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A Deontic Logic Framework Allowing for Factual Detachment

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Adaptive Logics for Defeasible Reasoning

Part of the book series: Trends in Logic ((TREN,volume 38))

Abstract

Since our ethical and behavioral norms have a conditional form, it is of great importance that deontic logics give an account of deontic commitments such as ?A commits you to do/bring about B?. It is commonly agreed that monadic approaches are suboptimal for this task due to several shortcomings, for instance their falling short of giving a satisfactory account of ?Strengthening the Antecedent? or their difficulties in dealing with contrary-to-duty paradoxes. While dyadic logics are more promising in these respects, they have been criticized for not being able to model ?detachment?: A and the commitment under A to do B implies the actual obligation to do B. Lennart ?qvist asks in his seminal entry on deontic logic in the Handbook of Philosophical Logic: ?We seem to feel that detachment should be possible after all. But we cannot have things both ways, can we? This is the dilemma on commitment and detachment.? In this chapter I answer ?qvist?s question with ?Yes, we can?. I propose a general method to turn dyadic deontic logics in adaptive logics allowing for a defeasible factual detachment while paying special attention to specificity and contrary-to-duty cases. I show that a lot of controversy about detachment can be resolved by analysing different notions of unconditional obligations. The logical modeling of detachment is paradigmatically realized on basis of one of Lou Goble?s conflict tolerant CDPM logics.

A former version of the content of this chapter has been elaborated in [37].

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Notes

  1. 1.

    See [2, 3], where Goble presents a family of logics, CDPM.xy where \({\mathbf{x}} \in \{\mathbf{1, 2}\}\) and \({\mathbf{y}} \in \{{\mathbf{a, b, c}}\}\). I use in this chapter CDPM as a generic term for this family of logics.

  2. 2.

    The inheritance principle, also often referred to as rule (K), is given by: If \(\vdash A \supset B\) then \(\vdash OA \supset OB\). It is validated by most deontic logics, above all by standard deontic logic.

  3. 3.

    It would go beyond the scope of this chapter to offer a detailed analysis of the example. However, this has been done before in the literature and I refer the interested reader to e.g. [1], Sect. 8. It should be mentioned that recent developments in reactive modal logics (see e.g., [5–10]) offer a way to overcome many of the usual shortcomings of monadic deontic logics in the context of Contrary-to-Duty examples such as the Chisholm Paradox.

  4. 4.

    I restrict the discussion in this chapter to the case that the arguments of the obligation operator are propositional formulas, i.e., formulas without occurrences of obligation operators. The handling of detachment concerning obligations such as \(\mathsf{O}(\mathsf{O}(A\ |\ B)\ |\ C)\) deserves a discussion in its own right. Such a discussion would have to answer questions concerning the proper way of dealing with nested obligations, e.g., should we infer \(\mathsf{O}(A\ |\ B\wedge C)\) from \(\mathsf{O}(\mathsf{O}(A\ |\ B)\ |\ C)\), or, to what extent, if any, can we make sense of detaching obligations from obligations.

  5. 5.

    There is also deontic detachment(cp. [13]). I will discuss it in more detail in Sect. 12.6.1.

  6. 6.

    This distinction is not exhaustive. There are other conceptions, such as Carmo and Jones’ “ideal” obligation (see [16] and Sect. 12.2.5).

  7. 7.

    Note that it is rather problematic to add \(\mathsf{P}(f\wedge a\ |\ \top )\) to the premises, since the latter is equivalent to \(\lnot \mathsf{O}(\lnot f \vee \lnot a\ |\ \top )\) and in most deontic logics \(\mathsf{O}(\lnot f \vee \lnot a\ |\ \top )\) is entailed by \(\mathsf{O}(\lnot f\ |\ \top )\) (due to modal inheritance).

  8. 8.

    This version of the Forrester Paradox (cp. [17]) is taken from McNamara ([18], p. 243).

  9. 9.

    It would be suboptimal to define (strong) CTD obligations by (i) together with \(\vdash A \supset \lnot C\), since then e.g. \(\mathsf{O}(f\ |\ f \wedge a)\) would be a CTD obligation to \(\mathsf{O}(\lnot f\ |\ \top )\). However, this is counter-intuitive since \(\mathsf{O}(\lnot f\ |\ \top )\) is excepted in the context \(f\wedge a\), as argued above.

  10. 10.

    Prakken and Sergot offer a similar view. A “key difference between contrary-to-duty and prima facie obligations” ([22], p. 224) is that, unlike prima facie obligations and opposite to Wang’s view, the former do not satisfy any form of (defeasible) factual detachment. The only form of detachment they satisfy is the following strong detachment principle: \(\models (\mathsf{O}(A\ |\ B) \wedge \Box B) \supset \) O \(A\), where \(\Box \) is the necessity operator of an adequate modal logic. Detaching the obligation to kill gently would cause an inconsistency which is “counter to our intuitions”. The obligation not to kill is not overridden, it is fully valid.

  11. 11.

    Foot in [23] introduces a similar concept. She dubs ‘obligations of type 2’ obligations which “tell us the right thing to do” ([23], p. 385). They answer the question “And what all things considered ought we to do?” (p. 386). Of course, it would have been the best thing to do for Doe not to kill his mother. However, interpreting the factual premises as unalterable facts, the best thing Doe may (still) do is to kill his mother gently (if he already kills her).

  12. 12.

    Of course, in an ‘ideal’ world there is no need to apologize. Thus, Prakken and Sergot in [12] call it a ‘pragmatic oddity’ if, in the case of \(b\), both obligations, to keep the promise and to apologize, are considered to be ‘actual’. Van Der Torre and Tan ([21], p. 63) call it counter-intuitive. Let us see if our proposal causes an ‘oddity’. An oddity could be given in two respects: (a) concerning ideality and (b) concerning a pragmatic aspect. Due to (a) there would be an oddity if we had \({\mathsf{O}}^{{\mathsf{p}}}A\) and \({\mathsf{O}}^{{\mathsf{p}}}B\) for incompatible \(A\) and \(B\), and the intended reading of \({\mathsf{O}}^{{\mathsf{p}}}A\) would be that \(A\) is true in all “ideal worlds”. This, however, is not the case with our analysis since for our proper obligation \({\mathsf{O}}^{{\mathsf{p}}}A\) the intended reading is not that this is an obligation ideal in the mentioned sense, but rather that it is an obligation actual in the given context that is neither excepted nor burdened. Furthermore, taking into account that, (i), some proper obligations may be violated while other proper obligations may still be realizable (the ones which are also instrumental obligations such as in our case the obligation to apologize) and, (ii), that the two obligations in question are consistent, I do not think that this is counter-intuitive. Due to (b) we would have a pragmatic oddity in case we would get two in some sense incompatible obligations which tell us what to do. This boils down to the concurrence of two incompatible instrumental obligations in our sense. However, again we do not have an oddity in the given example since we only have \({\mathsf{O}}^{{\mathsf{i}}}a\) and no other instrumental obligation with which it is incompatible.

  13. 13.

    Presupposing that the additional modal information modeled in Carmo and Jones’ system is expressed in terms of conditional obligations, “ideal” obligations can be incorporated within the formal framework presented in this chapter as I will remark later (see Footnote 20). For the sake of conciseness I will though focus on instrumental and proper obligations.

  14. 14.

    In this respect my approach is similar to the ones offered by defeasible deontic logics such as Rye’s [24] or Horty’s [25]. These accounts proceed in a similar way as default logic [26]: a given set of conditional obligations is enhanced to so-called extensions with respect to certain consistency criteria. One important feature is that by an analysis of the relationships between the conditional obligations the logics identify the “overridden” (in Horty’s terminology) resp. “defeated” and “violated” (in Rye’s terminology) obligations that are rejected as members of the extensions. In our terminology, excepted and violated obligations are sorted out. A consequence relation is then defined in terms of membership to these extensions.

  15. 15.

    The permission operator may also be defined by \(\mathsf{P}(A\ |\ B) =_{\mathrm{df}}\lnot \mathsf{O}(\lnot A\ |\ B)\).

  16. 16.

    There is a similarity between the presented approach and defeasible deontic logics such as Rye’s [24] or Horty’s [25] (see Footnote 14) concerning the fact that excepted or violated obligations are in a sense incapacitated via an analysis of the relationships between the obligations. Of course,the current approach does not “sort out” for instance overshadowed and overridden obligations by constructing extensions, but rather labels them by \(\bullet _\mathsf{i}\) and \(\bullet _\mathsf{p}\). Note that this allows us to stay within the (adaptively extended) standard proof theory of the given deontic base logic.

  17. 17.

    Nevertheless, as will be shown later (see Example 12.6.2), the ALs which are going to be presented in the following model detachment as desired for this example.

  18. 18.

    Again, the definition fails for cases that feature nested permissible contexts. For instance \(\mathsf{O}(f\wedge a\ |\ a)\) is a strong CTD obligation to \(\mathsf{O}(\lnot f\ |\ \top )\) according to the simplified definition above, although, according to the refined notions in Sect. 12.2, \(f\wedge a\) is an exceptional context to \(\mathsf{O}(\lnot f\ |\ \top )\).

  19. 19.

    In logics verifying \(\vdash \mathsf{O}(E\ |\ F) \supset \mathsf{P}(E\ |\ F)\) (e.g. standard deontic logic) the first condition of the antecedent, \(\mathsf{P}(D\ |\ B\wedge C) \vee \mathsf{O}(D\ |\ B \wedge C)\), can be simplified to \(\mathsf{P}(D\ |\ B \wedge C)\).

  20. 20.

    It should be remarked at this place that our framework can easily be enhanced such as to model a notion similar to Carmo and Jones’ “ideal” obligations. The language is enhanced similar as for instrumental and proper obligations by a unary “ideal” obligation operator \(\mathsf{O}^\mathrm{I}\) and by \(\bullet _\mathrm{I}\). The factual detachment rule is defined analogous as for the instrumental and proper case, \(\vdash \bigl (\mathsf{O}(A\ |\ B) \wedge B \wedge \lnot \bullet _\mathrm{I} \mathsf{O}(A\ |\ B) \bigr ) \supset \mathsf{O}^\mathrm{I} A\). An analogous rule to (Ep) is used where \(\bullet _\mathsf{p}\mathsf{O}(A\ |\ B)\) is replaced by \(\bullet _\mathrm{I} \mathsf{O}(A\ |\ B)\). The major change in comparison to the rules for the blocking of proper detachment is, as discussed in Sect. 12.2.4, with (CTDR) since in the “ideal” case we also block weak CTD obligations from detachment. Thus, the rule is altered to: If \(\vdash D \supset \lnot C\), then \(\bigl ( \mathsf{O}(A\ |\ B\wedge C) \wedge \mathsf{O}(D\ |\ B) \bigr ) \supset \bullet _\mathrm{I} \mathsf{O}(A\ |\ B\wedge C)\). In order not to unnecessarily increase the complexity of the discussion, I will not follow this option further in this chapter. As remarked in Sect. 12.2.5, the language of standard deontic logic lacks the modal expressiveness of Carmo and Jones’ proposal and the deontic implications of the additional modal information need to be explicitly expressed in terms of the conditions of obligations.

  21. 21.

    In logics verifying \(\vdash \mathsf{O}(E\ |\ F) \supset \mathsf{P}(E\ |\ F)\) (e.g. standard deontic logic) the first condition of the antecedent, \(\mathsf{P}(D\ |\ B\wedge C) \vee \mathsf{O}(D\ |\ B \wedge C)\), can be simplified to \(\mathsf{P}(D\ |\ B \wedge C)\).

  22. 22.

    Goble defines a dyadic version \(\mathbf{{SDDL}}\) of standard deontic logic in [27]. It is equivalent to van Fraassen’s \(\mathbf{{CD}}\) of [28] and David Lewis’ \(\mathbf{{VN}}\) of [29]. See Chap. 11, Sect. 11.2.

  23. 23.

    It validates all instances of \(\vdash \bigl (\mathsf{O}(B\ |\ A) \wedge \mathsf{P}(C\ |\ A) \wedge \mathsf{P}(B\ |\ A)\bigr ) \supset \mathsf{O}(B\ |\ A \wedge C)\) (PRatMono). That (PRatMono) is counter-intuitive can be demonstrated by means of our asparagus example (namely \(\mathsf{O}(\lnot f\ |\ \top )\), \(\mathsf{P}(a\ |\ \top )\) and \(\mathsf{O}(f\ |\ a)\)): if we add the intuitive premise \(\mathsf{P}(\lnot f\ |\ \top )\), then the counter-intuitive \(\mathsf{O}(\lnot f\ |\ a)\) is derivable. This defect is not fatal though: I proposed in [30] and in Sect. 11.4 a version of \(\mathbf{{CDPM.1}}\) which overcomes this shortcoming.

  24. 24.

    Rational Monotonicity (cp. [15]) can be stated in terms of the language of dyadic deontic logic used in this chapter as follows: \(\vdash \bigl (\mathsf{O}(B\ |\ A) \wedge \mathsf{P}(C\ |\ A)\bigr ) \supset \mathsf{O}(B\ |\ A \wedge C)\). It is verified in dyadic standard deontic logic. Goble is aware of the fact that his (WRM) leads to counter-intuitive, even explosive behavior in some cases. I offered an improvement based on the idea of conditionally applying SA within an AL which is able to avoid these problems (see [30] and Sect. 12.8).

  25. 25.

    The reader might further object that (CN) (in a similar way as (QR) is not very intuitive. Goble’s intention is to stay as close as possible to standard deontic logic. However, (CN) is neither an essential part of his logic nor in any way essential to the presented approach and may thus be disregarded as well.

  26. 26.

    Lines 1–5 are not stated in the form of an AL proof. It can be easily adjusted by adding the empty condition \(\emptyset \) in a fourth column.

  27. 27.

    That (DDP\(\top \)1) is a consequence of (DDP1) can easily be shown. Proofs for the validity of (DDP1) and (DDP\(\top \)1) in \(\mathbf{{CDPM.2d}}\) can be found in Appendix J.

  28. 28.

    Proofs can be found in Appendix J.

  29. 29.

    In \(\mathbf{{DCDPM.2e^+}}\) we would have to add another additional premise, \(\mathsf{P}(w \wedge f\ |\ d \wedge \lnot f)\), in order to derive \(\mathsf{O}(w \wedge f\ |\ d)\) analogously by (DDP2). Otherwise the proof is analogous.

  30. 30.

    In \(\mathbf{{CDPM.2d^+}}\) also the stronger “If \(\vdash {} B \supset \lnot A\) then \(\bigl ( \mathsf{O}(B\ |\ C) \wedge C \wedge \mathsf{O}(A\ |\ C) \bigr ) \supset \bullet _{{\mathsf{x}}}\mathsf{O}(A\ |\ C)\)” is valid. Note that in this version \(\mathsf{P}(C\ |\ C)\) is not part of the antecedent. In view of this, in \(\mathbf{{CDPM2.e^+}}\) one may want to add the following rule in order to have a stricter handling of deontic conflicts: If \(\vdash {} B \supset \lnot A\) then \(\bigl ( \mathsf{O}(A\ |\ C) \wedge \mathsf{O}(B\ |\ C) \wedge C \bigr ) \supset \bullet _{{\mathsf{x}}}\mathsf{O}(A\ |\ C)\).

  31. 31.

    Furthermore, many of the adaptive strengthenings of \(\mathbf{{CDPM}}\) defined in [30] are able to derive \(\mathsf{O}(a\vee b\ |\ \top )\) from \(\mathsf{O}(a\ |\ \top )\) and \(\mathsf{O}(b\ |\ \top )\). Thus, forming a combined AL with one of these systems, analogous to the way it is sketched in Sect. 12.8, \(\varphi = {\mathsf{O}}^{{\mathsf{p}}}(a\vee b) \wedge {\mathsf{O}}^{{\mathsf{i}}}(a\vee b)\) is derivable from \(\mathsf{O}(a\ |\ \top )\) and \(\mathsf{O}(b\ |\ \top )\), whereas for \(\mathbf{{DCDPM.2d^+}}\) and \(\mathbf{{DCDPM.2e^+}}\) we have to add the additional premise \(\mathsf{O}(a\vee b\ |\ \top )\) in order to derive \(\varphi \).

  32. 32.

    The semantics of \(\mathbf{{CDPM.2\alpha ^+_{\mathbb {P}}}}\) is defined by means of neighborhood frames similar as the semantics of \(\mathbf{{CDPM.2\alpha ^+}}\). This is spelled out in Appendix J.

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Acknowledgments

I thank Joke Meheus, Mathieu Beirlaen, Frederik Van De Putte, Dunja Šešelja and the anonymous referees of the Journal for Applied Logic for valuable comments which helped to improve the paper.

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  1. Department of Philosophy and Moral Science, Centre for Logic and Philosophy of Science, Ghent University, Sint-Pietersnieuwstraat 61, 9000, Ghent, Belgium

    Christian Straßer

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Straßer, C. (2014). A Deontic Logic Framework Allowing for Factual Detachment. In: Adaptive Logics for Defeasible Reasoning. Trends in Logic, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-00792-2_12

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