# Reasoning with dimensions and magnitudes

• John Horty
Original Research

## Abstract

This paper shows how two models of precedential constraint can be broadened to include legal information represented through dimensions. I begin by describing a standard representation of legal cases based on boolean factors alone, and then reviewing two models of constraint developed within this standard setting. The first is the “result model”, supporting only a fortiori reasoning. The second is the “reason model”, supporting a richer notion of constraint, since it allows the reasons behind a court’s decisions to be taken into account. I then show how the initial representation can be modified to incorporate dimensional information and how the result and reason models can be adapted to this new dimensional setting. As it turns out, these two models of constraint, which are distinct in the standard setting, coincide once they are transposed to the new dimensional setting, yielding exactly the same patterns of constraint. I therefore explore two ways of refining the reason model of constraint so that, even in the dimensional setting, it can still be separated from the result model.

## Keywords

Precedent Constraint Dimensions

## Notes

### Acknowledgements

This paper is a light revision and expansion of an earlier version (Horty 2017) that was presented at ICAIL 2017, where it won the Carole Hafner Best Paper Award. This recognition was especially meaningful to me since I have fond memories of working with Carole to organize ICAIL 1995. It was in helping Carole organize that conference that I first learned about the field of AI and Law, and how interesting it could be. I am deeply indebted to Trevor Bench-Capon for raising the issue that motivated this paper in the first place, for his enthusiasm, and for his generous comments on earlier drafts. Apart from providing some additional examples and explanations, along with formal proofs of observations, this revision differs from the earlier version appearing in ICAIL 2017 in only two ways. First, the method of interpreting standard information into dimensional information described here, in the Appendix, differs from the interpretation described in Sect. 4 of that earlier version—both interpretations work, and and they provide different insights. Second, this version of the paper includes a discussion of the first refinement of the reason model, found here in Definition 13, which was mentioned in my ICAIL 2017 presentation but not included in the earlier version for reasons of space. Since my intention in this paper is simply to present in a more careful and systematic way the material from the earlier ICAIL 2017 version, I have not considered work that has appeared since then, including Rigoni (2018) and Bench-Capon and Atkinson (2017). I hope to be able to discuss this more recent work in the future.

## References

1. Al-Abdulkarim L, Atkinson K, Bench-Capon T (2016) ANGELIC secrets: bridging from factors to facts in US trade secrets law. In: Proceedings of the twenty-ninth international conference on legal knowledge and information systems (JURIX 2016), pp 113–118Google Scholar
2. Aleven V, Ashley K (1997) Evaluating a learning environment for case-based argumentation skills. In: Proceedings of the sixth international conference on artificial intelligence and law (ICAIL-97). The Association for Computing Machinery Press, pp 170–179Google Scholar
3. Alexander L (1989) Constrained by precedent. South Calif Law Rev 63:1–64Google Scholar
4. Allen L (1957) Symbolic logic: a razor-edged tool for drafting and interpreting legal documents. Yale Law J 66:33–79Google Scholar
5. Ashley K (1990) Modeling legal argument: reasoning with cases and hypotheticals. The MIT Press, CambridgeGoogle Scholar
6. Bench-Capon T (1999) Some observations on modelling case based reasoning with formal argument models. In: Proceedings of the sixth international conference on artificial intelligence and law (ICAIL-99). The Association for Computing Machinery Press, pp 36–42Google Scholar
7. Bench-Capon T (2002) The missing link revisited: the role of teleology in representing legal argument. Artif Intell Law 10:79–94
8. Bench-Capon T, Atkinson K (2017) Dimensions and values for legal CBR. In: Legal knowledge and information systems: JURIX 2017, the thirtieth annual conference. IOS Press, pp 27–32Google Scholar
9. Bench-Capon T, Rissland E (2001) Back to the future: dimensions revisited. In: The fourteenth annual conference on legal knowledge and information systems (JURIX-2001). IOS Press, pp 41–52Google Scholar
10. Bench-Capon T, Sartor G (2003) A model of legal reasoning with cases incorporating theories and values. Artif Intell 150:97–143
11. Bench-Capon T, Robinson G, Routen T, Sergot M (1987) Logic programming for large scale applications in law: a formalisation of supplementary benefit legislation. In: Proceedings of the first international conference on artificial intelligence and law (ICAIL-87). The Association for Computing Machinery Press, pp 190–198Google Scholar
12. Berman D, Hafner C (1993) Representing teleological structure in case-based legal reasoning: the missing link. In: Proceedings of the fourth international conference on artificial intelligence and law (ICAIL-93). The Association for Computing Machinery Press, pp 50–59Google Scholar
13. Duxbury N (2005) The nature and authority of precedent. Cambridge University Press, CambridgeGoogle Scholar
14. Dworkin R (1977) Taking rights seriously. Harvard University Press, CambridgeGoogle Scholar
15. Dworkin R (1986) Law’s empire. Harvard University Press, CambridgeGoogle Scholar
16. Eisenberg M (1988) The nature of the common law. Harvard University Press, CambridgeGoogle Scholar
17. Goodhart A (1930) Determining the ratio decidendi of a case. Yale Law J 40:161–183
18. Hage J (1993) Monological reason-based logic: a low-level integration of rule-based and case-based reasoning. In: Proceedings of the fourth international conference on artificial intelligence and law (ICAIL-93). Association for Computing Machinery Press, pp 30–39Google Scholar
19. Horty J (2004) The result model of precedent. Leg Theory 10:19–31Google Scholar
20. Horty J (2011) Rules and reasons in the theory of precedent. Leg Theory 17:1–33
21. Horty J (2015) Constraint and freedom in the common law. Philos Impr 15(25):1–27Google Scholar
22. Horty J (2017) Reasoning with dimensions and magnitudes. In: Proceedings of the sixteenth international conference on artificial intelligence and law (ICAIL 2017). The Association for Computing Machinery Press, pp 109–118Google Scholar
23. Horty J, Bench-Capon T (2012) A factor-based definition of precedential constraint. Artif Intell Law 20:181–214
24. Lamond G (2005) Do precedents create rules? Leg Theory 11:1–26Google Scholar
25. Loui R, Norman J, Olson J, Merrill A (1993) A design for reasoning with policies, precedents, and rationales. In: Proceedings of the fourth international conference on artificial intelligence and law (ICAIL-93). The Association for Computing Machinery Press, pp 201–211Google Scholar
26. Prakken H (2002) An exercise in formalising teleological case-based reasoning. Artif Intell Law 10:113–133
27. Prakken H, Sartor G (1998) Modelling reasoning with precedents in a formal dialogue game. Artif Intell Law 6:231–287
28. Prakken H, Wyner A, Bench-Capon T, Atkinson K (2015) A formalization of argumentation schemes for legal case-based reasoning in ASPIC+. J Log Comput 25:1141–1166
29. Raz J (1979) The authority of law. Oxford University Press, OxfordGoogle Scholar
30. Rigoni A (2018) Representing dimensions within the reason model of precedent. Artif Intell Law 26:1–22
31. Rissland E, Ashley K (1987) A case-based system for trade secrets law. In: Proceedings of the first international conference on artificial intelligence and law (ICAIL-87). The Association for Computing Machinery Press, pp 60–66Google Scholar
32. Rissland E, Ashley K (2002) A note on dimensions and factors. Artif Intell Law 10:65–77
33. Rissland E, Skalak D (1989) Interpreting statutory predicates. In: Proceedings of the second international conference on artificial intelligence and law (ICAIL-89). The Association for Computing Machinery Press, pp 46–53Google Scholar
34. Schroeder M (2007) Weighting for a plausible Humean theory of reasons. Nous 41:110–132
35. Sergot M, Sadri F, Kowalski R, Kriwaczek F, Hammond P, Cory HT (1986) The British nationality act as a logic program. Commun Assoc Comput Mach 29:370–386Google Scholar
36. Simpson AWB (1961) The ratio decidendi of a case and the doctrine of binding precedent. In: Guest AG (ed) Oxford essays in jurisprudence. Oxford University Press, Oxford, pp 148–175Google Scholar