Skip to main content
SpringerLink
Log in

Contrary-to-Duty Reasoning: A Categorical Approach

  • Published:
Logica Universalis Aims and scope Submit manuscript

Abstract

This paper provides an analysis of contrary-to-duty reasoning from the proof-theoretical perspective of category theory. While Chisholm’s paradox hints at the need of dyadic deontic logic by showing that monadic deontic logics are not able to adequately model conditional obligations and contrary-to-duties, other arguments can be objected to dyadic approaches in favor of non-monotonic foundations. We show that all these objections can be answered at one fell swoop by modeling conditional obligations within a deductive system defined as an instance of a symmetric monoidal closed category. Using category theory as a foundational framework for logic, we show that it is possible to model conditional normative reasoning and conflicting obligations within a monadic approach without adding further operators or considering deontic conditionals as primitive.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abrusci V.M.: Non-commutative intuitionistic linear logic. Math. Log. Q. 36(4), 297–318 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abrusci V.M.: Phase semantics and sequent calculus for pure non-commutative classical linear propositional logic. J. Symb. Log. 56(4), 1403–1451 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Abrusci V.M., Ruet P.: Non-commutative logic I: the multiplicative fragment. Ann. Pure Appl. Log. 101, 29–64 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Alchourrón C.: Detachment and defeasibility in deontic logic. Stud. Log. 57(1), 5–18 (1996)

    Article  MATH  Google Scholar 

  5. Alchourrón, C., Makinson, D.: Hierarchies of regulations and their logic. In: Hilpinen, R. (ed.) New Studies in Deontic Logic, pp. 125–148. D. Reidel Publishing Company, Dordrecht (1981)

  6. Anderson A.R.: A reduction of deontic logic to alethic modal logic. Mind 67(265), 100–103 (1958)

    Article  Google Scholar 

  7. Anderson A.R.: On the logic of commitment. Philos. Stud. 10(2), 23–27 (1959)

    Article  Google Scholar 

  8. Anderson A.R.: Reply to Mr Rescher’s conditional permission in deontic logic. Philos. Stud. 13(1), 6–8 (1962)

    Google Scholar 

  9. Antonelli, G.A.: Non-monotonic logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2012)

  10. Åqvist L.: Good Samaritans, contrary-to-duty imperatives, and epistemic obligations. Noûs 1(4), 361–379 (1967)

    Article  Google Scholar 

  11. Åqvist, L.: Deontic logic. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 8, 2nd edn, pp. 147–264. Kluwer Academic Publishers, Dordrecht (2002)

  12. Asher, N., Bonevac, D.: Common sense obligation. In: Nute, D. (ed.) Defeasible Deontic Logic, pp. 159–203. Kluwer Academic Publishers, Dordrecht (1997)

  13. Baez, J.C., Stay, M.: Physics, topology, logic and computation: a Rosetta stone. In: Coecke, B. (ed.) New Structures for Physics. Lecture Notes in Physics, vol. 813, pp. 95–174. Springer, Berlin (2011)

  14. Beirlaen M., Straßer C.: Two adaptive logics of norm-propositions. J. Appl. Log. 11(2), 147–168 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  16. Blute, R., Scott, P.: Category theory for linear logicians. In: Ehrhard, T., Girard, J.-Y., Ruet, P., Scott, P. (eds.) Linear Logic in Computer Science, vol. 316, pp. 3–64. Cambridge University Press, London (2004)

  17. Bonevac D.: Against conditional obligation. Noûs 32(1), 37–53 (1998)

    Article  MathSciNet  Google Scholar 

  18. Carmo, J., Jones, A.J.I.: Deontic logic and contrary-to-duties. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 8, 2nd edn, pp. 265–343. Kluwer Academic Publishers, Dordrecht (2002)

  19. Chellas, B.F.: Conditional obligation. In: Stenlund, S. (ed.) Logical Theory and Semantic Analysis, pp. 23–33. D. Reidel Publishing Company, Dordrecht (1974)

  20. Chellas B.F.: Modal logic: An Introduction. Cambridge University Press, London (1980)

    Book  MATH  Google Scholar 

  21. Chisholm R.: Contrary-to-duty imperatives and deontic logic. Analysis 24(2), 33–36 (1963)

    Article  Google Scholar 

  22. Côté, Pierre-André.: Interprétation des lois, 3rd edn. Éditions Thémis (2006)

  23. Decew J.: Conditional obligation and counterfactuals. J. Philos. Log. 10(1), 55–72 (1981)

    Article  MathSciNet  Google Scholar 

  24. Eilenberg S., Lane S.M.: General theory of natural equivalences. Trans. Am. Math. Soc. 58, 231–294 (1945)

    Article  MATH  Google Scholar 

  25. Gentzen, G.: Untersuchungen über das logische Schliessen. Math. Z. 39, 176–210, 405–431 (1934)

  26. Girard J.-Y.: Linear logic. Theo. Comput. Sci. 50(1), 1–102 (1987)

    Article  MATH  MathSciNet  Google Scholar 

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

  28. Goble L.: Normative conflicts and the logic of ‘ought’. Noûs 43(3), 450–489 (2009)

    Article  MathSciNet  Google Scholar 

  29. Hansen, J.: Paradoxes of commitment. In: Meggle, G. (ed.) Actions, Norms, Values, pp. 255–263. Walter de Gruyter, Berlin (1999)

  30. Hintikka, J.: Some main problems of deontic logic. In: Hilpinen, R. (ed.) Deontic Logic: Introductory and Systematic Readings, pp. 59–104. D. Reidel Publishing Company, Dordrecht (1970)

  31. Horty J.: Moral dilemmas and non-monotonic logic. J. Philos. Log. 23(1), 35–65 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  32. Horty, J.: Non-monotonic foundations for deontic logic. In: Nute, D. (ed.) Defeasible Deontic Logic, pp. 17–44. Kluwer Academic Publishers, Dordrecht (1997)

  33. Jones A.J.I.: On the logic of deontic conditionals. Ratio Juris 4(3), 355–366 (1991)

    Article  Google Scholar 

  34. Jones A.J.I., Pörn I.: Ideality, sub-ideality and deontic logic. Synthese 65(2), 275–290 (1985)

    Article  MathSciNet  Google Scholar 

  35. Jørgensen J.: Imperatives and logic. Erkenntnis 7(1), 288–296 (1937)

    Google Scholar 

  36. Kanger, S.: New foundations for ethical theory. Stockholm (1957)

  37. Kanger, S.: New foundations for ethical theory, In: Hilpinen, R. (ed.) Deontic Logic: Introductory and Systematic Readings, pp. 36–58. D. Reidel Publishing Company, Dordrecht (1971)

  38. Lambek J.: The mathematics of sentence structure. Am. Math. Mon. 65(3), 154–170 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  39. Lambek J.: Deductive systems and categories I. Math. Syst. Theory 2(4), 287–318 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  40. Lambek, J.: Deductive systems and categories II. Standard constructions and closed categories. In: Hilton, P.J. (ed.) Category Theory, Homology Theory and their Applications I. Lecture Notes in Mathematics, vol. 86, pp. 76–122. Springer, Berlin (1969)

  41. Lawvere, F.W.: Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories. Ph.D. thesis, Columbia University (1963)

  42. Loewer B., Belzer M.: Dyadic deontic detachment. Synthese 54(2), 295–318 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  43. Lokhorst, G.-J.: Deontic linear logic with Petri net semantics, Tech. report, FICT (Center for the Philosophy of Information and Communication Technology) (1997)

  44. Lane, S.M.: Categories for the Working Mathematician, 2nd edn. Springer, Berlin (1971)

  45. Makinson D., van der Torre L.: Input/output logics. J. Philos. Log. 29(4), 383–408 (2000)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  47. Makinson D., van der Torre L.: Permission from an input/output perspective. J. Philos. Log. 32(4), 391–416 (2003)

    Article  MATH  Google Scholar 

  48. Makinson, D., van der Torre, L.: What is input/output logic? In: Foundations of the Formal Sciences II: Applications of Mathematical Logic in Philosophy and Linguistics. Trends in Logic Series, vol. 17, pp. 163–174. Kluwer Academic Publishers, Dordrecht (2003)

  49. Marquis, J.-P.: From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory. Springer, Berlin (2009)

  50. Marquis, J.-P.: Category theory. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Winter 2014 edn. http://plato.stanford.edu/archives/win2014/entries/category-theory/ (2013)

  51. Marquis, J.-P., Reyes, G.E.: The history of categorical logic: 1963–1977. In: Gabbay, D.M., Kanamori, A., Woods, J., (eds.) Handbook of the History of Logic: Sets and Extensions in the Twentieth Century, vol. 6, pp. 689–800. North-Holland, Amsterdam (2012)

  52. McNamara, P.: Deontic logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Winter 2014 edn. http://plato.stanford.edu/archives/win2014/entries/logic-deontic/ (2010)

  53. Morreau, M.: Reason to think and act. In: Nute, D. (ed.) Defeasible Deontic Logic, pp. 139–158. Kluwer Academic Publishers, Dordrecht (1997)

  54. Mott P.: On Chisholm’s paradox. J. Philos. Log. 2(2), 197–211 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  55. Nair S.: Consequences of reasoning with conflicting obligations. Mind 123(491), 753–790 (2014)

    Article  MathSciNet  Google Scholar 

  56. Niles I.: Rescuing the counterfactual solution to Chisholm’s paradox. Philosophia 25(1–4), 351–371 (1997)

    Article  Google Scholar 

  57. Nute, D. (ed.): Defeasible Deontic Logic. Kluwer Academic Publishers, Dordrecht (1997)

  58. Nute, D., Yu, X.: Introduction. In: Nute, D. (ed.) Defeasible Deontic Logic, pp. 1–16. Kluwer Academic Publishers, Dordrecht (1997)

  59. Peterson, C.: Analyse de la structure logique des inférences légales et modélisation du discours juridique. Ph.D. thesis, Université de Montréal (2014)

  60. Peterson, C.: A logic for human actions. Manuscript submitted for publication (2014)

  61. Peterson, C.: Categorical foundations for logic: Lambek’s legacy revisited, Manuscript submitted for publication (2014)

  62. Peterson C.: The categorical imperative: category theory as a foundation for deontic logic. J. Appl. Log. 12(4), 417–461 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  63. Prakken H., Sergot M.: Contrary-to-duty obligations. Stud. Log. 57(1), 91–115 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  64. Prakken, H., Sergot, M.: Dyadic deontic logic and contrary-to-duty obligations. In: Nute, D. (ed.) Defeasible Deontic Logic, pp. 223–262. Kluwer Academic Publishers, Dordrecht (1997)

  65. Prior A.: The paradoxes of derived obligation. Mind 63(249), 64–65 (1954)

    Article  Google Scholar 

  66. Reiter R.: A logic for default reasoning. Artif. Intell. 13, 81–132 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  67. Rescher N.: An axiom system for deontic logic. Philos. Stud. 9(1), 24–30 (1958)

    Article  Google Scholar 

  68. Rescher N.: Conditional permission in deontic logic. Philos. Stud. 13(1), 1–6 (1962)

    Article  Google Scholar 

  69. Robison J.: Further difficulties for conditional permission in deontic logic. Philos. Stud. 18(1), 27–30 (1967)

    Article  Google Scholar 

  70. Royakkers, L., Dignum, F.: Defeasible reasoning with legal rules. In: Nute, D.(ed.) Defeasible Deontic Logic, pp. 263–286. Kluwer Academic Publishers, Dordrecht (1997)

  71. Ryu, Y.U., Lee, R.M.: Deontic logic viewed as defeasible reasoning. In: Nute, D. (ed.) Defeasible Deontic Logic, pp. 123–137. Kluwer Academic Publishers, Dordrecht (1997)

  72. Seely, R.A.G.: Linear logic, *-autonomous categories and cofree coalgebras. In: Gray J.W., Scedrov A. (eds.) Categories in Computer Science and Logic, Contemporary Mathematics, vol. 92, pp. 371–382 (1989)

  73. Straßer C.: A deontic logic framework allowing for factual detachment. J. Appl. Log. 9(1), 61–80 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  74. Tomberlin J.: Contrary-to-duty imperatives and conditional obligations. Noûs 15(3), 357–375 (1981)

    Article  MathSciNet  Google Scholar 

  75. Tomberlin, J.: Contrary-to-duty imperatives and Casta neda’s system of deontic logic. In: Tomberlin, J.(ed.) Agent, Language and the Structure of the World, pp. 231–249. Hackett, Indianapolis (1983)

  76. van Benthem J., Grossi D., Liu F.: Priority structures in deontic logic. Theoria 80(2), 116–152 (2014)

    Article  MathSciNet  Google Scholar 

  77. van der Torre, L., Tan, Y.-H.: The many faces of defeasibility in defeasible deontic logic. In: Nute, D. (ed.) Defeasible Deontic Logic, pp. 79–121. Kluwer Academic Publishers, Dordrecht (1997)

  78. van der Torre L., Tan Y.-H.: Contrary-to-duty reasoning with preference-based dyadic obligations. Ann. Math. Artif. Intell. 27(1–4), 49–78 (1999)

    Article  MATH  Google Scholar 

  79. van Eck J.: A system of temporally relative modal and deontic predicate logic and it’s philosophical applications. Log. Anal. 25(99), 249–290 (1982)

    MATH  Google Scholar 

  80. van Fraassen B.C.: The logic of conditional obligation. J. Philos. Log. 1(3), 417–438 (1972)

    Article  MATH  Google Scholar 

  81. von Wright G.H.: Deontic logic. Mind 60(237), 1–15 (1951)

    Article  Google Scholar 

  82. von Wright G.H.: A note on deontic logic and derived obligation. Mind 65(260), 507–509 (1956)

    Article  Google Scholar 

  83. von Wright G.H.: Deontic logics. Am. Philos. Q. 4(2), 136–143 (1967)

    Google Scholar 

  84. von Wright G.H.: Deontic logic: a personal view. Ratio Juris 12(1), 26–38 (1999)

    Article  Google Scholar 

  85. Vorobej M.: Conditional obligation and detachment. Can. J. Philos. 16(1), 11–26 (1986)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Munich Center for Mathematical Philosophy, Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, 80539, Munich, Germany

    Clayton Peterson

Authors
  1. Clayton Peterson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Clayton Peterson.

Additional information

I would like to thank the anonymous referees for their helpful comments on previous versions of this paper. A special thank you goes to Jean-Pierre Marquis for his constructive comments, encouragements, support and, more importantly, for his trust. I would also like to thank Andrew Irvine, François Lepage and Yvon Gauthier for their involvement in the project. This research was financially supported by the Social Sciences and Humanities Research Council of Canada.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peterson, C. Contrary-to-Duty Reasoning: A Categorical Approach. Log. Univers. 9, 47–92 (2015). https://doi.org/10.1007/s11787-014-0111-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11787-014-0111-7

Mathematics Subject Classification

Keywords

Search

Navigation