Figuring Out Root and Epistemic Uses of Modals

This paper investigates how children ﬁgure out that modals like must can be used to express both epistemic and “root” (i.e. non epistemic) ﬂavors. The existing acquisition literature shows that children produce modals with epistemic meanings up to a year later than with root meanings. We conducted a corpus study to examine how modality is expressed in speech to and by young children, to investigate the ways in which the linguistic input children hear may help or hinder them in uncovering the ﬂavor ﬂexibility of modals. Our results show that the way parents use modals may obscure the fact that they can express epistemic ﬂavors: modals are very rarely used epistemically. Yet, children eventually ﬁgure it out; our results suggest that some do so even before age 3. To investigate how children pick up on epistemic ﬂavors, we explore distributional cues that distinguish roots and epistemics. The semantic literature argues they differ in “temporal orientation” (Condoravdi, 2002): while epistemics can have present or past orientation, root modals tend to be constrained to future orientation (Werner 2006; Klecha, 2016; Rullmann & Matthewson, 2018). We show that in child-directed speech, this constraint is well-reﬂected in the distribution of aspectual features of roots and epistemics, but that the signal might be weak given the strong usage bias towards roots. We discuss (a) what these results imply for how children might acquire adult-like modal representations, and (b) possible learning paths towards adult-like modal representations.


INTRODUCTION
Almost half of the world's languages have modal forms that can be used to express different "flavors" of modality (van der Auwera & Ammann, 2005).For instance, in English, a must sentence like (1) can express an epistemic necessity (Alex is probably a tofu eater), or various kinds of "root" necessities (i.e.non epistemic ;Hoffmann, 1966): deontic (Alex is required to eat tofu), bouletic (Alex wants to eat tofu), or teleological (Alex needs to eat tofu).This flavor flexibility seems to be particular to "functional modals", i.e., words from a grammatical category seemingly dedicated to modality (in English, modal auxiliaries like must and semimodals like have to), in contrast to "lexical modals", i.e., verbs (e.g.require), adjectives (e.g.likely) or adverbs (e.g.maybe), which also express possibilities and necessities, but typically in a single flavor.
(1) Alex must eat tofu.This paper investigates when and how children solve the difficult mapping problem of linking functional modals like must to both root and epistemic flavors on the basis of their input representation, by looking at how modals are used in naturalistic speech to and by young children.By laying bare the mapping problem for modals, and articulating and quantifying what cues are available to resolve it, this work paves the way for asking what capacities-linguistic, conceptual or pragmatic-must be in place, and what expectations learners must have about how modality is expressed in natural language. 1he previous acquisition literature shows an asymmetry in children's functional modal productions: children start with root meanings around age 2, but fail to produce epistemic meanings until at least age 3 (Kuczaj & Maratsos, 1975, Wells, 1979, Stephany 1979, Astington, 1993, Cournane, 2015a, 2015b, 2021 a.o.).This "epistemic gap" (Cournane, 2015a(Cournane, , 2015b) ) is often taken to reflect a conceptual lag (Sweetser 1982, Shatz & Wilcox, 1991, Astington, 1993, Papafragou, 1998), or grammatical development (Heizmann, 2006;Cournane, 2015aCournane, , 2015b;;Veselinović & Cournane, 2020).This literature tacitly assumes that once the concepts or grammar are in place, mapping modal words to the meanings they express is trivial.In this paper, we show that the mapping problem for functional modals is in fact far from trivial, and yet, we also show, that children resolve it even sooner than originally thought.We investigate how they might do so.
To appreciate the depth of the mapping problem for modals, let us first consider the case of their close relatives, attitude verbs, such as think and want.Because these verbs express abstract concepts, learners cannot rely on visual cues to learn their meanings, and their acquisition thus need to rely heavily on linguistic cues.As such, they have been hailed as parade cases for "syntactic bootstrapping" (Landau & Gleitman, 1985;Gleitman, 1990), the hypothesis according to which children learn a word's meaning by exploiting principled links between its meaning, which is closed to observation, and its syntactic distribution, which is more easily observable.In the case of want and think, children would observe that think takes finite complements, want nonfinite complements, expect finiteness to track a belief vs. desire split amongst attitude verbs, and from there infer that want expresses desire and think expresses belief (Hacquard & Lidz, 2018).Indeed, children do seem to exploit finiteness to figure out attitude verb meanings (Harrigan et al., 2019).
For modals, the situation is even more complicated, since the same word can express both desire (root) and belief (epistemic) meanings.Can children also rely on syntactic cues to figure out modal meanings?Given that modals take nonfinite complements, the bootstrapping strategy for attitudes might lead them to infer root (desire) meanings.This mapping to root meanings might be reinforced by two factors.First, the concepts underlying root meanings may be more easily accessible than those underlying epistemic ones (Bartsch & Wellman, 1995;Perner et al., 2003;De Villiers, 2005;Steglich-Petersen & Michael, 2015), even if both types of concepts may be in place in infancy (Onishi & Baillargeon, 2005;Southgate et al., 2007).Second, root meanings may be particularly salient in discourse: root modals can routinely be used to perform requests, a discourse function which young children seem particularly attuned to (Shatz, 1978;Spekman & Roth, 1985).Thus, various factors from syntax to pragmatic or conceptual salience may conspire for children to initially map modals to root meanings.And, once children have mapped a modal word to a root meaning, why not stop there?Why assume that modals can also be used to express epistemic flavors?Note that root and epistemic meanings are not mutually exclusive: if it's likely that Alex eats tofu, then she might well need to eat it.Thus, we cannot expect epistemicity to be self-evident because root meanings would be false whenever epistemic meanings are intended.How, then, do children pick up on epistemic flavors?
The answer to this question depends in part on whether children expect their language to have flavor flexible modals.After all, not all languages do.Children might get a head start and expect flavor flexibility for functional modals, if, as Hacquard & Cournane (2016) argue, there is a principled link between the functional status of modals (e.g., the fact that they are auxiliaries) and their ability to express both root and epistemic flavors.If true, realizing that a modal like must is functional from its auxiliary status (marked morpho-syntactically) would make available the possibility that it is flavor flexible.A learner equipped with this expectation would get flavor flexible semantics for free based on syntactic category, but would still need to ascertain whether both types of flavors actually occur for any given modal (some modals only express one type, e.g., can is not epistemic unless negated).Alternatively, children may not have any particular assumptions about functional modals with respect to flavor flexibility.They would then only postulate flavor flexibility on the basis of positive evidence.For example, they might first hypothesize that must is deontic, and would then have to revise this hypothesis when they discover that it can also have epistemic uses.Under either assumption, children need to realize which of their modals express epistemic flavors, in the absence of clear physical cues.Are there any linguistic cues that would give away epistemic flavors?We address this question by looking for environments in which root flavors are ruled out, or at least unlikely.
We first probe the depth of this mapping problem, using a larger and more densely sampled corpus than prior studies, by examining (i) children's input: do children hear modals with both root and epistemic uses?Which modals, in what proportions, and in what linguistic environments?And (ii) children's productions: do children produce both root and epistemic modals before age 3? Which modals, in what proportions, and in what linguistic environments?Our input results show that the mapping problem is made worse by the fact that parents overwhelmingly use modals with root flavors in speech to children, even more so than reported for adult speech (Ruppenhofer & Rehbein, 2012;Rubinstein et al., 2013;Marasović et al., 2016).And yet, our child productions results show that already at age 2, English-learning children show evidence of having learned that at least some functional modals can express epistemic flavors.
We investigate possible morphosyntactic cues that differentiate root and epistemic modals in principled ways, which children could exploit to infer epistemic meanings.After reviewing several candidates from the literature, we argue that the most promising is a constraint on the "temporal orientation" of modals (Condoravdi, 2002), according to which root modals tend to be future-oriented, while epistemics can be past or present-oriented (Werner 2006;Klecha, 2016;Rullmann & Matthewson, 2018).What makes this constraint particularly compelling from a bootstrapping perspective, is that first, it is supposed to be principled (it is motivated by informativity considerations: whether the modal makes a non trivial meaning contribution), and second, it has clear morphosyntactic correlates, as temporal orientation is largely determined by aspect (Condoravdi, 2002).When the modal's "prejacent" (the proposition that the modal combines with) consists of a bare eventive (i.e., with no overt aspect), as in (2), a modal can have both future and present orientation (the running follows or overlaps the time of possibility), and both root and epistemic interpretations are possible ('Alex is allowed to run'; Alex is a potential runner').However, with a stative prejacent, as in (3), or with a progressive, as in (4), the orientation is present; with a perfect, as in ( 5), the orientation is past: in such cases, only epistemic interpretations seem possible.
Future/Present TO root, epistemic (3) Alex may love running.
Past TO ?? root, epistemic This constraint suggests a possible avenue for discovering epistemic flavors: by observing modals with present or past orientation, that is, modals with perfect, progressive, or stative prejacents, a learner privy to the temporal orientation constraint could infer that the modal must be interpreted epistemically.We explore the viability of this proposal by examining the aspectual distribution of roots and epistemics.We find that while the constraint is well-reflected in child-directed modal usage, the signal is weakened by the strong bias towards root uses.We discuss ways in which children could still exploit these aspectual cues, through additional cues from different kinds of subjects with roots and epistemics, and contextual cues.We explore possible learning paths towards adult-like representations.
The rest of this paper is organized as follows.In section 2, we provide further background about how root and epistemic flavors differ in meaning and distribution, notably in their interactions with subjects and aspect.Section 3 presents an overview of the modals used by two-year-olds and their mothers in the Manchester Corpus (Theakston et al., 2001) on the CHILDES database (MacWhinney, 2000), and shows that adults and children produce both roots and epistemics with an overall bias towards root meanings, which is even more pronounced for children.Section 4 examines the aspectual profiles of roots and epistemics in the input, and shows that they differ significantly.However, because of the large frequency skew towards root uses, learners may only be able to use this distributional difference if they use it in conjunction with contextual cues.In section 5, we discuss the distribution of subjects and show that roots and epistemics differ in the kinds of subjects they take, though, again, the large frequency skew towards root uses makes it difficult to use reliably.In section 6, we discuss how children might be able to exploit aspectual and subject cues, and sketch possible learning paths towards adult-like modal representations.Section 7 concludes.

Modal flavors: Meaning differences
In a language like English, the same modal words can be used to express both root and epistemic flavors: (1) can express an obligation or a likelihood.In the adult grammar, this is a case of ambiguity rather than generality, as can be shown using Zwicky & Sadock (1975)'s Identity of the Senses tests.A sentence like Alex and Billy must eat tofu can either mean that both Alex and Billy are required to eat tofu, or that both are likely tofu eaters, but mixed readings are disallowed.In this way, modals pattern with ambiguous terms like bank, rather than general terms like teacher.The source of this ambiguity is however a matter of debate: is it lexical ambiguity (several musts), polysemy (several senses of must), or is there just one must, whose restriction is provided by context, and leads to different flavors, as in the classic Kratzerian account?Kratzer (1981) analyzes possibility and necessity modals as existential and universal quantifiers over possible worlds.In this framework, a modal's domain of quantification is determined by two conversational backgrounds, a modal base (MB), and an ordering source (OS): the modal quantifies over the best worlds of MB, i.e., those that most closely approach the ideal set by OS.
In this system, roots and epistemics differ in MB: epistemics take an epistemic MB, which picks out worlds compatible with the available evidence.Roots, which include ability, circumstantial, deontic, bouletic, and teleological modals, take a circumstantial MB, which picks out worlds compatible with relevant circumstances.Root modals are further differentiated via an OS, which provide different orderings: deontic, teleological, bouletic.3Some authors (e.g.Condoravdi, 2002) add a third "metaphysical" (or "historical") MB, which picks out worlds that share a common history but diverge in their future.Others reanalyze purported metaphysical modality as involving a circumstantial MB, like roots (Abusch, 2007(Abusch, , 2012;;Thomas, 2014).What flavor a modal expresses then depends on what MB and OS the context makes available.Yet, epistemic and root modals seem to differ systematically in their interactions with elements like tense and aspect, in ways that seem to go beyond contextual differences.This has led some to give up on the unified Kratzerian account, and postulate distinct epistemic and root lexical entries for functional modals like must. 4 Here we can remain agnostic as to whether there happens to be one or several musts in the adult grammar: under either view, the learning puzzle remains for how children figure out that the same string can be used to express distinct flavors, and not just general possibility or necessity.And under either view, the distributional differences between roots and epistemics provide potential avenues for children to pick up on epistemic flavors: if a modal appears in an environment where root meanings are illicit, it could cue them in that the modal expresses an epistemic flavor.We turn to these possible constraints next.

Constraints on modal flavors
There are two kinds of interactions for which roots and epistemics have been argued to systematically differ.These could potentially be exploited for a syntactic bootstrap of epistemic flavors: the first involves subjects, the other tense and aspect. 5Turning to subjects first, root and epistemic modals were initially thought to differ in argument structure, with roots treated as control predicates, and epistemics as raising predicates (Ross, 1969;Jackendoff, 1972;Lightfoot, 1979;Zubizarreta, 1982;Roberts, 1985;Brennan, 1993), based on an apparent thematic relation between root modals and their subjects.Were learners to expect a thematic relation between a root modal and its subject, the appearance of an expletive subject would be a strong indication that the modal is epistemic.However, it was later shown that root modals can occur with expletive subjects, and generally behave like raising predicates (Hackl, 1998;Bhatt, 1999;Wurmbrand, 1999).The special relation that roots seem to have with their subject need not be syntactic, but could stem from a root modal's need to be semantically anchored to one of the prejacent event's participants, not necessarily its subject (Hacquard, 2006;Kratzer, 2012).This is illustrated in (7).Weather it can appear with epistemic might (7a), but seems degraded with can, which can only receive root interpretations in positive contexts (7b).However, the sentence improves when a location is explicitly mentioned (7c).Thus, while there may not be a difference in terms of a syntactic requirement, we might still expect roots and epistemics to differ in the kinds of subjects they tend to combine with.We examine their distribution in section 5, and assess whether differences could potentially be suggestive to the learner.
b. ??It can rain.c.It can rain hard in this part of the world Hacquard (2006) Epistemic and root modals have also been argued to differ in temporal properties, both in terms of Temporal Perspective (the time at which the modality is evaluated) and Temporal Orientation (the time at which the prejacent is evaluated relative to the modal's time of evaluation).In terms of Temporal Perspective (TP), many argue that while root modals' TP is set by tense, epistemics always have a present TP (Groenendijk & Stokhof, 1975, Stowell, 2004, Abusch, 1997, Hacquard, 2006, a.o.).This is often taken to follow from a 5 Roots and epistemics are also argued to differ in their interactions with negation and with each other, but neither seems promising for acquiring English modals: (i) the scope of negation is not transparent and is subject to lexical idiosyncrasies (e.g., negation appears below modal auxiliaries, but can be interpreted above some: epistemic may scopes over negation, deontic may scopes under, must scopes over negation under both interpretations); (ii) double modals do not occur in the variety of English we examine.These interactions could potentially be more useful for other languages.
difference in scope: epistemics scoping above tense, and roots below, in accordance with Cinque's hierarchy (Cinque, 1999).This can be illustrated with semi-modal had to, which, when interpreted deontically, must refer to a past obligation, but which seems to refer to a current necessity when epistemic: (8), for instance, expresses that "it's necessary given what we know now that there were a hundred people there".
(8) There had to be a hundred people there.Stowell (2004) Not everyone agrees that epistemics must have a present TP (see von Fintel & Gillies, 2010 and Rullmann & Matthewson, 2018 for arguments they do not).But even assuming they must, it's unclear that this could help learners discover epistemic flavors.To make use of TP, learners would first have to notice that, despite surface appearances, the modal's time of evaluation in (8) is present.They would further need to realize that the current necessity is not a mere past necessity that still happens to hold at utterance time.This seems rather implausible, especially since epistemic had to is virtually absent in the input.6Thus, TP seems like an unlikely avenue for discovering epistemic flavors.
The difference in Temporal Orientation (TO) in roots and epistemics seems more promising, given its pervasiveness and robust morpho-syntactic correlates.As discussed in the introduction, root modals have been argued to be restricted to future TO, unlike epistemics, which can have present and past TO (Werner 2006;Klecha, 2016;Rullmann & Matthewson, 2018).7 Thus, while Alex may run can express a deontic possibility about a future run, Alex may have run and Alex may be running, which trigger a past and a present TO respectively, seem to only be epistemic.If past and present TO are constrained to epistemic modality, learners could in principle discover epistemic readings by noticing modal uses with present or past TO.
The exact underpinnings of the temporal orientation constraint (TOC) are debated in the literature, but it is generally taken to follow from informativity considerations preventing vacuous uses of modals, where a modal statement would be indistinguishable from its unmodalized counterpart.The first such principle, called the Diversity Condition (DC), was proposed by Condoravdi (2002) to explain why metaphysical modals, in contrast to epistemic modals, seem restricted to future TO.The DC requires that a modal's prejacent p not be "settled" amongst the worlds of the modal base (MB): the MB has to contain both p and non p worlds.Because the past and the present are settled, the same facts hold throughout worlds that share a history (Condoravdi's "metaphysical alternatives").This means that, with present or past TO, the worlds of a metaphysical MB cannot differ with respect to p, violating the DC.The future, on the other hand, is not settled, hence the worlds of the MB can differ as to whether p holds in the future.The DC, on the other hand, does not constrain epistemic MBs, which pick out worlds compatible with what is known.What we know about the past or the present may leave some uncertainty about p, hence, even with past or present TO, an epistemic MB can have both p and non p worlds.Condoravdi's proposal was later extended to all root modals (Werner 2006;Klecha, 2016;Rullmann & Matthewson, 2018): roots take a circumstantial MB, which picks out worlds compatible with a set of circumstances.Circumstances are supposed to be settled up to the present, thus circumstantially accessible worlds are indistinguishable until then, and therefore cannot differ with respect to p in the past and in the present, in violation of the DC.
While some take the TOC to be a semantic requirement that applies to all root modals (Werner 2006, Klecha, 2016, Rullmann & Matthewson, 2018), others point to possible counterexamples, and assume that the TOC is more of a pragmatic tendency (Matthewson, 2012;Thomas, 2014;Harr, 2019).One type of counterexample involves deontic modals, as in (9).8 (9) Sam ought to be at church.(Thomas, 2014) Another type of counterexample involves "actuality entailments" (Bhatt, 1999), which arise when a root modal combines with perfective aspect, and triggers an implicative inference that the prejacent occurred in the actual world.Actuality entailments were first shown in languages like Hindi or French which distinguish perfective and imperfective aspect overtly in the past, as shown in the French example in ( 10): (10) entails that Jane took the train.Given that circumstances are settled, Jane should take the train in all of the MB worlds, in violation of the DC (Matthewson, 2012).The TO is present, in violation of the TOC, as the time of the possibility and that of the prejacent event overlap exactly (Mari & Martin, 2007).
(10) Jane a pu prendre le train, #mais elle ne l'a pas pris.(Hacquard 2009) Jane could-pst-pfv take the train, #but she did not take it.
Importantly for our purposes, actuality entailments also seem to occur in English in the present, when an ability modal takes a prejacent with a perception verb (Dieuleveut, in progress), as illustrated in ( 11): (11) has an implicative reading according to which the speaker sees Venus.We will see that such actuality entailments are relatively frequent in speech to children, and present the biggest challenge to the exploitation of the TOC for the discovery of epistemic modality.
(11) I can see Venus from where I'm standing.(Dieuleveut, in progress) While examples like ( 9)-( 11) violate the original DC, some argue that they still obey some informativity principle that preserves its spirit, which essentially prohibits uses of modals that would be equivalent to their non-modal alternatives (e.g.Thomas', 2017 Modal Economy).For instance, despite their actuality entailments, ( 10) and ( 11) convey an additional meaning compared to their unmodalized counterpart, for instance, an inference that the ability described is not trivial (Bhatt, 1999).
Thus, while the exact nature and basis of the TOC are still a matter of active debate, there is general consensus that epistemic and root modality tend to differ in temporal orientation, and that this tendency is principled: root modality tends to be restricted to future orientation, because of informativity considerations ruling out trivial uses of modals.If learners expect speakers to avoid trivial meaning contributions, in particular of modals, they could use the TOC to infer epistemic meanings: observing a possibly trivial use of a modal, because of its present or past TO, could alert them to the possibility that the modal has a different flavor than the one they initially ascribed to it.But for the TOC to be used in such a way, a modal's TO would have to be observable from the way modals are used in children's input.In section 4, we examine the morphosyntactic footprints of the TOC in English, to assess whether its exploitation could be a viable strategy to uncover epistemic flavors of functional modals.

MODAL INPUT AND CHILD PRODUCTIONS
To get a sense of the kind of modal talk children hear and produce, we ask how frequently parents and children express epistemic vs. root modality, how frequently modality is expressed using lexical vs. functional modals, and among functional modals, how often are they used with root vs. epistemic meanings.

Methods
The Manchester Corpus consists of 12 child-mother dyads, recorded for one hour in play sessions, twice every three-week period, spanning an age range from 1;09 to 3;00.9We used the data from ages 2;00 to 3;0010 to get the results from the epistemic gap period specifically, with age 2 as the reported onset of functional modal usage.We chose this corpus for its relative density and uniformity of sampling sessions during the epistemic gap period.The density allows us to get a more accurate picture of rare early child uses of epistemics than any previous study, and the uniformity across 12 dyads allows us to generalize observed patterns above and beyond individual differences, particularly since the speakers are all from the same speech community and time period.
All utterances containing modal words were extracted (81,854 of 564,625 total utterances).Modals were coded for syntactic category, as in (2) (functional: auxiliaries, quasiauxiliaries; lexical: adverbs, adjectives, verbs), and for flavor (root, epistemic, future), by hand, by reading all contexts.With respect to epistemic flavor coding, there is controversy about what modality might expresses: does it only express epistemic possibility (as was assumed in Kratzer, 1981Kratzer, , 1991)), or can it also express metaphysical possibility (Condoravdi, 2002)?We coded might as epistemic, but we also provide numbers for potentially metaphysical uses, and return to this issue in section 4. Functional modals shall, will, and going to express future meanings; we have included them here for completeness but we will exclude them from further analyses in section 3 as our focus here is on those modals that express root and epistemic modality.The complete list of modals occurring in our sample is provided in appendix 1.
(12) Modal lemmas by syntactic category: Functional Aux = can, could, may, must, should, might, shall, will, would Quasi-Auxiliaries (QA) = have to, got to, ought to, supposed to, going to.Lexical V = epis: know, think, seem . . .; root: want, order, let us . . .Adv = epis: maybe, perhaps, probably . . .Adj = epis: sure, certain . . .root: able, capable . . .epis/root: possible . . .We do not differentiate amongst various root flavors (e.g.ability, teleological, deontic), and leave the question of how children figure out that modals can be used to express a variety of different non-epistemic flavors for future work, though see section 4 for a brief discussion.Repetitions (repeating oneself or what someone else said) from both children and adults were excluded except in the aggregate (Table 1 and 3).All data is available at osf.io/v9ure/.

Results
In the following subsections, we provide the results of our corpus study on lexical and functional modals, first for the adult data, followed by the child data.

Input: Mothers' modal production
The results by syntactic and semantic category are summarized in Table 1.We find that for lexical modals, both epistemic and root modality are well attested in the input (4.6% of all mother utterances contain a lexical epistemic vs. 3.7% for lexical root modals).Functional modals are well-represented in the input: 13% of all adult utterances contain a functional modal.Examples of input utterances for each category are given in (13).
(13) Examples of modal utterances from the input11 a. Lexical epistemic: Maybe there are no trousers.Mother (Ruth 2;00) b.Lexical root: why did you want to become a mouse?Mother (Aran, 2;06) c.Lexical root/epistemic:12 If that's possible which I do not think it is?Mother (Warren, 2;05) d.Functional epistemic: must epi have just fallen out, Nicole.Mother (Nicole, 2;09) e. Functional root: I'm sure polar bears can root ride in cars.Mother (Warren, 2;03) f.Functional future: I'll take it away Mother (Liz 2;00) To investigate how often functional modals are used for root and epistemic flavors in the input, we focused in on modals that can theoretically express root or epistemic flavors (can, could, may, must, should, have_to, got_to, supposed_to, ought_to and might). 13Table 2 shows the distribution of root vs. epistemic flavors for each modal.We find that, overall, functional modals are used much more frequently to express root (90.7%) over epistemic (9.3%) modality.This effect is driven by the fact that the most frequent modals (can, have to) are nearly always used to express root modality.Our results further show that modals that can express both root and epistemic flavors in principle are mostly used for one or the other: can, could, have to, got to, should, supposed to and ought to express root modality more than 90% of the time.Must, might and may are more often used with epistemic flavor (the latter is infrequent in this corpus).For might, we report in parentheses the number of potentially metaphysical readings.

Interim discussion on the input
To sum up, looking at both lexical and functional modal input, children hear a fair amount of epistemic modal vocabulary, and of utterances with functional modals.Whatever is responsible for the purported epistemic gap, it is not a lack of exposure to epistemic vocabulary.14However, for functional modals we see that epistemic and root flavors are not equally well-represented in the input.The higher frequency of root uses might make it challenging for learners to see that functional modals can also express epistemic modality.The proportion of epistemics is substantially lower than in the corpus studies on adult speech mentioned in the introduction (Ruppenhofer & Rehbein, 2012, Rubinstein et al., 2013, Marasović et al., 2016, see also Coates, 1983, Klages & Römer, 2002, Hacquard & Wellwood, 2012, Baker et al., 2012).For some of these studies, the difference might be due to the fact that they involve written language (Coates, 1983 andHaan, 2011 show that modal use differs substantially in written vs. spoken language).But even studies on spoken adult language show a higher proportion of epistemics: in the MASC corpus of American English, Marasović et al. (2016) find 26% epistemics in almost 2,000 functional modals (n = 508, their Table 5, p.17).The difference between these studies and ours could be a difference between American vs.British English.Or it could be a difference between child-and adultdirected speech: this study is the first to investigate child-directed speech.The Manchester corpus only includes speech to children (see fn. 8), thus, it's possible that children hear more epistemics from overheard speech.Note, however, that in a similar study probing modal use in Dutch input (van Dooren et al., 2019), which did include overheard speech, the proportion of epistemics was still extremely low (1.7% epistemics out of 10,903 modal sentences).
We turn next to children's productions.What modals do children produce and with what flavors?How well do children's productions mirror that of their parents?3.2.3.Child modal production Children produce a fair number of utterances with lexical and functional modals (4.6% and 3.4% of total utterances, respectively), though proportionally less so than their mothers.These results are summarized in Table 3, and examples are given in (13).Children also produce proportionally fewer lexical epistemics than their mothers (0.8% of total utterances vs. 4.6%), and less than they produce lexical root modals (the reverse pattern of adults).Thus, while young children may be less disposed to express epistemic modality, they do produce some lexical epistemics before age three, in line with O'Neill & Atance's (2000) and Cournane's (2015aCournane's ( , 2015bCournane's ( , 2021) )   must epis be gone.(John, 2;09)

Interim discussion on child production
To sum up, our results show that children overwhelmingly produce roots over epistemics, even more so than their mothers.However, they do produce some epistemics (both lexical and functional) at age 2, suggesting that the epistemic gap reported in the literature (Kuczaj & Maratsos, 1975, Wells, 1979, Stephany 1979, Astington, 1993, Cournane, 2015a, 2015b, 2021 a.o.) may be an effect of the lower sampling density of previous corpus studies.
The lower overall rate of epistemics confirms Cournane's (2015aCournane's ( , 2015b) ) finding that when children start to produce epistemic functional modals, they are at lower rates than their mothers.This suggests that children in this age range have a root bias.Specific support for this bias comes from the results on must.While mothers in our sample use must mostly with epistemic flavor (64.8%), children mainly use it with root flavor (84.6%).Yet, two-thirds of the children in our sample already use some functional modals with both root and epistemic flavors before age 3, suggesting they have already solved the complex mapping problem for these modals.This finding further bear on our understanding of children's conceptual maturity.The absence of epistemic modals in children's speech was originally attributed to a lack of the underlying concept (e.g., Papafragou, 1998 ties it to immature theory of mind).To the extent that mastery of the word reflects mastery of the underlying concept, our results suggest that the concepts underlying epistemic modality are in place before age 3.

Discussion of input and child production
Our corpus results show that the way adults talk about possibilities and necessities may make it challenging to see that functional modals can express both root and epistemic flavors, as they are mostly used to express the former.As we saw earlier, other factors already tip the scale towards root flavors: modals take nonfinite complements like verbs that express root meanings (e.g., want), the concepts underlying root meanings may be more accessible, and root meanings may be particularly salient to children attuned to (indirect) requests.Once children assign root meanings to modals, how do they discover that they can also express epistemic meanings?
Our child production results suggest that some children pick up on functional modals' epistemic flavors already at age 2, i.e., a year earlier than has been previously reported in the literature (see also Cournane, 2021).How do they manage to do this, given the low frequency of adult epistemic uses of functional modals?We next explore whether there are reliable morpho-syntactic cues that children could exploit that reflect the constraints on root modality discussed in section 2. We turn to aspectual features in section 4, and to subjects in section 5.

DISTRIBUTIONAL DIFFERENCES BETWEEN ROOTS AND EPISTEMICS: TOC AND ASPECTUAL CUES
As we saw in section 2, the most promising restriction on the availability of root flavors is the Temporal Orientation Constraint (TOC), according to which roots can only be future-oriented, while epistemics can be past-or present-oriented.The TOC could alert learners to a possible epistemic flavor, if they notice a non-future oriented use of a modal where a root interpertation should not be possible.But how can learners pick up on temporal orientation if they do not yet know the meaning of the modals?How 'observable' is temporal orientation?Fortunately, temporal orientation may largely be determined by aspectual features of the modal's prejacent.Given their morpho-syntactic nature, aspectual features should be easier to observe than temporal orientation itself, and could help learners pick up on temporal orientation first, assuming they understand the temporal contributions of these aspectual features, and on modals' epistemic meanings second, assuming they expect modal uses to be governed by something like the Diversity Condition and the TOC.
English modals take nonfinite complements, where temporal orientation (TO) is largely determined by both grammatical aspect (perfect, progressive), and lexical aspect (stative or eventive) (Condoravdi, 2002).As we saw in (2)-( 5), repeated below as ( 18)-( 21 We hypothesize that due to their morpho-syntactic nature, aspectual cues will be easier to observe than TO itself.Thus, as an operational assumption, we recast the TOC as a Stativity Constraint (SC).Note that both the perfect and the progressive can be viewed as stativizers.We can thus subsume the aspectual restrictions as follows: (22) Stativity Constraint (SC): root modals are incompatible with stative prejacents (bare stative prejacents or prejacents with progressive or perfect aspect) We thus want to see whether the SC is illustrated in children's input in a clear enough way to help them discover epistemic meanings.For the SC to be useful, two requirements must be met.First, epistemic modals should occur frequently with stative prejacents.Were epistemic modals to mostly take bare eventives, learners could not rely on the SC to discover epistemic uses, as these might be hard to distinguish from root ones.Second, the SC needs to be robust and track the TOC closely: there should be very few counterexamples to it (i.e., root modals with stative prejacents), and whatever counterexamples there are should somehow be differentiable from epistemic uses through other cues.
Importantly, the TOC and the SC are not equivalent.Indeed, it is possible for the prejacent of a root modal to be stative, but still future-oriented.This is illustrated in (23), whose use seems to grant a permission for a future state of having an apple.A further type of counterexample to the SC but not to the TOC, are counterfactual uses, illustrated in (24).With counterfactual uses, the modality is not epistemic, but either metaphysical or circumstantial, depending on one's theory.17Such uses are counterexamples to the SC because of the presence of the perfect, which, at least on the surface, seems to be part of the prejacent.However, counterfactual uses are not counterexamples to the TOC, because the resulting interpretation is one where the modal is interpreted with a past temporal perspective, but a future orientation.The perfect, which on the surface appears under the modal, seems to be interpreted above it, backshifting the modal's time of evaluation (Condoravdi, 2002): (24) expresses a past possibility for a (counterfactual) win in the future of that past.
(24) Alex could have won, but she did not.
future TO root/metaphysical Hence, various cases could threaten the usefulness of the SC, the operational counterpart of the TOC.First are counterexamples to the TOC itself, discussed in section 2: actuality entailments (e.g., I can see Venus from here), which are present-oriented, as well as other cases where root modals are present or past-oriented (e.g., Sam ought to be at church).Second are counterexamples to the SC, but not the TOC: future-oriented statives and counterfactuals, as we just saw.In the next section, we plumb the input to test the usefulness of both the TOC and the SC as a means to discover epistemic meanings, by looking at (i) whether counterexamples to the TOC and the SC occur, and with what frequency; and (ii) whether epistemic modals reliably take stative prejacents.We document counterexamples to the TOC and to the SC separately, because it can help us sort out what children need to exploit either: if the SC is not robust, but the TOC is, then learners should not be able to rely solely on aspectual cues.Furthermore, as the first corpus study examining the robustness of the TOC,18 naturally-occurring counterexamples may be useful independently of learnability considerations to understand their cause, be it the Diversity Condition, or one of its descendants.

Methods
To assess the distribution of aspectual cues with respect to modal flavor, all input sentences with a functional modal were coded for grammatical and lexical aspect.For grammatical aspect, we coded for presence of a perfect or a progressive in the prejacent.For all prejacents that lacked an overt grammatical aspect marker, we coded for lexical aspect: whether the prejacent was stative or eventive.The tests used were the classic tests from Dowty (1979).The full list of predicates is in Appendix 2.
available in complement of stop, eventive b.I see the car.
simple present as non-habitual, stative c.Suddenly, I saw the car.
inchoative, eventive Sentences with elided prejacents like (25) were excluded (total = 56 roots, 30 epistemics) as it could not be unequivocally determined from the sentence whether be or have was the main verb or the auxiliary, both options being possible in the Manchester dialect.

Results
Turning first to grammatical aspect, we expect that only epistemics should embed perfect or progressive aspect.Our results, reported in Table 5, show that this expectation is borne out: functional modals combine with embedded aspect ∼ 10% of the time when epistemic, but 1% of the time when root. 19To test for the distribution of grammatical aspect per flavor, we employed the glmer function in the statistical package lme4 in R (R Core Team, 2013).
The data was fitted into a generalized linear mixed (logit) model 20 using the maximum likelihood method (Laplace Approximation) (Baayen, 2008;Dixon, 2008;Matuschek et al., 2017).Fixed effects included flavor (epistemic vs. root) and aspect (grammatical aspect vs. bare), as well as the interaction between flavor and aspect.The 12 child/adult pairs were entered as a random effect (flavor ∼ grammatical aspect + (1| child)).The model shows a significant difference of the distribution of grammatical aspect depending on flavor, ß = 2.47, Z = 21.5, p < 0.001 * * * .Turning to lexical aspect, we expect root modals to mostly combine with bare eventives, and epistemics to combine with both eventives and statives.Our results are reported in Table 6.We find that epistemics combine with statives ∼ 68% of the time, and roots ∼ 13% of the time, putting aside perception verbs.To test for the distribution of lexical aspect per flavor, we employed the glmer function in the statistical package lme4 in R (R Core Team, 2013).The data was fitted into a generalized linear mixed (logit) model 21 using 19 An interrater reliability analysis using the Kappa statistic was performed to determine consistency among raters in grammatical aspect and lexical aspect.500 modal sentences were double-coded, and the interrater reliability for the raters for grammatical aspect was found to be κ = 0.91 and for lexical aspect κ = 0.97 (Landis & Koch, 1977).20 We also ran a chi-square test of goodness-of-fit with Yates continuity: grammatical aspect occurrence (with, without) differs significantly by flavor (root, epistemic), X2(1) = 705.87,p < .0001* * * .Chi-square tests are standard in child corpus research, so we provide this for comparison.However, these may be problematic because of lack of full independence of observations: each observation from the same speaker may be affected by being from that speaker (likewise from the same subcorpora dyad).21 Chi-square test of goodness-of-fit with Yates continuity: Lexical aspect type (eventive, stative) differs significantly by flavor (root, epistemic), X2(1) = 1052, p < .0001* * * .the maximum likelihood method (Laplace Approximation) (Baayen, 2008;Dixon, 2008;Matuschek et al., 2017).Fixed effects included flavor (epistemic vs. root) and lexical aspect (stative vs. eventive), as well as the interaction between flavor and aspect.The 12 child/adult pairs were entered as a random effect (flavor ∼ lexical aspect + (1| child)).The model shows a significant difference of the distribution of lexical aspect depending on flavor, ß = 2.48, Z = 395, p < 0.001 * * * .Finally, to see how frequently each flavor occurs with a stative/stativized prejacent, we combine the information from grammatical and lexical aspect.The results are reported in Table 7.The pattern we see is similar to what we see above: epistemics and roots show in fact the opposite distribution from each other (∼70% statives for epistemics and ∼ 70% eventives for roots).To test for the distribution of flavor (epistemic vs. root) by stativity, we employed the glmer function in the statistical package lme4 in R (R Core Team, 2013).The data was fitted into a generalized linear mixed (logit) model using the maximum likelihood method (Laplace Approximation) (Baayen, 2008;Dixon, 2008;Matuschek et al., 2017). 22In this and later models, we treat flavor of usage as the dependent variable, and observable features of the utterances as independent variables, essentially asking if the observable condition leads to a significantly greater likelihood of one flavor over the other; for the child, does what they can observe predict flavor of use?We treated aspect (stative vs. eventive) as a fixed effect and the 12 child/adult pairs as a random effect (Flavor ∼ stative + (1|child)).The model shows a significant difference of the distribution of flavor depending on aspect, ß = 2.58, Z = 41.5, p < 0.001 * * * .
Our results show a clear difference in the aspectual properties of the prejacents of epistemic and root modals: epistemics mostly combine with statives, roots mostly with eventives.However, a substantial number of roots still occur with stative prejacents, especially if we treat perception verbs as statives.In the rest of this section, we take a closer look at these statives, to assess whether they are counterexamples to the TOC, or just to  the SC.We then discuss ways in which learners might distinguish counterexamples to the TOC/SC from genuine epistemic uses.We first provide a breakdown of prejacent types by epistemic modal in Table 8.We find that uses of might account for the vast majority of epistemics with bare eventives (437/448).As mentioned earlier, the status of might is controversial: some treat it as unambiguously epistemic, others argue that it can be metaphysical.So far, we have treated all might as epistemic.If we exclude cases where might is arguably metaphysical, the correlation between epistemicity and stativity is extremely strong: practically all epistemic prejacents (98%) are stative (see Table 11).
We now turn to root modals with stative prejacents.Table 9 shows the distribution of aspectual properties by modal.
To assess whether these counterexamples to the SC are also counterexamples to the TOC, we examined each root modal use individually, to assess its temporal orientation.A summary is provided in Table 10.In a nutshell, we find that most root uses are future- oriented, except for a class of counterexamples involving perception verbs (whose stativity status is controversial).
Putting aside perception verbs, we find that the majority of stative prejacents are futureoriented (1,985 out of 2,092).Most of these (1,381 instances) are performative uses of root modals to issue commands, or give or request permissions, as illustrated in (27).Commands and requests can only be felicitous if the state requested or allowed does not already hold, and hence are typically future-oriented.( 27) a. May I have a spoon to eat it with?(Mother, Nicole 2;02) b.You have to have more than three.
(Mother, Aran 2;08) Another type of future oriented uses involves 'imaginary play' (Garvey & Kramer, 1989) (220 instances), where a speaker decrees that a new state is about to hold.This is illustrated in (28), where the area that the mother is pointing to is to become the carpark.(28) Mother: where's the carpark, Anne? [ . . . ] (Mother, Anne 2;02) Mother: I think this could be the carpark, could not it in here?
We also found 338 instances of counterfactuals, illustrated in (29).These include most instances of root modals with perfect prejacents, and most uses of modals should and supposed to.Finally, 46 cases contain an explicit future reference, illustrated in (28).
(29) a.You could have said hello.
(Mother, Anne 2;03) c.You're not supposed to be eating the stethoscope.
(Mother, Ruth 2;02) (30) We'll have to have the hoover out after this.(Mother, Ruth 2;00) This leaves a total of 107 counterexamples to the TOC out of the 2,092 instances of root modals with stative prejacents (other than perception verbs).Out of these 107 counterexamples, 20 arguably do not involve root modals per se, but rather quantificational modals (Heim, 1982, Brennan, 1993, Portner 2009), where the modal's contribution is to provide quantification either over individuals as in (31a), or over situations, as in (31b): (31a) seems to express that some men are teachers, (31b) that the subject is sometimes excitable.Of the remaining 97 cases, 23 involve idiomatic expressions like I cannot believe..., or I cannot stand.... Two are cases of actuality entailments.The remaining 61 cases seem to mostly involve necessity modals (25 instances), as in (31a), or negated possibility modals (20 instances), as in (32b).The full list can be found in Appendix 3. To sum up, putting aside perception verbs, counterexamples to the TOC are extremely rare (about 0.7% of all root modals).
(31) a. Men can be teachers.
(Mother, Joel 2;07) Turning now to the rather frequent perception verb prejacents, we find that the vast majority occur with can (2,418/2,526; see Table 9).When can combines with a perception verb, it triggers an actuality entailment (Dieuleveut, in progress), illustrated in ( 33): (33) seems to imply that the mother actually sees lots of footprints.Such actuality entailments are not only counterexamples to the SC (if we consider perception verbs as statives), but also to the TOC: the TO is present, since the time of the state has to coincide with the time of the ability.Table 11 summarizes our findings in terms of TO.We find a strong correlation between flavor and aspect: most roots are future-oriented (virtually all, if we exclude perception verbs), most epistemics are not (virtually all if we group metaphysical might with root modals).
Recall the two requirements of a syntactic bootstrapping strategy 23 relying on the SC, rather than the TOC: first, epistemics have to reliably combine with stative prejacents to be discoverable.Second, there need to be very few counterexamples, i.e., root modals with stative prejacents.We see that the first requirement is clearly met: epistemics mostly take stative prejacents.However, the second requirement is not: while most root modals combine with eventive prejacents (∼70%), 30% combine with stative prejacents, half of which are future-oriented, half of which involve perception verbs.Given the big skew towards root meanings, the number of stative prejacents of root modals is in fact twice as much as stative prejacents of epistemic modals.Hence, if learners were to rely strictly on aspectual cues, they could be misled into treating root modals with stative prejacents as epistemic, or treat epistemic uses as mere noise.
However, the strategy might still be exploitable if learners can supplement aspectual cues with contextual cues.As we saw, counterexamples to the SC fall into two categories: (i) actuality entailments with perception verbs, which violate both the SC and the TOC; and (ii) future-oriented statives, which only violate the SC.In the next section, we argue that both types may involve particularly salient contextual cues, which differentiate them from genuine epistemic uses.In a nutshell, one characteristic of epistemic modality is that its use requires contextual uncertainty-or at least a lack of direct evidence-about the truth of the prejacent.Crucially, the counterexamples to the SC that we find in our corpus differ clearly from epistemics in this respect: in the case of actuality entailments with perception verbs, the truth of the prejacent should be rather salient, since it involves direct perceptual evidence; in the case of future-oriented statives, which mostly consist of commands, imaginary play, and counterfactuals, the falsity of the prejacent should also be fairly obvious.

Can contextual cues distinguish epistemics from roots when aspectual cues fail?
Turning first to future-oriented statives, the context should be particularly clear about their future orientation, despite the potentially misleading aspectual cues: this is because in the types of cases found in the corpus, the state expressed by the prejacent can clearly be inferred not to hold.First, requests and permissions are felicitous only if the state requested does not already hold.In (27a), for instance, the speaker must clearly not have a spoon for her request to make sense.Similarly, 'imaginary play' involves proposals about the future.In ( 28), the mother is suggesting that from speech time onward, the child and her pretend that a certain object serves as something other than its usual function.As for counterfactuals, if children understand that counterfactuality is about future (unrealized) possibilities from a point in the past, they should be able to use the falsity of the prejacent at speech time to infer its futureorientation.A cursory examination of the context suggests that the falsity of the prejacent is particularly salient in these cases.For instance, in (29a), the child should be aware that he did not say 'hello'.Finally, cases with explicit reference to a future time like (30), wear their futurity on their sleeve.
Turning next to actuality entailments which occur when can combines with perception verbs, the context should be particularly clear that the state expressed by the prejacent holds, especially since these cases involve perception, making it likely that there are visual or other sensory cues in the situational context (Landau & Gleitman, 1985).In (33), for instance, both the mother and the child should be able to see actual footprints (the mother uttering look, further suggests that she is drawing the child's attention to the footprints).
Thus, for the vast majority of counterexamples to the SC, the situational context seems to make either the truth or the falsity of the prejacent particularly salient.In comparison, for epistemics with stative prejacents, illustrated in (34), the context should make neither the truth nor the falsity of the prejacent salient, given that epistemics typically require contextual uncertainty about the prejacent, and are infelicitous when there is direct evidence for it (von Fintel & Gillies, 2010;Mandelkern 2017, a.o.): asserting that it must or might be raining when standing in pouring rain is infelicitous (von Fintel & Gillies, 2010).
(34) a.I think it may be an apple.

Discussion
In this section, we focused on whether TO could help learners discern epistemic uses.If root modals are restricted to future-oriented prejacents, a past or present TO could alert learners that the modal expresses an epistemic meaning.For this strategy to work, the input has to provide clear cues that distinguish epistemic and root uses in terms of TO.Given that TO might be hard to identify in context, especially when the intended flavor is not known, learners may be able to exploit aspectual cues that correlate with TO and are easier to observe: for learners to exploit the TOC, there need to be clear cases of stative prejacents with epistemics, and none with roots.
Our results show that aspect does distinguish roots and epistemics: roots mostly combine with eventives, epistemics mostly combine with statives.However, we do find a significant number of roots with statives.While the numbers are proportionally low relative to all root uses, and proportionally significantly lower than stative prejacents of epistemics, they are rather frequent in terms of raw counts.In fact, because of the skewed distribution of roots and epistemics, the raw counts of roots with a stative prejacent (n = 2,208) is actually higher than the raw count of epistemics with a stative prejacent (n = 1,119), a number made much worse if we include perception verbs (1,132 epistemics vs. 4,734 roots).What this means is that children would roughly have only a 1 in 3 (or 1 in 5) chance of getting an epistemic when they hear a modal with a stative prejacent.Given the sheer number of roots, the counterexamples to the SC thus call into question its usefulness.
However, when we take a closer look at counterexamples to the SC, we find that many are not counterexamples to the TOC: they involve future-oriented statives, in the context of offers, requests, imaginary play, or counterfactuals.Whereas picking up on TO in general might be difficult, we believe that these instances are cases where the context should be particularly clear that the prejacent does not yet hold, making its future orientation salient.We do find a well-attested class of counterexamples to the TOC, and not just the SC, namely present-oriented cases of actuality entailments triggered by ability modals with perception verb prejacents.But here as well, there should be particularly clear contextual cues: given the perceptual nature of these examples, the truth of the prejacent should be particularly salient.In this way, both types of counterexamples differ from epistemic uses, for which the truth of the prejacent should be contextually uncertain.Thus, our results argue that aspectual cues could provide a promising avenue to help learners bootstrap epistemic meanings for their modals, but only if they consider them in conjunction with correlated situational cues.If children can pick up on these situational cues, they may be able to exploit the SC to discover epistemic flavors in the following way.If, for the purposes of identifying modal meanings, children ignore uses of modals where the prejacent is either clearly true or clearly false, then aspectual cues are highly predictive of flavor: root modals are virtually always future-oriented, but epistemic modals are not.That children would initially ignore modal uses where the prejacent is either clearly true or clearly false when inferring modal meanings seems plausible: after all such uses are cases where neither possibility nor necessity seems at issue, and where the modal's contribution might thus be particularly obscure.Once children are more secure in their modal meanings, they may be in a better position to figure out the stranger cases, like counterfactual uses.Acknowledging there are also grammatical factors at play (e.g., the complexity of if . . .then conditional constructions), children produce (Tulling & Cournane, 2019) and understand (see McCormack et al. 2018;Nyhout and Ganea 2019, i.a.) counterfactual possibilities relatively late compared to more basic modal possibilities.

Subject distribution of roots and epistemics
Recall that another possible distributional difference between roots and epistemics involves subjects: root modals seem to have a special relationship with their subjects, but not epistemics.This was originally thought to derive from roots being control predicates, but was later shown to be illusory, and perhaps due to root modality needing to be anchored to one of the prejacent's event participants.Because of this apparent connection with subjects, we might expect root modals to only rarely combine with expletive subjects.Hence, the presence of an expletive subject could potentially alert learners to the possibility (if not necessity) that the modal expresses an epistemic flavor. 24side from an asymmetry in availability of expletive subjects, we might further expect differences in subjects across root and epistemics both in terms of person and animacy.First, root modals be used to direct other people's actions, which might result in a tendency to have animate subjects, as these are typically the kind of entities that can act upon the wish, goal, or obligation expressed by the modal.Second, child-directed speech generally mainly contains verbs in 1 st and 2 nd person (Laakso & Smith, 2007:732).In conversations with others, root modals are expected to fall in line with this tendency as they encode wishes, goals, or obligations, typically those of one of the interlocutors.In contrast, epistemic claims about the speaker or the addressee may be marked, as speakers typically have direct access to the states or events they participate in ( ?? I must be eating).Given these considerations, we might expect roots to mostly combine with animate subjects, mostly 1 st and 2 nd person, while epistemics could more freely combine with 3 rd person subjects, inanimates, and even expletive subjects.

Methods
Subjects were coded for person (1 st , 2 nd , 3 rd ) for the 12 mothers in the input, and for animacy for a subset of 6 mothers.Animacy was coded as animate (35a) vs. inanimate (35b).Toys representing an animate being (dolls, animals) were coded as animate, as pretend play treats them as animate.For pronouns, we treated 1 st and 2 nd person pronouns as uniformly animate, and for 3 rd person pronouns, we coded for whether it referred to animate or inanimate agents (36a vs. 36b) if it was recoverable from the context.Finally, we coded for whether inanimate pronouns were expletive pronouns (37a) or pronouns referring to an event (37b).
(35) a. well can Mummy have a go on your seesaw?(Mother, Aran 2;00) b. no the xylophone must be somewhere else.

RESULTS
Turning to person first, we find that 77% of roots occur with a 1 st or 2 nd person subject while only 26% of epistemics do (Table 12). 25To test for the distribution of flavor by person, we employed a similar model as above. 26We treated person (1/2 vs. 3) as a fixed effect and the 12 child/adult pairs as a random effect (flavor ∼ person + (1|child)).The model shows a significant difference of the distribution of flavor depending on person, ß = 2.33, Z = 39.44,p < 0.001 * * * .
Turning next to animacy, the overall results are presented in Table 13. 27A breakdown per modal, can be found in Appendix 4. As expected, we find that while 37% of epistemics combine with a non-animate (inanimate, expletive, eventive) subject (n = 331), less than 5% of roots combine with these subject types (n = 377).To test for the distribution of flavor by person, we employed a similar model as above. 28Fixed effects included flavor (epistemic vs. root) and animacy (animate vs. non-animate), as well as the interaction between flavor and animacy.The 12 child/adult pairs were entered as a random effect (flavor ∼ stative + (1| child)).We treated animacy (animate vs. non-animate) as a fixed effect and the 12 child/adult pairs as a random effect (flavor ∼ animacy + (1|child)).The model shows a significant difference of the distribution of flavor depending on animacy, ß = -2.51,Z = 28.04,p < 0.001 * * * .The difference is not solely due to the majority of 1 st and 2 nd person subjects in roots (which are uniformly animate): if we focus in on 3 rd person subjects, we still find the contrast, as 54% of 3 rd person subjects with epistemics are non-animate, while 25% of 3 rd person subjects with roots are non-animate.

DISCUSSION
Our results show that epistemics and roots differ in the types of subjects they combine with for both person and animacy.Roots occur mostly with 1 st and 2 nd person subjects, which are also the most frequent subjects overall in this sample and in childdirected speech generally.Epistemics, on the other hand, mainly occur with 3 rd person subjects.For animacy, we see a similar pattern, which is unsurprising, as 1 st and 2 nd person subjects are animate.Roots combine overwhelmingly with animate subjects, epistemics combine most frequently with non-animate subjects.Looking more closely at 3 rd person subjects for animacy where the results are not confounded with person, we find that while most root 3 rd person subjects are animate, epistemics show more expletive and eventive subjects, and inanimate referents.Thus, we see an overall preference for animate subjects for roots, and a greater proportion of non-animate subjects with epistemics.
As with temporal orientation, we find a significant distributional difference between root and epistemic uses of functional modals.Can the learner make use of these distributional facts?A problem might be that given the overall usage bias towards root meanings, the raw number of root modals with a non-animate subject (n = 377) is actually higher than that for epistemics (n = 331).This asymmetry means that the likelihood of an epistemic interpretation when hearing a modal with a non-animate subject is about the same as that of a root interpretation.Hence, hearing a non-animate subject does not unambiguously direct the learner to an epistemic meaning.What is more, none of the combinations of subject, animacy, and flavor are actually ruled out.As discussed earlier, roots are acceptable with non-animate 3 rd person subjects.While rare (see Table 13), we did find instances of roots with inanimate (38), eventive (39), and expletive (40) subjects.While some could arguably count as animates (the tractor in (37b) might be personified by the speaker), ( 39) and ( 40 (Mother, Ruth 2;07) In sum, while the distribution of subject animacy over roots and epistemics clearly differ, the overall skew towards root meanings make animacy cues difficult to exploit on their own.

DISCUSSION: EXPLOITING ASPECTUAL AND PERSON CUES AND POSSIBLE LEARNING PATHS
Children need to figure out that modals can express both root and epistemic flavors, despite the fact that they mostly hear them with root uses, that the root concepts they express may be more accessible, and that modals' complements resemble that of verbs that express root meanings.How do children figure this out?Because modals express abstract concepts with few physical correlates, they are exactly the type of expressions that motivate syntactic bootstrapping, the hypothesis according to which learners exploit principled links between a word's syntactic distribution and its meaning.But modals present a challenging twist: the same word is used to express these different meanings.
In sections 4 and 5, we looked for syntactic features that could differentiate roots from epistemics, and found that they differ both in terms of the aspectual properties of their prejacents and their subjects: roots mostly combine with eventives, epistemics mostly combine with statives; roots mostly combine with 1 st and 2 nd person subjects, and epistemics with 3 rd person subjects, often non-animate.However, in both cases, the usage skew towards root interpretations calls into question the usefulness of these cues: even if roots mostly combine with eventives, the number of stative prejacents with roots is larger than the number of stative prejacents with epistemics.Similarly, roots mostly take animate subjects, but the number of non-animate subjects with roots is comparable to the number of non-animate subjects with epistemics.Can these cues nonetheless be exploitable, and if so, what would children need to exploit them?It's not entirely clear what learners might be able to infer from subject cues alone, given that the link between subject type and epistemic modality is rather indirect.It's possible that inanimate, or even 3 rd person, subjects would stand out, given their overall rarity in the input, and draw children's attention to the utterances in which they appear, even if it might not point them directly to an epistemic interpretation.
The aspectual cues might be more promising.While the number of counterexamples to the stativity constraint (SC) is significant, they largely fall into two categories: (i) statives that are nonetheless future-oriented, and thus in line with the temporal orientation constraint (TOC), which mostly occur in the context of requests, imaginary play, or counterfactuals; (ii) actuality entailments with perception verbs, which violate both the SC and the TOC.In both cases, however, we argued that there might be particularly salient contextual cues that distinguish them from epistemic uses, as either their truth (for actuality entailments) or falsity (for future statives) should be fairly obvious.In contrast, the prejacent should neither be obviously true nor obviously false with epistemics, as their use requires a lack of direct evidence.If children initially ignore modal uses when the prejacent is clearly true or clearly false, that is, where the modal's contribution is rather unclear, they should be able to exploit aspectual cues to infer modal flavor.
Can we really expect children to pay attention to temporal orientation, if the TOC is a violable pragmatic constraint, rather than a semantic requirement?If non future-oriented root modals are not semantically impossible, would learners necessarily postulate epistemic readings?We believe that even if the TOC is merely pragmatic (Matthewson, 2012;Thomas, 2014;Harr, 2019), it could still be useful in alerting learners to possible violations of the Diversity Condition (DC), especially in the absence of clear contextual cues that the prejacent is either true or false.And, given the lack of more reliable syntactic correlates, the TOC may well be the best pathway for discovering epistemic flavors of functional modals.
Could there be other, non-syntactic, routes to epistemicity?Again, because modals express abstract concepts, cues from the physical context are bound to be limited.However, children might be able to exploit pragmatic cues, which have been argued to play an important role in the acquisition of attitude verbs (Dudley et al., 2017;Lewis et al., 2017).From a very young age, children are sensitive to speakers' goals and their illocutionary intents.They notably seem to understand requests as such, whether direct or indirect (Shatz, 1978;Spekman & Roth, 1985).If children are attuned to illocutionary goals, they may be able to make inferences about a modal's underlying meaning.In particular, the ease with which deontic modals can be used to perform orders (Ninan, 2008, Portner, 2009) might alert learners that they express obligations.But, while this kind of pragmatic bootstrapping might be particularly useful for root meanings, it's not clear how much it would help pick out epistemic flavors.Indeed, the kind of indirect speech acts typically performed with epistemics are of an assertoric nature.But from a discourse function perspective, such acts may not stand out from direct assertions describing root possibilities or necessities: Noticing that a modal statement is used to assert might not readily give away its epistemicity.Thus, the morpho-syntactic route, and in particular, the one that exploits aspectual cues, might be the most promising to solve the mapping problem of modals.
What would it take for children to exploit aspectual cues, and is there evidence that the relevant linguistic, pragmatic and conceptual capacities are in place by age 2? First, learners should expect modals to be governed by something like the DC.Recall that the DC is often taken to fall from more general informativity considerations,29 so children would not necessarily need a modal-specific principle, but perhaps simply expect speakers to avoid trivial uses of (modal) expressions, and let such considerations guide their hypotheses about modal meanings.Second, by the time children are learning epistemic meanings for functional modals, they would need to be able to make certain aspectual distinctions, both at the grammatical and lexical level.The acquisition literature suggests that they likely have these abilities in place very early in syntactic-semantic development (Wagner, 2001;van Hout, 2016).Third, they should be able to exploit situational and pragmatic cues in conjunction with aspectual cues: being attuned to the truth or falsity of the prejacent when contextually salient, for instance, could help them pick up on counterfactuality or actuality entailments.Being sensitive to the request made with sentences like Can I have a cookie?could help them figure out the future orientation of the prejacent despite its stativity.Here again, there is experimental evidence that children understand indirect requests (Shatz, 1978, Spekman & Roth, 1985).Fourth, children would need to have access to the concepts underlying modal flavors, particularly the more-contested epistemic concepts.And here as well, there is growing evidence that they do (Onishi & Baillargeon, 2005;Southgate et al., 2007).Finally, for children to be able to exploit situational and pragmatic cues in conjunction with aspectual cues, they should be able to reliably track correlated statistical information in their input.None of the trackable properties of the input that we examined is categorically true, so children will need to learn possible meanings of modals from statistical tendencies.While the bulk of statistical learning work with children has focused on transitional probability and simple frequency of occurrence tracking (e.g., Saffran et al., 1996;Thompson & Newport, 2007;Smith & Yu, 2008;Vouloumanos, 2008), the results show that infants and toddlers are remarkably good at tracking statistical information (in language and elsewhere).Note that statistical learning alone would not garner the learning results that we see with children, as the assumptions we discuss for the TOC, and the grammatical combinations for root vs. epistemic interpretations, are necessary priors for the statistical learning to be fruitful.
How children make use of these cues depends on what assumptions they can make about modals.There are two possible starting points.First, children may not initially expect that modals can express multiple flavors.If so, they might first postulate root meanings for modals, and then discover, via aspectual cues that they must also express epistemic flavors, and update their lexicon accordingly.Alternatively, children may have a bit of a head start: they may expect functional modals to be able to express multiple flavors, if there is a principled link between a modal's functional status and flavor flexibility that children are privy to (Hacquard & Cournane, 2016).In that case, learners would have to discover which of their modals allow epistemic flavors.Even if there is a principled link between functional status and flavor flexibility, it may not always be obvious, as in the case of a semi-modal like have to, which syntactically resembles a regular verb.Thus, children may have differing expectations about different modals, and the acquisition of these modals may follow different paths.
When we look at individual functional modals, we see that some are more likely to be used epistemically, namely might and must: not only do these modals express epistemic modality more often, they further show a stronger signal for both aspectual and animacy properties.Previous corpus and production studies30 (Hirzel et al., in progress;Papafragou, 1998;Cournane, 2014Cournane, , 2015aCournane, , 2015bCournane, , 2021) ) show that might, and for some must, are the first functional epistemic modals children use.In our study, epistemic might and must comprise 98 of the 118 child functional epistemic productions.Once children pick up on the epistemicity of these modals, they may be able to use them to analogize epistemic meanings for other modals (Cournane, 2015a:94).Finally, realizing that a modal can express different subflavors of root modality, could make it easier to extend this one-to-many mapping to non-root flavors (but see Phillips & Knobe, 2018 and references therein for claims that children have trouble distinguishing modal concepts). 31inally, whether learners actually make use of aspectual cues remains an open question.A number of comprehension studies test children's sensitivity to aspectual properties of the prejacent when interpreting modals.The overall results show that while preschoolers access epistemic interpretations of necessity modals in Spanish, German, English, and Bosnian-Croatian-Serbian (BCS), the exact role of aspect in triggering epistemic readings is unclear because these studies: (a) only look at one prejacent type (Fond, 2003), (b) involve a small sample (Heizmann, 2006), (c) children had a weak deontic bias at age 3 and a strong epistemic bias by age 5 (Cournane, 2015a;Cournane & Pérez-Leroux, 2020).The best evidence for children using morphosyntactic cues32 for modal flavor comes from BCS, where 3-year-olds show adult-like preferences for root versus epistemic uses of morati (must) (Veselinović, 2019;Cournane & Veselinović, submitted), although older children show a non-adult epistemic bias.

CONCLUSION
In this paper, we investigated how children figure out that functional modals can express epistemic flavors in addition to root flavors.The previous acquisition literature suggests that while children produce root meanings early on, they are delayed in producing epistemic meanings (Kuczaj & Maratsos, 1975;Cournane, 2015aCournane, , 2015b)).Through this corpus study, we have explored child productions at a larger scale than previously done, to ask whether children really fail to produce epistemics before age 3. Our results show that the epistemic gap is in large part an input effect: the way speakers use functional modals makes it difficult to see that these modals can express epistemic meanings, as they mostly express root modality.Yet, children eventually pick up on epistemic meanings.Our results show that some seem to do so before age 3, a year earlier than was originally thought (see also Cournane, 2021).
We have argued that figuring out modal meanings-particularly for functional modals that can express different flavors-is no trivial task, especially given the way modals are used in the input.We have highlighted ways in which the linguistic context, and in particular, the way modals interact with aspects, could be helpful in providing cues that correlate with modal flavor.We however also showed that the large skew towards root uses make it impossible to exploit these aspectual cues on their own: children need to use them in conjunction with various pragmatic and situational cues.Thus, the acquisition of modal meanings seems to require a combination of sophisticated linguistic, pragmatic, conceptual, and statistical capacities.The fact that children eventually pick up on modal meanings, and might do so even before age 3, suggests that such capacities are in place very early in development.

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Context: mother and child are reading the book the Night before Christmas Mother: And I can see lots of footprints, look.The dog's made some footprints and the children have made some footprints.(Mother, Anne 2;04).

Table 1
Modal input by syntactic and semantic category (12 adults, % of total utterances)

Table 2
Functional modals by flavor, ordered by descending frequency (12 adults)

Table 4
Functional modals by flavor, ordered by descending frequency (12 children) age range of the purported epistemic gap.16The epistemic modals children produce are those that are most often used epistemically by adults: might, must, and may.Examples of epistemic child uses with context are given in (14).Some children (n = 8) produce the same modal with both root and epistemic uses within our sample, examples provided in (15)-(16).

Table 7
Aspect of the prejacent by modal flavor, input (n = 12 adults)

Table 8
Aspectual properties of the prejacent for epistemic uses by modal, input (n = 12 adults)

Table 9
Aspectual properties of the prejacent for root uses by modal, input (n = 12 adults)

Table 12
Subject person by modal flavor, input (n = 12 adults)

Table 13
Subject animacy by person and flavor, input (n = 6 adults) ) clearly show that roots can combine with non-animate subjects in child-directed speech).