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 formalism and a case study in a real legal domain, we show a new generalization and a new re nement 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 compared with general propositional formulas, and not only with factors. 2) Re nement: 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 contributions the role of factors in case-based reasoning can be re ned and compound formulas based on factors can be taken into account in casebased reasoning.
Case-based reasoning, one of the main legal reasoning types, has been discussed
in the Arti cial Intelligence and Law community for years. It allows for a form
of analogical reasoning [
Case-based reasoning with factors has been formalized using many di erent
approaches. For instance, abductive logic programming[
In [
In [
In this section, we present the precedent model formalism and apply it to
precedent comparison in case-based reasoning (also shown in [
The formalism introduced in this paper uses a propositional logic language L generated from a set of propositional constants. We x language L. We write : for negation, ^ for conjunction, _ for disjunction, $ for equivalence, > for a tautology, and ? for a contradiction. The associated classical, deductive, monotonic consequence relation is denoted .
Precedents consist of factors and outcomes. As explained in the introduction, our use of factors is related to but di ers 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 factors 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 ' is \A kills B", then its negation :' 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 ! is \A is guilty", its negation :! is \A is not guilty".
Following existing work in case-based reasoning, a precedent is a logical consistent 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 situations need to be decided upon.
De nition 1 (Precedents). A precedent is a logically consistent conjunction of distinct factors and outcomes = '0 ^ '1 ^ : : : ^ 'm ^ !0 ^ !1 ^ : : : ^ !n 1, where m and n are non-negative integers. We say that '0; '1; :::; 'm are the factors of , !0; !1; :::; !n 1 are the outcomes of . If n = 0, then we say that is a situation with no outcomes, otherwise 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 !0 ^ : : : ^ !n 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 0; 1 2 L are two precedents. 0 = f1 ^f2 ^f3 ^o,
1 = 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 [
P is a set of precedents such that for all ; 0 2 P with 6= 0, ^ 0 ?; and is a total preorder over P .
As customary, the asymmetric part of is denoted >. The symmetric part of is denoted . Let , 0 be two precedents, 0 means is at least as preferred as 0; > 0 means is more preferred than 0; and 0 means is as preferred as 0.
Example 2. Following Example 1, we assume 0 is more preferred than 1. In a precedent model with only precedents 0 and 1, the preference relation of this model is 0 > 1. 2.2
Notions of comparing precedents in case-based reasoning include analogies, distinctions and relevances, they are related to general formulas, not only the factors or outcomes. Analogies between two precedents are the formulas that follow logically from both two precedents. Distinctions are the unshared formulas between two precedents, that only follow logically from one of the precedents and its negation is logically implied by the other precedent. Relevances are the unshared formulas between two precedents, that are relevant to the analogies and distinctions between them. These formulas only follow from one of the precedents, but both themselves and their negation are not logically implied by the other one.
De nition 3 (Analogies, distinctions and relevances). Let ; 0 2 L be two precedents, we de ne: 1. a sentence 2 L is an analogy between and 0 if and only if and 0 . A most speci c analogy between and 0 is an analogy that logically implies all analogies between and 0. 2. a sentence 2 L is a distinction in with respect to 0 ( - 0 distinction) if and only if and 0 : . A most speci c - 0 distinction is a distinction that logically implies all - 0 distinctions. 3. a sentence 2 L is a relevance in with respect to 0 ( - 0 relevance) if and only if , 0 6 and 0 6 : . is a proper - 0 relevance if and only if is a - 0 relevance that logically implies the most speci c analogy between and 0. A most speci c - 0 relevance is a relevance that logically implies all - 0 relevances.
Both - 0 distinctions and 0- distinctions are called the distinctions between and 0. Both - 0 relevances and 0- relevances are called the relevances between and 0. When a most speci c analogy/distinction/relevance exists it is by de nition unique, and we can refer to it as the most speci c analogy/distinction/relevance.
Example 3. Following Example 1, we have: { Analogies between 0 and 1: e.g., f1, f1 _ f2, f1 _ f3; { 0- 1 distinctions: e.g., f2, f2 ^ o, f2 _ :f1; { 0- 1 relevances: e.g., f3, f2 _ f3; { Proper 0- 1 relevances: e.g., f1 ^ f3 ^ (:o _ f2) ^ (:f2 _ o).
Now we further discuss notions in De nition 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 between precedents and 0, the sets of and 0 worlds are subsets of the set of Analogy - 0 distinction
- 0 relevance and 0 0 0
0 and 0 : ; 0 6 and 0 6 : worlds; for any - 0 distinction , the worlds are a subset of the worlds, while the 0 worlds and the worlds are disjoint; for any - 0 relevance , the worlds are a subset of the worlds, while the 0 worlds and the worlds are not subsets of each other and the intersection of the 0 worlds and the worlds are always not empty. The following proposition shows the properties of analogies, distinctions and relevances between precedents.
Proposition 1. Let ; 0 2 L be precedents. Then the following holds: 1. The most speci c analogy between and 0 always exists and is logically equivalent to _ 0; 2. There exists a distinction between and 0 if and only if ^ 0 ?; If a - 0 distinction exists, then the most speci c - 0 distinction exists and is logically equivalent to ; 3. The most speci c - 0 relevance does not always exist; 4. If the most speci c - 0 distinction exists, then the most speci c - 0 distinction logically implies each proper - 0 relevance. Each proper - 0 relevance logically implies the most speci c analogy between and 0.
Proof. Let ; 0 2 L be precedents. For Property 1, by De nition 3, for any analogy , and 0 . By propositional logic it follows that any analogy is logically implied by _ 0. By De nition 3, _ 0 is therefore a most speci c analogy.
For Property 2, assume a - 0 distinction exists. By De nition 3, and 0 : . It follows by propositional logic that ^ 0 ?. If ^ 0 ?, then by propositional logic 0 : . By De nition 3 and propositional logic, is therefore the most speci c - 0 distinction.
For Property 3, assume language L is generated from a set of propositional constants ff1; f2g. If = f1, 0 = :f1, the most speci c - 0 relevance does not exist. - 0 relevances like f1 _ f2, f1 _ :f2 cannot be logically implied by a unique - 0 relevance.
For Property 4, by Property 2 if the most speci c - 0 distinction exists, then it is logically equivalent to . By De nition 3, logically implies all - 0 relevances, including proper ones, and proper - 0 relevances always logically imply the most speci c analogy between and 0.
As shown in Proposition 1, _ 0 is always the most speci c analogy between and 0. In legal case-based reasoning, the most speci c analogy between two precedents we de ne here may seem to be counterintuitive. However, as the de nition of precedent comparison is based on propositional logic, _ 0 is the only sentence that can logically imply all the analogies between two precedents.
The most speci c - 0 distinction is logically equivalent to 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 de ne 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 speci c 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 speci c distinction (if it exists) and the most speci c analogy.
The following corollary shows a property of precedents in precedent models.
Corollary 1. Let (P; ) be a precedent model, for all
speci c - 0 distinction always exists.
and 0 2 P , the most
Proof. The corollary follows from De nition 2 and Property 2 of Proposition 1.
Two precedents can be compared with a third precedent using the analogy
relation de ned below, which is similar to what is called on-pointness in HYPO [
De nition 4 (Analogy relation between precedents). Let , 0 and 00 2 L be precedents. We de ne:
00 0 if and only if _ 00 0 _ 00.
Then we say is at least as analogous as 0 with respect to 00. As customary, the asymmetric part of the relation is denoted as 00 0, which means is more analogous than 0 with respect to 00. The symmetric part of the relation is denoted as 00 0, which means is as analogous as 0 with respect to 00. If it is not the case that 00 0 and 0 00 , then we say and 0 are analogously incomparable with respect to 00.
Example 4. Comparing 0 and 1 in Example 1 in terms of precedent 2 = f1 ^ f2, we have 0 2 1; If 2 = f1 ^ :f2, then we have 1 2 0; If 2 = f1, then we have 0 2 1; If 2 = :f3, then 0 and 1 are analogously incomparable with respect to 2.
Proposition 2. Let , 0 and 00 2 L be precedents. Then the following holds: 1. The analogy relation is re exive and transitive, hence a preorder; 2. 00 0 if and only if 0 _ 00; 3. If 00 0, then 0 00 and vice versa; 4. For any 2 L, if 00 0, and is an analogy between 0 and 00, then is also an analogy between and 00.
Proof. For property 1, the analogy relation is re exive, since _ 00 _ 00. The relation is also transitive because of the transitivity of entailment in propositional logic. Assume = f1 ^ f2, 0 = f1 ^ f3 and 00 = f1 ^ f2 ^ f3, and 0 are analogously incomparable with respect to 00, hence the relation is not in general total.
For Property 2, from left to right, by De nition 4 we obtain _ 00 0 _ 00, and by propositional logic 0 _ 00. From right to left, from 0 _ 00 and propositional logic, we obtain _ 00 _ 00, and by De nition 4 00 0.
Property 3 then follows directly from Property 2 by the commutativity of _.
Property 4 follows directly from De nition 3 and 4.
Notice that if 00 0, then it is still possible that 6 0 and 6 00. For instance, if = f1, 0 = f1 ^ f2, 00 = f1 ^ :f2, then we have 00 0, but both 0 and 00 are not logically implied by . Also notice that if 00 0, it cannot be concluded that 0. For instance, = f1 ^ f2, 0 = f3 and 00 = f1. In this example, 00 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 di erent from similar notions in HYPO (as dimensions) and CATO (as binary factors), which always favor a speci c outcome. 3.1
We assume that all factors in F favor a speci c outcome in O, and O = f!; :!g, where ! stands for an outcome that one of the parties in the court wins the claim, :! stands for the other party winning the claim. F = F! [ F:!, such that for all ' 2 F , if ' 2 F!, then :' 2 F:! and vice versa. F! stands for the factors that support outcome !, F:! stands for the factors against outcome !, namely supporting outcome :!.
In HYPO, shared factors between two cases are called relevant similarity, while the unshared factors are called relevant di erence. Unshared factors can be used for pointing out the two cases should be decided di erently. When comparing 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 ; 0 and 00 be precedents, ! 2 O be an outcome of 0. Notions related to HYPO-style comparison can be formalized by the precedent model formalism as follows: 1. Relevant similarity s 2 L is the relevant similarity between and 0 if and only if s is the conjunction of all the factors that are analogies between and 0. 2. Relevant di erence against ! in 0 d 2 L is the relevant di erence against ! in 0 between and 0 if and only if d is the conjunction of all the factors ' that are distinctions or relevances between and 0, such that: (a) if ' is a - 0 distinction or - 0 relevance, then ' 2 F:!; and (b) if ' is a 0- distinction or 0- relevance, then ' 2 F!. 3. On pointness Precedents is on point to 0 if and only if the relevant similarity between and 0 is not an empty conjunction. Assume s 2 L is the relevant similarity between precedents and 00, s0 2 L is the relevant similarity between precedents 0 and 00, is more on point than 0 with respect to 00 if and only if both and 0 are on point to 00 and s s0, s0 6 s.
Comparing with the notions de ned in De nition 3, the relevant similarity between precedents is always an analogy between them, since the conjunction of factors as the relevant similarity can be logically implied by both of them. However, the relevant di erence can not be simply equal to distinctions or relevances between precedents.
Notice that in on pointness, only factors are compared and there is no outcome involved in the comparison, however, in the analogy relation, both factors and outcomes are taken into account. Therefore, if precedent is more on point than precedent 0 with respect to precedent 00, then it is not always that 00 0. For instance, if = f1 ^ f2 ^ o, 0 = f1 ^ :o and 00 = f1 ^ f2 ^ f3, then it is obviously that is more on point than 0 with respect to 00. However, since 6 0 _ 00, is not more analogous than 0 with respect to 00, and 0 are analogously incomparable with respect to 00. 3.2
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 [
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 ! 2 O is logically equivalent to Pla. As shown in Figure 2, Yokana favors defendant (:Pla) and American Precision favors plainti (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 model American Precision
Yokana
Situation
Mason F7 ^ F16 ^ F21 ^ Pla
F7 ^ F10 ^ F16 ^ :Pla
F1 ^ F6 ^ F15 ^ F16 ^ F21 Factors supporting plainti 's claim: F6, F7, F15, F21
Factors supporting defendant's claim: F1, F10, F16 { The relevant similarity between Mason and Yokana: F16; { The relevant di erence 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 speci c 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 incompatible.
When comparing American Precision and Yokana in (P; P P ) in terms of Mason, 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 di erent comparison relations (preference relation/analogy relation/on pointness), the selection of better precedent can be di erent. Notions in HYPO can be well-de ned with the precedent model formalism. The analysis also shows that HYPO-style precedent comparison is di erent from the comparison in Section 2.2. We will further discuss these points in Section 4. 4
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 [
The new generalization of precedent comparison Case-based reasoning models following HYPO often discuss comparison between precedents in terms of 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 di erence between our comparison approach and
formalizations that follow HYPO [
As shown in Section 3.2, when comparing American Precision and Yokana in terms of the analogy relation de ned in De nition 4, the result of this comparison is di erent 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 speci c 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 [
The new re nement of precedent comparison In [
As shown in Section 3.1, the relevant similarity and the relevant di erence between cases in HYPO can be formalized by analogies, distinctions and relevances de ned in De nition 3. Although the relevant similarity is an analogy between precedents, factors in the relevant di erence are not always distinctions between precedents, but also can be relevances. In this sense, our approach compares precedents in a more speci c way than HYPO. However, as we haven't de ned dimension of factors in the formalism, it is unable to discuss magnitude of factors in relevant di erences, which means we cannot compare precedents in terms of dimensions, such as nding a contrary case which has some factors with extreme magnitude. This needs further discussion in the future.
This re nement 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 [