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Maximality vs. Optimality in Dyadic Deontic Logic

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Abstract

This paper reports completeness results for dyadic deontic logics in the tradition of Hansson’s systems. There are two ways to understand the core notion of best antecedent-worlds, which underpins such systems. One is in terms of maximality, and the other in terms of optimality. Depending on the choice being made, one gets different evaluation rules for the deontic modalities, but also different versions of the so-called limit assumption. Four of them are disentangled, and compared. The main observation of this paper is that, even in the partial order case, the contrast between maximality and optimality is not as significant as one could expect, because the logic remains the same whatever notion of best is used. This is established by showing that, given analogous properties for the betterness relation, the same system is sound and complete with respect to its intended modelling. The chief result of this paper concerns Åqvist’s system F supplemented with the principle (CM) of cautious monotony. It is established that, under the maximality rule, F + (CM) is sound and complete with respect to the class of models in which the betterness relation is required be reflexive and smooth (for maximality). From this, a number of spin-off results are obtained. First and foremost, it is shown that a similar determination result holds for optimality; that is, under the optimality rule, F + (CM) is also sound and complete with respect to the class of models in which the betterness relation is reflexive and smooth (for optimality). Other spin-off results concern classes of models in which further constraints are placed on the betterness relation, like totalness and transitivity.

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Notes

  1. This semantics is in the “imperativist” tradition of deontic logic, which attempts to reconstruct deontic logic as a logic about imperatives. For a list of authors working in this tradition, see, e.g., [10, fn. 1].

  2. I follow Sen’s terminology in his [30]. The notions of optimization and maximization are respectively referred to as “stringent” and “liberal” maximization by Herzberger [14], and “greatestness” and “maximality” rationality by Suzumara [33].

  3. Both definitions can be found in the literature. Hansson [13], Makinson [22, Section 7.1], Prakken and Sergot [28] and Schlechta [29] use the max rule. Alchourrón [1, p. 76], Åqvist [2, 3], Hansen [11, Section 6], McNamara [23] and Spohn [32] work with the opt rule. Neither Goldman [8], nor Jackson [17], nor Hilpinen [16, Section 8.5] specifies what notion of best is meant. (The last one uses “best” and “deontically optimal” interchangeably, but leaves optimality undefined).

  4. Hansson [13] and Prakken and Sergot [28] use max-limitedness, while Alchourrón [1, p. 84], Åqvist [2, 3], Hansen [11, Section 6], McNamara [23] and Spohn [32] use opt-limitedness, and Schlechta [29] max-smoothness. I am not aware of any authors who have considered opt-smoothness explicitly.

  5. However, not all the authors have used it that way. For instance, for Hansson it seems to have been more a concern for non-emptiness, which is essential to validate the principle (given as D⋆ , Section 2.4 below) that ought implies permitted, for consistent (or possible) antecedents.

  6. The proof is by induction on A. (Note that the inductive hypothesis plays a role only for the cases in which A is ¬B, B ∧ C, B ∨ C or B → C.)

  7. The use of a settledness operator is in line with Hansson’s own interpretation of circumstances in [13, Section 13]. For more on strong factual detachment, see [27, 28].

  8. I owe this observation to the referee. A proof of completeness of SDL with respect to various classes of preference models applying the so-called Danielsson-type evaluation rule may be found in Goble [7, Section 1].

  9. Their “max” variant comes close to Hansson’s own picture of DSDL2.

  10. In Parent [26], the class of models applying the opt rule, with ⪰ reflexive and opt-limited, is axiomatized using an alternative language, which has the unary modal operator “Q” (“ideally”, ...) as main building block. The proof of completeness given there makes an essential use of the axiom Q A ∧ Q B → Q(A ∨ B), which corresponds to Sen’s property γ (see [30]) in modal logic notation. Although the axiom remains valid under the max rule, it has no obvious counterpart in terms of ○(−/−).

  11. Cf. Proposition 4, and the example immediately before.

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Acknowledgments

I am indebted to an anonymous referee for helpful comments and suggestions, which have substantially improved this article.

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  1. Université Aix-Marseille, CNRS, CEPERC UMR 7304, 13621, Aix-en-Provence, France

    Xavier Parent

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Parent, X. Maximality vs. Optimality in Dyadic Deontic Logic. J Philos Logic 43, 1101–1128 (2014). https://doi.org/10.1007/s10992-013-9308-0

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