Cross-categorization of legal concepts across boundaries of legal systems: in consideration of inferential links

Abstract

This work contrasts Giovanni Sartor’s view of inferential semantics of legal concepts (Sartor in Artif Intell Law 17:217–251, 2009) with a probabilistic model of theory formation (Kemp et al. in Cognition 114:165–196, 2010). The work further explores possibilities of implementing Kemp’s probabilistic model of theory formation in the context of mapping legal concepts between two individual legal systems. For implementing the legal concept mapping, we propose a cross-categorization approach that combines three mathematical models: the Bayesian Model of Generalization (BMG; Tenenbaum and Griffiths in Behav Brain Sci 4:629–640, 2001), the probabilistic model of theory formation, i.e., the Infinite Relational Model (IRM) first introduced by Kemp et al. (The twenty-first national conference on artificial intelligence, 2006, Cognition 114:165–196, 2010) and its extended model, i.e., the normal-IRM (n-IRM) proposed by Herlau et al. (IEEE International Workshop on Machine Learning for Signal Processing, 2012). We apply our cross-categorization approach to datasets where legal concepts related to educational systems are respectively defined by the Japanese- and the Danish authorities according to the International Standard Classification of Education. The main contribution of this work is the proposal of a conceptual framework of the cross-categorization approach that, inspired by Sartor (Artif Intell Law 17:217–251, 2009), attempts to explain reasoner’s inferential mechanisms.

Appendices

Appendix 1: Members of the concept clusters and the feature clusters

See Figs. 17, 18, 19, 20, 21, and 22.

Fig. 17

Labels for concept clusters obtained by the n-IRM: Jaccard

Fig. 18

Labels for feature clusters obtained by the IRM: Jaccard (BMG)

Fig. 19

Labels for concept clusters obtained by the n-IRM: BMG—JP as background knowledge (BMG)

Fig. 20

Labels for feature clusters obtained by the IRM: BMG—JP as background knowledge (BMG)

Fig. 21

Labels for concept clusters obtained by the n-IRM: BMG—DK as background knowledge (BMG)

Fig. 22

Labels for feature clusters obtained by the IRM: BMG—DK as background knowledge (BMG)

Appendix 2: Ontology construction methods

2.1 Formal concept analysis

The FCA (Ganter and Wille 1997) is the method that analyzes a relation connecting objects and their features. A context C where a formal concept occurs is defined as C = (G, M, I). In this definition, G and M represent a set of objects and a set of features, respectively. I refers to relations between G and M. In Fig. 12, the context Japanese educational system is represented as G: (J1, J2, J3, ….. J13), M: (Jf1, Jf2, Jf3,….. Jf12), and their relations I. Each element g (e.g., J2) of G is expressed as $g \in G$. If this $g \in G$ has a feature m (e.g., Jf2) that is a member of M (expressed as $m \in M$), this relation is represented as gIm. When all members of a set of objects A (J5, J8) that is part of G ($A \subseteq G$) shares a set of features (Jf5, Jf6, Jf7) in Fig. 12, it is defined as $\acute{A} = \left\{m \in M\;|\;gIm\;for\;all\;g\;\in\;A \right\}$. In the same way, when a set of objects (J5, J6, J7) are shared by all members of a set of features B (Jf7, Jf10, Jf12) that is part of M ($B \subseteq M$), it is expressed as $\acute{B} = \left\{g\;\in\;G\;|\;gIm\;for\;all\;m\;\in\;B \right\}$. A formal concept existing in the context (G, M, I) is expressed as (A, B) defined by $A \subseteq G, B \subseteq M, \acute{A} = B, \acute{B} = A$. Here, A and B are respectively called the extent and the intent of the concept (A, B). The set of all concepts existing in the context (G, M, I) is drawn as a Gallois lattice as shown in Figs. 12 and 13.

2.2 Terminological ontology

The method of Terminological Ontology (TO; Madsen et al. 2004) is originated from the theory of terminology. The theory of terminology was first introduced by Wüster (1959). The original objective of terminology by Wüster (1959) was to eliminate ambiguity from technical languages by means of standardization of terminology in order to make the terms efficient tools of communication (Cabré 2000). The traditional theory of terminology thus addresses the relation between concepts and terms, starting from concepts and focusing on the present state of the conceptual structure and its representation (Kageura 2002).

The uniqueness of TO is its feature specifications and subdivision criteria (Madsen et al. 2004). The principles and constrains defined for the applications of feature specifications are described in detail in Madsen et al. (2004). The most important principle is that a concept must inherit all feature specifications (i.e., features) of its superordinate concepts. Another important key point is that subdivision criteria are strictly defined as dimensions and dimension values (Madsen et al. 2004). It means that a given dimension can only occur for specifying features on sister concepts and a given dimension value can only appear on one of these sister concepts (Madsen et al. 2004). A dimension and its dimension values are registered as (DIMENSION : [value1, value2, …]). In the case of Fig. 23, one dimension specification under the concept “JP education” can be represented as (PHASE : [under school age, compulsory]). This dimension specification subdivides the concept “JP education” into two sub concepts “preschool education” and “compulsory” which respectively possess the features, [PHASE: under school age] and [PHASE: compulsory]. Finally, a concept must be distinguished from each of its nearest superordinate concepts as well as from each of its sister concepts by at least one feature specification (Madsen et al. 2004).

Fig. 23

Terminological ontology principles

These strict principles, however, generate some difficulties in constructing an ontology when some feature specifications are considered as very important in several places in the ontology. For example, features such as [FOUNDATION: self governing] [FOUNDATION: municipality] might occur in two different occasions such as under “elementary education” and under “lower secondary education” in Fig. 23. As a solution, Madsen et al. (2004) argue that this problem is solved by creating nodes, “e: private” and “e: municipality” respectively possessing the features [FOUNDATION: self governing] and [FOUNDATION: municipality] at a higher level of the ontology as depicted in Fig. 23. Accordingly, subordinate concepts, “public elementary school” and “public lower secondary school” can both inherit [FOUNDATION: municipality] because of the polyhierarchical structure.

These strict rules are not directly applicable to the present work, since the information extracted from the cross-categorization approach only consists of the concept clusters and the feature clusters, which are rather fuzzy sets of concepts and features. In order to apply these strict principles of TO to the present work, the rules have been modified as follows:

• 1. A feature cluster inherited from a superordinate concept cluster can only occur on its descendant concept clusters

• 2. A non-inherited feature cluster can only occur in one concept cluster and its descendant concept clusters in an ontology

• 3. A concept cluster must be distinguished from each of its nearest superordinate concept clusters as well as from each of its sister concept clusters by at least one feature cluster.

• 4. For fulfilling these rules, polyhierarchical inheritance of feature clusters and generation of pseudo concept clusters are allowed.