This paper investigates the ‘Diversity’ inferences (D-inferences) arising from disjunction embedded in the scope of a universal quantifier, e.g., Every X is A or B suggests Some Xs are A and Some Xs are B. It has previously been claimed (i) that D-inferences are independent from ‘Negative Universal’ inferences (NU-inferences), which are the negations of Every X is A and Every X is B, but (ii) that for disjunction in the scope of a universal modal the D-inferences cannot be observed independently of the NU-inferences ([3, 7, 9, 13]). Experiment 1 tested the availability of D-inferences in the absence of NU-inferences for the determiner every and the epistemic modal must. Experiment 2 followed up on Experiment 1 by testing the same two quantifiers, only this time the modal must expressed deontic necessity. The results show that, for both types of quantifiers, D-inferences could be derived independently of NU-inferences. While the results for every essentially replicate those reported in [7], the results for must are new and go against the aforementioned claim (ii). In addition, the response time results from both experiments show that D-inferences are associated with response delay effects in the opposite direction to those observed for regular scalar implicatures in similar tasks ([4, 5]). We argue that these findings about the time course of D-inferences raise a new challenge for an implicature-based approach to these inferences

Breheny, R

Marty, P

Ramotowska, S

Romoli, J

Sudo, Y

English

UCL Discovery

Diversity with Universality∗Sonia Ramotowska1, Paul Marty2, Jacopo Romoli1, Yasutada Sudo3, andRichard Breheny31 Heinrich-Heine-Universität Düsseldorf2 L-Università ta’ Malta3 University College LondonAbstractThis paper investigates the ‘Diversity’ inferences (D-inferences) arising from disjunc-tion embedded in the scope of a universal quantifier, e.g., Every X is A or B suggestsSome Xs are A and Some Xs are B. It has previously been claimed (i) that D-inferencesare independent from ‘Negative Universal’ inferences (NU-inferences), which are thenegations of Every X is A and Every X is B, but (ii) that for disjunction in the scopeof a universal modal the D-inferences cannot be observed independently of the NU-inferences ([3, 7, 9, 13]). Experiment 1 tested the availability of D-inferences in theabsence of NU-inferences for the determiner every and the epistemic modal must. Ex-periment 2 followed up on Experiment 1 by testing the same two quantifiers, only thistime the modal must expressed deontic necessity. The results show that, for both typesof quantifiers, D-inferences could be derived independently of NU-inferences. Whilethe results for every essentially replicate those reported in [7], the results for must arenew and go against the aforementioned claim (ii). In addition, the response time resultsfrom both experiments show that D-inferences are associated with response delay effectsin the opposite direction to those observed for regular scalar implicatures in similar tasks([4, 5]). We argue that these findings about the time course of D-inferences raise a newchallenge for an implicature-based approach to these inferences.1 IntroductionA sentence like (1), where disjunction occurs in the scope of a nominal universal quantifier, givesrise to the inference in (1-a), schematised in (1-b). Inferences like (1-a) are generally referred toas ‘Diversity’ or ‘Distributive’ inferences. We will use the more compact label ‘D-inferences’.(1) Every visible box contains a blue ball or a yellow ball. ∀x(Ax ∨ Bx)a. ; Some visible box contains a blue ball and some contains a yellow ballb. ∃xAx ∧ ∃xBxWhile the existence of D-inferences is uncontroversial, the status and source of these infer-ences is still very much debated. Traditionally, they are considered scalar implicatures, derivedthrough negating the alternatives to (1) in (2-a) and (2-b) ([8, 12]).(2) a. Every visible box contains a blue ball. ∀x(Ax)b. Every visible box contains a yellow ball. ∀x(Bx)∗We would like to thank Maria Aloni, Moysh Bar-Lev, Itai Bassi, Kyle Blumberg, Ivano Ciardelli, MarcoDegano, Melissa Fusco, Paloma Jeretic, Matt Mandelkern, Paolo Santorio, Florian Schwarz, and BenjaminSpector. This research was supported by the Leverhulme Trust Grant RPG-2018-425Diversity with Universality Ramotowska, Marty, Romoli, Sudo and BrehenyThis account of D-inferences makes a straightforward prediction: D-inferences shouldalways arise in combination with the additional ‘Negative Universal’ inferences (henceforth,NU-inferences) in (3-a), schematised in (3-b).(3) a. ; Not all visible box contain a blue ball and not all of them contain a yellow ballb. ¬∀xAx ∧ ¬∀xBxThere is experimental evidence that challenges this prediction ([7]): Sentences like (1) wererobustly judged as true, when their D-inferences were true while their NU-inferences werefalse (e.g., in situations where all of the visible boxes contain a yellow ball and some of themcontain also a blue one), and rejected when their D-inferences and NU-inferences wereboth false (e.g., in situations where all of the visible boxes contain a yellow ball and none of themalso contains a blue one). These results suggest that D-inferences can arise independently ofNU-inferences. In fact, no evidence for the presence of NU-inferences was found.In addition to this finding, it has also been considered, based on introspective judgments,that for disjunction in the scope of a modal quantifier the D-inference cannot be observedindependently of the NU-inference ([3, 7, 9, 13]). For example, the D-inference associatedwith a sentence like (4) is considered to always arise with its NU-inference, (4-b).(4) The mystery box must contain a blue ball or a yellow ball. □(A ∨ B); The mystery box might contain a blue ball and it might contain a yellow balla. ♢A ∧ ♢Bb. ¬□A ∧ ¬□BCompeting accounts have been proposed for the D-inferences of disjunction under univer-sal nominal quantifiers ([1, 2, 7, 10, 14]), but crucially, some of them predict disjunction underuniversal modals to give rise to D-inferences independently of NU-inferences ([14, 10, 1]),while others do not ([2, 3]).To adjudicate between these opposite predictions, we carried out two sentence-picture veri-fication experiments. The results replicate the earlier experimental finding that D-inferencescan be observed without NU-inferences for disjunction under nominal universal quantifiers,but they also show that for disjunction under universal modals, D-inferences can be observedindependently, contrary to what has been assumed in the literature. In addition, we observethat D-inferences trigger effects on response times that are the opposite of what is typicallyobserved for scalar inferences, which we take to be a challenge for accounts of D-inferencesas a special case of scalar inferences.2 Experiments2.1 ParticipantsFor each experiment, 100 native speakers of English were recruited online through Prolific.ac([11]) using the same pre-screening criteria (first language: English, nationality: UK/US, coun-try of birth: UK/US, prior approval rate: above 90%). Participants were paid £2.20, andaverage completion time was about 13 minutes. Participants gave written informed consent.Data were collected and stored in accordance with the provisions of Data Protection Act 2018.2Diversity with Universality Ramotowska, Marty, Romoli, Sudo and Breheny2.2 Materials and DesignExamples of sentence-picture items used in Experiment 1 and Experiment 2 are given in Table 1.In each experiment, sentences were presented as being uttered by one of two characters, Samor Mia, and the participants’ task was to decide whether or not the character’s utterance wasa good description of the picture accompanying that utterance.Example sentence Example picture ConditionExperiment 1nominalEvery visible box containsa yellow ball or a blue ball.TrueA AB B ?Target-1A AB A ?modal epistemicThe mystery box must containa blue ball or a yellow ball.Target-2A AC A ?FalseA CD C ?Experiment 2nominalEvery box containsa yellow ball or a blue ball.TrueA AB BTarget-1A AB Amodal deonticSam must pick a box witha blue ball or a yellow ball.Target-2A AC AFalseA CD CTable 1: Example items illustrating the experimental conditions for the target sentences inExperiments 1 and 2. There were 6 instances of each condition per quantifier type.In Experiment 1, target sentences were nominal sentences like (1), involving the determinerevery, and modal sentences likes (4), in which must was used as an epistemic modal. Picturesinvolved three open (‘visible’) boxes, each of which contained one or two colored balls, andone closed (‘mystery’) box, with a question mark on it. Participants were instructed that themystery box always has the same contents as one of the three visible boxes. They were asked todecide whether or not the sentence was a good description of what’s inside the relevant box(es).In the control conditions, target sentences were paired with pictures that make them either trueor false, regardless of the inferences under investigation. In the target-1 conditions, theywere paired with sentences that make their NU-inferences false, but their D-inferencestrue. In the target-2 conditions, they were paired with sentences that make both theseinferences false. Quantifier type (2 levels: nominal, modal) and picture type (4 levels: true,target-1, target-2, false) were crossed to obtain a 2× 4 within-subjects factorial design.Experiment 2 was built on Experiment 1 by adapting the materials and instructions fromExperiment 1 so as to extend our investigation to deontic modality. Target sentences werenominal sentences with the determiner every, similar to those tested in Experiment 1, andnovel modal sentences in which the modal must was used to express deontic necessity. Pictureswere similar to those used in Experiment 1, except that they involved only three boxes, all open.3Diversity with Universality Ramotowska, Marty, Romoli, Sudo and BrehenyIn the instructions, participants were told that the two characters were playing together andthat, depending on the game, one of them either had to describe what’s inside the boxes or hadto pick one of the boxes. The first game scenario was used to test nominal sentences and thesecond to test modal sentences. Depending on the game scenario, participants had to decidewhether the utterance was a good description of the box(es) or of the character’s options. Therest of the design of Experiment 2 was identical to that of Experiment 1 in all respects.2.3 ProcedureThe procedure was the same in both experiments, provided the differences in instructions wedescribed above. nominal trials and modal trials were presented in two separate blocks, theorder of which was counterbalanced between participants. Each block started with a shorttraining and included additional control items designed to check that subjects understood theinstructions properly and had no problem understanding the quantifiers and connectives usedin the target sentences. In each block, trials were presented in a random order. Participantsprovided their answers by clicking one of two response buttons, labelled ‘Good’ and ‘Bad’,respectively. Participants’ responses and response times were recorded on each trial.2.4 Data treatment8 participants in Exp. 1 and 6 participants in Exp. 2 were excluded prior to analyses becausetheir performance on the control items was below the pre-established threshold of 80% accuracy.For both experiments, participants’ responses and response times on the target nominal andmodal trials were analysed using mixed-effects regression models with the maximal randomeffect structure justified by the design and supported by the data. Figure 1 shows the meanacceptance rate to the target sentences by quantifier type and experimental condition. Figure 2shows the mean RTs by quantifier type, experimental condition and response type.2.5 ResponsesOverall, the response patterns for modal sentences were similar to those for nominal sentences.In both experiments, both sentence types gave rise to intermediate acceptability ratings in theirtarget-2 conditions (comparisons with true and false: all |β|s> 4.34, all ps< .001), witha bimodal distribution of by-subject mean ratings (unimodality tests: all Ds> 0.10, ps< .001)while they were both uniformly accepted in their target-1 conditions (comparisons with true:all |β|s< 0.42, ns; comparisons with false: all |β|s> 12.5, ps< .001; unimodality: all Ds< 0.05,ns). Responses to nominal and modal sentences in the target-1 and target-2 conditionswere further compared by testing the interaction between quantifier and picture types. Asignificant interaction was found in Experiment 1 (β = −1, 80, SE = .49, p < .001), showingthat the acceptability contrast between target-1 and target-2 was greater in the modalthan in the nominal cases; no such an interaction was found in Experiment 2 (β = −0, 42,SE = .32, p = 0.19). These results replicate the main results from [7] and establish that thekey contrasts previously observed for nominal quantifiers reproduce with modal quantifiers.2.6 Response timesIn both experiments, nominal and modal sentences showed response delay effects in the oppo-site direction to those commonly observed for regular scalar implicature in similar acceptabilityjudgement tasks. Specifically, in the target-2 conditions, participants took significantly longer4Diversity with Universality Ramotowska, Marty, Romoli, Sudo and BrehenyFigure 1: Mean acceptance rate by quantifier type and experimental condition. The distributionof by-participant mean ratings is visualised by a histogram, the grand mean by a thick bar withits value on top and the 95% CI around it, and the median by a cross.to accept than reject both nominal and modal sentences (all βs> 333, all ps< .01). By con-trast, in the control conditions, participants were significantly faster to accept than reject thesesame sentences (all βs< −241, all ps< .001) while they were equally fast at rejecting them intheir false and target-2 conditions in both experiments (all |β|s< 38, ns) and equally fastat accepting them in their true and target-1 conditions in Experiment 1 (all |β|s< 14, ns).Figure 2: Mean RTs (in ms) by sentence type, experimental condition and response type. Errorbars represent 95% CIs. RTs were analysed by considering correct responses in the control true(‘accept’) and false (‘reject’) conditions, ‘accept’ responses in the target-1 conditions, andboth ‘accept’ and ‘reject’ responses in the target-2 conditions.3 DiscussionWe experimentally investigated the D-inferences and NU-inferences associated with dis-junction embedded under the nominal universal quantifier every and the universal modal mustin its epistemic and deontic readings. Our results show that disjunction under both every andmust gives rise to D-inferences independently of NU-inferences. Also, the D-inferenceswere overall observed to the same degree of robustness. The results for every essentially repli-cate previous finding by [7], but the results for must go against the claim in the literature5Diversity with Universality Ramotowska, Marty, Romoli, Sudo and Brehenythat for disjunction under a universal modal, the D-inference is always observed in tandemwith the NU-inference ([3, 7, 9, 13]). Our results, therefore, present a challenge to accountsof D-inferences which cannot derive them without NU-inferences for disjunction underuniversal modals.1In addition, we found response delay effects in the opposite direction to those generally ob-served for regular scalar implicatures in similar verification tasks ([4, 5] among others). Specif-ically, we found that it takes more time to accept than reject nominal and modal sentenceswith disjunction in cases where their D-inferences and NU-inferences are false, comparedto cases where only their NU-inferences are. This processing slowdown, established acrossmultiple comparison points, suggests that, in contrast to what is usually observed with regularscalar inferences, it is the suspension rather than the derivation of the D-inferences that iscostly. This finding is thus challenging for accounts of D-inferences as a special case of scalarinferences.2Taken together, the present findings suggest that we need a theory of D-inferences thatcan derive them independently of NU-inferences for both disjunction under universal nominalquantifiers and disjunction under universal modals. The reaction time data also suggest thatthey should not be treated as a species of scalar inferences, but as a different type of inference,e.g., as proposed by [1].Lastly, we would like to remark on the discrepancy between our experimental finding andthe introspective judgments reported in the literature for disjunction under universal modals.Most notably, [7] observes that (5) feels misleading in a context in which there is a requirementto wear sneakers in the gym (while running shorts are optional).(5) You are required to wear sneakers or running shorts.Importantly, the oddness of (5) is unexpected if the sentence readily receives a reading witha D-inference but without an NU-inference. We do not mean to deny the validity ofthe observation regarding (5). Rather, we take it as suggesting that D-inferences cannotalways be observed independently of NU-inferences and that a theoretical account of D-inferences ultimately must be able to explain what the relevant restrictions consist in andwhy these restrictions exist.References[1] Maria Aloni. Logic and conversation: the case of free choice. Semantics and Pragmatics, 15:5,2022.[2] Moshe Bar-Lev and Danny Fox. Free choice, simplification, and Innocent Inclusion. NaturalLanguage Semantics, 28:175–223, 2020.[3] Richard Jefferson Booth. Independent alternatives: Ross’s puzzle and free choice. PhilosophicalStudies, 179(4):1241–1273, 2022.1All such accounts can also derive a reading without either inference. This reading, however, is compatiblewith both our target conditions and, thus, it cannot account for the differences we found between the two.2A similar challenge has been raised to implicature accounts of free choice inferences, which also do not giverise to the delay effects found with regular scalar implicatures ([6]). The main response to that challenge locatesthe source of this difference between free choice inferences and regular implicatures in the type of alternativesinvolved in their derivation. In particular, the idea is that the difference would be linked to the fact that regularimplicatures are based on alternatives constructed by replacing lexical items in the asserted sentence, while freechoice inferences only involve alternatives which are part of the asserted sentence. We note that this responsedoes not extended straightforwardly to the case of D-inferences. This is because, at least in some implicatureaccounts of D-inferences (e.g. [2]), the alternatives involved in their derivation are constructed with lexicalsubstitutions in the same way as in the case of regular implicatures.6Diversity with Universality Ramotowska, Marty, Romoli, Sudo and Breheny[4] Lewis Bott, Todd M Bailey, and Daniel Grodner. Distinguishing speed from accuracy in scalarimplicatures. Journal of Memory and Language, 66(1):123–142, 2012.[5] Lewis Bott and Ira Noveck. Some utterances are underinformative. Journal of Memory andLanguage, 51(3):437–457, 2004.[6] Emmanuel Chemla and Lewis Bott. Processing inferences at the semantics/pragmatics frontier:disjunctions and free choice. Cognition, 130(3):380–396, 2014.[7] Luka Crnič, Emmanuel Chemla, and Danny Fox. Scalar implicatures of embedded disjunction.Natural Language Semantics, 23(4):271–305, 2015.[8] Danny Fox. Free choice and the theory of scalar implicatures. In Uli Sauerland and Penka Stateva,editors, Presupposition and Implicature in Compositional Semantics, pages 71–120. Palgrave, 2007.[9] Melissa Fusco. Deontic modality and the semantics of choice. Philosophers’ imprint, 15, 2015.[10] Serge Minor. Implicatures and events: A study of distributive inferences. under review, 2022.[11] Stefan Palan and Christian Schitter. Prolific. ac—a subject pool for online experiments. Journalof Behavioral and Experimental Finance, 17:22–27, 2018.[12] Uli Sauerland. Scalar implicatures in complex sentences. Linguistics and Philosophy, 27(3):367–391, 2004.[13] Geoffrey Sayre-McCord. Deontic logic and the priority of moral theory. Noûs, pages 179–197,1986.[14] Yasutada Sudo. Scalar implicatures with discourse referents: A case study on plurality inferences.Linguistics and Philosophy, to appear.7

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