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An Implicature account of Homogeneity and Non-maximality


I provide arguments in favor of an implicature approach to Homogeneity (Magri in Pragmatics, semantics and the case of scalar implicatures, Palgrave Macmillan, London, pp 99–145, 2014) where the basic meaning of the kids laughed is some of the kids laughed, and its strengthened meaning is all of the kids laughed. The arguments come from asymmetries between positive and negatives sentences containing definite plurals with respect to (1) children’s behavior (Tieu et al. in Front Psychol, 2019., (2) the availability of Non-maximal readings, and (3) the robustness of neither-true-nor-false (‘gappy’) judgments (Križ and Chemla in Nat Lang Semant 23(3):205–248, 2015). I propose to avoid some problems of Magri’s analysis by modeling the Implicature account of Homogeneity after the Implicature account of Free Choice, based on a hitherto unnoticed analogy between the two phenomena. The approach that emerges has the advantages of Magri’s implicature account of Homogeneity (predicting asymmetries), while at the same time bears a close resemblance to recent approaches to Non-maximality (Malamud in Semant Pragmat 5(3):1–58., 2012; Križ and Spector in Interpreting plural predication: homogeneity and non-maximality, Ms., Institut Jean Nicod, 2017), which enables restating their account of Non-maximality as following from the context-sensitivity of implicature calculation.

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  1. A note about the terminology: the term Homogeneity is sometimes used in the literature to refer to the phenomena described in (1), and sometimes to refer to the claim that the sentences in (1) are judged neither-true-nor-false in a situation where some but not all of the kids laughed. I will only use the term Homogeneity for referring to the former, i.e., the phenomena in (1); the latter phenomenon will be termed Gappiness. I will discuss the relation between the two phenomena in Sect. 6.

  2. Of course, this does not mean that (1b) is not structurally ambiguous between negation taking scope above and below the definite plural; it does however lead to the conclusion that both of those structures miraculously end up having the same meaning in out-of-the-blue contexts.

  3. \(A \Rightarrow _{\textit{st}} B\) iff B is true whenever A is true and B is defined.

  4. Since Magri’s Implicature account (as well as what I will propose) is couched within a view where implicatures are a special case of ambiguity (i.e., an ambiguity between the basic meaning and the strengthened meaning), the Implicature account is in fact a special case of the Ambiguity account. I choose to distinguish between them since as we will shortly see the Implicature account makes substantially different predictions than the other Ambiguity accounts mentioned above.

  5. Malamud’s (2012) account of Non-maximality (in the footsteps of which we’ll follow in Sect. 5) is based on the idea that the interpretation of definite plurals should be treated in the same way implicatures are. Since accounting for Homogeneity isn’t her main goal, and since as Križ and Spector (2017) observed her account fails to derive the correct Homogeneity facts, I do not consider her account to be an Implicature account of Homogeneity.

  6. Of course, one may add auxiliary assumptions to the Ambiguity and Trivalence accounts in a way that would make them asymmetric. For example, within the Ambiguity account one could suggest that for a sentence containing a definite plural to be true the truth of both the universal and the existential readings is required in principle (as in Spector 2013; Križ and Spector 2017), but they are not on equal footing: we might be forgiving (i.e., allow for Non-maximality) in cases where the universal reading is false but not in cases where the existential reading is. However, such a move would be stipulative; If asymmetry is borne out, as I will argue, it is obviously more parsimonious and thus preferable to have an account in which it follows from the core assumptions. For this reason I will maintain the division between symmetric and asymmetric accounts. As Benjamin Spector pointed out to me, the Rational Speech Act implementation of Križ and Spector (2017) found in Spector (2018) predicts a slight asymmetry between positive and negative sentences due to differences in the cost assigned to their alternatives. I will not pursue here a detailed comparison with this implementation, but merely point out that such differences are unlikely to provide a satisfactory explanation for the asymmetry in children’s behavior to be discussed in Sect. 2.1, since children have been argued to lack access to the alternatives Spector’s analysis relies on (see Sect. 7).

  7. This will only be the expectation with the aid of an auxiliary assumption ensuring that the permissive kids do not have access to an LF where negation takes scope below the definite plural.

  8. Among all 78 test trials of this condition (3 such test trials for each child, including 2 children who were excluded from analysis) there was only one ‘yes’ response; the two other responses by the same child on the same condition were ‘no’ responses.

  9. The opposite is true for negative sentences: adults turned out to be more permissive than children with such sentences. I set this difference aside for the sake of the discussion here, though one might entertain the idea that the behavior of the minority of adults who are permissive in negative sentences results from allowing to compute an implicature under negation (perhaps reanalyzing the sentence the hearts aren’t red as involving ‘meta-linguistic’ negation).

  10. The Non-maximality data reported in this section and in Sect. 5.3 are based on judgments by several native speakers of English as well as several native speakers of Hebrew on parallel Hebrew sentences. Informants were told what the context of utterance is (trying to establish a question under discussion) and were then asked whether the sentence seems true in a certain situation (e.g., if 5 out of the 30 kids laughed). Informants were also asked about what they think would be a natural inference from the sentence given only a context of utterance (e.g., for (10b), whether they infer that no kid took any vitamin or not).

  11. The idea that there might be a difference between positive and negative sentences with respect to Non-maximality was first pointed out to me by Elitzur Bar-Asher Siegal (p.c.), and was also brought to my attention by Manuel Križ (p.c.).

  12. An alternative analysis of (10a) with no Non-maximality involved could be one where their vitamins is interpreted as the vitamins they were supposed to take, where this in turn is analyzed as an existential quantifier over pluralities of two vitamins. First, I would like to point out that it is not obvious to explain why given this analysis we should not have a parallel reading for (10b) where it means no kid took at least two vitamins. Second, note that other examples I use here to argue for asymmetry are not amenable to this kind of analysis, especially (14)–(15). I thank Kai von Fintel and Irene Heim for discussions of this issue.

  13. Note furthermore the effect of fixing the scope of the definite plural using a relative clause: In contrast with (14b), a non-maximal reading is accessible when the definite plural contains no relative clause as in (i). Unlike (14b), in this case the definite plural may refer to the plurality of all kids who are in any performance by this clown and take scope above negation.

    figure n

    This contrast may speak against responses which attribute the asymmetries discussed here to independent differences between positive and negative sentences. One such response has been proposed to me by Sophia Malamud (p.c.), suggesting that the typical Question Under Discussion is different with and without negation. Another response of this kind has been proposed to me by Manuel Križ (p.c.), suggesting that there is a difference between the conditions under which we conceive of the difference between no kid laughed and almost no kid laughed as irrelevant and those under which we conceive of the difference between every kid laughed and almost every kid laughed as irrelevant.

  14. A possible response to this objection against Magri’s (2014) account might be that those children compute implicatures only if the resulting meaning is a complete answer to a reasonable question: for instance, all the hearts are red is a complete answer to the question which hearts are red?, but some but not all of the hearts are red is not. See Singh et al. (2016) for a related proposal.

  15. Such an extreme case has been discussed by Malamud (2012, ex. (2)):

    figure r
  16. Free Choice effects have been argued to arise even when disjunction takes scope above the modal (Zimmermann 2000). See Klinedinst and Rothschild (2012) and Bar-Lev and Fox (2019) for the claim that such examples should be analyzed differently from the case discussed here where disjunction has narrow scope.

  17. See Aloni (2007, (2016), Barker (2010), Starr (2016), Willer (2017), a.o., for proposals relying on such a non-standard semantics and their treatment of the behavior under negation pointed out by Alonso-Ovalle.

  18. Note that the same point holds for (21) and (22) without exactly, in which case the ‘negative parts’ of the meaning in (21a) and (22b) are arguably cancellable implicatures. While I will not propose an explicit account of sentences like (21) or its counterpart without exactly, given the account to be proposed in Sect. 4 such an account will be dependent on an account for the parallel Free Choice data, an issue which is discussed in Bar-Lev (2018).

  19. Such data speak against Ambiguity approaches to Homogeneity (Krifka 1996) and Free Choice (Aloni 2007) on some plausible assumptions about ellipsis licensing. See Spector (2013) and Križ and Spector (2017) for a more involved Ambiguity account of Homogeneity which accounts for such data.

  20. Definite plurals and Free Choice disjunction in Strawson-Downward Entailing environments such as the antecedent of a conditional do not conform to the same inference pattern. The analogy however still persists in such environments: for both one may construct examples where a locally weak (existential/disjunctive) interpretation is preferred as well as examples where a locally strong (universal/conjunctive) interpretation is preferred:

    figure y
    figure z

    That said, there is a difficult to pinpoint intuition that Free Choice disjunction tends towards the weak interpretation in such contexts. If there’s something to that intuition, it could be due to the possibility of a scope construal where disjunction takes scope above the existential modal, a possibility which will deliver a disjunctive meaning on the Implicature account of Free Choice and one which will have no parallel in the case of definite plurals given the account I will propose. For discussions of these issues I thank Paolo Santorio and Frank Staniszewski.

  21. Of course, one may accept the argument that a unified account is desired but reject the choice of an Implicature account to this end. Such a move has been recently made by Goldstein (2019) who analyzes Free Choice based on Trivalence accounts of Homogeneity. I will take the Implicature route, based on the arguments for asymmetry discussed in Sect. 2.

  22. One-place predicates are assumed to be of type \(\langle e,st \rangle \) (since the exhaustivity mechanism I will introduce in Sect. 4.4 operates on propositions). The domain argument of \(\exists {\textsc {-pl}}_{}\) is assumed to be of type et for simplification.

  23. Malamud’s idea by itself does not guarantee it; on her proposal, the set of propositions is closed under conjunction.

  24. I allow myself here a common abuse of notation: D is a syntactic object and not a set. We can define the alternatives more carefully by assuming that D carries an index i, and that the alternatives carry other indices as in (i). Assuming that the assignment function g is surjective we will end up with all subsets of g(i) as alternatives (and potentially other sets as well, which however do not make any difference for our purposes). I thank Roger Schwarzschild for a discussion of this issue.

    figure ao
  25. Each of the subsets of D will either yield a proposition equivalent to one of those in (39), or lead to vacuous quantification (given any subset of D which contains neither Kelly nor Jane). One has to ensure that the latter kind of alternatives is somehow treated differently than the former: they may either not be considered as alternatives at all, or be trivially excluded by the exhaustivity mechanism. I do not intend to propose here an explicit way to ensure it, and henceforth simply ignore such alternatives. (A similar issue arises for Chierchia 2013, e.g., for the analysis of John is allowed to eat any cookie.)

  26. Matthew Mandelkern (p.c.) pointed out that structural complexity isn’t enough for the current purposes. In order to avoid alternatives like every kid laughed I will have to assume that a sort of semantic complexity plays a role: the kids cannot be replaced with every kid which is of a higher type.

  27. For further analyses relying on the same idea see Bar-Lev and Margulis’s (2014) treatment of Hebrew kol, Meyer’s (2016) treatment of English or else, and Oikonomou’s (2016) theory of imperatives.

  28. As a reviewer points out, given this view one might expect some cross-linguistic variation with respect to Homogeneity effects, assuming that some languages could have both existential and universal pluralization operators. However, I am not aware of any language which lacks Homogeneity effects altogether. In Sect. 8.3 I will entertain the possibility that even English has a universal pluralization operator, albeit one which is only licensed by certain DPs. Note however that the licensing conditions for this operator could vary from one language to another. In fn. 52 I consider such variation as a possible source for the difference between English and French in whether definite plurals containing numerals show Homogeneity effects.

  29. Bar-Lev and Fox (2017) argued that the direct derivation of Free Choice provided by assuming Innocent Inclusion is advantageous over Fox’s (2007) indirect derivation which relies on an iterative application of an Innocent Exclusion-based exhaustivity operator \({\mathcal {E}}{} \textit{xh}_{}^{\textsc {ie}}\): assuming Innocent Inclusion provides a global derivation for Universal Free Choice inferences (argued to be needed in Chemla 2009) which is unavailable with iterative applications of \({\mathcal {E}}{} \textit{xh}_{}^{\textsc {ie}}\), and furthermore can account for the interaction of Free Choice disjunction with only. In Bar-Lev (2018) and Bar-Lev and Fox (2020) further evidence for Innocent Inclusion is provided based on Simplification of Disjunctive Antecedents and cases of Free Choice where a universal quantifier intervenes between the existential modal and disjunction.

  30. The only exception to this generalization I’m aware of is Russian po (Križ 2017), a floating quantifier that maintains Homogeneity. An analysis of po one might consider based on the implicature view is that, just like \(\exists {\textsc {-pl}}_{}\), it is an existential quantifier with no universal counterparts. I do not aim to provide here a satisfactory analysis of po; further research is needed in order to determine whether such a direction could account for all the curious properties of po discussed by Križ.

  31. A reviewer further points out that the assumption that Homogeneity depends on the presence of a covert pluralization operator is at odds with the claim that some languages do not allow distributive readings with predicates which contain indefinites or numerals (e.g., earn exactly 100 euros, see Flor et al. 2017) in the absence of overt quantificational operators; this would seem surprising on the current view given that these languages aren’t exempt from Homogeneity effects. First, I would like to point out that this concern would apply equally well to the standard view of Homogeneity due to Schwarzschild (1994) and Gajewski (2005), which shares the assumption that covert pluralization is responsible for Homogeneity. Second, even English covert pluralization doesn’t allow distributive readings for such predicates very easily (see Dotlačil 2010, §2; Champollion 2020, §2.3 and references therein), a fact which is not well understood. A way to reconcile such facts with obligatory covert pluralization, following Schwarzschild (1994), is to stipulate that indefinites and numerals prefer to outscope covert pluralization (perhaps this preference is stronger in the languages discussed by Flor et al.). See however some issues with such a view in Dotlačil (2010, §3.4.1). I would have to leave a more in depth discussion of these important issues to another occasion.

  32. Further complication of our commitment to the absence of conjunctive alternatives comes from Homogeneity removal by quantifiers like all which, as we will discuss in Sect. 8.3, can be analyzed by assuming a universal counterpart of \(\exists {\textsc {-pl}}_{}\). I will entertain there the possibility that such an operator exists after all but is only licensed by quantificational DPs, which is why replacing \(\exists {\textsc {-pl}}_{}\) in the kids laughed with it would only be able to derive a licit LF if we at the same time replace the kids with the more complex each/all of the kids (or other quantifiers, which are at least semantically more complex, see fn. 26). The general picture, i.e., the claim that conjunctive alternatives are not derivable without moving to more complex structures, will still be maintained.

  33. The Ambiguity and Implicature accounts are on equal footing here essentially because the Implicature account is a sub-species of the Ambiguity account, see fn. 4.

  34. I should note that Malamud (2012) sets such reduction as one of her goals. However, the connection between a general theory of pruning and her analysis of Non-maximality isn’t made transparent.

  35. In reality, when the number of kids is 3, examples where we understand the kids laughed non-maximally are hard to come by and perhaps non-existent. Presumably, this is since it’s difficult to conceive of the difference between 2 kids laughing and 3 kids laughing as irrelevant, whereas it’s easier to think of the difference between say 29 kids laughing and 30 kids laughing as irrelevant.

  36. This is not a quirk of having 3 kids in the context. Given any set of kids \(\{x_1,\ldots ,x_n\}\) we’d have for the kids laughed the basic disjunctive meaning (\(x_1\text { laughed}\vee ...\vee x_n \text { laughed}\)). For any j, \(1 \le j \le n\), at least j kids laughed is equivalent to the conjunction of all alternatives built of \(n+1-j\) atomic disjuncts; hence exhaustifying over a set of alternatives containing these alternatives but no alternative stronger than any of them will yield the meaning at least j kids laughed.

  37. One can never object to the claim that the kids laughed by asserting that one of them did:

    figure bp
  38. See their paper for a possible intuition behind this idea. Crnič et al.’s constraint (their ex. (37)) is stronger than (63) for reasons which are irrelevant for our purposes.

  39. The reader may verify that given that no licit pruning choice can lead to IE alternatives, any licit pruning choice would lead to Inclusion of all the alternatives in \(\textit{SubAlt}\) which remain after pruning.

  40. Assuming that the kids are Kelly, Jane and Bill, and that \(a=\) Kelly laughed, \(b=\) Jane laughed, and \(c=\) Bill laughed, we get 12 distinct members in the set \(\textit{Readings}\):

    figure br

    As far as distributive predication is concerned, the set of possible readings we get turns out almost identical to Križ and Spector’s ‘candidate meanings’, which is simply \(\textit{SubAlt}^\wedge \) (the alternative view of pruning proposed in Bar-Lev 2018, §3.A makes identical predictions to theirs). It is indeed impossible to find cases where the kids laughed has a non-maximal reading equivalent to a member of \(\textit{SubAlt}\), for instance where it means that one specific kid, say Kelly, laughed. While other accounts (e.g., Križ 2015, 2016; Križ and Spector 2017) do predict such readings to be possible, it is not clear that this can be taken as an advantage of the current system: the kids laughed is still predicted to have a meaning according to which two specific kids, say Kelly and Jane, laughed, even when there are 20 kids. Such readings seem equally unavailable, and a more principled explanation for why they are is called for.

  41. See Katzir (2014) for an alternative way to ensure that context does not break generalized symmetry, which does relate to the context; Bar-Lev (2018, §3.A) develops an alternative to what I’m about to propose which is based on Katzir’s proposal.

  42. As such this constraint closely resembles the idea behind Križ’s (2016) account of Non-maximality.

  43. For the context in (57) in which the question made salient is who laughed?, it would suffice to ask for each alternative in isolation whether it’s relevant, namely the view initially entertained here would be enough and one would not need the more cumbersome constraint on pruning proposed here. This is however not in general a property of contexts forcing a maximal reading. If the salient question is did all the kids laugh?, a maximal reading will be the only one possible for the kids laughed but propositions built of one atomic disjunct (e.g., Kelly laughed) will not be relevant in isolation.

  44. There can be at most one such proposition given Q: Since Relevance as well as \(\textit{Readings}\) are closed under conjunction, the conjunction of all relevant members of \(\textit{Readings}\) is guaranteed to be the maximally strong proposition among the members of \(\textit{Readings}\) which are relevant to Q.

  45. The constraint on pruning in (67) guarantees that the largest set in \({\textit{Alt}}(S)\) which leads to a relevant meaning will be picked out. The reason why this also makes sure that the set which leads to the strongest meaning among the relevant propositions in Readings will be picked out is that the following holds for any two subsets of \(\textit{SubAlt}\), \(C'\) and \(C''\), which are licit choices of pruning given Crnič et al.’s constraint: if \(C' \supset C''\) then \(\llbracket {{\mathcal {E}}{} \textit{xh}_{C'}\hbox { the kids }\exists {\textsc {-pl}}_{D}\hbox { laughed}}\rrbracket \Rightarrow \llbracket {{\mathcal {E}}{} \textit{xh}_{C''}\hbox { the kids }\exists {\textsc {-pl}}_{D}\hbox { laughed}}\rrbracket \).

  46. Recall that for simple negative sentences which involve no fixing of the scope of the definite plural below negation (e.g., with a bound variable, as in (70b)) we would still expect non-maximal readings due to the possibility of negation taking narrow scope. Indeed we have seen that non-maximal readings for (9), repeated here, are readily available to some speakers.

    figure bz
  47. One might consider deriving Non-maximality by letting the kids refer to a plurality which doesn’t contain all the kids. This idea has however been forcefully and convincingly attacked by Križ (2015, §3.1.5), and I will hence not assume this plays any relevant role.

  48. The importance of covers for a theory of Homogeneity is elaborated on in Bar-Lev (2018, §4–5), Bar-Lev (2019). See also Sect. 8.1 below.

  49. Note that for this to work we’ll have to get rid of our assumption that we only consider atomic pluralities in the definition of \(\exists {\textsc {-pl}}_{}\), which we’ll do in Sect. 8.1.

  50. One might wonder whether applying \({\mathcal {E}}{} \textit{xh}_{}\) under negation could provide us with yet another route to Non-maximality in negative sentences, given that it has been argued to apply under negation under certain circumstances (see Magri 2009; Fox and Spector 2018). I will stick to a simplified view according to which it never does, because I do not think the examples discussed in this section which show clear non-maximal readings under negation (especially (72)) have the hallmarks of cases involving embedded \({\mathcal {E}}{} \textit{xh}_{}\) under negation discussed by Fox and Spector; however, it’s possible that an LF with embedded \({\mathcal {E}}{} \textit{xh}_{}\) is available but is extremely dispreferred, and its availability contributes to obscuring the asymmetry between positive and negative sentences.

  51. Manipulating covers has indeed been shown to be difficult (see the debate in Gillon 1987, 1990; Lasersohn 1989 about the limited availability of so-called intermediate readings which require non-default covers), though little is known about what determines the choice of a cover. Full discussion of this issue is outside the scope of this paper.

  52. Why there are no non-maximal readings for the 4 kids and War & Peace and Anna Karenina remains a mystery; let me however sketch a direction for an analysis, following insights by Križ (2015). Suppose mentioning the names of the individuals involved or their number in the 4 kids laughed and I read War & Peace and Anna Karenina makes it very likely that we think of the fine-grained questions how many of the kids laughed? or which books did I read? as being under discussion. Given the analysis in Sect. 5, such questions are indeed expected to only give rise to maximal readings. It should be noted that unlike most facts discussed in this paper, the Homogeneity properties of those expressions are subject to crosslinguistic variation. Križ (2015, §3.3.5) presents data showing that definite plurals containing numerals in French behave ‘non-homogeneously’, differently from the English behavior described in (74) and from French definite plurals without numerals.

    figure ch

    As Benjamin Spector pointed out to me, the French data may be seen as particularly problematic for the current approach, since we have in this case a default universal reading under negation which is surprising if the pluralization operator is existential. One response to this problem could be to assume that French numeralized definite plurals should be analyzed as quantificational, assuming a covert universal quantifier applies to the definite description such that it ends up meaning all 4 books (as suggested to me by Manuel Križ, p.c.). Another response could be that the presence of the numeral provides a locus for focus, which prompts exhaustification under negation (Fox and Spector 2018). A third possibility builds on an idea we will entertain in Sect. 8.3 according to which English has a universal pluralization operator which is only licensed by certain DPs. Perhaps a difference between French and English is whether definite plurals containing numerals license this operator or not. I admit that none of these responses is entirely satisfying, and leave it as an unresolved problem.

  53. The data reported here is taken from experiment C2 in Križ and Chemla (2015). I thank Manuel Križ for making the detailed (unpublished) data available to me.

  54. The difference presumably being whether they consider the possibility of having a Q for which the maximal reading is relevant (‘neither’) or not (‘completely true’).

  55. Note that the availability of such an LF does not predict (85b) to be acceptable, since on this LF it would be expected to behave just like simple disjunction for which a similar cancellation is impossible:

    figure cq
  56. One might entertain the possibility that this difference is due to the salience of deletion alternatives which are syntactically part of the prejacent, in contrast with substitution alternatives which are not.

  57. They do not clearly fall under substitution alternatives either, since they do not involve lexical replacements. Still, they involve substitution with elements which are not syntactically parts of the prejacent.

  58. See Champollion (2016a, ex. 32) for the equivalence between standard definitions of \(^{{\star }_{}}\) and the truth conditions in (89).

  59. Note that this move changes nothing if P is distributive, for the same reason that \(\llbracket ^{{\star }_{}}\,\rrbracket (P)(x) \Leftrightarrow \llbracket {\textsc {dist}}\rrbracket (P)(x)\) whenever P is distributive. This means that \(\textit{SubAlt}\) in (40b) remains the set of alternative propositions of the kids \(\exists {\textsc {-pl}}_{D}\) laughed under the current modification. Taking into account non-distributive predication, the set of alternative propositions of a sentence of the form DP \(\exists {\textsc {-pl}}_{D}\) VP, would be (again under the assumption that D contains all members of \(\textit{Part}(\llbracket \text {DP} \rrbracket )\)):

    figure cw

    Or, equivalently (as long as \(\textit{Part}(\llbracket \text {DP} \rrbracket )\) is the closure under \(\sqcup \) of \(\textit{Part}_{\textit{AT}}(\llbracket \text {DP} \rrbracket ))\):

    figure cx

    For example, here is the set of alternatives of the kids lifted the piano (the VP abbreviated as l.t.p) where \(\llbracket \text {the kids} \rrbracket =\text {Kelly} \sqcup \text {Jane}\), assuming that D contains Kelly and Jane (as well as \(\text {Kelly} \sqcup \text {Jane}\), though its presence is inconsequential), \(D'\) contains Kelly but neither Jane nor \(\text {Kelly} \sqcup \text {Jane}\), and \(D''\) contains Jane but neither Kelly nor \(\text {Kelly} \sqcup \text {Jane}\) (compare with (39)).

    figure cy
  60. Non-maximality is, as before, the result of pruning alternatives which yields a weaker meaning.

  61. Note that we’d end up with two domain variables: D which gives rise to subdomain alternatives, and a cover variable which doesn’t. This redundancy disappears in Bar-Lev (2018, ch. 5) where the cover variable is instead identified with the domain variable D, but that system is more complex in other respects which for the sake of brevity I do not present here.

  62. In Sect. 5.3 this was brought up as a possible source for Non-maximality in negative sentences.

  63. I further rely on the following common assumption (see, e.g., Krifka 1986; Kratzer 2007):

    figure dd

    The assumption in (96) may seem entirely ad hoc and at odds with the assumption in (i): since we defined the overlap relation based on parthood in (89), and since the parthood relation is generalized to pairs in (i), one could expect the overlap relation to automatically generalize to pairs; but this generalized overlap would give us something different from (96) (i.e., the disjunction in (96) would be replaced with conjunction).

    Perhaps surprisingly, a way to make (96) fall out from a general definition of overlap is to admit a null element 0 which is part of every plurality in the domain of individuals \(D_e\) (for reasons to do so see Landman 2011; Buccola and Spector 2016; Bylinina and Nouwen 2018). Once this is done, a generalized overlap relation between two entities could be defined as having a part in common which is not the bottom element (see Rothstein 2017, p. 33), as in (ii).

    figure de

    Importantly, if \(x \in D_e\), then \(\textit{Bot}(x)\) can only hold if \(x=0\), and if \(x \in D_e \times D_e\), then \(\textit{Bot}(x)\) can only hold if \(x=\langle 0, 0 \rangle \). Let me show now that assuming (i) while admitting a bottom element and assuming the notion of overlap in (ii), (96) follows (I will show this assuming the variables \(x,x',y,y'\) in (96) all range over members of \(D_e\), which is what’s relevant for our discussion). From right to left: Suppose one of the disjuncts on the right-hand side of (96) is true; for instance, assume \(x \circ x'\). Then (given (ii)) there must be some \(x''\) s.t. (a) \(x'' \sqsubseteq x\), (b) \(x'' \sqsubseteq x'\), and (c) \(x'' \ne 0\). Now for any arbitrary y and \(y'\): \(\langle x'', 0 \rangle \sqsubseteq \langle x, y\rangle \) because of (a) (and due to 0 being part of every member of \(D_e\) together with (i)), and similarly \(\langle x'', 0 \rangle \sqsubseteq \langle x', y'\rangle \) because of (b). Since \(\langle x'', 0 \rangle \) is not a bottom element of \(D_e \times D_e\) (because of (c)), it also follows that \(\langle x, y \rangle \circ \langle x', y' \rangle \). (Of course, the same conclusion can be reached assuming \(y \circ y'\).) From left to right: if \(\langle x, y \rangle \circ \langle x', y' \rangle \), then there must be a pair \(\langle x'', y'' \rangle \) distinct from \(\langle 0, 0 \rangle \) which is part of both \(\langle x, y \rangle \) and \(\langle x', y' \rangle \). This means that there is an \(x''\) and \(y''\) s.t. (a) \(x'' \sqsubseteq x \wedge x'' \sqsubseteq x'\); (b) \(y'' \sqsubseteq y \wedge y'' \sqsubseteq y'\); and (c) \(x'' \ne 0\) or \(y'' \ne 0\). If \(x'' \ne 0\), then given (a) it follows that \(x \circ x'\); and if \(y'' \ne 0\), then given (b) it follows that \(y \circ y'\). Given that result and (c), it follows that \(x \circ x' \vee y \circ y'\).

  64. The application of \(\exists \exists \textsc {-pl}_{}\) to danced with is mandated by our assumption that the sister of plural DPs are pluralized. This assumption further requires an \(\exists {\textsc {-pl}}_{}\) to apply to the whole VP (which is a sister of the girls); it is omitted here since it’s semantically vacuous.

  65. More adequately, what all should force is exhaustification only over the alternatives generated by the domain variable of the pluralization operator which attaches to the sister of the DP it associates with. Otherwise we would get undesired results for a sentence like not all the kids read the books: instead of the desired meaning ‘not every kid read some book’ we’d get ‘not every kid read every book’ (here I assume that the books could take non-surface scope below negation so that its sister is headed by \(\exists {\textsc {-pl}}_{}\) and there’s no \(\exists \exists \textsc {-pl}_{}\) in the structure; if \(\exists \exists \textsc {-pl}_{}\) applies to the verb, following Sect. 8.2, a more serious issue arises; see fn. 70).

  66. One would have to make sure that the domain variable abstracted over is the right one in case there’s more than one pluralization operator in the structure (see fn. 65). An indexing mechanism along the lines of Križ and Spector (2017) might be utilized to this end.

  67. Similar solutions can be utilized for Homogeneity removal by other upward monotone plural quantifiers as in five kids lifted the piano. We can posit a covert existential quantifier \(\exists _{\textit{all}}\) which quantifies over pluralities of five kids and incorporates the Homogeneity removal effect we implemented in the semantics of all, as in (ia). The sentence ends up true as long as there is a plurality of five kids that \(\llbracket ^{{\star }_{}} \hbox { lifted the piano} \rrbracket \) is true of. As pointed out by a reviewer, this line of thought does not extend well to non upward-monotone quantifiers, due to the existential quantification introduced by \(\exists _{\textit{all}}\).

    figure dk

    We can also define the semantics of floating all (e.g., in the kids all lifted the piano) based on (100b) with the minimal difference that it first takes the \(\langle et ,\langle e,st \rangle \rangle \)-type argument and then the e-type argument.

  68. It is straightforward to define \(\forall \forall \textsc {-pl}_{}\), a two-place counterpart of \(\forall \textsc {-pl}_{}\). Quantificational DPs could allow for co-distributive readings if we assumed that they require either \(\forall \textsc {-pl}_{}\) or \(\forall \forall \textsc {-pl}_{}\) to head their sister (much like our assumption in Sect. 8.2 that non-quantificational DPs require either \(\exists {\textsc {-pl}}_{}\) or \(\exists \exists \textsc {-pl}_{}\) to head their sister). See however fn. 70 for some complications with this view.

  69. For the purposes of this discussion we can assume that all is semantically vacuous and only serves as a way to license \(\forall \textsc {-pl}_{}\). This would be similar in spirit to the view of Homogeneity removal by all in Brisson (1998, (2003), though here the effect of all is changing the type of the pluralization operator involved, while for her it changes the type of the covers which can serve as restrictors of the pluralization operator.

  70. A reviewer points out that explaining how all can remove Homogeneity and at the same time allow for co-distributive interpretations is quite a complex matter, given the view of Homogeneity with two-place predicates from Sect. 8.2. Consider a sentence with a non-quantificational plural in object position and a quantificational plural in subject position, as in (i). Given our assumptions from Sect. 8.2, \(\exists \exists \textsc {-pl}_{}\) could attach to the verb read, and the result (simplified) would be the predicate in (ia) (there is also another possible LF, mentioned in fn. 65, which involves no \(\exists \exists \textsc {-pl}_{}\); this LF would however not derive co-distributivity).

    figure do

    While the existential quantification over parts of the first argument of the verb is precisely what we want in the basic semantics, since this would give us the desired existential quantification over books, for the second argument we would want something stronger. But since \(\exists \exists \textsc {-pl}_{}\) lumps both arguments together, it would be quite difficult to only strengthen the quantification over the second argument while leaving the quantification over the first argument intact. Indeed, all the views of Homogeneity removal considered in this section predict that it doesn’t matter whether we have all the girls or the girls in subject position in (i), we should expect the same truth conditions (that is, for LFs where \(\exists \exists \textsc {-pl}_{}\) applies to the verb). This is of course wrong.

    One of the directions pursued in this section faces a problem also with cases where there is a non-quantificational plural in subject position and a quantificational plural in object position, as in (ii): Given the entry for all in (100b), it would be impossible to abstract over the domain variable introduced by \(\exists \exists \textsc {-pl}_{}\) which is of type \(\langle e,et \rangle \) and get the right kind of argument for all the books (that is, even if all the books does not remain in situ).

    figure dp

    Since this section is only meant to sketch directions for accounts of Homogeneity removal I cannot seriously pursue here a solution to these problems. A possible direction is to move to an event semantics framework where \(\exists \exists \textsc {-pl}_{}\) (and possibly \(\forall \forall \textsc {-pl}_{}\), see fn. 68) does not apply directly to the verb, but rather to syntactically realized thematic roles; such a view would allow us to determine the quantificational force for each argument position independently of the others. See Bar-Lev (2018, appendix A.3) for a framework of this sort utilized in order to account for what Križ calls Upward Homogeneity.


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First and foremost, I would like to thank my Ph.D. thesis supervisors, Luka Crnič and Danny Fox, for endless discussions on this work. I also greatly benefited from talking to Itai Bassi, Elitzur Bar-Asher Siegal, Gennaro Chierchia, Kai von Fintel, Martin Hackl, Irene Heim, Roni Katzir, Manuel Križ, Daniel Margulis, Roger Schwarzschild and Benjamin Spector, among others. Feedback I got at SALT 28 and the LF Reading Group at MIT was very helpful. My work on this project was funded by the Language, Logic and Cognition Center at the Hebrew University of Jerusalem, as well as the Jack, Joseph and Morton Mandel School for Advanced Studies in the Humanities, which also funded my stay at MIT in the academic year 2017–2018 during which much of this work was developed.

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Bar-Lev, M.E. An Implicature account of Homogeneity and Non-maximality. Linguist and Philos 44, 1045–1097 (2021).

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