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Free choice, simplification, and Innocent Inclusion

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We propose a modification of the exhaustivity operator from Fox (in: Sauerland and Stateva (eds) Presupposition and implicature in compositional semantics, Palgrave Macmillan, London, pp 71–120, 2007. that on top of negating all the Innocently Excludable alternatives affirms all the ‘Innocently Includable’ ones. The main result of supplementing the notion of Innocent Exclusion with that of Innocent Inclusion is that it allows the exhaustivity operator to identify cells in the partition induced by the set of alternatives (assign a truth value to every alternative) whenever possible. We argue for this property of ‘cell identification’ based on the simplification of disjunctive antecedents and the effects on free choice that arise as the result of the introduction of universal quantifiers. We further argue for our proposal based on the interaction of only with free choice disjunction.

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  1. Here and throughout the paper we conflate logical forms and their truth conditions; e.g., \(\Diamond (a \vee b)\) will at times stand for a logical form in which an existential modal takes scope over disjunction and at other times for the corresponding truth conditions. We distinguish between the two only if we think things might be confusing otherwise.

  2. We refer to accounts of FC as “implicature accounts” when they propose the same mechanism for FC and for run-of-the-mill scalar implicatures. This terminological choice (which we adopt from much current literature) might be confusing at times, as some of the accounts, like our own, do not take scalar implicatures to be implicatures in the traditional Gricean sense (but rather logical entailments of a potentially ambiguous sentence under one of its parses).

  3. The problem Nouwen (2017) focuses on is FC with ability modals. As we will see, this problem will not be resolved under our approach without stipulations about the nature of the alternatives. See Sect. 6.

  4. Note that under an \(\mathcal {E} xh _{}\)-based theory of scalar implicatures, there is no reason to expect the nature of the operator to mimic neo-Gricean reasoning. In fact, if it did, this would be a suspicious coincidence which would cry for an explanation. Of course, it would be preferable for the properties of the operator to have some conceptual grounding. In Fox (2007, 2014, 2016, 2018b), it is suggested that the operator’s properties should follow from the function it plays, namely allowing speakers to efficiently convey all of their beliefs about a topic of conversation (characterized by a partition). Cell identification meets this design specification in many cases. Specifically, it allows opinionated speakers to convey all of their relevant beliefs in response to a question Q while meeting an independent formal constraint on answers (Fox 2018a, b). See also the discussion around (23) below.

  5. Implicature calculation as resulting in cell identification has been pursued in Franke’s (2011) game-theoretic approach, which is essentially a cell identification view of scalar implicatures. However, without various complications, Franke fails to derive several basic facts, such as FC disjunction with more than two disjuncts, universal FC inferences, and SDA with more than two disjuncts. Furthermore, he crucially relies on the assumption of equal priors for deriving FC (see Fox and Katzir 2018 for discussion).

  6. And, of course, they are also explained when weaker demands are made, as in Rooth (1992) and Heim (1996)—the ‘parallelism domain’ for ellipsis (the domain of \(\sim \) in Rooth 1992, Heim 1996) need not contain \(\mathcal {E} xh _{}\). If this operator were forced to be inside the parallelism domain, we would expect the implicature to be derived for the elided material as well; see Crnič (2015) for arguments that the presence of \(\mathcal {E} xh _{}\) inside the parallelism domain indeed has the predicted consequences. We will return to the issue of FC with ellipsis and the relevance of exhaustivity to parallelism in Sects. 5.3 and 9.2.

  7. A similar argument against ambiguity comes from non-monotonic contexts (see Bar-Lev 2018, Chap. 1):

    figure i

    One might consider the following fix for salvaging the ambiguity approach, in light of (7) and (i): allowed A or B gives rise to a special kind of ambiguity, which requires truth on all of its resolutions (along the lines of the ambiguity approach to homogeneity phenomena in Spector 2013; Križ and Spector 2017; Spector 2018). Since it does not seem to extend to every kind of ambiguity resolution we do not pursue this route.

  8. This is not strictly speaking true of Starr (2016), whose proposal nonetheless involves non-standard modifications of logical operators, a move we are trying to avoid.

  9. In addition, some of those approaches (Starr 2016; Aloni 2016) predict FC inferences when disjunction takes wide scope above the modal, a result argued against in Bar-Lev (2018, Chap. 2).

  10. This criticism is reminiscent of Schlenker’s (2009) criticism of dynamic semantics approaches.

  11. There are several reasons for assuming that disjunctive alternatives are generated: First, their existence provides an explanation for the fact that from a sentence like (i) we infer (ia) and (ib), i.e., the negation of the disjunctive alternatives (You are required to solve problem A, You are required to solve problem B). See Sauerland (2004), Spector (2007), Katzir (2007), Fox (2007); as well as Sect. 5.5.

    figure l

    Second, if FC inferences are to be treated as scalar implicatures, as we assume here, disjunctive alternatives are required. Third, the independently motivated structural approach to the generation of alternatives (Katzir 2007) predicts such alternatives to be generated, being the result of replacing a constituent (P or Q) with a sub-constituent (P, Q).

  12. To avoid clutter, we ignore here and throughout the paper alternatives generated by replacing \(\Diamond \) with \(\Box \) which do not interfere with the derivation of FC. We discuss whether such alternatives should be derived to begin with in Sect. 5.5.

  13. As pointed out by Simons (2005), FC inferences are not always accompanied by the inference that excludes the conjunctive alternative \(\Diamond (a \wedge b)\). Our derivation of FC in what follows does not depend on deriving the falsity of the conjunctive alternative (in fact it follows even if this alternative is omitted altogether from the set of alternatives). See Sect. 9.1 and Bar-Lev (2018, Chap. 2) for reasons why this alternative will be assigned false only if it’s taken to be relevant. Of course, one would then need to say why the same reasoning cannot apply to simple disjunction: why can’t the conjunctive alternative in this case be irrelevant, and thus lead to a conjunctive reading for simple disjunction? The answer that has been given in the literature (e.g., Fox and Katzir 2011) is based on the assumption that relevance is closed under Boolean operations (conjunction and negation). It is thus possible for \(\Diamond a\) and \(\Diamond b\) to be relevant without \(\Diamond (a \wedge b)\) being relevant; in contrast, once a and b are relevant, \(a \wedge b\) must be relevant as well. A simplifying assumption in the discussion here and in what follows is that all formal alternatives are relevant, but this should not be seen as an empirical claim that we always infer the falsity of the IE alternatives. See Fox and Katzir (2011) and Singh et al. (2016) for relevant discussion.

  14. Lack of closure under conjunction is a necessary, though not sufficient, requirement for compatibility with a conjunctive inference. There are cases that meet the requirement in which a conjunctive inference is nevertheless incompatible with the negation of all IE alternatives. To see this, consider Fox’s (2007) explanation of the fact that singular indefinites (in contrast to existential modals and plural indefinites) do not license conjunctive inferences: Some boy ate ice cream or cake does not lead to the inference that some boy ate ice cream and some boy ate cake. The explanation is based on the observation that singular indefinites lead to a potential only one scalar implicature. (Someone in the class did the homework can lead to the inference that only one person did the homework.) This potential scalar implicature demands populating the space of alternatives with alternatives whose negation amounts to the proposition that there aren’t two boys who ate ice cream or cake. This proposition, together with the negation of the conjunctive alternative (i.e., the proposition that no boy ate ice cream and cake) is incompatible with the conjunctive inference.

  15. To understand why we call this function \( Cell \), consider the partition induced by the set of alternatives, that is, the partition of logical space into sets of worlds that assign the same truth value to every member of C\(Partition (C)\). \(Partition (C)\) can be written in a format reminiscent of our definition of \( Cell (p,C)\) as the set of non-contradictory propositions, each of which is the conjunction of a subset \(C'\) of C and the conjunction of the negations of all remaining alternatives (i.e., alternatives in \(C \setminus C'\)):

    figure x

    It is easy to see that \( Cell (p,C)\) is either a contradiction or a cell in \(Partition (C)\).

  16. As long as \(p \in C\) and C is finite, the first conjunct p on the right-hand side of (20) is redundant. See fn. 20.

  17. See Fox (2018b) for the relevance of this notion of exhaustivity as cell identification to issues having to do with the semantics of questions.

  18. Why do we have to consider the set of IE alternatives for determining the set of II alternatives, and not vice versa? Let us consider what would happen if we first considered what’s II: take for example the sentence Some boy came and its alternative Every boy came. If we were to include first, we would derive that the alternative Every boy came is true, namely exhaustifying over Some boy came would yield a meaning equivalent to Every boy came. This would make for a very inefficient tool to use in conversation: by choosing an utterance from the set of alternatives \(\{Some boy came , Every boy came \}\) an opinionated speaker would only be able to convey one epistemic state she might be in (one cell in the partition); she would not be able to convey an epistemic state that entails Some but not all boys came. In other words, this procedure would be a bad tool for answering questions (see Fox 2018b) since, no matter what utterance we choose from the set of alternatives, we get the same cell in the partition. (And this problem, of course, would generalize to every case in which the set of alternatives, C, is consistent. In every such case each member of C would be mapped by exhaustification to the same cell: \(\bigwedge C\).) One can also prove that there are no cases with the reverse property, where point-wise exhaustification of the members of C would yield more distinctions if Inclusion is done before Exclusion. We can, thus, conclude that prioritizing exclusion over inclusion allows speakers to convey a greater number of epistemic states.

  19. An alternative one could attempt to pursue is to stick to the definition of \(\mathcal {E} xh _{}^{{\textsc {ie}}}\) and derive results parallel to those we derive using \(\mathcal {E} xh _{}^{{\textsc {ie+ii}}}\) by having different assumptions than Fox (2007) about the alternatives of exhaustified constituents. For example, one might assume that the only alternatives \(\mathcal {E} xh _{}^{{\textsc {ie}}}\) projects are its sub-domain alternatives, i.e., alternatives generated by replacing the set of alternatives \(\mathcal {E} xh _{}\) operates on with its subsets. Indeed, (ia) turns out to be semantically equivalent to (ib) for all choices of p and C we checked.

    figure ah

    However, even if these two formulas end up being semantically equivalent in general, as we suspect, a theory of cell identification based on \(\mathcal {E} xh _{}^{{\textsc {ie+ii}}}\) will allow us to capture facts that we cannot capture with \(\mathcal {E} xh _{}^{{\textsc {ie}}}\) when coupled with the specific assumption about projection in (ia) (the identity of \(C'\)). Specifically, our proposal in Sect. 4 for the semantics of only and the different treatment considered for IE and II alternatives in Sect. 9.1 and Bar-Lev (2018, Chap. 2) crucially rely on the distinction between IE and II alternatives, which is only available with \(\mathcal {E} xh _{}^{{\textsc {ie+ii}}}\). Furthermore, distributive inferences for sentences of the form \(\forall x (Px \vee Qx)\) can be derived with recursive application of \(\mathcal {E} xh _{}^{{\textsc {ie+ii}}}\) along the lines of Bar-Lev and Fox (2016) (see Sect. 5.5) using Fox’s assumptions about how alternatives project; recursive application of \(\mathcal {E} xh _{}^{{\textsc {ie}}}\) with the assumption about projection in (ia) won’t do in this case.

  20. Note that p (the prejacent) can never be in \({ IE }(p,C)\) and will always be in \({ II }(p,C)\), assuming that the prejacent p must be in C and that C is finite. Namely, p(w) in (i) would be redundant under these assumptions since it would be entailed by \(\forall r\in { II }(p,C) [r(w)]\). So (i) would be equivalent to (26), where p(w) is taken out.

    figure ai
  21. If \(Cell (p,C)\) is not a contradiction, then \(\{p\} \cup \{\lnot q:q\in { IE }(p,C)\} \cup \{r:r \in C\setminus { IE }(p,C)\}\) is consistent. And if this is the case, there is only one maximal subset of C consistent with the prejacent and the falsity of the IE alternatives, one that contains all the non-IE alternatives. The set of II alternatives is hence the set of all non-IE alternatives: \({ II }(p,C)=C\setminus { IE }(p,C)\). Since \(\mathcal {E} xh _{C}^{{\textsc {ie+ii}}} (p) \Leftrightarrow p \wedge \bigwedge \{\lnot q:q \in { IE }(p,C)\} \wedge \bigwedge II (p,C)\), it is also true that \(\mathcal {E} xh _{C}^{{\textsc {ie+ii}}} (p) \Leftrightarrow Cell (p,C)\).

  22. We say that a set \(C'\) is symmetric relative to a prejacent p iff \(\forall r \in C': p \wedge \lnot r \ne \bot \) and \(p \Rightarrow \bigvee C'\). We also say that an alternative q is symmetric to alternatives \(q_1,\ldots ,q_n\) given a prejacent p whenever the set \(C'=\{q_1,\ldots ,q_n\}\) isn’t symmetric relative to p but \(C' \cup \{q\}\) is.

  23. Since this is a strong presuppositional analysis, we will have to deal with arguments that only’s presupposition is weaker than its prejacent (Ippolito 2008). We hope that there is a way to deal with these arguments that will not destroy the picture we are trying to draw here.

  24. Example (33) might exemplify a broader generalization pertaining to all scalar implicatures under operators which are Strawson-DE but not DE, discussed in Gajewski and Sharvit (2012), Spector and Sudo (2017), Marty (2017), Anvari (2018). It is not clear to us, however, that FC disjunction gives rise to similar facts more generally, e.g., when embedded under sorry or surprised.

  25. As Chris Barker pointed out to us, a yes answer to (35) could lead (in certain contexts) to the inference that we are free to choose between ice cream and cake, and a no answer would naturally convey that we are allowed neither ice cream nor cake. Note that a similar situation arises in other cases in which a scalar implicature–generating sentence is used to form a yes/no question:

    figure ar

    In this case too, a yes answer can lead to the inference that John did not do all of the homework, whereas a no answer would mean that John did not do any of the homework.

  26. Alxatib suggests two possible accounts, both relying on the assumption that there is an exhaustivity operator other than only in the structure, along with additional assumptions. If we are right, none of this is needed.

  27. Our entry for only in (32) together with our analysis of SDA in Sect. 7 predicts (i) to presuppose the disjunctive alternatives of only’s prejacent, in parallel to (33).

    figure at

    The simplification inferences (i) gives rise to are, however, somewhat weaker than expected: (i) doesn’t seem to presuppose that if you work hard you succeed, but rather the weaker presupposition that if you work hard you might succeed. We believe the problem is more general, since the presupposition of only if sentences is weaker than expected on most accounts regardless—that is, even for simple sentences that do not involve disjunction (see von Fintel 1997). We hope this can be captured with a modification of only’s presupposition (see in this connection fn. 23).

    In parallel to our discussion of FC with only, we further expect simplification inferences to survive embedding in a question only in the presence of only. This seems to be borne out: while from (iia) we infer that if you work hard you might succeed, this is not an inference of (iib).

    figure au
  28. See Crnič (2015) for arguments showing that truly embedded \(\mathcal {E} xh _{}\) is taken into account for parallelism considerations.

  29. In fact, even if the parallelism domain contained \(\mathcal {E} xh _{}\), the correct result would still be derived under a global derivation of universal FC (when \(\mathcal {E} xh _{}\) has scope over the binder), since \(\mathcal {E} xh _{}\) could have scope above no in the second sentence.

  30. As in the case of unembedded FC disjunction (see fn. 13), the following derivation of universal FC does not depend on the exclusion of any of the IE alternatives. In many cases they would not be relevant and thus would not be assigned false.

  31. This is so since the set of exhaustified alternatives for the second level of exhaustification is as follows:

    figure bb

    The last five alternatives contradict the prejacent and hence can be trivially excluded. The only non-trivially IE alternatives are \(\mathcal {E} xh _{}^{{\textsc {ie}}}(\forall x \Diamond Px)\) and \(\mathcal {E} xh _{}^{{\textsc {ie}}}(\forall x \Diamond Qx)\); the negation of both yields \((\forall x \Diamond Px \rightarrow \exists x \Diamond Qx) \wedge (\forall x \Diamond Qx \rightarrow \exists x \Diamond Px)\). Taken together with the prejacent, this yields the result of the second application of \(\mathcal {E} xh _{}^{{\textsc {ie}}}\) in (ii):

    figure bc
  32. A reviewer asks what justifies taking not every to be an alternative to no, a crucial assumption for deriving parallel alternatives in the positive and negative cases of universal FC. First, we would like to point out that this is needed in order to explain how the scalar implicature in (ia) comes about for (i).

    figure bf

    Second, not every is expected to be an alternative to no if we combine the structural view of alternative generation in Katzir (2007) and Fox and Katzir (2011), together with a decompositional account of no assumed in much current literature (according to which it’s syntactically composed of negation and an existential quantifier; see, e.g., Sauerland 2000).

  33. Chemla’s (2009b) results show a significant difference in robustness between the universal FC inferences in (36) and (37). This might follow from the existence of another route to embedded FC in the positive case of universal FC, namely that of a local derivation of FC, which is unavailable in the negative case. The presence of negation might also play a role here by potentially introducing more alternatives; the universal FC inferences will not be derived if, as we propose in Sect. 8.2, alternatives where negation is replaced with \(\mathcal {E} xh _{}\) are generated.

  34. An extremely interesting data point, which we think is related to this issue, was brought to our attention by an anonymous reviewer:

    figure bk

    As the reviewer points out, (i) seems to show that a sentence of the form \(\Diamond (p \vee q)\) should not have the alternative \(\Box (p \vee q)\); for if \(\Box (p \vee q)\) was an alternative to \(\Diamond (p \vee q)\), the question in (ia) would make it relevant and hence excluded, and we would expect the inference that you don’t have to write an essay or give a presentation in order to get credit. In fact, we get the opposite inference, that you do have to do one of them. This data point lends further support to the idea that the alternatives of intervening modals don’t have to be generated to begin with. In the absence of the alternative \(\Box (p \vee q)\), the observed inference could be derived by considering alternatives like You may take an exam (to get credit for the course), etc., and negating all of them, to the effect that nothing other than writing an essay and giving a presentation would let you get credit for the course.

    Comparing (i) with the following case, which features a quantificational DP instead of a modal, is particularly telling: Here we do not infer \(\forall x (Px \vee Qx)\) but rather its negation, namely that not every boy solved problem A or B (and consequently the intuition that the question wasn’t fully answered). This is explained by assuming that in this case the alternatives of some boys must be generated.

    figure bl
  35. As pointed out to us by Gennaro Chierchia, (i) can be taken to further support Nouwen’s descriptive claim. Specifically, if we assume that any is an existential quantifier, it must be strengthened to a universal quantifier in this environment, and this can be done by exhaustification, as pointed out in Chierchia (2013), Crnič (2017).

    figure bp
  36. Since the entailment relations between the alternatives are the same in the cases of \(\forall x \Diamond (Px \vee Qx)\) and \(\Diamond \forall x (Px \vee Qx)\), the computation is the same as in fn. 31 modulo the relative scope of \(\Diamond \) and \(\forall x\).

  37. Nouwen’s more general claim is that standard implicature-based analyses of FC rely on distribution over disjunction as a necessary condition for deriving FC. Within such approaches, \(\phi (a) \wedge \phi (b)\) can only be derived from \(\phi (a \vee b)\) if \(\phi \) distributes over disjunction.

    figure bu

    However, the context surrounding disjunction does not distribute over disjunction in both universal FC, (iia), and the Nouwen (2017) case discussed in this section, (iib):

    figure bv

    As we have shown, Innocent Inclusion derives FC for both cases. In other words, our analysis does not depend on distribution over disjunction for the derivation of FC inferences.

  38. Of course, the set of worlds quantified over here and in what follows is not the set of all possible worlds but rather a subset thereof, determined, e.g., by a modal base and an ordering source.

  39. Note though that the presence of a universal quantifier in the structure might suffice, since disjunction could take scope above it (just as Nouwen pointed out for (51)).

  40. We remain agnostic regarding the proper variably strict analysis. In what follows we occasionally refer to “the closest world,” as if we assumed a Stalnaker ian analysis, but this is only done for ease of presentation. As the reader may verify, the results in this section hold for a Lewis ian system as well.

  41. Loewer (1976) writes: “Notice the similarity between the two situations. In both cases the surface form of an English sentence is ‘Modal operator (A or B)’, but its logical form seems to be ‘Modal operator A and modal operator B’” (p. 534).

  42. The excludability of the conjunctive alternative, which is shared between our account and Franke ’s, might seem objectionable since the falsity of does not seem to be a necessary implicature of . Note first that the falsity of is at least consistent with :

    figure ce

    Recall moreover that a similar objection was discussed in fn. 13 (and fn. 30) regarding the excludability of the conjunctive alternative in FC disjunction. Our response there applies here as well: since the set of alternatives is not closed under conjunction and the conjunctive alternative is not the conjunction of the disjunctive ones, it is possible for the disjunctive alternatives to be relevant without making the conjunctive alternative relevant. In contexts where it’s not relevant, it will indeed not be excluded.

  43. This is since the exhaustified disjunctive alternatives end up falsifying the prejacent (which, unlike in the case of FC, is not entailed by them); as a result their falsity on the second layer of exhaustification becomes vacuous. We illustrate this here for the disjunctive alternative :

    figure cg
  44. As Franke (2011, pp. 55–58) points out, he does not capture universal FC without stipulations; for precisely the same reasons he does not capture universal SDA. The inability of his system to derive universal FC/SDA persists even if we move from the set of alternatives he assumes (with no multiple replacements) to the one we do.

  45. As noted by Klinedinst , this local implicature would be derived in a non-monotonic context rather than a downward entailing one (given the non-monotonicity of conditionals within a variably strict semantics). Importantly, deriving implicatures in DE contexts normally leads to a weakening of the meaning at the global level; in the current case it does not. It is thus essentially different than computing an implicature in a DE context.

  46. Another argument by Santorio against implicature accounts, however, holds also for globalist accounts such as Franke (2011) and our own proposal. The argument comes from probably-conditionals:

    1. (i)

      If the winning ticket is between 1 and 70 or between 31 and 100, probably Sarah won.

      1. a.

        If the winning ticket is between 1 and 70, probably Sarah won.

      2. b.

        If the winning ticket is between 31 and 100, probably Sarah won.

    As Santorio argues, the sentence could be true if both simplifications are true but what is taken on our approach to be the prejacent is false. We suspect however that the problem is not unique to conditionals and thus the solution should not be hardwired into the semantics of conditionals as Santorio’s is. The same effect can be seen with most (the focus of Sect. 8.3). Suppose there are 7 kids, such that 3 of them are both on team A and team B, 2 of them are only on team A, and the remaining two are only on team B, and there are no other people on either team. The situation can be described as in (ii).

    1. (ii)

      Most kids on team A or team B are on both teams.

      1. a.

        Most kids on team A are on both teams.

      2. b.

        Most kids on team B are on both teams.

    Here too, it seems that the sentence can be true in virtue of (iia) and (iib) being true, even though it is false that most of the 7 kids are on both teams (which is equivalent to (ii)). We thank Paolo Santorio for pointing out this problem and wish there was more we could say. We return to simplification with most in Sect. 8.3.

  47. In Sect. 8.2 we assume that negation triggers \(\mathcal {E} xh _{}\) as an alternative. As the reader may verify, adding such alternatives in the case at hand would change nothing.

  48. Here too the exclusion of the conjunctive alternative is not a necessary inference of (70).

  49. The bulk of the ideas in this section came up in a discussion with Itai Bassi, to whom we are greatly indebted.

  50. See Lassiter (2018) for a more elaborate argument against the semantic validity of SDA based on examples of the following sort:

    figure cm

    In the following discussion we set aside such cases and the complications they give rise to in determining both the basic semantics and the alternatives generated.

  51. Nute (1980) further observed that the status of (72) improves if it appears as a continuation to (71), as in (i). Regrettably we have nothing to say about this effect.

    figure cn
  52. More generally, following Nute (1980), we assume that SDA fails whenever the conditional has the form (where \(p^+\) logically entails p). This generalization captures the acceptability of (71) as well as (i), also discussed by Nute , in which the consequent is strictly stronger than the first disjunct in the antecedent. We will not explicitly discuss such cases; the interested reader may verify that the explanation to be provided here for (71) extends to (i), the key fact being that entails . See Bar-Lev (2018, Chap. 1) for complete derivations.

    figure cp
  53. With this line of reasoning one would expect to find other cases, unrelated to simplification, where the tautological nature of one of the alternatives leads to other alternatives becoming IE. However, we did not yet find such cases which are not predicted to be bad for independent reasons.

  54. The exclusion inference in (75b) is contextually redundant for (71), given world knowledge that fighting with one side (usually) entails not fighting with the rival side. It would, however, be detectable if it wasn’t contextually entailed, as in (i) (uttered in a context where it is common ground that Mary studies physics).

    figure cs

    It is quite difficult to imagine (i) being uttered truthfully by a speaker who believes that if Mary had studied linguistics or history she would have definitely studied linguistics, but possibly also history. The exclusion inference predicted by applying \(\mathcal {E} xh _{}^{{\textsc {ie+ii}}}\) (together with the assumption that , which follows, for example, if Conditional Excluded Middle holds, as it does on Stalnaker ’s analysis) correctly precludes this possibility. This inference can be avoided though; Kai von Fintel (p.c.) pointed out that (iia) is felicitous. This behavior turns out to be predicted assuming that linguistics or both is parsed as \(\mathcal {E} xh _{}\)(linguistics) or both (see Sect. 8.2). While the infelicity of (iib) may make the exclusion inference look like an obligatory one, we believe this is an independent issue having to do with felicity conditions on the use of at least, as is suggested by the felicity of (iic). We thank Benjamin Spector for providing  and this perspective on (iib).

    figure ct
  55. This is not the case for Klinedinst (2007): for him the implicature leading to simplification is derived at an embedded level, within the antecedent, and subsequently cannot be affected by the identity of the consequent.

  56. We omit the alternative with the contradictory antecedent since, being a non-contingent proposition, it will not affect the result: if it is taken to be a tautology as in Stalnaker (1968), for example, it will be trivially II. It is possible that more alternatives are generated, ones with no \(\mathcal {E} xh _{}\): and . Since these are equivalent to other alternatives, they would change nothing.

  57. This can be understood more easily if we consider what the closest \(\bar{p} \vee \bar{q}\) world might be. It can be either a \(\bar{p} \wedge \lnot \bar{q}\) world, or a \(\bar{q}\wedge \lnot \bar{p}\) world, or a \(\bar{p} \wedge \bar{q}\) world. If we assign false to and , then the closest \(\bar{p} \vee \bar{q}\) world cannot be a \(\bar{p} \wedge \lnot \bar{q}\) world or a \(\bar{q} \wedge \lnot \bar{p}\) world (if the prejacent is going to be true); it must be a \(\bar{p} \wedge \bar{q}\) world—that is, the alternative must be true.

  58. The ultimate story is a bit more nuanced. As we know from fn. 42, the falsity of the conjunctive alternative is derived for (76) only if it’s relevant. The truth of the conjunctive alternative in the case of (78), by contrast, is an obligatory inference; see Sect. 9.1.

  59. Of course, this assumption should follow from a general theory of alternative generation such as Katzir (2007). A more in-depth discussion of this issue is something that we hope to return to in the future. Another issue is that so-called ‘indirect implicatures’, e.g., from Not all students came to Some students came, are not derivable if alternatives where negation is replaced with \(\mathcal {E} xh _{}\) are generated. As we mention towards the end of this section, we assume that whether such alternatives are generated depends on whether negation is contained in a focus-marked constituent or not; we’d hence expect indirect implicatures only if it’s not. See Chierchia (2004) and Romoli (2012) for conflicting opinions as to the status of indirect implicatures relative to standard ones.

  60. Another reason might be that unlike in the case of (78), where the conjunctive alternative can be arrived at by merely deleting parts of the prejacent, its parallel in the case of (77) cannot. See Sect. 9.1 for the possible relevance of this difference.

  61. Another potentially relevant case of a simplification problem is with superlatives:

    figure dd

    We have not yet fully investigated the applicability of Innocent Inclusion to such cases.

  62. For reasons we do not understand, embedding disjunction in a relative clause seems to make the sentence less odd in this scenario:

    figure df
  63. Here too we can generate a McKay and van Inwagen -style effect, as seen in (i). As with simplification in conditionals, we infer that it is false that most philosophy students are linguistics students (see fn. 54). For elaboration on this see Bar-Lev (2018, Chap. 1).

    figure dg
  64. Most would also trigger all-alternatives, of course, which we ignore since they only add more IE alternatives but have no effect on what’s II, which is the focus of our discussion.

  65. Since \(most (P \cup Q)(R)\), \(\lnot most (P)(R)\), and \(\lnot most (Q)(R)\) can all be true, for instance if \(P=\{a,c\}\), \(Q=\{b,c\}\), and \(R=\{a,b\}\).

  66. We thank an anonymous reviewer for suggesting this way of presentation.

  67. Similar examples, with inference patterns similar to those discussed in this section, can be constructed using conditionals with disjunctive antecedents instead of FC disjunction.

  68. We thank an anonymous SALT 27 reviewer for pointing out that FC disjunction under some still requires a local derivation.

  69. No parallel issue for a global derivation arises with singular indefinites like some boy: in this case, the only one implicature associated with singular indefinites (namely that no more than one boy is allowed ice cream or cake; see fn. 14), together with the inference that some boy is allowed ice cream and some boy is allowed cake, entails the desired inference that there is one and only one boy who’s both allowed ice cream and allowed cake.

  70. An alternative way to address this problem would appeal to a particular theory of wide scope indefinites (‘referential indefinites’; Kasher and Gabbay 1976, Fodor and Sag 1982, and much subsequent work). If some boys here is referential, the problem doesn’t arise.

  71. Note, however, that this would require assuming the existence of an existential alternative to the universal distributivity operator, raising questions similar in spirit to those that arose in our discussion of Nouwen (2017) and ability modals (Sect. 6.2).


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For helpful discussions and comments we thank Chris Barker, Brian Buccola, Gennaro Chierchia, Andreas Haida, Roni Katzir, Jacopo Romoli, Paolo Santorio, Benjamin Spector, William Starr, and especially Itai Bassi and Luka Crnič. Feedback we got at SALT 27, the MIT Workshop on Exhaustivity 2016, the LLCC seminar at The Hebrew University of Jerusalem, and Ling-Lunch at MIT was also very instructive. We would like to thank the editors of NALS and two anonymous NALS reviewers for their insightful comments and suggestions, and Christine Bartels for her excellent editorial advice. All mistakes and shortcomings are, of course, our own.

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Bar-Lev, M.E., Fox, D. Free choice, simplification, and Innocent Inclusion. Nat Lang Semantics 28, 175–223 (2020).

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