5 Scalar Implicature Theory

Spector (2007) puts forward a view that is similar in nature to Sauerland’s (2003), but instead of saying that the competition is always in the presuppositional domain, he analyzes plurality inferences of bare plurals as scalar implicatures.

Example 5.1

The customer bought books about climate change

Basic idea:

• By assumption, plural nouns are semantically number neutral, which straightforwardly accounts for unmarked plurals.
• The plurality inference is a scalar implicature.

When a plural noun phrase occurs in a presuppositional context, that’ll presumably feed into anti-presupposition, although Spector (2007) himself does not discuss such cases.

Example 5.2

The emergency exits of the building are clearly indicated.

5.1 ‘Exactly one’

If the plurality inference is a scalar implicature, it should be based on an alternative utterance that is somehow stronger in meaning. What is the competitor? Conceptually it would be nice if it were the singular version of the sentence.

Example 5.3

1. The customer bought a book about climate change
2. The customer bought books about climate change

But notice that these two sentences are semantically equivalent on the assumption that plural is semantically number neutral. Then, there is no way of deriving a scalar implicature!

Suppose that the following sentence is a competitor.

Example 5.4

The customer bought exactly one book about climate change.

Then we derive the correct inferences:

• Example 5.4 asymmetrically entails Example 5.3b. Suppose the speaker said Example 5.3b. Why didn’t they say Example 5.4? —Because they don’t believe that it is true.

• In Downward Entailing contexts, the entailment is reversed. So a bare plural does not trigger an scalar implicature.

• We can also account for the partial plurality inference of bare plurals in universal statements:
  Example 5.5

  1. Every student read books.
  2. Every student read exactly one book.

  The scalar implicature will be that not every student read exactly one book. Together with the semantics of 5.5, it means every student read at least one book and at least one of them read multiple books.

• This furthermore accounts for bare plurals in non-monotonic contexts.
  Example 5.6

  1. Exactly one customer bought books.
  2. Exactly one customer bought exactly one book.

  There is no entailment here, but suppose that a scalar implicature can be derived from logically independent alternatives. Then Example 5.6a will have a scalar implicature that 5.6b is not true. Then, the overall entailment is that there is exactly one customer who bought any books, and it’s not true that exactly one customer bought exactly one book.

5.2 Higher order implicatures

But there is something conceptually weird about assuming that books and exactly one book compete.

Instead, Spector (2007) proposes that Example 5.7b competes with Example 5.7a with its scalar implicature. That is, Example 5.7a can have a scalar implicature that the customer bought exactly one book about climate change, which makes it essentially equivalent to Example 5.4.

Example 5.7

1. The customer bought a book about climate change.
2. The customer bought books about climate change.

How to derive the scalar implicature of 5.7a is an interesting question. Spector (2007) suggests that several is an alternative to a:

Example 5.8

The customer bought several books about climbate change.

But it’s not so clear to me if the negation of this sentence means the customer bought at most one book.

In any event, the analysis makes different predictions for quantified examples. For Example 5.9, it works ok.

Example 5.9

Exactly one student read books.

By assumption, this competes with 5.10a, which has as a scalar implicature the negation of Example 5.10b.

Example 5.10

1. Exactly one student read a book.
2. Exactly one student read several books.

Example 5.10a with its scalar implicature means: Exactly one student read a book and that student read exactly one book. Example 5.9 has the negation of this as its scalar implicature, so it means: Exactly one student read a book and that student read more than one book.

However, the account makes a different prediction for bare plurals under universals. Example 5.11b has its scalar implicature the negaton of Example 5.11c. So it effectively means: Every student read at least one book and at least one student read exactly one book. The negation of this is the scalar implicature of Example 5.11a, so it effectively means: Every student read at least one book and no student read exactly one book, i.e. every student read multiple books.

Example 5.11

1. Every student read books.
2. Every student read a book.
3. Every student read several books.

This is perhaps a reading of Example 5.11a, but it’s not the reading we are after.

Spector (2007) attempts to fix this problem by considering some as an alternative to every.

5.3 Embedded scalar implicatures

Zweig (2009), Ivlieva (2013), and Mayr (2015) explore the possibility that the plurality scalar implicature can be computed at a subconstituent level. Recall that the overall meanings of Examples 5.12 are equivalent.

Example 5.12

1. The boy broke cups
2. The boy broke a cup.

Mayr (2015) points out that at the nominal level, there is a truth-conditional difference:

Example 5.13

1. 〖cup〗= λx. x is a singular cup
2. 〖cups〗= λx. x is a singular cup or a plurality consisting of cups

If a scalar implicature can be computed at this level, then we can derive a reading of the plural noun that it’s true of multiple cups:

Example 5.14

λx. x is a plurality consisting of cups

Similarly Zweig (2009) discusses a similar possibilty of drawing a scalar implicature at the event-level (see also Ivlieva 2013).

Example 5.15

1. 〖John broke a cup〗= λe. there is a singular cup that John breaks in e
2. 〖John broke cups〗= λe. there is a singular cup that John breaks in e or there is a plurality of cups each of which John breaks in a subevent of e

These two predicates of events are semantically distinct and we could derive the scalar implicature for Example 5.15:

Example 5.16

λe. there is a plurality of cups each of which John breaks in a subevent of e

These theories that rely on embedded computations of scalar implicature need to be augmented by a theory of the distribution of embedded implicatures, because in Downward Entailing contexts, bare plurals should not give rise to plurality inferences. There are some attempts to explicate the distribution of embedded implicatures (e.g. Fox and Spector 2018).

5.4 Pragmatics of Discourse Referents

In Sudo (2019), I propose that plurality inferences of bare plurals can be understood simply as scalar implicatures generated by discourse referents.

5.4.1 Discourse Referents

Indefinites feed into anaphora in a particular way.

Example 5.17

1. John has a wife. She is a lingiust.
2. John is married. ??She is a linguist.

Example 5.18

1. One of the ten marbles is not in the bag. It’s probably under the sofa.
2. Nine of the ten marbles are in the bag. ??It’s probably under the sofa. (Heim 1982)

To account for this, Karttunen (1976) introduced the idea of discourse referents.

• Indefinites introduce new discourse referents.
• They carry some information as to what they refer to.
• Anaphoric elements refer back to discourse referents.
• Certain operators (e.g. negation) shield discourse referents from being accessed from outside.

Example 5.19

1. Bill saw a unicorn. It had a gold mane.
2. No one saw a unicorn. #It had a gold mane.

Standardly discourse referents are formalized by variables in dynamic semantics (e.g. Heim 1982) and Discourse Referent Theory (e.g. Kamp 1981), but the idea of discourse referents is compatible with situation semantics as well (e.g. Elbourne 2005). As Karttunen (1976) stresses, discourse referents are something that any semantic theory wants to be able to account for.

The idea of discourse referents has been extended to various quantificational expressions beyond indefinites, especially in the context of dynamic semantics (Van den Berg 1996; Nouwen 2003; Brasoveanu 2007).

Example 5.20

1. Exactly one student passed. She solved difficult problems.
2. Most students passed. They studied very hard.

These dynamic theories discuss complex interactions between discourse referents in quantificational contexts (‘quantificational subordination’).

Example 5.21

1. Every PhD student of mine wrote a long paper. ??It was full of typos.
2. Every PhD student of mine wrote a long paper. They all submitted it to Linguistics & Philosophy.

5.4.2 Plurality Inference as Quantity Implicature

Once we have discourse referents in our semantics, we should wonder what kind of pragmatics we do with them. I claim that the plurality inference of bare plurals can be analyzed as a quantity implicature.

This is natural given that the Maxim of Quantity is very general and about informativity,

• Make your contribution as informative as is required (for the current purposes of the exchange).
• Do not make your contribution more informative than is required.

Now, consider simple examples. As with other Scalar Implicature Theories, I assume that a singular noun is semantically singular, while a plural noun is number neutral. What this means in this context is that a singular indefinite introduces a discourse referent whose possible referents are all singular entities, while a bare plural introduces a discourse referent whose possible referents are singular entities or puralities.

Example 5.22

1. The boy broke a glass.
2. The boy broke glasses.

These two sentences are truth-conditionally equivalent, but are not identical with respect to the discourse referents.

• Example 5.22a introduces a discourse referent whose possible values are \(g_1, g_2, g_3, \dots\).
• Example 5.22b introduces a discourse referent whose possible values are \(g_1, g_2, g_3, \dots, g_1\oplus g_2, g_1\oplus g_3, g_2\oplus g_3, \dots\).

Then Example 5.22a contains more information, because you have less possible values for the discourse referent (you know more about what the discourse referent represents). If the cooperative speaker utters 5.22b, then the hearer reasons that what the alternative, 5.22a, means is not what the speaker intends, as in ohter cases of scalar implicature. Consequently, the scalar implicature is drawn that the discourse referent refers to pluralities.

5.4.3 Embedding

This analysis makes good predictions for bare plurals occuring in various embedded contexts. I offer a formally detailed theory couched in dynamic semantics with selective generalized quantifiers in Sudo (2019).

Negation makes discourse referents inaccessible from outside, so the following two setnences have no discourse referent to reason about. Consequently they are semantically identical, so no scalar implicature is drawn.

Example 5.23

1. The boy didn’t break a glass.
2. The boy didn’t break glasses.

(Theoretically double negation introduces some issues, but since I’m not presenting the formal details here, I will skip it.)

Similarly, the theory seems to work fine for other connectives, but they involve some additional complications, e.g. conditionals give rise to modal subordination, disjunction has its own scalar implicature, so I will not discuss them here.

We can also deal with quantifiers.

Example 5.24

1. Exactly one student wrote a paper.
2. Exactly one student wrote papers.

• Example 5.24a introduces one discourse referent whose possible values are singular students, and another discourse referent whose possible values are singular papers.
• Example 5.24b introduces one discourse referent whose possible values are singular students, and another discourse referent whose possible values are singular papers or pluralities consisting of papers.

Then by the same reason as above, Example 5.24b will generate a scalar implicature that singular papers are not possible values of the second discourse referent.

And we derive the partial plurality inference with universal quantifiers.

Example 5.25

1. Every student wrote a long paper..
2. Every student wrote long papers.

Here, we would like to encode a dependency relation between the students and the papers (‘who wrote what’), so that we can correctly capture anaphora like the following.

Example 5.26

They all submitted it/them to Linguistics & Philosophy.

The sentences in Example 5.25 introduce two discourse referents, one for students \(d_s\), one for papers \(d_p\), and each possible value of \(d_s\) is paired with a possible value of \(d_p\) and each possible value of \(d_p\) is paired with a possible value of \(d_s\).

• In Example 5.25a, each possible value of \(d_s\) is a singular student and each possible value of \(d_p\) is a singular paper that is long.
• In Example 5.25b, each possible value of \(d_s\) is a singular student and each possible value of \(d_p\) is a singular long paper or a plurality of long papers.

What the discourse referents of Example 5.25a encode is possible ways of pairing each student with a singular paper. The scalar implicature that Example 5.25b generates then is that’s not what the speaker means, i.e. it’s not the case that each student is paired with a singular paper.

This analysis further accounts for an observation due to Crnič, Chemla, and Fox (2015) about disjunction under universal quantifiers.