The definition of coarse ethics
A concept must be defined before it is explored. CE is reflectively defined in this study by excluding the following premise that seems to be widely adopted in traditional ethics [14]: it is required to assume that ideals exist, and if one of two choices approximates an ideal more than the other, then the former should be evaluated higher than the latter. It is easy to illustrate this assumption citing the case that a person who lies only once per year should be evaluated higher than another who lies every day if the maliciousness of their lies is equal, because the former is closer to the ideal of honesty.
This premise can be found, for example, in Kantian ethics [16]. He explains his own methodology as follows:
Virtue is always in progress and yet always starts from the beginning.–It is always in progress because, considered objectively, it is an ideal and unattainable, while yet constant approximation to it is a duty. That it always starts from the beginning has a subjective basis in human nature, which is affected by inclinations because of which virtue can never settle down in peace and quiet with its maxims adopted once and for all but, if it is not rising, is unavoidably sinking. [29]
It is assumed in this text that an ideal represents the extreme of a smooth normative evaluation function and that humans are capable of asymptotically approaching it; rather, they are obligated to asymptotically approach it. Kant does not offer a specific example, but the above example of two liars is useful here: each person should approach the ideal of honesty even though the individual cannot become completely honest; hence, the person who lies only once every year should be appraised higher than the other who lies every day, because it is not acceptable to refer to the proverb that a miss is as good as a mile.
This asymptotic approach, however, runs into a problem; a smooth evaluation cannot be directly applied. In considering the issue of global warming, for example, it is easily understood that eating vegetables contributes more to the reduction of greenhouse gases than eating meat [42]. However, it is difficult to ascertain the type of vegetable combination that will reduce greenhouse gases the most, and ethical judgements will become impossible if people are instructed to weigh the daily increases and decreases of greenhouse gases to the nanogram every day. In short, while Kant states that people should endeavour to raise their ethical positions to stop falling, we cannot accurately estimate whether we are slightly rising or falling at any given moment.
Kant notes this difficulty and offers a short reflection on such matters:
But that human being can be called fantastically virtuous who allows nothing to be morally indifferent (adiaphora) and strews all his steps with duties, as with mantraps; it is not indifferent to him whether I eat meat or fish, drink beer or wine, supposing that both agree with me. Fantastic virtue is a concern with petty details which, were it admitted into the doctrine of virtue, would turn the government of virtue into tyranny. [29]
This description can be grounded in the daily experience that it is impossible to make uninterrupted ethical judgements about every trivial action. However, Kant does not mention here how a phenomenon can be identified as indifferent. Further, the example Kant cites of meat and fish is inappropriate for modern society because given global warming and veganism, the choice of meat eating cannot be said to be ethically indifferent. Such discrepancies can occur between AI and its users; that is, a choice could appear indifferent for users but different for AI. In such an instance, the trade-off would be problematic if AI explains something different as indifferent by sacrificing accuracy for human interpretability, because a potential risk embedded in the difference is kept hidden from the users. In other words, the concept of ‘indifferent’ (in Greek: adiaphora) is not self-evident; rather, this notion depends on what should be disregarded as indifferent.
CE can contribute to the resolution of the two problems of the asymptotic approach: the impossibility of smooth evaluation and the relativity of indifference by relinquishing the asymptotic approach and by permitting regulators, judges and agents to rough up their regulations, judgements or actions. This paper henceforth uses the term ‘evaluation’ for the abovementioned normative activities for the sake of simplicity. CE permits the evaluation of two objects as indifferent even if one of the items is actually or possibly closer to the ideal. The conception of CE is potentially related to every topic concerned with the limits of human understanding, for example, how to teach children or how to explain technology to non-experts. As the sections that follow evince, such a coarse approach facilitates answers to questions about the formal conditions within which XAI is allowed to sacrifice its own accuracy for human interpretability and to what degree.
Coarsening and refinement
Illustrative overview
There are discrete ways to perform rough evaluations. This section offers a simple example and introduces three special symbols. Teachers are obligated to assess students according to their scores. How the evaluation is achieved depends on decisions made by individual teachers as follows:
The first evaluation method (E
1)
Students take a 100-point test in one-point increments (i.e. without decimal) and are evaluated in the order of their scores Table 1.
The second evaluation method (E
2)
Students take a 100-point test in one-point increments and are evaluated as ‘inadequate’ if they score between 0 and 59, ‘fair’ if they secure tallies between 60 and 69, ‘good’ if their marks are computed between 70 and 79, ‘very good’ if they accrue points between 80 and 89 and ‘excellent’ if their scores are calculated between 90 and 100 Table 2.
The third evaluation method (E
3)
Students take a 100-point test in one-point increments, and those who attain 60 or more points pass the test, whereas those aggregating points under 60 must retake the test as a penalty Table 3.
In such a case, E2 is a rough version of E1, and E3 is a rough version of E2. For now, the following simple arrangement is introduced: if two students si and sj, who are evaluated differently in more fine evaluation (i.e. si > sj or si < sj according to their scores), are evaluated identically in coarse evaluation, then si is said to be coarsely equivalent or coarsely equated to sj, and this relation is denoted as si ≈ sj in the coarse evaluation. In CE, the symbol ≈ denotes normative coarse equation, and the following formulation is introduced:
$${o}_{i}\,\,\,\,{\approx }\,\,_{ \mathrm{\,\,Value}}^{ \mathrm{Evaluation}\_\mathrm{method}} \, \, {o}_{j}$$
For example, if a student (s1) scores 81 and another student (s2) scores 82, they are distinguished if E1 is applied but receive the same rating of ‘very good’ if E2 is employed. Therefore, in terms of E1, they should be assessed as s1 < s2, are deemed coarsely equivalent according to E2 conditions and denoted as follows:
$${s}_{1}\,\,{\approx }\,\,_{ \mathrm{\,\,VeryGood}}^{ E2} \, \, {s}_{2}$$
Further, if a student (s3) scores 79 and another student (s4) obtains 80, they are distinguished when both E1 and E2 are applied: s3 gets ‘good’ and s4 ‘very good’ according to E2; however, both are simply adjudicated as passing the examination if E3 is employed and may be represented as
$${s}_{3}\,\,{\approx }\,\,_{\mathrm{ \,\,Pass}}^{ E3} \, \, {s}_{4}$$
In addition, if the relation between oi and oj is unequal, a similar formulation may be applied as follows:
$${o}_{i} <\,\,_{ \mathrm{\,\,Comparative}\_\mathrm{value}}^{ \mathrm{Evaluation}\_\mathrm{method}} \, \, {o}_{j}$$
For example, s1 and s2 can be described as follows:
$${s}_{1} <\,\,_{ \mathrm{\,\,Higher}}^{ E1} \, \, {s}_{2}$$
Coarsening and refinement
In the three evaluation methods described above, E1, E2 and E3 are partially compatible with each other. For example, s1 can translate marks obtained in an examination from E1 (81) to E2 (very good) to E3 (pass). When an evaluation method Ej is a rough version of another, such as Ei, Ei is coarsened into Ej and is denoted as Ei ⇝ Ej (coarsening). Thus, the relationship between E1, E2 and E3 can be represented as E1 ⇝ E2 ⇝ E3, and the notation E1 ⇝ E3 is possible as well. Conversely, if an evaluation method Ei is a more honed version of another, such as Ej, Ej is refined into Ei and is denoted as Ej \(\curvearrowright\) Ei (refinement). For example, the above four students may be described as follows:
$${s}_{3}(79)<\,\,_{ \mathrm{\,Higher}}^{ E1} \, \, {s}_{4}(80)<\,\,_{ \mathrm{\,Higher}}^{ E1} \, \, {s}_{1}(81)<\,_{ \mathrm{\,Higher}}^{ E1} \, \, {s}_{2}(82)$$
$$\rightsquigarrow {s}_{3}<\,\,_{\mathrm{\,Higher}}^{ E2} \, \, {s}_{4}\,\,{\approx }\,\,_{ \mathrm{\,VeryGood}}^{ E2} \, \, {s}_{1}\,\,{\approx }\,\,_{ \mathrm{\,VeryGood}}^{ E2} \, \, {s}_{2}$$
$$\rightsquigarrow {s}_{3}\,\,{\approx }\,\,_{ \mathrm{\,Pass}}^{ E3} \, \, {s}_{4}\,\,{\approx }\,\,_{ \mathrm{\,Pass}}^{ E3} \, \, {s}_{1}\,\,{\approx }\,\,_{ \mathrm{\,Pass}}^{ E3} \, \, {s}_{2}$$
In this case, coarsening is possible, but refinement is impossible. The best student (s2), who scores 82 points, understands that the evaluation is ‘very good’ on E2 and ‘pass’ on E3 (i.e. the possibility of E1 ⇝ E2 and E1 ⇝ E3); however, a student only aware of having passed an exam cannot apprehend without additional information how many points were earned (the impossibility of E3 \(\curvearrowright\) E1). The possibility of refinement is not always guaranteed; thus, the initial assessment should be carefully selected. This study focuses on coarsening because of this unidirectional tendency.
Two requirements for adequate coarsening
The illustration offered thus far must be more rigorously explained to serve the purpose of academic contention. This study does not intend to argue that a null score is coarsely equated with a full score for lazy students. In other words, the concept of coarsening does not imply the disarrangement of an evaluation. Therefore, the question of under what formality the changing of an evaluation denotes adequate coarsening should be answered. This study posits two formal conditions for adequate coarsening: adequately high coverage and order-preservation.
The first requirement: adequately high coverage
Ej is said to cover Ei if and only if a rough evaluation method such as Ej encompasses one or more objects evaluated by the original Ei. The degree to which Ej encompasses Ei is called coverage. The coverage from Ej to Ei is said to be full if and only if Ej incorporates all the objects of Ei and does not include extra objects. Further, the coverage extended from Ej to Ei is labelled over-coverage if and only if Ej includes objects not evaluated by Ei. For adequate coarsening, the coverage offered by the rough evaluation method Ej to its original Ei is full or at least sufficiently high from some perspective. Adequately high coverage may include over-coverage in some instances.
Full coverage is desirable for adequate coarsening, which guarantees that every object evaluated by Ei can be found in Ej and vice versa. Such full coverage would free a judge from the trouble of not finding a rule for some objects. However, this study does not consider full coverage a necessary condition for adequate coarsening because of the costs of its satisfaction. Rather, coarsening is regarded as adequate when the coverage from a rough evaluation method to its original technique is sufficiently high from some standpoint. Thus, full coverage is characterised simply as the most evident case of adequately high coverage, and for the same reason, over-coverage (i.e. over-regulation) is also acceptable.
This mitigation helps several XAI models, such as Ribeiro et al.’s [40] LIME. LIME primarily trains local surrogate models to explain individual predictions and approximates the predictions through an interpretable surrogate model (e.g. by linear regression [35]). For instance, after a deep-learning model predicts the existence of a cat in an image, it can explain the reason for the prediction by focussing on contributory features such as long whiskers, triangular ears and distinctively tilted eyes, even though the original black box appears to use the whole image. The concept of adequately high coverage can make the local approximation ethically justified, and at the same time, it can help its users avoid deception by low coverage, as in the instance when a person wearing a cat mask is recognised as a cat.
Tjoa et al. [49] surveyed examples of LIME application to medical diagnosis. Once such identification concerned the prediction of influenza by highlighting the importance of certain symptoms (headache, sneeze, weight and no fatigue). This example can be formulated in CE as follows:
$$\mathrm{weight}<\,\,_{ \mathrm{\,More}\_\mathrm{contributes}}^{ \mathrm{Flu}\_\mathrm{diagnosis}} \, \, \mathrm{no\,fatigue} <\,\,_{ \,M.c.}^{ F.d.} \, \, 0<\,\,_{\, M.c.}^{ F.d.} \, \, \mathrm{headache}<\,\,_{ \,M.c.}^{ F.d.} \, \, \mathrm{sneeze}$$
This inequation demonstrates the degree to which each symptom contributes to the prediction of influenza. In this example, sneezing and headaches contribute positively, but no fatigue and weight contribute negatively to the likelihood. Does the diagnosis adequately encompass the original black box? It is difficult to establish a unique criterion for this estimation; thus, the present study will probe the question of who should bear the burden of proof of adequately high coverage. For example, if a transparent model with LIME does not mention a symptom to which not only a deep-learning model but also human doctors usually pay attention, then it should additionally explain why the important feature was ignored after the local approximation. The key point of this analysis is that such coarsening requires stronger justification than full coverage, and the burden of justification should be imposed on AI or its developer, not on users. Conversely, the diagnostic AI has satisfied its own black box accountability if human doctors also use the same four signs to diagnose influenza.
The second requirement: order-preservation
Assuming the order between an object oi and another object oj is oi = oj according to the original evaluation, this order may be termed properly preserved if and only if the order oi = oj is maintained by a rough evaluation. When the order of the two objects is oi < oj according to the original evaluation, this order may be called properly preserved if and only if the order is sustained as oi < oj by the rough evaluation. However, this order is called coarsely preserved if and only if the order is changed into oi ≈ oj. Adequate coarsening mandates that the order of all objects is properly or at least coarsely preserved; the order must not be reversed for any object.
The following example may be contemplated vis-à-vis the order-preserving stipulation for XAI. A fintech company financed B, but not A. Hence, A queried the rationale for the rejection from the financing AI of this company. The AI selected only one reason for the rejection and explained it as ‘Sorry, you have too little security to allow lending’. However, this explanation fails to preserve the order of clients if B has no security and was able to borrow money merely by evidencing talent. The original evaluation was that B had more investment suitability than A when all things considered (\(A \, <\,\,_{\mathrm{\,More}\_\mathrm{suitable}}^{\mathrm{Total}\_\mathrm{value}} \, B\)), but the rough explanation meant \(A \, >\,\,_{\mathrm{\,Less}\_\mathrm{suitable}}^{\mathrm{Security}} \, B\). This reversal makes A distrust the AI because the rough clarification that was tendered does not match the order between A and B for financing. The AI must thus enumerate all reasons until A is persuaded of the order, \(A<B\).
Definition: adequate coarsening
The exact definition of the special symbol ⇝ can now be tendered, as noted below. The original evaluation Ei is said to be adequately coarsened into Ej if and only if both the requirements of adequately high coverage and of order-preservation are satisfied; only this adequate relation is represented as Ei ⇝ Ej. This definition enables the avoidance of dishonest simplifications and relieves CE from arbitrary revaluations.