Legal Incoherence

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One can conceptualize a legal rule as a mapping from a set of facts (a "scenario") onto an outcome, which can be idealized as "affirmative" or "negative". Among courts with more than one judge, however, that legal rule may be thought of as a composite of (1) a set of functions by which each judge maps some subset of the scenario onto the judge's preferred outcome and (2) an aggregation rule (such as majority rule) that is used to determine the outcome of the case. This Demonstration shows how such a system results in decision rules—"the law"—that reflect the views of no single justice and creates cases that, while decided differently, are, from the perspective of many judges and observers, logically "indistinguishable".


You select the threshold fraction of total votes required for an "affirmative" court decision. By default, this fraction is set at 1/2, meaning that it takes more than 50% of the votes to achieve an affirmative decision. You select the number of "factions" on the court, which represents the number of decision functions used by all the judges. So, if a court has nine members, two of whom see the law one way, three of whom see the law another way, and four of whom see the law still another, there are three factions. You then select how many judges are in each faction. For each judge, you then select the "bits" of data within the scenario they deem relevant. A plot appears to the right showing your selection: black is relevant, white is irrelevant. For each faction, you also select a decision rule that maps the bits of data that faction believes to be relevant onto an outcome. A grid appears to the right of this control showing standard enumerations of the chosen decision rule. The left column shows the standard enumeration on the assumption that only the bits of data the judge actually uses are relevant (the "narrow rule"); the right column shows the enumeration of the equivalent rule on the assumption that all the data in the scenario is relevant (the "broad rule"). The "narrow rule" thus can range from 0 to , where is the number of bits that faction considers to be relevant. The equivalent "broad rule" can range from 0 to , where is the maximum number (five) of relevant bits. Hovering your mouse over these enumeration numbers produces a decision tree for the rule in which the symbols reflect the first through fifth factual bits.

The Demonstration computes and displays the enumeration of the actual decision rule produced when the judges use their individual decision rules to vote. It then produces a chart showing, for each faction, a list of perceived incoherencies. An "incoherency" exists when at least two scenarios that the judge thinks should have the same outcome end up having a different outcome because of the aggregation of votes. Each incoherency is visualized as an array showing the scenarios that the faction thinks should be treated the same and a pie chart showing the fraction of times the case will in fact be decided affirmatively and the fraction of times the case will in fact be decided negatively. If there are more than six incoherent scenarios for a faction, the display is truncated to show only the first six.


Contributed by: Seth J. Chandler (March 2011)
Open content licensed under CC BY-NC-SA



Snapshot 1: only the judges in the majority faction fail to see any incoherence in the law

Snapshot 2: in a highly factionalized court, all the judges see multiple instances of incoherence

Snapshot 3: a three-faction court requiring at least a 67% vote in which all of the factions see incoherence

Snapshot 4: a five-faction court in which all the judges see instances of incoherence

This Demonstration is partly motivated by the difficulties law students and legal practitioners often have reconciling bodies of precedent in order to determine the "true legal rule" that can be applied in a subsequent case. While such difficulty in reconciliation—"incoherence"—is sometimes simply the result of history or of disagreements between the judges and the person studying the law about the ways in which certain facts can be relevant, this Demonstration suggests that, at least in theory, the incoherence derives simply from the process of aggregating votes of judges whose own internal rules are indeed "coherent". That aggregation may result in only a small set of judges (or possibly no judges) seeing complete coherence between legal scenarios and outcomes.

If a faction does not consider all the facts in a scenario to be relevant, the "broad rule" will contain a number of "don't care" bits.

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