Abstract: In this paper we build on two recent attempts to formalise reasoning with dimensions. Both of these approaches effectively map dimensions into factors, which enables propositional reasoning, but we show that sometimes a balance between dimensions needs to be struck. To permit trade-offs we need to keep the magnitudes and reason more geometrically. We also revisit the link between dimensions and values, arguing that values play a number of distinct roles, not only explaining preferences between factors, but also ensuring that all the purposes of the underlying law are considered.
1.Introduction
One of the major tasks addressed by AI and Law over the last three decades or so has been to come to a good understanding of the representation of, and reasoning with, legal cases. This has also been central to the work of Professor Yoshino. In 1984 he organised the Legal Expert System Association (LESA) in Japan [36], and began to develop legal expert systems, (LES and LES-2 [35] in the domain of contract law). LES-2 was largely within the logic programming paradigm of [30]. Further developments included the language CPF (Compound Predicate Formula) [32], specifically to represent legal knowledge, and methods to allow meta inference [33]. Professor Yoshino continued to work on contracts, especially in the context of the Convention on the International Sale of Goods (he organised a series of workshops on this topic collocated with ICAIL in the 90s) and he came to recognise the need for case law, and the need to use fuzziness to handle this [20]. This paper describes our own current work on this topic, in particular how best to handle the problem of non-Boolean features of cases, necessary to capture certain kinds of reasoning with legal cases.
Our work is developed within the tradition of HYPO and its successors, a line of research described in [10], which has only recently started to again pay serious attention to non-Boolean features. Although such features were present as dimensions in HYPO, they have (with occasional exceptions such as [12]) tended to be simplified to Boolean factors in the drive to obtain a good understanding of the logic of precedential reasoning. Recently, however, as described below, interest in such features has undergone a significant revival.
As told in [10], the story begins with the dimension-based HYPO [7] and moves through the factor-based CATO [5], to expression as a set of rules [25] enabling a formalisation of factor-based reasoning by Horty in [22], which was refined by Rigoni in [27]. Factors can be seen as stereotypical patterns of facts, either present or absent in a case, and, if present, favouring either the plaintiff or the defendant. Dimensions, in contrast, are ranges of values (either numeric or enumerated), running from an extreme pro-plaintiff point to an extreme pro-defendant point. The applicability of dimensions to a case, and the point at which the case lies, is determined by the case facts, and the dimension may favour either party. The relationship between dimensions and factors is discussed in detail in [29]. Since the mid-90s, factor-based representations have been the main focus, and, although dimensions have always had their advocates [12], it is only more recently that dimensions have been revived as a way of connecting factors to the facts, and providing a way of capturing additional nuance (e.g. [26], [2] and [4]). More recently Horty has attempted to extend his formalism to accommodate dimensions [21], and in [28] Rigoni has critiqued this approach from the standpoint developed in [27]. In parallel with these developments, there has been exploration of the relationship between case law decisions and the purposes, or social values, they promote. The idea of its associated purpose as a measure of strength of a dimension was introduced in [14] and most fully expressed (in terms of values) in [13]. Recent discussions of the role of values, considering dimensions as well as factors, can be found in [1] and [2].
In this paper we will consider the role of values and their relation to dimensions in the light of [21] and [28]. After some background, we will consider how to argue with dimensions in legal CBR, and the role of values. We do so using the domain knowledge represented as an Abstract Dialectical Framework (ADF) [16] as described in [3]. In particular we will show that:
– Any legal CBR problem can be reduced to a series of steps involving at most two dimensions, so that higher dimensional spaces need not be considered;
– The non-leaf nodes of the ADF can be seen as being one of five kinds, as determined by the types of their children;
– For some nodes, dimensions cannot be reduced to factors and need to retain their magnitude, to permit trade-offs;
– Values are required to play several different roles, not just the expression of preferences between rules as in [13].
2.Formalising Factors and Dimensions
The formalisations of factor-based reasoning of both Horty and Rigoni are based on the method of expressing precedents as rules found in [25]. In that paper a case is considered to be a triple , where is the set of all pro-plaintiff factors present in the case, is the set of all pro-defendant factors present in the case and is the outcome, either plaintiff () or defendant (). Now will be the strongest reason to find for the plaintiff and will be the strongest reason to find for the defendant. We can therefore deduce that either or depending on the value of . These preferences permit only reasoning. Although [25] has a notion of as a dialogue move, a key insight of Horty is that may be stronger than is required and some subset of may be sufficient to defeat . Horty does not specify how exactly the subset is determined, but it could be interpreted as the of the case. In general the use of gives rise to what Horty terms the or model and attributes to Alexander [6], and the subset what he terms the model, and attributes to Lamond [23].
Figure 1. Dimension after 2 cases
Horty’s key idea in [21] is that dimensions can be mapped into factors, with the position on the dimension occupied by a particular case determining (through precedent cases) whether the corresponding factors are present or absent in that case. Note that the point at which the factor becomes present may not be the point on the dimension in the case facts of the precedent, which would be the result model. Instead the factor might become present at a point weaker for the side it favours so including the precedent and some other future cases, giving the reason model. In his example (taken from [25]), a person is attempting to show a change of fiscal domicile on the basis of absence from his home country. In the example a person is absent abroad for 36 months and change is found on the grounds that the absence is longer than a year. Thus the factor is present on the results model if the absence is at least 36 months and on the reason model if the absence is greater than a year. On his approach, however, Horty finds that, when using magnitudes, the result and the reason models collapse into one. Moreover, the reason does not constrain. Horty considers a second case in which the person has been absent for 18 months, but the court wishes to apply a stricter standard and rule against change on the grounds that it was less than two years. Horty wishes to say that the court can consistently decide in this way, although it cannot offer as a reason that the absence was below a threshold greater than three years, which would contradict the of the precedent case. Rigoni objects in [28] both to the collapse of the two models, and to allowing the court to decide that 18 months is not enough for change, which he claims is counter intuitive, given the reason expressed in the precedent. Essentially Rigoni is happy to adopt the reason given in the decision, and disregard the particular instantiation using the facts of the case in which the reason was stated. That might be expressed as Rigoni treating the reason as , whereas Horty treats it as .
The dimension and the two cases in the example are shown in Figure 1. There are a number of points of interest within this dimension. One set is the positions occupied by the precedent cases. Another set is the positions used to express reasons (if any) in the precedent cases. Finally we have the point, identified in [28] but not explicitly in [21], at which the dimension ceases to favour and begins to favour . Rigoni terms this the (SP). The question is where SP lies. For Horty it lies somewhere to the right of 36, whereas for Rigoni it cannot lie to the right of 12, if Case 1 is serving as a precedent. Rigoni then presents an alternative way of formalising dimensions which avoids the collapse of the two models and satisfies his intuition to disallow the decision in Case 2. Rigoni suggests that a dimension should be thought of as a of factors and uses magnitude to constrain their relative strength and SP to determine their polarity. This is essentially also the approach of [26] and [11], although presented less formally in those papers.
Here we will avoid reducing dimensions entirely to...
