A Lewisian taxonomy for deontic logic


Philosophers like G.H. von Wright and D. Makinson have pointed to serious challenges regarding the foundations of deontic logic. In this paper, I suggest that to deal successfully with these challenges a reconsideration of the research program of the discipline is useful. Some problems that have troubled this particular field of logical study for decades may disappear or appear more tractable if we view them from the perspective of a language game introduced by D. Lewis involving three characters: the Master, the Slave and the Kibitzer. The adoption of this perspective opens a natural approach to a new layout of the domain of deontic studies. I propose dividing deontic logic into six sub-areas which are distinguished (i) by their focus on the different idioms typical of the individual players, (ii) by conceiving the language game as either being static or as dynamic and (iii) by the aims of the logical inquiry. What kind of insights the proposed perspective provides is illustrated by an analysis of the so-called Ross paradox—a problem that has troubled deontic logic since its origins and, though it was many times pronounced solved, still keeps coming back ‘alive and kicking’.

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  1. 1.

    See von Wright (1991, p. 256, 1999, p. 15).

  2. 2.

    Among the contributions which address the foundational issues, we can mention Makinson (1999), Vranas (2010, 2011), Hansen et al. (2007), Broersen and van der Torre (2012) or Charlow (2014).

  3. 3.

    See Lewis (1979a, b).

  4. 4.

    Those who are well oriented in the area of deontic logic can skip this section.

  5. 5.

    In fact, precursors of deontic logic can be found among medieval logicians (cf. Knuuttila 1981).

  6. 6.

    The title of Mally’s monograph is Grundgesetze des Sollens. Elemente der Logik des Willens (Mally 1926). It is worth noting that Mally’s main goal was to provide an exact system of pure ethics, so he was primarily interested in the elucidation of the concept of the ethical Ought.

  7. 7.

    This was pointed out by Karl Menger, see Menger (1939). Unlike Lewis, who swiftly corrected his system (cf. Lewis 1920), Mally never reacted to the challenge.

  8. 8.

    Von Wright gives credit to his colleague C.D. Broad, who proposed the choice of the term (see von Wright 1951, p. 1).

  9. 9.

    This challenge was clearly articulated by Jørgen Jørgensen (1937–1938) and, thanks to A. Ross, it became known by the name Jørgensen’s dilemma (cf. Ross 1941).

  10. 10.

    Systems of legal norms of a particular country are the paradigmatic example here.

  11. 11.

    Rarely, however, does it get clearly articulated. In this disguised form it also appears in alternative approaches based e.g., on dynamic logic (see, for example, Meyer 1988; Dignum et al. 1996 or Pérez-Ramírez and Fox 2003). Philosophers who deny the possibility that norms (prescriptions, imperatives) might be bound by logical relations don’t, of course, share this idea (see e.g., Williams 1963; Wedeking 1970 or Hansen 2013). Explicit defence of the idea that there is no distinction between the logic of norms and the logic of normative propositions can be found in Hilpinen (2006).

  12. 12.

    Someone might suggest the name von Wright’s thesis, but this would not be completely appropriate as von Wright revised his views several times [in von Wright 1999 he, for example, explicitly says that “classical deontic logic, descriptively interpreted, cannot claim to be the (correct) logic of norm-propositions”] and he would surely have had provisos against the blunt formulation that can be found in Stenius’ Principle III. On the other hand, von Wright’s contribution to the spreading of this kind of approach to deontic logic is beyond doubt.

  13. 13.

    All quotations here are from von Wright (1963, p. 134). It should be stressed that von Wright later changed his views and opted for an independent logic of norms (see von Wright 1999). He was also aware that his attitude might have influenced the development of deontic logic in a problematic way: “In my first paper I thought the mere fact that one can construct a formal calculus with plausible-sounding axioms was all that was needed to satisfy the demands of logic. And I think it is right to say that this attitude still implicitly underlies much of the work that is being done in deontic logic. Its problematic nature, however, has to this day remained a thorn in my logical flesh, if I may use this metaphor” (von Wright 1999, p. 19).

  14. 14.

    We have, no doubt, such connectives in natural language. E.g., “if...(then)...” in the sentence “If it rains, (you should) keep the window closed!”, “...or...” in “(You should) shut up or I will shut you up” or “...and...” in “Stay here and I will go home” are connectives of this sort. Some linguistically oriented accounts of semantics—e.g., Starr (2013)—employ such connectives suitable for combining imperatives and declaratives, but I don’t know about a logical theory that introduces ‘versatile’ connectives of this kind as proper logical constants.

  15. 15.

    An interesting analysis of the different forms that derogation can take can be found in Cornides (1969).

  16. 16.

    It is worth noting that pointing to precariousness of the ‘parallelism assumption’ doesn’t amount to claiming that a logic based on a prescriptive language cannot be in principle mirrored in a logic based on a descriptive language. It just suggests that the ‘double interpretation’ strategy is much trickier than it seems and hasn’t yielded satisfactory results.

  17. 17.

    Lewis speaks about the Master, the Slave and the Kibitzer (“Kibitzer” was originally a Yiddish term designating a bystander offering (often unwanted) advice or commentary). Though his terminology has an advantage of transparency, the associations it evokes may be misleading. It should be noted that the role of the Prescriber is quite similar to the role that the sovereign called Rex has in Hart’s model of legislation (Hart 1961).

  18. 18.

    We can, of course, also assume that the players are capable of uttering common indicative sentences describing the factual world. Within the present discussion I will, however, keep things as simple as possible and leave the (no doubt very interesting) problems arising from the consideration of mixed language games combining prescriptions and/or deontic statements with factual statements aside.

  19. 19.

    When considering simple oral games I will normally assume (together with Lewis) that at each step of the game the Prescriber issues just one prescription, but it, of course, makes sense to also consider the more general (and complex) case in which at each step the Prescriber issues a set of prescriptions.

  20. 20.

    My conception of the SP and SA diverges slightly from the one outlined by Lewis, but they both presuppose that the actual world always belongs to SA (not necessarily to SP).

  21. 21.

    It is important to stress that nothing hinges on specifically using the imperative mood. Unlike authors focused on linguistically oriented semantic studies [e.g., Kaufmann 2012 or Starr (2013)], I am here indifferent as concerns the formulation of the prescriptions. They might be expressed all the way through by sentences like “it is obligatory that (you)...”, “(from now) you must (not)...” or “I hereby command (permit) you to...” as soon as it is clear that the sentences are understood as shaping (and not just describing) the SP. An interesting argumentation showing that if permissions are taken to have this prescriptive (action guiding) role then no possible worlds account of their truth-conditions can succeed, can be found in Fine (2014).

  22. 22.

    For the employment of a version of dyadic language (in which conditional prescriptions are primitive) and arguments in favour of this choice, see, e.g., von Wright (1961) or van Fraassen (1972).

  23. 23.

    Portner (2012) suggests that talking about entailment in case of sentences to which we don’t attribute truth or falsity is not felicitous; he instead speaks about the relation of warranting. I, however, don’t see any serious reason for avoiding the term “entailment” in this context.

  24. 24.

    In most systems of deontic logic, the operator O is taken as primitive and the other is introduced by the following definition \(\hbox {P}\upalpha =_{\mathrm{df} }\lnot \mathrm{O}\lnot \upalpha \). Acceptance of the definition amounts to adoption of the weak (or negative) concept of permission—permission as a ‘lack of prohibition’.

  25. 25.

    David Lewis straightforwardly adopts this conception of the game, see Lewis (1979b).

  26. 26.

    Such a strategy is sometimes applied in law in cases where priority to the earlier acts over the later ones is granted for certain reasons (lex prior derogat legi posteriori).

  27. 27.

    Typically, in such cases, we will have one Prescriber that addresses many Doers. We may either suppose that many ‘parallel’ games are being played in such a situation or that we are dealing with one game with a collective—distributively conceived—Doer. Classification of ‘games’ with various types of “sources and recipients” can be found in Rescher (1966, Chapter 2).

  28. 28.

    Of course, in the case of specific games some conflict-solving principles can be presupposed, e.g., adoption of the principle that more specific regulation is to be preferred against the general one (a version of the principle lex specialis derogat legi generali).

  29. 29.

    Being (explicitly or implicitly) laid down is thus a status (‘value’) of a prescription within a language game. Lewis boldly treats prescriptions as having truth values. In my view, this is problematic not only because it is at odds with common practice but, especially, because it obscures the crucial distinction between the descriptive language and the prescriptive language.

  30. 30.

    The denomination suggests that this logic should yield answers for the Doer who lost track of the situation and wants to know whether he is allowed (obliged, forbidden) to take some course of action in a given stage of the game. It is natural to assume that the Kibitzer is the player who should be competent to employ the logic and provide the answers.

  31. 31.

    It is worth noting that the question of what the Doer is obliged to do can also be interpreted in a more specific way, namely as asking what the Doer is at a given moment and situation described by a set of factual ‘premises’ obliged (permitted) to do. In such a case, reports about different conditional obligations as well as other deontic statements that don’t go to the point are irrelevant. The Kibitzer at the end of Game (3) is expected to just give ‘direct advice’ like “You are obliged to stay in the house”, “You may go to the cellar” (in a situation where a tornado is coming) or “You may smoke” (in the—inadmissible—situation where the Doer is outside of the house).

  32. 32.

    The logics for the Doer are obviously of a quite specific nature—they interconnect two different languages. Their ‘inputs’ are formulated in the language of the Prescriber while their ‘outputs’ are sentences belonging to the language of the Kibitzer.

  33. 33.

    Of course, due to the literature’s richness the survey will inevitably be sketchy.

  34. 34.

    Things, however, are not so simple. Dynamic games can be designed so that they don’t allow for making more moves ‘at once’. Also, it may be reasonable to conceive the scope of DyLoKi so that statements which are not suitable as straightforward descriptions of a deontic situation [such as \((\hbox {O}p \vee \hbox {O}q) \leftrightarrow \hbox {O}\lnot r\) or \(\hbox {O}p \rightarrow \lnot s]\) don’t figure in the updating sequences. In such a case, the problems addressed by StLoKi are not automatically ‘covered’ by DyLoKi.

  35. 35.

    Someone can, for example, conclude that while StLoKi should employ a monadic language, StLoPr requires a dyadic language.

  36. 36.

    This term originates, as far as I know, with Lewis (1979b).

  37. 37.

    Though, as I suggested, quite a large number of the theories had bigger ambitions since their creators often wanted to constitute a ‘universal’ deontic logic capturing logical properties of “ought sentences” all at once in both their descriptive and prescriptive interpretations.

  38. 38.

    Alchourrón and Makinson (1981), Meyer (1988), Brown (2004), Lindström and Segerberg (2006), or Stolpe (2010) can be seen as contributions to this area.

  39. 39.

    This doesn’t hold if we consider complex systems with hierarchized prescriptions. For discussion of such systems compare, e.g., Bulygin (1986), Hansen (2006b, 2006c).

  40. 40.

    We might, of course, consider other backgrounds of the game than the liberal one introduced by Lewis. For basic considerations concerning this possibility see Svoboda (2003).

  41. 41.

    Some of these issues are, in a specific form, discussed from the perspective of jurisprudence (recently, e.g., Governatori and Rotolo (2010). Other relevant ideas can be found in Boella and van der Torre (2008) or Broersen et al. (2012).

  42. 42.

    This gets manifested in the further development of his work. In articles by Makinson and van der Torre on Input-Output logic they conceive the logic as “the logic of conditional norms” (see Makinson and van der Torre 2003a, p. 163nn). As the norms are not truth-evaluable it seems proper to classify their logic as StLoPr. The logic, however, has features that are problematic. The Ross paradox, for example, turns out to be a valid inference pattern of the logic, and weak (negative) permissions are treated as kinds of implicit norms (see Makinson and van der Torre 2003b).

  43. 43.

    Belzer bases his solution on the assumption that possible worlds are (weakly) ordered—some are more highly ranked (more permissible).

  44. 44.

    See, e.g., Føllesdal and Hilpinen (1971), Åqvist (2002).

  45. 45.

    Cf., e.g., Ross (1941), Hare (1967), Meyer et al. (1994), Danielsson (2005), Hansen (2006a), Straßer and Beirlaen (2012).

  46. 46.

    It is presumable that at the moment of issuing the second prescription she knows or assumes that the Doer has not managed to fulfil the original order.

  47. 47.

    More controversial is the question whether all cases when a Prescriber issues a weaker prescription in the situation when she previously issued a stronger one (for example Don’t speak loudly! after Don’t speak! or Stay in Ireland until you finish your exams! after Stay in Dublin until you finish your exams!) should be generally interpreted as moves expanding the SP (and whether such moves should be seen as testifying that the Prescriber must have changed her intentions).

  48. 48.

    It is perhaps worth noting that Rescher (1966) admits that the inferential step might be admissible if the form !(\(p \vee q)\) represents what he calls alternative-indicating command (and not, as is standard, a choice-presenting command). This is not, however, a serious proviso as his alternative-indicating commands present rather marginal cases in which “or” is used to introduce a ‘punishing alternative’ (e.g., “Shut up or get out of here!”).

  49. 49.

    We will surely require that any respectable StLoDo or DyLoDo theory yields this result.

  50. 50.

    Her answer breaches conversational maxims discussed by Paul Grice (see Grice 1975).

  51. 51.

    It is perhaps worth stressing once again that ‘formula’ !\(c \vee \) !w does not come into consideration as an expression of the Prescriber’s language even though it seems to faithfully reflect the form of the sentence.

  52. 52.

    This would be even more obvious if the two actions in question were clearly incompatible (e.g., Stay in London or go to New York). Those who would defend the two answers as admissible would be forced to claim that two incompatible statements can be true at the same time.

  53. 53.

    The meaning of this sentence is not adequately captured by the formula \(\hbox {O}(p \vee q)\) employing Brown’s Type 2 interpretation of the operator O, which is introduced with the intention to avoid the Ross paradox. Brown’s Type 2 formula \(\hbox {O}(p \vee q)\) can be true even if there is no freedom of choice (cf. Brown 1996).

  54. 54.

    Hansson (2013) shows that the straightforward introduction of a free-choice operator \(\hbox {P}_{c}\) into the language of standard deontic logic by the definition \(\hbox {P}_{{c}}(p \vee q) \leftrightarrow \hbox {P}p \wedge \hbox {P}q \wedge \hbox {P}\lnot p \wedge \hbox {P}\lnot q \) proposed in Woleński (1980) has clearly unacceptable consequences, and it is obvious that introduction of a free-choice obligation operator by the definition \(\hbox {O}_{c}(p \vee q)\leftrightarrow \hbox {O}(p \vee q) \wedge \hbox {P}p \wedge \hbox {P}q \wedge \hbox {P}\lnot p \wedge \hbox {P}\lnot q\) would yield similar problems. Generally Hansson convincingly demonstrates that the free choice interpretation of formulas like \(\hbox {P}(p \vee q)\) is untenable under standard extensional interpretation of disjunction.

  55. 55.

    For more about the possible backgrounds of the game see Svoboda (2003).

  56. 56.

    For a detailed linguistic substantiation of this claim, see Portner (2012).

  57. 57.

    Admittedly inconsistencies arising between a strong prescription (an order) and a weak prescription (a permission) are not as ‘urgent’ as inconsistencies between two strong prescriptions (normally we tend to presume that orders ‘trump’ permissions). This, however, surely doesn’t mean that we should conclude that any permission is consistent with any order.

  58. 58.

    We can imagine a somewhat mischievous Kibitzer saying “You may borrow either the big or the small crowbar, try to guess which one”. And then, a little later: “I will give you a hint—you must not borrow the big one”. Any reasonable Doer would, of course, quickly conclude that he is allowed to borrow the small crowbar.

  59. 59.

    Roughly—if the prescription in the ‘premise’ is laid down in a context, the statement in the ‘conclusion’ is true in the context.

  60. 60.

    Of course, in the case of a specific game (meta)rules might be introduced that would ‘instantly’ remove the conflict.

  61. 61.

    Someone might suggest that moral judgements are, by their very nature, double-faced (both descriptive and prescriptive) and we should try to develop a specific logic that respects this. I don’t find such a view convincing. First, I think that it is important to clearly distinguish when one is making a claim about morals or about some particular ethical code and when one articulates a moral prescription. Second, I believe that ethics (resp. metaethics) as well as law (resp. jurisprudence) are areas in which one can profit from making use of the analytic tools developed within deontic logic. They, however, should be viewed as areas in which logic is applied rather than as areas that are governed by their own logics.


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I am grateful to Ansten Klev, Jaroslav Peregrin, and Vít Punčochář for valuable comments on drafts of the paper.

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Correspondence to Vladimír Svoboda.

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Work on this paper was supported by the research Grant No. 13-20785S of the Czech Science Foundation.

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Svoboda, V. A Lewisian taxonomy for deontic logic. Synthese 195, 3241–3266 (2018). https://doi.org/10.1007/s11229-017-1370-7

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  • Deontic logic
  • Logic of imperatives
  • Language games
  • Ross paradox
  • Free choice permission