Abstract

Under certain conditions, existential sentences have readings that are not predicted and cannot be modeled, given common assumptions about the construction. For example, the sentence There could be three outcomes to this election is normally taken to express that three things could be the outcome of the election, not that it is possible that the election will have three outcomes. This article proposes the first analysis of such “summative” readings of existentials, premised on a novel semantics for cardinal determiners as counting the values of an individual concept across indices. The analysis is shown to predict the observed descriptive generalizations about summative existentials, including the highly restricted circumstances that give rise to them. The unorthodox semantics proposed for cardinals is argued to be independently motivated.

1 Summative Readings of Existentials

This article describes and analyzes a phenomenon in existential sentences that none of the many analyses of such sentences in the literature (e.g., Barwise and Cooper 1981, Keenan 1987, 2003, McNally 1992, Zucchi 1995, Francez 2007a) accounts for, or can be extended to account for. Specifically, it describes a reading of existential sentences that arises systematically, but only under particular conditions.

Suppose there is a race, which only one participant will eventually win, and you are watching it with a friend. At the outset, all contestants stand a good chance of winning, given what is known about their abilities. As the race proceeds, however, some of the contenders, for various reasons, can no longer realistically be expected to win. Commenting on the race, you tell your friend:

(1) At this point, there can (only) be three winners in this race.

On one reading, paraphrasable as in (2a), your statement is false in the relevant context, since only one person can win the race. On a more salient reading, paraphrased in (2b), your statement might well be true.

(2)

  • At this point, it is possible that this race will have (only) three winners.

  • At this point, (only) three people are such that they might win this race.

Exactly the same situation is presented by (3), whose two readings can be paraphrased as in (3a) and (3b). Since world knowledge dictates that an election can only have one outcome, the sentence would normally be false on the interpretation in (3a), but on the interpretation in (3b) it might well be true.

(3) There can be three outcomes to this election.

  • It is possible that this election will have three outcomes.

  • There are three scenarios such that they might be the outcome of this election.

Borrowing terminology from Szabó (2006), I call these readings of existentials summative, since they involve summing the number of distinct entities satisfying a condition across a set of times or worlds. Szabó points out that sentences such as (4) have summative readings. The truth of (4) depends on how many distinct individuals satisfy the description Helen’s husband across the set of past times.

(4) Helen had three husbands.

Similarly, the truth conditions of (1) on the summative reading paraphrased in (2b) involve counting how many distinct individuals satisfy the description the winner in this race across the set of accessible possible worlds.

That existentials have summative readings is an utter surprise given what is believed about existentials. First, consider the summative interpretation of (1), which is used as a running example throughout the article. On the summative reading, the truth conditions of the sentence relative to a model (and a world of evaluation) do not involve a set of three winners, despite its containing the noun phrase three winners. All existing analyses of existentials predict precisely the opposite. Second, as Heim (1987) points out, when an existential sentence features a modal, the modal always takes wide scope over the postcopular noun phrase (henceforth the pivot). Heim demonstrates this with the contrast in (5).

(5)

  • There might be someone following me.

  • Someone might be following me.

On the summative reading of (1), however, the modal does not take wide scope, as shown by the contrast between (2a) and (2b). In fact, as I point out in Francez 2007a, 2009b, neither on the common assumption that pivots are quantificational nor on the assumption (made in McNally 1992 and much subsequent literature) that pivots are property-denoting is there a way to assign scope to the modal that yields a summative reading.

This article develops an analysis that derives the summative interpretation of existentials, and does so in a way that accounts for the restricted circumstances, detailed below, under which it arises. The key component of the proposed analysis is the assumption that cardinal determiners are lexically ambiguous (at least) between their standard generalized-quantifier meanings and a novel denotation, whereby they count the distinct values of an individual concept across a set of worlds—henceforth, a concept-value-counting denotation. The assumption that a concept-valuecounting denotation is available to cardinal determiners is shown to have the benefit of immedi ately accounting for an entirely unrelated phenomenon, namely, predicative uses of modal adjectives. On the basis of descriptive generalizations about the distribution of summative readings in existentials, I argue that summative existentials manifest a different phenomenon than the summative readings of other quantified sentences discussed and analyzed by Szabó (2006, 2010, 2011). Szabó argues that the latter arise from a general mechanism of scope displacement and are accordingly available whenever such mechanisms are expected to be operative. Summative readings of existentials, in contrast, show a much more restricted distribution, and I argue that the relevant restrictions follow from the proposed analysis—specifically, from the proposal that numerals have a concept-value-counting denotation.

2 Summative Existentials: The Descriptive Generalizations

Summative readings are not an idiosyncrasy of English existentials, but are a stable feature of their counterparts across a variety of unrelated languages, as demonstrated in (6a). Native speakers of the relevant languages find these sentences to be correct descriptions of a scenario like the one described for (1), that is, one in which the race can only have a unique winner, but there are three possibilities for who the winner might turn out to be. Summative readings of existentials are therefore systematic and arise in a grammatically principled way.

(6)

  • Italian

    Ci possono essere tre vincitori in questa gara.

    CI can.PL be.INF three winners in this race

    ‘There can be three winners in this race.’

    (Andrea Beltrama, pers. comm.)

  • Hebrew

    be-šlav ze, yexolim liyot šloša menacxim ba-meruc.

    in-stage this can.M.PL be.INF three.M winners in.the-race

    ‘At this stage, there can be three winners in the race.’

  • Hindi

    is samasyā-ke tin hal ho sakte hañ.

    this.OBL problem-GEN three solutions.NOM be can.M.PL be.PRES.3PL

    ‘There can be three solutions to this problem.’

    (Ashwini Deo, pers. comm.)

  • Basque

    Asterketa honetan, hiru irabazle egon daiteke.

    race this.in three winner be can

    ‘There can be three winners in this race.’

    (Karlos Arregi, pers. comm.)

Summative readings do not arise with all existentials that feature a modal. Their distribution appears to be constrained by two descriptive generalizations. The first is a restriction to relational nouns.

  • Restriction to relational nouns

    Summative readings of existentials arise only with so-called relational nouns, never with sortal nouns.1

For example, (7) does not have a distinct summative reading. It cannot describe a situation in which there are three things that might turn out to be books on the table in some worlds, but not in others.

(7) There could be three books on the table.

That this reading is unavailable can be diagnosed by the infelicity of a continuation enumerating the things that could turn out to be books. With summative existentials, such continuations are possible. While the discourse in (8) is perfectly coherent, the ones in (9) and (10) are contradictions.

(8) The race has ten participants, exactly one of whom will win. At this point, there could be three winners: the Italian, the German, or the Eritrean.

(9) The table is piled with things, exactly one of which is a book. #At this point, there could be three books on the table: the thing in the upper corner, the thing next to the vase, or the thing in the middle.

(10) This spy carries exactly one disguised camera in her purse. #There can be three disguised cameras in her purse: the lipstick, the lighter, or the cigarettes.

The second generalization is a restriction to existential modals.

  • Restriction to existential modals

    Existentials with universal modals do not have summative readings distinct from the wide scope reading for the modal.

For example, (11) is true if and only if in all accessible worlds, this race has three winners.

(11) There must be three winners in this race.

The generalization, in fact, is broader. Deontic modals, whether existential or not, never give rise to summative readings. There is no reading of (12a) on which it is equivalent to (12b).

(12)

  • There may be three participants in this meeting.

  • Three people may participate in this meeting.

Additionally, summative readings are more difficult to get with some possibility modals than with others. For example, many speakers do not intuit a summative reading with the modal might. Such speakers (myself included) cannot perceive a reading for (13a) on which it is equivalent to (13b). Later on, I will demonstrate that this restriction is not a restriction against epistemic modality.

(13)

  • There might be three winners in this race.

  • Three people might win this race.

Any successful analysis of summative readings of existentials must be able to explain these restrictions on their distribution. The analysis proposed below explains generalization A and the universal-modal aspect of generalization B. The observation that summative readings do not arise with deontic modals, while not explained, is shown to be part of a more general incompatibility between such modals and concept-value-counting denotations. The difficulty of getting summative readings with the modal might, to the degree that it is real, remains a challenge for further research.

3 The Analysis: Counting Possibilities

Structurally, existentials vary across languages in complicated ways. For the purposes of this article, I concentrate on English and assume that existentials consist of an expletive subject,2 a copular verb, the pivot, and a verb phrase or sentence modifier called a coda. Sentence (1), repeated in (14), has the structure in (15) (or a similar one in which the coda is a VP modifier).3

(14) There can be three winners in this race.

(15)

graphic

In intuitive terms, the analysis I propose is that (14) is true if and only if the individual concept the winner in this race has three distinct values across the set of (in this case, metaphysically or circumstantially) accessible possible worlds.

Articulating the composition in slightly more detail, the sentence’s truth conditions are that the familiar race r is such that, for three things x, the set of propositions p that are possibly true contains the proposition that x is the value of the concept the winner inr.

The analysis rests on three assumptions:

  1. Relational nouns like winner can denote (relational) individual concepts, that is, functions from worlds and individuals to individuals.

  2. Codas like in this race determine the value of one of the arguments of the relational noun.

  3. Cardinal determiners like three have available a concept-value-counting denotation, whereby they take an individual concept and a set of worlds as arguments.

These three assumptions, discussed in more detail below, lead to an analysis in which the expression three winners requires a modal argument and composes with the modal auxiliary can. The sentence There can be three winners denotes a context-dependent proposition, that is, a function from individuals to propositions (much like a sentence containing a pronoun in a variable-free semantics; see, e.g., Jacobson 1999). Given some contextually supplied individual d, the proposition is true if and only if across the worlds specified by the modal, there are three values to the concept the winner in d. Modeling the meaning of the sentence as a function from individuals to truth values is a way of capturing the fact that the truth of a summative existential depends on supplying the first argument of the relational concept provided by the head noun in the pivot, and that this argument is implicit, contributed either by context or else by the coda, a contextual modifier. This kind of context dependence is a general characteristic of all existentials (see Francez 2009a). In (14), the relevant individual is supplied by the coda in this race.

Of these three assumptions, the third is by far the most controversial and might seem, on the face of it, like a stipulation tailored to deal with summative readings. However, I show below that concept-value-counting denotations are independently required to account for data unrelated to existentials; furthermore, assuming them is what enables an explanation for generalizations A and B about the restricted availability of summative readings in existentials.

More formally, the analysis is built on the following denotations. Nouns like winner and solution denote relational individual concepts, functions that map individuals to functions from worlds to individuals, as in (16), where for any individual d in the domain of quantification, fwinner-of-d(w) is the unique (but potentially plural) individual who wins d in w.4

(16)

graphic
winner
graphic
= λuλw.fwinner-of-u(w)

That nouns like winner can have the denotation in (16) (or the related extensional functional reading in Löbner 1985) is a view often adopted in the literature on so-called concealed questions, exemplified in (17a) (e.g., Heim 1979, Romero 2005, Aloni 2008, Frana 2010), as well as in related literature on so-called specificational copular constructions such as (17b) (e.g., Romero 2005, Arregi, Francez, and Martinović 2013).5

(17)

  • Shirin knows the winner in this race.

  • The winner in this race is Shirin.

The concept-value-counting denotation I propose for cardinal determiners like three is given in (18). The numeral three takes a relational concept, a set of propositions, and an individual and returns True if and only if there are three entities x such that the proposition that x is the value of the concept is an element in the set. For example, in my running example, three takes (not in this order) a race r, the set M of propositions true in some possible world, and the relational concept the winner in x and returns True if and only if there are three individuals such that the proposition that they are the value of the concept the winner in r is in M.

(18)

graphic
three
graphic
= λ
graphic
e,seλ
graphic
st,tλy.∃3x :
graphic
w.
graphic
(y)(w) = x)

The concept-value-counting denotation certainly does not resemble familiar denotations for numerals in the literature. It is not a generalized quantifier, nor is it an adjective-like modifier of set-denoting expressions. It does involve quantification, however—in particular, what Szabó (2011) calls a bare quantifier, a quantifier whose domain is not restricted in any way by linguistic material. Instead, on this denotation what the arguments of the numeral three contribute is its scope. What three ends up quantifying over are the values of a concept across the possible worlds.

That the denotation in (18) involves quantification is merely a convenience. A concept-valuecounting denotation can equally well be given without invoking quantificational elements, as in (19).

(19)

graphic
three
graphic
= λ
graphic
e,seλ
graphic
st,tλy. | {x :
graphic
w.
graphic
(y)(w) = x)} | = 3

The denotation of the modal auxiliary can in (20) is standard. The modal denotes the set of propositions true in some world accessible from the actual world @ according to some accessibility relation R. The notation R@ designates the set of worlds that are R-accessible from @.

(20)

graphic
can
graphic
= λp.∃wR@ : p(w)

Finally, codas are treated as sentential (or perhaps VP) modifiers whose role is to provide a value for the implicit argument of the existential, which in this case is the argument of the relational noun. Like other modifiers, codas can be stacked, with interesting semantic effects (Francez 2009b). The same is true for summative existentials. Roughly, each coda provides a restriction for the previous coda.

(21)

  • There can be three winners in each race in this tournament.

  • There can be two outcomes to the elections in each of the neighboring countries.

For simplicity, however, I disregard stacking here and assume that a coda, in the case of summative readings, is simply a quantifier that takes an existential as its scope argument.6 The meaning of in this race, assuming the preposition is vacuous, is given in (22).7

(22)

graphic
in this race
graphic
= λPet.P(r)

Given these denotations, composition is completely unremarkable and proceeds by function application. The meaning of three winners is given in (23). This noun phrase takes a modal and returns a property that holds of any individual y if and only if there are three entities d such that the proposition that d is the winner of y is in the denotation of the modal.

(23)

  • graphic
    three winners
    graphic
    =

  • λ

    graphic
    st,tλy.∃3x :
    graphic
    w.fwinner-of-y(w) = x)

The meaning of the existential without the coda is then derived as in (24), by applying the meaning of the pivot to the modal.

(24)

  • graphic
    there can be three winners
    graphic
    =
    graphic
    three winners
    graphic
    (
    graphic
    can
    graphic
    ) =

  • λy.∃3x :

    graphic
    can
    graphic
    w.fwinner-of-y(w) = x) =

  • λy.∃3x : ∃wR@ : Fwinner-of-y(w) = x

Finally, the meaning of the coda is applied to the meaning of the existential, to yield the desired truth conditions, shown in (25).

(25) ∃3x : ∃wR@ : fwinner-of-r(w) = x

This analysis, then, derives the correct meaning for summative existentials, and does so in conformity with the more general assumption that codas are optional contextual modifiers saturating or restricting an implicit argument of the pivot (Francez 2009b). Its merit and insight, however, must be measured by the degree to which it can explain the restrictions on summative readings observed above, and the degree to which its key components—in particular, concept-value-counting denotations for cardinals—can be independently motivated. The next two sections are dedicated to these issues.

4 The Restricted Distribution of Summative Readings

The proposed analysis derives summative readings for all and only existentials that involve a cardinal determiner with a concept-value-counting denotation. Such a determiner requires, semantically, a (relational) individual concept-denoting noun and a modal expression as its arguments, and it is therefore already expected that not all existentials will have summative readings. It is also expected, however, that there should be nonexistential sentences in which such cardinal determiners can be found, giving rise to similar interpretations. Both of these expectations are met, as are a few others discussed in this section.

Generalization A above, that summative readings do not arise in the presence of a sortal noun, is precisely what the analysis predicts. This is because sortal nouns cannot contribute relational concepts to the compositional interpretation. Recall the contradictory discourse in (26), repeated from (10).

(26) This spy carries exactly one disguised camera in her purse. #There can be three disguised cameras in her purse: the lipstick, the lighter, or the cigarettes.

If the existential that is the second sentence in (26) had a summative reading, it would be perfectly consistent with the first sentence, which asserts the existence of a unique hidden camera in the spy’s purse. The proposed analysis attributes the unavailability of a summative reading for (26) to the inability of nouns like camera (and/or the noun phrase disguised camera) to denote relational individual concepts. However, this inability is not an absolute semantic property. When the context suggests such an interpretation strongly enough, it becomes available (see, e.g., Schwager 2008, Frana 2010). This can be readily observed in contexts thought to involve concept interpretations, such as the subject position of change-of-state verbs (see Deo, Francez, and Koontz-Garboden 2013 for extensive discussion). Out of the blue, example (27a) is naturally interpreted to involve a different winner in every race, whereas (27b) is, somewhat oddly, interpreted as describing a single camera that changes.

(27)

  • The winner will change next week.

  • The camera will change next week.

It is fairly easy, however, to imagine a context that makes (27b) perfectly natural. For example, suppose a spy’s purse must contain a camera at all times, and the cameras are changed once a week. Then (27b), uttered by a spy about the contents of her purse, is unremarkable. Similarly, existentials with sortal nouns can receive summative readings in a context that strongly suggests a concept interpretation of the pivot, as in (28).

(28) The syllabus for this class always contains one book and five articles. There is no final decision yet, but at this point there could only be three books on the syllabus: Ulysses, Beloved, or The Hour of the Star.

Generalization A, then, turns out to be more refined than was stated earlier. The restriction against sortal nouns is not absolute, and summative readings are tied to the presence of individual concept interpretations for pivots, which are not equally available for sortal and relational nouns.8 The dependence of summative readings on concept interpretation for the pivot is precisely what the proposed analysis predicts in locating the source of such readings in the concept-value-counting denotation of cardinal determiners.

Generalization B, that universal modals do not give rise to distinct summative readings, is an immediate consequence of the concept-value-counting denotation proposed for cardinals. To reiterate the generalization, the sentence in (29) can only be interpreted as asserting that it is necessary that three people win this race.

(29) There must be three winners in this race.

On the proposed analysis (as the reader is invited to verify), (29) receives the interpretation in (30).

(30) ∃3x : ∀wR@ : fwinner-of-r(w) = x

Clearly, this interpretation entails (i.e., is a special case of ) the wide scope reading for the modal. If there are three individuals that are the values of the concept the winner of this race in every world, then in every world there are three people who win the race.

What seems at first to be a stipulative aspect of the proposed analysis, the assignment of concept-value-counting denotations to cardinals, turns out upon further examination to have significant explanatory power, predicting generalizations about when summative readings arise. The next section discusses independent motivation for introducing concept-value-counting denotations for cardinal determiners, including evidence from sentences with predicative modal adjectives. The latter also demonstrate that the unavailability of summative readings with deontic modals, observed above, is predictable if such readings are tied to concept-value-counting cardinals.

5 The Case for Concept-Value-Counting Denotations

Independent motivation for concept-value-counting denotations as possible interpretations for cardinal determiners comes from copular sentences with modal adjectives in predicative position, such as (31a). As far as I am aware, such sentences have not been discussed in the literature. Assuming that predicative possible has the denotation in (20), denoting the set of propositions true in some possible world, the derivation of (31a) is trivial given the concept-value-counting denotation proposed in (18), proceeds by simple function application (

graphic
three
graphic
(
graphic
winners
graphic
)(
graphic
possible
graphic
)), and correctly captures the fact that the sentences in (31) are truth-conditionally equivalent.

(31)

  • Three winners are possible in this race.

  • There can be three winners in this race.

Modal adjectives having been brought in, it is worth considering briefly a temptation to favor an alternative analysis of summative existentials, deriving them in some way from existentials with attributive modal adjectives, such as (32), which is also truth-conditionally equivalent to the sentences in (31).

(32) There are three possible winners in this race.

A concept-value-counting denotation for the numeral will not account for (32). A numeral with that denotation must combine with a concept noun, and only then with a modal; it is not designed to combine with a modified concept noun.9 This fact might be taken to indicate that the proposed analysis is wrong and that summative existentials should rather be derived from sentences like (32), perhaps by some operation raising the adjective into the position of the modal, or by assuming that both positions are in fact occupied, due to some concord phenomenon, and only one is interpreted and pronounced. Perhaps the syntactic assumptions that such an account entails can be motivated. Even so, the idea of deriving the relevant summative existential from (32) can be dismissed, for the simple reason that the synonymity between existentials with a modal and existentials with an attributive adjective, unlike the synonymy in (31), does not hold generally. In particular, attributive modal adjectives can occur in existentials with sortal nouns, giving rise to meanings not available to equivalent sentences with a modal auxiliary. This can be seen by contrasting (33a) and (33b).

(33)

  • There are three possible cameras in this purse.

  • There could be three cameras in this purse.

(33a) asserts that there are three things in the purse that might turn out to be cameras.10 It entails that there are at least three things in the purse. (33b) entails no such thing and is compatible with the purse being potentially empty. Furthermore, (33a) commits the speaker to the existence of things suspected to be potential cameras, whereas (33b) does not. Both of these observations come out clearly in the continuations in (34) and (35).

(34)

  • There are three possible cameras in this purse, #but it might be empty.

  • There could be three cameras in this purse, but it could be empty.

(35)

  • There are three possible cameras in this purse, #but currently there are no possible cameras there.

  • There could be three cameras in this purse, but currently there are no possible cameras there.

Since there is no general synonymy between existentials with modal auxiliaries and existentials with corresponding attributive modal adjectives, there is no motivation for deriving one from the other, even if such a derivation can be shown to be syntactically plausible.

Going back to modal adjectives in predicative position, we can observe in (36) that such adjectives cannot be deontic. The sentences in (36a) cannot be interpreted as semantically equivalent to their counterparts in (36b).

(36)

  • Three winners are allowed / required in this race.

  • Three people are allowed / required to win this race.

This indicates that deontic modals are ruled out not just in summative existentials, but more generally in contexts that involve concept value counting. In other words, for reasons that are not clear to me, deontic modality is simply not compatible with concept value counting. Further reflection on why this might be is left for future research, but if it is indeed the case, as the data suggest, then the absence of summative readings with deontic modals is a direct prediction of an analysis that pins such readings on concept-value-counting denotations.

Another interesting piece of data that seems to support the existence of concept-value-counting denotations comes from how many questions. The question in (37) cannot be interpreted as asking how many people might win; it can only be interpreted as asking how many winners this race can have.

(37) How many winners can there be in this race?

This is predicted if summative existentials do not involve cardinal determiners with standard denotations. If the numeral in the noninterrogative, summative existential counterpart of (37) had its standard meaning, specifying the cardinality of a set contributed by its complement noun, the pied-piping in (37) would retain that reading, counterfactually. According to a widespread assumption in the literature on quantity questions like how many (see, e.g., Heycock 1995, Romero 1998, Hackl 2000, Rett 2006), they track the cardinality of a set contributed by the common noun. On the proposed analysis, the set whose cardinality the numeral is tracking in the summative existential version of (37) is the one in (38).

(38) {x : ∃wR@[x is the winner of this race in w]}

It is entirely unsurprising that a how many winners question cannot track the cardinality of this set, since it is not a set contributed by the noun winners (neither as its extension nor as its intension). The assumption that there are concept-value-counting numerals and that it is such numerals that are involved in summative readings therefore immediately predicts the unavailability of a summative question reading for (37).

6 Summative Readings and Quantifier Displacement

As mentioned in section 1 (and already in Francez 2007a), summative existentials closely resemble the more general phenomenon investigated by Szabó (2006, 2010, 2011). Szabó discusses summative readings of sentences like the following:

(39)

  • John believes that eleven terrorists live in his building.

  • This race could have three winners.

Szabó argues that the summative reading of these sentences is a genuine instance of bare quantification in natural language. His analysis invokes a special mechanism of quantifier displacement, wherein a determiner can move to a position where it is restricted by an empty noun whose denotation is the entire domain of quantification, leaving behind what he calls a “restricted trace.” A restricted trace carries as a presupposition the content of the noun that, before movement, acted as the restrictor of the determiner. Szabó demonstrates that this mechanism can capture the summative readings of the examples he analyzes. From the perspective of parsimony, clearly an analysis of summative existentials that subsumes them under this more general phenomenon of summativity is preferable to one that doesn’t. In this section, I argue that such a move would be a mistake, since the phenomena are only superficially identical.

Omitting some detail, applying this mechanism to my running existential example would generate the LF structure in (40) (ignoring the prepositional phrase for simplicity).

(40) [S[DP[D three] [N ]] [8 8 [S there [can [be [DP[Dt8] [N winners]]]]]]]

As mentioned, the empty noun slot in (40) is interpreted as denoting the domain of quantification E, and the constituent [[t8] [N ]], called a restricted trace, is interpreted, relative to an assignment ɡ, as ɡ(8) if ɡ(8) is in the extension of the noun winners, and is undefined otherwise. How exactly this LF structure is to be interpreted is not an entirely straightforward question, and it depends on what one assumes about the interpretation of an existential with a restricted trace in pivot position. Assuming (nontrivially; see below) that the interpretation of the constituent in (41a) is as in (41b), we get the truth conditions in (41c) for the sentence, whenever it is defined. These are the summative truth conditions.

(41)

  • [8 8 [S there [can [be [DP[Dt8] [N winners]]]]]]

  • λx.∃wR@[xE] if x ∈ winners in w, else undefined.

  • THREE(E)({x : ∃wR@[xE & winner(x)(w)]})

There are two reasons, however, not to assimilate summative readings of existentials to summative readings elsewhere. The first is that an analysis involving quantifier displacement is far too general for the data involved. Quantifier displacement is a general syntactic operation, not sensitive to the lexical semantics of nouns or to the force of modals. Therefore, an analysis based on quantifier displacement has no way to account for the descriptive generalizations A and B above, and it wrongly predicts summative readings for all existentials that contain a modal. This lack of descriptive adequacy clearly outweighs considerations of parsimony.

The second reason is that an analysis involving quantifier displacement goes against what is generally known about existential sentences. In particular, it goes against the well-known empirical generalization, discussed extensively by Heim (1987) and mentioned in section 1, according to which scoping out of the pivot position in an existential is impossible. Heim is interested in defending the theoretical claim that scoping out of existentials is ruled out when it leaves behind an individual variable, and she takes this to be an instance of the definiteness effect. Her generalization is stated in (42).

(42)

  • Heim’s (1987:22) generalization: No scoping out of existentials

  • *There be x, when x is an individual variable.

The LF structure in (40), generated using Szabó’s mechanism of displacement, clearly violates this generalization. The constituent [DP[Dt8] [N winners]] is an individual variable, denoting, relative to any assignment function ɡ, the individual assigned by ɡ to the index 8 if that individual is a winner, and denoting nothing otherwise. Heim’s generalization, however, is empirically quite robust. It is manifested in the absence of de re readings exemplified by (5) and in the ungrammaticality (in English) of sentences like (43) (Heim’s (2)) in which the pivot is a bound individual pronoun.

(43) *No perfect relationship is such that there is it.

Heim furthermore argues that her generalization is manifested in a range of other phenomena that have been analyzed as involving displacement and scope, including the oddity of the sentences in (44).11

(44)

  • ??Which actors were there in the film?

  • *The men, all of whom there were in the room, ate guavas.

An analysis that seeks to assimilate summative readings of existentials to the more general phenomenon of summativity analyzed by Szabó, then, is disadvantageous compared with the analysis proposed here on both empirical and theoretical grounds. Empirically, it overgenerates, missing important generalizations about the availability of summative readings and about scopal interactions in existentials. Theoretically, assuming a theory that involves mechanisms of scope displacement, it requires such mechanisms to apply in a context in which there is significant reason to believe they do not. Therefore, summative existentials are a different phenomenon, and the source of their summativity lies elsewhere, as I have argued.

7 Summary and Conclusion

This article introduced and provided an analysis of the puzzling phenomenon of summative existentials. Such readings were shown to be restricted to existentials in which the pivot is interpreted as an individual concept and that contain a nondeontic existential modal. The main argument of the article is that summativity in existentials arises from a special interpretation of cardinal determiners as concept value counters. It was shown that an analysis of summative readings of existentials as involving concept value summation explains the observed restrictions on their availability, and cardinal determiners with concept-value-counting denotations lead to insights into the properties of sentences involving predicative modal adjectives. The proposed analysis was compared with one that seeks to assimilate summative existentials to a more general phenomenon arguably due to scope and quantifier displacement; the conclusion was defended that for both empirical and theoretical reasons, the two phenomena should not be unified and the proposed analysis should be preferred. If the proposed analysis is correct, then, it uncovers the existence of concept-value-counting determiners and contributes to a growing body of literature (e.g., Aloni 2000, Condoravdi, Crouch, and van den Berg 2001, Zimmermann 2006, Schwager 2007, Löbner 2011, Deo, Francez, and Koontz-Garboden 2013, Sæbø 2014, Francez 2015) highlighting the ubiquity of individual concepts in the interpretation of nominal expressions in natural language.

Notes

1 Which relational nouns can give rise to summative readings in existentials is subject to crosslinguistic variation. The Hebrew example (i) has a summative reading, whereas the English equivalent in (ii) is ungrammatical.

  • la-yeled ha-ze yexolim lihiyot šloša avot.

    to.the-boy the-this can.M.PL be.INF three.M fathers

    ‘Three people could be this boy’s father.’

    (Lit.: There could be three fathers to this boy.)

  • *There could be three fathers to/of this boy.

This variation merits closer scrutiny, but my conjecture is that it is due to syntactic constraints on the distribution of relational nouns and their arguments between pivot and coda positions.

2 By expletive subject I mean a meaningless element that is present in order to satisfy a syntactic constraint specific to English: that every finite clause have a subject, that an EPP feature be checked, and so on. See Deal 2009 for discussion of the syntactic issues surrounding expletive selection.

3 That English existentials have roughly this structure has been widely assumed in the literature. It is argued for extensively in McNally 1992 and Francez 2007a, 2009b (though see McCloskey 2014 for detailed arguments against maintaining it for Irish). Many authors (see, e.g., Jenkins 1975, Barwise and Cooper 1981, Williams 1984, Hazout 2004) assume that any constituent to the right of the pivot is an internal noun modifier. This is an unwarranted assumption. An existential, like any other sentence, can feature sentential and VP modifiers as well as noun modifiers. (ib) shows that the PP in (ia) cannot be part of the pivot, and the nonsynonymy of (iia) and (iib) indicates the same.

    • There are three winners with red jerseys in every game.

    • *Three winners with red jerseys in every game are here.

    • How many winners could there be in every game?

    • How many winners in every game could there be?

As in Francez 2007a, I reserve the term coda for material that follows the pivot NP and is not internal to it. Here, I assume the parse of (1) in which the PP is a coda, though the proposed semantic analysis of summative existentials would work exactly the same for NP-internal PP modifiers (see also footnote 6).

4 Of course, the meaning in (16) by no means exhausts what must be said about the meaning of winner. At a minimum, a full account of nouns like this must say more about their apparent sortal uses, as in Shirin is a winner. Furthermore, winner is a deverbal noun that involves a derivational morpheme -er, about which there is much to say. However, the details of such a fuller account are immaterial to my arguments here.

5 As a reviewer notes, the concept analysis is not an uncontroversial one for concealed questions; see Nathan 2006, Schwager 2008, and Aloni and Roelofsen 2011 for discussion. However, the analysis developed here does not hang on the correct analysis of concealed questions. What matters here is that nouns contribute individual concepts to the compositional computation of meaning, whether they do so lexically, as I assume here, or otherwise—for example, through a typeshifting operation, through combination with the numeral (as Lasersohn (2005) assumes for the definite determiner), or contextually.

6 The data in (21) clearly argue against an analysis of codas as internal noun modifiers (and see Francez 2007b, 2009b for extensive discussion of quantified codas), but it is worth pointing out that analyzing the PP as a noun modifier would not affect its compositional contribution in any way.

7 Two reviewers express healthy skepticism about the semantic vacuity of the preposition, but the skepticism seems exaggerated to me. First, semantically vacuous occurrences of prepositions like in very likely occur in modifiers—for instance, in examples like An Italian won in the race or The hero dies in the movie. Second, the preposition in my running example is clearly not locative, since a race is not a location, and reference to its location requires the locative preposition at (e.g., I’ll meet you at the race). The preposition could be taken to express a relation of constitutive parthood, as in There are three Italians in the race or There are no wheels on a hoverboard. On an analysis in which in is interpreted this way, the noun winner would contribute not a relational concept but simply a concept, the winner, and the coda would specify that the individual who is the value of the concept is, in a relevant sense, part of the race. The plausibility of such an analysis depends on an explanation for the indefeasible inference that the relevant winner won the race, rather than something else. It also depends on an account of the preposition to in There can be three outcomes to the elections. In other words, the preposition in the modifier seems to be idiosyncratically determined by the relational noun. Be that as it may, the simpler (if perhaps simplifying) assumption that the preposition is semantically vacuous suffices to explain the generalizations this article is concerned with.

8 Why this asymmetry holds is, of course, a very intriguing question to explore, but falls far beyond the scope of this article.

9 The details of an analysis of (32) that does work are irrelevant to my argument here. One option is to assign possible a denotation that maps nouns to sets of individuals that satisfy the descriptive content of the noun in some possible world.

10 Note that (33a) is not a summative existential. It is not compatible with a scenario in which the purse can in fact contain only one camera, but there are three things that might turn out to be that camera.

11 See Heim 1987 for elaborate discussion of why these sentences are manifestations of the ban on scoping out of the pivot position. For example, under standard assumptions (44a) would have to be interpreted as a set of true propositions of the form there is x in the film, where x is some actor. Of course, the data in (44) are only examples of generalizations about scope if they are taken to involve movement and variable binding, which is a theoretical assumption rather than a descriptive fact.

Acknowledgments

I thank Cleo Condoravdi, Karlos Arregi, Chris Kennedy, Zoltán Gendler Szabó, Ashwini Deo, Ming Xiang, Kyle Rawlins, Ezra Keshet, Marcin Morzycki, audiences at SALT 26 at the University of Texas at Austin and the Workshop in Linguistics and Philosophy at the University of Michigan, and three anonymous reviewers for discussion, questions, suggestions, critical comments, and corrections. Thank you to Karlos Arregi, Ashwini Deo, Andrea Beltrama, Jason Moore, and Lydia Zoells for judgments.

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