Motivating an optimality theoretic account
We have seen that the extent to which experimental participants take modified numerals to convey ignorance depends on the task that they are asked to perform. Following a suggestion by Degen and Goodman (2014), who found a similar task-dependency in a different empirical domain, we take our findings to reflect an asymmetry between production and comprehension tasks. Degen and Goodman explain this difference by assuming that participants behave as rational Bayesian listeners in comprehension tasks, and that this results in a more noisy dependent variable (because uncertainties on the speaker model and on the prior add up), which gives rise to null effects. However, the results of our Experiments 2 and 4 appear to be qualitatively different from those of Experiments 1 and 3, not just noisier. For this reason and others detailed below, we will depart from the assumption that participants in a comprehension task are fully rational, and pursue a different model of comprehension tasks inspired by work in language acquisition.
While Degen and Goodman (2014) are the only ones, to our knowledge, to have empirically demonstrated production/comprehension asymmetries with adults, the literature on language acquisition has uncovered many such asymmetries in children (see, e.g., Hendriks and Koster 2010; Hendriks 2014). Of particular interest for us is a well-studied example in pronoun resolution. Unlike adults, who only allow a disjoint reference reading for sentences like (10), children up to the age of 6 often allow the coreference reading as well (Chien and Wexler 1990, among others). However, in production these children start behaving like adults much earlier on: when coreference is intended they use a reflexive pronoun (herself) and when disjoint reference is intended they use a non-reflexive pronoun (her). This has been found both in corpus studies and in elicited experimental data (Bloom et al. 1994; De Villiers et al. 2006, among others).
Hendriks and Spenader (2006) propose an account of this asymmetry couched in Optimality Theory (OT). In OT, a grammar is seen as an ordered set of constraints on possible form/meaning pairs. In comprehension, such a grammar selects the optimal interpretation(s) for a given expression, while in production it selects the optimal expression(s) to convey a given intended interpretation. Which interpretations/expressions are considered optimal depends on the ‘mode’ of optimization. We will first consider so-called unidirectional optimization, since it can account for production/comprehension asymmetries.
In production, a speaker who wants to express a certain meaning considers the candidate forms which could express it and selects the one which is optimal w.r.t. the constraints in the grammar (taking the ordering of these constraints into account). In comprehension, the listener considers candidate meanings for the form they heard and, again, selects the one which is optimal w.r.t. the constraints in the grammar. Unidirectional optimization in comprehension does not look at competing forms but only at competing meanings. Because of this, it occasionally results in production/comprehension asymmetries. Namely, it may be the case that the meaning m that is optimal for a given form f would be optimally expressed by a different form \(f'\). In this case, a listener would interpret f as m, but a speaker would not use f to express m.
Hendriks and Spenader (2006) assume that children’s non-standard comprehension of pronouns results from unidirectional optimization, while adults perform so-called bidirectional optimization. Bidirectional optimization (Blutner 2000) requires looking at both competing forms and meanings in production and in comprehension, eliminating potential asymmetries. It has been argued that this is more taxing than unidirectional optimization on executive functions, in particular working memory (Hendriks 2014). Bidirectional optimization also requires extra time compared to unidirectional optimization during actual communication (van Rij et al. 2010). The hypothesis is that the limited working memory capacity of children does not facilitate bidirectional optimization. Children therefore resort to unidirectional optimization, which results in the observed production/comprehension asymmetries.
We will pursue an OT analysis of modified numerals in order to explain the task effect found in our experiments, which, as discussed above, plausibly reflects an asymmetry between production and comprehension comparable to the asymmetries found in child language. Of course, such an approach is only worth pursuing if it can be argued that it is plausible to assume that not only children, but adults too, may resort to unidirectional optimization in some cases, in particular when interpreting modified numerals in a psycholinguistic task.
We believe that there are multiple reasons why this is indeed a plausible assumption to make. First, van Rij et al. (2010, 2013) argue that bidirectional optimization involves ‘proceduralization’: when it is practiced enough, the best expression-interpretation pair can be selected in one step, without the need to go through two selection procedures. This, however, requires practice. Pronouns are extremely frequent, so it is to be expected that adults manage to perform bidirectional optimization in this case, unlike children who have not yet been exposed to them enough. Given the much lower frequency of modified numerals compared to pronouns,Footnote 9 it is likely that in the case of modified numerals bidirectional optimization has not been proceduralized to the extent that it has in the case of pronouns. Indeed, it has been found that deriving ignorance inferences of modified numerals comes with a significant processing cost in psycholinguistic tasks (Alexandropoulou et al. 2016, 2017; Alexandropoulou 2018), which would be unexpected if the process of identifying their optimal interpretation had been fully proceduralized.
A second reason why adults may resort to unidirectional optimization when interpreting modified numerals concerns the range of alternative candidates that have to be considered in the selection procedure. Even though OT in principle does not restrict the number of candidates, it seems plausible that language users only consider a limited range of alternatives. In the case of pronoun resolution, there are not many alternative expressions to be compared: the pronoun itself, a reflexive form and perhaps a definite description. On the other hand, in the case of modified numerals, there are many more alternatives to be considered. For instance, at least eight is plausibly compared to expressions which involve a different base numeral (e.g., at least seven, at least nine,...), expressions that involve a different modifier (e.g., more than eight, more than nine,...), and numeral expressions that do not involve a modifier at all (e.g., eight, nine,...). Reinhart (2006, §2.7) already hypothesized that adults may fail to correctly apply procedures such as the one necessary for pronoun resolution if there are many candidates to be compared. Bidirectional optimization in particular would be much more taxing in the case of modified numerals than in the case of pronoun resolution. Note that the same considerations apply more generally to any comprehension mechanism which relies on the comparison of alternative expressions (van Rooij et al. 2011), including the kind of Bayesian reasoning at play in Rational Speech-Act models (Frank and Goodman 2012).
Finally, we know that in psycholinguistic tasks adults often fail to derive implicatures, precisely because they ignore lexical alternatives which are too costly to retrieve (Van Tiel and Schaeken 2017, and references therein). This effect becomes even clearer when participants’ working memory resources are restricted (De Neys and Schaeken 2007; Marty and Chemla 2013; Marty et al. 2013). Unidirectional OT is one of several ways to formalize pragmatic reasoning when ignoring lexical alternatives, but it comes with some advantages in comparison with models which simply take ‘literal participants’ to not derive any implicatures at all (e.g., the literal listener in RSA models). In particular, unidirectional OT will predict that some ignorance inferences survive even when alternatives expressions are not considered, which will turn out to play an important role in modelling the participants’ behavior in Experiments 2 and 4, and it offers an alternative explanation of the task effect reported in Degen and Goodman (2014).Footnote 10 In this sense it may represent a plausible heuristics that listeners may rely on in situations where full-fledged pragmatic reasoning is too costly.
While these considerations justify, in our view, the pursuit of a unidirectional OT account of our experimental findings, it is beyond the scope of this paper to provide a definitive, general answer to the question whether production/comprehension asymmetries are encountered in adult language ‘in the wild’. Outside psycholinguistic tasks, we suspect that listeners typically do consider alternative expressions and derive the full range of implicatures available, minimizing asymmetries with how speakers use modified numerals. Whether they achieve this with bidirectional OT or Bayesian reasoning is an open question. In any case, the unidirectional comprehension OT account explains the surprising contrast between, on the one hand, the robust introspective judgments reported in the literature that superlative modified numerals always give rise to ignorance inferences, and on the other hand, our experimental results (as well as Westera and Brasoveanu ’s) which show that participants sometimes don’t derive these inferences in demanding experimental settings.
Readers who are not particularly interested in the psycholinguistic debate regarding inference tasks may want to skip discussion of our “listener” model (Sect. 3.3.4 and Sect. 3.4.3 specifically), as it really is a model of participants’ behavior in true comprehension tasks only. We take the “speaker” model, on the other hand, to be applicable to natural communication more generally.
Semantic assumptions
Bare numerals
We assume that a bare numeral n is semantically ambiguous between a one-sided reading (n or more) and a two-sided reading (exactly n). Relevant proposals include that of Geurts (2006) and Kennedy (2015), who take the two-sided reading to be basic and derive the one-sided reading through type-shifting operations, as well as the grammatical implicature account (Chierchia et al. 2012, a.o.), which takes the one-sided reading to be basic and derives the two-sided reading by application of a grammatical exhaustivity operator. For our purposes, the differences between these two ambiguity approaches (and others) do not matter, since we are only concerned here with unembedded uses of bare numerals describing cardinalities (see Spector 2013 for further discussion).Footnote 11
Our proposal is not compatible with approaches that take numerals to be semantically unambiguous and derive the missing interpretation through pragmatics. These include the neo-Gricean account (Horn 1972; Schulz and van Rooij 2006), which assigns bare numerals a one-sided semantic denotation only, and Breheny’s (2008) account, which assigns them a two-sided semantic denotation only. Besides previously identified shortcomings (Geurts 2006; Breheny 2008; Spector 2013; Kennedy 2015), the former would not allow us to explain the difference between bare numerals and superlative modified numerals (they would be semantically equivalent). The latter would allow us to account for our experimental results concerning modified numerals, but would be incompatible with the asymmetry observed in Experiment 1 for bare numerals. Namely, on this account Six of my cards are clubs is false not only when the speaker has 4 or 5 clubs, but also when she has 7 or 8 clubs. We found, however, that the sentence is much more degraded in the former situation (i.e., when it is false both on a one-sided and on a two-sided reading) than in the latter (i.e., when it is false only on a two-sided reading). Furthermore, we found that judgments in the latter, but not in the former case, were sensitive to the QUD, which would be unexpected under a purely semantic account.
Modified numerals
We assume that modified numerals are unambiguous and receive their traditional naive interpretation:
Note that under an exhaustification account of the ambiguity of bare numerals, something must be said as to why modified numerals are not ambiguous between the proposed denotation and a two-sided reading. Such accounts usually assume that exhaustification is vacuous for modified numerals (Fox and Hackl 2007, among others) though see Enguehard (2018) for an alternative view.
Pragmatic assumptions
As discussed above, we will model pragmatic behavior using Optimality Theory (OT). More specifically, we will build on Cummins ’ (2011, 2013) OT account of modified numerals.
Cummins ’ main idea is that the distribution of modified numerals results from a trade-off between three factors: complexity (modified numerals are more complex expressions than bare numerals), salience of the base numeral (round numerals and numerals that have been primed by previous context are easier to process), and informativity (bare numerals can convey exact quantities, so they are usually more informative). The net result is that modified numerals are used in two kinds of situations: when the speaker doesn’t have enough information to use a bare numeral, or when she has precise knowledge of a specific non-round number, but decides that using a round or salient number is more important than conveying the exact quantity.
We will now present each of the constraints we consider, and the restrictions they impose on a tuple \(\langle \varphi ,s,Q\rangle \) where \(\varphi \) is an expression (utterance), s is the speaker’s information state (a set of worlds), and Q the Question under Discussion (QUD), modeled here for simplicity as a partition on a set of worlds representing the common ground. An expression \(\varphi \) may have several interpretations \(\varphi _1, \varphi _2\dots \) (due to different possible syntactic parses and/or optional type-shifting or exhaustivity operators). Given our semantic assumptions in Sect. 3.2, the only expressions which we take to be ambiguous are the bare numerals. We write \([\![\varphi _i]\!]\) for the denotation of \(\varphi _i\), which we take to be a set of worlds.
Quality
We assume the usual Gricean maxim of Quality: if a speaker utters \(\varphi \), her information state s should support \(\varphi \) under at least one interpretation:
Note that Quality phrased this way is rather lenient since it only rules out the use of an ambiguous expression when none of its interpretations are supported by the speaker’s information state. We propose a second constraint to further restrict the use of ambiguous expressions. If a speaker uses an ambiguous expression, she should make sure that it is true under all its different interpretations (even the ones she did not intend to convey).Footnote 12 This can be seen as a more stringent version of Quality (it prevents the speaker from unintentionally conveying false information), or as part of the Maxim of Manner (be clear, avoid confusion).Footnote 13
Quantity
The maxim of quantity requires that the speaker make her contribution “as informative as is required (for the current purposes of the exchange)” (Grice 1975). We will adopt an implementation of Grice’s maxim which requires that the speaker conveys all the information relevant to the QUD that she has access to:
In short, (15) states that at least one interpretation of the utterance should convey all the information available to the speaker that is relevant to the QUD. If the expression is ambiguous between an interpretation that is not supported by the speaker’s information state, and one that is supported by the speaker’s information state but under-informative, we take it to be a violation of Quantity, as the speaker could not possibly intend to resolve the QUD with the unsupported interpretation.
Our Quant constraint differs from Cummins’ Informativeness constraint in several respects (see Cummins 2011, §2.4.1). First, we directly encode relevance in Quantity by measuring informativeness with respect to the QUD (as first proposed in Matsumoto 1995). Second, we do not need to distinguish between varying degrees of violation. This would however become necessary if we were to look at a broader range of expressions or if we adopted Fox and Hackl’s (2007) Universal Density of Measurement hypothesis, as discussed in Sect. 3.4.4.
Manner
Grice’s maxim of manner addresses a variety of issues with the way speakers articulate their thoughts. We already mentioned how our second Quality constraint against misleading ambiguities could be related to Manner.
The maxim of manner also requires speakers to aim for the simplest expression that satisfies their communicative purpose. We will model this by penalizing more complex expressions (both in term of production and processing cost for the listener). Following Cummins (2011), we take superlative modifiers to be more complex than comparative ones (see Geurts et al. 2010; Cummins and Katsos 2010 for experimental validation of this assumption), and bare numerals to be the simplest form (this corresponds to quantifier simplicity in Cummins 2011).
While Simp affects the choice of a modifier, the choice of a numeral involves similar considerations. More specifically, “round” numbers are typically easier to process (Dehaene 2011, see also Jansen and Pollmann 2001 for a formal definition of roundness). Similarly, numbers mentioned in the previous context (primed) are salient and therefore also easier to process. While Cummins (2011) distinguishes between the two sources of salience for numerals (roundness and priming), we will model both with a single constraint NSal, since this will be sufficient to explain our experimental results. Unlike Cummins (2011), we do not need to distinguish between different levels of roundness.
Finally, we assume that speakers also obey a faithfulness constraint that favors the use of numerals that are internally salient (i.e., salient to the speaker, though not necessarily to the addressee). We assume in particular that if a speaker knows that the value of a given quantity lies within a certain range \([n\ldots m]\), then the boundaries of this range, n and m, are internally salient. For instance, if a speaker makes a statement about how many students smoke, and if she knows that between 7 and 12 students smoke, then 7 and 12 are internally salient for her.Footnote 14\(^,\)Footnote 15
Thus, for instance, if as above a speaker knows that between 7 and 12 students smoke, then at least 7 students smoke and at most 12 students smoke satisfy ISal but more than 6 students smoke and less than 13 students smoke do not.
We assume that ISal plays a role in speakers’ choices of numerical expressions, because, other things being equal, internally salient numerals are presumably easier to produce for speakers than non-salient ones. Additionally, ISal may be seen as a particular instance of a more general pressure to align what is externally salient with what is internally salient.
Cummins (2011, §2.5.2) also considers the possibility of including a faithfulness constraint in his OT system which would favor the use of numerals that are internally salient to the speaker. “One approach,” he writes, “would be to change the definition of ‘primed’ numerals and quantifiers to include those which are activated in the mind of the speaker as well as those which are present in the preceding discourse.” Something like this would be needed, he points out, in order for his system to license the use of numerals that are not salient in any external sense (i.e., round or contextually primed). However, he does not explicitly introduce such a constraint, and does not specify in more detail what it would mean for a numeral to be ‘activated in the mind of the speaker’. Since this is exactly what our ISal constraint does, we view it as being very much in the spirit of Cummins’ proposal, and one that is indeed necessary in any OT system of this kind in order to license the use of numerals that are not externally salient.
Linking theory and experimental results
With our constraints in place, we need to specify how they are ranked, and how the predictions of the OT system translate into predictions about the behavior of participants in each of the tasks in our experiments.
We assume the order in (19).Footnote 16
Note that the two salience constraints are not ranked. Following Boersma (1997), among others, we interpret ‘\(\approx \)’ in a probabilistic manner: if NSal and ISal are in conflict, they do not cancel each other but either of them can take precedence at evaluation time. The result is that ties between these two constraints are not broken by lower constraints (e.g., Simp here), but both candidates are possibly optimal. This contrasts with classical OT, where two forms can never be optimal at the same time.
In Experiments 1 and 3, participants had access to the information state of the speaker s, and had to evaluate the acceptability of candidate responses to a question (which we identify with the QUD Q). We assume that an utterance which is optimal in the OT system receives maximum acceptability. For utterances which are not optimal, we assume that the acceptability depends on the rank of the fatal violation (the highest violation that distinguishes the utterance from the optimal candidate). The higher this violation ranks, the more degraded the utterance.Footnote 17
In Experiments 2 and 4, participants were placed in the position of a listener who had access to Q and \(\varphi \), and needed to infer whether the speaker had precise knowledge or not (i.e., whether s specified a single value or not). Such a task can be modeled in several ways. Bayesian models and bidirectional OT are meant to capture optimal rational behavior in such circumstances. However, it is impossible that the behaviour of the participants in our experiments was completely rational in this sense. Otherwise, they should have always inferred ignorance from superlative modifiers, since at least was always degraded in Experiments 1 and 3 when the speaker had precise knowledge. We therefore propose that participants performed the task in Experiments 2 and 4 in a simpler, sub-rational way.
Specifically, we will model participants’ behavior in Experiments 2 and 4 using unidirectional OT: participants compare the different possible knowledge states a speaker could be in, and select the state(s) s in which the highest violation incurred by \(\varphi \) is lower than the highest violation incurred in any other state \(s'\) (i.e., they optimize s given \(\varphi \) and Q). This is sub-rational because some expression \(\varphi '\) may be more optimal than \(\varphi \) for a speaker with information state s, so it may very well be that \(\varphi \) would not actually be used to convey s. Nevertheless, it is a good heuristics because it allows participants not to take alternative expressions into consideration—a move that is known to be cognitively costly—without entirely ignoring pragmatic constraints such as those corresponding to the Gricean maxims. The main effect of this heuristics is to completely disregard constraints which, for a given \(\varphi \), do not depend on s. This includes so-called markedness constraints such as NSal and Simp, which only refer to properties of \(\varphi \), as well as ISal in the case of more than. Indeed, more than violates ISal no matter what s is, so ISal will have no effect on the interpretation of comparative modifiers. In fact, ISal will not affect the decision between knowledge and ignorance for other expressions either, because for any given n, ISal does not distinguish between the knowledge states \(\{n\}\) and \([n,\dots )\). The end result is that only Qual, Quant, and SQual will affect the predictions for Experiments 2 and 4, effectively neutralizing the contrast between at least and more than.
We further assume that participants behave as follows. In case the optimal state s is an ignorance state, participants pick the maximal degree of ignorance on the scale that they are presented with. In case the optimal state s is an exact knowledge state, they go for the other extreme of the scale. Finally, if ignorance and exact knowledge states are in a tie, we assume that participants fall back on their prior expectations about the speaker, which doesn’t depend on \(\varphi \).
Deriving predictions and capturing the experimental data
We will first present the main predictions of the model and give some insight into how the different parts of the model interact with each other. We then describe the predictions of the production model in greater detail, and how it captures the results of Experiments 1 and 3, followed by the predictions of the comprehension model for Experiments 2 and 4. As discussed in Sect. 3.1, we don’t take the comprehension model to faithfully represent listeners in natural conversations, but rather to explain the behavior of participants in a psycholinguistic comprehension task. The production model on the other hand, is assumed to generalize beyond the experimental setting of Experiments 1 and 3, and will be explored in greater detail and compared to the predictions of competing proposals in the next sections.
We will discuss the predictions of our account for at least, more than and bare numerals, but will not discuss at most and fewer than. The pattern observed for negative modified numerals is similar, but the effects are stronger.
Getting a sense of the model’s inner workings
Before showing in detail how the model captures our experimental findings, we will first state the two most basic predictions and explain how they come about:
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1.
Comparative modified numerals are only optimal in combination with round numerals. They do not require speaker ignorance but can only be used when conveying a precise number is not necessary (e.g., with polar or coarse-grained QUDs).
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2.
Superlative modified numerals require speaker ignorance and the numeral they combine with must match the exact minimum the speaker considers possible.
The distribution of comparative modified numerals is determined by an interaction between their semantics and the ISal/NSal constraints. Specifically, their use implies that the numeral they combine with cannot be part of the speaker’s knowledge: if someone says “More than twelve students smoke”, they immediately exclude the possibility that exactly twelve students smoke. This means that more than cannot ever satisfy ISal in plain affirmative sentences. To stand a chance in the competition with bare numerals and at least, more than must at least satisfy NSal, i.e., it must combine with a contextually salient/round numeral. If the QUD calls for a precise answer, however, it is often impossible to fall back on the closest round numeral without incurring a violation of Quant. We therefore predict that more than is only used in combination with round or salient numerals, usually in the context of coarse-grained QUDs.
Second, since superlative modifiers are more costly than comparative modifiers and bare numerals, at least is only used when (i) SQual rules out bare numerals (i.e. the speaker does not have exact knowledge), and (ii) at least beats more than by satisfying ISal (which more than always violates). This means that at least requires partial ignorance and must combine with the lowest numeral the speaker considers possible. We thus capture the usually accepted felicity conditions for at least k (see, e.g., Schwarz 2016a): the speaker must consider both k and some value above k possible, but does not need to consider each value above k possible.
To sum up, while most accounts attempt to derive the ignorance effects associated with superlative modifiers from their semantics or the alternatives they give rise to, the present account relies primarily on the fact that comparative modifiers always violate the ISal constraint. The distribution of at least simply reflects the gaps left by bare numerals and more than. At this point, the ISal constraint may seem like a one-trick pony whose sole function is to rule out more than in situations where at least is typically used. We will see in Sect. 3.5 that ISal in fact plays a crucial role in capturing a surprising range of empirical facts that go beyond ignorance implicatures. But first, let us now dive into the details of how the model captures our experimental findings.
Production: Capturing Experiments 1 and 3
The tableaux for how many and polar QUDs are presented in Tables 8 and 9, respectively. For simplicity, we identify s with the set of values the speaker considers possible. Note that Quant only has an effect on how many questions, since all expressions considered in Table 9 completely resolve the polar QUD (though see Sect. 3.4.4). Let us now explain how the tableaux translate into predictions for Experiments 1 and 3. Whenever relevant, we indicate the experimental conditions testing a given prediction.
Bare numerals: When the speaker has exact knowledge, the matching bare numeral is predicted to be acceptable with any QUD (Exp. 1: Exact; Exp. 3: Precise). Higher numerals violate Qual so they are maximally degraded (Exp. 1: False), while lower numerals violate Quant in a how many QUD context, and SQual in a polar QUD context, leading to mild degradedness and a QUD effect (Exp. 1: Exceed). When the speaker is ignorant, the numeral matching the lower-bound incurs a violation of SQual, which is always fatal because of the at least alternative (Exp. 1: Ignorance; Exp. 3: Approximate).Footnote 18
More than: Table 8 shows that more than is almost always predicted to be degraded in response to a (fine-grained) how many question. Our results confirm this: more than is indeed degraded with how many QUD, whether the speaker has exact knowledge or not.
Turning to polar questions, we see that more than n is now in a tie for optimality in every situation where the speaker’s knowledge excludes the salient n (whether the speaker knows a precise number above n, or only a range). This was the case in all our target conditions in Experiment 3, and in the PolarOverInf case in Experiment 1. We correctly predict that more than is fully acceptable in these conditions. The PolarRelevant case in Experiment 1—where more than was observed to be degraded—is predicted to be out because it violates both NSal and ISal.
At least: With precise knowledge, at least as a response to a (fine-grained) how many question always incurs a violation of Quant, and is therefore predicted to be clearly degraded (Exp. 1: Exact, Exceed; Exp 3: Precise). In ignorance cases, at least becomes optimal (or in a tie to be optimal), and is therefore fully acceptable (Exp. 1: Ignorance; Exp. 3: Approximate).
For at least, the ignorance effect is predicted to persist in polar questions, but due to violations of lower-ranked constraints with precise knowledge (ISal, NSal, or Simp).Footnote 19 The model thereby captures the observation that at least always shows an ignorance effect, but the amplitude of this effect is modulated by QUD.
Table 8 OT Tableaux for Experiments 1 and 3, how many question (understood as a fine-grained how many question). n is a round number, k is a non-round number between \(n+2\) and the next round number. This is a tableau for production, so s and Q are fixed and only expressions \(\varphi \) are compared and evaluated against each other. Blocks separated by a horizontal line should be considered separately
Table 9 OT Tableaux for Experiments 1 and 3, polar question. n is the threshold for the polar question and is round, k is a non-round number above \(n + 2\). This is a tableau for production, so s and Q are fixed and only expressions \(\varphi \) are compared and evaluated against each other. Blocks separated by a horizontal line should be considered separately
Table 10 OT Tableaux for Experiments 2 and 4, HowMany question (understood as a fine-grained how many question). n is a round number. Cases beyond \(n+2\) behave similarly, up to the next round numeral. This is a tableau for comprehension, so \(\varphi \) and Q are fixed, and only possible information states s are compared and evaluated against each other. Blocks are separated by a horizontal line. Each block should be considered separately
Comprehension: Capturing Experiments 2 and 4
The tableaux for how many and polar QUDs are presented in Tables 10 and 11, respectively. Only round numerals are considered since we did not test non-round numerals in these experiments. As these tableaux model the hearer’s perspective, \(\varphi \) is fixed and s needs to be inferred. Moreover, we take the QUD to be fixed by the overt questions preceding each utterance. In real-life conversations, the QUD isn’t always clearly set and speakers may decide to refine it before answering, so Q would sometimes have to be inferred as well (see Klecha 2018 for a mechanism of QUD-revision in an OT framework).
The predictions are straightforward. In how many questions, the only optimal knowledge states for modified numerals are ones involving ignorance. Both superlative and comparative modifiers therefore lead to equally strong ignorance inferences, as observed in the experimental results.
In polar questions, both types of modifiers lead to a tie between precise and imprecise knowledge states. As a consequence, we predict that participants fall back on their prior expectations with regard to the speaker’s knowledge. Given our experimental results, this must be leaning towards ignorance, but the conclusion is weaker than when participants can infer ignorance directly from the utterance. Crucially, the effect is again the same for at least and more than, so the model captures the absence of a difference between superlatives and comparatives in this experimental setting.
For bare numerals, the model correctly predicts no ignorance inference whatsoever, since their optimal knowledge state in both polar and how many contexts is always precise knowledge. The model also captures the absence of a QUD effect on bare numerals in these experiments, in contrast with Experiments 1 and 3.
Table 11 OT Tableaux for Experiments 2 and 4, polar question. n is the threshold for the polar question and is round. Cases beyond \(n+2\) behave similarly. This is a tableau for comprehension, so \(\varphi \) and Q are fixed, and only possible information states s are compared and evaluated against each other. Blocks are separated by a horizontal line. Each block should be considered separately Universal density of measurement
So far we have implicitly assumed a discrete scale for cardinalities. In particular, we have assumed that Quant never distinguishes between more than n and at least \(n+1\). However, Fox and Hackl (2007) have argued that all scales are dense (the so-called Universal Density of Measurement hypothesis, abbreviated UDM). Here we discuss how adopting the UDM would affect the predictions of our account.
The first effect of the UDM would be to make more than fully unacceptable with precise how many questions. Recall that the only situation in which more than n was predicted to be compatible with a precise how many QUD was one in which n is round and the speaker’s knowledge is a range with lower bound \(n+1\). This situation resulted in a tie between more than n and at least \(n+1\), but with the UDM more than would violate Quant. In Experiment 3, this situation was not tested, and in Experiment 1, we didn’t properly control for roundness and salience.Footnote 20 Nevertheless, we think that the prediction of the UDM here is correct. In response to “How many students registered for the course exactly?”, more than 30 intuitively does not convey as much information as at least 31. This prediction could be captured by other mechanisms however (for instance, Dehaene 2011 and Krifka (2009) show that round numbers tend to have an approximative meaning).
The second effect of the UDM would be to make more than \(n-1\) even more degraded as an answer to a polar QUD. Without the UDM, it was already non-optimal (because it violated both ISal and NSal), but it would now violate Quant as well. Experiment 1 includes cases corresponding to this situation (PolarRelevant QUD with Exact and Ignorance Situation), and they do appear to be as degraded as other violations of Quant, suggesting that the UDM is again on the right track.
Finally, the UDM would affect the predictions for Experiments 2 and 4. In particular, since any information state would violate Quant with more than after precise how many questions, all the information states that pass Qual in Table 10 would be equally possible. This would predict no effect of QUD on the ignorance inferred from more than, in contradiction with our results. If we were to adopt the UDM, this last issue could be worked around by assuming different levels of violations for Quant, as proposed in Cummins (2011, §2.4.1) or Klecha (2018) (so that more than incurs more violations if the speaker has exact knowledge and the QUD is precise).
To sum up, adopting the UDM would require a graded version of Quant to keep the good predictions of the comprehension model intact, but would not affect the predictions of the production model much.
Further predictions
We will now discuss several predictions that the proposed model makes beyond the empirical data it was designed to account for. Discussing these predictions is particularly important in order to provide further support for the ISal constraint. While the other constraints in our model are familiar from other work and have received much independent support in other empirical domains, the ISal constraint is new, and has not so far been motivated on independent grounds. The fact that, as we will see, it makes correct predictions for a range of other puzzles related to modified numerals provides such independent motivation.
The first prediction relates to the observation that speakers, when using at least with a non-numerical prejacent, do not always convey that they consider the prejacent itself possible (Mendia 2016). The second prediction regards the effect of polarity on modified numerals. It has been noticed that at least is degraded under negation. Our account captures this effect and makes a number of subtle predictions for other embeddings, assuming ISal is generalized to quantified sentences. We then show that the model may also shed new light on the behavior of modified numerals in the scope of quantifiers and modals, although a detailed account of this behavior is left for future work.
Mendia’s observation
Consider sentence (20) below. We have seen that such sentences can only be optimal if the speaker considers it possible that exactly two students completed the quiz, and therefore implicate that she is uncertain as to whether exactly two students completed the quiz or more.
However, Mendia (2016) argues that this inference depends crucially on the fact that, for numerals, the relevant comparison class is totally ordered. In cases which involve a partially ordered comparison class, such as (21), the speaker might in fact know that Ann and Bill are not the only students who completed the quiz.
To support this claim, Mendia provides experimental data showing that the answer in (22b) is regarded considerably more acceptable than the one in (23b).
This leads Mendia to propose the following generalization:
To capture this contrast, we first need to specify how ISal would extend to scales other than numerals, since this is the constraint mainly responsible for ignorance inferences concerning the prejacent (see Sect. 3.4.2). Given a partially ordered set, we can generalize ISal using the least upper bound and greatest lower bound of the speaker’s information state, to the extent that they exist. Consider the case of (22b): the domain of individuals is partially ordered by the i-part relation (Link 1983). If the speaker knows that Ann, Bill, and exactly one of Clara, David and Elliot completed the quiz, then her information state amounts to the set \(\{a\oplus b\oplus c,a\oplus b\oplus d,a\oplus b\oplus e\}\). While this set has neither a minimum nor a maximum in \(D_e\), it has a greatest lower bound \(a\oplus b\) and a least upper bound \(a\oplus b\oplus c\oplus d\oplus e\). Assuming as we did above that these are the boundaries ISal is concerned with, this means that the speaker can use “at least Ann and Bill” without violating ISal, even when she knows that not only Ann and Bill completed the quiz.
We therefore predict that ISal can be satisfied without considering the exhaustified prejacent possible, to the extent that the set of values compatible with the speaker’s information state has a greatest lower bound that is not a minimum. This is a necessary condition for the use of at least, but it is not sufficient. Indeed open intervals in a totally ordered dense scale also make it possible to have a greatest lower bound without a minimum, but the oddness of (25b) suggests that at least still conveys ignorance towards the exhaustified prejacent in this case.
Our account captures this fact as well. In order to satisfy ISal without considering exactly 3 km possible, the speaker would have to consider any distance greater but arbitrarily close to 3 km possible. But if that were the case, more than 3 km would be optimal since it would satisfy every constraint at least 3 km satisfies, while incurring one less violation of Simp, as the following tableau shows:
The important difference between (25b) and (22b) is that—at least in English—partially ordered comparison classes do not license comparative modifiers (more than Ann and Bill completed the quiz is ungrammatical for some reason). Without a comparative competitor, at least only needs to beat its non-modified alternative to become optimal. This requires some ignorance, but not necessarily ignorance with respect to the exhaustified prejacent.
To sum up, we derive Mendia’s generalization (24) from two facts: (i) comparison sets can have a greatest lower bound which is not a minimum, and (ii) comparative modifiers require a total ordered comparison set. The use of at least without speaker ignorance regarding the exhaustified prejacent is licensed just in case the comparison set has a greatest lower bound which is not a minimum (needed to satisfy ISal), and the set is not totally ordered (needed to block the competition with more than, which now also satisfies ISal).
Polarity effects
Geurts and Nouwen (2007) noted that superlative but not comparative modified numerals are degraded under negation, as shown in (26).
Nilsen (2007) further shows that at least is degraded in a number of other downward-entailing environments, as seen in (27), but not in the restrictor of a universal quantifier or in the antecedent of a conditional, as seen in (28).Footnote 21 These contrasts have been confirmed experimentally in Mihoc and Davidson (2019), Mihoc (2019, chap. 5).
Our account directly captures the unacceptability of at least under negation. Remember that, given its additional violation of Simp, at least can only win against more than by satisfying ISal. However, negation flips the strict and non-strict comparison: \(\textit{not}\dots \textit{at least three}\) mentions but excludes 3, while \(\textit{not}\dots \textit{more than three}\) is compatible with 3. This means that under negation, at least cannot ever satisfy ISal, thereby losing its natural advantage over more than.Footnote 22 We further predict that more than loses its roundness/salience requirement under negation, as it can now satisfy ISal. Our intuitions, together with a quick corpus analysis, suggest that this prediction is correct.Footnote 23
Turning to restrictors, our current definition of ISal falls short, since we’re not talking about a single quantity but a set of quantities, for instance, the set containing every n such that some individual cancelled n days in advance in Nilsen’s examples. Generalizing ISal to sets of quantities would be necessary to make fully explicit predictions about such cases, but we can already point out one essential difference between negation and the restrictor of every: while \(\textit{not}\dots \textit{at least}\) n excludes exactly n, (28a) does not exclude that someone who cancelled exactly three days in advance got their money back, and in fact it entails that everyone who did cancel exactly three days in advance got their money back. This suggests that any reasonable generalization of ISal would not in fact block at least in the restrictor of every or in antecedents of conditionals. By contrast, the sentence with no does exclude that anyone who cancelled exactly three days in advance had to pay the fees. Here the exact way we generalize ISal may matter. We also want to point out that Nilsen’s examples may not be ideal, because they involve very small numbers, so it could be that (28b) is only acceptable because 3 is in the subitizing range. Looking at higher numerals suggests that there is in fact a contrast between everybody and nobody:
According to the few native speakers we consulted, (29b) and (29c) are slightly degraded compared to (29a) and (29d) respectively, as would be expected if, in the restrictor of negative quantifiers, it was at least and not more than that violated ISal. However, the theoretical and empirical work that would be necessary to properly extend our account to these examples is beyond the scope of the present manuscript.
Quantified and modal sentences
Modified numerals are known to give rive to specific effects not only in the restrictor of a quantifier, but also in the scope of a quantifier or modal operator. In particular, superlative modified numerals in the scope of a universal quantifier do not necessarily give rise to ignorance effects, but can give rise to so-called variation effects instead (e.g., Alexandropoulou et al. 2015):
This is entirely expected if bare numerals can receive an exact reading in the scope of the quantifier. In this case, it is possible for the sentence with the bare numeral to fail SQual even when the speaker is fully knowledgeable. Our account then predicts the following felicity conditions for (30):
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(i)
Every street is guarded by three or more policemen (the literal reading), and
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(ii)
The speaker doesn’t know for sure that every street is guarded by exactly three policemen (bare numeral filtered by SQual), hence some streets may be guarded by more than three, and
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(iii)
Some streets must/may be guarded by exactly three policemen (depending on how ISal is generalized to quantified sentences).
We therefore predict that ignorance is possible but not necessary, and if we assume that the speaker is in fact knowledgeable about exactly how many policemen guarded each street, we capture the variation effect as described in Alexandropoulou et al. (2015): some streets are guarded by exactly three policemen, some by more than three.
Similarly, our account sheds new light on the interaction between modified numerals and modals, which motivated Geurts and Nouwen’s assumption that superlative modifiers are undercover modals. In particular, our account immediately captures the observation in (31) that at least under possibility modals only has an ignorance reading.
To see what happens in (31), let us show why a knowledgeable speaker cannot use this sentence to answer the QUD “Which lengths (in pages) are allowed for this assignment?”. Let us first set aside the inverse scope reading of (31): if the modified numeral has wide scope, it is unembedded and therefore necessarily conveys ignorance. For any number k, the speaker, by assumption, knows whether the paper is allowed to be k pages long or not, hence we can identify the smallest such k, \(k_{\min }\). By definition, for any \(k<k_{\min }\), the assignment is not allowed to be k-pages long, hence it follows that the paper is required to be at least \(k_{\min }\) pages long (here the assumption that the speaker is knowledgeable is crucial). If no other page length is allowed (i.e. if the assignment must be exactly \(k_{\min }\)-pages long), then “the paper can/must be at least \(k_{\min }\) pages” is out in favor of a bare numeral. Assuming that the paper can also be k-pages for some \(k\not =k_{\min }\), and that \(k_{\min } -1\) isn’t particularly salient (so that more than is out), “the paper can/must be at least \(k_{\min }\) pages long” become possible options, but the necessity modal is more informative than the possibility modal, hence at least can only be used under the necessity modal. In short, assuming that the speaker is fully knowledgeable, at least under a possibility modal only satisfies ISal when a necessity modal could be used instead. As a result, we capture the observation that at least can only be used under a possibility modal when the speaker is ignorant of the exact numbers allowed.
This is far from saying that our account captures the interaction between modified numerals and modals. In particular, the account says nothing about the second puzzle that motivated Geurts and Nouwen (2007), namely that at most under possibility modals always imposes a strict upper bound, as illustrated in (32) (see Blok 2015, 2019 for additional puzzles). Nevertheless, we hope to have shown that pragmatic solutions should be explored in greater detail before departing from a plain semantics for modified numerals.