¹ Assuming she is unwilling or unable to overrule the precedent. For the view that this strict rule model is the normatively best doctrine of precedent, the one the courts ought to adopt, see, e.g., Larry Alexander, Constrained by Precedent, 63 S. Cal. L. Rev. 1 (1989–1990); Larry Alexander & Emily Sherwin, The Rule of Rules (2001).

² At least concerning horizontal precedents. For an argument that the strict rule model actually does capture the doctrine of vertical precedent in the United States, see Gabriel L. Broughton, Vertical Precedents in Formal Models of Precedential Constraint, 27 A. I. & L. 253 (2019). For criticism of the strict rule model on grounds of descriptive adequacy, see, e.g., Ronald Dworkin, The Model of Rules I, in Taking Rights Seriously 14 (1977); Grant Lamond, Do Precedents Create Rules?, 11 Legal Theory 1 (2005); Karl Llewellyn, The Common Law Tradition (1960). See also Rupert Cross & J. W. Harris, Precedent in English Law 98 (4th ed. 1991); Bryan A. Garner, et al., The Law of Judicial Precedent §§3, 8 (2016).

³ See, e.g., Benjamin N. Cardozo, The Nature of the Judicial Process (1921); Karl Llewellyn, The Bramble Bush 67 (2008); Joseph Raz, Law and Value in Adjudication, in The Authority of Law 180 (2nd ed. 2009); A. W. B. Simpson, The Ratio Decidendi of a Case and the Doctrine of Binding Precedent, in Oxford Essays in Jurisprudence 148 (A. G. Guest, ed. 1961). On some of the vices of rule-based decision-making in general, see, e.g., David A. Strauss, On the Origin of Rules (with Apologies to Darwin), 75 U. Chi. L. Rev. 997 (2008); Cass R. Sunstein, Problems with Rules, 83 Cal. L. Rev. 953 (1995).

⁴ The term “result model” is due to Alexander, supra note 1. Some version of the model is endorsed in, e.g., Lewis A. Kornhauser, Modeling Collegial Courts. II. Legal Doctrine, 8 J. L. Econ. & Org. 441 (1992); Edward H. Levi, An Introduction to Legal Reasoning (1949); G. W. Paton, A Textbook of Jurisprudence 159 (1946). For criticism, see, e.g., Larry Alexander & Emily Sherwin, Judges as Rule Makers, in Common Law Theory 27, 35–40 (Douglas E. Edlin, ed. 2007).

⁵ The result model is plausibly connected to some form of moral particularism, along the lines of Jonathan Dancy, Moral Reasons (1993).

⁶ Some of the key ideas underlying the reason model were introduced by Lamond, supra note 1. John Horty has developed an especially precise version in a series of papers beginning with Rules and Reasons in the Theory of Precedent, 17 Legal Theory 1 (2011) (hereinafter Horty, Rules and Reasons). See also John Horty, Constraint and Freedom in the Common Law, 15 Philosophers’ Imprint 1 (2015) (hereinafter Horty, Constraint and Freedom); John Horty & Trevor J. M. Bench-Capon, A Factor-Based Definition of Precedential Constraint, 20 A. I. & L. 181 (2012). For an alternative interpretation of Lamond’s ideas, see Robert Mullins, Protected Reasons and Precedential Constraint, 26 Legal Theory 40 (2020). For criticism, see Mark McBride, No Reasons, in New Essays on the Nature of Legal Reasoning 99 (Mark McBride & James Penner, eds. 2022).

⁷ See John Horty, The Result Model of Precedent, 10 Legal Theory 19 (2004); Horty, Rules and Reasons, supra note 6; Horty, Constraint and Freedom, supra note 6.

⁸ The framework was developed by Kevin Ashley and Vincent Aleven for CATO, an intelligent learning environment used to teach legal reasoning and argumentation to law students. See Vincent Aleven & Kevin D. Ashley, What Law Students Need to Know to Win, in Proc. Fourth Int’l Conf. A. I. & L. 152 (1993); Vincent Aleven & Kevin D. Ashley, Doing Things with Factors, in Proc. Fifth Int’l Conf. A. I. & L. 31 (1995); Vincent Aleven, Teaching Case-Based Argumentation through a Model and Examples (1997) (unpublished Ph.D. thesis, University of Pittsburgh). CATO was itself a successor to the HYPO system, due to Kevin Ashley and Edwina Rissland. See Kevin Ashley, Toward a Computational Theory of Arguing with Precedents, in Proc. Second Int’l Conf. A. I. & L. 93 (1989); Kevin Ashley, Modeling Legal Argument (1990).

⁹ Horty, Rules and Reasons, supra note 6.

¹⁰ See, e.g., Trevor Bench-Capon and Katie Atkinson, Dimensions and Values for Legal CBR, in Proc. Thirtieth Int’l Conf. Legal Knowledge & Info. Sys. 11 (2017); Trevor Bench-Capon and Katie Atkinson, Lessons from Implementing Factors with Magnitude, in Proc. Thirty-First Int’l Conf. Legal Knowledge & Info. Sys. 11 (2018).

¹¹ See Henry Prakken & Giovanni Sartor, Modelling Reasoning with Precedents in a Formal Dialogue Game, 6 A. I. & L. 231 (1998).

¹² See Trevor Bench-Capon, Some Observations on Modeling Case-Based Reasoning with Formal Argument Models, in Proc. Seventh Int’l Conf. A. I. & L. 36 (1999).

¹³ Dimensions first appeared in Ashley, supra note 8, and Edwina L. Rissland & Kevin D. Ashley, A Case-Based System for Trade Secrets Law, in Proc. First Int’l Conf. A. I. & L. 60 (1987).

¹⁴ John Horty, Reasoning with Dimensions and Magnitudes, 27 A. I. & L. 309, 329 (2019) (Observation 3).

¹⁵ Id.

¹⁶ See id. at §4.2; John Horty, Modifying the Reason Model, 29 A. I. & L. 271 (2021); Henry Prakken, A Formal Analysis of some Factor- and Precedent-Based Accounts of Precedential Constraint, 29 A. I. & L. 559 (2021); Adam Rigoni, Representing Dimensions within the Reason Model of Precedent, 26 A. I. & L. 1 (2018).

¹⁷ Since the basics of these models are well known, I move briskly through the initial set-up. Readers who would appreciate a more leisurely introduction might consult Horty, Constraint and Freedom, supra note 6, or Horty & Bench-Capon, supra note 6.

¹⁸ Definitions for the result model follow Horty, supra note 7.

¹⁹ See Horty, Rules and Reasons, supra note 6, at 8. See also Lamond, supra note 2, at 15–16.

²⁰ We decline to define the relative strength of reasons for the same side directly in terms of set inclusion in order to make things easier on ourselves when we eventually generalize these definitions to the dimensional setting. See Horty, supra note 14, at 314.

²¹ Recall that $ {c}_1=\left\langle {X}_1,{r}_1,{s}_1\right\rangle $ , where $ {X}_1=\left\{{f}_1^{\pi },{f}_2^{\pi },{f}_3^{\pi },{f}_1^{\delta },{f}_2^{\delta}\right\} $ , $ {r}_1=\left\{{f}_1^{\pi },{f}_2^{\pi}\right\}\to \pi $ , and $ {s}_1=\pi $ .

²² Horty, Rules and Reasons, supra note 6, at 23–24 (Observation 5).

²³ Horty, supra note 14, at 319 (Observation 1).

²⁴ Prakken, supra note 16, employs the notion of a particular decision’s being forced (i.e., required) in the context of a case base without explicitly appealing to the idea of a forcing case. Ilaria Canavotto’s notion of a supporting case is similar to that of a forcing case as defined above, except that a supporting case, in Canavotto’s sense, requires not only that a new dispute Y be decided for s. It also requires that Y be decided for s on the basis of a particular rule. This is only a rough description of the notion of a supporting case, however, as it elides certain complications that arise in the context of case bases that are inconsistent (which this paper ignores). For the details, see Canavotto, Reasoning with Inconsistent Precedents, 33 A. I. & L. 137 (2023).

²⁵ See supra note 16.

²⁶ Following Horty, supra note 14, at §3.

²⁷ Id. at 321.

²⁸ Id.

²⁹ Id. at 321–322.

³⁰ Id. at 321–322.

³¹ Following Horty, supra note 14, at §3.2.

³² See id. at 325.

³³ Again, this follows from Horty, supra note 14, at 319 (Observation 1).

³⁴ Id. at 329 (Observation 3).

³⁵ Horty, supra note 16, at 284.

³⁶ See Horty, supra note 14, at 330 (Definition 13).

³⁷ This same analysis applies straightforwardly to Horty’s second proposed refinement of the reason model as well, on which courts are required to decide cases based on acceptable rules. See id. at 332 (Definition 14). In more recent work, Horty abandons the idea of adding a further requirement to the reason model beyond maintaining the consistency of the case base. Instead, he proposes to strengthen the priority ordering on reasons that can be derived from a single decision for s, so that even where a fresh dispute offers new reasons to decide for $ \overline{s} $ , a decision for $ \overline{s} $ based on those reasons will sometimes be counted inconsistent with the precedent decision. See Horty, supra note 16, at 282 (Definition 9). Like the other proposals, this modification effectively recovers the possibility of removing reasons to decide for one side without adding reasons to decide for the other. That is, it recovers the possibility of removing reasons to decide for s without adding reasons to decide for $ \overline{s} $ that courts are actually permitted to rely upon.

³⁸ Rigoni, supra note 16.

³⁹ For a full formalization of the ideas in Rigoni, supra note 16, at least for the special case of “complete” dimensions, see Prakken, supra note 16, at §5.3.

⁴⁰ See Prakken, supra note 16, at §5.3.

⁴¹ Cf. Bench-Capon, supra note 12, at 17 (making this assumption explicit).

⁴² “Negative” factors have not received a great deal of attention in the literature, but see Horty, Rules and Reasons, supra note 6, at §7; Prakken, supra note 16, at §4.

⁴³ See, e.g., Sloan v. 216 Bedford Kings Corp., 208 A.D.3d 1192, 1193–1194 (N.Y. App. Div. 2022) (defendant had notice); Vickers v. Parcells, 198 A.D.3d 1160, 1164 (N.Y. App. Div. 2021) (defendant lacked notice).

⁴⁴ See, e.g., Campaniello v. New York State Div. of Tax Appeals Tribunal}, 75 N.Y.S.3d 706, 709 (2018) (defendant kept his condo); McKone v. State Tax Comm’n of State of N.Y., 490 N.Y.S.2d 628, 629 (1985) (defendants sold their house).

⁴⁵ See, e.g., Gomillion v. Lightfoot, 364 U.S. 339 (1960) (discriminatory purpose); Washington v. Davis, 426 U.S. 229 (1976) (no discriminatory purpose).

⁴⁶ Trevor Bench-Capon, Representing Popov v Hayashi with dimensions and factors, 20 A. I. & L. 15, 16 (2012).

⁴⁷ Trevor Bench-Capon & Edwina L. Rissland, Back to the Future: Dimensions Revisited, in Proc. Fourteenth Int’l Conf. Legal Knowledge & Info. Sys. 41, 43 (2018); Edwina L. Rissland & Kevin D. Ashley, A Note on Dimensions and Factors, 10 A. I. & L. 65, 69 (2002).

⁴⁸ 793 F.2d 875 (7th Cir. 1986).

⁴⁹ Id. at 885 (emphasis added).

⁵⁰ Id. at 886 (emphasis added).

⁵¹ For further examples, see, e.g., Life Spine, Inc. v. Aegis Spine, Inc., 8.F.4th 531, 541, 542 (7th Cir. 2021) (noting that neither the plaintiff’s patent nor its product displays disclosed the relevant trade secret to outsiders); A. L. Laboratories, Inc. v. Philips Roxane, Inc., 803 F.2d 378, 381 (8th Cir. 1986). See also Kubik, Inc. v. Hull, 56 Mich. App. 335, 347 (1974) (suggesting that, in order to establish the existence of a legally protectable trade secret, “[p]laintiff must prove that secrecy has been maintained either by non-disclosure or [by] disclosure in confidence”) (emphasis added).

⁵² The idea that some “positive” factors lack “negative” counterparts can still seem paradoxical, though, I think. If a possible reason to decide for the plaintiff is absent, how can that be anything but bad news for the plaintiff? And given the zero-sum nature of litigation, how could bad news for the plaintiff be anything but good news for the defendant? I suspect that the way to remove the air of paradox is to distinguish between the strength of the reasons to decide for s, on the one hand, and the probability that s will win, on the other. If we learn that the defendant in a trade-secrets case did not bribe an employee at her former company, for example, then this will generally raise the probability that the defendant will win, even if it does not necessarily provide a reason for the court to decide for the defendant.

⁵³ Horty, Rules and Reasons, supra note 6, at 26.

⁵⁴ For a suggestion along these lines, see, e.g., Horty, Rules and Reasons, supra note 6, at 26–27.

⁵⁵ Consider a dispute involving an innocent misrepresentation by one party to a contract, for example. If the court doesn’t say whether the plaintiff relied on the misrepresentation, then it is probably reasonable for future courts to assume that she did, since any court that found otherwise would almost certainly have said so. See Lamond, supra note 6, at 127.

⁵⁶ Actually, we don’t even have to assume there is a default rule. It may be that whether it is reasonable to assume $ f $ or to assume $ \neg f $ depends on further details about the precedent dispute. Given one context, it might make sense to respond to silence by treating the case as one of $ f $ ; given a different context, it might be more reasonable to treat it as one of $ \neg f $ . All that Definition 8 requires is that each particular instance of judicial silence calls for either one move or the other.

⁵⁷ Horty, supra note 14 (Observation 1).

⁵⁸ Id.

⁵⁹ Id.