Distance effects (also called locality effects) occur when processing complexity is affected by the distance between words that are involved in some kind of dependency. An example of such a dependency is the relation between the head noun and the verb of a relative clause, as the administrator – supervised in (2) (based on Grodner & Gibson, 2005).

The additional material in (2b), compared to (2a), results in processing difficulties. The difficulty manifests itself as increased reading times in eye-tracking while reading studies and self-paced reading studies (Bartek et al., 2011; Grodner & Gibson, 2005; Nicenboim et al., 2016, among others) or as a decrease in accuracy in studies using the speed-accuracy tradeoff paradigm (e.g. McElree, 2000).

Assuming a certain view of how memory is involved in language processing (Anderson, 1991; Lewis et al., 2006), there are two possible sources for such findings: decay and interference (see also Bartek et al., 2011). Both of these can play a role in processing. As the distance to the retrieval target grows, the activation of that target decays. At the same time, with greater distance comes an increase in the number of similar elements that compete with the antecedent for retrieval. This competition reduces accuracy, but can also have an impact on speed when faulty resolutions need to be rectified.

Distance effects are accounted for in different ways by various theoretical frameworks. For instance, in dependency locality theory (Gibson, 2000), the integration cost of connecting some current structure to a previous structure depends directly on the number of discourse entities (discourse referents, events, times) that have been introduced in the meantime.

Another dominant theory of how memory is involved in sentence processing posits that dependents in long-distance dependencies are retrieved via a so-called cue-based mechanism (Jäger et al., 2017; Lewis & Vasishth, 2005; McElree, 2001; McElree et al., 2003; Nicenboim et al., 2015; Nicenboim & Vasishth, 2018, and others). This means that when an item previously seen is needed to parse a dependency, it is retrieved from memory using cues determined by the word being processed. Consider (3) (taken from Van Dyke, 2007).

When reading the phrase was complaining, the parser has to find the subject of the clause (the resident) to arrive at the correct interpretation. To retrieve the subject from memory, the parser uses cues. In this way, the search is restricted to the elements that satisfy the constraints introduced by the verb phrase. We can assume that the parser would consider the cues {noun phrase}, {local subject} and {animate}, among others.

In examples like (3), there are several elements that match the cue {animate} and {noun phrase} without matching the cue {local subject}: the worker, and the neighbour are such cases. Such partially-matching elements are commonly called distractors. The interaction of phrases carrying partially matching sets of features with the retrieval process is called cue interference and is usually taken as evidence in favour of the cue-based retrieval mechanism (Jäger et al., 2017). Due to cue interference, distractors can in some environments cause misretrieval and slowdown in reading times at the moment of dependency resolution (e.g. Jäger et al., 2017; Nicenboim & Vasishth, 2018).

There is a plethora of research studying the cue-based retrieval mechanism in the syntactic domain (see Jäger et al., 2017, for an overview and a metaanalysis). This body of work extends also to syntactic phenomena involving retrieval of semantic information, such as the licensing of negative polarity items (Parker & Phillips, 2016) and to the role of semantic information in resolving syntactic dependencies (e.g. Cunnings & Sturt, 2018; Laurinavichyute & von der Malsburg, 2022).

In contrast to this research, the investigation of retrieval mechanisms for discourse dependencies is much more limited. Most studies in psycholinguistics that go beyond the domain of syntax consider pronoun resolution (Kush & Eik, 2019; Kush & Dillon, 2021; Schmitz et al., 2024). Yet, it is clear that there are other dependencies in discourse that could be investigated from this perspective.

In what follows, we will show in a theory-independent way that the processing of too relies on memory. This fact will allow us to derive experimental predictions using cue-based theories of memory retrieval. We note here that the predictions we are to discuss and test are likely shared with various other theories of memory use in language comprehension, like dependency locality theory. Our goal here is not to decide which of the theories is right. Rather, we want to show that under quite general conditions on memory, which we will make explicit using the cue-based retrieval model as our illustrative framework, we can establish whether there is a cost to processing too due to retrieval.

In contrast to studies concerned with syntactic dependencies, the phenomenon we are interested in is largely independent of syntax. Yet, the additive presuppositions we focus on in what follows do clearly qualify as dependencies. So, in this sense, they are quite close to what has been studied in the syntactic domain (see also Brasoveanu & Dotlačil, 2020; Chen & Husband, 2018, for a similar argument).

To see this, consider a sentence like (4). This triggers a presupposition that is connected to the so-called associate of the additive too, here assumed to be Sue: namely, that someone other than Sue lives in France.

Given this presupposition, we can now demonstrate that additives like too are involved in dependencies and that processing them relies on retrieval. Too is subject to a so-called strong contextual felicity constraint (Tonhauser et al., 2013), which means that it is infelicitous whenever the presupposition is not satisfied in the preceding context.1 Consider the example in (5) (intended as an utterance out of the blue).

The fact that we can recognise that (5) is infelicitous means that we use our knowledge of the preceding context when we process the additive. To be more specific, our capacity to judge the second sentence in (5) infelicitous after encountering too entails that upon encountering the additive, a retrieval process is triggered to match the presupposition. The result of this process is known to have an effect on sentence processing. Schwarz (2007) shows that reading times are longer in conditions where a trigger lacks an antecedent, compared to a condition where an antecedent is available. Presumably, such effects, as well as the infelicity judgment of (5), are caused by a failure of the match following the retrieval process.

The basis for this retrieval process is information-structural rather than syntactic. There is a general consensus that the associate of the additive is in focus (e.g. Geurts & van der Sandt, 2004; Ruys, 2015) or is a contrastive topic (e.g. Krifka, 1998). The semantic import of bearing focus (and being a contrastive topic) is the activation of alternatives (Krifka, 2008; Rooth, 1992). Such alternatives usually trigger an exhaustification implicature, the inference that all alternatives except for the asserted one are false (Fox & Katzir, 2011; Kiss, 1998; Rooth, 1992). For instance, Sue lives in France (with the subject bearing focus) triggers an implicature that none of the other contextually relevant individuals (i.e. alternatives to Sue) live in France. The additive in (4) overtly cancels that implicature, by presupposing that one of the alternatives is true (Bade, 2014; Krifka, 1998; Sæbø, 2004).

The additive’s presupposition can be obtained by looking at the alternative propositions triggered by the associate bearing focus. For instance, focus on Sue in the second sentence of (5) creates alternatives like Peter lives in France, Bob lives in France, Anne lives in France. The presupposition of too says that at least one such alternative is true. In other words, if the presupposition is satisfied textually, then the preceding discourse should entail one of these alternatives. Since Anne lives in Paris entails Anne lives in France, the presupposition is met in an example like (6), but since Anne lives in Brussels does not have that entailment, (5) ends up being infelicitous.

Given this picture, any proposition entailed in the preceding discourse is a potential antecedent to the additive. Importantly, this means that the number of potential antecedents increases as the preceding discourse gets larger. This is not dependent on the number of preceding clauses, but rather on the number of constituents. Consider, for instance, (7):

Examples like this show that the addition of any constituent has the potential to be responsible for an entailment that could satisfy some presupposition down the line. Focus on the subject of the second sentence in (7) activates alternatives of the form X is 14 years old. The additive triggers the presupposition that one of these alternatives is true, where X is not Sue’s cat. The first sentence entails that the bottle of wine given to John is 14 years old and, as a consequence, the presupposition is satisfied. However, obviously, without the addition of the AP 14 year old modifying bottle of wine, the presupposition would not be met and the discourse in (7) would be infelicitous. This observation has consequences for how we may understand the retrieval process involved in additive presuppositional dependencies. As examples like (7) show, the number of potential dependents (and, so, the number of distractors) increases not just with the number of intervening clauses, but also with the number of intervening constituents.

There are obvious parallels between the retrieval process involved in syntactic dependencies and that involved in presuppositions. Just as the number of syntactic attachment points increases as the sentence proceeds, there is a comparable accumulation of entailments as the discourse progresses. Consequently, just as we expect processing complexity to increase with increased distance within a syntactic dependency, all else being equal, we expect a similar effect of complexity to occur with presuppositions whenever the antecedent entailment is triggered earlier in the discourse. Put simply, if we assume that distance effects are due to inherent constraints on the memory processes involved in language processing, and given that additive presuppositions involve retrieval, we come to expect that the processing of additive presuppositions will display distance effects.

It should be noted, though, that it is possible to satisfy an additive presupposition without identifying a single word or phrase responsible for this entailment. Take the following example:

Clearly the presupposition is satisfied here; the set of sentences in the preceding discourse seems to entail the proposition Buck was changing for some relevant meaning of changing. However, the word itself does not appear in that discourse. So, in contexts like this, it would be hard to pinpoint a location in the preceding text where the entailment originates. This is not surprising; since additive presuppositions involve dependencies built on entailments, not all such dependencies will be reducible to a dependency between too and a word or a phrase.2 We purposefully avoid examples like this in our stimuli, which will allow us to match the antecedent of the presuppositional dependency (an entailment) with a position in the preceding text. This will allow us to test the expectation that presuppositional dependencies are subject to distance effects.

This expectation is dependent on presuppositions being triggered rapidly. If presuppositions are slow inferences, the comparison to syntactic processing becomes moot. In the literature, presupposition processing speed has been connected to the question whether presuppositions are more like semantic entailment (Heim, 1982) or more like pragmatic implicature (e.g. Schlenker, 2008). This is because, for a while, it was thought that pragmatic inferences are typically slow and costly (e.g. Huang & Snedeker, 2011, views tend to be more nuanced now, e.g. Degen & Tanenhaus, 2016). It has proven to be difficult to use acceptance/rejection reaction time studies to compare presupposition and implicature (Chemla & Bott, 2013; Romoli & Schwarz, 2015), but there is clear evidence from other paradigms that presuppositions are not delayed. Using the visual world paradigm, Romoli et al. (2015); Schwarz (2015) show that fixation shifts to pictures representing potential antecedents of additive presupposition triggers very soon after such a trigger is encountered. This suggests that the retrieval process is started rapidly. That is, antecedent retrieval is like the rapid processing of syntactic information. As a result, comparison with the resolution of syntactic dependencies makes sense from a processing point of view.

At the same time, there are also obvious differences between retrieval of syntactic material and retrieval involving semantics. It is much less evident what kind of cues could guide the retrieval of entailments. Entailments do not always coincide with linguistic events, such as the processing of a word or constituent. In this sense, it is harder to translate them into cues than, for instance, connecting a phrase to the feature {subject}. Still, such connections are proposed in representational theories of (discourse) semantics, most prominently in Discourse Representation Theory (DRT; Kamp, 1981; Kamp & Reyle, 1993). In DRT, an incremental semantic representation of the discourse is formed as the discourse progresses. The ensuing representations impose constraints on what information is accessible where in the discourse. As such, it offers a representational basis for semantic notions like entailment and, therefore, it could potentially act as a framework for imagining what a cue-based mechanism for presupposition retrieval looks like. In fact, DRT has been an influential framework for theories of presupposition (e.g. Geurts, 1999; van der Sandt, 1992).

The items used in the experiments below allow us to simplify, to some extent, the complex picture introduced above. Consider, for instance, (9), which is a stimulus from Experiment 1.

When encountering the verb dances followed by the additive too, the parser needs to find the information that makes the additive licensed. In particular, some part of the discourse (representation) needs to be found that entails X dances for something other than X = the waiter. The fully matching part of the discourse is The cook is a dancer, since this clause entails that the cook dances, but between that and the additive there are many constituents that introduce predicate-argument propositions that fail to match the presupposition. For instance, a great boxer has the same subject as dances and is thus unsuitable. The same goes for from southern Amsterdam. Both these predicates also fail to subsume the property of dancing, which is another reason why they are unsuitable antecedents. Still, they do qualify as distractors, since they introduce entailments (that the waiter is a great boxer, that the waiter is from southern Amsterdam) that potentially could satisfy upcoming presuppositions. This means we could simplify things by matching distractors to predicates appearing somewhere else in the text. Such predicates can be realised as a VP, an appositive noun, a PP, etc.3 What matters is the semantic content present in the discourse. We assume that {predicate} can be used as a cue when searching for a relevant item to retrieve. Since predicates are ubiquitous in language, we believe that this makes it theoretically possible to investigate whether the cue-based retrieval mechanism is a plausible theory of the on-line interpretation of additives. Under this theory, we expect interference due to the number of predicates, which can be approximated by the number of words intervening between the antecedent and the additive. Based on this assumption, the cue-based retrieval mechanism is compatible with the observation of more errors and/or slower reading times as the number of words intervening between the antecedent and the additive is increased.

Let us spell out the prediction in more detail. Assuming that predicates act as distractors, they interfere with the retrieval process. The distractors could be retrieved erroneously. If readers need to correct the misretrieval by attempting retrieval again, or they notice that the information retrieved does not match what they wanted to retrieve and end up with an infelicitous discourse, an increase in their reading time is expected (Nicenboim & Vasishth, 2018). Alternatively, according to some theories of cue-based retrieval, the presence of distractors should lower the activation of the antecedent, which is consequently harder to retrieve, so its retrieval should take more time (Jäger et al., 2017; Lewis & Vasishth, 2005). Either assumption would derive a slowdown when more words intervene between the antecedent and too. However, even outside of the cue-based retrieval mechanism, we predict an interfering role for intervening words. Under models of memory in which memory traces decay over time (Brown et al., 2007, base-level activation; Anderson & Lebiere, 1998), a similar prediction is made. Since increasing the number of words between the antecedent and the additive should increase the time elapsed, it is expected that the antecedent should be harder to retrieve when more words intervene.

In summary, given the assumptions about the various ways retrieval processes work in language processing and given that presupposition resolution involves retrieval, just as syntactic dependency resolution does, we predict that the greater the distance between the trigger and the antecedent, the more difficult it will be to retrieve the antecedent. This difficulty should manifest itself in longer reading times observed on the relevant regions. Besides reading times, we also examine whether the predicted difficulty will manifest itself as diminished acceptability of sentences with greater distance between the trigger and the antecedent.

The first experiment was designed to test whether linguistic distance has an influence on the perceived acceptability of sentences with presupposition carried by an additive too.

It is usually assumed that processing difficulty has an influence on acceptability (Miller & Chomsky, 1963, Staum Casasanto, Hofmeister, & Sag, 2010 and many others). The relation between the two is not so clear when it comes to distance (locality) effects. For example, Konieczny (2000) reports conflicting results from acceptability judgments and self-paced reading experiments on locality effects. This suggests that, in this particular domain, processing difficulty and acceptability are not connected in a straightforward way. We decided to collect acceptability judgment data as a sanity check and a first step in investigating the processing of too. In case distance effects were observed in the acceptability judgment task, we would have evidence that distance effects play a role even in off-line measures. If we do not observe such effects, the natural next step is to investigate on-line processing of too.

We compared sentences like (10a), in which an additive presupposition is triggered by too, with sentences like (10b), where there was no such trigger and no presupposition.

Apart from varying the presence or absence of a presupposition trigger, we also manipulated the length of the sentences used. This varied the distance between the presuppositional antecedent and the trigger, in a way designed to help us observe the differences in acceptability (or reading times) between the trigger/no trigger conditions.

In the experiment, four levels of distance (labelled 0, 4, 8, or 12 for extensions of 0 words, 4–5 words, 8–10 words, or 12–15 words; 4 words change between the conditions at minimum) were crossed with two levels of trigger (too/nil – for trigger present or no trigger, respectively). This results in a within-participants design with 4 × 2 = 8 conditions. The distance factor was manipulated by increasing the number of words between the trigger site and the antecedent. The trigger factor was manipulated by replacing the trigger with a non-triggering word, usually an adverb, and aligning the sentence meaning with this change. This usually involved changing the nominal predicate to a different noun. An example of an item is shown below.

The target items were sentences consisting of two parallel clauses. In the first clause, some property was asserted of the subject using a noun (e.g. dancer in (too)). In the second clause, this property was recalled using a verb followed by too (e.g. dances too in (too)). In the nil condition, the noun in the first clause was unrelated to the verb in the second clause, and too was replaced by an adverb (e.g., swimmer and dances often in (nil)).

We constructed 32 items of this kind. The target items were combined with 22 filler items. The fillers were grammatical sentences of similar length to the target items. Additionally, we used 10 items as controls. Their purpose was to check on the engagement of the participants. These were sentences that clearly elicit judgments of unacceptability, because they violate some syntactic or semantic constraint. All the stimuli are available in the supplementary material.

32 participants (female = 18 (56%), male = 12 (38%), other/don’t want to say = 2 (6%)) were recruited using Prolific (https://www.prolific.co). Participants were compensated 1.97 USD for taking part, which was calculated assuming that the experiment should take around 10 minutes to complete, and given that the effective minimum wage in the United States was 11.80 USD at the time of the experiment (16  ×  11.80  =  1.97) \documentclass[10pt]{article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ ({\textstyle{1 \over 6}}\,\, \times \,\,11.80\,\, = \,\,1.97) \] \end{document} . The participants were self-reported English native speakers and self-reported US nationals.

The experiment was conducted using Ibex.4 The 31 item sets were distributed across 8 lists in a Latin Square design, and participants were assigned at random to each list.5 Each of the target sentences was interspersed with either a filler item or a control item. The order of items across lists was randomized. The experiment was run from the participants’ internet browser.

Before starting the experiment, there was an introduction containing a consent form, a question regarding native language (or languages), and a question about the participant’s gender with three possible answers: male, female, other/don’t want to say. After the introduction, there was an explanation of the task, followed by a training session with two training items.

During the actual task, participants saw a sentence in the middle of the screen. Below the sentence was a question asking: How does the sentence sound? The question was accompanied by a 5-point scale labelled unnatural near 1 on the left, and natural near 5 on the right. On the top of the screen, a progress bar was displayed. After choosing a score either with a mouse cursor or with a keyboard, the participant saw a screen with the message: Please wait for the next sentence, and after one second, they were presented with the next item.

There are two specific predictions for this experiment. First, we predict that longer items will be judged as worse than the shorter ones. This prediction is in line with the findings of Dillon et al. (2014), who found that longer restrictive relative clauses are judged as worse than shorter ones.

The prediction which is of greater importance to us, concerns the difference in processing difficulty between the sentences with too and without too. We predict that increasing distance in the sentences with too will make them more difficult to process. However, if we test this prediction only on sentences with too, we cannot be sure if this difficulty is not just a general effect of the greater distance, independent of too. To overcome this difficulty, we are comparing sentences with too to sentences without too, and we will look for an interaction of the trigger with distance. The difference in difficulty between the long and short sentences should be larger for sentences with too than for those without it. We expect this difficulty to be visible as a difference in ratings across relevant conditions. That is, if distance effects in retrieval are visible in an off-line study, we predict a larger decrease in ratings across distance in the too conditions than in the nil conditions.

After visual inspection, two participants were removed from the analysis, since their median response to the control sentences was above 3, indicating that they treated purposefully unnatural sentences as quite natural.

A Bayesian hierarchical model was fit to the data obtained. In Bayesian data analysis, one specifies the likelihood, and the prior distributions over the parameters of interest (Gelman et al., 2003; Gelman & Hill, 2006; McElreath, 2018; Nicenboim et al., 2021). The analysis results in posterior probability distributions of plausible values for a given model and data. We report means and 89% credible intervals (CRI), i.e. the range of values for which we can be 89% certain that the true effect lies therein. The mean and the range of credible intervals are supposed to convey the shape of the posterior distribution. We specifically avoid using 95% credible intervals to discourage the readers from conducting unconscious hypothesis tests (see also McElreath, 2018, p. 58).

We used ordinal regression models, which are particularly well suited to the analysis of discrete data (Bürkner & Vuorre, 2019; Veríssimo, 2021). In particular, we used a cumulative model with a logit link function and so-called flexible thresholds (i.e. the estimated distance between the different scores can vary). Models used 4 sampling chains, with 4,000 samples drawn from each chain. Half of these samples were discarded for warm-up; hence each model had 8,000 samples available for the analysis. Trace plots and plots of posterior predictive distribution were visually inspected to identify convergence issues. Additionally, only the models with all ˆR ≤ 1.01, which suggests convergence, were used in the analyses.

All the models used were fit with a full variance-covariance matrix, i.e. they were so-called maximal models. The analysis was conducted in R software for statistical computing (R Core Team, 2021), and, in particular, with the use of the brms package (Bürkner, 2017), which uses the Stan probabilistic language (Stan Development Team, 2021).

The distance, levels of which were labelled with numbers closely matching the number of words between the trigger and its antecedent, was treated as a numeric variable which was scaled and centered before fitting the model. The trigger was sum-contrast coded (nil = –1, too = 1). Fixed effects of interest were: distance, trigger and their interaction. Participants and items were used as random effects.

We used regularising prior distributions for slopes and intercepts. For the intercept, we used a normal distribution with μ = 0 and σ = 2. This means that any of the intercepts (i.e. the thresholds for different Likert scale outcomes) should fall with 89% probability between 0.04 and 0.96 on the probability scale. For the slopes, we used a normal distribution with μ = 0 and σ = 0.5.

This assumes that if we fix the intercept at 0 log-odds (i.e. 0.5 on the probability scale), and the input at 1, then the values of the outcome will be, with 89% probability, between 0.69 and 0.31 on the probability scale. Or, to put it differently, the priors used for the slopes mean that when the intercept is at 0.5 probability, we assume, with 89% probability, that the slope coefficient will change the outcome by at most 0.197 on the probability scale. The priors were additionally examined using prior predictive checks.

Note that the size of the effect depends on the intercept, because of the logit link function. We report the results on the log-odds scale.

The 89% credible intervals for the slopes of interest are summarised in Figure 1.

The estimate of the distance parameter indicates that the longer items were judged as less natural (89% CrI = (–0.4, –0.2), mean = –0.3, sd = 0.08).

In the case of the trigger, the sign of the estimate is rather uncertain, with the 89% (–0.09, 0.18) interval lying on the both sides of 0. The mean (= 0.04, sd = 0.09) is positive (i.e. the sentences with too were judged as slightly more natural), but the spread of the posterior distribution indicates that the sign of estimate is not conclusive.

For our purposes, the most interesting parameter is the slope of the interaction. The posterior distribution of the slope of the interaction (mean = –0.07, sd = 0.08) has the 89% credible interval (–0.2, 0.05), that is, we can be 89% certain that the value of the interaction lies in this interval. The interval is more negative than positive, but the fact that it spans zero suggests that zero (no effect) is a possible value for the interaction term.

In sum, there is a clear effect of distance (longer items are less natural), but the model does not show that the presence of the trigger and the interaction have an effect on acceptability.

In the self-paced reading task, we used a subset of the conditions and items from the acceptability judgment task. These were chosen based on our intuition, i.e. one of the authors went through all the items and selected the ones they thought sounded good. Some of the remaining items which we thought could be improved were edited for the self-paced reading experiment. The exact changes between the items can be seen at the OSF repository.

With respect to the experimental manipulation, two levels of distance (short and long, originally 0 and 8) were crossed with two levels of trigger (too and nil), which resulted in a 2 × 2 within-participant design. We used 24 target items in the self-paced reading task. These were accompanied by 25 filler items.

82 participants (female = 44 (54%), male = 37 (45%), other/don’t want to say = 1 (1%)) were recruited using Prolific (https://www.prolific.co). They were compensated 3.00 GBP for taking part in the experiment. The experiment took, on average, 12 minutes and 38 seconds to finish; therefore, the participants would get 14.25 GPB per hour, on average. The participants were self-reported English native speakers.

The experiment was conducted in a noncumulative self-paced reading moving window paradigm (Just et al., 1982): during reading, only one word appeared and participants had to press the space bar to reveal the next word and hide the previous word. In the initial position, every word was masked by a line which was roughly equal to the word’s length. When the words disappeared, they went back to the initial position. A comprehension question followed each item.

As was the case for the acceptability judgment experiment, there are also two specific predictions for this experiment. The first prediction follows from the observed relationship between reading speed and distance. It is commonly observed (Demberg & Keller, 2008; Ferreira & Henderson, 1993; and others) that readers speed up with the length of a sentence. Therefore, we predict that the relevant regions in the longer sentences will be read faster than in the shorter sentences.

The prediction which is of greater importance to us concerns the difference in processing difficulty between the sentences with too and without too. We predict that increasing distance in the sentences with too will make them more difficult to process. Following the reasoning from the previous experiment, we should note that if we test this prediction only on sentences with too, we cannot distinguish the difficulty due to retrieval from the general effect of longer texts on reading. To overcome this, we are comparing sentences with too to sentences without too, and we will look for an interaction. The difference in difficulty between the long and short sentences should be larger for sentences with too than for those without it. We expect this difficulty to be visible as a difference in reading times across relevant conditions. That is, we predict that compared to the effect of distance in the nil conditions, there should be a slowdown effect due to distance in the too conditions.

Before the analysis, we investigated the quality of the responses to the comprehension questions. Out of 82 participants, 1 with the worst ratio (≤ 0.7 correct) of correct responses to the comprehension questions was removed from the results. The median correct response ratio among the remaining participants was equal to 0.9.

Additionally, time measurements which were under 50 ms or above 3,000 ms were removed from the results. Similar data trimming was done, for example, by Futrell et al. (2021). This resulted in the removal of a further 0.2% of the data.

Next, we fitted a Bayesian mixed-effects model to the response time data. This model assumed log-normal likelihood, since reading time data are approximately log-normally distributed and, therefore, our model can resemble the data-generating process more closely (see also Nicenboim et al., 2016, 2018; Rouder et al., 2008; for a more detailed discussion). Besides the choice of priors and likelihood, the details of model fitting were similar to those in the analysis of the acceptability judgment task. We analyzed the critical region (the word too and the corresponding word in the nil condition), the post-critical region (the word immediately following the critical region), and the post-post-critical region (two words after the critical region). The experimental factor trigger was coded using sum contrasts, with the nil condition as –1 and the too condition as 1. The experimental factor distance was also coded with sum contrasts. The long condition was coded as 1 and the short condition as –1. Both of these variables were entered into the model as fixed effects. Participants and items were entered as random effects.

Besides models quantifying the effects of interest on the entirety of the data, we also fit so-called nested models to quantify the effect of distance for each of the trigger conditions. These models were implemented by specifying two predictors: too_distance and nil_distance. For too_distance, we coded the conditions where the trigger was too and distance was long as 1, the conditions where the trigger was too and the distance was short as –1, and the nil conditions as 0. Analogously, for nil_distance, the conditions where the trigger was nil and the distance was long were coded as 1, the conditions where the trigger was nil and the distance was short were coded as –1, and the too conditions were coded as 0. Both of these predictors were then used to fit varying slopes and varying intercepts models. These models provide a comparison of the effect of distance separately for each trigger condition. They used 4 sampling chains, with 6,000 samples drawn from each chain. 2,000 samples from each chain were discarded for warm-up; hence each model had 16,000 samples available for the analysis.

For the intercept, we used a normal distribution with μ = 6 and σ = 1.5. This means that the intercept (i.e. the grand mean in our models) is assumed to be between 36.7 ms and 4435 ms, with 89% probability. For the slopes, we used a normal distribution with μ = 0 and σ = 0.1. This assumes that if we fix the intercept at 6 (which is e⁶ = 403.4 ms), and the input at 1, the output will be between 473.3 and 343.8 ms, with 89% probability. Note that the degree to which slope coefficients influence the outcome depends on the intercept, because of the log-normal likelihood.

If the prior distribution of the slopes seems too constraining, one should consider that it is in line with the estimates of the meta-analysis of interference effects studies (Jäger et al., 2017). In that meta-analysis, the estimate of the largest relevant effect (p. 328, Table 4: target mismatch configurations, AND/OR prominence effect) was 39.3 ms (95% CrI (10.8, 67.9)). One can use the credible interval to guide the selection of the prior distribution for the standard deviation. Following Schad et al. (2021), we come up with an estimate for the σ parameter on the log scale equal to 0.017. Since we cannot be sure whether effects related to the processing of semantic/pragmatic dependencies are of a similar order of magnitude, we settle on a much more uncertain prior of 0.1. Furthermore, we evaluated the prior distributions with prior predictive checks. The visual summaries of the checks are available at the OSF repository. As a prior distribution for both the standard deviations of the random effects, and for the residual standard deviation, we used a truncated normal with μ = 0 and σ = 1. For the random effects correlation between the intercept and the slope, we used the LKJ distribution (Lewandowski et al., 2009; Stan Development Team, 2021) with η = 2.

The results are reported on the log-ms scale.

The descriptive summaries of the reading times across conditions from one word before to two words after the critical region (the additive too and the corresponding word in the nil condition) are shown on Figure 2.

Let us turn to the findings of the Bayesian mixed-effects model. First, we discuss the results for the critical region (i.e. too or the parallel word), then the post-critical region (i.e. the word after the too or the parallel word), and finally the post-post-critical region (i.e. the two words after too or the parallel word).

For the critical region, the main effect of trigger was –0.02 log-ms (89% CrI: (–0.04, –0.007)), which suggests that there was a difference between the nil and too conditions, in particular, the nil condition took longer to read than the too condition. The main effect of distance was –0.02 log-ms (89% CrI: (–0.04, –0.008)), which means that the participants read the word faster when the distance increased. The interaction effect was 0.003 log-ms (89% CrI: (–0.01, 0.015)). The mean value of the interaction being very close to 0 and the spread of the CrI across both positive and negative values suggest that its direction is inconclusive.

Using the nested model, we further inspect the effect of distance in the too and the nil conditions. Within the too condition, the effect of distance is –0.02 log-ms (89% CrI: (–0.04, –0.001)), which means that the long condition was read faster than the short condition. The effect of distance within the nil condition was –0.03 log-ms (89% CrI: (–0.05, –0.006)), which also means that the long condition was read faster than the short condition.

Figure 3 shows the posterior probability distribution of the relevant parameters in the post-critical region. The main effect of trigger was –0.02 log-ms (89% CrI: (–0.04, –0.004)), which again suggests that there was a difference between the nil and too conditions, and that the nil condition took a little longer to read than the too condition. The main effect of distance was –0.02 log-ms (89% CrI: (–0.03, –0.004)), which suggests that participants read the word faster when the distance increased. The interaction effect was 0.02 log-ms (89% CrI: (0.005, 0.031)). The positive interval shows that the negative difference in distance, when comparing the long and short conditions, was diminished and maybe even reversed in the too condition.

In the nested model, the effect of distance within the too condition was 0.001 log-ms (89% CrI: (–0.02, 0.02)), that is, there was no difference in reading times between the conditions. In the same model, the effect of distance within the nil condition was –0.04 log-ms (89% CrI: (–0.06, –0.02)), which means than the long condition was read faster than the short condition.

Finally, for the post-post-critical, region the main effect of trigger was –0.01 log-ms (89% CrI: (–0.03, –0.002)) which, as in the previous regions, suggests that the nil condition took longer to read than the too condition. The main effect of distance was 0.005 log-ms (89% CrI: (–0.007, 0.02)), which suggests almost no distance effect. The interaction effect was positive, just as in the post-critical region, even though the credible interval included zero this time (mean 0.008 log-ms, 89% CrI: (–0.003, 0.02)).

In the nested model, we see that the effect of distance within the too condition was 0.01 log-ms (89% CrI: (–0.005, 0.03)), which means that the short condition was read faster than the long condition, even though the null effect is not excluded among possible values in the credible interval. The effect of distance within the nil condition was –0.003 log-ms (89% CrI: (–0.018, 0.012)), which means that there was no difference in reading times between the conditions.

The posterior distribution of the trigger was (in the 89% credible interval) negative in the post-critical region. This suggests that the nil condition took longer to read than the too condition, possibly because the trigger was more predictable than the corresponding element in the nil condition (an adverb). The main effect of distance was also negative in the same region, which suggests that the post-critical region was read faster when the distance was greater. Crucially, the effect of distance interacted with the trigger: there was a positive interaction in the post-critical region. As we mentioned, one way to interpret the positive interaction is that it shows that the negative effect of distance is diminished and maybe even reversed in items with too. The pattern of the interaction is easier to understand when we turn to the nested model, shown in Figure 3. We see that there was a speed-up in reading due to distance in the nil condition, but no such effect in the too condition. In other words, the interaction comes about due to the interplay of two effects: the speed-up effect due to distance in the nil condition (longer items are read faster in the post-critical region), and the absence of this speed-up in the too condition.

We interpret the finding as follows. First, the fact that the long condition is read faster than the short condition in the nil case is in line with the general observation that readers speed up during reading texts or experimental items, as seen in experimental research (Ferreira and Henderson, 1993; see also Vasishth & Lewis, 2006, p. 780) and the analysis of psycholinguistic reading corpora (Demberg & Keller, 2008; Dotlačil, 2021). It is also in line with the fact that we observe a (general) speed-up on the critical region in this experiment. Second, in contrast to the nil condition, something nullifies the speed-up in the too condition. This, we submit, is the retrieval process due to too. Since readers have to resolve the presupposition, they have to recall the antecedent. Such a recall is harder and slower in the long condition compared to the short condition. Under this interpretation, our findings would represent a novel instance of semantic inhibitory interference (Parker & Phillips, 2016; Van Dyke, 2007), one that is observed and can be studied in texts and long discourses, not just sentence-internally.

We can think of three possible explanations for the proposed slow-down in too compared to the baseline (the nil condition). First, it is possible that readers retrieve a distractor and subsequently correct their misretrieval, which takes extra time. Since misretrievals are more likely to happen with increased distance, the discussed reading times pattern is observed. Alternatively, readers might not correct their wrong retrieval but stick to it and consider the presupposition as unresolved. This would result in an infelicitous sentence, which would likely be read slower compared to a fully grammatical, felicitous sentence. Finally, it is possible that with greater distance, the antecedent decays in memory, resulting in longer retrieval time.

It should also be noted that the interaction effect persists to some extent at the post-post-critical region. If this is related to the observed effect, this might suggest that the retrieval difficulty affected the reading process for longer than just immediately after retrieval.

We should mention one possible confound in this study. As we explained in Section 1, additive presuppositions depend on information structure. In particular, the associate of the additive trigger has focus. Given that our stimuli are not spoken, there is no guarantee the information structure we intended for our stimuli matches how participants perceived them. Take, for instance, the following example:

Here, we assume that when participants read this sentence, they interpret waiter as being in focus. The additive then associates with the subject. Potentially, however, participants read (11) with focus on the verb dances. In that case, the presupposition becomes that the waiter has properties besides being someone who dances. Given that such properties are provided in the appositive clause intervening between the subject and the verb, our manipulation of distance would stop working under this interpretation. This is even more so because this interpretation is fully unavailable in the short distance condition, where no alternative properties of the subject are given.

While we acknowledge that this alternative understanding of our stimuli is available, we doubt that it played a major role in the results of Experiments 1 and 2. This is because informal intuitions that we gathered about our stimuli suggest that our own intended understanding is much more prominent than the understanding with focus on the verb. However, to be sure, the follow-up experiments were designed to avoid this possible confound.

Experiments 3 and 4 were designed to further investigate the retrieval of the antecedents of additives. We wanted to see whether the effect of distance on retrieval can be observed by manipulating the number of sentences in a discourse, not by manipulating the number of words in an appositive, i.e. a non-restrictive relative clause, as was done in Experiments 1 and 2. One reason to consider this is that a non-restrictive relative clause might be processed differently from, and independently of, the rest of the discourse (Dillon et al., 2014), adding unnecessary complications and orthogonal issues to the first two experiments. Another reason is that this allows us to exclude the possible confound in Experiment 1 and 2, that is, the possibility that in the long condition, the presupposition was resolved with a different information structure than in the short condition.

In the same way as in Experiment 2, we used two factors in Experiment 3: distance and trigger. Two levels of distance were crossed with two levels of trigger, resulting in a 2 × 2 within-participant design.

The distance factor was manipulated by changing the number of clauses between the items. In the short case, there was 1 additional clause introduced between the first sentence and the critical sentence. In the long case, there were 3 such clauses, see (short) and (long) in too and nil below. The sentences were always in the past tense.

The way these discourse fragments were constructed permits us to treat VPs as predicates.6 Then, the contrast of distance can be interpreted as the number of these VP-predicates that appear in the short and long conditions. In the case of (short), only one extra clause, introducing one such extra predicate, appeared between the predicate with too (won too) and the antecedent ((Anne) came first in her race). In case of (long), three extra sentences, introducing three extra predicates, interfered between the predicate with too and the antecedent. We hypothesised that the number of predicates between the antecedent and the trigger should influence how quickly the participants will be able to retrieve the information required by the trigger.

The trigger factor was manipulated by replacing the trigger too with a non-triggering word, often a noun, as in the illustrative item given above. The corresponding word in the nil condition (gold above) had to be of similar length as too (between 3 and 5 characters). As in Experiments 1 and 2, the items in the nil condition also differed in the first sentence. Care was taken to ensure that sentences that were changed between the conditions had similar lengths and numbers of words.

Notice that the items above no longer allow for the possible confound we discussed in connection with Experiment 2. There, some stimuli had more than one candidate for the additive’s associate. In this experiment, however, the only felicitous associate for too is the subject. In particular, assuming the predicate won as the associate would make the presupposition unresolved in both short and long contexts. What’s more, the subject of the target sentence is easily recognized as a contrastive topic. In the first two sentences, Anne is the topic. The shift of subject in the target sentence means that Caroline becomes a contrastive topic (see Büring, 1999, section 1.3). As such, it is the natural associate of the additive (Krifka, 1998; Sæbø, 2004).

We created 24 items of the type described above. The experiment furthermore included 24 filler items. These were simple and short grammatical discourses of a similar form as the target items. Additionally, 10 control items were added to the stimuli. These items were short discourse fragments which violated various semantic and syntactic constraints. They were similar to but not the same as, the ones used in Experiment 1.

Participant recruitment and compensation were carried out in the same way as in Experiment 1. 32 participants (female = 14 (44%), male = 18 (56%)) were recruited.

The procedure was very similar to Experiment 1. Some modifications were needed to accommodate the different design of the experiment. The experiment consisted of 4 conditions so the 24 target items were distributed across 4 lists and participants were assigned at random to each of these lists. Each participant saw 6 items from every condition.

The predictions are the same as in Experiment 1, with the caveat that the stimuli are discourse fragments and not sentences.

No participants were removed from the data. In the same way as in Experiment 1 we fit a cumulative ordinal Bayesian model to the data. All the model details were the same as in Experiment 1. The distance was sum-contrast coded with long conditions coded as 1 and short conditions coded as –1. The trigger was sum-contrast coded (nil = –1, too = 1). Fixed effects of interest were: distance, trigger and their interaction.

We used regularising prior distributions for slopes and intercepts. For the intercept, we used a normal distribution with μ = 0 and σ = 2. For the slopes, we used a normal distribution with μ = 0 and σ = 0.5. See also Experiment 1 for a more extensive discussion of the prior structures. We report the results on the log-odds scale.

The 89% credible intervals for the slopes of the interest are summarised on Figure 4.

As in the previous experiments, the crucial parameter here is the slope of the interaction. The posterior distribution of the slope of the interaction (89% CrI = (–0.2, 0.1), mean = –0.04, sd = 0.09) suggests that the effect of distance on acceptability does not clearly differ between the trigger conditions.

The estimate of the distance parameter indicates that the longer items were judged as less natural (89% CrI = (–0.3, –0.04), mean = –0.2, sd = 0.09).

In the case of the trigger, the fragments with too were more natural to the participants than those without it (89% CrI = (0.1, 0.5), mean = 0.3, sd = 0.1).

We conducted a self-paced reading study using the same design as in Experiment 3. Some nonessential changes were applied to the stimuli, and the control items were removed. Out of 48 items, 28 (17 target items and 11 filler items) were followed by a comprehension question.

The list of items is provided at the OSF repository.

79 participants (female = 45 (57%), male = 31 (39%), other/don’t want to say = 3 (4%)) were recruited for the self-paced reading study using Prolific (https://www.prolific.co).

The participants were compensated 3.00 GBP for taking part in the experiment. The experiment took, on average, 16 minutes and 46 seconds to finish; therefore the participants would get 10.7 GPB per hour, on average. The participants were self-reported English native speakers.

The recruitment of the participants, randomization, and data collection mimicked that in Experiment 2. The experiment was conducted in a noncumulative self-paced reading moving window paradigm, as in Experiment 2. Since the stimuli were constructed of quite long text fragments, we inserted line breaks manually to ensure that the regions of interest would not end up before or across the line break.

The predictions are the same as in Experiment 2, with the caveat that the stimuli are discourse fragments and not sentences.

Every participant answered more than 70% of comprehension questions correctly. For this reason, no participant was removed. Time measurements which were under 50 ms or above 3,000 ms were removed from the results. Similar data trimming was done, for example, by Futrell et al. (2021). This resulted in the removal of 0.46% of the data.

Data analysis of the final data from 79 participants was similar to the data analysis conducted for Experiment 2.

We fitted a Bayesian mixed-effects model to the response time data, measured at the critical, post-critical, and post-post-critical regions. The experimental factor trigger was coded using sum contrasts, with the nil condition as –1 and the too condition as 1. The factor distance was also coded using sum contrasts, with the short condition coded as –1 and long condition as 1. Both of these variables were used as fixed effects in the model. Participants and items were entered as random effects.

We also fit two additional models to the data. The first of these was the same nested model as in Experiment 2. With the second of those, we wanted to check for the effect of distance in regions other than the critical and post-critical. Here, we first pooled the data from the regions starting at the critical region –5, i.e. the first region at which the distance manipulation could be detected, up to the pre-critical region and the post-post-critical region (excluding the critical and post-critical region). The post-post-critical was the last region at which all the items had measurements. To these data, we fit a similar model as before, but without the trigger predictor. After this, we also examined the effect of distance at each of those particular regions. This was done to check for secondary support for our initial explanation, i.e. that the retrieval of the antecedent is erasing the general speed-up. The reasoning was that, if there is a general speed-up, we should also be able to observe it in other regions besides the critical and the post-critical regions. These models used 4 sampling chains with 6,000 samples drawn from each chain. 2,000 samples from each chain were discarded for warm-up; hence, each model had 16,000 samples available for the analysis.

The priors used were the same as the ones used Experiment 2. For the intercept, we used a normal distribution with μ = 6 and σ = 1.5. For the slopes, we used a normal distribution with μ = 0 and σ = 0.1. The prior distribution used for the standard deviations of the random effects and for the residual standard deviation was a truncated normal distribution with μ = 0 and σ = 1. For the random effects correlation between the intercept and the slope we used the LKJ distribution (Lewandowski et al., 2009; Stan Development Team, 2021) with η = 2. See also Section 4.5 for a more extensive discussion of the prior structures.

Model estimates are reported on the log-ms scale.

The descriptive summaries of the reading times across conditions for one word before and two words after the critical region (the additive too and the corresponding word in the nil condition) are shown on Figure 5.

At the critical region, the main effect of trigger was –0.02 log-ms (89% CrI: (–0.04, –0.008)). This suggests a negative difference between the conditions, i.e. the too condition was read faster than the nil condition. The main effect of distance was –0.009 log-ms (89% CrI: (–0.02, 0.003)). Since the probability density of the 89% credible interval is on the both sides of zero, one cannot conclusively infer the direction of the effect from the estimate. However, the region has much more density below zero which means that it is more likely that the participants were reading the word faster in the long condition than in the short one. The interaction effect was 0.005 log-ms (89% CrI: (–0.01, 0.02)). This suggests that the difference between the long and short conditions was smaller in the too conditions than in the nil conditions, although a caveat similar to the distance results applies here also. All the results observed at the critical region had the same direction as in Experiment 2. The magnitude of the results was also similar excluding the effect of distance, which, in Experiment 2, was more negative.

Let us turn to the nested model for the critical region. The model shows that the effect of distance within the too condition was –0.006 log-ms (89% CrI: (–0.02, 0.01)) and that the effect of distance within the nil condition was –0.01 log-ms (89% CrI: (–0.03, 0.007)). The results suggest that long was read faster than short in both conditions, even though the credible intervals include zero and, thus, the speed-up should not be taken as conclusive. Both of these effects were much closer to 0 than in the Experiment 2, but the fact that more probability density is below 0 aligns with the results observed in Experiment 2.

At the post-critical region, the main effect of trigger was –0.02 log-ms (89% CrI: (–0.03, –0.005)), i.e. the too condition was read faster than the nil condition. The main effect of distance was –0.01 log-ms (89% CrI: (–0.02, 0.001)). As in the case of the critical region, the probability density of the 89% credible interval is on both sides of zero, but most of the density is below zero. Therefore, it is more likely that the participants were reading the word faster in the long condition than in the short one. The interaction effect was 0.01 log-ms (89% CrI: (0, 0.02)). This suggests that the difference between the long and short conditions, which was negative as the main effect, was diminished or even (as we indeed will see) reversed in the too condition. The effects of trigger and interaction were virtually the same as in Experiment 2, the effect of distance was diminished compared to Experiment 2. The posterior probability distribution of the parameters is summarized in Figure 6.

The interaction can be more easily understood in the nested model, which shows that the effect of distance within the too condition was 0.01 log-ms (89% CrI: (–0.007, 0.03)). That is, it is more likely that the too condition took more time to read in the long than in the short condition. However, since the credible interval crosses zero this interpretation should not be taken as conclusive. The effect of distance within the nil condition was –0.02 log-ms (89% CrI: (–0.04, –0.006)), which means that the long condition was read faster than the short condition. Both of these effects were more positive than in Experiment 2.

For the post-post-critical region, the results are very similar. The main effect of trigger was –0.02 log-ms (89% CrI: (–0.04, –0.008)), that is, the too condition was read faster than the nil condition. The main effect of distance was –0.004 log-ms (89% CrI: (–0.01, 0.006)). The interaction effect was 0.007 log-ms (89% CrI: (–0.003, 0.02)). The effects of trigger and interaction were virtually the same as in Experiment 2, the effect of distance was more negative compared to Experiment 2.

In the nested model of the post-post-critical region, the effect of distance within the too condition was 0.001 log-ms (89% CrI: (–0.01, 0.02)), that is, there was no clear difference in reading times between the distance conditions. The effect of distance within the nil condition was –0.01 log-ms (89% CrI: (–0.03, 0.003)), which means that it is more likely that the participants were reading the long condition faster than the short condition, but since the credible interval crosses zero, the interpretation is inconclusive. These effects were the only ones that were reversed compared to the results from Experiment 2. The effect of distance within too condition was much smaller in this experiment, and the effect of distance within nil condition was much larger in this experiment.

The 89% credible interval of the estimate of the interaction parameter is predominantly situated on the positive side in the critical and the post-post-critical regions, and it is fully situated on the positive side in the post-critical region. The finding in the post-critical region replicates what we observed in Experiment 2. In the nested comparison of the post-critical region, we see that the interaction is driven by the speed-up in long items in the nil condition, which is crossed with a slowdown in long items in the too condition. The slowdown due to distance in the too condition, however, was not fully conclusive, since the credible interval crossed zero. In Experiment 2, we suggested that this can be explained as an interplay of two effects: readers speed up during reading, hence the speed-up due to distance in the nil condition, whereas the retrieval process is more effortful when the antecedent being retrieved is further away, hence the lack of speed-up, or even slowdown, in the too condition. We assume the same explanation accounts for the interaction in this experiment.

We can explore this explanation further since the short and long conditions were matched in several words preceding the critical region. Somewhat abstractly, we can represent the situation in this experiment as in (12). Sentence1, Sentence2 and Sentence3 are matched across the short and long conditions. Sentence3 includes the crucial manipulation, the additive particle too, which triggers retrieval. We can go up to 5 words back to the beginning of Sentence2 to explore the effect of distance on matched words before too was read.

If our explanation is right, we expect to find a speed-up in these regions. In fact, the findings are mixed. When we pool regions in Sentence2 and Sentence3 before the critical region, but including the post-post-critical region, we see a negative effect, corresponding to the expected speed-up: –0.002 log-ms (89% CrI: (–0.01, 0.008)). However, as we can see, the credible interval does not exclude zero. When we look at individual words, we see a clear negative effect (i.e. faster reading in longer distance) at region –4, four words back from the critical region: –0.02 log-ms (89% CrI: (–0.04, –0.006)). It is probably relevant that in this region a clause (Sentence2) ended in 18 out of 24 items, and there was a line break in the majority of test items after this word. In this respect, it might be worth pointing out that the critical region, which is the next region to show a quite clear speed-up effect of distance, is also the end of a clause, which is signalled by a comma, and followed by a gerund clause. However, it is not clear why we would see a spillover of the speed-up effect to the post-critical and the post-post-critical regions, but no comparable spillover from the region –4. Thus, while we have some evidence that the speed-up effect of distance can be found outside of the critical and the post-critical regions, the evidence is not entirely conclusive, since only one region shows this effect clearly.

The results of the post-critical region replicate the results from Experiment 2 even though Experiment 4 looks at retrieval across a series of sentences, while Experiment 2 investigated the effect of distance within a clause. This suggests that the memory processes involved in the retrieval of the antecedent of too might be similar both at the sentence level and at the discourse level.

As can be seen in Figure 5, the data are noisier than in Experiment 2. This can be related to the stimuli being discourse fragments instead of simple sentences.

For each experiment type (i.e. acceptability judgment and self-paced reading), we took all the collected data and treated them as if they were gathered in one experiment in which subsets of participants saw subsets of items.

For the self-paced reading tasks, we only considered data from the post-critical regions. To such datasets we fit a model with random intercepts and slopes for participants and items. The fixed effects were distance, trigger and their interaction. The predictor trigger was sum-coded with nil = –1, and too = 1. The predictor distance was the number of words between the antecedent and the trigger z-normalized, i.e. rescaled to a mean of 0 and standard deviation of 1.

For the acceptability judgment experiments, we treated the data in a similar way, but since the distance factors differed between the experiments (4 levels in the first experiment and 2 levels in the second), we took a subset of items from the first experiment (levels labelled 0 as short and 8 as long) and fit models to this subset. The distance was sum-coded with the long distance coded as 1 and the short distance as –1.8

First, we consider the self-paced reading data. We report the estimates of the models with the interaction. The main effect of the trigger was –0.02 log-ms (89% CrI: (–0.03, –0.008)). The negative credible interval shows that the post-critical regions are more likely read faster in the too condition than in the nil condition. The main effect of distance was –0.016 log-ms (89% CrI: (–0.03, –0.006)). This means that the participants read the post-critical region faster when the distance was greater. Finally, the interaction effect was 0.017 log-ms (89% CrI: (0.008, 0.03)). This shows that the negative effect of distance was diminished, or even reversed, in the too condition. The interaction estimate replicates the findings from individual experiments and supports our hypothesis that increased distance is problematic for presupposition resolution, because the retrieval of the antecedent is more effortful.

For the acceptability judgment data, the main effect of the trigger was 0.2 log-odds (89% CrI: (0.08, 0.35)), suggesting that the too condition was judged as more natural than the nil condition. The main effect of distance was –0.3 log-odds (89% CrI: (–0.5, –0.2)). This means that the participants judged longer sentences as worse on average than short sentences. The interaction effect was –0.08 log-odds, showing that the effect of distance was greater (more negative) in the too condition, but the credible interval included zero (89% CrI: (–0.19, 0.019)).

We now turn to the comparison of the two hypotheses H0 and H1 using the Bayes factor. Since, as mentioned above, the Bayes factor is sensitive to assumptions about prior distributions (Gelman et al., 2003; Navarro, 2019; Schad et al., 2021), we provide a sensitivity analysis. This means that we check several priors for the part of the model in which H0 and H1 differ, i.e. the interaction term distance × trigger.

The summary of the sensitivity analysis for Bayes factor BF₁₀ regarding the self-paced reading studies is provided in Table 1. In this sensitivity analysis, the prior distribution for the interaction term is normal, with μ centered at zero and σ one value from the list 0.01, 0.05, 0.10, 0.20, 0.50. We chose these values for σ since small values (0.01 and 0.05) restrict the interaction term within the most plausible range and should thus lead to a BF₁₀ that shows evidence in favor of the model with the interaction term, H1 (provided the data support H1). The remaining values (0.1, 0.2, 0.5) are unreasonably large and overestimate the effect size of the interaction term, and, thus, it is likely that somewhere among these values, we will observe that BF₁₀ should become smaller than 1 and thus reveal evidence in favor of H0, the model without the interaction term.

A similar reasoning could be applied to the sensitivity analysis of the Bayes factor regarding the acceptability judgment task. Again, we decided to use a normal prior distribution, with μ set to 0 and σ taking values from the list 0.005, 0.009, 0.05, 0.1, 0.3, 0.5. We assumed that the smaller values of σ should provide more evidence for the model with the interaction term and the larger values, for the model without the interaction term, but note that here we do not really have a good reference point for what effect size should be expected.

In Table 1, we see that when the standard deviation of the prior distribution is ≤ 0.2, Bayes factor estimates show evidence in favour of the model with the interaction, and only when the standard deviation is quite large, σ = 0.5, is BF₁₀ smaller than 1, showing evidence in favour of the null model (the model without the interaction term). This decrease of evidence in favor of H1 is not surprising, given that other studies of interference effects in memory retrieval found clear evidence in favor of H1 only when the effect was small (Schad et al., 2021).

Among the values considered, there is a maximum, (BF₁₀ = 13.3), when the standard deviation is 0.01, i.e. the observed data in the pooled model are 13.3 times more likely to occur under H1 than H0 in this case. This size of BF₁₀ is sometimes labeled as moderate-to-strong evidence in favor of the full model (van Doorn, et al., 2021).

Regarding the acceptability study, we see in Table 2 that no values of σ provide evidence for the model with the interaction. In fact, the estimates for most values of σ are close to 1, which means that there is no change in evidence for either of the models, as far as acceptability judgment studies are concerned.

In the pooled reading studies, we see that the 89% credible interval for the interaction parameter lies on the positive side of 0, indicating that for the pooled model, it is likely that the effect is positive. Moreover, the Bayes factor estimates indicate that there is clear evidence (moderate-to-strong in the range of most reasonable priors) in favor of the H1 model. There was no evidence in favor of the interaction model in case of the pooled acceptability studies, that is, the detectable slowdown in reading does not translate to decreased acceptability in our experiments.

The sensitivity analysis of the Bayes factor shows the relationship between the standard deviation of the prior distribution on the interaction parameter, and the strength of the evidence for the full model (the model with the interaction term).

The interaction term expresses the difference between the effect of distance in the too condition and the effect of distance in the nil condition. The model fitted to the pooled data predicted the mean estimate of the interaction term as 0.0173 log-ms. It is easier to interpret these estimates on the ms scale. At the intercept estimate of the model, this corresponds to a difference of 14 ms. The difference is calculated as follows:

This means that at the intercept (here, the grand mean reading time), we expect 14 ms of difference per one unit of distance. Recall that the distance is measured in the scaled number of words between the beginning of the text and the trigger. On this scale, one unit of distance corresponds to 6.2 words.

The results of these experiments provide evidence that reading is slowed down when the antecedent is further away from the trigger, the additive particle too. However, given that retrieval of the antecedent is needed, and given that it seems clear that this retrieval should be more taxing with increased distance, it should not be difficult to find evidence for this distance effect. Yet, the effects in both experiments are rather small. The strength of this evidence, as quantified by the Bayes factor, is conditional on the prior distribution for the slope of the interaction used in the model. We saw moderate to strong evidence for models with prior distributions that assume rather small effects sizes.

Notice, however, that memory retrieval effects seem to be small in general (Jäger et al., 2017). Their small effect size is also what makes them difficult to establish. As an example, consider facilitatory interference effects in antecedent-reflexive constructions. A study by Dillon et al. (2013) found no facilitation, but a later experiment which collected data from many more participants (Jäger et al., 2020) did find some such effect.

The size of the effect also provides clues as to how we should reason about its source. It is usually assumed that retrieval can happen either via a serial search mechanism or via a direct-access, cue-based mechanism (McElree et al., 2003). The former is a costly mechanism; the parser has to attend to every possible item in search for a match. To believe that a similar mechanism is responsible for retrieval in the experiments presented, we would expect a drastic slowdown with every additional candidate. We did not observe such a slowdown. Rather, the approximate size of the effect is comparable to the findings on retrieval in syntax. Assuming that only these two possibilities are on the table, we are compelled to conclude that the mechanism responsible is direct access.

The slowdown observed in the experiments could be interpreted as an effect of decay: the greater distance resulted in a slower retrieval of the relevant information, because its representation in memory weakened with time. Interpreting the findings in terms of decay is compatible with cue-based theories of retrieval, which can be cast in the framework in which temporal decay is one of the factors affecting retrieval time (e.g. ACT-R Anderson & Lebiere, 1998; Anderson, 2007).

Another possibility is that the effect is driven by a different kind of cue from the ones we assumed so far. The experiments assumed that all the cue-value pairs other than the ones directly manipulated remained constant across the conditions. However, there are cue-value pairs the status of which might change with distance. For example, the salience of the licensing element might decrease with distance.9 If the retrieval of the antecedent is guided by a cue {salient}, and salience diminishes with distance, we would expect the parser to have trouble retrieving the target.

We started this work arguing that additive presuppositions trigger a retrieval process, on the basis of the fact that language users have the capacity to judge the felicity of additive too, given the presence or absence of appropriate material earlier in the discourse. That is, the contrast between (15) and (16) shows that the processing too relies on recall of earlier parts of the discourse.

The way we have presented the reliance on memory so far is in line with the treatment of presupposition as an anaphora-like phenomenon, as per quite a few proposals from semantic theory (Kripke, 2009; van der Sandt & Geurts, 1991; van der Sandt, 1992;). However, one aspect of the contrast between (15) and (16) may be relevant for processing, yet does not follow from the anaphoric character of additive presuppositions. Not only is the additive felicitous in (15), it is also obligatory.10 That is, upon having processed Sue lives in France in (15), anything other than an additive will be deemed infelicitous. This obligatory presence of too is usually explained along the following lines. If Sue is a contrastive topic in the second sentence and the additive is left out, then the presence of alternatives to Sue would trigger the implicature that (out of the contextually salient set of individuals) only Sue lives in France. But this implicature is contradicted by what we just learned about where Anne lives. By presupposing that some alternative is true in addition to the asserted one, exhaustification is cancelled. This is because only one operation can associate with the focus/contrastive topic at a time. So, the presence of too precludes exhaustification from taking place and this is why the additive is obligatory in (15) (e.g. Bade, 2016; Krifka, 1998; Sæbø, 2004).

We do not believe this aspect of additive presuppositions drastically changes the retrieval picture, though it may suggest a slightly different narrative about what is behind our results.11 As explained above, the obligatory nature of too makes it very predictable. This predictability is probably supported by various factors, one of which is the parallelism in the items of our first two experiments, as, for example, in (17).

When processing such a sentence, after arriving at the site of the verb in the second clause (dances), participants can retrieve the predicate (is a dancer) from the first clause and, as a result, predict too. This prediction is likely to result in an increase in reading speed, but such a speed-up is preceded by a retrieval effort, which is less costly for shorter sentences (since retrieval of the previous predicate is less inhibited by the distracting predicates). Because the items in Experiments 3 and 4 do not include the kind of parallelism found in (17), it could be that the prediction of too is less profound in those experiments, which could account for the smaller effect.

On this view, one may expect to see the effect of distance earlier, since the retrieval happens at the verb, but it is likely such differences cannot be captured in the self-paced reading paradigm. In short, then, we think that although the predictability of too allows for a narrative that is slightly different from the anaphoric processing framing we used elsewhere, the overall predictions are not so different, since the kind of retrieval involved is the same.

Our starting point was to combine insights from semantic approaches to additive presuppositions with psycholinguistic findings on memory retrieval. The combination lead us to the hypothesis that an increase in distance between the additive too and the antecedent should result in more costly retrieval, which should translate into a slowdown in reading. In two self-paced reading experiments, this slowdown in too was observed, but, crucially, the slowdown was observed with respect to the baseline, that is, the sentences that lacked the additive. Furthermore, as we saw, an increase in distance caused a speed-up in the baseline. A slowdown in sentences with too relative to this baseline means, in the absolute measures studied in the nested comparisons, no speed-up/slowdown (Experiment 2) or a slowdown which, however, is not fully conclusive (Experiment 4).

We suggested that the speed-up in the baseline is expected independently, based on previous findings that show that readers speed up as they proceed in reading text (Demberg and Keller, 2008; Dotlačil, 2021; Ferreira and Henderson, 1993; see also Vasishth & Lewis, 2006, p. 780). Furthermore, we claimed that the fact that we observe a slowdown compared to the baseline but no absolute slowdown, in sentences with too should be understood to arise through the combination of two factors. On one hand there is a slowdown caused by more costly retrieval when the distance from the antecedent increases. On the other hand, readers read longer texts and, therefore, they should speed up. The net result is potentially no observable absolute slowdown in sentences with too.

Such a trade-off between speed-up due to increased sentence length and slowdown due to harder retrieval was observed in other studies on retrieval and was discussed, among others, by Levy & Keller (2013); Futrell et al. (2020). In those studies, however, the focus was on how specific readers’ expectations, modeled in Surprisal Theory, interact with the distance effects observed in retrieval.

One way to strengthen the results of the current study would be to design an experiment in which the distance in too does not cause just a relative speed-up but in which a slowdown can be seen irrespective of any baseline. Nicenboim et al. (2016) present an extended discussion of problems that arise when researching distance effects which could potentially be useful in constructing novel experiments that would further strengthen the results in this way.