Modus Ponens Under the Restrictor View

Abstract

There is a renewed debate about modus ponens. Strikingly, the recent counterexamples in Cantwell (Theoria, 74, 331–351 2008), Dreier (2009) and MacFarlane and Kolodny (The Journal of Philosophy, 107, 115–143 2010) are generated by restricted readings of the ‘if’-clause. Moreover, it can be argued on general grounds that the restrictor view of conditionals developed in Kratzer (1986) and Lewis (1975) leads to counterexamples to modus ponens (cp. Charlow Synthese, 190, 2291–2323 2013; Khoo Philosophical Studies, 166, 153–64 2013). This paper provides a careful analysis of modus ponens within the framework of the restrictor view. Despite appearances to the contrary, there is a robust sense in which modus ponens is valid, owing to the fact that conditionals do not only allow for restricted readings but have bare interpretations, too.

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Notes

  1. 1.

    This presentation follows closely the one in [54, Chap. 5].

  2. 2.

    The observation there is essentially the same as the argument for restricted quantifiers in constructions like ‘Most people enjoy watching TV’ (cp. [4]).

  3. 3.

    Square brackets indicate the supposed phrase structure of a sentence.

  4. 4.

    See in particular the collection of papers in Kratzer [33].

  5. 5.

    More precisely, the ordering is defined in the following way, where A is a set of propositions A:

    $$w_{1} \leq_{A} w_{2} \equiv_{\text{df.}} \forall p \in A: (w_{2} \in p) \supset (w_{1} \in p). $$

    In words: if (and only if) w1 makes true all propositions in A which w2 makes true, then w1Aw2.

  6. 6.

    As Kratzer acknowledges, the present definition is modeled closely after a similar definition given by Lewis [35, 16] for counterfactuals.

  7. 7.

    Since Kratzer wants to allow for worlds getting closer and closer without there being some closest worlds, the definition is a little more complex. For a proposition to be necessary it requires that there is to any world u determined by the conversational background a world vg(w)u such that the proposition holds at all worlds closer than v or at least as close as v.

  8. 8.

    Following MacFarlane and Kolodny [43, 127ff.], I set aside the possibility that the problematic argument reveals that something is wrong with the classical inference rules for disjunction.

  9. 9.

    MacFarlane and Kolodny [43, 129] mention that the problem generalizes to epistemic ‘must’. Carr [9] discusses an extension of the original miners case in terms of ‘probable’. Moreover, the case is similar to a potential counterexample to modus tollens put forward in ut forward in [68]. See also MacFarlane and Kolodny [43, 128] for a deontic variant of Yalcin’s puzzle. Cantwell [7] takes modus tollens to fail for similar reasons.

  10. 10.

    See Lewis [37, 39]. Hájek and Hall [24] contains an overview about various similar results. For means of escape, see Bacon [3], McGee [45], Schulz [53, Chp. 8] and van Fraassen [60]. Cp. also Rothschild [51] for a more informal attempt to explain the data contextually. Note, however, that restricted readings are available not only for epistemic modals but also for adverbial quantifiers, deontic modals and nested conditionals. It is an open question whether the available positive results can be extended to substitute the restrictor view in all its realms of application.

  11. 11.

    The problematic reading of the conditionals in the Probabilified Miners seems to be due to a construal of the ‘if’-clause as a restrictor of the probability operator. The same is likely to be true of the original miners scenario, which proceeds in terms of ‘ought’ on a subjective interpretation. According to the latter, what an agent ought to do is sensitive to what the agent knows. For lack of space, I will refrain from arguing that a restrictor analysis is plausible for the problematic conditionals in the original miners case, but see the discussion in Carr [9], Charlow [10], MacFarlane and Kolodny [43] and Silk [55].

  12. 12.

    For further discussion, see Égré and Cosiz [15].

  13. 13.

    The situation changes if we make the minor premise, A, epistemically necessary. For instance, if we consider instead □A as the minor premise on an epistemic interpretation of □, then the argument pattern is valid for epistemic ‘must’, ‘probably’ and subjective ‘ought’. Hence, the counterexamples essentially require a context of epistemic uncertainty. This is somewhat obscured in the miners scenario due to the fact that it proceeds from a disjunction of two minor premises which are uncertain. And a disjunction can be certain while both its disjuncts are uncertain.

  14. 14.

    Wide scoping is a prominent response to puzzles involving deontic conditionals (cp. e.g. [5, 6]). See Kratzer [33, 67f.] and Silk [56] for comparison with the restrictor view.

  15. 15.

    Yalcin [68] considers this problem in the context of modus tollens.

  16. 16.

    On this view, ‘if’-clauses would be monsters in Kaplan’s sense.

  17. 17.

    Thanks to an anonymous referee for helping me to get clear about this issue.

  18. 18.

    I am setting aside here defenses of modus ponens which couple a shifty semantics (or a dynamic semantics) with an alternative notion of logical consequence. See Willer [66], Gillies [22] and Starr [59].

  19. 19.

    This notion might well be in need of further refinements. For our purposes, it merely labels conditionals for which the restrictor view has to tell a special story.

  20. 20.

    One may ask where a default necessity operator could come from. One possibility close to Kratzer’s suggestion would be that ‘if’-clause come with a necessity operator “build in” which can, in appropriate contexts, be overridden by overt operators (cp. [67]). Another possibility would be to develop the idea that virtually all declarative sentences come with an implicit necessity operator at matrix level, following work in linguistics on modals (e.g. [20]) and on implicatures (e.g. [2], [11] and [46]).

  21. 21.

    The same seems to be true for generic and habitual interpretations (I owe this observation to an anonymous referee of this journal). If so, the amount of flexibility in interpreting bare conditionals may be limited to a small class of episodic, generic and habitual readings. I restrict my attention to episodic readings, for they are the ones which matter for the following.

  22. 22.

    Lycan [41, 42, ch. 3] argues that modus ponens fails across the board. I set discussion of Lycan’s examples aside as this would take us too far astray. But see Nolan [47, 224f.] for a critique of Lycan’s theory.

  23. 23.

    That ‘must’ is strong is not uncontroversial. See Lassiter [34] for a response to von Fintel and Gillies [62]. Note, though, that Lassiter acknowledges that his overall conclusion that ‘must’ does not imply knowledge is compatible with ‘must’ being factive.

  24. 24.

    See also Frank [17, 46-55], Frank [69], and the discussion in Kratzer [33, 106f.]. Cp. further Fintel and Iatridou [63].

  25. 25.

    Geurts [18] suspects that the restricted reading being tautologous pragmatically explains why the bare reading is preferred. But see Kratzer [33, 106f.] for a reason why this might not be the best possible explanation.

  26. 26.

    See Schnieder [52] focusing on the problem discussed in Yalcin [67] why certain modals do not felicitously embed under ‘if’.

  27. 27.

    A case with this structure has been described by Regan [50, 264f., n.1] in the context of co-operation problems. Parfit [48] slightly modifies the example and illustrates it with the story of the miners. MacFarlane and Kolodny [43] use it to argue against modus ponens.

  28. 28.

    See Dowell [12] and Kiesewetter [29] for discussion of how to best characterize the pertinent notion of epistemic reach in this context. For our purposes, we can make the simplifying assumption that ‘ought’ simply targets the body of knowledge the agent has.

  29. 29.

    Semantic accounts of subjective ‘ought’ along these lines are developed in Carr [9], Charlow [10], Mac-Farlane and Kolodny [43] and Silk [55]. See in particular the detailed discussion of the various options for a restrictor analysis by Cariani et al. [8]. Cp. also Cantwell [7] focusing on ‘ought’ as it occurs in the gentle murder paradox (cp. [16]).

  30. 30.

    Cariani et al. [8, sec. 2.3.3] highlight the possibility of bare readings for subjective ‘ought’.

  31. 31.

    The first example is attributed to Kratzer by von Fintel [61], the second one is von Fintel’s own. It should be pointed out that Kratzer and von Fintel actually deny—for reasons having to do with the fact that the miner’s puzzle requires ‘ought’ to be seriously information-dependent in the sense of MacFarlane and Kolodny [43, 133]—that the conditionals have a restricted reading of the kind needed to set up the miners puzzle. But see Carr [9] for a response to von Fintel’s worries. Cp. also Carr [55].

  32. 32.

    The structure of the problem is already described in Adams [1, 33].

  33. 33.

    This describes the semantic role of already complex expressions of the form \(\ulcorner {\text {if} \phi }\ulcorner \). The semantics of ‘if’ would, on this picture, be taken to be a function from propositions, the semantic value of ‘ϕ’, to functions whose domain and range consists of the semantic values of operators, that is functions from propositions to propositions.

  34. 34.

    McGee [44, 465] provides a similar theorem to the same effect. Cp. also the discussion of Gibbard’s proof in Gillies [22].

  35. 35.

    Of course, if we follow Lewis and take bare conditionals to have the material truth conditions, then the conclusion of McGee’s example would—despite appearances to the contrary—simply be true on the most likely turn of events. Cp. the solution to McGee’s puzzle offered in Sinott-Armstrong et al. [57] and Katz [27], but see also Piller [49].

  36. 36.

    In effect, the present reconstruction amounts to the solution of McGee’s puzzle advocated in Lowe [40].

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Acknowledgements

An earlier version of this paper has been presented at a research seminar in Barcelona 2012, at the MCMP-colloquium in 2013 and at a research colloquium in Milan 2014. I would like to thank all the participants for their helpful comments. Special thanks are due to Thomas Krödel, Sven Rosenkranz, Giuliano Torrengo, Richard Woodward and an anonymous referee of this journal. The paper profited from the generous support of the DFG-funded project “Knowledge and Decision” (SCHU 3080/3-1).

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Schulz, M. Modus Ponens Under the Restrictor View. J Philos Logic 47, 1001–1028 (2018). https://doi.org/10.1007/s10992-018-9459-0

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Keywords

  • Conditionals
  • Modus ponens
  • Restrictor view