2.1 Variable quantificational force

Yoon’s clean/dirty contrast is connected to homogeneity (Fodor 1970, Löbner 1987, 2000, Križ 2016, 2019, Champollion et al. 2019, Bar-Lev 2021): the idea that sentences with definite plural subjects are typically taken to describe a set of individuals who all match (i.e., are homogeneous) with respect to the relevant predicate. In (6), by default, either all the kids laugh or none of them do. The universal interpretation (6a) is a strong default for positive (non-negated) sentences with definite plural subjects.²

Homogeneity means that the truth conditions of the positive and negated forms of (6) do not exhaust the logical space – that is, they are contraries rather than contradictories in the sense of Aristotle (Horn 2025). We are left with a gap of murky judgments (“varied” across speakers and contexts, and “guarded” with hesitation, according to Rooth 1987, p. 254) in non-homogeneous “mixed” scenarios where some kids laugh and others don’t. These murky judgments, in turn, create the potential for existential/non-maximal interpretations like those observed above, whereby sentences like (7) may be judged true, depending on the context, when only some of the kids laughed.

Homogeneity (6) and non-maximality (7) can thus be seen as “two sides of the same coin” (Križ & Spector 2021: 1132) because both properties reflect delicate judgments, sensitive to the Question Under Discussion, about sentences with plural subjects in mixed scenarios. To capture such judgments, the literature offers three families of analyses, characterized by Bar-Lev (2021: 1047) as 1–3.

1. The Ambiguity Approach (Krifka 1996, Lasersohn 1999, Malamud 2012, Križ & Spector 2021): The kids laughed has family of potential meanings, filtered down in context (inspired by the Strongest Meaning Hypothesis of Dalrymple et al. 1994, 1998) to those that are logically strongest and/or most pragmatically informative in light of the QUD.

2. The Trivalence Approach (Križ 2016, Champollion et al. 2019): The kids laughed is undefined when some kids laughed and others didn’t, but can be used anyway when it resolves the QUD in the correct way.

3. The Implicature Approach (Bar-Lev 2021, Chierchia 2022): The kids laughed has essentially existential truth conditions, but can be pragmatically strengthened to a universal interpretation when that interpretation would resolve the QUD.

Rather than choosing among these approaches, this paper explores how the potential for non-maximality is linked to the lexical meaning of the predicate, as shown in Yoon’s clean/dirty contrast.

As Chierchia (2022) notes, definite plurals constitute just one example among many where we observe variable quantificational force in the absence of any overt quantifier; other examples include bare plurals (Carlson 1977, Leslie 2008), locative prepositions (Mador-Haim & Winter 2015), markers of free choice and negative polarity (Giannakidou 2001), and more. Representing a further instance of variable quantificational force in the absence of a quantifier, plural predication also raises questions about distributive versus collective interpretations of the predicate (Link 1983, Roberts 1987, Champollion 2010). On a distributive interpretation (8a), a predicate is inferred to apply individually (to ‘distribute’ down) to each member of a plural subject. On a collective interpretation (8b), the predicate applies to the plural subject as a whole, but not to each member individually.

Most adjectival predicates (including clean, dirty, healthy, sick, and so on) require a distributive interpretation (i.e., are “stubbornly distributive” in the sense of Schwarzschild 2011), presumably because they describe properties that inherently characterize individuals (Roberts 1987, Glass 2018). Collective interpretations of adjectival predicates (8b) are rarer, restricted to certain predicates that can characterize groups above and beyond their individual members, and require more contextual support (Scontras & Goodman 2017, Glass 2018). But as Brisson (2003: 131) observes, both distributive and collective interpretations of a predicate can involve universal or existential (“non-maximal”) interpretations of the definite plural subject.

2.2 Adjectives

Variable quantificational force is also found in other facets of adjective meaning. Gradable adjectives are commonly taken to map individuals to their degree on a scale (Bartsch & Venneman 1972, Cresswell 1977, Kennedy 1997, Kennedy & McNally 2005) – an ordered dimension, such as height for the adjective tall. As observed by Unger (1975), Cruse (1980), Rotstein & Winter (2004), Kennedy & McNally (2005) and Kennedy (2007), some scales are open-ended: height extends infinitely, yielding vague relative gradable adjectives such as tall, which can be modified by relative adverbs such as very and fairly, but resist those describing maxima (perfectly) or minima (slightly). Tall’s antonym short also arguably extends indefinitely (Kennedy 2007: 35), even though an entity could in principle have zero height, as evidenced by the fact that short also cannot easily be modified by perfectly or slightly.

Other scales have a clear maximum standard: a cup can be so clean that it could not be cleaner. Maximum-standard adjectives such as clean can therefore be modified by adverbs describing maximal degrees (perfectly, totally). On the other hand, dirty is a minimum-standard adjective: a cup counts as dirty once it has any noticeable dirt on it. So dirty can be modified by adverbs describing minimal degrees (slightly, partially). While pairs such as clean/dirty have one maximum standard and one minimum standard, there are other pairs such as full/empty where both ends show a maximum standard. Table 1 presents a typology of scale structures for gradable adjectives adapted from Kennedy (2007), as diagnosed by their compatibility with maximum- versus minimum-standard modifiers – although, as observed by Kamoen et al. (2011), such judgments may not be as clear-cut as the literature suggests.

Table 1: Typology of scale structures for gradable adjectives, adapted from Kennedy (2007: 33–36).

Scale structure	Left-side antonym	Right-side antonym
Open at both ends:	(✗slightly, ✗perfectly) short, light, cheap, sad (relative)	(✗slightly, ✗perfectly) tall, heavy, expensive, happy (relative)
Closed at one end:	(✗slightly, ✓perfectly) clean, healthy, straight, safe, dry (maximum-standard)	(✓slightly, ✗perfectly) dirty, sick, bent, dangerous, wet (minimum-standard)
Closed at both ends:	(✗slightly, ✓perfectly) empty, transparent (maximum-standard)	(✗slightly, ✓perfectly) full, opaque (maximum-standard)

We saw above (1) that clean leads to an inference of universal quantification across individuals when applied to a plural subject (all the cups are clean). When applied to a singular subject, it similarly evokes a maximal (universal) degree of cleanliness (9a). On the other hand, dirty leads to an inference of existential quantification across individuals when applied to a plural subject (some of the cups are dirty), as well as an existential degree of dirt when applied to a singular (9b).

As for their semantic representation, Kennedy (2007) proposes that all gradable adjectives (relative adjectives such as tall, absolute maximum-standard adjectives such as clean, and absolute minimum-standard adjectives such as dirty) denote measure functions, mapping an individual to its degree along the adjective’s associated scale. In the positive (i.e., non-comparative, non-superlative) form, gradable adjectives are argued to combine with a silent morpheme POS (Bartsch & Venneman 1972), which requires that this degree exceeds a contextual standard s for what “counts as” tall, clean, or dirty in the context.

All three types of adjectives (relative, maximum-standard, and minimum-standard) have the same semantics, even though they differ in their interpretation: tall is vague, while by default, clean describes a maximal degree of cleanliness and dirty describes a minimal degree of dirt. To explain these differences, Kennedy (2007) points to their scale structure (Table 1). He argues that vague relative adjectives such as tall are anchored only to the discourse context, while absolute maximum-standard adjectives such as clean are anchored to the maximal point of the scale, and absolute minimum-standard adjectives such as dirty are anchored to the minimum. The unified semantics in (10) interacts with the typology of scale structures (Table 1) to explain the varying quantificational force of these adjectives in the realm of degrees.

I adopt this analysis as far as it goes. But it remains an open question, or a matter of lexical idiosyncrasy, which adjectives are associated with what sort of scale and why.

Another quantificational dimension of adjective meaning is found among adjectives describing properties that comprise multiple dimensions³, such as healthy, which involves the state of all one’s organs and biometrics – in contrast to unidimensional adjectives such as clean/dirty, which describe the single dimension of cleanliness/dirt. By default, someone who is healthy is taken to satisfy all these dimensions (11a), while someone who is sick may be deficient in only one (11b).

To represent these adjectives semantically, Sassoon (2013: ex. 9) suggests that healthy by default quantifies universally over dimensions of health, while its antonym sick asserts the existence of at least one dimension lacking health. Sassoon’s quantificational analysis can be synthesized with Kennedy’s semantics for gradable adjectives in the positive (i.e., non-comparative) form to arrive at denotations such as (12).

As observed by Sassoon (2013), the facets of quantification noted in (9) and (11) are connected: someone who is healthy shows a maximal degree of health across all dimensions thereof. Moreover, Yoon (1996) classifies healthy as a “total” predicate like clean, meaning that it is taken to quantify universally across members of a definite plural subject (13a). Thus healthy quantifies universally across degrees, dimensions, and individuals. On the other hand, someone who is sick exceeds a minimum standard of sickness on at least one dimension, and Yoon (1996: 227) argues that (13b) can be used when only some of the kids are sick. Thus sick quantifies existentially across degrees, dimensions, and individuals.

Across the domains of individuals, degrees, and dimensions, it remains an open question why the quantificational force of healthy is universal while that of sick is existential. Yoon (1996) frames it as a lexical idiosyncrasy, which is not explanatory; Krifka (1996) shows that it is not purely lexical in the first place. We are left to wonder: Why then do healthy and sick behave as they do? Which further predicates will pattern in which ways?

My answer will leverage the structure of pairs such as clean/dirty and healthy/sick. Such pairs illustrate that the lexicon of adjectives is organized not just around scales/degrees, but also around antonyms (Sapir 1944, Boucher & Osgood 1969, Cruse 1980, Lehrer & Lehrer 1982, Lehrer 1985) that frame the underlying scale from opposite perspectives. One half of the pair is typically considered “unmarked” (a term critiqued by Haspelmath 2006 for being confusingly polysemous) – in the sense of describing a scale along its increasing dimension (tall describes increasing height, while short describes decreasing height); being morphologically basic rather than complex (even versus uneven); learned earlier by children (Clark 1993: 32); listed earlier in binomial constructions (happy or unhappy is more common than unhappy or happy; Cooper & Ross 1975: 65); more flexible in terms of the contexts that it can occur in (how tall are you? can be used in a range of contexts, while how short are you? is restricted to contexts where shortness is presupposed); more frequent overall; and/or – I argue, crucially – more positive in terms of sentiment. This paper posits that sentiment helps to explain all the adjective-related quantificational inferences described here.

2.3 Sentiment

Researchers have long observed that words carry emotional as well as denotational meaning (Osgood et al. 1957). In psychology, the opposition between good and bad (linked by Boucher & Osgood 1969 to the yin/yang of Eastern philosophy) is thought to organize the human experience at a fundamental level (Taylor 1991, Rozin & Rozyman 2001, Baumeister et al. 2001, Barrett 2006), including the meanings of words. Unlike denotational meaning, sentiment is narrow enough to plot on a single axis, but also connects to broader topics in emotion and cognition.

In natural language processing, a long tradition of work aims to automatically identify the sentiment of text (Pang & Lee 2008). Adjectives in particular are noted as a locus of textual sentiment (Wiebe 2000, Taboada 2016), and the lexicon of adjectives is organized around antonym pairs with opposing sentiment: clean and healthy are desirable, dirty and sick are undesirable.

Sentiment is long recognized in linguistics too. In corpus linguistics, scholars study semantic prosody: patterns of collocational sentiment spanning multiple words, as when set in tends to describe unpleasant states of affairs (Sinclair 1991: 73–74).

In the study of adjectival antonyms in particular, Horn (1989: 274) credits Zimmer (1964) with the observation that negative affixes such as un- tend to attach to positive-sentiment adjectives to form negative-sentiment antonyms (happy/unhappy), connecting morphological markedness to negative sentiment. Lehrer (1985: 399) similarly counts a positive connotation as one of several attributes that tend to characterize the unmarked half of an antonym pair.

Sentiment is important in cognitive linguistics, where researchers aim to connect language to psychology. Paradis et al. (2012) use the Implicit Association Test (Greenwald et al. 1998) to probe the emotional valence of eight antonym pairs of relative gradable adjectives describing physical properties that are not explicitly sentiment-coded, such as large/small, heavy/light, and fast/slow. They find that each antonym pair is strongly divided by sentiment, one half being emotionally positive and the other negative, consistent with the idea from Boucher & Osgood (1969) that antonyms are organized around sentiment opposition. They also find that the positive-sentiment half of the pair is usually the one describing more of the relevant property (more size, more speed), further connecting positive sentiment to the unmarked half of an antonym pair. But they also find exceptions: heavy and thick describe greater weight/size but are found to be more emotionally negative, at least out of context, than their antonyms light and thin.

To complicate matters further, a given adjective type can describe a property that is desirable for some objects but undesirable for others: cold beer is good while cold pizza is bad (Fahrni & Klenner 2008); a thick blanket is cozy while thick fog is dangerous. Using classifications of adjectives from Bierwisch (1989: 88) and Dixon (2004: 3-4), it is especially common among dimensional adjectives (i.e., those describing size) for a given property to vary in desirability depending on what it is applied to. In contrast, among adjectives describing value judgments (beautiful/ugly), non-dimensional physical properties (clean/dirty), and human propensities (clever/stupid), desirability is more consistent across contexts.

Cruse (1980) uses sentiment to explain the scale structure of maximum/minimum-standard pairs such as clean/dirty (Table 1). “The desirable state is zero value of the property” (i.e., zero dirt), he writes (Cruse 1980: 23), “and this is signaled by one of the terms of the opposition” (i.e., clean). Moreover, “any positive value” – of dirt – “represents an unsatisfactory state, and thus is signaled by the other term of the opposition” (dirty). Cruse (1980: 21) describes clean as e-positive (e- for evaluative), dirty as e-negative (using terms from Boucher & Osgood 1969: 1). Cruse (1980: 15) uses the term q-positive (q- for quantity) for adjectives describing a greater quantity of the relevant property. Thus clean is q-negative (describing less dirt) but e-positive (desirable); dirty is q-positive (describing a greater quantity of dirt) but e-negative (undesirable).

As Cruse explains, maximum/minimum-standard pairs such as clean/dirty show different grammatical and inferential patterns from relative gradable antonyms such as tall/short. Relative gradable antonyms such as tall/short allow an intermediate “zone of indifference” of objects that are neither tall nor short (Sapir 1944: 98), while pairs such as clean/dirty arguably exhaust the logical space by describing zero versus any amount of dirt.

As for scales such as full/empty that are closed at both ends (Table 1), Cruse (1980: 15) says these are “relatively infrequent” and would “require quite a separate treatment.” Such pairs are often divided by sentiment (full is more positive than empty), but sentiment does not drive any difference in quantification over degrees, because both antonyms equally describe maximum standards.

Within formal semantics, Cruse 1980 is credited by Rotstein & Winter (2004) for many of the observations defining the class of absolute gradable adjectives like clean/dirty. Rotstein & Winter (2004) inspire the influential analysis by Kennedy (2007) of the lexical semantics of relative versus absolute gradable adjectives, and Kennedy (2007) lays the foundation for the work of Sassoon (2013) on multidimensional absolute adjectives such as healthy/sick. But none of these papers, nor Yoon 1996, mention Cruse’s observation that antonym pairs of absolute gradable adjectives such as clean/dirty and healthy/sick are organized around sentiment. In other words, while clean/dirty and healthy/sick clearly contrast in sentiment, sentiment has not featured prominently in the semantics literature about such adjectives. The effect of sentiment on variable quantification among degrees is discussed by Cruse, but hardly pursued in later work; and its effect across individuals and dimensions has not been noted at all.

Sassoon (2010: 144) does mention sentiment, but only to dismiss it as irrelevant to her point about which half of an antonym pair can appear with measure phrases or ratio comparisons (five feet {tall/*short}, twice as {tall/*short}). Similar to Paradis et al. (2012), she observes exceptional cases such as old/young (or thick/thin) where the adjective describing more of the relevant property is arguably less desirable in some contexts (i.e., q-positive but e-negative in Cruse’s terms), and says that she aims to explain linguistic facts about measure phrases and ratio comparisons that pattern with the adjective describing more of the relevant property, rather than cultural facts about which property is more desirable.

Recently, though, some work in the formal semantics of adjectives has started to confront the role of sentiment. Glauch (2024) studies adjectives carrying both denotational and emotional meaning (“mixed expressives”; Gutzmann 2011) such as shitty, arguing that negative appraisal is key to explaining their felicity conditions. Nouwen (2024) strives to explain why the sentiment of an adjective predicts the meanings of adverbs built from it, as when pleasantly warm denotes moderate warmth while unpleasantly warm denotes an extreme. Gotzner & Mazzarella (2024) show experimentally, as predicted by Ducrot (1973: 125), that negated positive-sentiment adjectives are more likely to be pragmatically strengthened to entail their antonym: not very clean is taken to convey dirty, whereas not very dirty does not necessarily mean clean. As discussed below (Section 3.3.3), Beltrama (2025) explores adjectives describing mildly positive evaluations such as adequate, arguing that they describe a minimally sufficient degree of the relevant property given some goal or norm.

Adjectives with strong sentiment are also often subjective in the sense of Lasersohn (2005), allowing faultless disagreement (Kölbel 2004): two people can disagree about whether roller coasters are fun or scary without either of them being wrong (see, e.g., Stojanovic & Kaiser 2022 for discussion of how subjectivity and sentiment overlap). Subjectivity is found to shape syntax (Scontras et al. 2017, 2019), in that subjective adjectives occur further away from the head noun in multi-adjective attributive strings (fun blue toy is preferred over ?blue fun toy), and more likely to appear in predicative position (The red dress is cute is a more likely utterance than The cute dress is red; Glass 2025). Such work has begun to uncover both grammatical and inferential consequences of the emotional side of the lexicon.

While it has only blossomed recently, this connection has deep historical roots. In work published after his death in 1939, discussing comparatives such as more than versus less than, Sapir (1944: 111-112) observes:

“Here, as in every other phase of linguistic inquiry, we find that the more closely we study actual linguistic forms, the more we are driven to realize that they never express merely static, affectively neutral, concepts and judgments, but classes of concepts and judgments in which nuclear notions, capable of logical definition, are colored by unavowed dynamic and affective determinants.”

Over eighty years later, this paper takes inspiration from Sapir’s insight that the logical and emotional dimensions of language are linked.

3.1 The Anna Karenina Principle

In particular, I argue that such inferences can be rooted in the insight – discussed under many names in diverse fields, sometimes called the Anna Karenina Principle (14) – that for a composite entity or situation to be good, all of its components must be good; equivalently, the entity/situation goes wrong if any component is wrong. This principle links quantification to sentiment.

Ranging from the renowned first line of Tolstoy’s novel Anna Karenina, to insights dating back to Aristotle (15), various thinkers have noted that happiness and success require a conjunction of multiple favorable elements, such that the (disjunctive) absence of any of them suffices for unhappiness/failure. The human geographer Jared Diamond (1997) named the Anna Karenina Principle⁵ and used it to explain why many animals are unsuitable for domestication, and thus why some societies developed faster than others.

While the literary allusion may seem esoteric, I use the name Anna Karenina Principle for consistency with prior work across many fields.

In social psychology, the Anna Karenina Principle is known as negativity dominance (Peeters & Czapinski 1990, Taylor 1991, Rozin & Rozyman 2001, Baumeister et al. 2001): the finding that the overall assessment of a composite object is not just a sum or average of its traits, but is as bad as its worst element. Negativity dominance is reflected in folk sayings such as “a chain is as strong as its weakest link.” It is viscerally illustrated in the domain of food: “one rancid ingredient spoils the finest soup” (Kanouse 1984: 705); an elaborate meal is ruined by a single cockroach, while there is no parallel anti-cockroach whose presence would redeem it (Rozin & Rozyman 2001: 296). Along the same lines, a person who murders once is a murderer, while one must worry often to be considered a worrier (Leslie 2008: 15), because murder is appalling while worrying is benign. A faithful spouse is faithful always, while a single affair qualifies them as unfaithful.⁶ Echoing the literary roots of the Anna Karenina Principle, Rozin & Rozyman (2001) link negativity dominance to the “tragic flaw” from Ancient Greek tragedies: any single defect overcomes all of a hero’s advantages.

Negative events trigger cognitive processing as well as gut emotion, perhaps because people are motivated to figure out how the event could be avoided next time (Taylor 1991: 73). Weiner (1985) finds that bad outcomes prompt greater causal reasoning than good ones, and Knobe (2003) shows that people attribute greater intention to agents who cause bad outcomes rather than good ones.

Negativity dominance can also be seen as the flip side of the Pollyanna Hypothesis of Boucher & Osgood (1969), named after a famously optimistic fictional heroine. Boucher & Osgood (1969: 1) observe that positive-sentiment word tokens are more common in text (a widely replicated finding; Dodds et al. 2015), which they take to show that people tend to “look on the bright side of life.” While this finding may seem to conflict with negativity dominance, Taylor (1991: 77) weaves both observations together by suggesting that the normal state of affairs is mildly positive, which is why positive-sentiment word tokens are more frequent, but also why any single bad element dominates our attention by disrupting this homeostasis.

Negativity dominance is connected to negative differentiation: the observation that negative experiences are more diverse and distinctive than positive ones (Rozin & Rozyman 2001, Baumeister et al. 2001), as presaged by Tolstoy and Aristotle. In astrobiology, it is noted that the emergence of life requires various parameters – heat, atmospheric chemicals, and so on – to all fall within a habitable “Goldilocks zone” (Gonzalez et al. 2001): not too much, not too little, representing the “doctrine of the [golden] mean” from Aristotle (Kraut 2022) and Confucius (Legge 1861). A pleasant meal must likewise be neither be too spicy nor too bland (Unkelbach et al. 2020). Even on a single dimension such as temperature or spice, there are two ways to fail (too much, too little), but only one way to succeed (just right). As previewed above, Nouwen (2024) uses the “Goldilocks zone” to explain why adverbs built from positive-sentiment adjectives denote intermediate degrees (pleasantly warm) while those built from negative-sentiment adjectives denote extremes (unpleasantly warm).

Moving to multiple dimensions, if any bad element suffices for a composite object to be deemed bad, then bad things are also more distinctive because each bad thing may have its own unique reason for badness.

In pragmatics, Horn (2022: 827) invokes the Anna Karenina Principle to explain why, when two clauses are conjoined, it is often the negatively-valenced one that occupies the position of information-structural prominence at the end of the sentence. He spotlights the order of conjuncts in Tolstoy’s (15a) as well as another famous literary opening:

Horn suggests that negative information receives structural prominence because it is more distinctive and interesting. Social psychologists might add that negative information dominates our assessment of a composite.

In semantics, Glass (2023) uses the Anna Karenina Principle to explain why the seemingly neutral verb cause tends to occur with negative-sentiment complements in corpora (cause problems, cause cancer). Glass (2023) draws on findings from social psychology (Kun & Weiner 1973) that successes are taken to require multiple necessary-but-individually-insufficient factors (hard work and luck), such that the absence of any of them suffices for failure, and casts the Anna Karenina Principle in terms of causal models inspired by Pearl (2009). But the key insight extends beyond events and causal relations: more generally, good things – composite objects as well as events – require a conjunction of good components, such that the (disjunctive) absence of any of them suffices for badness.

3.2 The Anna Karenina Principle across individuals

This paper aims to use the Anna Karenina Principle to explain Yoon’s clean/dirty contrast in the realm of individuals, via pragmatic reasoning about how the QUD is addressed by a sentence attributing a predicate to a definite plural subject.

Beginning with the semantic representation of such sentences, we saw above (Section 2.1) that the literature offers various possible analyses for predicates applied to definite plurals. But all approaches agree that such sentences allow both universal and existential interpretations as determined by the QUD. For concreteness, I illustrate with the analysis of Champollion et al. (2019), which builds on Križ (2016). These authors argue that a sentence such as (17) is true in a world w_{all open} in which all the doors are open, false in a world w_{none open} in which none of them are open, and undefined in a “mixed” scenario w_{some open} where some are open and others closed. Here, of course, some must read as “some but not all.”

Although its truth value is undefined in a mixed scenario, Champollion et al. (2019) argue, (17) can be used felicitously when it’s “true enough” (echoing Lasersohn 1999, p. 523) – when the worlds are indistinguishable with respect to the QUD (w_{some open}≈w_{all open}), so that (17) answers the QUD in the same way whether it’s actually true, or undefined but “true enough.”

Returning to the example from Krifka (1996), let us imagine that the QUD asks whether the speaker is able to reach the safe. If the safe is behind three side-by-side doors, such that the speaker can pass through any of them to reach it, then w_{some open}≈w_{all open} because the QUD about reaching the safe is resolved affirmatively in both scenarios. On the other hand, if the safe is behind three sequential doors, then w_{some open}≉w_{all open} because these situations would resolve the QUD in different ways: the speaker can’t reach the safe in w_{some open}, but can reach it in w_{all open}. This analysis explains why (17) can be interpreted as universal or existential depending on the arrangement of the doors. It also explains why, as noted above, the homogeneous w_all interpretation is strongly preferred, because the sentence is actually true in that scenario. In contrast, the heterogeneous/mixed interpretation w_some requires more context and may be met with confusion because speakers have to recover the QUD to decide whether the sentence counts as true enough in context even though it is technically undefined.

The same intuition could be re-worked in other frameworks. If (17) is compatible with a variety of meanings as determined by the QUD, then such reasoning could explain which interpretations are available. And if (17) has essentially existential truth conditions that can be strengthened to universal ones in light of the QUD, then this reasoning could explain when such strengthening is needed. Most important is the observation that the some-but-not-all interpretation may be available or unavailable, depending on whether w_some and w_all would give equivalent or distinct answers to the QUD. Secondarily, one would have to explain why the homogeneous w_all interpretation is favored above the mixed w_some interpretation, which on the theory of Križ (2016) and Champollion et al. (2019) arises because the sentence is true in the w_all scenario but undefined in the w_some one. If the sentence is instead analyzed to be true in the w_some scenario, then these facts would need a different explanation.

In any case, such an analysis captures how universal versus existential interpretations of definite plurals depend on the QUD. But here the QUD is specific to the imagined context about the safe, and the variability hinges on the stipulated position of the doors. Next, I propose to generalize this analysis to contexts such as Yoon’s clean/dirty example, where the QUD is not stated explicitly and the variability hinges on the sentiment of the adjective.

For Roberts (2012), conversations circle around a Big Question, “What is the way things are?,” raising a series of nested sub-questions that interlocutors pursue as guided by their domain goals. Roberts thus elaborates what Grice (1989: 45) means when he refers to the “current purposes of the exchange.” The sub-question, “Is the current situation okay?,” fits naturally under the overarching Big Question and arguably serves various domain goals. Assuming via the Anna Karenina Principle that a good thing must be all good while any bad element makes it bad, asking whether the current situation is okay amounts to asking whether everything is okay, or whether something is wrong.

In contrast, the opposite set of questions – whether something is okay; whether everything is wrong – are less useful. It is under-informative to know whether something is okay, because this still does not tell us whether the whole situation is okay. It is over-informative to know whether everything is wrong, because a situation can go wrong even without every part of it being so.

As previewed above, the questions (18) are thus predicted to be vastly more common than their opposites (19) in any corpus, and presumably also far more likely as implicit Questions Under Discussion across a variety of contexts.

As a result, when readers scour their background knowledge to interpret a decontextualized example such as (20), they may reason that “Is the current situation okay?” (18) is a likely QUD, and that this question is answered affirmatively by a sentence with a positive-sentiment predicate. The sentiment of the adjective offers a clue as to how it may address this common QUD.

The Anna Karenina Principle tells us that a situation is only okay if it is all okay. A scenario with all clean cups is therefore not equivalent to one with some clean cups (w_all≉w_some) because these would provide different resolutions to the QUD of whether the current situation is okay (Table 2). Thus (20a) is only true in the w_{all clean} scenario, not “true enough” in the mixed w_{some clean} scenario. So (20a) is interpreted universally, which is also the default interpretation for sentences with definite plural subjects overall.

On the other hand, readers may assume that this QUD (“Is the current situation okay?”) is answered negatively by a sentence with a negative-sentiment predicate. According to the Anna Karenina Principle, the situation is not okay if any part of it is not okay. Thus, a situation with some dirty cups is equivalent to one with all dirty cups (w_{some dirty}≈w_{all dirty}) because the situation is not okay in either case (Table 2). (20b) is thus “true enough” in the mixed w_{some dirty} scenario, so (20b) can receive an existential interpretation. As noted above, speakers are more hesitant to give judgments in such scenarios when the cups are “mixed” with respect to cleanliness, so the existential interpretation is more fragile (“varied and guarded” per Rooth 1987) than the universal one. But this analysis at least explains why this existential interpretation is more likely for negative-sentiment adjectives.

This proposed explanation begins by positing “Is the current situation okay?” as the QUD. One might wonder, what if the QUD instead asked about the mathematical facts of the situation (21a)? In fact, (21b) is a very awkward response to that (explicit) QUD, perhaps because the answer to the QUD typically carries prosodic focus (Roberts 2012), whereas the quantificational effect of (21b) cannot bear focus because it arises implicitly.

This explanation is intended to capture the behavior of clean/dirty in a typical context in which clean is desirable, dirty undesirable, consistent with the sentiment-linked dimension of their meaning. For example, while setting up for a party, we want clean cups ready for guests. Such typical contexts are most likely to be imagined by hearers/readers out of context, explaining the inference patterns observed by Yoon (1996). But in a highly atypical context such as (22) in which dirtiness is for some reason desired (inspired by Lasersohn 1999, who also discusses the interpretation of definite plurals in medical testing contexts), then we predict the inferences to flip: dirty should be interpreted as universal, while clean should be more likely to receive an existential interpretation.

Such contextual variability is predicted on any theory that derives the interpretation of definite plurals from pragmatic reasoning about the QUD. Originally, Krifka’s point about the importance of the QUD was cast in tension with Yoon’s observation about the role of the lexical predicate. But on my proposal, both elements work together: in typical contexts, the type-level sentiment of the lexical predicate usually determines the desirability of the property described by a given token of that adjective, which in turn interacts with the Anna Karenina Principle to shape the quantificational force of the answer to the QUD about whether the current situation is okay. This analysis captures the clean/dirty contrast observed by Yoon (1996) in typical contexts as well as its exceptions (22) in highly atypical ones.

Beyond maximum- versus minimum-standard pairs such as clean/dirty, the same explanation is predicted to extend to relative-standard adjectives as in (23), for example in a context where a CEO is asking how ten employees feel about a new policy. Here too, the positive-sentiment adjective answers “Yes” to the question of whether the current situation is okay, which – according to the Anna Karenina Principle – means it is all okay, favoring a universal interpretation. The negative-sentiment version answers “No” to that question, which – according to the Anna Karenina Principle – means it is at least somewhat bad, yielding a greater potential for an existential interpretation.

Beyond adjectival antonym pairs, the same logic extends to predicates of any lexical category that can be divided into desirable/undesirable pairs, either based on the sentiment of their type-level meaning (succeed/fail, win/lose, love/hate, spare/damage – the latter of which is mentioned by Yoon 1996: 217) or on the particular discourse context of a given token (walk/run in a situation where one or the other is desired). While the overall prediction is broader, this paper focuses on adjectives to explain the original data from Yoon (1996), and to use antonym-swapped minimal pairs to explore at scale how various inferences are grounded in the desirability of the properties that they describe.

But when two antonyms do not clearly differ in the desirability of the property that they describe, the Anna Karenina Principle cannot explain their interpretation. Both awake and asleep have middling ratings in the Hedonometer sentiment lexicon of Dodds et al. (2011, 2015): 5.74 and 6.14, respectively, on a 1-to-9 scale. Here, it may be less clear out of context which antonym describes a more desirable property, and thus less clear how each half of the antonym pair would address the QUD “Is the current situation okay?”, yielding no clear prediction about how people will interpret such sentences. In other words, my proposed analysis of Yoon’s clean/dirty contrast makes predictions about the hundreds of predicate pairs that clearly diverge in sentiment, but not about pairs describing properties whose desirability is indistinguishable. In such cases, one would need to know which one is more desirable in context.

In sum, we predict:

The statistical effect may be subtle, because the universal interpretation is a strong default; because the existential interpretation describes a “mixed” scenario in which speakers hesitate to provide clear judgments; because human judgment studies are almost never categorical in the first place; because some antonym pairs (awake/asleep) do not clearly differ in sentiment; and because speakers may be somewhat uncertain about the implicit QUD. But the direction of the effect is predicted clearly by the Anna Karenina Principle in combination with pragmatic reasoning about how the sentence answers the QUD of whether the current situation is okay or not.

3.3 The Anna Karenina Principle across degrees

Beyond individuals, the Anna Karenina Principle also yields predictions about the inferential behavior of adjectives in the domains of degrees and dimensions, this time through the mechanism of lexicalization – the mapping between concepts and word types (Talmy 1985, Brinton & Traugott 2005, Levin & Rappaport Hovav 2019) – rather than pragmatic reasoning about how the utterance of a particular token answers the QUD.

3.4 The Anna Karenina Principle across dimensions

Regarding dimensions, the same lexicalization pressure would predict that speakers find it useful to discuss maximal health (across all dimensions) versus its absence (on any dimension), but might find it less useful to discuss health along at least one dimension. Here too, assuming that the lexicon is shaped by communicative utility, it is argued to encode the distinctions that speakers find most useful following the Anna Karenina Principle combined with the common QUD of whether things are okay.

Sassoon (2013) suggests that her “conjunctive” adjectives such as healthy are also maximum-standard adjectives, while their “disjunctive” antonyms such as sick are minimum-standard adjectives. She offers (in her Section 1.3.2) various reasons that adjectives show matching quantificational force across both degrees and dimensions (universal/maximal for healthy, minimal/existential for sick), but she does not invoke the desirability of the property denoted by the adjective, nor does she explain which adjectives would show which type of quantificational force and why.

Sassoon (2013) also notes that some relative-standard adjectives can vary contextually as to whether they quantify universally/conjunctively or existentially/disjunctively across dimensions: a person can be intelligent/stupid in every way, or intelligent/stupid in a single area such as mathematics. Just as these adjectives pick out a vague, relative standard for what counts as intelligent/stupid in context, they also vary contextually in their quantificational force across dimensions. But here, because both antonyms equally can be conjunctive or disjunctive, there is no asymmetry to be explained.

Regarding quantification across dimensions, the prediction stemming from the Anna Karenina Principle thus applies only to the small handful of adjectives such as healthy/sick for which one antonym describes a maximal degree of a multidimensional property such as health – not a unidimensional one such as cleanliness or height, nor one that can be multidimensional or unidimensional depending on the context, such as intelligence/stupidity. The semantic representation (sketched in Section 2.2) does not matter as much as the phenomenon itself: that some adjectives quantify universally/conjunctively across dimensions, while their antonyms quantify existentially/disjunctively.

Representing this limited set of adjectives, My grandma is healthy is easily taken to provide an affirmative answer to the common QUD of whether the current situation is okay. Following the Anna Karenina Principle, the current situation is only okay if it is all okay; that is, Grandma is healthy along all dimensions. With respect to this QUD, a scenario in which Grandma is healthy across all dimensions is not equivalent to one in which she is healthy on some dimensions but sick on another (w_{all healthy}≉w_{some health}) because the QUD is resolved differently in these scenarios. Therefore, My grandma is healthy conveys that she is healthy along all dimensions.

In contrast, My grandma is sick is taken to provide a negative answer to the common QUD of whether the current situation is okay. Following the Anna Karenina Principle, the current situation is not okay if anything is wrong. Here, the QUD is resolved in the same way in a scenario in which Grandma is sick along some dimension as when she is sick along all dimensions (w_{some sickness}≈w_{all sick}; Table 6). So My grandma is sick conveys that she is sick along at least some dimension.

On this analysis, conjunctive positive-sentiment adjectives and disjunctive negative-sentiment ones are therefore more useful than their alternatives (hypothetical conjunctive negative-sentiment adjectives or disjunctive positive-sentiment ones), thus more worthy of being lexicalized. We therefore predict:

In sum, when combined with pragmatic reasoning about how sentences answer the common QUD of whether the current situation is okay, the Anna Karenina Principle grounds a series of hypotheses that aim to explain why sentiment shapes quantification among adjectives across individuals, degrees, and dimensions. The next section tests these predictions empirically. To foreshadow, these hypotheses are all indeed consistent with the data reported below.

4.1 Quantification over individuals

In the domain of individuals, I elicited binary-choice inference judgments of the form in (27) – where each item was randomly assigned to one half or the other of the antonym pair (clean or dirty):

These items were constructed from 55 adjectival antonym pairs (110 adjectives) drawn from the literature (Lehrer 1985, Yoon 1996, Rotstein & Winter 2004, Kennedy 2007, Kamoen et al. 2011), given in full in the Appendix. These adjectives comprise not just maximum/minimum-standard pairs such as clean/dirty, but also relative/relative (happy/sad) and maximum/maximum pairs (full/empty). Any antonym pair mentioned in the semantics literature was included, as long as both halves of the pair have sentiment ratings in the Hedonometer lexicon. I excluded pairs including quite rare words, such as ferocious/meek (mentioned by Lehrer 1985) and pure/impure (Rotstein & Winter 2004, Kennedy 2007), because at least one member was absent from the Hedonometer.

For each antonym pair, I chose a plural noun that makes sense with both halves of the antonym pair as something that an individual might be responsible for keeping in good condition – typically artifacts (clean/dirty predicated of cups) and social roles (happy/sad predicated of customers). As in Yoon’s original example, (27) thus evokes the QUD of whether the cups are in good condition or not – that is, whether the current situation is okay. This QUD is not stated explicitly, because my hypothesis is that it is commonly inferred out of context, as in Yoon’s original example.

These items were intermixed with fourteen fillers using sentiment-neutral non-gradable adjectives. Seven used colors such as orange, seven used materials such as wooden. Combining the 55 target items with these fillers, there were 69 questions overall. To prevent fatigue, each experimental participant judged 40 randomized items, of which approximately 8 (20%) were fillers.

With approval from the Institutional Review Board at Georgia Tech, these items were presented in a random order (with target items randomly assigned to one half or the other of the antonym pair) to 40 participants (self-identified native English speakers located in the United States) recruited for pay on the Prolific web platform.

To test the hypothesis (28), I ran a mixed-effects logistic regression using the lme4 package (Bates et al. 2015) in R (R Core Team 2012) predicting binary response (all or some) as a function of sentiment, using random intercepts for each participant and each antonym pair.

As predicted (Figure 1), adjectives with higher sentiment (clean) are more likely to receive the universal all interpretation (β = 0.14,z = 2.60, p < 0.01), meaning that every one-point increase in sentiment corresponds to a 15% increase (i.e., exp(0.14)=1.15) in the odds of a universal interpretation. The universal interpretation is a strong default overall, chosen in 80% of all target items, reflecting the fact (Section 2.1) that readers/hearers prefer to imagine all members of the definite plural to be homogeneous with respect to the property described by the adjective, and are reluctant to assign “mixed” interpretations. The effect size of the predicate’s sentiment is relatively small (Figure 1), especially in comparison to the huge variability found across participants (random effect variance = 7.94 on a log-odds scale), again perhaps because people resist interpretations describing “mixed” scenarios and because they may be somewhat uncertain about the QUD or how it is answered by adjectives of middling sentiment. But the default universal interpretation is less likely for negative-sentiment adjectives, as predicted by (28).

Figure 1: Percentage of all (versus some) responses for each predicate as a function of sentiment.

Overall, the data are consistent with this paper’s proposed explanation for Yoon’s clean/dirty contrast: that the sentiment of an adjective combines with the Anna Karenina Principle to shape our expectations about how an assertion answers the QUD of whether the current situation is okay.

4.2 Quantification over degrees

In the domain of degrees, I explored modifiers of the same 55 antonym pairs (110 adjective types) in corpus data. Beyond a few clear headline examples such as clean/dirty, it is not easy to label these adjective types for whether their associated scales are minimum-standard (dirty), maximum-standard (clean), or open (tall). Studying the acceptability of adjectives with modifiers said to indicate maximum-standard scale structure (e.g., almost/completely easy), Kamoen et al. (2011) find that acceptability varies widely depending on the surrounding textual context of the token, which makes it difficult to label decontextualized adjective types. If scale structure were labeled by the author, the data might be further confounded by confirmation bias. So instead of labeling scale structure by hand, I took inspiration from Sassoon (2013) in approximating it from the frequency of modifiers in corpus data.

Using one hundred million words of English text from the AskReddit web forum from January 2018 (Baumgartner et al. 2020)¹¹, I extracted all tokens (tagged as adjectives by the part-of-speech tagger built into the NLTK Natural Language Toolkit of Loper & Bird 2002) of these 110 adjective types. For each adjective token, I checked its syntactic dependents according to a dependency parser (Honnibal & Johnson 2015) and recorded its modifiers: dependents part-of-speech-tagged as adverbs; those linked to the adjective by the npadvmod dependency to capture (a) bit, which is tagged as a noun phrase; and the adjective-modifying adjective little. Of all modifiers, I counted those that the literature classifies as indicating a maximum, minimum, or relative standard (Rotstein & Winter 2004, Kennedy & McNally 2005, Kennedy 2007, Kamoen et al. 2011, Sassoon 2013):

Adapting an idea from Sassoon (2013), I computed the number of minimum-standard modifiers as a percentage of all (maximum-, minimum-, and relative-standard) modifiers. There are 24 tokens of dirty with minimum-standard modifiers, out of 239 tokens of dirty with any of the modifiers listed in (29), so dirty appears with minimum-standard modifiers 24/239=10% of the time (versus 5/440=1% for clean). For comparison, I also computed the number of maximum-standard modifiers as a percentage of all modifiers (82/440=19% for clean, 2/239 ≤ 1% for dirty).

To test the hypothesis (30), I conducted a linear regression predicting the percentage of minimum-standard modifiers as a function of sentiment.

As predicted, positive-sentiment adjectives are less often modified by minimum-standard modifiers (Figure 2; β = –1.6%, t(108) = –3.53,p < 0.001, adjusted R² = 0.095). In other words, every one-point increase in a predicate’s sentiment means that its percentage of minimum-standard modifiers is predicted to decrease by 1.6 on a 100-point scale (the meaning of β), and sentiment explains almost ten percent of the variability in the percentage of minimum-standard modifiers that we find for each predicate (the meaning of adjusted R²).

The hypothesis (30) makes no predictions about the behavior of maximum-standard modifiers. And indeed (Figure 2), in another linear regression predicting the percentage of such maximum-standard modifiers as a function of sentiment, sentiment has no significant effect; some adjectives favoring maximum-standard modifiers are positive in valence (clean) while others are negative (worthless, impossible, empty).

Overall, the data are consistent with the hypothesis (30) that the lexicon is shaped by communicative utility, as shaped in turn by the Anna Karenina Principle in combination with the common QUD of whether the current situation is okay.

Figure 2: Frequency of minimum- and maximum-standard modifiers for each predicate as a function of sentiment.

4.3 Quantification over dimensions

In the domain of dimensions, I gathered six adjectives (good, healthy, identical, normal, typical, similar) that Sassoon (2013) deems to be conjunctive multidimensional adjectives, and four more (bad, sick, abnormal, different) that she classifies as disjunctive.¹² Healthy is conjunctive, quantifying universally over all dimensions of health (all organs, all biometrics), while sick is disjunctive, quantifying existentially over any dimension thereof.

To test (31), I compared the sentiment ratings of Sassoon’s conjunctive adjectives versus disjunctive adjectives.

As predicted (Figure 3), the conjunctive adjectives such as healthy are far higher in sentiment (mean = 6.35 out of 9) than the disjunctive ones such as sick (mean = 3.43). These results are significant in Welch’s (Welch 1947) unpaired¹³, two-sided t test (t = 3.43 with 5 degrees of freedom; p = 0.019).

Figure 3: Sentiment of multidimensional adjectives classified by Sassoon (2013) as conjunctive versus disjunctive.

These findings are consistent with the hypothesis (31) that among adjectives describing multidimensional properties, the lexicon encodes the distinctions that are most useful according to the Anna Karenina Principle alongside the common QUD, “Is the current situation okay?”

4.4 Discussion

Aiming to explain Yoon’s clean/dirty contrast in the realm of individuals described by definite plurals, this paper leverages the Anna Karenina Principle, alongside the QUD of whether the current situation is okay, to predict that positive-sentiment adjectives should be interpreted as universal while negative-sentiment adjectives should be more likely to allow an existential interpretation. Consistent with this hypothesis, a judgment study of 55 antonym pairs (110 adjective types) finds that existential interpretations of definite plurals are more likely for negative-sentiment adjectives: that is, The cups are dirty is more likely to be taken to describe some-but-not-all cups.

The proposed explanation for Yoon’s clean/dirty contrast makes predictions that extend beyond definite plurals. Among degrees, I find (among the same 110 adjective types) that negative-sentiment adjectives are more often modified by minimum-standard modifiers in corpus data (e.g., slightly dirty). These results are consistent with the suggestion from Cruse (1980) that the lexicon encodes the distinctions that are most useful according to the Anna Karenina Principle with respect to the common QUD of whether the current situation is okay. Among dimensions, I find (in a study of ten adjective types) that the adjectives that are interpreted as conjunctive/universal (healthy) are positive in sentiment, while those that are interpreted as disjunctive/existential are more negative (sick). These results are consistent with the same hypothesized lexicalization pressure.

In other words, the Anna Karenina Principle is argued to shape both token-level pragmatic inferences (the interpretation of definite plurals), and type-level semantic features (the distinctions between maximum- versus minimum-standard adjectives, and between conjunctive versus disjunctive multidimensional ones). It is a point in favor of the proposed analysis that it explains such diverse phenomena.

Beyond adjectives, the same reasoning might explain observations from Leslie (2008) about generic sentences describing dangerous properties, as when (32) is judged true even though only a tiny percentage of mosquitoes are infected with West Nile.

Leslie (2008: 15) suggests that (32) is judged true because it describes a “harmful, dangerous, or appalling” property that one would aim to avoid. To generalize this insight, I would suggest that (32) answers “No” to the implicit question, “Are mosquitoes safe/okay?” even though it describes a tiny minority of mosquitoes; following the Anna Karenina Principle, a safe situation is perfectly safe, while any amount of danger makes it unsafe.

More broadly, I suggest, clean and dirty are just the tip of the iceberg. Future work may find that further inferences studied in semantics are also connected to the emotional valence of the properties described in language.