent model is a pair (P, ≥) where
P is a set of precedents such that for all π, π 0 ∈ P with π 6= π 0 , π ∧ π 0 ⊥; and
≥ is a total preorder over P .
As customary, the asymmetric part of ≥ is denoted >. The symmetric part of
≥ is denoted ∼. Let π, π 0 be two precedents, π ≥ π 0 means π is at least as
preferred as π 0 ; π > π 0 means π is more preferred than π 0 ; and π ∼ π 0 means π
is as preferred as π 0 .
4 H. Zheng et al.
Example 2. Following Example 1, we assume π0 is more preferred than π1 . In a
precedent model with only precedents π0 and π1 , the preference relation of this
model is π0 > π1 .
2.2 Comparing precedents in the formalism
Notions of comparing precedents in case-based reasoning include analogies, dis-
tinctions and relevances, they are related to general formulas, not only the fac-
tors or outcomes. Analogies between two precedents are the formulas that follow
logically from both two precedents. Distinctions are the unshared formulas be-
tween two precedents, that only follow logically from one of the precedents and
its negation is logically implied by the other precedent. Relevances are the un-
shared formulas between two precedents, that are relevant to the analogies and
distinctions between them. These formulas only follow from one of the prece-
dents, but both themselves and their negation are not logically implied by the
other one.
Definition 3 (Analogies, distinctions and relevances). Let π, π 0 ∈ L be
two precedents, we define:
1. a sentence α ∈ L is an analogy between π and π 0 if and only if π α and
π 0 α. A most specific analogy between π and π 0 is an analogy that logically
implies all analogies between π and π 0 .
2. a sentence δ ∈ L is a distinction in π with respect to π 0 (π-π 0 distinction) if
and only if π δ and π 0 ¬δ. A most specific π-π 0 distinction is a distinction
that logically implies all π-π 0 distinctions.
3. a sentence ρ ∈ L is a relevance in π with respect to π 0 (π-π 0 relevance) if
and only if π ρ, π 0 6 ρ and π 0 6 ¬ρ. ρ is a proper π-π 0 relevance if and
only if ρ is a π-π 0 relevance that logically implies the most specific analogy
between π and π 0 . A most specific π-π 0 relevance is a relevance that logically
implies all π-π 0 relevances.
Both π-π 0 distinctions and π 0 -π distinctions are called the distinctions between
π and π 0 . Both π-π 0 relevances and π 0 -π relevances are called the relevances be-
tween π and π 0 . When a most specific analogy/distinction/relevance exists it is
by definition unique, and we can refer to it as the most specific analogy/distinc-
tion/relevance.
Example 3. Following Example 1, we have:
– Analogies between π0 and π1 : e.g., f1 , f1 ∨ f2 , f1 ∨ f3 ;
– π0 -π1 distinctions: e.g., f2 , f2 ∧ o, f2 ∨ ¬f1 ;
– π0 -π1 relevances: e.g., f3 , f2 ∨ f3 ;
– Proper π0 -π1 relevances: e.g., f1 ∧ f3 ∧ (¬o ∨ f2 ) ∧ (¬f2 ∨ o).
Now we further discuss notions in Definition 3. Figure 1 illustrates analogies,
distinctions and relevances using Venn diagrams representing sets of worlds in
which sentences are true. As shown in Figure 1, for any analogy α between
precedents π and π 0 , the sets of π and π 0 worlds are subsets of the set of α
Precedent Comparison in the Precedent Model Formalism 5
Analogy π-π 0 distinction π-π 0 relevance
α δ ρ
π π0 π π0 π π0
π α and π 0 α π δ and π 0 ¬δ π ρ, π 0 6 ρ and π 0 6 ¬ρ
Fig. 1. Precedent comparison illustrated by worlds in which sentences are true
worlds; for any π-π 0 distinction δ, the π worlds are a subset of the δ worlds,
while the π 0 worlds and the δ worlds are disjoint; for any π-π 0 relevance ρ, the π
worlds are a subset of the ρ worlds, while the π 0 worlds and the ρ worlds are not
subsets of each other and the intersection of the π 0 worlds and the ρ worlds are
always not empty. The following proposition shows the properties of analogies,
distinctions and relevances between precedents.
Proposition 1. Let π, π 0 ∈ L be precedents. Then the following holds:
1. The most specific analogy between π and π 0 always exists and is logically
equivalent to π ∨ π 0 ;
2. There exists a distinction between π and π 0 if and only if π ∧ π 0 ⊥; If a
π-π 0 distinction exists, then the most specific π-π 0 distinction exists and is
logically equivalent to π;
3. The most specific π-π 0 relevance does not always exist;
4. If the most specific π-π 0 distinction exists, then the most specific π-π 0 distinc-
tion logically implies each proper π-π 0 relevance. Each proper π-π 0 relevance
logically implies the most specific analogy between π and π 0 .
Proof. Let π, π 0 ∈ L be precedents. For Property 1, by Definition 3, for any
analogy α, π α and π 0 α. By propositional logic it follows that any analogy
α is logically implied by π ∨ π 0 . By Definition 3, π ∨ π 0 is therefore a most specific
analogy.
For Property 2, assume a π-π 0 distinction δ exists. By Definition 3, π δ
and π 0 ¬δ. It follows by propositional logic that π ∧ π 0 ⊥. If π ∧ π 0 ⊥,
then by propositional logic π 0 ¬π. By Definition 3 and propositional logic, π
is therefore the most specific π-π 0 distinction.
For Property 3, assume language L is generated from a set of propositional
constants {f1 , f2 }. If π = f1 , π 0 = ¬f1 , the most specific π-π 0 relevance does
not exist. π-π 0 relevances like f1 ∨ f2 , f1 ∨ ¬f2 cannot be logically implied by a
unique π-π 0 relevance.
For Property 4, by Property 2 if the most specific π-π 0 distinction exists,
then it is logically equivalent to π. By Definition 3, π logically implies all π-π 0
relevances, including proper ones, and proper π-π 0 relevances always logically
imply the most specific analogy between π and π 0 .
As shown in Proposition 1, π ∨ π 0 is always the most specific analogy between
π and π 0 . In legal case-based reasoning, the most specific analogy between two
precedents we define here may seem to be counterintuitive. However, as the
6 H. Zheng et al.
definition of precedent comparison is based on propositional logic, π ∨ π 0 is the
only sentence that can logically imply all the analogies between two precedents.
The most specific π-π 0 distinction is logically equivalent to π if it exists.
From the perspective of case-based reasoning with factors, such as HYPO, it is
odd that the precedent itself is a distinction, but notice that here we represent
precedents as conjunctions of factors and outcomes, hence precedents themselves
can be considered as a formula. Since precedent comparison we define in this
section aims at comparing all formulas in precedents, in this sense, we can say
the precedent itself, as a propositional formula, is the most specific distinction.
Property 4 in Proposition 1 shows why we have singled out proper relevances:
in the formally precise sense of the proposition, they are logically ‘in between’
the most specific distinction (if it exists) and the most specific analogy.
The following corollary shows a property of precedents in precedent models.
Corollary 1. Let (P, ≥) be a precedent model, for all π and π 0 ∈ P , the most
specific π-π 0 distinction always exists.
Proof. The corollary follows from Definition 2 and Property 2 of Proposition 1.
Two precedents can be compared with a third precedent using the analogy rela-
tion defined below, which is similar to what is called on-pointness in HYPO [4].
The analogy relation is based on the shared formulas between precedents. When
comparing precedents π and π 0 in terms of precedent π 00 , if the most specific
analogy between π and π 00 logically implies the most specific analogy between
π 0 and π 00 , then we say that π is at least as analogous as π 0 with respect to π 00 .
Formally we define the analogy relation as follows.
Definition 4 (Analogy relation between precedents). Let π, π 0 and π 00 ∈
L be precedents. We define:
π π00 π 0 if and only if π ∨ π 00 π 0 ∨ π 00 .
Then we say π is at least as analogous as π 0 with respect to π 00 .
As customary, the asymmetric part of the relation is denoted as π π00 π 0 , which
means π is more analogous than π 0 with respect to π 00 . The symmetric part of
the relation is denoted as π ∼π00 π 0 , which means π is as analogous as π 0 with
respect to π 00 . If it is not the case that π π00 π 0 and π 0 π00 π, then we say π
and π 0 are analogously incomparable with respect to π 00 .
Example 4. Comparing π0 and π1 in Example 1 in terms of precedent π2 =
f1 ∧ f2 , we have π0 π2 π1 ; If π2 = f1 ∧ ¬f2 , then we have π1 π2 π0 ; If
π2 = f1 , then we have π0 ∼π2 π1 ; If π2 = ¬f3 , then π0 and π1 are analogously
incomparable with respect to π2 .
Proposition 2. Let π, π 0 and π 00 ∈ L be precedents. Then the following holds:
1. The analogy relation is reflexive and transitive, hence a preorder;
2. π π00 π 0 if and only if π π 0 ∨ π 00 ;
3. If π π00 π 0 , then π π0 π 00 and vice versa;
Precedent Comparison in the Precedent Model Formalism 7
4. For any α ∈ L, if π π00 π 0 , and α is an analogy between π 0 and π 00 , then α
is also an analogy between π and π 00 .
Proof. For property 1, the analogy relation is reflexive, since π ∨π 00 π ∨π 00 . The
relation is also transitive because of the transitivity of entailment in propositional
logic. Assume π = f1 ∧ f2 , π 0 = f1 ∧ f3 and π 00 = f1 ∧ f2 ∧ f3 , π and π 0 are
analogously incomparable with respect to π 00 , hence the relation is not in general
total.
For Property 2, from left to right, by Definition 4 we obtain π ∨ π 00 π 0 ∨ π 00 ,
and by propositional logic π π 0 ∨ π 00 . From right to left, from π π 0 ∨ π 00 and
propositional logic, we obtain π ∨ π 00 π ∨ π 00 , and by Definition 4 π π00 π 0 .
Property 3 then follows directly from Property 2 by the commutativity of ∨.
Property 4 follows directly from Definition 3 and 4.
Notice that if π π00 π 0 , then it is still possible that π 6 π 0 and π 6 π 00 . For
instance, if π = f1 , π 0 = f1 ∧ f2 , π 00 = f1 ∧ ¬f2 , then we have π π00 π 0 , but both
π 0 and π 00 are not logically implied by π. Also notice that if π π00 π 0 , it cannot
be concluded that π π 0 . For instance, π = f1 ∧ f2 , π 0 = f3 and π 00 = f1 . In this
example, π π00 π 0 but f1 ∧ f2 6 f3 .
3 Application: HYPO in the precedent model formalism
In this section, we formalize notions of comparison in HYPO with the precedent
model formalism, and discuss them by a case study in a real legal domain. Factors
in our approach are different from similar notions in HYPO (as dimensions) and
CATO (as binary factors), which always favor a specific outcome.
3.1 Formalizing notions of comparison in HYPO
We assume that all factors in F favor a specific outcome in O, and O = {ω, ¬ω},
where ω stands for an outcome that one of the parties in the court wins the
claim, ¬ω stands for the other party winning the claim. F = Fω ∪ F¬ω , such
that for all ϕ ∈ F , if ϕ ∈ Fω , then ¬ϕ ∈ F¬ω and vice versa. Fω stands for the
factors that support outcome ω, F¬ω stands for the factors against outcome ω,
namely supporting outcome ¬ω.
In HYPO, shared factors between two cases are called relevant similarity,
while the unshared factors are called relevant difference. Unshared factors can
be used for pointing out the two cases should be decided differently. When com-
paring two precedents in terms of a current situation, HYPO always makes sure
that the precedents are on point to the situation, namely they share at least one
factor. If a precedent shares more factors with the situation than a second one,
then this precedent is more on point than second with respect to the situation.
Let π, π 0 and π 00 be precedents, ω ∈ O be an outcome of π 0 . Notions related
to HYPO-style comparison can be formalized by the precedent model formalism
as follows:
8 H. Zheng et al.
1. Relevant similarity ψs ∈ L is the relevant similarity between π and π 0 if and
only if ψs is the conjunction of all the factors that are analogies between π
and π 0 .
2. Relevant difference against ω in π 0 ψd ∈ L is the relevant difference against
ω in π 0 between π and π 0 if and only if ψd is the conjunction of all the factors
ϕ that are distinctions or relevances between π and π 0 , such that:
(a) if ϕ is a π-π 0 distinction or π-π 0 relevance, then ϕ ∈ F¬ω ; and
(b) if ϕ is a π 0 -π distinction or π 0 -π relevance, then ϕ ∈ Fω .
3. On pointness Precedents π is on point to π 0 if and only if the relevant
similarity between π and π 0 is not an empty conjunction. Assume ψs ∈ L is
the relevant similarity between precedents π and π 00 , ψs0 ∈ L is the relevant
similarity between precedents π 0 and π 00 , π is more on point than π 0 with
respect to π 00 if and only if both π and π 0 are on point to π 00 and ψs ψs0 ,
ψs0 6 ψs .
Comparing with the notions defined in Definition 3, the relevant similarity be-
tween precedents is always an analogy between them, since the conjunction of
factors as the relevant similarity can be logically implied by both of them. How-
ever, the relevant difference can not be simply equal to distinctions or relevances
between precedents.
Notice that in on pointness, only factors are compared and there is no out-
come involved in the comparison, however, in the analogy relation, both fac-
tors and outcomes are taken into account. Therefore, if precedent π is more on
point than precedent π 0 with respect to precedent π 00 , then it is not always that
π π00 π 0 . For instance, if π = f1 ∧ f2 ∧ o, π 0 = f1 ∧ ¬o and π 00 = f1 ∧ f2 ∧ f3 ,
then it is obviously that π is more on point than π 0 with respect to π 00 . However,
since π 6 π 0 ∨ π 00 , π is not more analogous than π 0 with respect to π 00 , π and π 0
are analogously incomparable with respect to π 00 .
3.2 Case study
In this section, we use an example from a real legal domain to discuss precedent
comparison introduced in Section 2.2 with notions of comparison in HYPO and
comparison with preference relation in precedent models. This example has also
been discussed in [1, 20].
Let (P, P × P ) be a precedent model containing two precedents (the Yokana
case3 and the American Precision case4 ). Notice that P × P denotes the trivial
preference relation where all precedents are as preferred as each other. The
current situation is adapted from the Mason case5 . We suppose outcome ω ∈ O
is logically equivalent to Pla. As shown in Figure 2, Yokana favors defendant
(¬Pla) and American Precision favors plaintiff (Pla).
When comparing Mason with Yokana in (P, P × P ) according to the notions
in HYPO, we have:
3
Midland-Ross Corp. v. Yokana, 293 F.2d 411 (3rd Cir.1961)
4
American Precision Vibrator Co. v. National Air Vibrator Co., 764 S.W.2d 274
(Tex.App.-Houston [1st Dist.] 1988)
5
Mason v. Jack Daniel Distillery, 518 So.2d 130 (Ala.Civ.App.1987)
Precedent Comparison in the Precedent Model Formalism 9
Precedent model Situation
Mason
American Precision Yokana
F7 ∧ F16 ∧ F21 ∧ Pla F7 ∧ F10 ∧ F16 ∧ ¬Pla F1 ∧ F6 ∧ F15 ∧ F16 ∧ F21
Factors supporting plaintiff’s claim: F6, F7, F15, F21 Factors supporting defendant’s claim: F1, F10, F16
Fig. 2. Precedent model for the Mason case
– The relevant similarity between Mason and Yokana: F16;
– The relevant difference between Mason and Yokana against ¬Pla in Yokana:
F6 ∧ F15 ∧ F21 ∧ F10.
When comparing Mason with Yokana through the notions in Section 2.2:
– Analogies between Yokana and Mason: e.g., F16, F16 ∨ F21;
– The most specific analogy between Yokana and Mason: (F7 ∧ F10 ∧ F16 ∧
¬Pla) ∨ (F1 ∧ F6 ∧ F15 ∧ F16 ∧ F21).
– Mason-Yokana relevances: e.g., F6 ∧ F15 ∧ F21, F1 ∧ F21;
– Yokana-Mason relevances: e.g., F10, F16 ∧ ¬Pla;
– There is no distinction between Mason and Yokana, as they are not incom-
patible.
When comparing American Precision and Yokana in (P, P × P ) in terms of Ma-
son, American Precision is more on point than Yokana with respect to Mason,
however, American Precision and Yokana are analogously incomparable with
respect to Mason.
According to different comparison relations (preference relation/analogy re-
lation/on pointness), the selection of better precedent can be different. Notions
in HYPO can be well-defined with the precedent model formalism. The analysis
also shows that HYPO-style precedent comparison is different from the compar-
ison in Section 2.2. We will further discuss these points in Section 4.
4 Discussion and conclusion
In this paper, we discuss precedent comparison in case-based reasoning with the
precedent model formalism, which is described in a formal propositional logic
language. Unlike other case-based reasoning models, in which precedents are
represented as dimensions [4], sets of rules [11], sets of factors [15], combinations
of rules, facts and outcome [16] and hierarchies [6, 18]. The formalism we present
here represents precedents and current situations with conjunctions of factors
and outcomes. Comparing with the case model formalism in [21], we give a more
concrete account of precedents. The representation we use here allows us to
discuss case-based reasoning from a perspective that is closer to logic, thereby
allowing the comparison of precedents in terms of general formulas. In this way,
we are able to present a new generalization and a new refinement of precedent
comparison in case-based reasoning with the precedent model formalism.
The new generalization of precedent comparison Case-based reasoning mod-
els following HYPO often discuss comparison between precedents in terms of
10 H. Zheng et al.
factors. In the formalism we present here, we generalize the comparison approach
in case-based reasoning, namely comparing precedents not only with factors, but
also with more general propositional formulas.
Section 3.1 shows a key difference between our comparison approach and
formalizations that follow HYPO [4, 6, 16]. Factors in HYPO-style comparison
typically favor a side in the court case, showing which factors can strengthen or
weaken the arguments given by the parties involved, hence constraining possible
argument moves. However, the more general formulas used in our comparison
approach may not favor a specific side in the court (as shown in Section 2). The
formulas we discuss are more general logical expressions than factors.
As shown in Section 3.2, when comparing American Precision and Yokana in
terms of the analogy relation defined in Definition 4, the result of this comparison
is different from other comparison relations, namely the preference relation in
precedent models and the on pointness in HYPO. According to the preference
relation in (P, P × P ), American Precision and Yokana are as preferred as each
other; according to the on pointness, American Precision is more on point with
respect to Mason; and according to the analogy relation, these two precedents are
analogously incomparable with respect to Mason. This is because the analogy
relation discusses comparison in terms of the most specific analogy between
precedents, while other comparison relations are in terms of other notions.
The new comparison approach allows us to discuss general formulas beyond
factors in case-based reasoning, such as conjunctions or disjunctions of factors,
which can bring new discussion on case-based reasoning with the precedent
model formalism. For instance, for future research we can discuss hierarchical
factors shown in CATO [6], as higher level factors can be represented with com-
pound formulas based on base-level factors. Therefore, it seems possible to com-
pare abstract factors between precedents directly, and discuss argument moves
like downplaying a distinction in the formalism.
The new refinement of precedent comparison In [20], we don’t distinguish
the distinctions and relevances between precedent, nor do HYPO [4] and other
case-based reasoning models [6, 15]. In the formalism we present here, relevances
between precedents are distinguished from analogies and from distinctions. While
analogies between two precedents refer to formulas that hold in both precedents,
and distinctions to formulas that hold in one precedent and are negated in the
other, relevances are formulas that are not yet determined in a precedent and
hence have the potential to turn out as an analogy or distinction once deter-
mined. Although both distinctions and relevances are related to the unshared
factors, these formulas cannot be considered as distinctions directly, since if
such relevant formulas in a precedent can be found in a situation, they will be
considered as analogies rather than distinctions between the precedent and the
situation.
As shown in Section 3.1, the relevant similarity and the relevant difference
between cases in HYPO can be formalized by analogies, distinctions and rele-
vances defined in Definition 3. Although the relevant similarity is an analogy
between precedents, factors in the relevant difference are not always distinctions
Precedent Comparison in the Precedent Model Formalism 11
between precedents, but also can be relevances. In this sense, our approach com-
pares precedents in a more specific way than HYPO. However, as we haven’t
defined dimension of factors in the formalism, it is unable to discuss magnitude
of factors in relevant differences, which means we cannot compare precedents in
terms of dimensions, such as finding a contrary case which has some factors with
extreme magnitude. This needs further discussion in the future.
This refinement also points to case-based reasoning in a dynamic scenario,
in which the situation can change accordingly when new facts are found. For
instance, in the example shown in Figure 2, when comparing Yokana with Mason,
F1 ∧ F21 is a relevance between them. If these two factors can be found in
Yokana, then it will be more on point than American Precision with respect
to Mason. However, if the ¬F21 is implied by Yokana, then F21 and ¬F21 will
become the distinctions between these two cases.
Continuing from the preliminary report [20] and the technical note [23], in
this paper, we applied the approach of precedent comparison to a real legal do-
main, and discuss it with the case comparison in HYPO [4]. With the precedent
model formalism, we provide a way that both generalizes and refines case-based
reasoning with factors. We discuss not only the shared factors between prece-
dents, but also other compound formulas based on factors, which allows us to
compare precedents from a more logical perspective and discuss other features
among precedents. In this way, we show a new generalization of case-based rea-
soning with factors. We further distinguish the unshared formulas as distinctions
and as relevances, i.e., unshared formulas between two precedents that are rele-
vant for the analogies and distinctions that can arise in a discussion. In this way,
we show a new refinement of case-based reasoning with factors. These ideas show
that the precedent model formalism has the potential to help analyze argument
moves in case-based reasoning and support the selection of good precedents to
cite in a court discussion, these still need further research in the future. It would
also be interesting to investigate computational mechanisms to discover good
argument moves in a legal discussion using precedent models.
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