From Hercules to Ariadne

There is a curious feature of the most famous quarrel in modern jurisprudence. For half a century, the question whether hard cases have right answers has been argued with great subtlety by thinkers who agreed, in outline, about what the disagreement would look like if one could see it. Hart's judge in the penumbra and Dworkin's Hercules confront the same kind of case. Where the law seems to give out, one says the judge must choose; the other says the judge must look harder. And yet the quarrel has been conducted largely as a dispute about concepts, not about anything that might be counted.

Ross Brodskiy and Nathan Pokov's Ariadne's Thread is, among other things, an attempt to count. Its ambition is not to win the argument but to change its register. To see why that is a significant move, and why it is also a limited one, it helps to recall how the argument came to be stuck.

The quarrel

H.L.A. Hart's The Concept of Law, first published in 1961, gave positivism its most durable image of adjudication. General rules have a core of settled meaning in which any competent decider reaches the same result, and a penumbra of open texture in which the rule does not dictate the outcome. In the penumbra, on Hart's account, the judge exercises a genuine discretion.

Ronald Dworkin denied that conclusion across a series of works, from "Hard Cases" in the Harvard Law Review in 1975, through Taking Rights Seriously in 1977, to Law's Empire in 1986. His central claim was that judges do not possess strong discretion even in hard cases, because the interpretation of the legal record that best fits the settled materials and best justifies them as a matter of political morality supplies an answer. To make the claim vivid he imagined Hercules, a judge of superhuman skill and patience who could actually construct that interpretation and rule accordingly.

Around this central exchange gathered an older and a younger skepticism. The legal realists had observed that doctrine tends to supply a thrust and a parry for any contested question, so that the citation of authority understates how much the decider chose; Duncan Kennedy's "Form and Substance in Private Law Adjudication" carried a version of that observation into 1976. Joseph Raz, writing on the relevance of coherence in 1992, pressed a skepticism about coherence that cut against the coherentist picture on which Hercules depends. By 2007, Scott Shapiro could write a short guide to the Hart-Dworkin debate for the perplexed, which is itself a sign that the debate had become a landscape rather than a road.

Why did it not end? One reason is structural. Each side could absorb every observation the other offered. When judges divided, the positivist saw discretion and the Dworkinian saw an answer not yet found. When judges converged, the Dworkinian saw the right answer and the positivist saw the core of a rule doing its ordinary work. A debate in which every datum can be redescribed by both parties is not one that ordinary observation will close. It needed a different kind of observation, or a different question.

The ideal that decides

Hercules was never meant as a blueprint. He was a thought experiment about the existence of right answers, a way of saying what law would have to be like for adjudication to be something other than disguised legislation. But ideals have a way of becoming specifications when the tools arrive. Once language models could write fluent legal prose, it became natural to ask whether a machine Hercules might be built. Brodskiy's Russian-language essay for continental lawyers, From the Supreme Court Machine to Ariadne's Thread, names the temptation directly: the dream of a Supreme Court machine.

The paper declines the dream, and its reasons are worth dwelling on, because they are not the obvious ones. It notes, first, that no one has built Hercules and that the artificial intelligence and law tradition has not tried. Its landmark systems modeled fragments of legal reasoning: case-based argument in the HYPO and CATO line, formal models of argument and precedential constraint, outcome prediction. The nearest approach to Hercules, the theory construction with values of Bench-Capon and Sartor, builds the single most coherent value theory that explains a body of cases and uses it to decide the next one. It is a convergence machine, and it is Herculean in exactly that respect.

The deeper reasons are two. Hercules decides, and an instrument that issues verdicts inherits every legitimacy objection that attaches to automated adjudication, as it should. And Hercules presupposes the thing in dispute. His method makes sense only if a single best interpretation exists, and that is precisely what the half-century quarrel has not settled. An instrument built in his image would be built on a contested premise.

Here is the paradox the paper exposes, though it does not put it this way. The most sophisticated philosophical defense of legal determinacy turns out to be the worst possible design brief for an instrument meant to study determinacy. An ideal of judgment is measured by the quality of its verdicts. An instrument is measured by the honesty of its readings. To build the first is to assume an answer; to build the second is to leave the answer open long enough to observe it.

The paper locates the object of observation in a seam. Every contested ruling, it argues, fuses an analyzable part, what follows mechanically from a fixed method, fixed authorities and a fixed view of the facts, with a committed part, the method and the weighting of values the decider actually holds. Hart's penumbra is the place where the two fuse. Dworkin's denial of strong discretion is the claim that the seam can in principle be un-welded, so that the value-choice is exposed and defended rather than smuggled in. Positivist and Dworkinian agree that both parts exist; they disagree about whether the committed part is ever genuinely unconstrained. The paper's quiet insight is that an instrument needs only the agreement.

There is a question of legitimacy folded into that seam. Law transforms power into authority partly through a grammar of necessity: the law requires this. When the law does require it, the grammar is honest. When a differently disposed but equally lawful decider could have ruled the other way, the grammar conceals a choice, and a choice concealed as compulsion cannot be contested, because one cannot argue with a compulsion. The paper's anti-pretext output, which marks the points where a discretionary setting is presented as a legal command, is the instrument's answer to that old concealment. It does not accuse. It locates.

From premise to quantity

In place of Hercules the authors set Ariadne, who in the myth neither fights the Minotaur nor walks the labyrinth on Theseus's behalf. She hands him the thread. The dispute is the labyrinth, the lawful postures available on the question are its corridors, and the metric Z is the thread. She guides; she does not rule.

The move itself can be stated in a sentence. Take the question the Hart-Dworkin debate fights over, whether a contested case has a determinate answer, and convert it from a premise to be assumed into a quantity to be measured: the systematic dispersion of qualified, differently disposed adjudicators confronting the same case, net of the instrument's own inconsistency. If lawful postures converge, the merits overdetermine the outcome and the identity of the judge does not predict it. If they diverge, the case is open and posture breaks the tie. If the divergence is no larger than the instrument's noise, the instrument abstains, because it cannot tell openness from error.

A posture, in this vocabulary, is a filter and an ordering over one shared list of interpretive and procedural moves. Continental readers will recognize the family resemblance, and the same Russian essay makes the point for them: the civilian tradition already knows grammatical, systematic, teleological and historical interpretation as distinct ways of reading shared materials, and its ideal of legal certainty is one that, as the essay observes, has never been measured.

Measuring disagreement among adjudicators is not, in itself, new. Asylum grant rates within a single immigration courthouse have been reported to range from five to eighty-eight percent, and noise audits in the sense of Kahneman, Sibony and Sunstein already price the spread of human judgment. What the paper adds is control. It holds the legal materials fixed by requiring every posture to reason over the same verified, good-law substrate; it varies only the posture; and it subtracts the spread that the instrument produces when nothing about the posture has changed. It then proves that the resulting quantity behaves as a ruler should. The authors put the discipline in a sentence that deserves to be quoted exactly: "we prove properties of the ruler; we do not prove facts about what we measure with it."

The philosophical consequence is subtle. The instrument does not need to win the debate before it can be used. It reports case by case. Run over a representative population of disputes, the distribution of Z would say which picture the evidence favors. If right answers are pervasive, most cases read low, with a thin tail of hard ones. If the law is pervasively open, most cases read high. A metaphysical question has been converted, at least in part, into a demographic one: not whether the law runs out, but how often, and where.

That is the whole trick. It is not a small one.

What the measurement cannot settle

And yet the number, even a well-made one, does not end the quarrel. The paper is candid about this, and its candor is instructive.

Consider a gate where Z is zero. The positivist reads a core case in Hart's sense: the law settles the matter. The Dworkinian reads a case in which the merits so dominate that even imperfectly instantiated postures converge on the answer Hercules would also find. Now consider a gate where Z is high. The positivist reads penumbral discretion. The Dworkinian reads a place where Hercules would still find an answer that a cheaper instrument cannot see. The number is the same; only the gloss differs. The paper calls the method, in this sense, jurisprudentially ecumenical, and the later versions say plainly that it is not a proof of Hart and not a refutation of Dworkin.

What changes, then? Not the verdict of the debate, but its terrain. A Dworkinian who accepts the measurement must now say that Hercules finds answers precisely where every lawful posture, reasoning over the same verified law, parts company. That is a coherent position. It is also a more exposed one than before, because it has been located. The positivist, for his part, must accept that the penumbra, if it is measured, will have a size and a shape, and that its size may turn out to be smaller or larger than the theory's rhetoric implied.

There is a second limit, and it has an irony worth savoring. Litigated cases are a selected sample. Parties settle the cases whose outcomes they can predict, a point Priest and Klein made in 1984, so the disputes that reach decision are biased toward the hard ones. The paper turns this into support: the cases that settle are, in its terms, low-Z cases, and the settlement rate is itself an estimator of how much of the law is closed. But the jurisprudential implication runs further. The philosophy of adjudication has been written largely from the hard cases, because those are the cases that reach opinions. The law's easy cases are nearly invisible precisely because they are easy. The courtroom shows us the densest part of the labyrinth and invites us to mistake it for the whole.

The third limit is the most serious, and it is empirical. The instrument works only if its simulated adjudicators can genuinely hold different postures. Recent experiments by Posner and Saran report that a frontier model applied the legally correct outcome in every tested case, where human judges in the comparison did so roughly half the time, and that it resisted being steered toward the human pattern. The machine that popular anxiety imagined as Hercules turns out, on this evidence, to resemble something more modest: a formalist clerk of the positivist's core. The paper's answer is that the finding concerns the closed, rule-like region of the law, where Ariadne is built to read near zero. Whether the same models can hold distinct postures in the open region is the question on which the whole program depends, and the authors defer it to a calibration experiment they have named and not yet run.

Beneath all three limits lies the deepest objection, that adjudication is a normative practice of justification and cannot be reduced to the measurement of a proxy. The paper does not refute this. It answers by scope. The instrument does not adjudicate and does not claim to measure justice. It measures one proxy and hands it to a human who retains the decision, in keeping with the principle, stated in the European ethical charter on artificial intelligence in judicial systems, that such tools remain under the user's control.

That, finally, is why the corrected ideal is Ariadne and not Hercules. Hercules reads the whole law and rules the right answer. Ariadne reads the verified law and rules nothing. She measures how open the maze is and gives the thread to the one who must walk it.

For half a century, jurisprudence asked whether the law has an answer. The better question, it may turn out, was how often it does, and where, and whether we could learn to tell the difference before the judge rises to speak.