Skip to main content
SpringerLink
Log in

Avoiding Deontic Explosion by Contextually Restricting Modal Inheritance

  • Chapter
  • First Online:
Book cover Adaptive Logics for Defeasible Reasoning

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

Abstract

In order to deal with the possibility of deontic conflicts Lou Goble developed a group of logics (DPM) that are characterized by a restriction of the inheritance principle. While they approximate the deductive power of standard deontic logic, they do so only if the user adds certain statements to the premises. By adaptively strengthening the DPM logics, this chapter presents logics that overcome this shortcoming. Furthermore, these ALs are capable of modeling the dynamic and defeasible aspect of our normative reasoning by their dynamic proof theory. This way they enable us to have a better insight into the relations between obligations and thus to localize deontic conflicts.

A former version of the content of this chapter has been elaborated in the article “Avoiding Deontic Explosion by Contextually Restricting Modal Inheritance” [1]. It is co-authored by Joke Meheus and Mathieu Beirlaen.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    SDL is obtained by adding the principle (D), \(\vdash {{\mathsf {O}}}A\supset \lnot {{\mathsf {O}}}\lnot A\) to the normal modal logic \({\mathbf {K}}\). There are various alternative axiomatizations of SDL, cfr. footnote 5.

  2. 2.

    In view of the definition of \({\mathsf {P}}A\), \({{\mathsf {O}}}A \wedge {\mathsf {P}}A\) expresses that the obligation \(OA\) is unconflicted.

  3. 3.

    See also [17] where the authors define consequence relations for rank-1 modal logics in this way and prove strong completeness.

  4. 4.

    We slightly adjusted the criteria (DEX-1)–(DEX-3) (the latter two will be introduced in a moment) offered by Goble since his criteria were formulated in terms of theoremhood while we focus on the consequences of premise sets.

  5. 5.

    Goble axiomatizes SDL by adding (D), (N), (RE), (RM), and (AND) to full propositional logic. A consequence relation \(\vdash _\mathbf{{SDL}}\) can be defined analogous to \(\vdash _\mathbf{{DPM.1}}\). We slightly adjusted Goble’s (\(\star \)) since he is mainly interested in theoremhood, while we focus on consequence relations.

  6. 6.

    A restricted inheritance principle following the intuition of (PAND’) would be: If \(\vdash {} A \supset B\), then \(\vdash {} {\mathsf {P}}B \supset ({{\mathsf {O}}}A \supset {{\mathsf {O}}}B)\). Inheritance is applied to \({{\mathsf {O}}}A\) in order to derive \({{\mathsf {O}}}B\) if it does not result in a deontic conflict \({{\mathsf {O}}}B \wedge {{\mathsf {O}}}\lnot B\).

  7. 7.

    In [11], van der Torre and Tan presented a sequential system which, in a first phase, disables the application of (RM) and allows for the application of a restricted aggregation rule. In a second phase, it disables this aggregation rule and allows for the application of (RM). Although this system overcomes this problem, it can do so only by introducing two different \({\mathsf {O}}\)-operators and by requiring that (RM) is never applied before the restricted aggregation rule. As the authors themselves admit, this is rather strange from an intuitive point of view (see also [6], pp. 470–471).

  8. 8.

    To stay in line with Goble’s (\(\star \)), we formulate the strengthened requirement in terms of SDL. If one’s preferred logic is different from SDL, the requirement may easily be adapted. The basic idea is that, where one’s preferred deontic logic (for conflict-free premise sets) is \({\mathbf {L}}\), one expects from a conflict-tolerant deontic logic on the basis of \({\mathbf {L}}\) that it leads to the same consequence set as \({\mathbf {L}}\) for all \({\mathbf {L}}\)-consistent premise sets. This is exactly what ALs allow for.

  9. 9.

    \(\varGamma \) is SDL-consistent iff \(\varGamma \nvdash _{\mathbf{SDL}} \bot \).

  10. 10.

    We will discuss the case of \({\mathbf {DPM.2}}\) being a lower limit logic shortly in Sect. 10.7.

  11. 11.

    In Section 10.6 in the context of the lower limit logic \({\mathbf {DPM.1}}\) we will have to make a slight adjustment when defining \(\varOmega \). However, the main idea stays the same.

  12. 12.

    This can easily be seen: from \(\vdash _\mathbf{CL} \lnot (A\wedge B)\), \({{\mathsf {O}}}A\) and restricted inheritance we get \({{\mathsf {O}}}\lnot B \vee {!}A\). The latter together with \({{\mathsf {O}}}B\) results in \(!A \vee {!}B\).

  13. 13.

    Note that line 8 follows by (N) and applying aggregation to lines 1 and 2.

  14. 14.

    The proofs of \({{\mathsf {O}}}(M \vee F)\) and \({{\mathsf {O}}}T\) for \({\mathbf {ADPM.1}}\) are left to the reader.

  15. 15.

    A subset \(\varGamma '\) of \(\varGamma \) is maximally consistent iff \(\varGamma '\) is consistent and for every consistent \(\varGamma '' \subseteq \varGamma \), if \(\varGamma ' \subseteq \varGamma ''\) then \(\varGamma '' = \varGamma '\).

  16. 16.

    In the deontic logic literature on maximally consistent subsets, skeptical operators are contrasted with more ‘credulous’ operators. The latter allow one to derive the obligation \({{\mathsf {O}}}A\) as soon as \(A\) is derivable from some maximal consistent subset of \(\varGamma \) (see e.g. [20, 22] or the “full join constraint output” of Input/Output logics in [21]) while the former allow to derive \({{\mathsf {O}}}A\) iff \(A\) is derivable in all maximal consistent subsets of \(\varGamma \).

  17. 17.

    One such counterexample is given by the premise set \(\{{{\mathsf {O}}}A, {{\mathsf {O}}}B, {{\mathsf {O}}}\lnot (A \wedge B), {{\mathsf {O}}}(\lnot (A\wedge B) \wedge D), {{\mathsf {O}}}(C_1 \wedge C_2)\}\). Here it is not possible to derive \({{\mathsf {O}}}C_1\) by means of \({\mathbf {ADPM.1}}\) or \({\mathbf {ADPM.2'}}\) since \(!(C_1 \wedge C_2)\) is involved in a minimal \(\mathsf{{Dab}}\)-consequence. Note that we can derive \({{\mathsf {O}}}C_1\) on the condition that \({{\mathsf {O}}}(C_1 \wedge C_2)\) is not conflicted. By applying aggregation we get \({{\mathsf {O}}}(C_1 \wedge (A \wedge B))\) (in \({\mathbf {ADPM.2'}}\) we also need the condition that both \({{\mathsf {O}}}A\) and \({{\mathsf {O}}}B\) are not conflicted). Similarly, we can derive \({{\mathsf {O}}}\lnot (C_1 \wedge (A \wedge B))\) from \({{\mathsf {O}}}(\lnot (A \wedge B) \wedge D)\) by means of inheritance on the condition that the latter is not conflicted. Altogether this shows that \(!(C_1 \wedge C_2)\), \(!(\lnot (A \wedge B) \wedge D)\) and \(!(C_1 \wedge (A\wedge B))\) are involved in a \(\mathsf{{Dab}}\)-consequence. It is not difficult to see that they are indeed involved in a minimal \(\mathsf{{Dab}}\)-consequence.

  18. 18.

    In a non-deontic setting, this approach was taken up by Rescher & Manor in their definition of an ‘argued’ consequence relation [24]. Adaptive characterizations of Rescher & Manor’s consequence relations are given in [25, 26].

References

  1. Straßer, C., Meheus, J., Beirlaen, M.: Tolerating deontic conflicts by adaptively restricting inheritance. Logique at Analyse 219, 477–506 (2012)

    Google Scholar 

  2. Beirlaen, M., Straßer, C., Meheus, J.: An inconsistency-adaptive deontic logic for normative conflicts. J. Philos. Logic 2(42), 285–315 (2013)

    Article  Google Scholar 

  3. Da Costa, N., Carnielli, W.: On paraconsistent deontic logic. Philosophia 16(3), 293–305 (1986)

    Google Scholar 

  4. Fraassen, B.C.V.: Values and the heart’s command. J. Philos. 70(1), 5–19 (1973)

    Article  Google Scholar 

  5. Goble, L.: A proposal for dealing with deontic dilemmas. In: Lomuscio, A., Nute, D. (eds.) DEON. Lecture Notes in Computer Science, vol. 3065, pp. 74–113. Springer, Heidelberg (2004)

    Google Scholar 

  6. Goble, L.: A logic for deontic dilemmas. J. Appl. Logic 3, 461–483 (2005)

    Article  Google Scholar 

  7. Horty, J.F.: Reasoning with moral conflicts. Noûs 37(4), 557–605 (2003)

    Google Scholar 

  8. Meheus, J., Beirlaen, M., Putte, F.V.D.: Avoiding deontic explosion by contextually restricting aggregation. In: Governatori, G., Sartor, G. (eds.) DEON (10th International Conference on Deontic Logic in Computer Science). Lecture Notes in Artificial Intelligence, vol. 6181, pp. 148–165. Springer, Heidelberg (2010)

    Google Scholar 

  9. Priest, G.: In Contradiction: A Study of the Transconsistent. Oxford University Press, Oxford (2006)

    Google Scholar 

  10. Schotch, P.K., Jennings, R.E.: Non-kripkean deontic logic. In: Hilpinen, R. (ed.) New Studies in Deontic Logic: Norms, Actions, and the Foundations of Ethics, pp. 149–162. Reidel Publishing Company, Dordrecht (1981)

    Google Scholar 

  11. van der Torre, L., Tan, Y.H.: Two-phase deontic logic. Logique at Analyse 171–172, 411–456 (2000)

    Google Scholar 

  12. Gowans, C.W. (ed.): Moral Dilemmas. Oxford University Press, New York (1987)

    Google Scholar 

  13. Sinnott-Armstrong, W.: Moral Dilemmas. Basil Blackwell, Oxford (1988)

    Google Scholar 

  14. Goble, L.: Multiplex semantics for deontic logic. Nord. J. Philos. Logic 5, 113–134 (2000)

    Article  Google Scholar 

  15. Batens, D.: A universal logic approach to adaptive logics. Logica Universalis 1, 221–242 (2007)

    Google Scholar 

  16. Batens, D.: Adaptive Logics and Dynamic Proofs. A Study in the Dynamics of Reasoning. (201x)

    Google Scholar 

  17. Schröder, L., Pattinson, D.: Rank-1 modal logics are coalgebraic. J. Logic Comput. 5(20), 1113–1147 (2010)

    Google Scholar 

  18. Batens, D.: Towards the unification of inconsistency handling mechanisms. Logic Log. Philos. 8, 5–31 (2000). Appeared 2002

    Google Scholar 

  19. Goble, L.: Deontic logic (adapted) for normative conflicts. Logic Journal of IGPL (2013). Online First

    Google Scholar 

  20. Hansen, J.: Conflicting imperatives and dyadic deontic logic. J. Appl. Logic 3, 484–511 (2005)

    Article  Google Scholar 

  21. Makinson, D., van der Torre, L.: Constraints for input/output logics. J. Philos. Logic 30(2), 155–185 (2001)

    Google Scholar 

  22. Van Fraassen, B.C.: Values and the heart’s command. J. Philos. 70, 5–19 (1973)

    Article  Google Scholar 

  23. Meheus, J., Beirlaen, M., Putte, F.V.D., Straßer, C.: Non-adjunctive deontic logics that validate aggregation as much as possible (forthcoming)

    Google Scholar 

  24. Rescher, N., Manor, R.: On inference from inconsistent premises. Theor. Decis. 1, 179–217 (1970)

    Article  Google Scholar 

  25. Batens, D.: A strengthening of the Rescher-Manor consequence relations. Logique at Analyse 183–184, 289–313 (2003). Appeared 2005

    Google Scholar 

  26. Batens, D., Vermeir, T.: Direct dynamic proofs for the Rescher-Manor consequence relations: the flat case. J. Appl. Non-Class. Logics 12, 63–84 (2002)

    Article  Google Scholar 

  27. Putte, F.V.D., Straßer, C.: A logic for prioritized normative reasoning. J. Logic Comput. (2012)

    Google Scholar 

  28. Putte, F.V.D., Straßer, C.: Extending the standard format of adaptive logics to the prioritized case. Logique at Analyse 55(220), 601–641 (2012)

    Google Scholar 

  29. Brink, D.: Moral conflict and its structure. Philos. Rev. 103, 215–247 (1994)

    Article  Google Scholar 

  30. Beirlaen, M., Straßer, C.: Two adaptive logics of norm-propositions. J. Appl. Logic 11(2), 147–148 (2013)

    Google Scholar 

  31. Beirlaen, M., Straßer, C.: A paraconsistent multi-agent framework for dealing with normative conflicts. In: Leite, J., Torroni, P., Agotnes, T., Boella, G., van der Torre, L. (eds.) Computational Logic in Multi-Agent Systems. Lecture Notes in Computer Science, pp. 312–329. Springer, Heidelberg (2011)

    Google Scholar 

  32. Beirlaen, M., Straßer, C.: Nonmonotonic reasoning with normative conflicts in multi-agent deontic logic. J. Logic Comput. (2013)

    Google Scholar 

  33. Straßer, C., Beirlaen, M.: Towards more conflict-tolerant deontic logics by relaxing the interdefinability between obligations and permissions (in preparation)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sint-Pietersnieuwstraat 61, 9000, Ghent, Belgium

    Christian Straßer

Authors
  1. Christian Straßer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Straßer .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Straßer, C. (2014). Avoiding Deontic Explosion by Contextually Restricting Modal Inheritance. In: Adaptive Logics for Defeasible Reasoning. Trends in Logic, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-00792-2_10

Download citation

Publish with us

Policies and ethics

Search

Navigation