Intermediate factors and precedential constraint

The first case is that of MaxMonday. Max asks Jo for ice cream. The facts are that he folded his clothes (F1), but was inattentive in class (F5). Jo has to balance the good behaviour at home with the bad behaviour at school. In this case she decides that Max can have his ice cream, on the grounds that the good behaviour outweighed the bad. This will set a precedent, but questions arises as to the ratio of the case.

There are (at least) two possible interpretations.⁶ We could see the case as giving rise to the H-constraint that behaviour at home (Q) is more important than behaviour at school (R). This is quite a sweeping judgement.

Or we could argue for the more limited finding, the F-constraint, that folding clothes (F1) is more important than inattention in class (F5), remaining silent as to the relative importance of other forms of behaviour.

So we have two interpretations:

Jo may give clues as to her interpretation. If she says ok, you tidied your room, which meant you behaved at home, so you can have ice cream even though you misbehaved at school, she is clearly thinking in hierarchical terms. If, on the other hand she says ok, you folded your clothes so you can have ice cream even though you were inattentive in class, she is clearly thinking in terms of MMF. But she might say ok, your behaved at home by folding your clothes, so you can have ice cream even though you misbehaved at school by being inattentive, in which case it is less clear which constraint is intended. The ratio of cases are often not explicitly stated: they are identified by the subsequent judges when citing them as precedents.

The next case is EmmaMonday, in which Emma asks Jack for ice cream. Like Max, Emma had also shown mixed behaviour. She had made her bed (F2), but had interrupted her teacher (F6). Emma will offer the interpretation MMH to argue that she should get ice cream because, like Max, she had misbehaved at school but behaved at home. In contrast MMF suggests that MaxMonday sets no precedent because there are no base level factors in common. This is a Type C case, H-constrained, but not F-constrained.

Jack must first consider the relative strengths of F1, the good home behaviour in MaxMonday and F2, the good behaviour in EmmaMonday. Suppose he decides they are of similar strength, and establish the intermediate factor TidiedRoom with the same force. He must then consider the relative strengths of F5, the factor representing misbehaviour at school in MaxMonday, and F6, the misbehaviour at school in EmmaMonday. If he also considers these to be of similar strength he should follow MaxMonday and assert the new F-constraint:

This also yields the H-constraint MMH, but does not yet establish its validity. For that we would need further decisions confirming that \(F1 \succ F6\) and \(F2 \succ F5\). Even with MMF and EMF\(_\pi\), a future judge is free to decide that F1 is weaker than F2, and F6 stronger than F5 so that \(F6 \succ F1\) holds, which means MMH will give the wrong answer. Only when all possibilities where we have to balance good behaviour at home with bad behaviour at school have been tested can we say that the H-constraint is valid.⁷ But then all the possibilities will be covered by F-constraints, so that F- and H-constraint will coincide and no Type C cases will arise for that H-constraint.

Jack, however, is not constrained to find for Emma. He may well consider that the general disruption resulting from interrupting the teacher is more serious that inattention, which affects only Emma, and therefore outweighs the bed making. He therefore denies ice cream, adopting the more restrictive interpretation MMF of MaxMonday and setting the new constraint,

The decision in this case can be simply no, even though you made your bed, you interrupted your teacher. In order to remove any suspicion that he is overruling Jo’s decision in MaxMonday, however, Jack may clarify by saying that interrupting the teacher is serious misbehaviour (R+) whereas inattention is only simple misbehaviour (R) and thus can also offer a justification in terms of H-constraint:

This time the decision will be no, even though you behaved at home by tidying your room, interrupting the teacher is serious misbehaviour at school, worse than just being inattentive in class. Note, however, that this move involves accepting that intermediate factors can have different strengths, depending on which base level factors established them.⁸

On Tuesday Emma folds her clothes (F1), hands in her homework (F4), but interrupts her teacher (F6). This is like the case c3 in Canavotto and Horty (2023a). Emma pleads MMH as a precedent, distinguishing EmmaMonday by having handed in her homework. Here the question before Jack is whether handing in the homework is sufficient to mitigate the interruption. Jack may decide that the homework reduces the seriousness of the behaviour so that R rather than \(R+\) applies, and then he can simply follow MMH. Or he may think that homework is so important that it negates entirely the interruption, in which case there is no reason at all to deny the ice cream, so that R does not apply at all. In discussing their case 3 in Canavotto and Horty (2023a), Canavotto and Horty, seem to adopt the latter position, giving the decision:

“Emma tidied up her room because she folded her clothes and she behaved at school because she turned in her homework. Since she tidied up her room, Emma behaved at home. Because of this, my decision is that Emma can have ice cream” (p19).

In this case the H-constraint is simply

What of the F-constraint in this case? Canavotto and Horty proposed a method for flattening cases in Canavotto and Horty (2023b), which produces the F-constraint:

In Bench-Capon (2023) Bench-Capon reasons differently. He argues that the plaintiff reason from EmmaTuesday cannot be simply F1, giving the preference ETF\(_{ch}\), because F1 was never tested against F6 in EmmaTuesday, since F6 had been neutralised by F4. There is no certainty that Emma would have got her ice cream had F4 been absent: indeed EmmaMonday strongly suggests otherwise. Therefore, since F1 could be defeated by F6 unless F4 is present to neutralise F6, Bench-Capon argues that the F-constraint from EmmaTuesday should have both pro-ice cream factors:

Let us suppose Jack allows Emma her ice cream, leaving open for the moment whether he is thinking in terms of ETF\(_{ch}\) or ETF\(_{bc}\).

EmmaTuesday does not H-constrain the case in favour of Max, since F6 was cancelled by F4, so that R was not present in that case. But given the flattening of Canavotto and Horty (2023b), yielding the F-constraint ETF\(_{ch}\), Jo is F-constrained to find for Max, simply because he folded his clothes. This is consistent with EMF\(_\delta\), although it does suggest that clothes folding in more important than bed making. This is the example of a Type B case in Canavotto and Horty (2023b), F-constrained but not H-constrained.

If, however, we adopt Bench-Capon’s interpretation of the reason for EmmaTuesday, ETF\(_{bc}\), in the absence of F4 Jo is not F-constrained and can find against Max, adding the constraint:

Note that Jo is not constrained to decide this way. She is entitled to find for Max, establishing the constraint

The possibility of adopting MTF\(_{\pi }\) would mean that, since F2 is not (from EMF\(_\delta\)), but F1 might be, preferred to F6, we have to recognise that tidied room (P) cannot have uniform strength since, given the possibility of MTF\(_{\pi }\), it may, or may not, defeat F6. The strength of P would then depend on which of the base level factors established it, which makes it unreliable for use in constraints: it needs to be replaced by the appropriate base level factor. Moreover, even if Jo does decide against Max, asserting MTF\(_\delta\), it would still be possible for the combination of F1 and F2 to defeat F6, so we still may need the idea that P may be established with different strengths.

Most plausible is that Jo will follow MTF\(_{\delta }\), although we may conjecture that in a future case (MaxWed, see Sect. 5) it may be established that F1 and F2 can, in combination, defeat F6, giving the constraint:

showing that there is potentially a stronger version of P, \(P+\), and this in turn will mean that Q also can have different strengths. In hierarchical terms the constraint from MaxWed is:,Whether MWF and MWH hold or not will be resolved in a case presented in Sect. 5. If we did want to represent intermediate factors with different strengths in the hierarchy in such a way as to allow H-constraint, we could add a number of additional nodes to the hierarchy in Fig. 1 to provide a different intermediate factor for each strength. This is shown (accepting MWF) in Fig. 2. Note that every intermediate factor can be rewritten as a specific combination of base level factors, so that every H-constraint has an equivalent F-constraint. In the hierarchy shown in Fig. 2 we can, if we wish, use H-constraints, and it may help our explanations if we do. But we can use the H-constraint only because we are confident that H-constraint and F-constraint will coincide because in this hierarchy precedents have shown that all the children of intermediate factors establish them with equal strength. And, as discussed in Sect. 3.2, we need to have precedents showing us that this condition holds before we can use the H-constraint.

If, in the case of EmmaTuesday we ask why Emma could have ice cream, a good answer would indeed be because she behaved at home or because she folded her clothes, that is either the intermediate factor, or the flattening suggested in Canavotto and Horty (2023b), ETF\(_{ch}\). But if the question is why was she not denied ice cream after having interrupted her teacher, the answer is rather because she handed in her homework. This seems to support the contention of Canavotto and Horty that the correct reason is ETF\(_{ch}\), but that the enabler F4 also needs to be considered.

However, from the point of view of logic, leaving all considerations of cognitive aspects such as satisfying explanation aside, enablers achieve little.

Enablers suggest a logical statement like

But this is logically equivalent to

the reason ETF\(_{bc}\) identified in Bench-Capon (2023).

There remains, however, a distinction between enablers and other factors, albeit not a logical one. This is that enablers cannot be used, on their own, as the reason for a decision. As Canavotto and Horty say in Canavotto and Horty (2023a), “parents often expect their children to turn in their homework and they would not give the children a treat simply because they did what they were expected to do.” So in a case with only F4 and F6, we should find against ice cream. But this has nothing to do with the relative merits of F4 and F6. This is because no ice cream is the default: the purpose of the rules is to ensure that a positive reason for the treat is required, and, as conceived in Canavotto and Horty (2023a), handing in homework is not such a positive reason, even though it can neutralise the adverse effects of F6. Indeed even if F4 were the only factor present, the finding would still be against ice cream because no behaviour worthy of reward would have been performed. Of course, our view that homework is expected rather than worthy of reward has not yet been tested in the courts. In a future case with just F4 present Jack would be free to decide for ice cream showing that F4 had been a bona fide reason rather than an enabler all along.⁹

The ratio of the case is fundamentally part of the justification and explanation of the decision. This suggests that the ratio should incorporate the cognitive aspects, such as intermediate factors and enablers. But does this mean that we need to modify the reason model? Instead we can see the situation as similar to the difference between an argument providing a justification and a proof.

In providing an argument to justify a claim we often omit information that we believe to be known by our audience or too trivial to need stating. Suppose I say “Rover is old” and am asked to justify this. I will typically say “He is 20”, knowing that my questioner knows that dogs become old around the age of 12. Such an explanation will normally satisfy.

But it would not do as a proof. 20 is not old for a person, let alone a house or a city, so we would need to explicitly state that Rover is a dog. Moreover, for a proof we would need to explicitly state that 20 is greater than 12. And if there were a breed of dog, say the Himalayan Yetihound found in Shangri-La, which typically lives to the age of 50, we would have to add that Rover is not an Himalayan Yetihound.

Once we accept this distinction between what is required for justification and explanation and what is required for logical constraint, we can see that the ratio of the case need not coincide with the reason required by the reason model, the subset of winner’s factors preferred to the loser’s set of factors used to constrain future cases. It seems that, in Canavotto and Horty (2023a), an enabler is not part of the reason but an external consideration that allows the reason to support a decision. The ratio, however, would have to include enablers since they are not assumed, but discussed in course of reaching the decision. It may be that the ratio can use the abstractions offered by intermediate factors while the reason model can continue to operate using only F-constraints.

Restricting intermediate factors to an explanatory role means that we need not ensure that all children of an intermediate factor have the same strength. Because their role is only to group factors cognitively rather than to apply constraints, we can accept, for example, that both inattention in class and interrupting the teacher are bad behaviour at school, even if only one of them can defeat making the bed. This enables us to keep the simpler hierarchy of Fig. 1 rather that the complicated hierarchy of Fig. 2, but prevents us from using hierarchical constraint to determine the outcome.

With these additional cases we can produce the Angelic model of intermediate factors shown in Table 2. Acceptance conditions without sources (mainly the defaults) are suggested by the expert. We have followed the expert opinion of Canavotto and Horty (2023a) and modelled F4 as an enabler, so that it never provides a reason for ice cream. Note also that we can say F4 neutralises F5 or F6 because EmmaTuesday shows that it neutralises F6, and we have reason to believe that F5 is weaker than F6.

The ADM readily supports implementation as a logic program (e.g. Al-Abdulkarim et al. 2016b), more procedural Prolog (e.g. Bench-Capon and Atkinson 2018), an imperative program (e.g. as a Java implementation Atkinson et al. 2021), a web-based application (e.g. using Javascript Collenette et al. 2023), a set of argumentation schemes (e.g. using Carneades Bench-Capon and Gordon 2022), or a mobile phone application (using Logiak Atkinson et al. 2019). A Prolog implementation of the model in Table 2 is given in the Appendix.

We will now illustrate the operation of the program. We will use the four cases from Sect. 3 so that the output can be related to the discussion in that Section. The program takes as input cases described using the factors in Table 1.

First, MaxMonday:

As this was the first case in the domain, it is the citation for all the points established. Here the decision makes no reference to the behaviour at school, since the home behaviour is what merited the ice cream. Next, EmmaMonday:

Again the decision cites itself as the case for the points it established. Also note that the decision this time contains no reference to the behaviour at home, since the school behaviour was the reason for the decision.

Next EmmaTuesday:

Next, MaxTuesday:

Finally we have modelled F4 as an enabler. Thus if we have a case with only F4, say MaxThursday, we get simply:

No case is cited for this: it is part of the conception of the scheme that treats must be earned by good behaviour at home, and that handing in homework is expected rather than meritorious.

The explanations of the decisions produced in this way have a number of featuresIs the last point a strength or a weakness? It does explain why the winner won, and it can be inferred that whatever the reasons for the other side, the winner’s reasons were preferred to them, so it has the advantage of economy. It corresponds to the how? explanations of early logic programs such as (Sergot et al. 1986). On the other hand, the loser may feel dissatisfied that there is no explanation of why the case was lost. The explanation presented is the one that falls out naturally from the Angelic design. It would, however, be possible to write an explanatory program that processes the output to give an explanation in terms of the Issue-Rule-Application-Conclusion (IRAC) methodology as described in Bench-Capon (2020) and Bench-Capon (2024) if a more expansive explanation were required.¹⁰ An IRAC explanation of EmmaMonday would look like:

The issue in EmmaMonday is whether the seriously bad behaviour at school is sufficient to outweigh the good behaviour at home. The rule is that seriously bad behaviour at school is sufficient to deny ice cream, even if home behaviour is good. Interrupting her teacher was an example of seriously bad behaviour and so ice cream is denied.

and in MaxWed, which established the strength of excellent behaviour:

The issue in MaxWed is whether the seriously bad behaviour at school is sufficient to outweigh the excellent behaviour at home. The rule is that excellent behaviour at home is sufficient to allow ice cream, even when school behaviour is seriously bad. Max tidied his room completely, both folding his clothes and making his bed, so his behaviour at home was excellent. Therefore ice cream is allowed.

Note that the issues concerns intermediate factors, even though the base level factors determine the decision. Perhaps the most interesting case is EmmaTuesday, which established the role of homework:

The issue is whether Emma handing in homework was sufficient to outweigh the interruption of the teacher. The rule is that behaviour at school is neutral if homework was handed in, even if the teacher was interrupted. Emma did hand in her homework. Therefore, given also that behaviour at home was good, ice cream is allowed.

Here the issue is the conflict concerning F4, the enabler in Canavotto and Horty (2023a), and F6. There are no issues relating to behaviour at home, and so the reasons why behaviour at home were good are not stated. However, since behaviour at school was only neutral, the good behaviour at home is needed to justify the granting of ice cream. This very closely corresponds to the distinction between enablers and reasons in Canavotto and Horty (2023a).

Some cases remain to be tested. Suppose a child tided the room and handed in the homework, but was both inattentive and interrupted the teacher. As the domain is currently modelled, this would deny ice cream, unless the room had been completely tidied (i.e \(F5 \wedge F6 \succ F1 \wedge F4\) but \(F1 \wedge F2 \wedge F4 \succ F5 \wedge F6\)). If this proves to be incorrect when such a case arises, the acceptance conditions will need to be modified to reflect the actual decision (Al-Abdulkarim et al. 2016a).