1.1 Exhaustification

The aim of this squib is to discuss some issues regarding the hypothesis about zero put forward by Bylinina & Nouwen (2018). These authors assume that numerals have the weak, ‘at least’ meaning as basic and the strengthened, ‘exactly’ meaning as derived by way of exhaustification, and propose that zero be considered no different from other numerals in this respect.¹ This is illustrated by (1a) and (1b).

Before we go on, let us be explicit about the notion of exhaustification which we will adopt for the purpose of this discussion. First, [EXH φ] is true iff φ is true and every excludable alternative of φ is false. Second, an alternative of φ is a sentence derivable from φ by replacing certain scalar items in φ with their scale mates. And third, ψ is an excludable alternative of φ iff ψ is an alternative of φ, ψ is stronger than φ, and φ ∧ ¬ψ does not entail any other alternative of φ.² These premises, together with the standard view that numerals form a scale (cf. Horn 1989), will derive (1b).³

Coming back now to the sentences in (1), we can see not only that (1a) is entailed by (1b), but also, that (1a) is entailed by every sentence. In other words, (1a) is trivially true: the cardinality of the set of students who smoked, or in fact, of any set, is necessarily 0 or greater than 0. Given this consequence of their analysis, and given the assumption that trivial sentences are “semantically defective,” Bylinina and Nouwen claim that zero, unlike other numerals, always requires semantic strengthening by exhaustification.⁴ We will call this claim EXHAUSTIFICATION.

1.2 Prevention

EXHAUSTIFICATION may be said to receive supporting evidence from the deviance of (3) which, to the best of our knowledge, is a novel observation.

Here is how the reasoning may go. It has been noticed that modification of numerals by at least prevents semantic strenghthening of the numeral (cf. Krifka 1999; Nouwen 2010; Kennedy 2015; Schwarz 2016; Buccola & Haida 2017). For example, (4a) cannot be read as (4b).

Let us state the generalization, which we will call PREVENTION.

Although what is crucial for our discussion is that PREVENTION holds, not how it is explained, here is one way this generalization can be accounted for. Suppose that the alternatives of a sentence containing at least are derived from it by replacing at least with its scale mates exactly and more than (cf. Kennedy 2015; Buccola & Haida 2017). This means the alternatives of φ = at least two students smoked are ψ = exactly two students smoked and χ = more than two students smoked. And because (φ ∧ ¬ ψ) ⟹ χ and (φ ∧ ¬ χ) ⟹ ψ, neither ψ nor χ is an excludable alternative of φ, which means that [EXH φ] ⟺ φ. Thus, modification of a numeral by at least makes EXH vacuous, thus preventing the numeral from being semantically strengthened by exhaustification.⁵

Given that modification of a numeral by at least prevents semantic strengthening by exhaustification of the numeral, EXHAUSTIFICATION, which says zero requires such strengthening, leads to the prediction that modification of zero by at least will result in deviance. This prediction is confirmed by the deviance of (3). Thus, the deviance of (3) constitutes supporting evidence for EXHAUSTIFICATION.⁶

1.3 Pragmaticism

Now, starting from the supposition that EXHAUSTIFICATION is correct, a question that arises is (6).

The answer given by Bylinina and Nouwen is that it is pragmatic. To quote from Bylinina & Nouwen (2018: 10): “Unlike other numerals, zero invokes exhaustification obligatorily. This is for purely pragmatic reasons.” We will call this claim PRAGMATICISM.

Under the perspective of PRAGMATICISM, one would say that the sentence in (8) cannot be parsed as (8a) because (8a) is trivially true, hence uninformative, hence pragmatically odd. The reason (8) is acceptable is because there is another parse, (8b), which is informative and thus pragmatically appropriate.

With respect to the sentence in (9), one would say, in light of PRAGMATICISM, that it is perceived as deviant because both of its parses, (9a) and (9b), are trivial, hence uninformative and pragmatically odd.⁷

We believe that PRAGMATICISM can be challenged. If the deviance of (9), repeated in (10a), comes about by it being uninformative, then all sentences expressing the same meaning, and hence are equally uninformative, should be perceived as deviant in the same way. This is not the case, as evidenced by the acceptability of (10b), which is semantically equivalent to (10a).

Both (10a) and (10b) are trivial and thus equally uninformative, but only (10a) is perceived as deviant.⁸ This suggests that the contrast between these two sentences is grammatical in nature. The question now is (11).

1.4 L-triviality

To address this question, let us consider (12a) and (12b), which are possible parses of (10a) and (10b), respectively.⁹¹⁰

Note the underlining in (12). Its purpose is to distinguish between the logical expressions, which are underlined, and the non-logical expressions, which are not underlined. For the purpose of this squib, we take logical expressions to be those whose semantic contribution depends only on facts about language, and the non-logical expressions to be those whose semantic contribution depends on both facts about language and facts about the world. To illustrate, suppose we are to know whether (12b) is true in a given possible world w.¹¹ For EXH, zero, or and more than, we would have to know what they mean in English. For students and smoked, however, we would have to know not only what they mean in English, but also, who the students are in w and among them who smoked in w.

Given the classification of expressions into the logical and the non-logical, we now have a way to describe a crucial difference between (12a) and (12b) regarding the source of their triviality: (12a) is trivial by virtue of its logical vocabulary alone, while (12b) is trivial by virtue of both its logical and non-logical vocabulary. Here is what we mean in more concrete terms. Suppose we replace the word students in the second disjunct of (12b) with the word professors, the result, which is (13), is not trivial: (13) is true in case no students smoked or some professors smoked, and false in case some students smoked and no professors smoked.

In other words, had the non-logical part of (12b) been different, the sentence would not have been trivial. The same, however, does not hold for (12a): there is no way to make (12a) non-trivial by changing its non-logical part. It remains trivial no matter which noun replaces students and which verb replaces smoked.

Let us call sentences such as (12a) “L-trivial,” borrowing from Gajewski (2009). Now, L-triviality has been argued to cause deviance (Barwise & Cooper 1981; Fintel 1993; Gajewski 2003; Abrusán 2007; Gajewski 2009). To the extent that that argument is convincing, the fact that (12a) is L-trivial while (12b) is not can be taken to explain the contrast between these two structures, and hence, explain the contrast between (10a) and (10b).¹² Thus, we have come to a tentative answer to the question in (11), given below.¹³

1.5 Circumvention

The argument for (14) relies, crucially, on the assumption that EXH can rescue a sentence from L-triviality. To see this, consider (15a), which has the truth condition in (15b).

The first disjunct is trival. Moreover, it is L-trivial, as substituting ⟦students⟧ or ⟦smoked⟧ with any other predicate will not change the fact that it is trivial. This means that the whole disjunction is L-trivial. But (15a) is just (12b) without EXH, and since the argument for (14) depends on the claim that (12b) is not L-trivial, this means that that argument depends on the claim that EXH can circumvent L-triviality. Let us call this claim CIRCUMVENTION.

We believe CIRCUMVENTION can be challenged. Consider (17) and its possible parses in (17a) and (17b).

This sentence is perceived as deviant in the same way (10a) is. The parse without EXH, (17a), is L-trivial. However, the parse with EXH, (17b), is not. Assuming, again, that at least alternates with exactly and more than, the truth condition of (17b) will be (18a), which is equivalent to (18b), a contingent proposition.¹⁴

If CIRCUMVENTION is correct, we predict (17) to be non-deviant, because it has a parse which is not L-trivial. Thus, the fact that (17) is perceived as deviant suggests that CIRCUMVENTION is not correct, i.e. that EXH cannot rescue a sentence from L-triviality. This is perhaps not a surprising result. We expect a sentence to be ungrammatical if it contains a constituent which in isolation would itself be ungrammatical. If we take L-triviality to be a grammatical criterion, we expect a sentence with an L-trivial constituent to be deviant regardless of what other linguistic materials it may contain. This means that we cannot append EXH to an L-trivial sentence φ to “save” it, even if [EXH φ] is not L-trivial.

Another fact which speaks against CIRCUMVENTION is that (19) is perceived as deviant (Barwise & Cooper 1981).

As every expresses the subset relation, the truth condition of (19) can be stated as ⟦student⟧ ⊆ ⟦there⟧, where ⟦there⟧ = D_e, i.e. the set of all things that exist. Since every set is a subset of D_e, the sentence is L-trivial, hence deviant. Now, given the standard assumption that every has some as its scale mate, a parse of (19) with EXH, (20a), would express the proposition in (20b), which is equivalent to (20c).

Thus, appending EXH to (19) results in the proposition that there is no student, which is not trivial. If CIRCUMVENTION is correct, (19) should be acceptable and should express this contingent proposition. This is not what is observed.

1.6 The dialectical situation

Let us recap. We start with a claim about the meaning of the numeral zero, made by Bylinina & Nouwen (2018), which says that zero has the basic weak meaning of ‘zero or more.’ The dialectics then goes as follows. If Bylinina and Nouwen’s claim is correct, then zero requires EXH. This requirement is either pragmatic or grammatical. Observation suggests that it is not pragmatic, hence must be grammatical. Saying that the requirement for EXH by zero is grammatical, however, implies that EXH can rescue a sentence from L-triviality. But observation suggests that EXH cannot rescue a sentence from L-triviality. Thus, we are lead to the conclusion that Bylinina and Nouwen’s claim is not correct, i.e. that zero does not mean ‘zero or more.’ The conclusion is tentative, pending alternative explanations for the facts that Bylinina and Nouwen provided to support their claim.

We would note, ending this section, that all of the data we discussed would be consistent with the view that zero has the strong meaning as basic and that EXH cannot circumvent L-triviality. Specifically, (21a) would mean ‘no students smoked,’ (21b) would be trivial but not L-trivial, and both (21c) and (21d) would be L-trivial and therefore deviant.¹⁵

However, noting such facts is far from proposing a theory of zero. What we hope to have achieved is to raise some issues, present some novel observations, and share some ideas which can stimulate and provide some guiding intuition for further research.

2.1 Existential sentences

To corroborate the intuition that zero, unlike other numerals, cannot be modified by the adverb at least, we conducted an experiment, hosted on Amazon MTurk, in which participants were asked to rate the naturalness of eight English sentences. The sentences were derived from the sentence frames in (22) by replacing the placeholder n with the numeral zero, for a set of four deviant sentences, and the numeral two, for a set of four non-deviant sentences.

We asked participants to rate the resulting sentences, presented in pseudo-randomized order, on the following four-point Likert scale: 4 (natural), 3 (relatively natural), 2 (relatively weird), 1 (weird). To illustrate, in advance of the trials, how these values may be associated with English sentences, we gave participants the four sentences in (23) and commented: “You may agree that (23a) is natural, (23b) is weird, while (23c) and (23d) are possibly somewhere in between.”¹⁶

We used an even-numbered scale to force participants to discriminate between sentences which they may be unable to judge as 4 (‘natural’) or 1 (‘weird’), and avoided suggesting that there is agreement on the score of such sentences in order not to stifle participants’ trust in their own judgments. We believe that the use of a four-point scale, excluding a neutral value, does not have an adverse effect on the interpretation of the expected findings of our study: even if there was perceived social or other pressure (not) to classify sentences as ‘(relatively) natural’ or ‘(relatively) weird’, potential skewing would not have a bearing on the expected difference in the score for sentences with at least zero and sentences with at least two.

We conducted our experiment with 32 Amazon MTurk workers who, after the trials, identified as being native speakers of English. Thus, overall we received 32 scores for each of our eight sentences and hence 128 scores per sentence type. Figure 1 shows that sentences with at least two received the highest score 4 (‘natural’) by ≥50% of all subjects, while sentences with at least zero received the two lowest scores 2 (‘relatively weird’) and 1 (‘weird’) by ≥50% of all subjects. The difference in the means of the scores (3.4 v 2.0), depicted in Figure 2, is highly significant (p < 2.2^–16).

Figure 1

Boxplot of at least two & at least zero.

Figure 2

Means of at least two & at least zero.

This result supports our empirical claim that numerical statements containing at least zero, unlike statements containing other superlative modified numerals, are deviant.

2.2 Universally quantified sentences

If the deviance induced by the triviality of zero is pragmatic, it should be alleviated by exhaustification, and at least zero should be acceptable under universal quantification, as such quantification renders exhaustification non-vacuous. Example (17) suggests that the deviance of at least zero persists under universal quantification. To corroborate this intuition, we conducted another experiment, again hosted on Amazon MTurk. This experiment targeted the contrast between #every … at least zero and its counterpart containing zero or more as well as the non-zero counterparts of the former two. We used the eight sentences in (24) as stimuli.

Two of these sentences, viz. the sentences in (24a), are deviant, while the others are non-deviant. Abstracting from the deviance of (24a-i) and (24a-ii),¹⁷ we can characterize the meaning of the sentences in (24) in the following way. All sentences license a distributive inference.¹⁸ For instance, the sentence in (24b-i) licenses the inference that some humans have (exactly) zero children and some humans have one or more children. The distributive inference renders (24b-i), as well as the other sentences (24a) and (24b), non-tautological. The distributive inference of (24b-i) does not, however, render this sentence false in the actual world. In this regard, it differs from e.g. the sentence in (24a-ii): in the actual world, it is false that some humans have (exactly) zero biological mothers. Overall, as indicated, if we take the distributive inference into account four of the eight sentences in (24) are true in the actual world, while the other four are false.¹⁹

We instructed participants to classify the sentences in (24) as either ‘true’, ‘false’, or ‘weird’²⁰ by giving them the following instruction in advance of the trials: “Imagine you are helping an alien who wants to learn English and also learn about humans in general. Your job is to say for several English sentences whether they are true, false, or sound weird.” The scenario we sketched in this instruction served to introduce an addressee without common knowledge. With this, we aimed for subjects to be less drawn to classify non-deviant sentences such as the sentences in (24b-i) or (24c-ii) as weird despite the fact that they express common knowledge with little informational content.²¹ At the same time, the instruction aimed at making subjects more inclined to classify the deviant sentences in (24a) as weird by pointing out that the addressee also wants to learn English.

We expect that the proportion of ‘weird’ responses is higher for the sentences in (24a) than for the sentences in (24b), in accordance with our empirical claim that the former, in contrast to the latter, are deviant. Furthermore, we expect that the proportion of ‘weird’ responses is higher for the sentences in (24a) and (24b) than for the sentences in (24c) and (24d). This expectation is based on two factors: (i) the assumption that tautological sentences are more prone to being judged as weird than contingent sentences, and (ii) the experimental finding that there tends to be a substantial subpopulation of subjects that do not to compute distributive inferences (Crnič et al. 2015). Such a subpolulation derives tautological truth conditions for the sentences in (24a) and (24b), while the sentences in (24c) and (24d) are contingent with or without the distributive inference.

We conducted our experiment with 157 Amazon MTurk workers who, after the trials, identified as being native speakers of English. Thus, we received 157 scores for each of our eight sentences (1256 observations in total). Figure 3 shows the proportion of ‘true’, ‘false’, and ‘weird’ responses for the degree phrases in (24a), (24b), (24c), and (24d), respectively.

Figure 3

Proportion of True-False-Weird judgments.

In the subsequent statistical analysis, we consider as dependent variable the binary distinction between ‘weird’ responses and ‘true’ or ‘false’ responses (‘non-weird’ responses). We used binomial logistic regression to analyze the relationship between this binary response variable and the following four independent variables: modifier with values ‘at least’ and ‘or more’, numeral with values ‘zero’ and ‘non-zero’, at least zero with values ‘+’ and ‘–’, and truth value with values ‘true’ or ‘false’. These variables describe whether the stimulus sentence contains the modifier at least or the modifier or more, whether the stimulus sentence contains the numeral zero or the numeral one or two, whether the stimulus sentence contains the degree phrase at least zero or any other degree phrase, and whether the stimulus sentence is true or false in the actual world, respectively. All four variables turn out to be significant predictors of the response variable with p values of 0.01449, 0.00273, 0.01984, and 0.00995, respectively. Specifically, the modifier at least and the truth value ‘true’ decrease the odds of the ‘weird’ response (reduction of the odds by 43% and 28%, respectively), while the numeral zero and the degree phrase at least zero increase the odds of the ‘weird’ response (increase of the odds by 78%, and 110%, respectively). The results of the ANOVA between this model and the null model show that, as suggested by the p values of the logistic regression, the numeral variable is by far the biggest contributor to the fit of the model (reduction of deviance by 44.4 from 1347.7 null deviance, p = 2.686^–11), followed by the at least zero variable (reduction of 6.7, p = 0.0094), and truth value (reduction of 5.2, p = 0.0221), while the modifier variable does not contribute significantly to the fit of the model (p = 0.5).

These results confirm the expectation that the proportion of ‘weird’ responses is higher for the sentences in (24a) and (24b) than for the sentences in (24c) and (24d), and higher for the sentences in (24a) than for the sentences in (24b). Thus, overall the results are compatible with our empirical claim that the sentences in (24a) are deviant, while those in (24b), (24c), and (24d) are non-deviant.²² Questions remain about the influence of the truth or falsity of the stimulus sentence on the odds of the ‘weird’ response and about the reason for the overall tendency for the modifier or more to raise the odds of the ‘weird’ response compared to the modifier at least.²³ We hope to address these questions in future research.