Experiments on the comprehensibility ofcomparative- vs. superlative-modified numerals

under downward-entailing operators

Teodora Mihoc and Kathryn Davidson

Draft of July 27, 2019

Abstract

The literature on comparative- and superlative-modified numerals often notes that they are pair-wise truth-conditionally equivalent but differ in (at least) two major ways, with respect to igno-rance effects and acceptability under negation (although not necessarily in other types of downward-entailing environments). Both empirical contrasts have been claimed for roughly the same length oftime. However, while the ignorance contrast has been investigated experimentally and taken seri-ously by every theory of modified numerals, the negation contrast has been largely neglected – fewtheories engage with it, and there has been no experimental investigation. In this paper we aim tohelp fill this gap by investigating the negation contrast (and related patterns) in three experiments.Overall the results provide strong support for the claimed negation contrast, showing that no the-ory of modified numerals can ignore it going forward. The results also show that the interaction ofsuperlative-modified numerals with negation depends on many factors – in particular, it does notalways result in a penalty.

1 Introduction

1.1 Motivating data

Naively speaking, given a discrete scale, comparative-modiffied numerals (CMNs; more/less than three)and superlative-modified numerals (SMNs; at least/most three) are pairwise truth-conditionally equiv-alent.

(1) John called more than two / at least three people. (=1 iff John called 3 or 4 or more people)

However, in spite of this apparent similarity, CMNs and SMNs are often claimed to differ in thefollowing ways. First, SMNs are claimed to be degraded in a context of certainty, (2), although bothCMNs and SMNs are compatible with a context of ignorance, (3), and with a how many? QUD bothCMNs and SMNs clearly give rise to ignorance, (4) (Geurts and Nouwen 2007, Büring 2008, Nouwen2010, 2015, Geurts et al. 2010, Coppock and Brochhagen 2013, Mayr 2013, Kennedy 2015, Westeraand Brasoveanu 2014, Schwarz 2016, Cremers et al. 2017, Alexandropoulou 2018).

(2) John called three people. Therefore, he called 3more than two / # at least three.

(3) I don’t know how many people John called, but it was 3more than two / 3at least three.

(4) How many people did John call? More than two. / At least three. ( the speaker is ignorant)

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Second, SMNs are claimed to be worse than CMNs in the scope of negation, (5), although not in otherdownward-entailing environments such as the antecedent of a conditional or the restriction of a univer-sal, (6)-(7) (Nilsen 2007, Geurts and Nouwen 2007, Cohen and Krifka 2011, Coppock and Brochhagen2013, Mayr 2013, Cohen and Krifka 2014, Spector 2014, 2015).

(5) John didn’t call more than two / # at least three people. (meaning: he called 0, 1, or 2)

(6) If you call 3more than two / 3at least three people, you lose.

(7) Everyone who called 3more than two / 3at least three people lost.

These claims have been around for roughly the same length of time. For example, both are men-tioned in Geurts and Nouwen (2007). However, they have received very different attention. The claimsrelated to ignorance have been investigated experimentally and also taken seriously by every theoryof modified numerals since. In contrast, the claims related to negation have not been investigated ex-perimentally, and have been taken seriously only by a few theories (Cohen and Krifka 2011, 2014 andSpector 2014, 2015).

1.2 Goal

The goal of this paper is to provide a first experimental investigation of the negation contrast, andrelated patterns, as a way to check if the empirical claims are supported and, if so, to encourage moreengagement with these data in the theory.

1.3 Preview of results

Altogether we conducted three experiments. The experiments provide strong support for (5)-(7), show-ing that these patterns can no longer be neglected by theory. At the same time they also underscorethe complexity of the pattern, showing that the interaction of SMNs with negation depends on manyfactors, and is not simply a matter of adding up processing costs.

1.4 General notes on methodology

Participants Participants were self-reported native speakers of English, recruited on Amazon’s Me-chanical Turk, and paid $2 (Exp. 1) and $1 (Exps. 2 and 3) for their participation. There was nooverlap of participants between experiments.

Task Our data posed specific design challenges: (1) Sentences with CMNs/SMNs can be clumsy, par-ticularly when further combined with negation or other factors. (2) (At least in plain episodic contexts,)(non-)ignorance can be an issue. (3) An SMN under negation can be scoped out to get a specific reading,but that’s not the reading we want to investigate. (4) The nature of the contrast between CMNs/SMNsunder negation is unclear – it doesn’t seem to be a matter of good/grammatical/acceptable vs. not (theSMN version of (5) seems syntactically well-formed, and semantically we are able to compute its truthconditions) or natural vs. not (the CMN version of (5) is pretty unnatural also) but rather a matter ofeasy vs. not (when we present friends with a sentence like the SMN version of (5), the typical reactionis “It just melted my brain!”). To address (1)-(3), we decided to present our CMN/SMN sentences inthe context of a card game, to control ignorance by keeping it constant between trials within each ex-periment, and to make it implicitly clear that the CMN/SMN sentences are always about the number of

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cards of some suit, not about specific cards. All these points were enforced both in the way the contextwas described (at the outset) and through pictures (in each trial) depicting the information state (handof cards) against which the statements were truthfully being made (following a similar design fromCremers et al. 2017). To address (4), we decided to ask participants for comprehensibility judgments:Do you think x will understand what y said? If asking a question about comprehensibility seems to buildprocessing cost into the probe, note that, for the purpose of our data, it is not obvious that a more tradi-tional question about acceptability avoids this. On the other hand, for the reasons we outlined earlier,such a question might have left participants confused given our sentences, whereas, as we will see, thequestion we chose managed to bring out our contrast of interest very well.

Trial summary Our sentences were obtained by crossing factors such as environment type, polarity,modifier type, and modifier monotonicity, which we explain below. Aside from these factors, trials onlydiffered in minor ways – the suit of cards (always different but always matching the sentence), the nameof the character uttering the sentence (always different, random), and the suit of cards (counterbalancedacross the items); to minimize variations due to numeral complexity, the numeral was always the same.The contrast between CMNs and SMNs under negation seemed fairly subtle, so in order to maximizechances of detecting it, and because we had no reason to be concerned about conscious access to thetask affecting responses, we chose not to use any fillers. The questionnaire was prepared in Qualtrics(Qualtrics Labs 2016). Each participant saw every trial in a different, random order.

Results The results were analyzed in R (R Core Team 2015). First we summarize the raw means, andthe binomial confidence intervals associated with these means (binom; Dorai-Raj 2014), in the form ofa plot (ggplot2; Wickham 2009). Then we report the results of fitting logistic mixed effects regressionmodels (models fitted with lme4, Bates et al. 2015; detailed comparisons with attached measures –odds ratios (OR), confidence intervals (CI), z, and p values – extracted with lsmeans, Lenth 2016). Ineach case the model includes fixed effects for all the factors used to generate the items, and all theirinteractions, and a random effects structure including a random intercept for participant and randomslopes for the maximal principled structure for which the model converged. (We didn’t include randomeffects for items – as we said, aside from the factors, the trials differed only in minor ways, e.g., thename of the character, that we don’t expect to interfere with the judgments.) Our data and analysisscripts for each experiment are available online at [to be added later].

2 Exp. 1: Negative declarative, positive antecedent, positive restriction

2.1 Question

Are SMNs worse than CMNs in a negative declarative but not in other downward-entailing environmentssuch as a positive antecedent of a conditional or a positive restriction of a universal, as claimed in (5)-(7)?

2.2 Methods

Participants 99, of which 3 excluded for cause prior to analysis.

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Task Participants were introduced to the task as follows:

In this survey you will answer questions about a group of friends playing a game. At thebeginning of the game each player gets dealt a hand of seven cards. After taking a quicklook at them, they must place the cards face down and try to remember their hands. Thenthey take turns giving clues about their hands to the other players in the form of statementsdescribing their hands. You will see what a player remembers about his/her cards and thestatement s/he makes, then you will be asked if you think the other players will understandwhat s/he said.

Note: a or a means that the player doesn’t remember if a particular card in his handwas a club or a spade, or a diamond or a heart, respectively.

Sample trial As below.

Figure 1: Exp. 1 example trial.

Trial summary Each participant saw 24 trials, obtained by crossing the following factors: Env =embedding environment (DECL = declarative; ANTCOND = antecedent of conditional; RESTUNIV =restriction of universal); Pol = polarity of Env (POS = positive, NEG = negative); and ModType =modifier type (COMP = comparative, SUP = superlative) x ModMon = modifier monotonicity (UE =upward-entailing, DE = downward-entailing), which yield Mod = modifier (MORETHAN, LESSTHAN,ATLEAST, ATMOST). See Table 1.

Env Pol ModType (COMP, SUP) x ModMon (UE, DE) = Mod

DECLPOS I have Mod 3 [suit]NEG I don’t have Mod 3 [suit]

ANTCONDPOS If you have Mod 3 [suit], then we have something in commonNEG If you don’t have Mod 3 [suit], then we have something in common

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Table 1 (Continued)

RESTUNIVPOS Everyone who has Mod 3 [suit] has something in common with meNEG Everyone who doesn’t have Mod 3 [suit] has something in common with me

Table 1: Exp. 1 trial summary.

2.3 Expectations

SMNs should be worse than CMNs in DECL-NEG but not in ANTCOND-POS or RESTUNIV-POS.

2.4 Results

As in Figure 2.

Figure 2: Exp 1 raw means, by Modifier. Bars represent 95% binomial confidence intervals.

R1 Notice that, for all environment types, downward modifier monotonicity and negative polaritynegatively affected comprehensibility for both CMNs and SMNs. Regression analysis confirms thesetrends – there was a significant effect of ModMon = DE (β = −1.98, z = −2.536, p = 0.0112 *) andof Pol = NEG (β = −2.08, z = −2.677, p = 0.0074 **).

R2 Notice also that, for the same level of monotonicity, SMNs (a) in a positive declarative are the sameas CMNs; (b) in a negative declarative are much worse than CMNs; (c) in a positive antecedent/restrictionare largely (except for at most in a restriction) the same as CMNs; and (d) in a negative antecedent/restrictionare worse than CMNs, but less so than in a negative antecedent. Statistical analysis again supports thesetrends, cf. Table 2.

Env Pol ModType by ModMon OR CI z pDecl Pos MoreThan-AtLeast 1.00 [0.09, 11.20] -0.000 1.0000Decl Pos LessThan-AtMost 1.44 [0.52, 4.03] 0.856 0.3918

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Table 2 (Continued)Decl Neg MoreThan-AtLeast 6.41 [2.57, 15.98] 4.872 <.0001Decl Neg LessThan-AtMost 31.49 [12.01, 82.56] 8.569 <.0001AntCond Pos MoreThan-AtLeast 0.49 [0.03, 9.33] -0.579 1.0000AntCond Pos LessThan-AtMost 1.63 [0.66, 4.00] 1.304 0.3843AntCond Neg MoreThan-AtLeast 2.19 [1.00, 4.76] 2.402 0.0163AntCond Neg LessThan-AtMost 3.33 [1.50, 7.38] 3.618 0.0003RestUniv Pos MoreThan-AtLeast 0.82 [0.18, 3.72] -0.318 1.0000RestUniv Pos LessThan-AtMost 2.73 [1.15, 6.50] 2.771 0.0168RestUniv Neg MoreThan-AtLeast 2.88 [1.33, 6.23] 3.278 0.0021RestUniv Neg LessThan-AtMost 4.00 [1.81, 8.84] 4.181 0.0001

Table 2: Exp. 1 predicted contrasts for levels of ModType, given same level of ModMon.

R3 Finally, notice that CMNs under negation degrade from a declarative to an antecedent/restriction,but SMNs don’t. This too is supported, cf. Table 3.

Env Pol Mod OR CI z pDecl-AntCond Neg MoreThan 2.93 [1.15, 7.48] 2.752 0.0118Decl-RestUniv Neg MoreThan 3.28 [1.30, 8.33] 3.060 0.0066Decl-AntCond Neg LessThan 4.62 [2.02, 10.55] 4.434 <.0001Decl-RestUniv Neg LessThan 3.62 [1.58, 8.28] 3.728 0.0004Decl-AntCond Neg AtLeast 1.00 [0.47, 2.11] 0.000 1.0000Decl-RestUniv Neg AtLeast 1.48 [0.70, 3.11] 1.246 0.6382Decl-AntCond Neg AtMost 0.49 [0.19, 1.23] -1.853 0.1305Decl-RestUniv Neg AtMost 0.46 [0.18, 1.16] -2.019 0.1305

Table 3: Exp. 1 predicted contrasts for levels of Env, given Pol = NEG.

2.5 Discussion

R2(b)-(c) confirm our starting expectations (except for at most in a positive restriction): as claimed in(5), SMNs are worse than CMNs under negation, and, as claimed in (6)-(7), they are no different fromCMNs in a positive antecedent/restriction. We confirm that theories of modified numerals must accountfor the difference between SMNs and CMNs under negation, while showing that it doesn’t hold for someother downward-entailing operators.

Do our results shed any light on what kind of a theory this theory of SMNs might be? Let’s considera few possibilities, and how they fit with our results overall.

On one theory, let’s call it T1, the difference between DECL-NEG and ANTCOND/REST-POS is that theformer contains a negation and the latter do not, and the explanation has to do with processing cost, asfollows: SMNs incur a processing penalty (cf. Geurts et al. 2010, Alexandropoulou 2018; often takenfor granted, though we believe the jury is still out); negation incurs an additional processing penalty(cf. Wason and Johnson-Laird 1972; although much following literature has shown this is context-dependent, e.g. Nieuwland and Kuperberg 2008); and processing costs add up non-linearly. T1 capturesR2(b) but not (R2(a) and) R2(c), unless we grant that there is in fact a penalty for superlativity in those

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conditions but it is too small for our task to detect. T1 fails to capture R2(d) and R3 – it captures whyCMNs degrade between DECL-NEG and ANTCOND/RESTUNIV-NEG (presumably ANTCOND/RESTUNIV aremore complex than DECL) but not why SMNs don’t.

The problem with T1 seems to be that it takes the cost of an SMN under negation to be fixed, whereasit may itself vary depending on other factors. This possibility is explored in T2 and T3, which at thesame time seek to identify semantic reasons for why these costs exist, and why their cumulative costmight vary, as follows:

T2 (Cohen and Krifka 2011, 2014) says that the difference between SMNs in DECL-NEG vs. ANTCOND/RESTUNIV-POS goes back to two separate lexical meanings of SMNs. The meaning in DECL-NEG is one where thetruth conditions of SMNs arise pragmatically, via scalar implicature. Under the assumption that im-plicatures embedded under downward-entailing operators are unavailable because they would lead toweakening, this captures R2(b). The meaning in ANTCOND/RESTUNIV is one where the truth condi-tions of SMNs arise semantically, and do not depend on the monotonicity of the environment. It ishowever an evaluative meaning that cares about whether there is positivity/negativity match betweenthe antecedent/restriction and the consequent/scope (where the polarity of the continuation is un-derstood loosely, e.g., lose counts as negative). Since in our experiment the continuation was alwaysneutral/positive (we have something in common/has something in common with me), this captures theabsence of a penalty for SMNs in R2(c) and the presence of a penalty in R2(d). Moreover, since on thisaccount SMNs in a negative declarative and in a negative antecedent/restriction go back to differentmeanings, the fact that the penalty in the latter case isn’t worse isn’t surprising, capturing R3.

T3 (Spector 2015) says that the difference between DECL-NEG and ANTCOND/RESTUNIV-POS lieswith the fact that, in addition to their downward-entailing truth conditions, the latter also carry anupward-entailing presupposition. This difference matters for SMNs because they require exhaustifi-cation relative to a set of domain alternatives, and this exhaustification takes into account both truthconditions and alternatives and must crucially lead to strengthening. As it turns out, this conditioncannot be met when exhaustifying a DECL-NEG but, thanks to their presupposition, can be met forANTCOND/RESTUNIV-POS, capturing R2(b)-(c). ANTCOND/RESTUNIV-NEG defines an upward-entailingenvironment, just like a positive declarative, so SMNs aren’t expected to incur any special penalty forbeing under negation in these environments. This fails to capture why in ANTCOND/RESTUNIV-NEG

SMNs were still found worse than CMNs, cf. R2(d), but could offer a partial explanation for why in thiscondition they were at least not worse than in DECL-NEG, cf. R3.

To sum up, Exp. 1 supports existing claims that SMNs are worse than CMNs in negative declarativesbut not in positive antecedents/restrictions. All of T1-3 can capture these basic patterns but they differin their predictions for cases of embedding under two downward-entailing operators, and here thecomplexities of T2-3 seem to have an advantage. To elucidate this further, in Exp. 2 and 3 we investigateSMNs under two downward-entailing operators again, in the first case testing a commitment from T2and in the second – from T3.

3 Exp. 2: The additional polarity

3.1 Question

Does T2 correctly predict that an SMN in a negative antecedent/restriction will be worse with a positivecontinuation and better with a negative (in a loose sense, e.g., lose counts as negative) continuation?

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3.2 Methods

Participants 45, of which 5 excluded for cause prior to analysis.

Task Similar to Exp. 1, adapted to support positive/negative continuations.

In this survey you will answer questions about a group of friends playing a game. At thebeginning of the game each player gets dealt a hand of seven cards. They are not allowed tosee their own cards but they are allowed to take a quick look at their neighbor’s hand. Theytry to remember their neighbor’s hand as well as they can because in the next step they haveto come up with a rule that would make that neighbor (and possibly other players too) loseor win. You will see what a player remembers about their neighbor’s hand and the rule theymake up, then you will be asked if you think the other players will understand what theysaid. Note, we’re not asking you if it is a good rule or a bad rule, but whether it is a rulethat is going to be understandable for the other players to follow.

Note: a or a means that the player doesn’t remember if a particular card in his handwas a club or a spade, or a diamond or a heart, respectively.

Sample trial As below.

Figure 3: Exp. 2 example trial.

Trial summary Each participant saw 32 trials, obtained by crossing the following factors: Env =embedding environment (ANTCOND = antecedent of conditional; RESTUNIV = restriction of univer-sal); Pol1 = polarity of Env (POS = positive, NEG = negative); Pol2 = polarity of predicate in conse-quent/scope (POS = positive; NEG = negative); and ModType x ModMon =Mod (as before). See Table4.

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Env Pol1 Pol2 ModType (COMP, SUP) x ModMon (UE, DE) = Mod

ANTCOND

POSPOS If you have Mod 3 [suit], you winNEG If you have Mod 3 [suit], you lose

NEGPOS If you don’t have Mod 3 [suit], you winNEG If you don’t have Mod 3 [suit], you lose

RESTUNIV

POSPOS Everyone who has Mod 3 [suit] winsNEG Everyone who has Mod 3 [suit] loses

NEGPOS Everyone who doesn’t have Mod 3 [suit] winsNEG Everyone who doesn’t have Mod 3 [suit] loses

Table 4: Exp. 2 trial summary.

3.3 Expectations

SMNs in ANTCOND/RESTUNIV-NEG-NEG should be better than in ANTCOND/RESTUNIV-NEG-POS.

3.4 Results

As in Figure 4.

Figure 4: Exp 2 raw means, by Modifier. Bars represent 95% binomial confidence intervals.

R1 As in Exp. 1, downward modifier monotonicity and negative polarity in the antecedent/restrictionseem to negatively affect comprehensibility, although the trend for monotonicity is not always clear (inpositive antecedents/restrictions less than was on a par with more than; this is possibly due to differencesin the overall flow of Exp. 2 vs. Exp. 1). Matching this, regression analysis doesn’t detect a significanteffect of ModMon = DE, although it does detect a significant effect of Pol1 = NEG (β = −1.80, z =−2.167, p = 0.0303 *).

R2 For both environment types, (a) the upward-monotonic SMN (at least) was generally rated simi-larly to its CMN counterpart, while (b) the downward-monotonic SMN (at most) was generally rated

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worse, although to varying degrees, as follows: POS-NEG >> POS-POS >> NEG-POS >> NEG-NEG.Statistical analysis supports these trends, cf. Table 5.

Env Pol1 Pol2 ModType by ModMon OR CI z pAntCond Pos Pos MoreThan-AtLeast 1.00 [0.10, 10.20] -0.000 1.0000AntCond Pos Pos LessThan-AtMost 7.08 [1.11, 45.22] 2.365 0.0180AntCond Pos Neg MoreThan-AtLeast 0.64 [0.08, 5.43] -0.468 0.6400AntCond Pos Neg LessThan-AtMost 62.48 [12.05, 323.99] 5.631 <.0001AntCond Neg Pos MoreThan-AtLeast 3.18 [0.99, 10.23] 2.222 0.0526AntCond Neg Pos LessThan-AtMost 4.74 [1.43, 15.73] 2.904 0.0074AntCond Neg Neg MoreThan-AtLeast 0.55 [0.13, 2.31] -0.938 0.3484AntCond Neg Neg LessThan-AtMost 3.20 [0.99, 10.34] 2.222 0.0526RestUniv Pos Pos MoreThan-AtLeast 1.00 [0.10, 10.22] 0.000 1.0000RestUniv Pos Pos LessThan-AtMost 45.15 [4.08, 500.05] 3.551 0.0008RestUniv Pos Neg MoreThan-AtLeast 7.38 [0.61, 89.96] 1.793 0.1461RestUniv Pos Neg LessThan-AtMost 40.53 [8.14, 201.71] 5.171 <.0001RestUniv Neg Pos MoreThan-AtLeast 2.07 [0.68, 6.30] 1.464 0.1431RestUniv Neg Pos LessThan-AtMost 2.67 [0.86, 8.26] 1.944 0.0518RestUniv Neg Neg MoreThan-AtLeast 0.40 [0.12, 1.28] -1.768 0.1542RestUniv Neg Neg LessThan-AtMost 1.99 [0.60, 6.56] 1.296 0.1949

Table 5: Exp. 2 predicted contrasts for levels of ModType, given same level of ModMon.

R3 Chances of comprehensibility are quite high for all the modifiers when both polarities are positive(with the notable exception of at most in a universal), but for at most they drop dramatically when thesecond polarity becomes negative (including in the case of a universal). And chances of comprehensi-bility are generally lower for all modifiers when the first polarity is negative (cf. R1), but for at leastthey improve dramatically when the second polarity becomes negative also. Statistical analysis confirmsthis, cf. Table 6.

Env Pol1 Pol2 Mod OR CI z pAntCond Pos Pos-Neg MoreThan 1.56 [0.18, 13.24] 0.468 1.0000AntCond Pos Pos-Neg LessThan 1.56 [0.18, 13.26] 0.468 0.6398AntCond Pos Pos-Neg AtLeast 1.00 [0.10, 10.22] 0.000 0.9999AntCond Pos Pos-Neg AtMost 13.80 [3.97, 47.99] 4.719 <.0001AntCond Neg Pos-Neg MoreThan 0.85 [0.24, 3.05] -0.286 0.7750AntCond Neg Pos-Neg LessThan 1.26 [0.43, 3.66] 0.477 0.6332AntCond Neg Pos-Neg AtLeast 0.15 [0.04, 0.56] -3.214 0.0026AntCond Neg Pos-Neg AtMost 0.85 [0.23, 3.08] -0.286 1.0000RestUniv Pos Pos-Neg MoreThan 0.48 [0.03, 7.98] -0.585 1.0000RestUniv Pos Pos-Neg LessThan 3.25 [0.23, 46.43] 0.995 0.6397RestUniv Pos Pos-Neg AtLeast 3.55 [0.51, 24.63] 1.464 0.2863RestUniv Pos Pos-Neg AtMost 2.92 [0.96, 8.92] 2.153 0.0314RestUniv Neg Pos-Neg MoreThan 1.64 [0.54, 5.03] 0.992 0.6423RestUniv Neg Pos-Neg LessThan 1.80 [0.60, 5.37] 1.205 0.4562RestUniv Neg Pos-Neg AtLeast 0.31 [0.10, 1.01] -2.222 0.0263

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Table 6 (Continued)RestUniv Neg Pos-Neg AtMost 1.35 [0.40, 4.58] 0.542 1.0000

Table 6: Exp. 2 predicted contrasts for levels of Pol2.

3.5 Discussion

What we found was much richer than expected. Each of our explanations T1-3 fails for multiple reasons,but the general problem for T1 and T3 is that they don’t predict any variation due to the polarityof the continuation, while the general problem for T2 seems to be that, although it does anticipatesuch variation, it doesn’t explain why it differs between for at least vs. at most. The polarity of thecontinuation seems to interact with modifier monotonicity non-trivially.

4 Exp. 3: The additional downward-entailing operator

4.1 Question

Does T3 correctly predict that an SMN under negation improves with further embedding under anadditional downward-entailing operator?

4.2 Methods

Participants 45.

Task Similar to Exp. 1, adapted to support clausal embedding for matrix negation.

In this survey you will consider a commentator for a televised card-playing game, and an-swer questions about how understandable the commentator is.

At the beginning of the game each player gets dealt seven cards, two of which are hidden.Then in each round some rule is issued, and players can choose whether or not to bet ontheir own hand. A commentator, who knows what the hidden cards are for each player,discusses the player’s move.

You will see a player’s hand and the commentator’s comment, then you will be asked if youthink the viewers will understand what the commentator said.

Note: In the hands that you will see, cards with a white background such as representcards that are visible to the player, while cards with a grey background such as representhidden cards, that is, cards that are not visible to the player but visible to the commentator.

Sample trial As below.

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Figure 5: Exp. 3 example trial.

Trial summary Participants saw 16 trials, obtained by crossing the following factors: Env=matrix em-bedding environment (MATRIXNEG = scope of matrix negation, ANTCOND = antecedent of conditional);Pol = polarity of the embedded clause (POS = positive, NEG = negative); and ModType x ModMon =Mod (as before). See Table 7.

Env Pol ModType (COMP, SUP) x ModMon (UE, DE) = Mod

MATRIXNEGPOS [name] doesn’t know that s/he has Mod 3 [suit]NEG [name] doesn’t know that s/he doesn’t have Mod 3 [suit]

ANTCONDPOS If [name] knew that s/he has Mod 3 [suit], s/he would bet differentlyNEG If [name] knew that s/he doesn’t have Mod 3 [suit], s/he would bet differently

Table 7: Exp. 3 trial summary.

4.3 Expectations

SMNs in ANTCOND-NEG and MATRIXNEG-NEG should be the same.

4.4 Results

As in Figure 6.

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Figure 6: Exp 3. raw means, by Modifier. Bars represent 95% binomial confidence intervals.

R1 As for Exp. 1 and (largely also) Exp. 2, we notice a negative effect of downward modifier mono-tonicity and polarity of the immediate embedding clause being negative. Regression analysis confirmsthis: there was a significant effect of ModMon = DE (β = −2.34, z = −2.481, p = 0.0131 *) and of Pol= NEG (β = −2.15, z = −2.186, p = 0.0288 *).

R2 Given the same level of monotonicity, SMNs seem to be on a par with CMNs in every condition.Statistical analysis confirms this, except for at least in MATRIXNEG-NEG, which was found significantlyworse than more than, cf. Table 8.

Env Pol ModType by ModMon OR CI z pAntCond Pos MoreThan-AtLeast 1.30 [0.10, 16.05] 0.233 0.8161AntCond Pos LessThan-AtMost 1.33 [0.30, 5.96] 0.428 0.6687AntCond Neg MoreThan-AtLeast 2.96 [0.69, 12.71] 1.672 0.0946AntCond Neg LessThan-AtMost 0.62 [0.16, 2.37] -0.793 0.5065MatrixNeg Pos MoreThan-AtLeast 2.47 [0.34, 17.98] 1.023 0.6131MatrixNeg Pos LessThan-AtMost 3.13 [0.84, 11.63] 1.946 0.1033MatrixNeg Neg MoreThan-AtLeast 4.38 [1.17, 16.40] 2.505 0.0245MatrixNeg Neg LessThan-AtMost 2.09 [0.49, 8.85] 1.143 0.5065

Table 8: Exp. 3 predicted contrasts for levels of ModType, given same level of ModMon.

R3 In general ratings for each modifier appear similar between the two environment types, but more sofor CMNs than SMNs, which, given the same level of polarity, seem to degrade somewhat from ANTCOND

to MATRIXNEG. Statistical analysis indeed reveals a qualitatively small but statistically significant trend,cf. Table 9.

Env Pol Mod OR CI z pAntCond-MatrixNeg Pos MoreThan 1.68 [0.20, 14.37] 0.539 0.5898AntCond-MatrixNeg Pos LessThan 1.56 [0.38, 6.38] 0.706 0.9598

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Table 9 (Continued)AntCond-MatrixNeg Pos AtLeast 3.19 [0.35, 29.04] 1.179 0.2386AntCond-MatrixNeg Pos AtMost 3.66 [1.02, 13.11] 2.282 0.0450AntCond-MatrixNeg Neg MoreThan 2.32 [0.56, 9.64] 1.320 0.3738AntCond-MatrixNeg Neg LessThan 1.07 [0.29, 3.89] 0.111 0.9598AntCond-MatrixNeg Neg AtLeast 3.42 [1.10, 10.66] 2.426 0.0305AntCond-MatrixNeg Neg AtMost 3.57 [0.90, 14.23] 2.065 0.0450

Table 9: Exp. 3 predicted contrasts for levels of Env.

4.5 Discussion

What we found is yet again richer than expected, and our T1-3 are yet again inadequate. It is not clearwhat T1 and T2 would predict for each of these conditions, but they seem to face an insurmountablechallenge in MATRIXNEG-POS because they would predict SMNs in this condition to be bad for thesame reasons that we discussed for a negative declarative, contrary to what we found. Although T3’spredictions as simplified earlier might seem to fail also, there are complications. In all our conditions inthis experiment the embedded clause is introduced by the factive attitude predicate know. Now, it hasbeen noticed that an item that is bad in the scope of negation improves with an intervening factive, andsuggested that on an exhaustification approach such as the one sketched in T3 one might derive this bystudying the effect of exhaustifying a prejacent with a factive (Spector 2014). T3 might thus be able tocapture MATRIXNEG-POS. Of course, a challenge to T3 remains the fact its main prediction here was notupheld – SMNs were a little worse in MATRIXNEG-NEG than in ANTCOND-NEG, showing that the choiceof higher downward-entailing operator might also slightly matter.

5 Conclusion and outlook

In this paper we investigate claims from the literature that SMNs are worse than CMNs in a negativedeclarative but not in the antecedent of a conditional or the restriction of a universal. We find supportfor these patterns but we also uncover many other patterns that paint a much more complex picturethat we might have suspected at the outset. The results suggest several interactions of SMNs withdownward-entailing operators which no current theories predict in full. If our modest investigation intothe negation contrast has proven so stimulating both in generating empirical puzzles and in identifyingeven more gaps in the theory, we believe there is significant potential for future research into thiscontrast to uncover much more. On the empirical side the claims we tested can be tested again usingdifferent methods to check if the method makes a difference (as has been the case for the ignorancecontrast, where the result of testing with a variety of methods has also led to a more nuanced pictureof this contrast over time). These claims can also be tested together with similar claims for items calledpositive polarity items, to which SMNs are often descriptively likened. On the theoretical side, a theoryof SMNs must make sense of these patterns – not just the ignorance contrast and the starting negationcontrast, but also the additional variability we uncovered, and also of the claimed similarity betweenthese patterns in SMNs and other items.

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