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What difference might and may make

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Abstract

How does your information change when you learn that something might be the case, where the modal “might” is epistemic? On the orthodox view, a proposition is added to your information base; on the view defended here, no propositions are added to your information base but some are removed from it. I (i) argue that Stephen Yablo’s recent attempt to define this removal operation as a kind of propositional subtraction fails, (ii) offer a definition of my own in terms of the part–whole relations between the truthmakers of the propositions one accepts, and (iii) argue that a deontic analogue of this account solves a problem about permission posed long ago by David Lewis.

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Notes

  1. The idea that this exhausts the semantic role of “might”—that “might” is just a speech act modifier—goes back to Kant (1781), p. 74, and Frege (1879), p. 5. For a recent proposal along these lines, see Westmoreland (1995). For criticism, see Fintel and Gillies (2010). Note that, unlike these authors, I am not making a general claim about the meaning or semantic role of “might”. My topic is the epistemic effect of stand-alone might-sentences—the epistemics, not the semantics, of might. Whether this effect is to be accounted for by the semantics, the pragmatics, or both, is a question I want to remain neutral on in this paper.

  2. See, for instance, Rott (1989), p. 109, Gillies (2006), p. 240, Yalcin (2007), p. 1010, Swanson (2011), p. 254, and Yablo (2011). More on Yablo’s view below.

  3. Among those who have offered a truth-conditional semantics for epistemic modals are both contextualists (e.g., DeRose (1991), Kratzer (1977, 1981), and Fintel and Gillies (2009)) and relativists (e.g., MacFarlane (2009) and Egan (2007)).

  4. Note that I am talking here about the effect of accepting a might-sentence and not about the effect of hearing and understanding an utterance of such a sentence. Hearing and understanding an utterance of a sentence always leads to the acceptance of propositions, such as the proposition that the utterance took place, but the acceptance of these propositions is a side-effect and not the intended, or essential, effect of uttering the sentence. By framing the discussion in terms of the effect of accepting a might-sentence and not in terms of the effect of hearing and understanding an utterance of, a might-sentence, I am abstracting away from the unintended or inessential effects of such utterances. Cf. Stalnaker (1979) on the essential effect of assertions.

  5. The dynamic approach to meaning goes back to Gärdenfors (1984), Gärdenfors (1988), Chapter 6, Heim (1982, 1983), and Kamp (1981). See also von Fintel and Gillies (2007).

  6. While Veltman models information states as sets of possible worlds, Willer models them as sets of sets of possible worlds. This allows Willer to distinguish between those possibilities that are live possibilities for the agent and those that are merely consistent with his evidence. Might-sentences can turn a possibility that is merely consistent with the agent’s evidence into a live possibility, thereby expanding the set of live possibilites. However, since this is accomplished by eliminating from the agent’s information state those elements that are inconsistent with the prejacent, the overall effect is a strengthening of the agent’s information state.

  7. Veltman (1996), p. 229; Willer (2013), p. 60. See also Yablo (2011), p. 274, footnote 16.

  8. Thus Gillies (2006), p. 118, writes: “...although our topic is epistemic change and although we are investigating epistemic change in the context of solipsistic epistemic modals like might, let us assume that the impetus for such change is always non-modal. So, although agents will have epistemic commitments like It might be raining, they only ever revise their picture of the world with respect to “plain facts” like It is raining.

  9. I write “any prior assertions” because I am not assuming that there always are such assertions. “Any” is existentially non-committal here, as in “Any trespassers will be prosecuted.” And just as a sign that says “Trespassers will be prosecuted” can be a device for keeping out trespassers even if there aren’t any trespassers, so a might-sentence can be a device for cancelling the effects of prior assertions even if there aren’t any prior assertions.

  10. One might object that if the examples show that might-claims are devices for cancelling the effects of prior assertions, then exactly analogous examples show that simple (non-modal) assertions are devices of canceling the effects of prior assertions. But this is clearly absurd. Therefore, the examples don’t show that might-claims are devices of canceling the effects of prior assertions. But, first, the examples are supposed to motivate, not establish, the view, and, second, the claim that might-sentences are devices for canceling the effects of prior assertions is plausible since this is their only essential effect, whereas the analogous claim about simple assertions is not plausible since they always have another essential effect—they expand one’s information state by the asserted proposition.

  11. So, just as an assertion that \(\varphi \) leaves one’s information state unchanged if one already believes that \(\varphi \), so a might-\(\varphi \) claim leaves one’s information state unchanged if one already allows that \(\varphi \). This is not to deny that both speech acts can have other, inessential, effects, such as reinforcing a belief or reminding one of a possibility.

  12. Besides deontic “may” there is also epistemic “may”. However, to facilitate exposition, I will use “might” as my exemplar of an epistemic possibility modal and “may” as my exemplar of a deontic possibility modal.

  13. I will omit this qualifier henceforth.

  14. Note that \(S- [\![\lnot P]\!]^{}\) is the result of subtracting \([\![\lnot P]\!]^{}\) from \(S\), not the relative complement of the set \([\![\lnot P]\!]^{}\) in the set \(S\).

  15. Cf. Jaeger (1976), p. 320; Hudson (1975), p. 130, Yablo (2011), Figures 9.1, 9.2, pp. 280, 281.

  16. This intuition probably stems not from arithmetic but from the physical operations on collections of objects that we describe with arithmetic, such as taking a stone out of a bowl of stones. Cf. Hudson (1975), p. 130.

  17. The situation is different in the belief revision literature on the closely related concept of contraction. There condition (i) is known as Recovery and is highly controversial. It is assumed in the original AGM-model (Alchourrón et al. 1985) but not in later variants such as the theory of severe withdrawal ofRott and Pagnucco (1999) or the theory of belief base contraction of Hansson (1999). However, while the concepts of contraction and logical subtraction are closely related (both concern weakening operations), they are nevertheless distinct concepts. Contraction is (a model of) a kind of belief change: it is (a model of) the change one undergoes when giving up a belief without acquiring a new one. It is thus subjective and epistemic. Logical subtraction, on the other hand, is supposed to be an entirely objective and metaphysical matter; it is supposed to be as objective and non-epistemic as set-theoretic complementation. Recovery (condition (i)) might therefore be defensible as a condition on logical subtraction even if it is indefensible as a condition on contraction. I will follow Jaeger, Hudson, and Yablo and reserve the term “subtraction” for those weakening operations that satisfy (i). But this is a purely terminological decision; no substantive questions are begged thereby. If it turns out (as I will argue) that updates with might are not cases of logical subtraction in my sense of an operation meeting condition (i), it may still turn out that they are cases of subtraction in a different sense of subtraction, such as the one in Fuhrmann (1999), on which subtraction need not meet condition (i). More on contraction below.

  18. See Hudson (1975) for an argument that the additional condition is that \(X\) is the logically weakest proposition meeting conditions (i) and (ii). For discussion, see Jaeger (1976) and Humberstone (2000).

  19. The reader is invited to test this claim against his own intuitions.

  20. This is again the assumption known as Recovery in the belief revision literature (see, e.g., Gärdenfors 1984 and Fuhrmann 1999).

  21. See, for instance, (Fuhrmann 1999), who calls an operation “subtraction” that does not satisfy this principle.

  22. Yablo (2011), pp. 282–283.

  23. It is usually assumed that the common ground is closed under logical consequence, but even if we relax this condition and require only that the common ground is closed under obvious logical consequence, we are still left with a set of propositions that is larger than necessary to determine the context set.

  24. Likewise, for ¡P. To avoid repetition, I will focus on the epistemic case in this section, but everything I say applies, mutatis mutandis, to the deontic case.

  25. The right-hand side is essentially full meet contraction (cf. Gärdenfors 1988, p. 78), what follows the union-sign maxichoice contraction (Gärdenfors 1988, p. 76).

  26. The name is not meant to suggest any relation to the relevant implication of Anderson and Belnap (1975).

  27. Notice that we cannot in this way avoid reference to non-actual situations. If \((A\wedge B)\vee C\) happens to be false, it is vacuously true that every minimal (actual) situation in which \((A\wedge B)\vee C\) is true could have been part of some minimal situation in which \(A\wedge B\) is true.

  28. E.g. in Kratzer (2011).

  29. These are, of course, only intuitions. Without clear criteria for situations and parts of situations, we can’t be sure that there are such propositions; hence the hedge. Thanks to Angelika Kratzer for pointing this out.

  30. Cf. Rosen (2010), p. 116. Notice that my use of “truthmaking” differs from the standard one (found, for instance, in Mulligan et al. 1984), on which truthmaking is a monotonic relation.

  31. At least this is so with respect to basic propositions—propositions that attribute a property (or an \(n\)-ary relation) to an individual (or an \(n\)-tuple of individuals). It may not hold with respect to universal generalizations. Propositions which satisfy this monotonicity requirement are called “persistent”. Cf. Kratzer 1989, p. 616.

  32. Whether the converse is also true is something we don’t need to decide for the purposes of this paper. Apparent counterexamples are such states as \(a < b\) and \(b > a\), which don’t involve the same relations but seem identical nonetheless, or states that are related as \(a \parallel b\) is to direction(\(a\)) \(=\) direction(\(b\)) (Cf. Frege (1988), §64).

  33. Here I am following Kratzer (1989), p. 616.

  34. Notice that this is not the same as the sum (union) of whatever is necessitated by some part that makes the first conjunct true and whatever is necessitated by some part that makes the second conjunct true, for the two parts may jointly necessitate something that neither necessitates individually.

  35. Compare the definitions of lumping in Kratzer (1989) and of content-part in Yablo (2012).

  36. I.e., that it might be the case that it is not the case that both Antony and Lepidus will be honored by the senate.

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Acknowledgments

I would like to thank Stephen Yablo, Peter Vranas, Kevin Toh, Angelika Kratzer, and two anonymous referees for helpful comments and discussion.

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    Gerhard Nuffer

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Nuffer, G. What difference might and may make. Synthese 192, 405–429 (2015). https://doi.org/10.1007/s11229-014-0576-1

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