The effects of aspect on meaning interpretation in the online and offline processing of modal constructions

2.1 Modal verbs and grammatical aspect

Due to the “exceptionally complex” (Abraham and Leiss 2012: 1) nature of modality, a broad spectrum of perspectives exists regarding “the relations and interactions between modality and other linguistic categories, such time/tense and aspect, evidentiality, and negation” (Squartini 2016: 50). In English, the distinction between the progressive and simple is an aspectual one (Greenbaum 1996: 81). As Anthonissen et al. (2016: 2) point out, in addition to the prototypical use of the English progressive as an aspectual marker, it can also have various other functions, such as expressing futurity or subjective attitudes of the speakers. In functionalist theory, it is no longer disputed that grammatical structures can carry semantic-pragmatic meaning. As Wright (1995: 155–157) notes, the progressive in certain contexts can play an epistemic role instead of being an actual aspect marker. Thus, the fact that the progressive can have expressive functions such as signaling subjective attitudes of irritation or annoyance (often accompanied by adverbials always, constantly, etc.) (Leech et al. 2009: 134) can be seen as another indicator of the subjective nature of the progressive. Its increasing use with stative verbs that also expresses subjective attitudes (e.g., I am loving this show) (Leech et al. 2009: 129) provides further evidence of this.

The subjective meaning of the progressive has been discussed not only in the context of the English language. For instance, Anthonissen et al. (2016) conduct a corpus study and investigate the German am-progressive. Even though German does not have a fully grammaticalized progressive form, the am V-inf sein ‘at V-inf be’ construction (e.g., Ich bin am Kochen ‘I am cooking’) can be used to indicate that the event is ongoing (Anthonissen et al. 2016: 1, 23–24). The study shows that, similar to other languages, the am-progressive conveys not only temporal and aspectual meanings but also (inter)subjective ones (Anthonissen et al. 2016: 4). The authors suggest that the use of the am-progressive in German is motivated by “an urge to express (inter)subjective qualifications” (Anthonissen et al. 2016: 24).

Given the inherently subjective nature of the progressive, when it occurs with a modal auxiliary, the resulting expression tends to convey a predominantly subjective meaning. Accordingly, in the sentence Kelly must be working, the modal must is more likely to be interpreted as an epistemic one as opposed to Kelly must work, which is more likely to evoke a deontic interpretation. Frajzyngier et al. (2008: 83) observe that when modal auxiliaries such as must are used with the progressive, they express an epistemic meaning. Similarly, Szymański (2020), building on Abraham’s (2008: 6) hypotheses on the compatibility of perfective aspect with root modality and imperfective aspect with epistemic modality,^[3] hypothesizes that the progressive in combination with should also evokes an epistemic reading (Szymański 2020: 376). However, based on the results of his corpus study on the semantics and the predicate structure of should, Szymański (2020) concludes that the modal should is not sensitive to the progressive (Szymański 2020: 389). These two diverging results raise the question of whether a change in aspect systematically results in a change of the meaning expressed by a modal. Different modals might show differential sensitivity to aspect in their interpretation, thus suggesting the existence of additional modal-specific, mid-level constructions (e.g., [SBJ must BE V_ing]_epistemic).

2.2 Modal verbs: a constructionist approach

The definition by Goldberg (2006: 5) on how to identify a construction suggests that linguistic units can be considered constructions if they are either non-predictable (or non-compositional) with regard to their form or function or if they are sufficiently frequent in a language. Not only lexical items but also grammatical constructions can be form-function pairings (Croft and Cruse 2004: 257; Goldberg 2013: 17). Even the most schematic constructions “are associated with a specific function that directly contributes to the interpretation of the utterance in which they occur” (Leclercq 2023: 68). This function and the semantic-pragmatic meaning of a construction can transfer onto the lexical items in a specific instantiation of the construction (Leclercq 2023: 68). This process of coercion (Hilpert 2019: 17–18) helps speakers to interpret even novel words in already familiar syntactic constructions. For instance, on the example of expressions as in (1), Goldberg (1995: 152) illustrates that the novel use of the verb help in the construction [SUBJ [V OBJ OBL]] is interpreted by speakers according to the caused-motion meaning conveyed by this schematic construction.

In an experimental study conducted by Kaschak and Glenberg (2000), 80 % of the participants attributed a meaning of transfer to innovative denominal verbs in ditransitive constructions such as He crutched her the apple, whereby crutch was interpreted as a ditransitive verb conveying the meaning of literal transfer (Kaschak and Glenberg 2000: 515). It can therefore be suggested that when modal verbs which are polysemous across epistemic and non-epistemic meanings (Depraetere and Cappelle 2023: 35) occur in combination with constructions that are mainly associated with a certain semantic-pragmatic meaning (e.g., the schematic progressive construction [SBJ BE V_ing] that mainly expresses an epistemic meaning; see Section 2.1), they will take on the meaning of that construction unless further contextual information is provided.

While a great amount of research has been published on modality (Depraetere et al. 2023: 1), it is understudied within CxG (Hilpert and Flach 2023: 254). Depraetere and Cappelle (2023: 44) discuss whether “contextual information [can] be considered part of the meaning of a lexical item” and argue that since context is important for the speakers’ accurate interpretation of modals, “the crux of a full understanding of modal meaning lies in the realization that we need to acknowledge more than just the actual modal verbs as carriers of modal meanings” (Depraetere and Cappelle 2023: 45). However, while the current study investigates the role of the syntactic context of modals, Depraetere and Cappelle (2023) discuss the lexico-pragmatic patterns in which they occur.

Wärnsby (2002) also employs a CxG approach to investigate the epistemic meanings of English and Swedish modal verbs (Wärnsby 2002: 1), attempting to find answers to the question as to how speakers correctly interpret modal meaning in terms of it being epistemic or deontic. Based on her corpus analysis, Wärnsby (2002) comes up with 13 schematic constructions that trigger epistemic over deontic meaning (e.g., [NP + AUX + V_state] as in John must love Mary or [NP + AUX + PROG + Vevent] as in John may be going out with Mary) (Wärnsby 2002: 5–6, 9). Wärnsby (2002) concludes that while the application of CxG to the investigation of modal verbs is possible, its fruitfulness is questionable as it suggests an extremely confusing and complex constructional network (Wärnsby 2002: 7–8). Commenting on Wärnsby (2002), Leclercq (2023: 75) notes that modal verbs, “like the rest of our linguistic knowledge, can be analysed in terms of complex networks of interconnected constructions”. Using modal auxiliaries, Hilpert and Flach (2023) also argue that “grammatical constructions can be fruitfully understood as networks of associative connections” (Hilpert and Flach 2023: 254). For example, according to a corpus-based analysis conducted by Hohaus (2020), the choice between different core modal verbs can be influenced not only by certain lexical elements but also by more abstract syntactic structures. More specifically, Hohaus (2020) finds that the modal should is most strongly associated with complement clauses, while the modal might occurs more often in relative clauses (Hohaus 2020: 102). Thus, according to the CxG approach, modal verbs should not be regarded as purely atomic and fully specific constructions (Hilpert 2016; Leclercq 2023: 71), as in the construct-i-con (i.e., “a large network of constructions” [Hilpert 2019: 2]), items can be stored redundantly in a myriad of constructional nodes and networks (Leclercq 2023: 72). For instance, based on the results from his corpus analysis on the modal verb must, de Haan (2012) argues that the uses of must with the progressive on the one hand and with a simple main verb on the other are stored as two separate constructions to represent the primarily epistemic meaning of the former and the primarily deontic meaning of the latter (de Haan 2012: 723). In addition to that, modals can be stored as fixed expressions like I can’t complain, which, on the pragmatic level, has a conventionalized and non-compositional meaning of ‘all in all, the situation is not bad’ (Cappelle and Depraetere 2016: 24). This non-compositional meaning is why I can’t complain can be considered a construction. Simultaneously, can can be represented in a maximally abstract way as a modal verb that expresses the general semantic meaning of possibility (Cappelle and Depraetere 2016: 28–29).

2.3 The scope of the study

Most studies on English modals either adopt a diachronic approach and describe their development from or in different periods of the English language (e.g., Plank 1984; Fischer 2004; Castillo 2022) or they synchronically describe the morphosyntactic and semantic features of modal verbs using a qualitative approach e.g., Mortelmans et al. 2009; Nuyts 2016). As for the use of CxG for the analysis of modal verbs, studies have mostly employed a corpus-based approach (e.g., Wärnsby 2002; de Haan 2012; Hilpert 2016; Hohaus 2020). Only a few papers have used experimental and quantitative approaches to synchronically investigate English modal auxiliaries (e.g., Papafragou and Ozturk 2007; Mifka-Profozic 2017; Flach et al. 2023). However, studies primarily explore the lexical environment of modals (e.g., Hilpert 2016; Depraetere and Cappelle 2023), while the current study is interested in their grammatical-aspectual context.

The present paper conducts an experimental study designed to investigate how aspect affects meaning interpretation of the broadly synonymous English modals must and should and the semi-modal have to (see Section 3.1). The results of this experiment are interpreted from the perspective of CxG, which allows us to make inferences about the schematicity degree and node structure of modal constructions. First, the study aims to provide confirmatory evidence that, contrary to the simple aspect, the progressive adds a subjective (i.e., epistemic) meaning to modals. Second, it investigates whether the strength of this effect varies across different modals. This provides insights into whether the epistemic meaning of modals is processed exclusively via a highly abstract constructional schema subsuming all modals (i.e., the higher-level construction [SBJ V_mod be V_ing]_epistemic)^[4] or whether language users rely on additional modal-specific subschemas (e.g., the mid-level construction [SBJ must be V_ing]_epistemic). Based on previous research, it is hypothesized that the latter is more likely, as modals exhibit varying degrees of frequency and subjectivity (e.g., must is more subjective than have to [Collins 2009: 15]). Only those modals that show idiosyncrasy with regard to either form or function or exhibit high frequency with the progressive are likely to have additionally formed mid-level constructions to express epistemic meaning. Thus, the paper seeks to emphasize the role of the grammatical-aspectual structure (in this case, the progressive) in the emergence of modal-specific, mid-level constructions.

To examine whether these assumptions are cognitively realistic, we conducted a psycholinguistic study. We suggest that the overarching research questions can be addressed by measuring the following variables: word-by-word RTs in an SPR task (online processing) combined with DTs and forced choices for judgment questions (offline processing) as indicators of the entrenchment of a certain modal with a certain grammatical structure and meaning as a construction. Online tasks can measure how the mental representation for a stimulus unfolds incrementally, tapping into more spontaneous and less controlled cognitive processes, whereas offline methods reveal the final interpretation, also taking into account metalinguistic, more strategic judgements.^[5]

On the one hand, if the hypothesis that the progressive adds an epistemic meaning to modals holds true, this should result in shorter RTs and DTs for progressive epistemic sentences compared to both progressive deontic and simple epistemic sentences as well as a higher number of epistemic interpretations for progressive sentences compared to simple ones in the judgement questions. On the other hand, if the second hypothesis that the strength of this effect varies across different modals is true, these results should show significant differences across modals (e.g., for the sentences Alex must be practicing more and Alex should be practicing more). This would suggest that the epistemic meaning of some modals in progressive sentences is processed via a higher-level schematic construction [SBJ V_mod be V_ing]_epistemic), exclusively, while others have developed their individual, mid-level constructions (e.g., [SBJ must be V_ing]_epistemic), which can be accessed more quickly and readily than other combinations of epistemic modals with the progressive. Drawing on insights from previous research (e.g., Krug 2000; Collins 2009; Hohaus 2020; Szymański 2020), it is hypothesized that the three modals in question will exhibit variations across all three variables.

3.1 Stimuli

The aim of the study is to compare the interpretation (i.e., deontic and epistemic) of three broadly synonymous auxiliaries must, should, and have to across two grammatical structures (i.e., simple and progressive). While must and should represent the core modals of the English language, have to is considered a so-called prototypical semi-modal (Quirk et al. 1985: 136–137; Leech et al. 2009: 91–92). The decision to explore exactly these three verbs was motivated by the fact that they are all necessity modals in PDE (Leclercq et al. 2023: 118), and, accordingly, can be used interchangeably in the constructed stimuli for full comparability. However, they still exhibit enough differences (i.e., are not absolute synonyms), especially in the case of epistemic modality. As Leclercq et al. (2023: 118) note, “epistemic and root necessity appear to be different with respect to which modals are suitable to encode these categories”. While root meaning can be expressed by each of the necessity modals, they exhibit variation in their capability to convey “(certain shades of) epistemic modality” (Leclercq et al. 2023: 118–119). Similarly, Collins (2009) writes that despite the general semantic similarity between several core and semi-modals, “there are shades of difference between them” (Collins 2009: 15). For instance, the semi-modal have to can be semantically affiliated to the core modal must; however, while the former is considered to express a more objective meaning, the latter is perceived to be more subjective (Collins 2009: 15).

In total, 96 sentences were created based on personal intuition and proofread and edited by two native speakers of American English (AmE). The design involved eight sets of stimuli, referred to as quads, as illustrated in Table 1. Each quad was constructed based on a 2 × 2 factorial design, crossing two independent variables: intended interpretation (epistemic vs. deontic) and aspect (simple vs. progressive). This design resulted in four distinct conditions within each quad:

1. Epistemic-Simple

2. Epistemic-Progressive

3. Deontic-Simple

4. Deontic-Progressive

Each condition featured a modal sentence (e.g., Alex must practice more), which was followed by a disambiguating sentence designed to clarify the intended meaning of the modal verb (e.g., He has lost his skills over the past few weeks). To further explore the influence of different modal verbs on interpretation, each quad was presented in three different versions, depending on the specific modal employed: must, should, or have to. This design was motivated by Traugott’s (1989: 32) examples where she illustrates the epistemic and deontic meanings of must by constructing two sentences with almost identical onsets containing the modal must but different offsets that clarify the deontic or the epistemic meanings conveyed by must (John must speak Japanese if he wants to be effective in his work to express deontic meaning; John must speak Japanese well, he is so successful to express epistemic meaning) (Traugott 1989: 32).

To allow conversion of the sentences into the progressive form, only non-stative verbs were selected as the infinitives following the modals. To control for confounding variables that might have an effect on the interpretation, the subjects in all sentences are animate, proper nouns in 3rd person singular form, and the sentences are all in present tense. The advantage of this design lies in its complete comparability across all levels of the collected results.

To determine whether the participants interpreted the modal auxiliary in its deontic or epistemic meaning, each sentence was followed by a judgment question, which offered two possible paraphrases of the modal, one with a deontic and the other with an epistemic meaning. The examples in (2) are the paraphrased versions for all the sentences in Table 1, i.e., for all three verbs (must, should, and have to) and for both structures (simple and progressive), with (2a) representing the deontic meaning and (2b), the epistemic meaning. The interpretation only refers to the modal sentence. They are all in present simple in order to not prompt any preference. The paraphrases of the deontic modals are based on the meaning explanation provided by the Oxford English Dictionary (OED) (s.v. “must, v.¹, sense II.2.a”), where the meaning of must is explained through expressions had to, was obliged to, it was necessary that (I etc.) should. However, given that should is one of the target words of the current study, the decision was made to exclude should from it is necessary that somebody should in the paraphrased sentences. Brief research was conducted in the corpus News on the Web (NOW) (Davies 2016) with the search strings ‘it BE necessary that PRON VERB+’ and ‘it BE necessary that NAME VERB+’ to ensure that the use of the phrase without should is sufficiently frequent and does not render the sentence unnatural.

3.2 Participants

In total, 87 participants took part in the experiment. They were recruited using the online crowdsourcing platform Prolific (www.prolific.com) and were paid for their participation. The participants were required to be native speakers of English and have no cognitive impairment, dementia, or dyslexia. Moreover, their approval rate on Prolific needed to be at least 95 %. Participant age ranged from 21 to 73 (mean: 36.9, median: 35.0, standard deviation [SD]: 11.4). The study was balanced across gender (female: 43, male: 44). From the CxG perspective, speakers of various English varieties are assumed to “store different sets of constructions” (Leclercq 2023: 85). Additionally, modals exhibit regional variation (Siemund 2013: 159). Therefore, since the investigation of regional variation of modal auxiliaries lies outside the scope of the current study, the decision was made to limit the study to only one variety, namely, AmE. Thus, the participants were required to be located in the United States (US) to take part in the study. However, it is important to note that individuals currently residing in the US, even if they are not originally from there, may still be part of the study, as long as they are (self-reported) native speakers of English.

The results of four participants were not considered in the analysis as they did not meet the accuracy threshold of at least fourteen correct answers (87.5 %) out of the 16 comprehension questions to the distracting sentences. The mean of the score for the distractor questions before excluding the four participants was 15.3 (SD: 1; min: 10, max: 16) and 15.5 (SD: 0.5; min: 14, max: 16) after the exclusion. All participants provided their consent before proceeding with the experiment.

3.3 Experimental procedure

The experiment was presented via PCIbex farm (Zehr and Schwarz 2018) and took a median time of 9.41 min to complete. The abbreviated demonstration version of the experiment can be accessed at the following link: https://farm.pcibex.net/r/ktoGxW/. Before the actual experiment began, the participants read the instructions and were asked to provide demographic information about themselves. They were informed that the sentences were unrelated and did not make up a story or a dialogue. Additionally, they were asked to read at a speed that was natural for them in order to fully grasp the meaning of the sentences since they would need to answer follow-up questions.

Each experimental trial consisted of two parts, with the first aiming at exploring online processing and the second, offline processing. The online processing part was a noncumulative, linear (Jegerski 2014: 22–23), word-by-word SPR task. When the spacebar was pressed, the current word of the sentence disappeared, and the next word appeared. Immediately after each sentence, the offline processing was tested using forced-choice judgement questions.

All else being equal, if the RTs in an SPR experiment to the identified critical segments are longer than for a control condition, this usually means enhanced difficulty in cognitive processing, while shorter RTs indicate relative ease of interpretation (Kaiser 2013: 137; Jegerski 2014: 24). Modulations of cognitive effort can be caused by various factors, such as the level of familiarity and thus predictability of subsequent segments.

In order to account for carryover effects, where participants’ subsequent performance might be influenced by their prior one (Myers et al. 2010: 14; Abbhul et al. 2013: 120), and to ensure that the participants did not see highly similar sentences, they received different subsets of the stimuli. Specifically, they saw only two sentences from the same quad, each time with a different modal and a different proper name. Additionally, the stimuli were presented in a randomized order for each participant, which was coded in the experiment on PCIbex.

Each participant received twelve target sentences and 36 unrelated distractors. This high number of distractors (almost 67 % of the experiment) was used to counterbalance the repetitions in the target stimuli and make the target questions less salient. Out of the 36 distractors, sixteen were followed by a comprehension question, the results of which were used to eliminate participants who did not pay attention to the study. The target sentences were followed by judgment questions with two possible interpretations, whereby the participants were asked to choose the paraphrase that better fit the sentence. In case of the forced-choice judgment questions, the sentence that was being interpreted was also shown on the screen to enable a more precise judgment. The DTs of the judgment question were also measured. After each comprehension question related to the distracting sentences, the participants were given feedback regarding the accuracy of their choice (Correct. Keep it up! or Sorry, that’s incorrect.), whereas the judgment questions related to the target sentences were simply followed by Thanks for your opinion. Before the actual experiment began, participants practiced three sentences, each followed by a comprehension question, to familiarize themselves with the procedure.

4.1 Statistical method

Data wrangling and statistical analysis were performed in RStudio (version 2024.9.0.375). In total, data from 83 participants were analyzed. In the online processing task, reaction times to the critical word, the first spillover, the second spillover, and the final word were analyzed together (n = 3,984). The critical region for RT analyses was determined based on the observation where the meaning conveyed by the modal (i.e., epistemic or deontic) can be clearly identified. Given the lack of generally acknowledged heuristics to determine the critical region in reading studies on modals (e.g., Mifka-Profozic 2017; Flach et al. 2023), it was decided to apply strict criteria for the sake of maximal comparability across all sentences (e.g., the critical word is always a noun). Additionally, since the processing of the critical word can spill over the following segments (i.e., so-called spillover effects) (Kaiser 2013: 141–142; McConnell 2023: 11), the two words following the critical word are also considered in the analysis and identified as spill1 and spill2. To account for potential wrap-up effects (Just and Carpenter 1980: 345; Stine-Morrow and McCall 2022: 267), the final word was also included in the analysis.^[6]

Raw RTs under 100 ms or above 1,500 ms were identified as outliers and removed from the analysis, excluding only 35 data points (final n = 3,949, loss of 0.89 %). For calculating the DTs and the final choices of the judgment questions, 996 data points were collected. Values above 20,000 ms were removed, resulting in 2.5 % data loss (final n = 971). All numerical variables (i.e., RTs, DTs, word position, word length, and normalized frequency)^[7] were log-transformed and standardized, whereas categorical variables (i.e., modal, structure, and meaning) were sum-coded.

All three analyses were conducted using linear mixed-effects models (generalized in the case of the model predicting final choices). Fixed effects were specified based on the study hypotheses, with covariates of no interest (e.g., word position, word length, and normalized frequency) included when relevant. Model selection followed a stepwise backward elimination procedure using the step() function, with fixed effects retained when statistically significant. Random effects structures were initially specified justified by the fixed effects in the model. When models did not converge or yielded singular fits, the random structure was simplified iteratively by removing the random slope with the lowest standard deviation, as identified using the summary() output, until the model converged and was no longer singular. Models were compared using anova(), and the final models were selected based on the lowest AIC and BIC among non-singular, converging models. When models were equally good, we applied Occam’s razor and selected the simpler one. The relevant model assumptions (i.e., collinearity, normality of residuals, and homoscedasticity) were visually checked. In the case of the generalized model, DHARMa (version 0.4.7; Hartig 2024) diagnostics confirmed no significant deviations in residual distribution, dispersion, or variance homogeneity. Table 2 presents the final models.

4.2 Online processing: reaction times to the critical regions

Table 4 presents the ANOVA output of the linear mixed-effects model with log-transformed and standardized RTs. Overall, the significant three-way interaction (p = 0.00612) highlights the interplay among modals, grammatical structures, and meanings. Particularly when used with deontic meaning in progressive sentences, must exhibits longer RTs, with a positive coefficient of 0.0196 (95 % CI [0.0052, 0.0340], p = 0.0077) (see Table 3). Figure 1 reveals a more nuanced overview of how individual modals interact with structure and meaning. Specifically, have to and must exhibit similar patterns, with longer RTs in progressive sentences when expressing deontic meaning, while in simple sentences, they show longer RTs for epistemic meaning. The starkest contrast between deontic and epistemic meanings across the two grammatical structures is evident for must, which exhibits the shortest overall RTs when used epistemically in progressive sentences. Notably, should deviates from these two modals, displaying shorter RTs for deontic progressive sentences compared to epistemic ones, with the latter yielding the longest overall RTs across all conditions. In simple sentences, the difference between deontic and epistemic RTs for should becomes neutralized.

Figure 1:

Marginal effects from the linear mixed-effects model for the log-transformed and standardized RTs to the critical regions across all conditions.

4.3 Offline processing: forced-choice judgment questions

4.3.1 Decision times

As shown in the output from the linear mixed-effects model (see Table 5), deontic meaning is associated with an overall decrease in DTs (p = 0.0072) with a negative estimate of −0.0464 (95 % CI [−0.0799, -0.0130]). However, the ANOVA output (see Table 6) reveals a significant influence of the interactions between modal and meaning (p = 0.01896) and meaning and structure (p = 0.03) on DTs. Specifically, the estimated positive coefficient of 0.0573 (95 % CI [0.0100, 0.1046]) suggests an increase in DTs when must is used with deontic meaning (p = 0.0183). Contrary to that, have to and should show a decrease in DTs when paired with deontic meaning (see Figure 2). Furthermore, the significant interaction of deontic meaning with progressive structure leads to an increase in DTs (β = 0.0372 (95 % CI [0.0037, 0.0706], p = 0.03) (see Figure 3).

Table 5:

Output of the linear mixed-effects model for the log-transformed and standardized DTs for the judgment questions across all conditions.

Mixed-effects model predicting DTs of questions (log-transformed and standardized) n = 971
Term	Estimate	Standard error	t-Value	Df	p-Value	95 % Confidence interval
						Lower bound	Upper bound
(Intercept)	0.0135	0.0343	0.3924	87.6156	0.6957	−0.0548	0.0817
Modal[S.have to]	−0.0448	0.0237	−1.8896	68.7210	0.063	−0.0920	0.0025
Modal[S.must]	−0.0248	0.0237	−1.0472	68.3737	0.2987	−0.0721	0.0225
Meaning[S.deontic]	−0.0464	0.0168	−2.7682	68.7045	0.0072	−0.0799	−0.0130
Structure[S.present progressive]	0.0030	0.0168	0.1805	68.5984	0.8573	−0.0304	0.0365
Modal[S.have to]:meaning[S.deontic]	0.0044	0.0237	0.1875	68.6220	0.8518	−0.0428	0.0517
Modal[S.must]:meaning[S.deontic]	0.0573	0.0237	2.4170	68.4156	0.0183	0.0100	0.1046
Meaning[S.deontic]:structure[S.present progressive]	0.0372	0.0168	2.2160	68.6442	0.03	0.0037	0.0706

1. Random effects structure: Random intercept for subject and sentence.

Table 6:

ANOVA table of the linear mixed-effects model for the log-transformed and standardized DTs for the judgment questions across all conditions.

Mixed-effects model predicting log-transformed and standardized DTs of questions (n = 971)
	Sum Sq	Mean Sq	NumDF	DenDF	F value	Pr(>F)
Modal	1.47513	0.73757	2	68.58388	4.4115	0.01577
Meaning	1.28115	1.28115	1	68.70445	7.66273	0.00724
Structure	0.00545	0.00545	1	68.59838	0.03259	0.85726
Modal:meaning	1.40565	0.70283	2	68.5713	4.20371	0.01896
Meaning:structure	0.82102	0.82102	1	68.64424	4.91063	0.03001

Figure 2:

Marginal effects from the linear mixed-effects model for the log-transformed and standardized DTs for the judgment questions displaying the two-way interaction between modals and meanings.

Figure 3:

Marginal effects from the linear mixed-effects model for the log-transformed and standardized DTs for the judgment questions displaying the two-way interaction between structures and meanings.

4.3.2 Final choices

The results from the generalized linear mixed-effects model, presented in Table 7, show a significant effect of structure on participants’ final choices. Particularly, the estimated positive slope of the progressive indicates that, compared to the simple structure, the progressive is associated with an increase of 0.8797 in the preference for epistemic over deontic interpretations (p = 0.0195) (see Figure 4).

Table 7:

Output of the generalized mixed-effects model predicting the probability of epistemic interpretation by structure in the judgement questions.

	Estimate	Std. error	z value	Pr(>|z|)
(Intercept)	−1.9792	0.4413	−4.485	7.3e–06
Structure[S.present progressive]	0.8797	0.3766	2.336	0.0195

Figure 4:

Marginal effects from the generalized mixed-effects model predicting the probability of epistemic interpretation based on structure in the judgement questions (Note: On the y-axis, a value of 0 % indicates that subjects exclusively chose deontic interpretations, while a value of 100 % indicates an absolute preference for epistemic ones).