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Legal Conventionalism pp 127–144Cite as

On Identifying the Law and Its Supposed Conventional Foundations. A Set-Theory Approach

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Part of the book series: Law and Philosophy Library ((LAPS,volume 126))

Abstract

The paper explores some possible applications of some elementary notions of set theory (such as “intersection”, “union”, and “difference”) to two age-old and intertwined problems of legal philosophy: the identification of law and the rational reconstruction of an ultimate rule of recognition. The main results of the inquiry are the following. (1) Contrary to a widespread view, “law” does not denote a unitary set of rules, but rather a family of sets, the main relations among which are examined in the paper. (2) Contrary to what the strong version of legal conventionalism advocates, there is a conceptual space to regard the rule of recognition as the overlapping of several sets of criteria of identification that are used within a certain community. Amongst other things, this thesis renders the famous Dworkin’s critique against conventionalism harmless.

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Notes

  1. 1.

    Here I shall use “norm” and “rule” synonymously.

  2. 2.

    Set theory is one of the fundamental parts of mathematical logic and is the product of some investigations seeking satisfactory definitions of “number” and “infinity”. For my discussion of the basic elements of set theory, I am highly indebted to Berto (2007, pp. 145 ff.); Lemmon (1965, appendix B); Quine (1961, 1963), Quine (1974, pp. 235 ff.); Russell (1918); Suppes (1960).

  3. 3.

    Alchourrón and Bulygin (1971, 1979); Rodríguez (2006), pp. 242–276; Ferrer Beltrán and Rodríguez (2011).

  4. 4.

    This apparently intuitive and innocuous principle leads to the well-known Russell’s paradox. Since in what follows I will not deal with questions which generate such a paradox, I will use this principle as a simple means of analysis. However, I am well aware that some more complex versions of this principle may avoid such paradoxical outcomes. For one of these versions, see Suppes (1960), p. 7. For discussion on Frege’s reply to Russell’s paradox, see Quine (1955), pp. 145–159.

  5. 5.

    An important concept of set theory, which will not be used in this essay, is that of “inclusion”. Inclusion is the relation between two sets α and β, such that every element of α is also an element of β. One should distinguish improper inclusion, which admits the possible identity of both sets, from proper inclusion, which excludes it. In other words, a set α improperly includes another set β, when the latter is a subset of the former, i.e. all the elements of β belong to α, which may (but need not) contain other elements. By contrast, a set α properly includes another set β when the latter is a subset of the former, which, in turn, needs to contain other elements (since, as has been affirmed, both sets cannot be identical).

  6. 6.

    This operation brings about a partition in the universe of discourse. Suppes (1960), pp. 83 ff.

  7. 7.

    As is known, for the laws of De Morgan, the negation of the conjunction of two propositions is equivalent to the disjunction with both original propositions denied, and the negation of the disjunction of two propositions is equivalent to the conjunction of both original propositions denied.

  8. 8.

    By a “wholly codified legal order” I mean one in which the legal sources are all written down in legal documents. By “legal sources”, I mean, following Ross (1958), p. 77: “the aggregate of factors which exercise influence on the judge’s formulation of the rule on which he bases his decision”.

  9. 9.

    Bulygin (2015), pp. 243–246.

  10. 10.

    Obviously, this list of criteria is far from being complete: many other criteria might be formulated.

  11. 11.

    Cfr. Alchourrón and Bulygin (1979) and Bulygin (2015), ch. 9.

  12. 12.

    For a paradigmatic example of a neo-formalistic theory of legal interpretation, see Marmor (2005).

  13. 13.

    Kelsen (1992), p. 80.

  14. 14.

    Guastini (2011b), pp. 227–230.

  15. 15.

    Ferrer Beltrán and Rodríguez (2011), p. 80.

  16. 16.

    It is worth observing that some, or even all, of the formulations used in deciding a legal issue may not belong to the legal system. The officials that should decide the issue may sometimes resort to normative systems which are not even legal (e.g. moral systems). In such cases, the analysis based on set theory allows one to show that, in the ordinary jargon, some elements which are dubbed as “legal” belong, actually, to its complement, that is the non-legal.

  17. 17.

    For discussion cf. Raz (1994), ch. 10.

  18. 18.

    Tarski (1983), pp. 346–350.

  19. 19.

    On rules logical consequences (or, as he dubs them, “juridical consequences”) see also MacCormick (1983), pp. 249–254.

  20. 20.

    For instance, the statute that provides that one is entitled to inherit if declared an heir by a valid will, entails the norm, derived by adding an irrelevant property, that if one has been declared an heir by a valid will and is blue-eyed (irrelevant property), then that person is entitled to inherit. I draw the example from Dworkin (1986), p. 351.

  21. 21.

    On the defeasibilist turn, see generally Ferrer Beltrán and Ratti (2012a, pp. 11–38); Ferrer Beltrán and Ratti (2012b).

  22. 22.

    This entails the rejection of the rule of inference usually called “enrichment” or “strengthening of the antecedent”. Enrichment is the rule of inference according to which, given a certain conditional “p→q”, one can derive an entailed conditional in which the original antecedent is enriched by adding whatever sentence “p&r→q”. An example can clarify it. If we have the conditional “If I dive into the pool, then I get wet”, then we can derive the conditional “If I dive into the pool and it’s Friday, then I get wet”.

  23. 23.

    Caracciolo (2012), pp. 218 ff. As Jorge Rodríguez has pointed out to me, here two conceptually different problems are intertwined: (1) if one accepts enrichment, from a certain normative basis one could derive many irrelevant consequences, not extensionally identifiable. (2) If one assumes that relevant properties are indeterminate, then enrichment is rejected, and no consequence can be univocally derived from the verification of certain facts. In both cases, the set of consequences is not extensionally determinable, but for different reasons.

  24. 24.

    Cf. Ferrer Beltrán and Ratti (2010); Ferrer Beltrán and Ratti (2012a, pp. 11–38); Ferrer Beltrán and Ratti (2012b).

  25. 25.

    See Guastini (2011a, b).

  26. 26.

    A set-theory notion which perhaps is useful to capture the idea of different normative systems derived from the same normative basis is that of “symmetrical difference”, which consists of the operation that allows one to identify the elements that belong, alternatively, to one or the other of two sets.

  27. 27.

    See Green (1996), p. 1697.

  28. 28.

    Coleman (2001a), p. 118.

  29. 29.

    See Jori (2010), p. 55.

  30. 30.

    Marmor (2001), p. 193.

  31. 31.

    It is worth recalling at least four shortcomings affecting the RR that can be found in Hart’s work. First, there is the famous paradox that consists in determining who are the officials whose practice produces or constitutes the RR, for the concept of “official” is parasitic to that of RR (for an attempt to solve the paradox, see Coleman (2001a), p. 121). Secondly, Hart introduces the concept of RR to distinguish between primitive and advanced legal systems: however, as Bobbio (2012), p. 160 has shown, all normative systems (the most primitive included) need a RR. Moreover, if the RR, qua social rule, is a duty-imposing rule, then the very distinction between primary and secondary rules collapses, for at least a secondary rule—the RR, in fact—has the same characteristics as the primary rules. Finally, as Bobbio (1994), pp. 241–242 has also shown, the RR, in so far as it allows one to identity the rules of change (which are secondary rules regarding the primary rules of conduct), is a third-order norm, which is at a higher level than secondary rules in general.

  32. 32.

    A paradigmatic formulation of this stance is found in Lamond (2013), p. 115: “the rule of recognition is a duty-imposing rule: it not only provides the criteria for the identification of the rules of the system but imposes a duty on officials to use those criteria in identifying the laws of the system”. See also Raz (1979), p. 93. Partially contrariwise, see Coleman (2001b), pp. 74 ff.

  33. 33.

    Now collected in Bulygin (2015), pp. 117–123.

  34. 34.

    Coleman (2001a, b), pp. 119–121 tries to reply to the argument at hand with one that reaches the conclusion that the RR is “a reason for officials because an aspect of it—the internal point of view—is a cause for them” (p. 120). This argument—if correct—avoids the fallacy, but needs to explain how norms can be actually generated by “a system of stable reciprocal expectations” (p. 119).

  35. 35.

    See Marmor (2001), pp. 215–216.

  36. 36.

    Coleman (2001b), pp. 78–81.

  37. 37.

    See Dworkin (1986), pp. 34–35.

  38. 38.

    But see Pino (2012), pp. 265–299 who conceives it as a tool to determine the “applicable law”.

  39. 39.

    Here Bulygin quotes Hart (1994), p. 101. In Bulygin’s formulation, the definition shared by the members of the community seems to be the content of a lexical definition (not of a stipulation). However, Bulygin probably intends to state that members of that society adhere to a stipulation, from the internal point of view, and in so doing they use the definition to guide their “identification practice” and criticize those who do not adhere to it.

  40. 40.

    This view is developed in Dolcetti and Ratti (2013), pp. 313–314.

  41. 41.

    See Marmor (2001, p. 216).

  42. 42.

    Other legal theoretical insights provided by a set-theory approach can be found in Ratti (2012, ch. 1), Ratti (2018).

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Authors and Affiliations

  1. Department of Law, Tarello Institute for Legal Philosophy, University of Genoa, Genoa, Italy

    Giovanni Battista Ratti

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  1. Pompeu Fabra University, Barcelona, Spain

    Lorena Ramírez-Ludeña

  2. Pompeu Fabra University, Barcelona, Spain

    Josep M. Vilajosana

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Ratti, G.B. (2019). On Identifying the Law and Its Supposed Conventional Foundations. A Set-Theory Approach. In: Ramírez-Ludeña, L., Vilajosana, J. (eds) Legal Conventionalism. Law and Philosophy Library, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-030-03571-6_8

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