Co-predications and the quantificational force of summative predicates

Abstract

In the literature on homogeneity, summative predicates have been described as quantifying universally over their argument’s parts in positive sentences, while being negated existentials in negative sentences. In this article, I provide a fuller picture of these predicates’ quantificational force in positive sentences through various ‘co-predications’—sentences in which two summative predicates are predicated of the same individual. In some co-predications, summative predicates are universal; in others, they are weaker, while remaining stronger than existential. In light of this new empirical paradigm, I suggest that summative predicates are lexically existential, but are exhaustified so as to exclude other same-class predicates. In addition to making this proposal, I also show that no other theory of homogeneity can capture the co-predicational paradigm.

1 INTRODUCTION

A topic of much recent interest in semantics has been the observation that predicates composing with pluralities have universal force in positive sentences (1a), but are negated existentials in negative sentences (1b).

This homogeneity effect is observed not only with pluralities, but also within atoms with so-called summative predicates, such as colour adjectives (Löbner 2000; Spector 2013; Križ 2015). These are predicates like green or wooden that are true of an individual by virtue of being true of that individual’s material parts; they are opposed to integrative predicates (like flag or happy) that hold of an individual as an undivided whole.

Call the paradigms in (1) and (2) plural and subatomic homogeneity effects.

This article makes two general contributions to the study of subatomic homogeneity. The first contribution is to outline a new empirical paradigm expanding empirically on the quantificational force of summative predicates in positive sentences. This new paradigm is a set of constructions I will call co-predications, in which more than one summative predicate is attributed to the same individual. A puzzle emerges due to the different quantificational force obtained in different co-predications: certain co-predications maintain the universal force of summative predicates even if this creates inconsistency within a single sentence (3a), but in other in co-predications, summative predicates are intuited as weaker and therefore consistent (3b).  

While the literature on homogeneity has discussed the existence of weak quantificational force in positive sentences as arising from particular discourse contexts (so-called non-maximality, discussed in section 2), (3b) shows weak quantificational force of a discourse-independent nature. This is not something that the literature has attempted to capture in positive sentences. Clearly, the fact that this weak meaning is not always observed in co-predications (3a) makes this explanandum all the more challenging.

The second contribution of this article is more theoretical. I will both suggest a new way to understand subatomic homogeneity, building on work by Harnish (1976) and Levinson (1983), and use the new co-predicational paradigm to evaluate various other theories of homogeneity. I will show that only the modified Harnish–Levinson proposal can account for co-predications. These authors suggest that summative predicates are lexically existential:

This obtains the truth conditions of negative sentences immediately. For positive sentences, Harnish and Levinson suggest that colour terms are strengthened through the exclusion of other colour terms (5); if the flag is partly green and has no other colour, it must be all green. I will refer to this as the Exclusion theory of subatomic homogeneity, and model exclusion through the grammatical Exh operator of Chierchia et al. (2012).

I will modify the Harnish–Levinson account in two substantial ways. First, the Exclusion account can only work with a stipulation that the exhaustification is necessarily computed locally to the colour term, something I will model through an Agree relation between colour terms and Exh. Second, I will use the trivalent exhaustivity operator (‘Pexh’) of Bassi et al. (2021) in order to capture heterogeneity-gaps, the truth-value gaps that arise when an individual is non-homogeneous vis-à-vis a summative predicate. Following this endorsement of this Exclusion theory, I will spend the remainder of the article showing that no other currently existing theory of homogeneity can capture all the co-predications in (3).

This article is organized as follows. Section 2 discusses the classic homogeneity paradigm and non-maximality with summative predicates. Section 3 then expands on this with the co-predication paradigm—the central empirical contribution of this article. From there, I turn to three sets of theories in sections 4–6. The Exclusion theory just described is taken up in section 4. Then, section 5 discusses two theories deriving universal quantification in positive sentences semantically (e.g. Löbner 2000, Bar-Lev 2021); both create more contradictions among co-predications than observed. Section 6 discusses theories involving semantic underspecification (e.g. Krifka 1996, Križ and Spector 2021); they predict too few contradictions. Finally, section 7 concludes with comments on what unites plural and subatomic homogeneity, and on additional benefits of the Exclusion theory beyond quantification in summative predicates.

2 BACKGROUND ON SUMMATIVE PREDICATES’ QUANTIFICATION

We have already observed the homogeneity paradigm with summative predicates; (6) is repeated from (2). This section expands empirically on the paradigm in (6) in two ways, both of which are well established in the literature. Section 3 will then add new observations to this discussion.

2.1 Heterogeneity-gaps

The first way that (6) oversimplifies the empirical picture is that it only describes the truth conditions of the sentences. Truth conditions will be the main focus of this article, but a theory of homogeneity would be incomplete without capturing that these sentences’ meanings are more complicated than that; in particular, their truth and falsity conditions are not complementary. One way to appreciate this comes from Löbner’s (2000) suggestion that the falsity conditions of a sentence are the same as the truth conditions of its negation; from there, it follows that (6a) has the falsity conditions that the flag is not green at all, and (6b) has the falsity conditions that the flag is entirely green. Thus, if the flag is partly but not fully green (what I will refer to as ‘non-homogeneous cases’), the sentences in (6) will be neither true nor false (see Križ 2015 for a recent defense of this, and Bar-Lev 2021 for an alternative view). I will refer to these sentences as having heterogeneity-gaps—truth-value gaps in non-homogeneous cases.

2.2 Non-maximality

The second way that (6) by itself is an oversimplification is that it does not show how malleable summative predicates’ quantificational force can be. To see this, first consider a colour adjective being predicated of something other than a flag; the empirical picture can become considerably more complex (see Kennedy and McNally 2010 and citations therein). Indeed, colour terms are often used to refer only to some salient component of their subject; this is something that cannot be observed with examples involving flags, because flags lack components other than those created by the colours themselves. For example, the two sentences in (7) could both be true of the same grapefruit, if its skin is yellow (7a) and its flesh is pink (7b).

Thus, summative predicates can be less than universal even in positive sentences. I write ‘less than universal’ because the colour terms in(7) are still stronger than existential; (7a) means the skin is entirely yellow, while (7b) means the flesh is entirely pink.

The observation of less-than-universal quantificational force in positive sentences does not only arise based on the nature of the argument (whether it is a flag or a grapefruit, in the above examples), but also on discourse context. Out of the blue, (8) is most readily understood as involving universal quantification by red, but this can change with some context (9).

In this example, it happens to be that, unlike the examples in (7) (which both require a certain part of the grapefruit—its skin or flesh—to be entirely yellow/pink), (9) is outright existential. Either way, in both cases, the meaning is not universal.

The observation of less-than-universal quantification in positive sentences with predicates displaying homogeneity is called non-maximality. This phenomenon has received considerable attention in the literature on plural homogeneity (e.g. Dowty 1987, Lasersohn 1999, Brisson 1998, 2003, Malamud 2012, Schwarz 2013, Križ 2015, Križ and Spector 2021, Bar-Lev 2021). Križ (2015) points out that (10) could be true and felicitous in many discourse contexts even if only most (e.g. eight out of ten) of the professors smiled.

In addition to observing non-maximality in positive sentences, we also find non-minimality in negative ones:

Here, not white means ‘not entirely white’ rather than ‘not white at all’. Presumably, this non-minimality, being conditioned by discourse context, is the mirror image of the non-maximality observed in (9) (but see Bar-Lev 2021). To keep things simple, I will focus on non-maximality in positive sentences (see Paillé 2023b on non-minimality with summative predicates).

From the observation that non-maximality can arise due to discourse factors (9), I assume that the same holds for the examples with grapefruits (7), even if no discourse context was necessary to observe it there. The idea is that one can utter (7a) if what matters for the conversation is the colour of the exterior of the grapefruit, while one can utter (7b) if what matters is the edible interior. Hence, even though observing non-maximality requires explicit discourse context in some cases but not others, we can safely assume non-maximality is generally conditioned by discourse. The flip-side is that, since this article will not generally focus on non-maximality (qua discourse-conditioned phenomenon), we can control for it by using examples with objects like flags or shirts that, without particular discourse contexts, resist non-maximality.

What causes non-maximality to arise in discourse? Prima facie, the most obvious way to think about it would probably be as domain-restriction. However, Križ (2015) argues against such an analysis for plural homogeneity, and his arguments extend to summative predicates. One of his arguments comes from discourse anaphora. Building on the discourse in (10), which is felicitous even if not all the professors smiled, we can go on to refer to all of the professors with they:

The same goes for summative predicates. Consider (13), where red is not universal vis-à-vis the entire car (for instance, the tyres and steering wheel are presumably not red) but only its painted exterior.

Much like with plural predication (12), the entire car can still be referred to by it in a discourse continuation (14). The crucial property of (14) is that the predicates in the second sentence are all summative, and they can be true of all the parts of the car (including the steering wheel, for instance) even while red in the first sentence is not.

Likewise, exceptives can be used to make explicit what the speaker has in mind in their non-maximal use of a summative predicate (15a), while exceptives result in a non-sequitur in real cases of domain-restriction (15b) (Križ 2015).

Rather than taking non-maximality to be a case of domain-restriction, Križ (2015) suggests that non-maximality arises from the existence of heterogeneity-gaps. The connection between non-maximality and heterogeneity-gaps can be appreciated from the fact that both of these phenomena disappear with all (Križ 2015):

The basic insight of Križ’s (2015:76ff) theory is that sentences that are neither true nor false in the world of utterance |$w_{0}$| can be used felicitously if |$w_{0}$| is, for the purposes of the conversation, equivalent to a world in which the sentence was true. The first ingredient of this theory is the standard assumption (e.g. van Rooij 2003) that a question under discussion (QUD) partitions worlds by how they resolve it. Let us consider the sentence (17) with a toy model of three worlds (18) corresponding to different amounts of red on the shirt:

If the QUD is ‘How much red does the shirt have?’ or ‘What does the shirt look like?’, all of these worlds are in their own cell—they all correspond to different answers to the QUD. On the other hand, if the QUD is ‘Does the shirt have any red on it?’, |$w_{1}$| and |$w_{2}$| both correspond to the answer ‘yes’, so they are in the same cell.

From here, Križ (2015) suggests to take to the letter Grice’s (1975:75) maxim of Quality, which Grice phrased as ‘Do not say what you believe to be false’ rather than ‘Say what you believe to be true’—these are not equivalent in a trivalent semantics. Križ (2015) suggests that speakers can utter sentences that are neither true nor false in the world of utterance, as long as they are true in some of the worlds in the cell of the partition containing the world of utterance. That is, a sentence must correctly identify the cell containing the real world, but does not need to identify the real world as such. Thus, QUD permitting, speakers may say things that are neither true nor false in the real world.

2.3 Section summary

This section has elaborated on the basic homogeneity paradigm (6) with two observations taken from the homogeneity literature. These observations show that it is an oversimplification to describe summative predicates simply as universal in positive sentences and negated existentials in negative ones. First, they display heterogeneity-gaps (their truth and falsity conditions are not complementary), and second, discourse factors can affect their quantificational force. The rest of this article will control for discourse factors by using sentences with flags as their subjects, which do not readily lend themselves to non-maximality out of the blue.

Even with these complications, if we take for granted Križ’s (2015) theory of non-maximality, it remains that on the standard view of summative predicates, all that one needs to explain about their quantificational force is the following: in positive sentences, summative predicates have universal truth conditions and negated-existential falsity conditions, with a heterogeneity-gap for non-homogeneous cases; and the reverse holds for negative sentences. But we now turn to new data showing that the empirical picture is in fact much more complicated.

3 TWO TYPES OF CO-PREDICATIONS

This section expands empirically on the quantificational force of summative predicates in positive sentences. We will see that, when two summative predicates from the same class (e.g. colour terms) are predicated of the same individual, or are co-predicated, their observed quantificational force depends on the manner in which they are co-predicated. Indeed, in some co-predications, summative predicates are universal just as in the basic homogeneity paradigm; as such, they are inconsistent with one another, and a contradiction emerges. In other cases, namely with and or also, co-predication is consistent, because the summative predicates are intuited as existential-plus: weaker than universal, but stronger than merely existential (together, they entail that all parts of the individual are covered by one or the other predicate). We take these two classes of co-predications in turn.²

3.1 When co-predicated summative predicates are inconsistent

In some co-predications, both summative predicates are intuited as universal, and a contradiction arises. Empirically, there are two such cases: when one summative predicate is attributive while the other is predicative (19a), and when two summative predicates are stacked on a single noun (19b).

In both cases, a sentence-internal contradiction is intuited due to the meanings clashing with world knowledge; we do not conceptualize of surfaces as possibly being entirely of two different colours.

While it is clearly in line with intuitions that the data in (19) are semantically deviant due to the quantificational force of the summative predicates, a sceptic could try to explain the deviance as arising from some notion of asymmetry between the predicates. Perhaps the problem for (19a) is that it is odd to place one colour term in the subject and the other in the predicative position, and the problem for (19b) is that green and white are at different syntactic heights. However, this cannot be maintained as a sufficient explanation. For (19a), the presence of one colour term in the subject and another in the vP cannot be the cause of the deviance, because there are other sentences with similarly ‘distributed’ colour terms that are acceptable; this will be shown in section 3.2. As for (19b), certain non-summative predicates from a given conceptual class (much like white and green come from the conceptual class of colour terms) can in fact be stacked in this way without a contradiction, so it is not the stacking as such that should be blamed:

The deviance of stacked colour terms is maintained even in discourses similar to (20):

Thus, the semantic deviance of the examples in (19) is due to the quantificational force of the summative predicates, rather than some accidental property of these sentences. The important conclusion is that co-predicated summative predicates are at least sometimes intuited as universal, and therefore inconsistent.

3.2 When co-predicated summative predicates are consistent

There are also at least two cases where co-predication is consistent, with the summative predicates interpreted as non-universal and no contradiction arising. This occurs when co-predications are mediated by additive particles or and; we take these in turn.

3.2.1 Additive particles

I just discussed deviant examples like (22) as an example of inconsistent co-predication:

What is striking is that an additive particle like also can remove the inconsistency of (22). In (23), the additive is intended to refer anaphorically (Kripke 2009) to the clause-internal antecedent white, rather than to any material from the preceding discourse.

The meaning of (23) is that the flag is partly white and partly green, and has no other colour. As such, while the quantificational force of the colour terms is not universal, it is more than just existential. Call it existential-plus; another term would be ‘exclusive existential’.

Examples like (23) may require a bit of work to explain why one predicate is in the subject position and the other is in the vP. Some speakers may find the example easier with an indefinite:

Either way, it is not difficult to come by scenarios that make it natural to put one colour term in the subject and the other in the vP. For example:

An important empirical puzzle touching on (23), about which I will have nothing to say, is why not all summative predicates behave in the same way. While colour terms can be intuited as existential-plus when co-predicated via also (26a), this is not the case with material terms (26b). This is in spite of both colour terms and material terms otherwise patterning similarly (they both display classical homogeneity and pattern together for all other types of co-predications).³

It is not clear why the two classes of summative predicates should pattern differently. I leave this for future work, and will focus on colour terms in this article. But whatever the nature of the class of summative predicates that can compose consistently via also, it includes more than just colour terms. The summative predicates live-action and animated are one example:

Without the additive, (27) means that the film is both (just about) entirely live-action throughout and entirely animated throughout—an impossible state of affairs, of course.

Many speakers report that sentences with colour adjectives joined by also need the colour term in the subject to cover a larger part of the flag than the colour term in the vP. That is, the felicity/truth conditions of (28) (for these speakers) are not quite identical, as suggested in the right-hand parentheses.

But in reality, greater coverage of parts is only one of many reasons why a speaker could choose to put one colour term in the subject, and the other in the vP. Consider (29), for example, which motivates this ‘distributed’ use of colour adjectives without implying that the subject-internal colour term covers more of the flag:

Similar asymmetries hold with integrative predicates too (see section 7 and Paillé 2022b on also with integrative predicates):

Since it does not touch directly on the quantificational force of summative predicates, I will put aside this asymmetry for the remainder of this article.

3.2.2 Conjunction

The second kind of co-predication that leads to a consistent, existential-plus meaning for summative predicates is conjunction, as in (31).

The description that (31) is intuited with the colours not covering the entire flag is non-controversial; but what I claim is specifically that this is due to the colour terms themselves being interpreted as weaker than universal, as shown informally in (32a). This is in contrast to a competing analysis by Krifka (1990), sketched out in (32b), where the colour terms are universal and quantification introduced by a non-Boolean and is the cause of the consistency in (31).

We spend the rest of this subsection defending the view in (32a) against (32b).

Krifka’s claim is due to an apparent parallel between (31) and conjunction with plural subjects. With plurals, conjoined predicates can be interpreted either as modifying all parts of their plural argument (33a) or as existential-plus (33b), each predicate |$P$| and |$Q$| being true of some proper part such that |$P(x)$| or |$Q(x)$| is true of each atomic part x. This latter interpretation can be brought out most clearly by conjoining predicates that cannot both hold of a single individual (at a given time).

Krifka (1990) suggests to understand the existential-plus meaning in (33b) not as arising from the predicates not distributing universally, but as being due to and dividing the subject the planets in two parts, with the predicates then distributing universally over those subpluralities. For our purposes, we can simply claim (incorrectly; see e.g. Schmitt 2021) that there are two lexical meanings for and, one of which is Boolean and one of which is not:⁴

On this view (adopted for simplicity), and is lexically ambiguous and speakers choose with which and to conjoin predicates based on some notion of naturalness or contradiction-avoidance when two predicates are mutually exclusive.⁵

Since, when a conjunction has a plural subject, and can break up the subject into two parts (34b), Krifka (1990) suggests that (35) is essentially the same phenomenon. The summative predicates are universal, but and|$_{2}$| (34b) breaks up the flag into two parts.

Thus, (35) is consistent due to quantification brought in by and|$_{2}$|⁠, not due to the colour terms being weaker than universal:

I now turn to showing that this is not the right analysis; in fact, there is no non-Boolean and available with atomic subjects like in (35). To see this, let us begin by manipulating the conjunction itself. There are morphologically overt ways of forcing conjunction to be Boolean. At least when it composes with conjoined predicates, both forces a Boolean interpretation of a conjunction; this is also the case with as well as (see e.g. Schwarzschild 1996:149, Szabolcsi 2015:199, Schmitt 2021:24; but Schmitt (2013:138–139) argues otherwise). (37a) shows that these ‘marked’ conjunctions are acceptable with a Boolean interpretation, while (37b) shows that they rule out a non-Boolean interpretation.

As such, if colour conjunctions were necessarily non-Boolean, both and as well as should be incompatible with them. This is not the case:

(38) suggests that the intuited meaning of (35) is actually compatible with a Boolean interpretation of and.⁶

This is enough to conclude that conjunction allows the co-predication of colour terms in a way that lets them have an existential-plus, rather than universal, meaning. For good measure, let us take the argument one step further. This time, we will manipulate the conjuncts rather than the conjunction, and arrive at the conclusion that colour conjunctions with atomic subjects are only ever interpreted as Boolean. We will conduct a simple test based on the following intuition: if the colour terms in conjunction data were possibly universal, we should be able to modify them to bring out their universal force explicitly without this causing inconsistency. For example, we could modify them with completely.

I assume that completely asserts that a predicate is true of all parts of the subject (39); the contribution of completely may be semantically vacuous, if the predicate is already universal.

Therefore, if colour terms are already universal, modifying them with completely makes no semantic difference, and the consistent meaning in (40) is predicted with and|$_{2}$|⁠:

But this prediction does not carry through; making the conjuncts explicitly universal in a colour conjunction (with an atomic subject) leads to a sharp contradiction:

(41) can only mean that the entire flag is simultaneously of both colours, contrary to the expectation if a non-Boolean and was available (40).

One way to try and avoid (41) as a falsification of Krifka’s theory is to claim that (39) is incorrect. Rather, one could hypothesize that completely P not only means that P is true of all parts of some argument x, but also that P is true of all the parts of the maximal individual (plural or atomic) that x is a part of. A contradiction would obtain in (41) even if and was non-Boolean because each conjunct would end up predicating a colour term of all parts of the entire flag. However, this alternative view does not hold up due to (42), where we would now expect a contradiction too: completely green and completely white would mean that all parts of the entire plural individual are green/white.

Thus, (39) is the right definition for completely, and it was correct to conclude from (41) that a non-Boolean and is not available there, and more generally that it is not possible for a non-Boolean and to refer to parts of atomic individuals.

There is another way one could problematize (41) as an argument against the possibility of a non-Boolean and with atomic subjects: an apparently similar test does not hold with plurals (43). Recall that with plurals, I am not arguing against the possibility of non-Boolean interpretations of conjunctions. In (43), each conjunct contains a token of each instead of completely because each quantifies universally over individuals rather than subatomic parts.

(43) might be expected to be consistent given the availability of a non-Boolean and with plural subjects: it would mean that each atomic flag of one subset of the flags is fully green, while each atomic flag of another subset of the flags is fully white (44). In (44), subscript at stands for ‘atomic’—‘Part_at|$(x)$|’ is the set of atomic parts of x, and ‘|$\sqsubseteq $|_at’ is the ‘atomic-part-of’ relation.

If so, there is either something wrong with my test, or with my suggestion that a non-Boolean and exists with plural but not atomic subjects.

However, floating each has independently been argued to necessarily take as its first argument the DP subject, its ‘associate’. (44a) and (44b) are incorrect. Or rather, on the proposal by Doetjes (1997) and Fitzpatrick (2006), floating each takes a pro DP co-indexed with the associate. This view is motivated by a restriction against A′-movement of the associate, agreement with the associate on the floating quantifier in languages like French, and overt clitic doubling of the associate on the floating quantifier in languages like Hebrew. Concretely, Fitzpatrick (2006:81) gives (45a) the LF in (45b) (and assumes the same syntax for each).

In (43), then, each token of each has as its first argument a pro, each of which is co-indexed with the associate the flags. This means that each conjunct carries the entailment that it is every individual flag that is green/white. We therefore expect a contradiction, even if a non-Boolean and is available with plurals. Finally, all of this is in contrast to the data with completely (41), since completely takes as its first argument a predicate rather than a pro co-indexed with the predicate’s argument (39).

In sum, not only is a Boolean interpretation available with colour conjunctions, it is in fact the only possibility. Thus, what we are observing in the co-predication in (46) is a pair of summative predicates that are non-universal and conjoined intersectively.

The flag is in the set of partly white things and the set of partly green things, and furthermore, it has no other colours.

Of course, there is an important question about why a non-Boolean and is unavailable for examples like (46). There is a literature on conjunction and whether it is underlyingly Boolean or non-Boolean; some authors (e.g. Schmitt 2021) claim that it is underlyingly non-Boolean. Notably, most of these works focus on conjoined predicates with plural subjects. It could be that and is only sensitive to the atomic parts of plurals, and not the subatomic parts of atoms; even if it is underlyingly non-Boolean with plurals, this could cause it to act as if it was Boolean with atoms. Either way, what matters for our purposes is that, descriptively, and is Boolean in (46) and, therefore, we know we are observing non-universal summative predicates.⁷

3.3 Section summary: the co-predication paradigm

In this section, I elaborated on the quantificational force of summative predicates in positive sentences. Beyond the initial simple observation that summative predicates are universal in sentences like (47), their quantificational force shows surprising behaviour in co-predications.

In particular, co-predicated colour terms sometimes maintain the universal force observed in (47), giving rise to sentence-internal contradictions (48a). But other co-predications result in colour terms’ quantificational force being weaker than in (47); it is existential-plus rather than universal (48b).

All of this places a new empirical burden on theories of homogeneity. It is not enough to describe the quantificational force of summative predicates as being universal in positive sentences and a negated existential in negative sentences (49) together with heterogeneity-gaps rather than falsity for non-homogeneous cases.

In fact, summative predicates can also be existential-plus rather than universal in positive sentences.

There is a third type of co-predication that I do not discuss in this article, namely disjunctions. These are consistent, while having universal colour terms.

Naturally, disjunctions’ consistent meaning is due to or rather than the quantificational force of the colour terms—hence why I do not focus on these.

We now turn to three sets of theories of homogeneity, with a focus on how and whether they can handle co-predications. I start with the theory I will defend, namely the Exclusion account (section 4), before turning to alternatives in sections 5 and 6. Other than the Exclusion theory, all currently existing theories fail to capture all the co-predications; the ones discussed in section 5 predict co-predications to be contradictory across-the-board, while those discussed in section 6 predict them to be consistent across-the-board.

4 THE EXCLUSION THEORY

In this section, I build a theory of subatomic homogeneity that can capture the co-predication paradigm. I start by overviewing how Harnish (1976) and Levinson (1983) suggest to capture colour terms’ universal meaning in positive sentences; they accomplish this through the exclusion of other colour terms (section 4.1). I immediately reframe their discussion in the terms of the grammatical theory of exhaustivity (Chierchia et al. 2012) rather than pragmatic reasoning.

The Harnish–Levinson approach captures parts of the homogeneity and co-predication paradigms, but has some important empirical deficits. For this reason, much of this section is dedicated to modifying the theory. First, to capture the parts of the co-predication paradigm that the Harnish–Levinson account cannot obtain (as well as a number of independent datapoints around colour terms’ quantificational force), I show in section 4.2 that the theory needs to claim that there is a constraint on the exhaustification of colour terms, such that exclusion must be computed ultra-locally to the adjective. I suggest as a possibility to derive such a constraint through an Agree relation between a⁰ and Exh. Second, to capture heterogeneity-gaps, I suggest in section 4.3 to capture exhaustification through the trivalent exhaustivity operator of Bassi et al. (2021) rather than the standard Exh of Chierchia et al. (2012). With these two important changes to the Harnish–Levinson account, we can successfully capture the quantificational force of summative predicates.

Despite these improvements, I will also show in section 4.4 that the account straightforwardly cannot be extended to plural homogeneity. While my conclusion in this article will ultimately be that this is not a problem (subatomic and plural homogeneity could reasonably be united only in being locally computed exhaustification effects, as I will claim in section 7), this will nonetheless motivate the search for alternative accounts of subatomic homogeneity, which will be the focus of sections 5 and 6.

4.1 Summative predicates exclude related predicates

Harnish (1976) and Levinson (1983) focus on the paradigm in (51):

In line with the discussion in section 3.2, these authors suggest that (51b) shows that the lexical meaning of colour terms is existential, not universal:⁸

From (52), Harnish and Levinson take (51a) to be the result of a quantity implicature. In a Gricean framework, (51a) can only be strengthened through the exclusion of stronger alternatives, in this case alternatives with conjoined colour terms (‘The flag is green and white’, and so on). Recent work (e.g., Katzir 2007, Fox and Katzir 2011) has suggested that alternatives cannot be more syntactically complex than the assertion, making this claim suspect. On the other hand, on the grammatical view of strengthening (e.g. Chierchia et al. 2012), it is often taken that Exh not only excludes stronger alternatives, but also non-weaker ones (e.g. Fox 2007):

Thus, the Harnish–Levinson account can be recast in terms of Exh excluding alternatives formed through other bare (non-conjoined) colour terms:

The meaning in (54c) is that the flag has at least one green piece, and does not have a piece that is of any other colour. Given world knowledge that all areas of a surface must have a colour, this is pragmatically strengthened to mean that the flag is entirely green, in the same way that (55) with an overt only means that the flag is entirely green.

It may be that the set of alternatives (54b) includes more than just sentences obtained by replacing green with other colour terms. Picture a window, part of which is made of green stained glass, and the rest of which is entirely transparent. (56) is not a good description of such a window.

Given the truth conditions in (54c), this is not predicted, because it is indeed the case in (56) that the window has no colour other than green. Postulating transparent as part of the set of alternatives to green is of no help: given that green stained glass is still more or less transparent, we would then predict that (56) would be false/infelicitous of a window which was entirely green stained glass, contrary to fact. On the other hand, there is the adjective clear which implies colourless transparency. If clear is an alternative to green along with other colour terms (57), the oddness of (56) uttered of a half-green, half-clear window is captured: the window does have a clear part, but Exh excludes that the window has any clear parts.

See Paillé 2023a for discussion of which kinds of predicates are alternatives to one another for exhaustification effects like the one being posited in this article (i.e. ultra-local ones on predicates, as described in section 4.2).

The Harnish–Levinson proposal immediately obtains the truth conditions of negative sentences (58), as non-exhaustified negations of the existential lexical meaning of colour terms. We will return to heterogeneity-gaps in negative sentences in section 4.3.

What about co-predications? Let us start with the consistent ones, returning to the inconsistent ones in section 4.2.

In the case of conjoined colour terms, if there is a single Exh operator above both colour terms, existential-plus quantification is exactly what is predicted (59). In (59), I pretend that the only colour terms are green, white, and red, and assume that alternatives are obtained by replacing conjuncts by other colour terms and/or removing conjuncts entirely.

(59) means that the flag has a green part, has a white part, and does not have parts of any other colours. All pieces of the flag must therefore be green or white.

As for the co-predications with also (60), it turns out that there is already a literature claiming that also interacts with exhaustification.

Indeed, in empirically different examples like (61) where additive particles are also obligatory, it has independently been claimed that the additive’s obligatoriness arises due to the interaction between additives and exhaustivity (Bade 2016, Aravind and Hackl 2017, and Paillé 2022a, following a similar claim by Krifka (1998) and Sæbø (2004)). On this view, also is required in (61) because without it, the second sentence would be exhaustified to mean that only Jade sang, contradicting the previous discourse. In (61), Adam and Jade bear alternatives due to being contrastive topics (Büring 2016).

Bade (2016) compares the hypothesis that also is obligatory in order to avoid unwanted exhaustification effects to an alternative view according to which they are obligatory whenever their presupposition is met (e.g. Chemla 2008, Singh 2008)—that is, due to the ‘Maximize Presupposition’ principle of Heim (1991). Since presuppositions project past negation, the Maximize Presupposition account predicts that additives would still be obligatory in negative sentences; but since exhaustivity standardly disappears under negation, the exhaustivity account predicts they should be optional (Bade 2016). The exhaustivity account is the one whose prediction is borne out; (62) does not need an additive (Bade 2016).

We conclude that additives interact with Exh in a non-trivial way; this means that (60) is expected from the Exclusion account of subatomic homogeneity.

Of course, one needs an account of how additives circumvent unwanted exhaustivity effects. This article is not the place to discuss the interaction between additives and exhaustivity at length. Let me simply follow the claim in Paillé 2022a that additives interact with Exh by pruning its alternatives (cf. Aravind and Hackl 2017). First, we will see in section 4.2 that (60) without also is best analysed with a local predicational Exh operator on each colour term:

I suggest that (60) is acceptable with also because, even though both colour terms are exhaustified independently of the other (i.e. the syntactic placement of Exh is no different than in (63)), when also is present, white is not an alternative to green and green is not an alternative to white. No contradiction arises.

The flag is still entailed not to be of any other colour, so that the colour terms are existential-plus.⁹

Despite these successes, there are aspects of colour terms’ quantificational force that the Exclusion theory, as it stands, does not capture. This includes a variety of data where colour terms are more consistently universal than predicated by the Harnish–Levinson account (including the inconsistent co-predications), and the classic observation (e.g. Löbner 2000) that colour terms lead to a truth-value gap in non-homogeneous situations. I take up these points in sections 4.2 and 4.3 respectively.

4.2 The exclusion of colour terms is always computed locally

If the Exclusion account is to be maintained, it must come with the stipulation that Exh is always local to the colour term. In all cases where a difference in meaning is predicted to be intuited according to whether Exh takes scope globally or locally to the colour term, the meanings that would be obtained from a global Exh are not intuited. Most of the examples in this section involve monoclausal sentences, so ‘locally’ has stronger meaning than just ‘in the same clause’. For our purposes, it will suffice to understand this locality constraint as Exh necessarily being in the maximal projection (aP) of the colour term. Such an ‘ultra-local’ Exh is shown in (65), putting aside the question of whether Exh is necessarily ultra-local in this particular example (where local or global Exh operators would produce the same meaning):

(65) has an Exh operator taking a colour term, and nothing else, as its prejacent—despite Exh having been defined as a propositional operator. I therefore define a predicational Pred-Exh operator (cf. Mayr 2015), based on a generalized notion of entailment. A predicate P entails another predicate Q if for all x, |$P(x)$| entails |$Q(x)$|⁠. For instance, dog entails animal, and scarlet entails red.

Thus:

I will suggest in section 4.2.3 to derive the locality constraint by having colour terms Agree with Exh; since there is no upward Agree (Chomsky 2001), Exh must be in the colour term’s projection. First, however, let us focus on the empirical observations about the locality of Exh.

4.2.1 Inconsistent co-predications: initial motivation for a constraint on Exh’s syntax

The initial motivation for a locality constraint comes from the half of the co-predication paradigm that the Exclusion theory, as stated, cannot explain: inconsistent examples like (68).

Let us start with the the predicational colour term green in (68a), returning below to colour terms internal to definite descriptions; for now, pretend for the sake of argument that only green has alternatives in (68a), not white. If Exh could take scope anywhere, it would be possible for Exh to scope globally (69). From this global position, Exh’s prejacent entails both the whiteness and greenness of the flag. Since Exh does not exclude alternatives that are entailed by its prejacent, neither colour is excluded, and no contradiction results:

In order for green in (68a) to be strengthened irrespective of white, Exh must necessarily appear locally to it, as in (70) (where I continue to overlook that white presumably has alternatives and a local Exh too):

In fact, the need for an ultra-local Exh is only motivated on the assumption that Exh can see entailment relations among colour adjectives, and therefore knows what it can and cannot exclude. To obtain a contradiction in (69) without forcing Exh to be ultra-local, one could try to claim that Exh is so blind to the content of predicates that it does not know that The white|$_{\exists }$| flag is white|$_{\exists }$| is entailed by The white|$_{\exists }$| flag is green|$_{\exists }$|. Exh would therefore negate The white|$_{\exists }$| flag is white|$_{\exists }$|, obtaining the semantic deviance. But there is independent evidence that Exh does know which colour predicates can be excluded.¹⁰ Consider the following minimal pair:

In (71b), scarlet does not exclude red because it entails it. Crucially, while red is a basic-level colour term and scarlet is a subordinate-level term, we know that basic-level colour terms are alternatives for subordinate-level colour terms from data like (72):

If scarlet only had subordinate-level colour terms as alternatives, (72) would not be universal on the Exclusion theory. Only excluding subordinate-level colour adjectives would not actually exclude the entire colour wheel; English and other languages cover the entire colour wheel with their basic colour terminology (Berlin and Kay 1969), but their subordinate colour terms are more of a patchwork. For instance, in my English, I have very few subordinate colour terms for types of pink or orange (and most of them are structurally complex, like hot pink), and they certainly do not cover all hues of pink/orange. (72) would therefore be compatible with the flag having pink and orange parts if scarlet only excluded subordinate-level colour terms. Thus, in (71b), we can infer that scarlet has basic colour terms including red as alternatives. But Exh apparently does not exclude red; it must therefore be that Exh sees entailment relations among colour predicates.¹¹ Hence, the only way to obtain clause-internal contradictions is through a locality constraint on Exh.

Let us now turn to the attributive adjectives in both (68a) and (68b). Attributive colour terms must be exhaustified just as much as predicative ones; otherwise, (68b) would be consistent. But these too must be subject to a locality constraint. Quite outrageously, with a colour adjective that is part of a definite description, a global Exh would create entailments about other flags; (73) shows this for (68b), but the same goes for (68a).

Note that the truth conditions in (73c) are not aligned with intuitions in two ways: there are entailments about other flags, and no contradiction is created. On the other hand, the observed meaning can be created by having an Exh operator local to each attributive colour term:

(74c) is contradictory and lacks entailments about other flags, as desired.

The claim that colour terms constrain Exh’s syntax may come across as a post-hoc complication only posited to capture the inconsistent co-predications. However, in the following subsection, I show that the locality constraint is observed even outside of inconsistent co-predications. Following that, I will show that the constraint can simply be modelled as an Agree relation between colour terms and Exh.

4.2.2 More evidence for a locality constraint on Exh

On the Exclusion account, the need to constrain Exh’s syntax is general. When a colour term co-occurs with another scope-bearing element, Exh must scope locally to the colour term. Intuitively, (75) means that exactly one flag is entirely green.

But a global Exh, scoping above both exactly one and the colour term, does not actually create this strong meaning:

In (76), green is not universal; it means that there is exactly one flag with any green on it.¹² This problem is avoided by having Exh scope below exactly one flag—that is, ultra-locally to the colour term:

The same goes to capture the meaning of colour terms in downward-entailing (DE) environments. Exhaustivity computed below a DE operator leads to global weakening, and is therefore strongly dispreferred (e.g. Chierchia et al. 2012, Fox and Spector 2018). This is what leads Križ and Spector (2021) to criticize exhaustivity accounts of homogeneity: neither plural nor subatomic homogeneity disappears in DE environments:

Putting aside the plural for now, (78a) is in fact far from unexpected given that we have already observed other instances where colour terms, even outside of DE environments, require a local Exh. The right meaning for (78a) is obtained by placing an Exh operator below if.

To be sure, neither summative predicates nor plural predication are consistently interpreted as universal below if, as in (79) (cf. Križ 2015:27 for similar examples with plural predication):

A possible way to analyse this is to claim that Exh is non-local or entirely absent in the non-maximal (79b-ii); this line of reasoning would rely on the presence of non-maximality below if as evidence that Exh is not necessarily present locally with summative predicates. Naturally, nothing forces such an analysis, because in section 2, we already posited mechanisms obtaining non-maximality that are independent of Exh’s presence/absence or its locality (see also section 4.3 below).¹³

There are two issues with the view that non-maximality under if shows that Exh is scopally flexible. The most important is that it cannot explain why (78a) is most readily intuited out-of-the-blue as involving a universal below if; the standard view is that Exh is dispreferred below DE operators, so without a locality contraint on Exh, an existential meaning under if is expected to be the default reading. A universal reading would have to be motivated by discourse context. But what we observe in (78a) and (79b) is the opposite: the universal is default, while the non-maximal reading requires an explicit scenario.

The second, weaker issue with the view that non-maximality should be analysed as evidence for flexible exhaustification is that not all cases of non-maximality (whether under if or elsewhere) are purely existential like (79b-ii). Many instances of non-maximality require a local Exh together with a local application of Križ’s mechanism for non-maximality. For example:

Here, we have non-maximality below if, with Exh still necessarily appearing below if, since the meaning is stronger than existential. Since non-maximality can often be shown to involve a local Exh (80), it is appealing to analyse the existential non-maximality in (79b-ii) in the same terms: a local Exh with local non-maximality.

We conclude that the quantificational force of summative predicates is always computed locally. One apparent exception to this comes from sentential negation; we return to this issue in section 4.3.

4.2.3 Formalizing Exh’s locality constraint as an Agree relation

Is this locality requirement just a stipulation? It would certainly be hard not to view it this way if it was only found with colour terms or summative predicates. In section 7 (see also Paillé 2022b), I suggest that Exh is generally found locally with predicates. I do not elaborate this idea in the present paper, but what I suggest is that language does not just exhaustify clauses/propositions, but also predicates themselves, independently of the clauses they are in. Postulating the existence of predicate-strengthening is not inherently more stipulative than postulating the (well-established) existence of sentence-strengthening.

Since the purpose of this article is to compare different theories of the quantificational force of summative predicates, I will not dwell too long on what causes the locality constraint, leaving this for other work. I will simply show as a proof of concept that the locality can be modelled through a well-established syntactic mechanism. What I tentatively suggest is that colour terms (or rather: the derivational morpheme a⁰ that merges the roots into the syntax) Agree with Exh; they therefore require an Exh operator to be present in the clause and specifically within their aP. This is essentially taking a proposal by Chierchia (2013), according to which Exh Agrees with alternative-triggering expressions, and turning it on its head; what we have here is an alternative-triggering expression (or the morpheme selecting it) Agreeing with Exh.

To start with, a theory of the locality constraint based on a syntactic interaction between Exh and a⁰ cannot be built around selection. Selection is an appealing mechanism for data like (81), where each colour terms has its own Exh (within their aP, I assume, although that’s not the only logical possibility).

But other data involve a single Exh for more than one colour term, which a theory built around selection would not be able to capture. This is the case for (82), where a single Exh must scope above both colour terms.

Colour terms therefore cannot be taken to select Exh, in which case each would have its own Exh in (82). But if they merely Agree with Exh, the contrast between (81) and (82) can be captured.

In the basic (non–co-predicational) case, we can derive the locality constraint via Agree as follows. I follow the theory of Agree in Chomsky 2001, but augment it with Spec–Head agreement (see e.g. Clemens and Coon 2018). On this approach, a⁰ bears a |$[$|uExh|$]$| feature which probes for an expression bearing an |$[$|iExh|$]$| feature, which in most cases will be Exh.¹⁴ There is no truly upward Agree, where a head H Agrees with an expression above its maximal projection HP; but there is agreement within the HP, including Spec–Head agreement. So if Exh is adjoined to H (here, a⁰), Agree can proceed (83) and the derivation converges syntactically.

As for the conjunction data, the Agree theory derives that a single Exh can scope above both colour terms due to the classic proposal that conjoined phrases inherit the category of the conjuncts (rather than projecting as a ConjP). In our case, conjoining two aPs creates a bigger aP (84). This is in fact predicted if the head and is acategorial; according to Chomsky (1995), acategorial roots do not project (see Paillé 2022b:ch. 6 for more discussion).

If the |$[$|uExh|$]$| feature on each a⁰ in (84) projects together with the category label, we expect the entire phrase to only require one Exh. In (85), feature projection (presumably happening in tandem with category projection) is shown with a dotted arrow.

From there, Exh is merged and Agree takes place between the aP_{|$[$|uExh|$]$|} label and Exh (see Béjar and Rezac 2009) on Agree from labels):

The |$[$|uExh|$]$| feature is taken care of, and the syntax converges.

In sum, Exh’s locality constraint with colour terms can be derived through an Agree relation between a⁰ and Exh. Given that Exh is an operator present in the syntax, there is no reason for it not to engage in operations like Agree. In section 7 and Paillé 2022b, I suggest that this does not only occur with colour terms and other summative predicates, but with predicates generally.

4.2.4 Section summary

In this subsection, I have modified the Exclusion account of Harnish (1976) and Levinson (1983) by placing a locality constraint on the Exh operators associating with colour terms, as required to capture not only the inconsistent half of the co-predication paradigm, but also a number of independent datapoints where colour terms are interpreted as universal more consistently than predicted by Harnish and Levinson. I suggested to model this constraint through an Agree relation whereby colour terms (or a⁰) probe for Exh, ensuring that the syntax only converges if there is an Exh operator within the colour term’s maximal projection.¹⁵

4.3 Deriving heterogeneity-gaps through Pexh

We now turn to a second modification of the Harnish–Levinson Exclusion account. As spelled out so far, the Exclusion theory does not predict heterogeneity-gaps as in (87).

However, as stated, the Exclusion theory predicts that positive sentences like (87) would in fact be false in non-homogeneous cases: (88) is expected to be true if the flag is green and no colour, and to be false otherwise.¹⁶

Can the Exclusion theory obtain the heterogeneity-gap in (87)?

It can, with one change: we need to follow Bassi et al. (2021) in taking Exh to be trivalent. Due to different empirical concerns entirely, Bassi et al. (2021) argue that semantic exhaustification only affects truth conditions, not falsity conditions. They suggest the trivalent operator in (90) as a substitution for the bivalent Exh repeated from (53) in (89).

A sentence exhaustified with Pexh is true if the prejacent is true and all the non-entailed alternatives are false. But the falsity conditions depend only on the prejacent.¹⁷

By replacing Exh with Pexh, the Exclusion theory obtains heterogeneity-gaps:

Consider the logical value obtained if the flag is half green, half white. In such a case, it is neither the case that |$\neg \mathsf{white}_{\exists }(f)$| holds (as would be needed for the sentence to be true), nor that |$\neg \mathsf{green}_{\exists }(f)$| holds (as would be needed for the sentence to be false). Therefore, the sentence is undefined, as desired.

There is another benefit to adopting the trivalent exhaustivity operator. On the face of it, the truth condition of negative sentences can be obtained on the Exclusion account by stating that such sentences are not exhaustified at all. However, in section 4.2, I claimed that there is always a local Exh with colour terms, even if this leads to global weakening. This makes it unclear why sentential negation would be the exception to this. On the particular theory of locality tentatively suggested above (based on Agree), Exh is strongly predicted to always be obligatory and necessarily local with colour terms, even under negation. But putting a bivalent Exh operator below negation does not lead to the intuited meaning:

The truth conditions in (92) do not actually require the flag not to have any green parts. With Pexh, however, nothing goes wrong if exhaustivity is computed under negation. Pexh only affects truth conditions, not falsity conditions—and not reverses these.

Thus, as far as the truth conditions are concerned, Pexh is vacuous under not. We can therefore claim, following the logic of section 4.2, that Pexh is present under negation:

Note that Pexh still affects the falsity (and undefinedness) conditions, so it is not entirely vacuous. This is as desired, since summative predicates can lead to heterogeneity-gaps (and non-maximality) in both positive and negative sentences (see Paillé 2023b for more discussion of non-minimality).

Pexh therefore allows the Exclusion account to both capture heterogeneity-gaps, and capture that negation behaves differently from other DE environments in lacking a strengthening (globally weakening) effect in the truth conditions. While I adopt the Pexh operator of Bassi et al. (2021), I will continue writing Exh and having bivalent truth conditions in the rest of this article.

On the surface, the very properties that have just led to the adoption of Pexh may appear to problematize the proposal that trivalent Pexh should replace bivalent Exh: Pexh is proposed by Bassi et al. (2021) to compute scalar implicatures, but these have not been described as having truth-value gaps, for example. In fact, postulating truth-value gaps for scalar implicatures might not be undesirable. On the pragmatic theory of strengthening via Gricean implicatures, utterances come with a theoretically substantial distinction between their ‘assertion’ and their ‘implicatures’. This makes it possible to describe sentences as having a true assertion but a false implicature, and therefore (on the whole) being neither true nor false. The semantic theory of exhaustification treats implicatures as part of the assertion, however, and therefore (with a bivalent Exh) cannot make this distinction; if Adam ate all of the cookies, (95) (with Exh) would be simply false.

But Pexh makes it possible to capture this kind of under-informativity—with Pexh, (95) is expected to be neither true nor false if Adam ate all the cookies.¹⁸ Note that, for our purposes, the presuppositional status of the undefinedness stemming from Pexh is not what matters (see section 6 of Bassi et al. 2021 for some discussion).

Capturing heterogeneity-gaps through Pexh immediately leads to the expectation of observing non-maximality with colour predication, assuming Križ’s (2015) theory of non-maximality described in section 2. Let me briefly comment on why this is preferable on the Exclusion theory to an alternative way to weaken utterances, namely by excluding fewer alternatives (‘pruning’ them), as Bar-Lev (2021) suggests to obtain non-maximality in his Inclusion theory (which I discuss in section 5.2). Let us first consider a sentence displaying non-maximality:

Hypothetically, this non-maximality could be obtained by pruning alternatives with green or brown:

However, on the Exclusion theory, the ‘pruning’ path to non-maximality differs from Križ’s empirically, since the pruning approach can only create existential force for colour terms. (97b) means that the leaf has an orange part, but this part may be very small and the leaf might be mostly green or brown. In contrast, Križ’s approach can distinguish between many different quantificational forces, predicting that non-maximality can be stronger than mere existential meaning. For this reason, pruning cannot be all there is to non-maximality: there are some clear cases where non-maximal sentences are more than existential, including (98) as well as e.g. (7) and (80).

On the Exclusion theory of subatomic homogeneity, (98) requires Križ’s theory, and therefore the heterogeneity-gaps created by Pexh.¹⁹

This concludes my modifications to the Harnish–Levinson proposal. On my Exclusion account, the assertion of a lexically existential colour term is intuited as universal because it involves the exclusion of other colour terms. This exclusion is necessarily computed locally, because colour terms Agree with Exh. The Exh operator is trivalent, resulting in heterogeneity-gaps.

4.4 No extension of the Exclusion account to plural homogeneity

The rest of this article will focus on showing that various theories of plural homogeneity cannot be extended to explain the quantificational force of summative predicates, due to the data from co-predications. In this section, I do the opposite, showing that the Exclusion account of subatomic homogeneity cannot be extended to plural homogeneity.

The Exclusion account crucially relies on world knowledge, so that ‘having a green part, and not having a part of any other colour (or a clear part)’ is strengthened to meaning ‘entirely green’. But this cannot carry over to plural homogeneity; world knowledge does not dictate anything about the parts of pluralities. Attempting to capture plural homogeneity through the exclusion of related predicates would look as in (99) (where, crucially, singing is not intonationally/contrastively focused). In (99), I am marking singing as ‘existential’ to create a parallel with my account with colour terms. We can put aside how this arises compositionally (see section 5.2 for Bar-Lev’s (2021) proposal); what matters is simply that the non-exhaustified meaning of the children are singing is that at least one of them is singing, in the same way that the non-exhaustified meaning of the flag is green is that at least one part of the flag is green. On the Exclusion account as extended to plural homogeneity, this would then be exhaustified to exclude conceptually related predicates:

There are two problems here. First, unlike the subatomic parts of surfaces which must all have a colour, it is in fact possible for individuals to do nothing. World knowledge does not make (99) entail that all children are singing. The utterance means that at least some children are singing and no children are doing anything else—but as far as world knowledge is concerned, there could be children who are neither singing nor doing anything else. Second, the entailments about children not doing other things are not in fact intuited without contrastive focus on singing. The children could be both singing and dancing, for example.

To salvage this, one could try to constrain the alternatives in (99) to only being mutually exclusive predicates (that is: mutually exclusive sentences obtained by replacing the verb with inconsistent predicates); sing would not have dance as an alternative, but it would have talk and keep quiet, for example. If at least one child in the plurality is singing, none is talking, and none is keeping quiet, they must all be singing. However, on this proposal, we lose the ability to explain the subatomic homogeneity data, where alternatives are crucially consistent predicates—for colour terms, they are lexically existential colour terms. If only inconsistent predicates could create alternatives, colour terms would not meet this requirement, and would therefore not be alternatives to one another. To defend this alternative view, one would need to explain why this mutual-exclusivity requirement among alternatives would hold for plural homogeneity but not subatomic homogeneity. There is also the problem that, if we take the view that alternatives are no more syntactically complex than the assertion (Katzir 2007; Fox and Katzir 2011), keep quiet cannot be an alternative to sing—so nothing ends up forcing the meaning that more than one child is singing.

In sum, the Exclusion account of subatomic homogeneity does not extend to plural homogeneity.

4.5 Section summary

Harnish (1976) and Levinson (1983) propose that colour terms are lexically existential, but are intuited as universal in positive sentences because their assertion involves the exclusion of other colour terms. In this section, I reformulated their account according to the grammatical theory of strengthening of Chierchia et al. (2012), and modified it in two ways. First, to capture the quantificational force of colour terms across a variety of examples, there must be a constraint on the syntax of Exh such that it necessarily occurs locally to the colour term. Second, to capture that non-homogeneous cases are undefined rather than false, I suggested to use the trivalent exhaustivity operator of Bassi et al. (2021) rather than the standard bivalent operator. Let us recap the ingredients of this approach and why they are necessary:

An important feature of this account, of course, is that nothing creates semantically universal meaning for summative predicates, in any sentence.

Despite all this work, we saw that the theory goes nowhere for plural homogeneity. This is not necessarily undesirable: the subatomic and plural homogeneity paradigms could have some common cause while differing in other ways. They could both be exhaustification effects but involve different kinds of alternatives, for example. Nevertheless, we now turn to considering other theories of plural homogeneity and seeing how they fare in the subatomic domain, with particular attention to the co-predication paradigm. In section 5, I focus on theories positing semantically universal meaning for positive sentences. These create too many contradictions; they cannot capture the existential-plus meaning of summative predicates in consistent co-predications. Then, in section 6, I focus on underspecification accounts of plural homogeneity. These create too few contradictions; they predict all co-predications to be consistent. Thus, none of the other theories of homogeneity can capture the quantificational force of summative predicates across-the-board, giving significant weight to the Exclusion account.

5 THEORIES OF HOMOGENEITY THAT CREATE TOO MANY CONTRADICTIONS

In this section, we consider two alternative theories of subatomic homogeneity, both of which derive universal quantification for positive sentences in the semantics and wrongly predict all co-predications to be inconsistent; I will focus on conjunctions like (101), which I showed in section 3 to be Boolean.

The first theory is the classic suggestion that summative predicates come with an ‘excluded-middle’ (‘all-or-nothing’) presupposition. The second theory is much closer to the Exclusion account, in that it also relies on an Exh operator appearing in positive sentences to strengthen weak lexical meaning (Bar-Lev 2018, 2021); but on this alternative theory, the alternatives Exh takes are of a different nature, and Exh does not do the same thing with them.

5.1 The EMP theory

A classic way to derive the homogeneity effect is to postulate an ‘excluded-middle presupposition’ (EMP), which ensures that predication only results in the assignment of a truth-value if the predicate holds of all or no parts of its argument (Löbner 2000; cf. Schwarzschild 1994, Gajewski 2005). In what follows, I follow Gajewski’s (2005) formalization of this approach for pluralities, and suggest two different ways to carry it over to subatomic homogeneity—neither of which can capture consistent co-predications.

5.1.1 The EMP with pluralities

Löbner (2000) hypothesizes that predication introduces a presupposition that the predicate holds of all or no parts of its argument. This is the case for both atomic parts of pluralities and subatomic parts of atoms. The EMP straightforwardly results in truth-value gaps for non-homogeneous cases.

An influential formalization of this idea for plural homogeneity comes from Gajewski (2005). He proposes that the EMP is introduced by an obligatory distributivity operator. dist is defined in (102), where |$\sqsubseteq $|_at refers to atomic parthood; (103) provides a sample LF.

In the positive case, dist asserts that the predicate holds of all atoms in the plurality. This is consistent with the first disjunct in the presupposition. Given the assertion, the only noticeable effect of the presupposition is that, if only some of the children sang, the sentence would be undefined rather than false.

In the negative case, even if the assertion is only that not all the children sang, the presupposition projects past negation and effectively strengthens the assertion to mean that none of them sang. The assertion is incompatible with the first presuppositional disjunct, so the only remaining possibility is for the second disjunct to hold—that is, for none of the children to have sung.

Given the all-or-nothing presupposition, it does not matter that the output condition of dist is written with |$\forall $| rather than |$\exists $|⁠. Having an existential output condition would yield identical definedness and truth conditions, because dist is only defined for predicates that hold of all or none of their plural argument’s atomic parts.

The EMP account of homogeneity has come under various types of criticism in the literature (Spector 2013; Križ 2015), both in terms of the core proposal that a presupposition is at work in creating the homogeneity paradigm, and in terms of Gajewski’s linking of this presupposition to distributivity—Križ (2015) shows that homogeneity is also observable with non-distributive plural predication. For the sake of argument, I put these criticisms aside, to focus exclusively on whether the account can capture the quantificational force of summative predicates.

5.1.2 Consistent co-predications as a problem for the EMP account

For Löbner (2000), subatomic homogeneity results from the same EMP as with pluralities. There are two ways to do this: either the presupposition is present in the lexical entry of summative predicates, or there is a subatomic dist_subat operator in addition to the atomic dist operator posited by Gajewski (2005).

The first option would mean that summative predicates are lexically defined for arguments only if they are true of all or none of their subatomic parts:

Postulating that the EMP is part of green’s lexical meaning could be motivated by the fact that the presence of this presupposition is regulated lexically, in light of the summative–integrative distinction. But to deal with Boolean conjunctions of colour terms (107), we need colour terms to have existential lexical meanings.

This is not compatible with the EMP. Writing out the assertive component of (106) as existential changes nothing, as described above for plural homogeneity.

Perhaps this problem could be solved if the EMP was contributed by an operator scoping above colour terms, with these being lexically existential. In conjunctions, such an operator could scope above both colour terms at once, as Exh does on the Exclusion account.²⁰ To see if this works, we first define a subatomic dist_subat operator, identical to Gajewski’s (102) but with reference to subatomic parts:

Moving the EMP from colour terms’ lexical meaning to an operator creates perfectly acceptable results for non–co-predicational sentences. We will need colour terms to be lexically existential for the conjunction data, so let us assume this right away:

We get the following meaning once green and dist_subat compose:²¹

(110) is a good result as far as non–co-predicational sentences are concerned. Despite the existential lexical entry for green, the positive disjunct means that all pieces are all green: if there was a piece which was not entirely green, the non-green piece of that piece would itself lack a green piece, contrary to the meaning of this disjunct. As for the negative disjunct, this straightforwardly means that there are no pieces with any green on them.

Let us now attempt to obtain consistent conjunctions by having dist_subat scope over both colour terms at once:

Before the functional application of dist_subat, the meaning of the conjunction is:

dist_subat then takes the entire conjunction as its argument, producing (113).

This is not a welcome result. The first disjunct requires all pieces of the flag to themselves have both a green and a white piece. As such, however small a piece you choose, it would have to be made up of a white piece and a green piece, and these white pieces and green pieces are themselves divisible between white pieces and green pieces, and so on infinitely. We would have needed this disjunct of the presupposition to contain a disjunction (‘all parts are green|$_{\exists }$|  or white|$_{\exists }$|’) rather than a conjunction. We could try to turn to the second disjunct to see if this somehow gets the right result for positive sentences, but it does not: a white and green flag can have a piece that has both white and green on it.

The conclusion is that there is at least one co-predication, namely conjunction, that the EMP theory of homogeneity cannot capture.

5.2 The Inclusion theory

The second account of homogeneity we consider in this section is Bar-Lev’s (2018, 2021); it is also based on exhaustivity, giving it some common ground with the Exclusion theory (see also Magri 2014). The basic premise is that the lexical meaning of plurals is existential, immediately capturing the truth conditions of negative sentences. In positive sentences, Exh strengthens the existential to a universal. Bar-Lev uses the notion of Innocent Inclusion (Bar-Lev and Fox 2017; Bar-Lev 2018) to have Exh assert the truth of (‘include’) the domain alternatives of the existential plural. I will focus exclusively on the theory Bar-Lev builds for distributive plural homogeneity, which is all that is needed to try to carry over his account to summative predicates.

5.2.1 Bar-Lev’s theory for plural homogeneity

In Bar-Lev’s theory, the meaning of plurals is existential: prior to exhaustification, the meaning of the children laughed is that at least one laughed. In what follows, assume there are two children, Adam and Jade.

This existential plain meaning comes about from an existential plural operator, |$\exists $|-pl, which takes as its first argument a domain variable ((115)–(117) are from Bar-Lev 2021:1062):

(114) therefore has the LF in (116), where the domain D is presented as a subscript on |$\exists $|-pl.

(116) obtains the meaning in (117), which (assuming that |$[\![ $|the kids|$]\!] $| = |$a \oplus{} j$| and |$D =$|  |$\{a,j\}$|⁠) is equivalent to (114).

Naturally, (117) immediately obtains the intended meaning for negative sentences, which mean that there is no individual that (i) is in the domain, (ii) is part of the denotation of the kids, and (iii) laughed. Thus, we get the intended meaning that no child laughed. In the positive, the sentence must be strengthened; for Bar-Lev (2018, 2021), the alternatives for Exh are obtained by replacing the sentence’s domain with subdomains:

The subdomain alternatives ‘Adam laughed’ and ‘Jade laughed’ are not innocently excludable (Fox 2007). Excluding them would dysfunctionally result in the sentence meaning that Adam or Jade laughed, but neither Adam nor Jade laughed. Innocent Exclusion is defined as in (119):

Further, the set of alternatives (118) is not closed under conjunction: there is no strong alternative of the form ‘Adam laughed and Jade laughed’ for Exh to exclude.

From here, Bar-Lev relies on the notion of Innocent Inclusion. This is the idea that Exh includes all alternatives that are not excluded and which can be included consistently:

Thus, Exh asserts the alternatives ‘Adam sang’ and ‘Jade laughed.’ This results in the meaning that all the children laughed.

5.2.2 Carrying Bar-Lev’s theory over to subatomic homogeneity: a first attempt

Let us start by seeing how Bar-Lev’s theory for plural homogeneity can be carried over to non–co-predicated summative predicates. First, we need existential meaning for colour terms. (121) would be the familiar way to do this, but it lacks the domain variable that Bar-Lev relies on to create subdomain alternatives.

Instead, let us imitate Bar-Lev’s |$\exists $|-pl in postulating an operator taking a domain variable—call it |$\exists $|-sg:

D in (122) must be populated not just by individuals (atoms) and pluralities, but also subatomic pieces. For presentation, let us assume there are two subatomic pieces to the flag, A and B. But keep in mind that, unlike the toy model used above for the kids, this toy model is truly misleading, since unlike pluralities, atoms are divisible between an infinite number of arbitrary, overlapping pieces.

From here, the idea for a sentence like (123a) is that green is an argument of the |$\exists $|-sg operator, whose domain argument creates subdomain alternatives. Prior to the merger of Exh, (123a) has the LF in (123b).

Still following Bar-Lev, (123b) is exhaustified in positive sentences (124a) but not negative ones (124b).

As for the meaning of colour terms, since we will need them to be existential for the conjunction data, let us only consider that possibility (but see section 5.2.3). If green is existential, the meaning we get from (124) for negative sentences is that there is no piece with any green on it, as desired.

Prior to exhaustification, the meaning obtained in the positive is that there is a piece of the flag which is partly green:

In our toy model where the pieces are A and B, (126) is equivalent to (127):

The alternatives triggered by the D variable in (124a) are the following:

Of course, the set of alternatives is actually infinite because the subject (the flag) can be cut up in an infinite amount of (possibly overlapping) pieces of arbitrary sizes. Following the Innocent Inclusion of all the alternatives in (128), we obtain the meaning that all pieces are partly green. Given that pieces can be subdivided into further pieces, this means that all pieces are in fact entirely green.

However, conjoined summative predicates pose the same problem for this theory as they did for the EMP. Given our toy model, the pre-exhaustification meaning of (129a) is in (129b) (‘some part of the flag is both white and green’).

The alternatives are obtained by replacing the domain with subdomains:

Now we exhaustify (129). No alternatives are excludable. Whether any are includable depends on how much Exh takes world knowledge about colours and surfaces into account. If we include all the alternatives, the meaning we end up with is that A is partly white and partly green, and B is also partly white and partly green. In other words, all pieces of the flag are themselves divisible between a partly green piece and a partly white piece, which are themselves divisible between white and green pieces, and so on—the same problem we had with the EMP. If we tried to get around this problem by claiming that the alternatives are not innocently includable, this would create another problem: no strengthening would take place at all, and the sentence would only mean that some piece of the flag is partly white and partly green.

5.2.3 A second attempt at using the Inclusion theory for summative predicates

So far, it does not appear that the Inclusion theory can capture co-predications of summative predicates. However, a Journal of Semantics reviewer suggests a way that Bar-Lev’s (2021) Inclusion theory as applied to summative predicates might work for conjunctions (131) after all.

My discussion in section 5.2.2 did not consider using or as an alternative to and—doing so makes it possible to capture (131). In this subsection, I first explain how this is so, then point out that the theory makes one of the same incorrect predictions as Krifka’s (1990) claim outlined in section 3.2.2 that and in (131) is non-Boolean.

Let us assume that colour terms are universal, and that Exh knows that they therefore cannot both hold of a particular part of a surface.²² Conjunctions come with an |$\exists $|-sg operator on each conjunct:

Prior to exhaustification, the basic meaning of this sentence is that there is a part of the flag which is entirely white, and there is a part which is entirely green.

If the two parts of the flag are A and B, this is equivalent to:

Since it does not have to be the same part that is white/green, the meaning is consistent. But the result so far is only that each colour term is existential; we must still strengthen them to being existential-plus.

The alternatives for Exh are obtained both by replacing D with subdomains, and replacing P |$\land $| Q with the normal alternatives for conjunction: P, Q, and |$P \vee{} Q$|⁠. That is, in addition to domain-restriction, (134) has the following alternatives:

The alternatives in (135) multiply once we also take domain-restriction into account (136).²³ I assume again for convenience that the flag has two parts, A and B. To make it clear where the alternatives come from, there is a guide on the right-hand side of (136), referring both to the size of the domain and whether the meaning comes from and, or, or a left/right conjunct.

(136a) is the basic meaning of the assertion. As for the other alternatives, which ones are excluded or included? Only the internally inconsistent (136b) and (136c) can be excluded. (136i)–(136l) are neither excluded nor included; including them together would result in inconsistency. As for (136d)–(136h), these cannot be excluded, but they can be included. Thus, the meaning of (132) is (137):

The first line in (137) entails that some part is white and some part is green; the second line entails that all parts are either white or green.²⁴ We have obtained a consistent conjunction with existential-plus meaning for the colour terms.²⁵

However, notice that we have just created meaning similar to what is obtained with a non-Boolean and, as discussed in section 3.2.2. In that section, I criticized the view that and in colour conjunctions (with an atomic argument) is non-Boolean in part due to the following datapoint:

We saw that Krifka (1990) predicts (138) to be acceptable due to his suggestion that colour terms are lexically universal, with and breaking up the flag in two. But something similar goes with the Inclusion theory we have just seen. The colour terms are universal, so nothing should go wrong if we replaced them with explicitly universal expressions like completely white. I assume that completely is a universal:

Therefore, if colour terms are lexically universal, completely white and white have the same meaning, and the truth conditions for (138) (on this Inclusion theory) are wrongly predicted to be identical to the basic conjunction:

I end this subsection by pointing out that, even if it worked for conjunction, this alternative approach does not carry over to the co-predications with also (141). Also, unlike and, does not have a disjunctive alternative.

One could try to claim that also is capable of pruning domain-alternatives. Assuming again that there are two parts, A and B, (141) would mean that part A of the flag is white (the B alternative from ‘|$\exists $|-sg_D white’ being pruned), while part B is green (the A alternative being pruned from ‘|$\exists $|-sg_D green’). But this would in effect end up creating a ‘non-Boolean also’, which is not in fact intuited. For instance, this would predict that also should also be able to make plural predication with inconsistent predicates consistent, which is not the case:

Assume there are four planets, A–D. If also could prune domain-alternatives as needed for (141), (142) would be expected to have truth conditions compatible with planets A and B (but not C or D) being 3 billion years old, and planets C and D (but not A and B) being 5 billion years old. The additive could prune the alternatives corresponding to planets C and D from ‘|$\exists $|-pl_D 3-billion-year old’, and the alternatives corresponding to planets A and B from ‘|$\exists $|-pl_D 5 billion years old’. Since this apparently cannot occur in (142), such pruning presumably also cannot occur in (141).

5.3 Section summary

In this section, I have overviewed two theories of homogeneity, which both posit semantically universal quantification in positive sentences: the EMP and Inclusion theories. While both can predict the quantificational force of colour terms in non–co-predicational sentences, they fall on the same problem for consistent co-predications: they create the meaning that each conjunct must be true of all arbitrary pieces of the subject. At least, this is the case on my initial attempt at using the Inclusion theory for summative predicates; the problem for the Inclusion theory is different if or is an alternative to and. Thus, even if the Exclusion theory of summative predicates cannot be extended to plural predication, it is empirically preferable for summative predicates to the two alternatives seen so far.

The Inclusion and Exclusion accounts share an important component: they both involve weak lexical meaning together with covert exhaustification in positive sentences, but not in negative sentences.²⁶ Moreover, as hinted at in section 4 and discussed in section 7, it is not only the Exclusion theory (for subatomic homogeneity) that must posit a locality condition on exhaustification; the Inclusion theory (for plural homogeneity) must do so too. The fact that the Exclusion account cannot be extended to plural homogeneity does not make it incorrect for subatomic homogeneity, and vice-versa for the Inclusion account. Given how much they have in common, each theory can be taken to correctly explain one side of the paradigm. If so, homogeneity (subatomic and plural) is a local and obligatory exhaustification effect whereby the weak lexical meaning of certain quantificational elements (summative predicates and the plural operator) is strengthened in positive sentences; but this strengthening occurs in different ways for summative predicates and the plural operator, due to the different nature of the alternatives.

6 THEORIES OF HOMOGENEITY THAT CREATE TOO FEW CONTRADICTIONS

In this section, we turn to a second set of theories on homogeneity, in particular work suggesting that the homogeneity paradigm involves underspecification between existential and universal quantificational force, with some mechanism determining which is actually intuited in a given sentence. Such theories cannot explain the co-predicational paradigm, but for different reasons from the EMP and Inclusion accounts. One of them (a pragmatic account based on the Strongest Meaning Hypothesis) clearly creates too few contradictions; inconsistent co-predications are predicted to be consistent. The other, a theory based around the co-assertion of ‘candidate interpretations’, does not (as currently stated) make clear predictions, but it either predicts inconsistent co-predications to be consistent or vice-versa. I present it in this section because its basic insight essentially builds on the first underspecification theory.

6.1 The Pragmatic Underspecification theory

Recall the classical homogeneity paradigm:

One theory about this builds on the simple intuition that each sentence in (143) has the quantificational force that provides the strongest global meaning. Krifka (1996), Lasersohn (1999), Winter (2001), and Malamud (2012) propose that grammar does not fix whether a predicate is interpreted universally or existentially when it takes a plurality as an argument. Rather, a pragmatic principle ensures that speakers prefer whichever interpretation results in the strongest meaning:

This is inspired by the Strongest Meaning Hypothesis (SMH) that Dalrymple et al. (1994) propose for reciprocals.

We might attempt to carry over the Pragmatic Underspecification theory of homogeneity to the subatomic case. On this view, green is lexically ambiguous (145), and whichever meaning is strongest in a particular sentence will be the one chosen.²⁷

On this pragmatic theory of homogeneity, computing the strength of predicates should never lead to inconsistency. After all, the quantificational strength of predicates results from a pragmatic preference for strong meanings over weaker ones; this preference would be overridden if the strongest meaning was inconsistent. Clearly, this is impossible to maintain in light of the inconsistent co-predications:

Such examples straightforwardly counter the prediction of the Pragmatic Underspecification account; the pragmatics would not create inconsistency out of potentially consistent lexical material.²⁸

One could try to defend this pragmatic theory by claiming that the pragmatic principle/SMH is blind to world knowledge; it would function like Magri (2009) claims that Exh functions, looking at logical entailment alone. Since there is no logical contradiction in a flag being entirely white and entirely green, the SMH would always apply to colour terms even if this creates inconsistency. While this would let the SMH capture the inconsistent half of the co-predication paradigm, we would lose the consistent half: summative predicates co-predicated via and or also would now be expected to involve universal colour terms.

6.2 The Co-Assertion theory

Križ and Spector (2021) also provide an underspecification theory, but not based in the SMH. They point out that the SMH cannot capture the meaning of pluralities in non-monotonic contexts:

Instead, Križ and Spector (2021) suggest that the possible meanings of plurals arise from the conjunction of candidate interpretations (CIs). The meaning of (147) is predicted if the two CIs in (148) are conjoined.

Call this the Co-Assertion account of homogeneity. The CIs posited by this theory quickly become far more complex than those in (148), in part because Križ and Spector (2021) aim to account for plural homogeneity with both distributive and collective predication; this is not a complication we need to linger on here. In a nutshell, Križ and Spector (2021:§3.1) suggest that, rather than those in (148), CIs are disjunctions in which the predicate is predicated of various parts of the plurality. Consider (149a). For each individual y that is part of a plurality x, there is a CI where one disjunct is of the form |$P(y)$|⁠, and for each sum z that contains y, there is also a disjunct of the form |$P(z)$|⁠. There are also CIs corresponding to the disjunction of those CIs. Hence, if there are two children, A and B, the CIs for (149a) are in (149b) (Križ and Spector 2021:1160).

For distributive predication, (149b) is equivalent to:

What is more, to capture non-maximality, Križ and Spector (2021) posit that not all CIs actually end up being co-asserted; this depends on the QUD. It is only the CIs that are ‘strongly relevant’ (they are ‘strongly relevant candidate interpretations’, SRCIs) to a QUD which are actually co-asserted. Assume (cf. section 2) that worlds are partitioned according to how they resolve a QUD. SRCIs are CIs that correspond exactly to a cell or a set of cells in a partition of worlds:

That is, p is strongly relevant iff there is a cell (corresponding to a single cell or a set of cells) in the partition of worlds such that p denotes all the worlds in that cell, and no other. What matters for us is that, if the QUD is Who sang?, for each atomic or plural individual who is a child, there will be a cell in the partition corresponding to that child singing. In fact, all the CIs in (149b) are SRCIs on this QUD.²⁹ Conjoining them yields the meaning that every child sang.

Turning now to subatomic homogeneity, let us consider the sentence (152) in the context of a QUD making each part of the flag relevant (such as ‘What colour is the flag?’ or ‘What does the flag look like?’—corresponding to the QUDs one accommodates upon hearing (152) out of the blue).

For simplicity of presentation, take the flag to have two parts, A and B—although as with my toy model used for the Inclusion theory (section 5.2), we have to keep in mind that this is quite misleading since we are dealing with subatomic pieces. The SRCIs are the following:

This is equivalent to:

For (152), conjoining the SRCIs in (153)/(154) creates the meaning that the entire flag is green (regardless of whether green is lexically existential or universal).

The theory cannot deal with the co-predication paradigm, however. In particular, it predicts the consistent and inconsistent halves of the paradigm to behave the same way. Due to a question about the nature of the co-assertion mechanism left open by Križ and Spector (2021), it is not clear whether both are predicted to be consistent, or both inconsistent. Let us focus on the following subset of the co-predication paradigm:

We begin with the inconsistent (155b); if we make the assumptions necessary for the Co-Assertion theory to capture (155b), we will then be unable to explain (155a).

(155b) cannot be explained through a single SRCI, since there is no world in which a flag is all white and all green. On the other hand, it would be possible to obtain (155b) by having a set of SRCIs which are internally consistent, but which result in inconsistency when co-asserted. We would need SRCIs as in (156) (simplifying somewhat in irrelevant ways, e.g. there is nothing about flags in (156)), where the colour terms must be existential for each SRCI to be internally consistent:

By conjoining the SRCIs, we obtain the impossible flag already discussed for the EMP and Inclusion theories of homogeneity, where each piece must somehow have a piece of each colour. Recall once again that it is only in my presentation that there are two subatomic pieces A and B—in reality, white and green would be predicated of every one of the infinity of possibly overlapping subatomic parts making up the flag.

This obtains (155b). But crucially, for this to occur, the mechanism whereby SRCIs are co-asserted must be blind to the creation of inconsistency. There is no equivalent of Bar-Lev and Fox’s (2017) ‘Innocent Inclusion’ in co-assertion—no Innocent Co-Assertion, which would have barred the SRCIs in (156) from being co-asserted. Yet, claiming there is no Innocent Co-Assertion wrongly predicts (155a) to be inconsistent, too. The SRCIs for (155a) look exactly as in (156) (the logical conjunction surfaces for (155b) due to Predicate Modification rather than the lexical entry for and, but they are semantically the same), and inconsistency is predicted just as for (155b). For (155a) to be consistent, we need Innocent Co-Assertion, both bringing back the puzzle of (155b) and failing to capture the existential-plus meaning of the colour terms in (155a), since no strengthening would occur.

6.3 Summary of sections 5 and 6

The quantificational force of summative predicates is not aligned with the predictions of underspecification theories of homogeneity, at least in the domain of subatomic homogeneity. The pragmatic approach predicts all co-predicated summative predicates to be weak, contrary to fact; the Co-Assertion approach predicts all co-predicated predicates to pattern together (as consistent or inconsistent, depending on assumptions about co-assertion), rather than making the cut between consistent and inconsistent co-predications.

What have we accomplished in sections 5 and 6? In section 4, I built a theory of the quantificational force of summative predicates that can capture both the homogeneity and co-predicational paradigms. But this Exclusion theory cannot be extended to plural predication. I therefore attempted to find a theory of plural homogeneity that can be translated to subatomic homogeneity. We have just gone through four different theories to see that none of them are able to handle the co-predication paradigm, making the Exclusion account the only currently existing option. This exercise has highlighted the importance of the co-predication paradigm in finding the right way to capture homogeneity.

7 CONCLUSION

In this article, I laid out an empirical paradigm touching on the quantificational force of summative predicates that provides new ways to evaluate various theories of homogeneity. I developed a new theory that can obtain the facts observed, and showed that the other currently existing theories of homogeneity cannot do so.

The paradigm, a set of co-predications, involves summative predicates sometimes being interpreted as strictly universal in positive sentences, even if this leads to sentence-internal inconsistency; but at other times being interpreted as existential-plus and consistent:

I showed that the Presuppositional (Löbner 2000, Gajewski 2005), Inclusion (Bar-Lev 2018, 2021), Pragmatic Underspecification (Krifka 1996 and others), and Co-Assertion (Križ and Spector 2021) theories of homogeneity, as translated from the domain of plural predication to atomic summative predication, all fail to capture this paradigm.

I therefore suggested to update the Exclusion theory of Harnish (1976) and Levinson (1983), according to which the meaning of summative predicates results from the exclusion of related predicates (colour terms exclude other colour terms, material terms other material terms, and so on):

But I modified this theory in two ways. First, I adopted the Pexh operator of Bassi et al. (2021) in order to capture heterogeneity-gaps:

Second, I showed that the exhaustivity operator must be subject to a locality constraint in order to capture that summative predicates are exclusive of one another in some co-predicational positive sentences and all non–co-predicational positive sentences (modulo non-maximality), e.g.:

One criticism of exhaustivity accounts of homogeneity (e.g. Križ and Spector 2021), namely that homogeneity does not preferably disappear in downward-entailing contexts, no longer holds as such. It is in fact a general fact of homogeneity that it is computed locally, whether the environment is DE or not.

While the Exclusion account does not carry over to plural homogeneity, I suggest this is not problematic. It is reasonable to state that what subatomic and plural homogeneity have in common is that they are both locally computed exhaustivity effects, but with alternatives of a different nature and, therefore, a difference in whether Exh includes or excludes them. Indeed, plural homogeneity behaves like subatomic homogeneity in necessarily being computed locally; for example, it does not preferably disappear in DE environments (see section 4.2.2):

If they are both locally computed exhaustification effects, this means that subatomic and plural homogeneity have substantial theoretical common ground, even if they arise in partly different ways.

I conclude by pointing out something important about how the meaning of summative predicates fits among the meanings of predicates generally (Paillé 2022b). The Exclusion theory can do something that the other theories of subatomic homogeneity cannot: it collapses the judgments for co-predications of summative predicates with co-predications of non-summative predicates, even though subatomic quantification is only a factor in the former case. Indeed, the co-predication paradigm carries over to integrative predicates completely, suggesting that summative and integrative predicates are in fact subject to the same Exclusion effect. Compare (157) and (162):³⁰

A play is a comedy or tragedy not by virtue of its parts being comedies or tragedies, but by virtue of meeting a set of requirements in its entirety; the predicates are integrative, and one cannot capture the pattern in (162) with a theory about part-structure. Most of the theories of homogeneity discussed in this article, however, are explicitly about part-structure, so that they would need to explain (157) and (162) through entirely different mechanisms. The exception is the Pragmatic Underspecification theory (section 6.1), whose basic insight (that predication is ambiguous and the strongest consistent meaning is chosen) is not exclusively made for part-structure, but which would fail to capture the entire paradigm in (162): (162a) would be predicted to be consistent, for the reasons given in section 6.1.

On the other hand, the pattern in (162) can be understood on the Exclusion account. The Exlusion theory only involves part-structure insofar as part-structure percolates from the lexical meanings of summative predicates. As such, this theory can be extended to a general theory of exclusion in predication even for predicates without part-quantification. The Exclusion theory as extended to integrative predicates would posit that the predicates comedy and tragedy (163a) are strengthened through local Exh (Pexh) operators (163b), creating contradictions in (162a) but not (162b).

Of the theories of homogeneity reviewed here, only the Exclusion theory can collapse the co-predication pattern touching on the quantificational strength of summative predicates (157) and the conceptual strength of integrative predicates (162).

Acknowledgements

I thank Bernhard Schwarz and Luis Alonso-Ovalle for help and support from the first day of this project, as well as Aron Hirsch for many helpful meetings. I also thank Nina Haslinger, Uli Sauerland, Viola Schmitt, and Benjamin Spector; three thorough and helpful Journal of Semantics reviewers; Yasutada Sudo, my editor for this paper; and conference participants and reviewers who gave me feedback on various aspects of this work.

Funding

This research has received funding from a Vanier doctoral scholarship from the Social Sciences and Humanities Research Council of Canada (SSHRC), an Eyes High postdoctoral fellowship from the University of Calgary, a SSHRC research grant (#435-2018-1011) to Elizabeth Ritter and Martina Wiltschko, and a postdoctoral fellowship from SSHRC.

Footnotes

References