In this section I discuss various ways in which the above models can be and have been adapted to dimensions. I first adapt the above factor-based definition of relevant differences to Horty’s dimension-based result model (Horty
2019). Then I will not do the same for Horty’s dimension-based reason models, for two reasons. First, as Horty himself shows, his (
2019) dimension-based reason model collapses into his result model, which arguably fails to capture the distinction between
ratio decidendi and
obiter dicta familiar from common-law jurisdictions. Although Horty (
2020) presents a revision of his reason model that does not collapse into his result model, it shares a feature with his original model (which is even more strongly present in Rigoni’s
2018 model) which raises some concerns about practical applicability. For these reasons I will first formally reconstruct Rigoni’s (
2018) model for so-called complete dimensions and then present an alternative which, although less expressive than Horty’s and Rigoni’s reaosn models, is arguably easier to apply in practice.
5.1 Relevant differences in Horty’s dimension-based result model
I adopt from Horty (
2019) the following technical ideas (again with some notational differences). A
dimension is a tuple
\(d = (V,\le _o,\le _{o'})\) where
V is a set (of values) and
\(\le _o\) and
\(\le _{o'}\) two partial orders on
V such that
\(v \le _o v'\) iff
\(v' \le _{o'} v\). A
value assignment is a pair (
d,
v). The functional notation
\(v(d) = x\) denotes the value
x of dimension
d. Then a (dimension-based)
case is a pair
\(c = (F, outcome (c))\) such that
D is a set of dimensions,
F is a set of value assignments to all dimensions in
D and
\(outcome (c) \in \{o,o'\}\). Then a (dimension-based)
case base is as before a set of cases, but now explicitly assumed to be relative to a set
D of dimensions in that all cases assign values to a dimension
d iff
\(d \in D\). Likewise, a (dimension-based)
fact situation is now an assignment of values to all dimensions in
D. As for notation,
v(
d,
c) denotes the value of dimension
d in case or fact situation
c. Finally,
\(v \ge _s v'\) is the same as
\(v' \le _s v\). The assumption that all dimensions have a value in each case is, of course, a genuine limitation, but since it is also made by Horty (
2019), I leave a study of models without this assumption to future research.
In HYPO (Rissland and Ashley
1987; Ashley
1990), two of CATO’s factors from our running example are actually dimensions.
Security-Measures-Adopted has a linearly ordered range, below listed in simplified form (where later items increasingly favour the plaintiff so decreasingly favour the defendant):
Minimal-Measures, Access-To-Premises-Controlled, Entry-By-Visitors-Restricted,
Restrictions-On-Entry-By-Employees
(For simplicity I will below assume that each case contains exactly one security measure; generalisation to multiple measures is straightforward by defining the orderings between sets of measures.) Moreover,
disclosed has a range from 1 to some high number, where higher numbers increasingly favour the defendant so decreasingly favour the plaintiff. For the remaining four factors I assume that they have two values 0 and 1, where presence (absence) of a factor means that its value is 1 (0) and where for the pro-plaintiff factors we have
\(0 <_{\pi } 1\) (so
\(1 <_{\delta } 0)\) and for the pro-defendant factors we have
\(0 <_{\delta } 1\) (so
\(1 <_{\pi } 0)\).
Accordingly, I change the running example as follows.
$$\begin{aligned} \begin{array}{ll} c_1 (\pi ): &{} deceived _{\pi 1}, measures = {\text{Entry-By-Visitors-Restricted}}, obtainable\!-\!elsewhere _{\delta 1}, \\ &{} disclosed = 10\\ c_2 (\delta ): &{} bribed _{\pi 2}, measures = {\text{Minimal}}, obtainable\!-\!elsewhere _{\delta 1}, disclosed = 5 \\ F_1: &{} bribed _{\pi 2}, measures = {\text{Access-To-Premises-Controlled}}, reverse\hbox {-}eng _{\delta 2}, \\ &{} disclosed = 20 \end{array} \end{aligned}$$
In Horty’s result model a decision in a fact situation is forced iff there exists a precedent
c for that decision such that on each dimension the fact situation is at least as favourable for that decision as the precedent. Horty formalises this idea with the help of the following preference relation between sets of value assignments.
In our running example we have for any fact situation \(F'\) that \(F(c_1) \le _{\pi } F'\) iff \(F'\) has \(\pi _1\) but not \(\delta _3\) and \(v({measures},F') \ge _{\pi }\) Entry-By-Visitors-Restricted and \(v({disclosed},F') \ge _{\pi } 20\) (so \(\le 20\)). Likewise, \(F(c_2) \le _{\delta } F'\) iff \(F'\) has \(\delta _1\) but not \(\pi _1\) and \(v({measures},F') =\) Minimal and \(v({disclosed},F') \ge _{\delta } 10\) (so \(\ge 10\)).
Then adapting Horty’s factor-based result model to dimensions is straightforward.
In our running example, deciding \(F_1\) for \(\pi\) is not forced, for two reasons. First, \(v(c_1, deceived ) = 1\) while \(v(F_1, deceived ) = 0\) and for deceived we have that \(0 <_{\pi } 1\). Second, \(v(c_1,{measures}) = \textit{Entry-By-Visitors-Restricted}\) while \(v(F_1,{measures}) = \textit{Access-To-Premises-Controlled}\, and \,\textit{Access-To-Premises-Controlled} <_{\pi }\) Entry-By-Visitors-Restricted. Moreover, deciding \(F_1\) for \(\delta\) is also not forced, since \(v(c_2,{measures}) =\) Minimal while \(v(F_1,{measures}) = \textit{Access-To-Premises-Controlled} and \textit{Minimal} <_{\delta }\) Access-To-Premises-Controlled.
I next adapt Definition
6 of differences between cases to dimensions. Note that unlike in the case of factors, there is no need to indicate whether a value assignment favours a particular side in the case, since the
\(\le _s\) ordering suffices for this purpose.
Let c be a precedent and f a focus case. Then clause (1) says that if the outcomes of the precedent and the focus case are the same, then any value assignment in the focus case that is not at least as favourable for the outcome as in the precedent is a relevant difference. Clause (2) says that if the outcomes are different, then any value assignment in the focus case that is not at most as favourable for the outcome of the focus case as in the precedent is a relevant difference.
In our running example, we have:
$$\begin{aligned} \begin{array}{ll} D(c_1,f) = \{( deceived ,1), ( measures , \textit{Entry-By-Visitors-Restricted})\}\\ D(c_2,f) = \{( measures , Minimal )\} \end{array} \end{aligned}$$
We have the following counterpart of Proposition
2 for Horty’s dimension-based result model.
The counterpart of Proposition
3 can be proven as a new result.
5.2 A dimension-based reason model with complete rules
I next discuss how Horty’s dimension-based result model can be turned into a dimension-based reason model. There are two features on which this can be done: by ‘relaxing’ an individual value assignment or by leaving some assignments out from a set of value assignments. In both ways a case is a triple \((c = (F(c),R(c), outcome (c))\), where F(c) is as in the result model a value assignment to a given set D of dimensions and where R(c), the rule of the case, is a set of value assignments that is in some way constrained by F(c). In the first way, rule R(c) consists of value assignments to each dimension in D such that for each element (d, v) in R(c) and each element \((d,v')\) in F(c) it holds that \(v(d) \le _s v'(d)\). In other words, in this approach a rule of a case assigns to each of the case’s dimensions a value that is at most as favourable for the case’s outcome as its value in the case. Below I will call such a rule a complete rule.
It is easy to see that this definition includes Horty’s result model as the special case where for all cases the rule equals the case’s fact situation. The counterpart of Proposition
6 can easily be obtained for this reason model by defining the relevant differences between a precedent and a focus case in Definition
9 relative to the precedent’s rule instead of to its fact situation.
Note that because of this result, Definition
10 does not have to quantify over cases with different rules as in Definition
5.
Since Bench-Capon and Atkinson (
2017), Horty (
2019,
2020) and Rigoni (
2018) all discuss the merits of their models in terms of the fiscal-domicile example introduced by Prakken and Sartor (
1998), I will from now on use that example as a running example. The issue is whether the fiscal domicile of a person who moved abroad for some time has changed. Let us consider two dimensions
\(d_1\), the duration of the stay abroad in months and
\(d_2\) the percentage of the tax-payer’s income that was earned abroad during the stay. For both values, increasingly higher values increasingly favour the outcome
change and decreasingly favour the outcome
no change. So, for instance,
\((d_1,12m) <_{change} (d_1,24m)\) and so
\((d_1,24m) <_{no~change} (d_1,12m)\). An example of a fact situation is
\(F = \{v(d_1) = 30m, v(d_2) = 60\%\}\) and an example of a case is
\(c = (F', change )\) where
\(F' = \{v(d_1) = 12m, v(d_2) = 60\%\}\).
This example can be used to show that Definition
10 does not collapse into the dimension-based result model. Suppose that
c has a fact situation
\(\{(v(d_1) = 30m), (v(d_2) = 60\%)\}\) and outcome
change and consider again fact situation
\(F = \{(v(d_1) = 24m), (v(d_2) = 75\%)\}\). Suppose the court in
c ruled that with a percentage earned abroad of
\(60\%\) a stay abroad of at least 12 months suffices for change of fiscal domicile. The rule of
c then is
\(\{(d_1,12m), (d_2,60\%)\}\). Then in the reason model deciding
F for
change is forced, even though the stay abroad in
F is shorter than in
c, so even though
F is weakened for the outcome along that dimension, since the stay is still longer than its value in
c’s rule. By contrast, in the result model this difference suffices to make
c distinguishable and deciding
F for
no change allowed.
The model of Definition
10 also avoids an arguably counterintuitive feature of Horty’s (
2019) dimension-based model (which Horty (
2017) also himself noted and which he avoids in his revised (
2020) model). In the tax example, if the rule of
c is
\(\{(d_1,12m)\}\) then in a new case in which the stay abroad is 24 months and the percentage of income earned abroad is
\(75\%\) deciding for
change is in Horty’s model not forced by the precedent, since it is weaker for
change than the precedent in that the stay abroad is not 30 but 24 months. However, this seems counterintuitive given that the court in the precedent ruled that 12 months abroad suffice for
change and given that the new case is stronger for this outcome in its only other dimension. With Definition
10 deciding for
change is instead forced by
c, since
\(\{(d_1,12m), (d_2,60\%)\} \le _{change} \{(d_1,24m), (d_2,75\%)\}\).
One issue remains. Horty’s factor-based reason model requires that courts select a rule in the new case that leaves the case base consistent when the case is added to it. In Horty’s model consistency is defined in terms of a preference relation between sets of reasons pro and con a decision (cf. Definition
3 above). However, the present model does not distinguish between pro and con value assignments, while still a notion of consistency is needed. Consider again the tax example with the two dimensions
\(d_1\) and
\(d_2\) and consider two precedents
\(c_1\) with rule
\(R_1 = \{(d_1,12m), (d_2,60\%)\}\) and outcome
change and
\(c_2\) with rule
\(R_2 = \{(d_1,8m), (d_2,60\%)\}\) and outcome
no change. Consider next a fact situation
F with
\(d_1 = 15\) and
\(d_2 = 60\%\). Then deciding
F for
change is forced. Suppose the court does so but formulates the rule
\(R_3 = R_2 = \{(d_1,7m), (d_2,60\%)\}\). Then in a new fact situation equal to rule
\(R_2\) both deciding
change and deciding
no change would be forced, so adding
\(f = (F,R_3, change )\) would make it inconsistent in that for the same fact situation two opposite outcomes are forced. So a constraint on rule selection should be that it should leave a consistent case base consistent in this sense.
5.3 A reason model with dimensions with switching points and incomplete rules
The second way in which the result model can be refined into a reason model is by allowing that the rule
R of a case assigns a value to a subset of its fact situation
\(F_D\), while still adhering to the constraint that the rule’s values of dimensions are at most a favourable to the case’s decision as their actual values in the case. How can precedential constraint then be defined? If for each relevant value assignment in a case it is known which side is favoured by it, that is, if each such assignment is turned into a factor, then this is relatively easy, by combining ideas of Rigoni (
2018) and Horty (
2019). As a matter of fact, neither of these papers contains such a combination, although it seems implicitly assumed by Rigoni. So the present section can be regarded as a full formalisation and a formal exploration of Rigoni’s ideas, although restricted to his special case of so-called ‘complete’ dimensions, which have a fixed ‘switching point’ and a total ordering on its values, which are all known.
Unlike Horty, Rigoni assumes that each dimension has a switching point, which is a subset of ‘adjacent’ values, to one side of which every value assignment favours one side and to the other side of which every value assignment favours the other side. Then Rigoni extracts ‘dimensional factors’ from dimensions, which like ‘standard factors’ favour one side or another except that they are not just given (as standard factors are) but derived from dimensions. I formalise these ideas as follows.
Let
o and
\(\overline{o'}\) be two outcomes. A
dimension with a switching point is a tuple
\(d = (V,\le _s,\le _{o'}, Pro (d), Con (d))\) where
V,
\(\le _o\) and
\(\le _{o'}\) are as before and
\(P ro(d)\) and
\(Con (d)\) are disjoint subsets of
V such that
-
for all \(v \in Pro (d)\) and \(v' \not \in Pro (d)\) it holds that \(v' <_o v\);
-
for all \(v \in Con (d)\) and \(v' \not \in Con (d)\) it holds that \(v' <_{o'} v\);
-
for all \(v, v' \in V\): if \(v \in Pro (d)\) and \(v \le _o v'\) then \(v' \in Pro (d)\);
-
for all \(v, v' \in V\): if \(v \in Con (d)\) and \(v \le _{o'} v'\) then \(v' \in Con (d)\).
This implicitly defines a dimension’s switching point
S(
d) as
\(V \setminus Pro (d) \cap Con (d)\). Note that the definitions imply that for all
\(v \in S, v' \in Pro (d)\) and
\(v'' \in Con (d)\) it holds that
\(v <_o v'\) and
\(v <_{o'} v'\). Thus the definitions formalise Rigoni’s idea that the switching point consists of all values that are ‘between’ the pro-
o and pro-
\(o'\) values. Note that any of the sets
S,
\(Pro (d)\) and
\(Con (d)\) can be empty. This allows to capture HYPO-style dimensions that ‘generally favour a particular side’, namely, by leaving
S and one but not both of
\(Pro (d)\) and
\(Con (d)\) empty. Finally, given a set
D of dimensions with switching points,
fact situations can now be
partial in that they can consist of value assignments to any nonempty subset of
D. For any fact situation
F we say that
\(F' \subseteq F\) favours
s iff either
\(s = o\) and
\(F' \subseteq \{ (v,d) \mid d \in D\) and
\(v \in Pro (d)\}\) or
\(s = o'\) and
\(F' \subseteq \{ (v,d) \mid d \in D\) and
\(v \in Con (d)\}\). For any fact situation
F, the set
\(F^s\) is the subset of all value assignments in
F favouring side
s.
In our tax example, we could have the following three dimensions, with
\(o = change\).
\(d_1 =\) duration; \(V =\) any number of months in natural numbers; \(v \le _o v'\) iff \(v \le v'\); \(v \in Pro (d_1)\) iff \(v \ge 18\); \(v \in Con (d_1)\) iff \(v \le 6\)
\(d_2 =\) earned; \(V =\) any percentage in natural numbers; \(v \le _o v'\) iff \(v \le v'\); \(v \in Pro (d_2)\) iff \(v \ge 60\); \(v \in Con (d_2)\) iff \(v \le 30\).
\(d_3 =\) gave up house; \(V = \{ yes , no \}\); \(no <_{o} yes\); \(Pro (d_3) = \{ yes \}\); \(Con (d_3) = \{ no \}\).
Given a set
D of dimensions with switching points as just defined, a case
c is now a tuple
\((F, pro (c), con (c),s)\), where
F is a fact situation,
\(pro (c)\), the case’s
reason, is a value assignment to a subset of
D such that for each
\((d,v) \in pro (c)\) there exists a
\((d',v') \in F^s\) such that
\(d = d'\) and
\(v \le _s v'\), and
\(con (c) = F^{{\overline{s}}}\). This notion of the reason of a case is essentially the same as the one of Horty’s factor-based reason model, in that it consists of a subset of all factors in the case favouring its outcome. The only difference is that now the factors favouring the two outcomes are not simply given but derived from a specification of dimensions with switching points. Then, just as for the factor-based reason model, we can say that a case decision expresses a preference for the case’s reason
\(pro (c)\) over its set
\(con (c)\). It is left to formally define this case-related preference relation. Horty (
2019) in fact defines such a preference relation, but on a different notion of the reason of a case, which I will briefly discuss below in Sect.
5.4. For now I simply adapt Horty’s definition to the just defined notion of a reason.
Let (
d,
v) and
\((d,v')\) be two value assignments both favouring side
s. Then (
d,
v) entails
\((d,v')\) iff
\(v' \le _s v\). Generalising, a set
X of value assignments favouring
s entails a value assignment (
d,
v) favouring
s iff some value assignment in
X entails (
d,
v), and a set
X of value assignments favouring
s entails a set
Y of value assignments favouring
s iff each value assignment in
Y is entailed by a value assignment in
X. Note that for each case the set
\(F^s\) entails the case’s reason
\(pro (c)\). Then Definition
3 can be adapted to the case with dimensions as follows:
Definition
4 then applies provided the superset relation between factor sets is replaced by the entailment relation between sets of value assignments. Then Definition
5 can be applied to sets of value assignments as follows.
To show that this reason model does not collapse into the result model, consider in our running example a case c with fact situation \(\{(v(d_1) = 30m), (v(d_2) = 75\%),(v(d_3) = no) \}\) and reason \(\{v(d_1) = 24m\}\) and outcome change, and a new fact situation \(F = \{(v(d_1) = 27m), (v(d_2) = 50\%),(v(d_3) = no) \}\). Then deciding F for change is forced in the reason model but not in the result model. In the reason model the outcome of c expresses that \(\{v(d_3) = no \} <_c \{v(d_1) = 24m\}\). Moreover, we have that \(v(d_1) = 27m\) entails \(v(d_1) = 24m\), so we have \(\{v(d_3) = no \} <_c \{v(d_1) = 27m\}\). By contrast, in the result model deciding F for change is not forced since, firstly, \(v(d_1,F) = 27m <_{ change } v(d_1,c) = 30m\) and, secondly, \(v(d_2,F) = 50\% <_{ change } v(d_2,c) = 75\%\).
The counterpart of Proposition
8 can be obtained for this adapted reason model by again redefining the relevant differences between a precedent and a focus case, now also taking missing and new dimension-based factors into account. Essentially, this definition combines the factor-based Definition
6 with the dimension-based Definition
11.
This completes the formalisation and investigation of Rigoni’s (
2018) dimension-based account of precedential constraint with complete dimensions.
5.4 An alternative dimension-based reason model
While the reason model of the previous section is well-behaved, it has an important pragmatic limitation, since it requires that for each dimension it is defined in general, abstracting from specific cases, whether a particular value assignment favours a side and if so, which side it favours. The problem is that, unlike with factors, it may in practice be hard for knowledge engineers to identify the side favoured by a particular value assignment, since often this will be context-dependent. See also Bench-Capon (
1999) and Rissland and Ashley (
2002); the latter remark at p. 69 that the direction of a value assignment is not for the knowledge engineer to decide but for the judge in a case.
Consider, for example, in our tax example a case with fact situation
\(\{v(d_1) = 30m, v(d_2) = 60\%\}\) and with outcome
change. Are both value assignments pro this outcome, or is one pro and the other con
change? And if the latter, then which is pro and which is con? This is not easy to say in general. On the other hand, what is uncontroversial is that increasingly higher values for these dimensions increasingly support
change and decreasingly support
no change. In Horty’s (
2019;
2020) approach this problem arises to a lesser extent, since he works with so-called magnitude factor expressions, which are of the form ‘the actual value of dimension
d in the case favours side
s at least as much as reference value
p’; this expression is meant to favour side
s. Such expressions are, unlike Rigoni’s switching points, not part of the general specification of dimensions but of individual case formulations, so they do not need to be chosen in general. Yet in this approach the problem still arises to some extent. For instance, in the just-given example still a reference value has to be known, as well as a side favoured by it for both dimensions. Such information may be hard to identify in case decisions.
For these reasons I will in this subsection explore an alternative approach, in which all that is needed is general knowledge about which side is favoured more and which side less if a value of a dimension changes, as captured by the two partial orders \(\le _s\) and \(\le _{s'}\) on a dimension’s values. These partial orders will usually be obvious from the domain. Of course, a potential downside of my alternative approach is that it sacrifices some expressivity of Horty’s approach, as I will further illustrate below after defining the approach.
Dimensions and cases are now again defined as in Sect.
5.2. Below for any two sets
X and
Y of value assignments,
\(Y^{\mid X}\) is the subset of
Y that consists of value assignments to any dimension that is also assigned a value in
X.
So deciding F for s is forced iff there is a precedent for s such that F is at least as favourable for s on all dimensions in the precedent’s rule.
Moreover, like with the reason model with complete rules, the constraint on rule selection is needed that adding a new case to a consistent case base should leave the case base consistent in that for no fact situation two opposite outcomes are forced.
To see how this definition works, consider again the tax example with dimensions \(d_1\) and \(d_2\) and consider precedent c with fact situation \(\{(v(d_1) = 30m), (v(d_2) = 60\%)\}\), with rule \(R = \{(d_1,12m)\}\) and with outcome change. Consider next a fact situation F with \(v(d_1) = 24m, v(d_2) = 50\%\). Then deciding F for change is forced since \(F^{\mid R} = \{(d_1,24m)\}\) and we have that \(R = \{(d_1,12m)\} <_{ change } \{(d_1,24m)\}\). Note that deciding F for change is forced by c even though F is in both dimension weaker for change than c. The point is that with respect to \(d_1\) the rule declares a stay of 12 months sufficient and that \(d_2\) is not in c’s rule.
Since a rule that assigns a value to all dimensions in
D is a special case, the above example that shows that Definition
10 does not collapse into the dimension-based result model also holds for this definition. It is also easy to show that Definition
15 just as its counterpart for cases with complete rules avoids the above-mentioned counterintuitive consequences. Moreover, a counterpart of Proposition
6 can be obtained for this reason model by redefining the relevant differences between a precedent and a focus case as follows. They are now relative to the cases’ rules instead of to its
\(F_D\), and differences can now only be relevant on dimensions that are assigned values in both cases.
Clause (1) says that if the outcomes of the precedent and the focus case are the same, then any value assignment in the focus case to a dimension in the precedent’s rule that is not at least as favourable for the outcome as in the precedent is a relative difference. Clause (2) says that if the outcomes are different, then any value assignment in the focus case to a dimension in the precedent’s rule that is not at most as favourable for the outcome of the focus case as in the precedent is a relative difference. Thus any value assignment in the focus case to a dimension that is not in the rule’s precedent is irrelevant for whether the precedent constrains the focus case, but all value assignments in the focus case to dimensions in the precedent’s rule are relevant for this question, regardless of whether these value assignments are in the focus case’s rule.
On the other hand, this approach also has limitations. Since it does not allow to indicate for a particular value assignment that it favours a particular outcome, some fine-grained distinctions in Horty’s and Rigoni’s approaches between cases where a decision is, or is not forced cannot be made. Consider again the last example. We saw that deciding fact situation F for change was forced by precedent c even though F is weaker for change than c on dimension \(d_2\), since \(d_2\) is not in c’s rule. This might be regarded a problem since the fact that the percentage of income earned abroad was less in the new situation F than in the precedent might by a decision maker be regarded as an exception to the precedent’s rule. In Rigoni’s approach as formalised above this can be formalised by letting, for instance, all values of \(d_2\) greater than or equal to \(60\%\) be the switching point, so that every value less than \(60\%\) favours no change. Then \(d_2 = 50\%\) in F is a relevant difference with precedent c and deciding for change is not forced.
Or consider a variant of this example with the same precedent
c with
\(v(d_1) = 30m, v(d_2) = 60\%\), with outcome
change and with rule
\(\{(d_1,12m), (d_2,60\%)\}\), but with a focus case
f with
\(v(d_1) = 24m, v(d_2) = 75\%\). Next, consider a third dimension
\(d_3\) on whether the tax payer kept his original house or gave it up, with the ordering
kept \(<_{ change }\) gave up, and suppose that in
c we have
\(v(d_3) =\) gave up while in
f we have
\(v(d_3) =\) kept. Then
f is weaker for
change on
\(d_3\) than
c. Yet according to Definition
10 precedent
c cannot be distinguished and deciding
f for change is forced. In Rigoni’s approach as formalised above,
\(v(d_3) =\) gave up can be defined to be a factor favouring no change, and then
\(v(d_3) =\) gave up is a relevant difference between
c and
f.
It seems that the only sensible way to change the definition to allow c to be distinguished is to allow any value assignment to any dimension not mentioned in c’s rule to be a relevant difference, since we cannot indicate whether such a value assignment favours a particular outcome.
Let us discuss this in more general terms. In the factor-based reason model the idea of a rule has a clear intuition, namely, that the pro-decision factors in the rule are sufficient to outweigh the con-decision factors in the case. However, with dimensions, where it cannot always be said that a value assignment favours a particular side, this intuition does not apply, since the value assignments outside the rule do not necessarily favour the opposite outcome. All that can said is that by stating the rule the court has decided that, given the rule, the case’s value assignments to the other dimensions are irrelevant. The question then is whether such a ruling is defeasible. If it is not, then every new case in which the dimensions in the precedent’s rule have values that are at least as favourable to the decision as in the rule is constrained by the precedent regardless of possible differences on the other dimensions. If that is regarded as too rigid, then there are two options. The first is that value assignments to dimensions not in a precedent’s rule can be a reason for distinguishing just in case in the new fact situation they are less favourable for the precedent’s outcome than in the precedent. But then the model collapses into the reason model with complete rules. The second option is that
every value assignment to a dimension that is not in the rule of the case can override the case’s outcome. In other words, every such value assignment can be an exception to the original ruling that the dimensions outside the rule are irrelevant. But then a problem similar to the problem with Horty’s (
2019) reason model arises: in our last example any income percentage, even a percentage higher than
\(60\%\), would suffice to distinguish
c. A possible solution to this problem might be refining the model by allowing but not requiring to indicate that a particular value assignment favours a particular outcome. Then in the last example, indicating that
\(60\%\) is a pro-
change assignment would suffice to prevent
c from being distinguished on higher percentages. I leave the development of this idea to future research. For now it can be concluded that a Rigoni-style approach in which value assignments are always pro a particular outcome leads to finer-grained distinctions between forced and not-forced decisions than the present approach but is arguably harder to apply in practice (although this should be verified in realistic applications). To a lesser extent this also holds for a Horty-style approach with magnitude factors.