In this paper, I provide a normative model of explanation for the output of AI algorithms used in legal practice. I focus on binary classification algorithms due to their extensive use in the field. In the last part of the paper, I examine the model's compatibility with causal explanations provided by Answer Set Programming (ASP) causal models. The motivation for proposing this model is the necessity for providing explanations for the output of legal AI. From the multiplicity of arguments supporting that necessity, the proposed model addresses the argument that legal AI's output should be objectionable. That can be achieved only if the explanation of the output has a form that makes it amenable to evaluation by legal practitioners. Hence, I firstly provide a normative model for the explanations used by legal practitioners in their practice (CLMLP) and then I provide the normative model for the explanations of legal AI's outputs (EXPBC) that I base on CLPLP. CLPLP in its turn is based on the Classical Model of Science (CMS) which is the normative model of explanations that every “proper” science should follow according to philosophers throughout history. Following the introduction of EXPBC, I propose three degrees of explainability regarding binary classification explanations according to their fidelity to EXP BC. I further argue that machine learning can not satisfy even the lowest degree of explainability, while rule-based AI - like ASP-based AI - can satisfy the highest degree. Concluding, I propose an ASP methodology in-progress that can use EXPBC to provide causal explanations. In the proposed ASP methodology, I am using causal graphs as models of causal inference as well as a metaphysically neutral interventionist account of causation. CMSLP and EXPBC are normative models that are based on the derivation relation of subsumptivedeductive inference among norms and propositions. On the other hand, the proposed ASP methodology is based on causal - and hence non-subsumptive-deductive - relations among norms and propositions that supervene on the subsumptive-deductive ones. Consequently, the motivation behind proposing this specific ASP methodology is to establish a precedent for a unification of diferent types of explanations (e.g., deductive and causal explanations) as well as to bridge the gaps among computational modelling, actual practice, and the philosophical underpinnings of a domain of expertise (law in this case).
1. Introduction
1AI has already multiple applications in legal practice. Public judicial institutions employ AI
to assess the possibility of the defendant recidivating [
A large amount of the legal AI literature is devoted on predicting the outcome of a case
(e.g., whether the Court will rule in favour of the prosecution [
Legal experts advocate that AI used in legal practice must provide an explanation for its output. One of the central arguments is that the outcome of such an algorithm should be objectionable [10]. That can be achieved only if the explanation of the output has a form that makes it amenable to evaluation by legal practitioners. As a result, any normative model of explanations of legal AI’s output should be constructed based on a normative model of explanations used in legal practice so as to justify its normativity. That also means that any normative model of explanations of legal AI’s output should be constructed prior to its implementation to actual AI models since the latter should follow the former. This is why, in this paper, I do not propose a normative model derived from the current state-of-the-art legal AI, but on the contrary, it is a model that the state-of-the-art legal AI should follow. The characterisation “state-of-the-art” itself can not be attributed without a prior listing of all the normative explainability requirements from the prospective of the legal practitioner that will actually use the proposed AI model.
The paper is structured in 5 sections with §1 being the introduction and §5 the conclusion. In §2, I construct a normative model of legal practice. Specifically, in §2.1, I clarify the meaning of the term “legal practice” in the context of this paper. In §2.2, I present the Classical Model of Science (CMS), i.e., a normative model of every “proper science” according to philosophers throughout history. Finally, in §2.3, I apply the CMS to legal practice in order to construct a normative model of legal practice that I name CMSLP.
In §3.1, I use the normative model of legal practice from §2.3 to finally construct a normative model of explanation for the output of binary classification legal AI. I name this model EXPBC. In §3.2, I use this model to classify the explanations provided by current legal XAI methods in three categories. Those categories are ordered in degrees of explainability. I argue that rule-based
2https://donotpay.com/ (last visit: 10/04/2022) AI can satisfy the highest degree of explainability while machine learning can not even satisfy the lowest degree.
So far, all proposed normative models (CMSLP, EXPBC) are based on subsumptive-deductive inference - the principal reasoning method used in legal practice. In §4 - the final section of the paper, I argue how Answer Set Programming (ASP) - as a paradigmatic case of rule-based programming - can be employed to: (i) model causal inference based on subsumptive-deductive inference;3 (ii) use causal inference’s supervenience on subsumption-deduction to satisfy the highest degree of explainability presented in §3.2. The motivation for this approach to causal explanations is to establish a precedent for a unification of types of explanations (e.g., deductive and causal explanations) and to bridge the gaps among computational modelling, actual practice, and the philosophical underpinnings of a discipline (the discipline of law in this case). Due to the latter motivation, the paper is written in a way that makes it accessible to all addressed audiences. The endeavour of addressing multiple disciplines is an emerging challenge for the practical use of AI and ergo, a secondary issue this paper tries to indirectly tackle not explicitly but by example. E.g., by avoiding unnecessary AI or philosophical jargon and by providing examples and intuitions where necessary all the while retaining the content’s quality.
Before constructing a normative model of legal practice, I have to clarify what legal practice is in the context of this paper. By “legal practice”, I mean the totality of the activities of the legal experts that are authorised to take part in the process of deciding a judgement (e.g., the lawyers of the defense and the prosecution, the judges of the authorised court). Those activities can be for instance the arguments the defence makes during a trial and all of the defence’s preparatory work before the trial. The authorisation of the legal experts is stipulated by law. E.g., the European Convention of Human Rights (the Convention) stipulates that the authorised court for deciding whether there has been a violation of the Convention’s articles is the European Court of Human Rights (ECtHR).4
I borrow this construal of legal practice from [11]. My motivation is that according to this construal, the produced results of legal practice are legally binding. And being legally binding will be a necessary property for the conception of truth I will employ in the proposed normative model of legal practice.
The normative model of legal practice I will provide will be based on the Classical Model of Science (CMS). CMS is a posterior reconstruction of how the ideal scientific explanation for every “proper science” should look like according to philosophers throughout history [12]. One may 3My choice of causal inference models is causal graphs based on a metaphysically neutral interventionist account of causation. 4Article 19 of the Convention. object to my decision to provide a normative model of legal practice using the CMS since the latter is about sciences and not about a practice. However, even if legal practice is not a science whatever that means - my arguments supporting that legal practice fits into the CMS will still hold since none of those arguments is based on the premiss that legal practice is a science.
Let’s move to the presentation of the CMS. I will provide a less detailed account of the CMS which I will call CMSsum. For the construction of EXPBC, there is no need to argue how legal practice fits into the more detailed version of CMS sum found in [12]. Specifically, EXP BC will be a minimal model. By “minimal”, I mean that one can further expand and/or precisify parameters of that model5 using the more detailed version of the CMSsum.6 However, for that to be done properly, I would have had to exceed the word limit. (SUM.1) CMSsum is a system S of propositions, concepts (or terms) which satisfy clauses (SUM.2) to (SUM.6). (SUM.2) CMSsum has a domain . The propositions, concepts and terms of S are about the entities of . (SUM.3) The propositions form a hierarchy based on their derivation: the propositions situated higher in the hierarchy are derived from the propositions situated lower. (SUM.4) The concepts and the terms form hierarchies and respectively based on their derivation: the constituents of a hierarchy situated higher in the hierarchy are derived from the constituents situated lower. (SUM.5) All propositions of S are true. (SUM.6) The domain experts know that the propositions of S are true and they have adequate knowledge of all concepts and terms of S. For each hierarchy ,∈{,, }, knowledge about one of its constituents is justified from knowledge about other constituents situated lower in that hierarchy. 2.3. Applying CMSsum to legal practice
As noted by [12], for each application of the CMS, CMS’s parameters may have to be modified to fit the particularities of that application. Below, I perform such modifications to apply CMSsum to legal practice. In §2.4, I sum up the outcome of those modifications to introduce a normative model of legal practice which I will name CMSLP. ∙ Applying (SUM.3): Legal reasoning can be construed as rule-based reasoning [13]. One way to perform such a construal is to view a legal inference as an inference whose premisses consist of particular facts and general rules [14] like the following example borrowed from [15]: 5By “parameters”, I refer to all the objects and relations that the CMS refers to: propositions, concepts, terms, derivation relations, the domain , the concepts of truth and knowledge, etc. 6An example of such a pacification is to narrow down the derivation relations among propositions to those of deduction and causality. That way we exclude all other derivation relation like reasoning by analogy or metaphysical grounding.
(1) If lives in Italy for more than 183 consecutive days over a 12-month period, (general rule) then is obliged to pay taxes in Italy on their worldwide income. (1) Alice lives in Italy for more than 183 consecutive days over a 12-month period. (particular fact) (2) Alice is obliged to pay taxes in Italy on their worldwide income. (conclusion) A particular fact is a proposition while a (general) rule is a norm. Intuitively, the diference is that propositions describe which state of afairs is the case (e.g., “Alice is paying taxes.”) while norms stipulate which state of afairs should be the case (e.g., “Alice should pay taxes.”). Since legal inferences - like Example 1 - contain both propositions and norms, we can not explain the inferences’ conclusions without appealing to the referred norms. Consequently, CMSLP has to contain norms apart from propositions. Moreover, Example 1’s conclusion is also a rule - not a general, but a particular one - and hence a norm. In other words, we can derive new norms from other norms and propositions. Consequently, the hierarchy in CMSLP contains both norms and propositions which are ordered based on derivation relations; ’s constituents situated higher are derived from those situated lower. E.g., norm 2 is situated higher in than norm 1 and proposition 1. The derivation relations whose conclusions are norms are those of subsumptive-deductive inferences: the particular fact (2) is subsumed by rule (1) and via deduction the conclusion (3) follows.
In the context of this paper, the output of a binary classification algorithm is essentially a judgement. E.g., a court’s judgement about whether there has been a violation of article X or not. As such, a judgement is the conclusion a legal inference. In Example 1, that conclusion has the form of a norm. However, we can always construe a judgement as a proposition and not as a norm . Specifically, := is a descriptive meta-analysis of whose existence is “parasitical” to that of [16]. E.g, instead of =“Alice should pay taxes.”, we can always say =“Alice is obliged by law to pay taxes.”. Having said that, whenever I discuss about a judgement as part of legal inference (e.g., Example 1) I will construe it as a norm unless explicitly saying otherwise.
But what are the norms of ? In §2.1, I construed legal practice as “the totality of the activities of the legal experts that are authorised to take part in the process of deciding a judgement”. The foregoing authorisation of the legal experts is authorised by the laws of a specific legal system. E.g., a court’s judgement about Alice’s taxes to the Italian State will be based on the norms of the Italian legal system. At the same time, if Alice’s activities lie outside of the jurisdiction of French law, then the norms of the French legal system are not applicable to them. Consequently, diferent legal systems may regulate diferent sets of entities (domains). As a result, the practice in each domain will be diferent. In other words, to every legal system corresponds a diferent CMS LP whose norms are: (i) those that make up the legal system; (ii) those that can be derived by the norms of (i) in a similar manner with Example 1. ∙ Applying (SUM.4): Subsumption is essentially a decision performed by the authorised legal experts; the experts decide inter alia whether terms appearing in the particular fact fall under concepts appearing in a rule. E.g., whether the term “Alice” belongs to (is subsumed by) the concept of “living in Italy for more than 183 consecutive days over a 12-month period”[14]. Consequently, CSMLP includes hierarchies and of the concepts and terms participating in the subsumption.
For the application of (SUM.5) to legal practice described right afterwards, it is important to highlight that the concepts of are interpretive concepts: an interpretive concept of a domain is a concept for which the domain experts can not concede on a specific list of criteria about whether an entity of belongs to that concept. The concepts for which domain experts do concede on such criteria are called criterial concepts. For instance, the domain experts of biology agree that there is a a unique criterion for whether an entity belongs to the concept of tiger: the entity has tiger DNA. In law, there is no such thing as a DNA of justice, freedom, dignity and the rest of legal concepts. To make things worse, the decision of whether an entity belongs to a legal concept is influenced by the background beliefs (ethical and political) of the authorised legal practitioners [17]. Hence, the extensions of the concepts of are decided on subjective criteria. Taking that into consideration, one could argue that CMSsum is not applicable to legal practice. Specifically, according to [ 12], the hierarchy reflects “ real or objective grounds (aitiai) of things” (ordo essendi). Therefore, since interpretive concepts’ extension are depended on subjective criteria they can not be grounded on “real or objective” aitiai.
To respond to that objection, I adopt the construal of legal interpretive concepts found in [18]. In brief, their argument is that in each tradition of legal practice (e.g., in the tradition of Italian legal practice) at a given point in time a unique conception of a legal concept emerges. The legal practitioners interpret that concept based on their background beliefs. Then, via their disagreement which is performed according to the methods and customs of that tradition, they concede at a decision of whether an entity belongs to . Ideally, their consensus coincides with the ordo essendi. Note that the decision about the extension of a legal concept is vital for legal practice since the truthfulness of a subsumption depends on that decision. ∙ Applying (SUM.5): The propositions of CMSLP refer to state of afairs among CMSLP’s concepts and terms. Since there is an ordo essendi about these concepts and terms, there is also an ordo essendi about their state of afairs and hence, all propositions of CMSLP are true.
Regarding the truthfulness of CMSLP’s norms, we have to deal with the objection that norms can not take truth values since they do not attempt to describe actual states of afairs. From all available responses to that problem, I adopt the response that a norm is true in a given normative system in virtue of that system including that norm.7 E.g., an Italian law is true in the Italian jurisdiction in virtue of belonging to the Italian legal system. Apart from the norms that are included in a normative system, we can infer new norms via subsumptive-deductive inference as is done in Example 1. Those norms are true in virtue of a truthful subsumption-deduction.
I will notate the conception of truth I have used so far as truth1. It is the truth reflected in the ordo essendi. However, next to truth1, we need to include another notion of truth: truth2. Specifically, as elaborated in §2.1, a judgement in the context of legal practice is decided by a group of authorised domain experts. Hence, it may be the case that the experts’ judgement does not correspond to the ordo essendi. A usual cause of such diferentiations is the mismatch between the consensus of the authorised legal experts on the extension of a legal concept and the ordo essendi. Despite that, judgement is still established as true in virtue of the authority of the authorised legal experts since their judgements are legally binding [16]. This is truth2. 7To argue in favour of that response, I would have to exceed both the scope and the word limit of this paper. Hence, I take it as a given.
In contrast with legal practice, in the practice of empirical sciences, no group of experts has the authority to establish truths; the domain experts do not hold any authority over the physical world [16]. For instance, Alice has cancer independently of the experts’ conclusion, while O. J. is guilty in virtue of the experts’ conclusion (truth2) and independently of whether he indeed killed his wife (truth1). There are of course institutional ways to challenge truth2 which are still though appeals to another group of authorised experts - like appealing to a higher court. On the contrary, in empirical sciences, a conclusion can always be challenged, and by any one of the domain experts [16].
In case that the legal concepts of CMSLP have been interpreted by the authorised legal experts (truth2) in a diferent way than the ordo essendi (truth1), they may also compose a diferent hierarchy than the hierarchy of the ordo essendi. I will notate that hierarchy as ,2. Similarly, since the concepts of ,2 play a decisive role in the subsumption of particular facts (propositions) by general rules (norms) in legal inferences, those propositions and norms may compose a diferent hierarchy than . I will notate that hierarchy as ,2. For instance, it may be the case that according to the ordo essendi, a particular fact is subsumed by a norm 1 allowing us to infer another norm 2. In that case, 2 is situated higher in than 1 and . However, if the experts interpret the concepts of 1 diferently than the ordo essendi, it may be the case that can not be subsumed by 1 and subsequently, we can no longer infer 2. In that case, 2 is not situated higher than 1 and . ∙ Applying (SUM.6): Since legal practitioners must accept the judgements (conclusions) of the authorised domain experts any conception of knowledge and justification must be based on truth2. ∙ Applying (SUM.2): As mentioned in ¶4 of (SUM.3)’s application to legal practice, CMSLP’s domain consists of all the entities that CMSLP’s norms regulate. The concepts, terms and propositions of CMSLP are about the entities of . 2.4. CMSLP: a normative model of legal practice
In this subsection, I introduce CMSLP, i.e., a normative model of legal practice. It is the result of the application of CMSsum to legal practice as elaborated in §2.3. (LP.1) CMSLP is a system S of propositions (particular facts), norms (general rules), concepts and terms which satisfy clauses (LP.2) to (LP.6). (LP.2) CMSLP’s domain consists of all the entities that the norms of CMSLP regulate. The propositions, concepts and terms of CMSLP are about the entities of . (LP.3) The propositions and norms of CMSLP form a hierarchy based on their derivation: the propositions and norms situated higher in the hierarchy are derived from the propositions and norm situated lower. The derivation of norms from other norms and propositions has the form of subsumptive-deductive inference. (LP.4) The concepts and the terms form hierarchies and respectively based on their derivation: the constituents of a hierarchy situated higher in the hierarchy are derived from the constituents situated lower. The concepts are interpretive and objective (i.e., there is an ordo essendi). (LP.5) There are two notions of truth: (i) truth1 that reflects the ordo essendi; (ii) truth2 which is established by a group of authorised domain experts. The authority of the authorised experts is authorised by the norms of . In legal practice, the truthfulness of a judgement is true in the sense of true2.8 truth2 induces new hierarchies ,2, ,2. truth1 coincides with truth2 only if ,2 coincides with . (LP.6) The domain experts know that the propositions and norms of ,2 are true (truth2) and they have adequate knowledge of all concepts and terms of ,2 and respectively. For each hierarchy ,2, ,2 and , knowledge about one of its constituents is justified from knowledge about other constituents situated lower in that hierarchy.
Assume a set of norms that are legally binding for a set of data (e.g., hateful tweets). Then, based on the norms , is segmented into two disjoint sets: = ∪ . Data are elements of if and only if they do not violate the norms of . Equivalently, are elements of if and only if they do violate the norms of . Since are norms that are legally binding for , any judgement on the violation of by ’s data is true if and only if it coincides with the judgments of the authorised legal practitioners. Since I want to provide a normative model and not a descriptive one, I assume that there is an algorithm ℱ that segments (or filters ) to the two disjoint sets = ∪ with 100% accuracy (ideal - normative case). Propositions of the form := ∈ - where ∈ { , } - are judgements for the legality of . Since represents a judgement, we can substitute it by a norm that represents the same judgement.9
Since is a judgement it is part of legal practice. Hence, its truthfulness corresponds to truth2. In this normative context, truth2 is the judgment to which the authorised legal experts would have conceded if the judgement had been decided by them and not by ℱ . Moreover, since is part of the legal practice, its explanation should be based on the normative model CMSLP. According to CMSLP, the truthfulness of is grounded on the hierarchy ,2: there are propositions and norms situated lower than in the hierarchy ,2 such that is derived from them via subsumption-deduction. Consequently, an explanation of should consist of the subsumptivedeductive arguments whose conclusion is . Finally, the user whose data are filtered knows the normative explanation of ℱ ’s output when they know those subsumptive-deductive arguments. (BC.1) Assume a dataset , a set of norms which are legally binding for , and an algorithm ℱ that segments (filters ) to = ∪ with 100% accuracy. Then, EXPBC is the CMSLP whose norms include and the norms , where := ∈ and ∈ { , }. (BC.2) EXPBC’s domain is the dataset .
8Since CMSLP is normative, one could argue that 2 should coincide with true1; that is the ideal (normative) case. However, since I want to provide a model of legal practice, I want it to be pragmatically useful. I.e., I want a model that is applicable to every group of actual legal practitioners. Therefore, the ideal state of afairs in every trial is the ideal state of afairs relevant to the capabilities of the authorised legal practitioners. Consequently, it may be the case that even in their ideal (normative) performance, those practitioners are not capable of discovering truth1. 9See ¶3 of (SUM.3)’s application to legal practice in §2.3. (BC.3) The truthfulness of the partition = ∪ is that of truth2. (BC.4) An explanation of a judgement consists of: (BC.4.a) all the propositions and norms that are the premisses of the subsumptive-deductive inference whose conclusion is (BC.4.b) the way those propositions and norms are structured (argumentative structure) to form the subsumptive-deductive inference of (BC.4.a). (BC.5) A user whose data are filtered by
ℱ knows: (BC.5.a) the propositions and norms described in (BC.4.a) (BC.5.b) the argumentative structure described in (BC.4.b). 3.2. EXPBC and the current state of AI: machine learning vs. rule-based programming
EXPBC is compatible with the the review of the current forms of explanation for the output of legal AI found in [10]. Based on that review, we can classify diferent explainable legal AI (or legal XAI ) models into three categories of diferent degree of explainability:10 (degree 1) Explanation that consist only of a subset of the propositions of (BC.4.a). (degree 2) Explanation that consist only of a subset of the propositions and norms of (BC.4.a). (degree 3) Explanation that consists of parts of the argumentative structure described in (BC.4.b).
From the above classification, we can infer that the content of explanations of degree 1 is contained in the content of explanations of degree 2 whose content is contained in the content of explanations of degree 3. Hence, explanations of degree 3 contain the same and more information of explanations of degree 2 which contain the same and more information than explanations of degree 3. The ideal would be for the XAI model to provide maximum information in its explanations. I.e., provide an explanation of degree 3 that contains all the propositions and norms of (BC.4.a) and their complete argumentative structure described in (BC.4.b).
Although I find Adrien et al.’s 2021 clustering of types of explanation quite useful, there is an important divergence between my interpretation of that classification and theirs. In contrast with them, I do not consider machine learning (ML) algorithms capable of satisfying any of the three degrees of explainability.
Take for instance the example of [
The binary classification algorithm that [
Article 3 (Prohibition of torture): “No one shall be subjected to torture or to inhuman or degrading treatment or punishment.” By doing so, the algorithm identified words that appear with high frequency in those cases (e.g., “injury”, “ukraine”, “detainee”, “food”). I.e., the patterns the algorithm was using to distinguish the two categories “violation” and “no violation” were patterns of words. Afterwards, during the testing phase, they “fed” the algorithm with new cases of alleged violations of Article 3. Whenever the algorithm encountered one of the aforementioned words in a new case, it was raising the probability of that case being judged by the ECtHR as a violation of Article 3. Consequently, when the algorithm was classifying a new case as a violation of Article 3, the explanation was the particular fact =“The words 1, 2, ..., appear in and they also appear with high frequency in past cases violating Article 3.”.
Now facts of the form of are facts about Article 3 itself. Since the term “Article 3” is mentioned in , for (a particular fact) to stand in a subsumptive relation with Article 3 (a norm), “Article 3” should either be a term appearing in Article 3 itself (self-reference) or it should belong to a concept mentioned in Article 3. Neither of the two is the case. In other words, unless a norm is self-referential - either by including the norm itself as a term or by including a concept to which the norm belongs to - the possibility of a fact of the form of standing in a subsumptive relation with that norm is excluded.11 Hence, such a fact can not be part of the propositions of the hierarchy ,2 since those propositions stand in subsumptive-deductive derivation relations with norms of ,2. Consequently, an explanation that includes facts like is not an explanation that adheres to the normative model EXPBC at all - not even in degree 1.
In contrast with ML algorithms, ruled-based models can provide explanations of degree 3. A rule-based model is a model that consists of rules () ⇒ () and facts (), where , , are propositional functions,12 is a variable and a term without free variables. Whenever () is true, then for every rule of the form () ⇒ () we have that () is true [15].
When they appear in the code of a programme Π , rules can be interpreted as norms and facts as propositions [15]. Specifically, () ⇒ () can be interpreted as “Whenever () is the case, then () should be the case.” and the fact () can be interpreted as “() is the case.”. Usually, the output has only facts like (). When in the output, facts can also be interpreted as norms: “According to the programme Π , () must be the case.”. This flexibility in the interpretation of the output is on par with the remark about being able to construe a judgement as both a norm and a proposition mentioned at ¶3 of (SUM.3)’s application to legal practice in §2.3. 11The generalised version of for all classification ML algorithms is: =“The patterns 1, 2, ..., appear in data and they also appear with high frequency in past cases of category . Hence, belongs to .” 12Their arities can vary.
Let’s see now a formalisation of Example 1 using rule-based programming. Assume that () :=“x pays taxes in Italy on their worldwide income.”, () :=“ lives in Italy for more than 183 consecutive days over a 12-month period.”. Then, we can formalize Example 1 as follows: line 1. () ⇒ () line 2. () output: { ()} Clearly, this formalisation provides an explanation of degree 3: it contains the norms (line 1, output), propositions (line 2), and the derivation relation among them. 4. ASP causal explanations supervening on EXPBC
ASP is a rule-based programming method that uses first-order logic (FOL) language. As such, an ASP programme Π is a set of rules of the form ⇒ ℎ, where ℎ is an atom and the body consists of combinations of literals , where each can either be an atom or its default negation ∼ . An atom is said to be proven whenever it appears in the head of rule whose body is satisfied. When we have the edge case where the body of a rule is empty (i.e., ⇒ ℎ), the head is considered proven under any circumstances. Hence, we call it a fact [20]. The notion of a proven atom is central for ASP since the output of any ASP programme Π is logical models that include only those atoms which have been proven. Those models are called answer sets and hence the name “answer set programming”. Finally, as a FOL programming method, ASP syntax includes relations (e.g., ) and functions (e.g., ) of arities and ′ respectively symbolised as / and /′. For instance, a common way of representing graphs in ASP is to employ the following two relations: (i) a predicate /1 which singles out the atoms which are nodes; (ii) a binary relation /2 which signifies the existence of a directed edge from its first to its second argument (e.g., (, ) represents → ). For a more detailed & quick introduction to the basics of ASP see [20].
Let’s see now how the rule-based model of Example 1 proposed in §3.2.2 can be realised via ASP. Assume that the predicate 183/1 stands for , the predicate 2/1 stands for . Then the programme Π 1 = {183() ⇒ 2(), 183()} has only one answer set: 1 = {2(), 183()}. Specifically, since 183() is a fact it has to belong to every answer set of Π 1 . At the same time, since the body of the rule 183() ⇒ 2() is satisfied for = , then its head 2()|= must also belong to every answer set of Π 1 . Since there are no more proven atoms, the only answer set compatible with Π 1 ends up being that of 1 . 4.2. Causal models of causal structures supervening on ,2
An argument against CMSLP could be that in the actual legal practice the derivation relations among the propositions and norms of ,2 are not exhausted in subsumptive-deductive inference. Indeed, by looking in any book on legal reasoning (see e.g., [21], [22]) one can see that there is a plurality of reasoning methods used in the actual practice: analogical/evidential reasoning, deontic logic, counterfactuals, etc. However, CMSLP is not a normative model of how legal practitioners should reason in their practice (e.g., by using abduction [22]). Instead, it is a normative model on how legal practitioners should explain the outcome of their practice. I.e., it is a meta-analysis of how they actually reason to conclude to that outcome.
Having said that, one can still reconstruct the proposed normative explanations to reflect a reasoning method diferent than subsumption-deduction in a way that it is still clear which is the hierarchical relation ,2 among the propositions and norms involved in that reconstruction. In what follows, I will do so for the case of causal inference using ASP. Note that causal inference is of prime importance for explanations of legal judgements. The most characteristic such explanation is the alleged causal relation between the defendants’ actions (alleged cause) to the applicants’ alleged harm (alleged efect ).
Before reconstructing the proposed subsumptive-deductive explanations to causal explanations, I need to decide on the the definition of causal inference; I have to know what I model before modelling it. There is a diverse plethora of available definitions motivated by diferent metaphysical conceptualisation of causation. Considering that, I am choosing a metaphysically neutral definition so as to be compatible with many such conceptualisations.
Definition 4.1 (Cause). is a cause of if: (i) it is possible to intervene on ; (ii) under some such possible intervention on , changes in the value of are associated changes in the value of . [23, p.3583]
Since the desideratum is a conceptualisation of causal inference to reflect the hierarchy ,2 the variables and of Definition 4.1 will be propositions and norms that are part of that hierarchy. More precisely, considering clauses (i) and (ii) of Definition 4.1, a proposition/norm will be the cause of another proposition/norm if 4.1.i′ it is possible intervene to the value of : The values of and in this situation are truth values - those of truth2. Moreover, “possibility” in this context is a conceptual possibility; it does not reflect the actual state of afairs, but it is a counterfactual case of another non-actual state of afairs. For instance, it may be the case that in the actuality Alice has lived in Italy for more than 183 consecutive days over a 12-month period, but it is also possible to conceptualise a counterfactual situation in which Alice left on day 182 for a conference on logical programming in Haifa, Israel. 4.1.ii′ by intervening in the value of , we intervene in the value of : This is the point where the hierarchy ,2 comes into play. ,2 is induced by derivation relations: norms situated higher are derived from norms and propositions situated lower via subsumption-deduction. See for instance the left graph of Figure 4.1. The direction of the edges 2 → 1 and 2 → 1 exhibits that 2 is derived from (or grounded on) 1 and 1. Since 2 is “derived” it is the true conclusion of an argument whose premisses are also true. In other words, all the variables in the current state of afairs (that of ,2) have the value “true”. Hence, the only intervention to any such variable that we can make is that of setting it false. Consequently, will cause if whenever we set its truth value to false we have that ’s truth value also become false.
The foregoing conceptualization of causation reflects the hierarchical structure ,2 since being the cause of implies that : (i) is situated lower in the hierarchy than ; (ii) is one of the premisses from which is derived via subsumption-deduction.
Let’s proceed with modelling this conceptualisation of causality. Assume a causal structure C that exists in the actual world. As a causal structure I define a collection of causal relata and the causal relations among them. From 4.1.ii′, we know that such a structure supervenes on the hierarchy ,2 and that the causal relata are the involved propositions/norms. A causal graph (C) = ⟨ (C), (C)⟩ is a graph whose nodes (C) are representations of C’s causal relata and whose edges (C) are representations of the direct causal relation13 among those relata and hence, it can serve as a model of C.
2 1
1 ,2 of Example 1.
C supervenes on ,2 2 (C) 1
1
A prominent method of constructing (C) is to bookeep a list C of all the independencies between the causal relata of C and then, construct (C) in such a way that expresses those and only those independencies. I.e., assuming that we also have a list (C) of the independencies among (C)’s nodes, the ideal is that for every set of independent relata in C their representations in (C) are also independent and vice versa. That requirement presupposes at least two distinct formal notions of independence - one for C’s causal relata and one for (C) - and that both notions explicate formally the same concept of independence.
Let’s start with the concept of independence we want to formalise. Assume three disjoint sets of causal relata , , . Then, an independency is a proposition of the form: “Knowing renders irrelevant to .” [25, p.5]. We symbolise this proposition as C |. Similarly, for three disjoint sets of nodes ′, ′, ′ an independency is a proposition of the form “Knowing the value of the variables in ′ renders the values of the variables in ′ irrelevant to the values of the variables in ′.” symbolised as (′C) ′|′. Verma & Pearl [26, p.354] provide a more “formal” definition which can be adjusted to the context of this paper as follows: Definition 4.2 (Independency IXY|Z). Assume an ordered set of variables and three disjoint subsets of = {1, 2, ..., }, = {1, 2, ..., }, = {1, 2, ..., } such that all variables of ∪ are situated lower than ’s variables in that ordering. Then, is independent of given if when assigning specific values to ’s variables ({1 = 1, 2 = 2, ..., = }) the value for every variable of ∈ will be the same for any set of values {1, 2, ..., } we assign to the variables of . 13An informal definition of “ direct cause” is the following: is a direct cause of if there is no causal relatum that mediates between and [24, p.20]. A paradigmatic example of that is the following: billiard ball 1 hits billiard ball 2 which hits billiard ball 3. 1 causes 3 to move, but it is not a direct cause since there is another causal relatum - that of 2 - that mediates between them.
Although this definition is more “ formal”, it is still not complete since it leaves many open questions - which are not in the scope of this paper to be answered - like what kind of orderings of the set are acceptable. Despite that, it is still quite an insightful definition since it reveals two important prerequisites for explicating formally the concept of independence: (I.1) there is a functional dependence of to both and : = (, ) (I.2) there needs to be an ordering of the variables such that the variables situated higher in that ordering can potentially be functionally depended in the variables situated lower in that ordering.
From 4.1.ii′, we infer that the ordering of (I.2) is that of the hierarchy ,2 and that the functional dependence of (I.1) is that of a truth assignment function. In the literature of causal modelling, C | is usually probabilistic independence and the functional dependence = (, ) is asymmetric. I.e., although is functionally dependent on both and , the inverse does not hold. The latter needs to be a requirement for our model as well since it represents an inherent asymmetric property of causal relata: the efect depends on the cause but not the inverse [27]. 4.3. ASP modelling of causal graphs & EXPBC
The construction of causal graphs using ASP14 is performed in the following three steps: (CM.1) the existence of a direct edge between two nodes , ∈ (C) is encoded as an atom (, ) (CM.2) we place restrictions on those atoms to force them respect the independencies (C) (CM.3) an answer set solver returns all stable models that satisfy those restrictions. The collection of edges in each such model is the required causal graph.
The ASP programme described in those three steps does not output a binary classification, but causal graphs. However, with a few extra steps we can induce from it a binary classification algorithm. Assume for instance that we want to use ASP to perform the same binary classification as in the example used in §3.2.1: the input is the facts of a case and the output is whether or not there has been been a violation of the Convention’s Article 3. Firstly, we construct an ASP programme Π (C) that outputs causal graphs based on the process described above. For that, we use past cases of violations of Article 3 to identify the independencies C |. Let’s postpone the discussion on the process of identifying those independencies for the conclusion of the paper. Now from 4.1.ii′ and Definition 4.2, we can infer that an independency of the form (C) |, means that if values of ’s variables change to false the values of all ’s variables will still remain true as long as the values of variables of remain true. Having that in mind, we can construct a binary classification ASP programme Π in the following way: Π = Π (C),2 ∪ Π ∪ Π • where Π (C),2 is an ASP programme that consists of the edges outputted by Π (C) • Π is an ASP coding of the facts of a case 14See e.g. [28, 29, 30] and [31, §7.2.1].
• Π is a set of rules which given an independence | requires the variables to be in the answer set in case that the variables of are in Π . A naive such set of rules could be that if all direct causes of are true then has to be true as well. The required output - the violation of Article 3 - is encoded in Π as an atom. In case that it appears in the answer sets, then Π has outputted that we have a violation of Article 3. It is part of my Thesis - an ongoing project - to come up with rules for Π and eventually construct an example of Π regarding hate speech cases of the ECtHR.
What I would like to remark in the conclusion is that in terms of explainability the most dificult black box to whiten in the foregoing proposal of causal modelling is the process of deciding the independencies. The common practice is to consider C to be a probabilistic independence. However, that leaves room for the same kind of criticism than the one against machine learning I used in §3.2.1; C is based on frequentistic arguments of the form of statistical independence tests: “In out of the examined cases, was independent of when we condition on and according to the statistical test they should be conditionally independent.”.
In order to overcome this problem we are left without any other choice than inquiring for different forms of functional (in)dependencies than the probabilistic ones. Conveniently, a plurality of important theorems that allow probabilistic independencies to be translated successfully to graph-theoretical independencies have been proven for functional dependencies that satisfy certain properties (e.g., symmetry | ↔ | and decomposition ∪ | → | [32]) and not for probability functions in particular. Hence, a good point to start is to find functions that still satisfy those properties and whose semantic interpretation is compatible with EXPBC. Since the variables of (C) are norms and propositions, it is rather intuitive that truth functions - not per se those of classical logic - have such semantic interpretations.
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