Participants. 600 participants with U.S. IP addresses and at least 99% of previous HITs approved were recruited on Amazon’s Mechanical Turk platform (ages: 18–73, median: 38.5). They were paid $2.20.

Materials and procedure. The prior probability, at-issueness, and projection of the contents of 20 clauses were measured in three separate blocks. Prior probability was manipulated by pairing each of the 20 clauses (e.g., Julian dances salsa) with two facts between participants: The content of each clause was expected to have a higher prior probability in the presence of one fact (e.g., Julian is Cuban) than of the other (e.g., Julian is German). See Supplement A for the full set of 20 clauses and facts, and Supplement C for evidence that the facts successfully manipulated the prior probability of the respective contents.

In the prior block, which was the prior block in Degen and Tonhauser (2021), the 20 clauses were realized as the complements of How likely is it that…? questions. As shown in Figure 1a, each target item consisted of one of the two facts for that clause and the How likely is it that…? question. Participants read the fact and assessed the prior probability of the content, given the fact. They gave their responses on a slider marked ‘impossible’ at one end (coded as 0) and ‘definitely’ at the other (coded as 1).

In the at-issueness and projection blocks, target items consisted of a fact and a polar question that was uttered by a named speaker, as in Figure 1c and Figure 1d, respectively. The polar questions were formed by realizing the 20 clauses as the complements of the 20 clause-embedding predicates in Figure 1b, for a total of 400 combinations. The predicates are the same as in Degen and Tonhauser (2021, 2022).5 Participants were told to imagine that they are at a party and that, on walking into the kitchen, they overhear somebody ask somebody else a question. At-issueness was measured using the ‘asking whether’ diagnostic (as in Tonhauser et al., 2018): Participants were asked to rate whether the speaker was asking about the CC, taking into consideration the fact. They gave their responses on a slider marked ‘yes’ at one end (coded as 0) and ‘no’ at the other (coded as 1). Greater not-at-issueness of the CC with respect to the implicit QUD should result in higher slider ratings. Projection was measured using the ‘certain that’ diagnostic (as in, e.g., Degen & Tonhauser, 2022; Mahler, 2020; Tonhauser et al., 2018): Participants were asked to rate whether the speaker was certain of the CC, taking into consideration the fact. They gave their responses on a slider marked ‘no’ at one end and ‘yes’ at the other; ‘no’ was coded as 0 and ‘yes’ as 1, so that greater projection of the CC should result in higher slider ratings.

The at-issueness and projection blocks also included 6 control trials each, which functioned as attention checks: The contents of these items were expected to be at-issue and not to project. The same 6 contents were also used to form 6 filler trials in the prior block. These filler items were not used to assess participants’ attention. For the full set of control and filler items see Supplement A.

Each participant’s set of items was semi-randomly generated: First, the 20 clauses were randomly paired with the 20 predicates. Then, one random half of the items was assigned the respective clause’s higher-probability fact, and the other half its lower-probability fact. Participants completed a total of 78 trials: 20 target trials in each block, 6 control trials in the projection and at-issueness blocks each, and 6 filler trials in the prior block. Each participant completed the same filler and control trials. To measure the prior probability of the contents, the prior block was presented first to all participants. The projection and at-issueness blocks were then presented in random order. Within blocks, trial order was randomized.

After completing the experiment, participants filled out a short optional demographic survey. To encourage truthful responses, participants were told that they would be paid no matter what answers they gave in the survey.

Data exclusion. Data was excluded based on self-declared non-native speaker status and other criteria shown in Supplement B, leaving 10,100 data points from 505 participants (ages 20–73; mean age: 39.5).

To address the research questions in (2), we want to know for each lexical meaning (contributed by a clause-embedding predicate) whether there is an effect of prior probability on projection (research question I), whether there is an effect of at-issueness on projection (research question II), whether there is an effect of the interaction of prior probability and at-issueness on projection (research question IIIa), and whether there is an effect of prior probability on at-issueness (research question IIIb). To address these questions, we fit two types of models to the data. All analyses were conducted in R (R Core Team, 2016), version 4.3.1) using RStudio (version 2023.12.0+369).

First, to address questions (I), (II) and (IIIa), we fit 20 Bayesian mixed-effects beta regression models, one for each clause-embedding predicate, using the ‘brms’ package (Bürkner, 2017). The models predicted certainty ratings6 (measuring projection) from a centered fixed effect of asking-whether rating (measuring at-issueness), a centered fixed effect of prior probability rating, and their interaction.7 Priors on all fixed effects were flat. The models included a random by-item intercept (where an item is a complement clause) and a random by-item slope for the fixed effects of asking-whether and prior probability rating.8 The outputs of these models are expected means and their 95% highest posterior density intervals (HDIs).9 Full model outputs are provided in the repository for Exp. 1.

Second, to address question (IIIb), we also fit 20 Bayesian mixed-effects beta regression models, one for each clause-embedding predicate. The models predicted asking-whether ratings (measuring at-issueness) from a centered fixed effect of prior probability rating, again with a flat prior on the fixed effect. The models included a random by-item intercept (where an item is a complement clause) and a random by-item slope for the fixed effect of prior probability rating. As with the models described above, the outputs of these models are expected means and 95% highest posterior density intervals. Full model outputs are provided in the repository for Exp. 1.

To facilitate interpretation, the models did not include a fixed effect of block or an interaction between a fixed effect of block and the other fixed effects. To investigate the role of block order, we conducted auxiliary analyses on the two subsets of the data that differ in whether the projection block preceded the at-issueness block (referred to as the ‘proj/ai’ subset) or followed it (the ‘ai/proj’ subset). (Recall that the prior block always came first.)

Hypothesis testing was conducted to investigate the effects of interest, that is (I) whether there was an effect of prior beliefs on projection for each of the clause-embedding predicates/lexical meanings, (II) whether there was an effect of at-issueness on projection for each of the clause-embedding predicates/lexical meanings, (IIIa), whether there was an effect of the interaction between prior beliefs and at-issueness on projection for each for each of the clause-embedding predicates/lexical meanings, and (IIIb) whether there was an effect of prior beliefs on at-issueness for each of the clause-embedding predicates/lexical meanings. Figure 3a reports, for each of the four hypotheses, whether there was evidence for an effect, whether the effect was positive or negative, and the Bayes factor associated with the effect.

Figure 2 shows participants’ ratings for the full dataset. The corresponding figures for the two subsets ‘proj/ai’ and ‘ai/proj’ are provided in Supplement D.

(I) The relation between prior beliefs and lexical meaning in modulating projection inferences. Recall that Degen and Tonhauser 2021 observed a positive effect of prior probability on projection: The higher an interpreter’s prior belief in a content, the more the interpreter takes the speaker to believe in the content. This result was replicated in the current experiment: As shown in Figure 2a, which shows participants’ certainty ratings (measuring projection) against their prior probability ratings by clause-embedding predicate,10 there was a positive effect of prior for each predicate. This suggests that the more likely interpreters are to believe a content a priori, the more that content projects.

These observations were supported by the models fitted to the full dataset and to the ‘proj/ai’ subset: As shown in Figure 3a, there is at least strong evidence for a positive effect of prior probability on certainty ratings (measuring projection) for each predicate in the full dataset as well as in the proj/ai subset. In the ai/proj subset, there is at least strong evidence for a positive effect for 14 of the 20 predicates. However, for 4 predicates, namely discover, acknowledge, reveal and say, there is only weak to moderate evidence for a positive effect. Furthermore, there is no evidence for an effect for be annoyed and weak to moderate evidence for a negative effect for inform.

One possible explanation for this difference between the ‘proj/ai’ and ‘ai/proj’ datasets is that the effect of prior beliefs on projection is an artifact of the within-participant design of Exp. 1. On this explanation, participants’ certainty ratings on the projection block were artificially influenced by their prior belief ratings, but only when the prior block immediately preceded the projection block. Another possible explanation is that the effect of prior beliefs on projection is real, but when the at-issueness block directly precedes the projection block, participants’ attention is drawn to the question of what the speaker is asking about, thereby drawing resources away from integrating information from the prior. One way of distinguishing these explanations is to collect certainty ratings from a set of participants who do not provide prior belief ratings and to use a separate group’s prior belief ratings as a predictor of these certainty ratings. This is what we did in Exp. 2, reported in the next section. Preliminary support for the latter explanation comes from Degen and Tonhauser’s (2021) Exp. 2, which suggested that there is an effect of prior beliefs on projection even when the prior belief ratings are collected from a different set of participants than the certainty ratings.

(II) The relation between at-issueness and lexical meaning in modulating projection inferences. Recall that Tonhauser et al.’s (2018) Gradient Projection Principle leads us to expect a positive effect of not-at-issueness on projection for all predicates. Djärv and Bacovcin (2017, 2020) and Mahler et al. (2020), on the other hand, lead us to expect a positive effect only for factive predicates and either the opposite or no effect for nonfactive ones. Figure 2b shows participants’ certainty ratings (measuring projection) against asking-whether ratings (measuring at-issueness) by clause-embedding predicate: There was a positive effect for most predicates (e.g., know, inform, announce, acknowledge), which suggests that for these predicates, the more not-at-issue the embedded content is, the more it projects. There was a negative effect for other predicates (e.g., pretend, think). Finally, there was no evidence for an effect of not-at-issueness on certainty ratings for a third group of predicates (e.g., prove, confirm). These observations suggest that there is by-predicate variation in thes effect of at-issueness on projection.

The observations were supported by the models. As shown in Figure 3a, there was very strong to extreme evidence for a positive effect of asking-whether rating (measuring at-issueness) on certainty ratings (measuring projection) for 13 of the 20 predicates in the full dataset and the two subsets, namely be annoyed, know, inform, see, hear, discover, acknowledge, reveal, confess, admit, announce, confirm and be right. For demonstrate, there was very strong to extreme evidence for a positive effect in the full dataset and the proj/ai subset, but no evidence for an effect in the ai/proj dataset. These results (that is, at least strong evidence in the full dataset and one subset) suggest that, for these 14 predicates, the more not-at-issue the CC is, the more it projects.

For think, suggest and pretend, there was at least strong evidence for a negative effect of at-issueness on projection in the full dataset, though the three predicates differed in the strength of this evidence in the two subsets: For suggest it was merely weak to moderate in both subsets, for think it was weak to moderate in one subset and very strong to extreme in the other, and it was very strong to extreme in both subsets for pretend. This suggests that, for pretend, think and possibly also suggest, the more at-issue the CC is, the more it projects. For the remaining 3 predicates (establish, prove, say), the models fit to the three datasets only provided at most weak to moderate evidence for a (positive or negative) effect of at-issueness in the full dataset, and no clear support in the two subsets. This suggests that, for these 3 predicates, there is no effect of at-issueness on projection.

These results are not predicted by Tonhauser et al.’s (2018) Gradient Projection Principle, which does not lead us to expect predicates with negative or no effects. The results also do not entirely match the results of Djärv and Bacovcin (2017, 2020) and Mahler et al. (2020): Although there was a positive effect for factive predicates, as well as a negative or no effect for some nonfactive predicates, there were also several nonfactive predicates with a positive effect (e.g., inform, acknowledge, confess, admit, be right).

(IIIa) The relation between prior beliefs and at-issueness in modulating projection inferences. Recall that most projection analyses in the RSA framework assume no relation between prior beliefs and at-issueness in modulating projection, that is, they assume that prior beliefs and at-issueness independently modulate projection. Figure 2c shows participants’ certainty ratings (measuring projection) against asking-whether ratings (measuring at-issueness) by prior probability and by predicate. There does not seem to be an interaction for most predicates (e.g., be annoyed, discover), but some predicates seem to exhibit an interaction (e.g., see, pretend).11

The models fit to the full dataset and the two subsets did not support the assumption of a systematic relation between prior beliefs and at-issueness in modulating projection. As shown in Figure 3a, there was at most weak to moderate evidence for an effect of the interaction in all three datasets for 12 of the 20 predicates (namely know, discover, reveal, admit, confess, announce, demonstrate, confirm, establish, prove, think, suggest). For five predicates, there was strong to extreme evidence for an effect in one of the three datasets, but at most weak to moderate evidence in the other two (inform, acknowledge, be right) or very strong to extreme evidence for an effect of the opposite polarity in another dataset (be annoyed, hear). Thus, for 17 of the 20 predicates, the data do not support the assumption of a systematic relation between prior beliefs and at-issueness in modulating projection. The only three predicates for which the models support the assumption of an effect are see (at least strong evidence for a negative effect in the full dataset and one subset), say (strong evidence for a positive effect in the full dataset and one subset), and pretend (at least strong evidence for a positive effect in all three datasets).

These results provide some support for the assumption made in extant RSA models that there is no relation between prior beliefs and at-issueness in modulating projection (Pan & Degen, 2023; Qing et al., 2016; Scontras & Tonhauser, 2025; Stevens et al., 2017; Warstadt, 2022). However, if the effects for see, say and pretend replicate in Exp. 2, such analyses will need to allow for individual predicates to exhibit an interaction between prior beliefs and at-issueness in modulating projection.

(IIIb) The relation between prior beliefs and at-issueness. Recall that Tonhauser et al.’s (2020) Non-redundancy Principle for At-issue Content predicts a positive effect of prior beliefs on not-at-issueness. With respect to Figure 2d, which shows participants’ asking-whether ratings (measuring at-issueness) by their prior probability ratings, this principle leads us to expect a positive effect for all of the predicates. While there may be a weakly positive effect for hear and demonstrate, there does not seem to be an effect for many of the 20 predicates, and a negative one for think, say, suggest and pretend.

These observations were supported by the models. As shown in Figure 3a, there was either no evidence for an effect of prior beliefs on at-issueness or only weak to moderate evidence in all three datasets for 8 of the 20 predicates (namely inform, see, discover, acknowledge, reveal, admit, announce, be right). For hear and demonstrate, there was at last strong evidence for a positive effect of prior beliefs on at-issueness in the full dataset and one of the subsets. For be annoyed, know, admit, think, say, suggest and pretend, there was at least strong evidence for a negative effect in the full dataset and one of the two subsets. These results suggest that the effect of prior beliefs on at-issueness exhibits by-predicate variation. That there was either no evidence for an effect or at least strong evidence for a negative effect for 15 of the 20 predicates does not support Tonhauser et al.’s (2020) Non-redundancy Principle for At-issue Content., which predicts a positive effect across the board.

Participants. 600 participants with U.S. IP addresses and at least 99% of previous HITs approved were recruited on Amazon’s Mechanical Turk platform (ages: 19–73, median: 37.5). They were paid $2.20.12

Materials and procedure. Exp. 2 differed from Exp. 1 only in that there was no prior block. As in Exp. 1, each participant’s set of items was semi-randomly generated: First, the 20 clauses were randomly paired with the 20 predicates. Then, one random half of the items was assigned the respective clause’s higher-probability fact, and the other half its lower-probability fact. Participants completed a total of 52 trials: 20 target trials in each block, and 6 control trials in the projection and at-issueness blocks each. Each participant completed the same control trials. The projection and at-issueness blocks were presented in random order.

Data exclusion. Data was excluded based on self-declared non-native speaker status and other criteria shown in Supplement B, leaving 10,160 data points from 508 participants (ages 19–73; mean age: 38.3).

To address the research questions in (2), we fit the same two types of models to the data as in Exp. 1 (see Section 2.2). However, in contrast to the models for Exp. 1, where there was a centered fixed effect of prior probability rating, there was a centered fixed effect of mean prior probability rating in the models for Exp. 2. This is because the prior probability ratings were aggregated by content/fact combination, as they came from a different set of participants than the participants of Exp. 2. The outputs of these models are expected means and 95% highest posterior density intervals. Full model outputs are provided in the repository for Exp. 2.

Hypothesis testing was conducted to investigate the effects of interest, that is (I) whether there was an effect of mean prior beliefs on projection for each of the clause-embedding predicates/lexical meanings, (II) whether there was an effect of at-issueness on projection for each of the clause-embedding predicates/lexical meanings, (IIIa) whether there was an effect of the interaction between mean prior beliefs and at-issueness on projection for each of the clause-embedding predicates/lexical meanings, and (IIIb) whether there was an effect of mean prior beliefs on at-issueness for each of the clause-embedding predicates/lexical meanings. Figure 3b reports, for each of the four hypotheses, whether there was evidence for an effect, whether the effect was positive or negative, and the Bayes factor of the evidence.

Figure 4 shows participants’ ratings for the full dataset. The corresponding figures for the two subsets ‘proj/ai’ and ‘ai/proj’ are provided in Supplement E.

(I) The relation between prior beliefs and lexical meaning in modulating projection inferences. Figure 4a shows, for each predicate, participants’ certainty ratings for the content/fact combinations against the mean prior probability ratings for the content/fact combinations that were collected from a different set of participants. There is a positive effect for each predicate. These observations were supported by the models fit to the full dataset and for the ‘proj/ai’ dataset. As shown in Figure 3b, there was at least strong evidence for a positive effect of mean prior probability ratings on certainty ratings (measuring projection) for each predicate in the full dataset as well as in the proj/ai subset (except for admit, where the evidence was weak to moderate). In the ai/proj subset, the evidence was merely weak to moderate for a positive effect for see, discover, announce, confirm and be right, and for a negative effect for know.

These results suggest that the effect of prior beliefs on projection observed in Exp. 1 was not an artifact of the within-participant design of that experiment. Rather, the results of Exps. 1 and 2 together support the assumption that participants integrate their prior beliefs about content when providing certainty ratings, regardless of whether they were asked to give prior probability ratings in an earlier block (Exp. 1) or not (Exp. 2). The fact that in both Exps. 1 and 2 the evidence for a positive effect of prior beliefs on projection is weaker in the ‘ai/proj’ block than in the ‘proj/ai’ block further suggests that when participants’ attention is drawn to the question of what the speaker is asking about, integrating information from their prior beliefs into their certainty ratings is more challenging than if they give certainty ratings before giving asking-whether ratings. Overall then, the results of Exps. 1 and 2 support the hypothesis that participants’ prior beliefs about content modulate their projection ratings for each of the 20 predicates under investigation.

(II) The relation between at-issueness and lexical meaning in modulating projection inferences. Recall, again, that Tonhauser et al.’s (2018) Gradient Projection Principle predicts a positive effect of not-at-issueness on projection for all predicates. Djärv and Bacovcin (2017, 2020) and Mahler et al. (2020), on the other hand, predict a positive effect only for factive predicates and either the opposite or no effect for nonfactive ones. Figure 4b shows participants’ certainty ratings (measuring projection) against asking-whether ratings (measuring not-at-issueness) by clause-embedding predicate: There was a positive effect of not-at-issueness on projection for most predicates (e.g., know, inform, announce, acknowledge). There were also predicates with no effect (e.g., think, suggest, pretend). In contrast to Exp. 1, there were no predicates with a clearly negative effect.

These observations were supported by the models. There was at least strong evidence for a positive effect of asking-whether rating (measuring at-issueness) on certainty ratings (measuring projection) for 12 of the 20 predicates in the full dataset and the two subsets: be annoyed, know, inform, see, hear, discover, acknowledge, admit, confess, announce, demonstrate and confirm. For an additional 5 predicates, there was at least strong evidence for a positive effect in the full dataset, and at least strong evidence in one of the subsets: reveal, establish, prove, be right and say. This suggests that, for these 17 predicates, the more not-at-issue the CC is, the more it projects. The models supported the assumption of a negative effect of projection on at-issueness only for suggest, where there was very strong to extreme evidence for a negative effect in the full dataset and in the ‘ai/proj’ subset. For think and pretend, the results of Exp. 2 did not support the assumption of an effect of at-issueness on projection. These observations again suggest that there is by-predicate variation in the effect of at-issueness on projection, though the variation is not identical to that of Exp. 1.

Comparing the results of Exp. 2 to those of Exp. 1, we found evidence for a positive effect of at-issueness on projection in both experiments for 14 predicates, namely be annoyed, know, inform, see, hear, discover, acknowledge, reveal, admit, confess, announce, demonstrate, confirm and be right. These predicates include those with the highest certainty ratings (that is, the predicates whose CCs are most projective), but also be right, a predicate whose CC is only weakly projective. Among the remaining six predicates, there were two for which the polarity of the effect of at-issueness on projection was the same across the two experiments, but stronger in Exp. 2 than Exp. 1, namely prove (positive effect) and suggest (negative effect). For the remaining 4 predicates (establish, think, say, pretend), the results of Exp. 1 did not replicate in Exp. 2, as either the polarity of the effect changed between the two experiments, or there was no effect in Exp. 2.

Overall, then, we take the results of Exps. 1 and 2 to suggest that there is by-predicate variation in the effect of at-issueness on projection: For 14 of the 20 predicates, the CC is more projective when it is more not-at-issue. For suggest, the CC is more projective when it is less at-issue. And for the remaining 5 predicates (think, pretend, establish, say, prove) there is no consistent effect of at-issueness on projection across the two experiments. These results are not completely predicted by the Gradient Projection Principle (which does not lead us to expect predicates without a positive effect). The results also do not entirely match the results of Djärv and Bacovcin (2017, 2020), who did not observe nonfactive predicates with positive effects or verbal nonfactive predicates with no effects, or the results of Mahler et al. (2020), who did not not observe nonfactive predicates with positive effects.

What are relevant properties of the six predicates that did not display a positive effect of at-issueness on projection (suggest, think, pretend, establish, say, prove)? As noted above, it is not appropriate to characterize these predicates as having weakly projective CCs because the CC of be right is on par with these predicates when it comes to projection, but does exhibit a positive effect of at-issueness on projection (see Supplement F for a visualization). For three of these predicates, namely pretend, think and suggest, we hypothesize that they do not show a positive effect of at-issueness on projection because their lexical meanings facilitate an inference that the speaker does not believe that the CC is true. This is most easily motivated for pretend, which lexically entails that the speaker believes that the CC is false (#Cole is pretending that Julian dances salsa, and Julian (indeed) dances salsa). For think and suggest, in turn, it is plausible that utterances with these predicates may give rise to the scalar implicature that the speaker does not believe the CC to be true by virtue of the speaker not having used know, the stronger alternative that entails that the speaker believes the CC to be true.13 Thus, even though the CCs of these predicates are at least moderately not-at-issue in both Exps. 1 and 2 (see Supplement F), the CC is not equally projective due to the lexical meanings of these predicates.

Unlike suggest, think, and pretend, the predicates establish, say, and prove cannot plausibly be taken to contribute to an inference that the speaker does not believe that the CC is true. These three predicates also differ from the other three in that Exp. 2 provided evidence for a positive effect of at-issueness on projection, but not Exp. 1. What might explain this difference between Exps. 1 and 2? An observation that may be relevant is that asking-whether ratings were not as stable across the two experiments as certainty ratings: The Spearman rank coefficient was only .72 for the mean asking-whether ratings of the 800 by-predicate/content/fact combinations, but .85 for the mean certainty ratings. (See Supplement G for details.) This might be taken to suggest that asking-whether ratings are more susceptible than certainty ratings to the ways in which participants might enrich the context in which they interpret the stimuli or to the (implicit) prosody with which they read the stimuli. If the lexical meanings of establish, say, and prove are such that such factors may modulate the at-issueness of the CC more than its projection, this might offer a path to an explanation of the differences in results between the two experiments.

(IIIa) The relation between prior beliefs and at-issueness in modulating projection inferences. Figure 4c shows participants’ certainty ratings (measuring projection) against asking-whether ratings (measuring at-issueness) by mean prior probability and by predicate. For some predicates, there does not seem to be an interaction (e.g., inform, see, hear), while there appears to be one for others (e.g., be annoyed, admit, announce).

The models fit to the full dataset and the two subsets did not support the assumption of a systematic relation between mean prior beliefs and at-issueness in modulating projection across the 20 predicates. For 9 of the 20 predicates, there was either no evidence for an effect or only weak to moderate evidence for an effect in all three datasets (namely know, see, hear, discover, reveal, demonstrate, establish, prove, pretend). For the three predicates be annoyed, confess, and suggest the overall evidence was inconclusive. Finally, there were 8 predicates for which there was at least strong evidence for an effect in the full dataset and one of the subsets: This effect was positive for inform, acknowledge, announce, be right and think, and negative for admit, confirm and say. While the results for these 8 predicates may be taken to suggest that there is an interaction between mean prior beliefs and at-issueness in modulating projection, it is important to note that none of these effects were observed in Exp. 1 (with individual-level prior ratings as a predictor). Furthermore, none of the effects observed in Exp. 1 replicated in Exp. 2: Whereas the results of Exp. 1 suggested a negative effect for see and a positive effect for say and pretend, the results of Exp. 2 do not support an effect for see and pretend, and they support a negative effect for say. We therefore take the results of Exps. 1 and 2 to suggest that there is no systematic relation between prior beliefs and at-issueness in modulating projection inferences for the CCs of the 20 clause-embedding predicates we investigated. This, in turn, provides support for the assumption made in extant RSA models of projection inferences that prior beliefs and at-issueness independently modulate projection (Pan & Degen, 2023; Qing et al., 2016; Scontras & Tonhauser, 2025; Stevens et al., 2017; Warstadt, 2022).

(IIIb) The relation between prior beliefs and at-issueness. Figure 4d, which shows participants’ asking-whether ratings (measuring at-issueness) by the mean prior probability ratings, does not suggest that there is an effect for most of the 20 predicates. Recall that Tonhauser et al.’s (2020) Non-redundancy Principle for At-issue Content leads us to expect positive effects.

This observation was supported by the models. For 15 of the 20 predicates, there was at most weak to moderate evidence for an effect of mean prior beliefs on at-issueness (namely know, inform, hear, discover, reveal, confess, announce, demonstrate, confirm, establish, prove, be right, think, say, pretend). For the remaining 5 predicates, there was at least some evidence for an effect, but none of these were observed in Exp. 1: For instance, whereas there was very strong to extreme evidence for a positive effect for see in the full dataset and the ‘ai/proj’ subset of Exp. 2, there was at most weak to moderate evidence for a negative effect in Exp. 1. Similarly, whereas there was at least strong evidence for a positive effect for acknowledge in the full dataset and the ‘proj/ai’ dataset of Exp. 2, there was no evidence for an effect in Exp. 1. Furthermore, none of the effects observed in Exp. 1 for hear, demonstrate, be annoyed, know, admit, think, say, suggest, and pretend replicated in Exp. 2: For all of these predicates, there was at most weak to moderate evidence for an effect of the same polarity in Exp. 2 in the full dataset.

We suggest that the results of Exps. 1 and 2 taken together provide support for the assumption that there is no systematic effect of prior beliefs on at-issueness, contrary to the positive effect that is predicted by Tonhauser et al.’s (2020) Non-redundancy Principle for At-issue Content.

The analysis proposed in Djärv and Bacovcin (2020) is an extension of Heim (1983), where presuppositions are specified in the lexical entries of the triggering expressions. For clause-embedding predicates, this means that the CCs of factive predicates are lexically specified as having to be satisfied in the input context before the context is updated with the utterance, whereas the CCs of nonfactive predicates do not carry this specification. Consequently, these analyses predict that the CCs of factive predicates project from under entailment-canceling operators, either because they are already entailed by the input context or because they are accommodated into the input context. Presuppositions are locally accommodated (in the local context of the entailment-canceling operator) only if global accommodation into the input context would result in inconsistency. No predictions are made for the CCs of nonfactive predicates.

Djärv and Bacovcin (2020) supplemented Heim’s (1983) analysis by proposing that prosody provides a cue to the QUD addressed by the utterance. Specifically, they proposed that utterances of clause-embedding predicates with focus on the subject of the embedded clause, as in (3a), give rise to the QUD inference that somebody has the property denoted by the verb phrase of the embedded clause but it is not known who. On the other hand, the variant with focus on the embedding predicate, as in (3b), gives rise to the QUD inference that the attitude holder has some relationship to the embedded proposition.

Importantly, these two QUD inferences interact differently with the projection inferences contributed by factive predicates (which are taken to required to be entailed by the input context) and nonfactive predicates (which do not contribute such a requirement). Specifically, in the probabilistic model assumed in Djärv and Bacovcin (2020: §4), the probability that the CC is true in utterances of sentences with focused clause-embedding predicates, as in (3b), provides a baseline against which the effect of focus on the embedded subject, as in (3a), can be evaluated. This baseline differs for utterances of sentences with factive and nonfactive predicates because the former but not the latter require the CC to be entailed by the input context. Their analysis predicts that the effect of subject focus differs for factive and nonfactive predicates: For factive predicates, the QUD inference contributed by subject focus (that somebody left town but it is not known who) lowers the probability that the CC is true relative to the predicate focus baseline (which presupposes the truth of the CC). By contrast, for the version of (3a) with a nonfactive predicate, the QUD inference contributed by subject focus increases the probability that the CC is true relative to the baseline (a probability of .5 that Anna left town).14 See Djärv and Bacovcin (2020: §4) for a detailed exposition of the probabilistic model.

Djärv and Bacovcin (2020) did not explicitly relate their analysis to at-issueness, but one might assume that the CC is at-issue in utterances of sentences in which the embedded subject is focused, as in (3a), whereas the CC is not-at-issue when the embedding predicate is focused, as in (3b). Under this interpretation, Djärv and Bacovcin’s (2020) extension of Heim’s (1983) analysis predicts that when the CC of a factive predicate is at-issue, it is less projective, whereas the opposite effect is expected for nonfactive predicates. We now consider whether Heim (1983) and Djärv and Bacovcin (2020) capture the results of our experiments.

(I) The relation between prior beliefs and lexical meaning in modulating projection inferences. Neither analysis predicts the effect of prior beliefs on projection across the 20 clause-embedding predicates, for two reasons. The first reason is that neither analysis makes predictions about the projection of the majority of the 20 predicates, namely the 15 nonfactive ones (see Degen and Tonhauser 2022 for extensive discussion). The second reason is that prior beliefs do not play a role in either analysis. One could imagine an extension on which presupposition accommodation is sensitive to the strength of the prior belief in the presupposition. Specifically, one might assume that the lower the prior belief in the presupposition, the more likely it is not accommodated globally, but locally. This extension would, however, constitute a major change for these analyses, where global accommodation taken to be the default, and local accommodation is only licensed if global accommodation of the presupposition leads to inconsistency.

(II) The relation between at-issueness and lexical meaning in modulating projection inferences. Djärv and Bacovcin’s (2020) extension of Heim (1983) does not predict the observed by-predicate variation in the effect of at-issueness on projection. This is because our experiments suggest that greater not-at-issueness results in greater projection not just for the CCs of factive predicates but also for several nonfactive predicates. While Djärv and Bacovcin’s (2020) analysis correctly predicts the pattern observed for factive predicates and for suggest, where we found a negative effect, they predict the opposite of what is observed for many of the nonfactive predicates we investigated (namely a negative effect where a positive one is observed).

In sum, Heim (1983) and the extension in Djärv and Bacovcin (2020) correctly predict that the CCs of factive predicates project and that greater not-at-issueness leads to greater projection for these predicates. These analyses do not, however, predict that the CCs of nonfactive predicates are also projective, that greater not-at-issueness may also lead to greater projection for nonfactive predicates, and they also do not predict the effect of prior beliefs on projection for the CCs of clause-embedding predicates.

Abrusán (2011, 2016) and Simons et al. (2017) do not assume that presuppositions are lexically specified. In Abrusán (2011, 2016), a lexical entailment of an uttered sentence is a presupposition if it is about a time that is not the event time of the matrix predication and it is not at-issue with respect to the QUD addressed by the utterance. For instance, the CC of B’s utterance in (4), that Phil’s ballet class is canceled, is predicted to be a presupposition (and therefore to project) because it is a lexical entailment of the sentential prejacent of the epistemic modal adverb perhaps and it is not at-issue with respect to A’s interrogative utterance.

In Simons et al. (2017), the CC of a clause-embedding predicate projects if it is entailed by the Current Question of the utterance (where the Current Question is the question that is congruent with the utterance).15 In (4), the Current Question of B’s utterance might be the set of propositions {Phil forgot that his ballet class is canceled today, Phil is aware that his ballet class is canceled today}. If so, the Current Question entails that Phil’s ballet class is canceled today, and the CC may therefore project under Simons et al.’s (2017) analysis.

(I) The relation between prior beliefs and lexical meaning in modulating projection inferences. Neither of these analyses predicts the effect of prior beliefs on projection, for the same reasons as the analyses discussed in the previous section: Neither analysis makes predictions about the projection of the majority of the 20 predicates, namely those whose CCs are not entailed, and prior beliefs do not play a role in either analysis.

(II) The relation between at-issueness and lexical meaning in modulating projection inferences. Projection is sensitive to at-issueness in Abrusán (2011, 2016) and Simons et al. (2017). Specifically, both analyses predict that content that is at-issue with respect to the question addressed by the utterance does not project, and that content that is not-at-issue with respect to said question may project. As such, both analyses correctly predict that entailed CCs that are not-at-issue are more projective than entailed CCs that are at-issue. What neither analysis predicts, however, is that at-issueness may modulate projection not only for entailed CCs but also for nonentailed ones.

In sum, the analyses in Abrusán (2011, 2016) and Simons et al. (2017) predict that entailed CCs may project and that their projection is sensitive to at-issueness. These analyses do not, however, predict the effect of prior beliefs on projection, that nonentailed CCs may also project, and that their projection may also be sensitive to at-issueness.

On the analysis proposed in Schlenker (2021), the CC of a sentence S like (5a) that is uttered in a context c is presupposed if the CC is presupposed by the sentence S' under the entailment-canceling operator, that is (5b), in the local context c'. For the CC to be presupposed in (5b), two conditions must be met: i) S' contextually entails the CC relative to c'; and ii) If we consider a “generic agent” who believes the propositions in c' and who has now learned about the truth of S', then the probability that this generic agent already believed the CC is above a contextual threshold α (§6.2). More colloquially, condition ii) requires that the generic agent “typically antecedently believes” the CC (p.6) upon interpreting S' in c'. Applied to (5), Schlenker’s (2021) analysis predicts that the CC of know in (5) is presupposed if i) (5b) contextually entails that Julian dances salsa, and ii) if a generic agent would typically antecedently believe that Julian dances salsa upon interpreting (5b) in the minimal contexts we provided to our participants. (For Schlenker 2021, the local context under negation c' is identical to c.)

Condition i) is met under the assumption that the CC of (5b) is a contextual entailment (it is, in fact, an entailment). Schlenker (2021) also assumes the condition ii) is met: “in many cases, one’s knowledge of facts will precede one’s knowledge of [Cole’s] beliefs about them…believing that [Julian dances salsa] is often an epistemic precondition for believing that” Cole knows that Julian dances salsa (p.6). One might, however, challenge this assumption on the basis of the corpus study presented in Spenader (2002), which showed that the CCs of the majority of the utterances of sentences with factive verbs (namely 81 of 109) had to be accommodated (i.e., were not contextually entailed). In other words, utterance of sentences with the factive predicates investigated by Spenader (2002) were “generally used to communicate information the speaker thought was hearer-new” (p.99); know was one of the predicates considered in Spenader (2002). This result suggests that one cannot assume that a generic agent typically antecedently believes the CC of know.

In the interest of evaluating the analysis, let’s assume that it is an open, empirical question which predicates are such that the probability of a generic agent antecedently believing the CC is above the contextual threshold α (and, of course, what that threshold might be). Schlenker (2021) appears to assume that there are two classes of predicates: those where the probability is above the contextual threshold (which includes know, announce, and inform), and those where it is not (which includes demonstrate and establish); see p.12 and appendix I in Schlenker (2021). Thus, one advantage of Schlenker’s (2021) analysis over the analyses reviewed in sections 4.1 and 4.2 is that it is able to predict the projection of the CCs of nonfactive predicates (modulo the open questions about condition ii)). The analysis correctly predicts that the CCs of know, inform, and announce are more projective than the CCs of demonstrate and establish, by virtue of the CCs of the former being possibly presupposed, in contrast to the CCs of the latter.

(I) The relation between prior beliefs and lexical meaning in modulating projection inferences. It is not clear that the analysis in Schlenker (2021) is able to predict the observed by-predicate projection variation: Even though the analysis does not divide predicates into factive and nonfactive ones, it nevertheless suggests a binary, categorical distinction between predicates, such that the CCs of predicates with probabilities above the threshold may be presupposed (depending on the context), and those below the threshold are not. As such, the analysis does not appear to predict the projection variation between the CCs that are analyzed as presupposed (e.g., that the CC of be annoyed is more projective than that of reveal). Furthermore, as discussed in Degen and Tonhauser (2022), the CCs of nonfactive predicates like establish and demonstrate are projective when compared to nonprojective main clause content. Schlenker’s (2021) analysis does not make predictions for these CCs. Finally, in Exp. 1, the mean certainty rating of the CC of announce, whose CC is assumed to possibly be presupposed, is .45, and that of demonstrate, whose CC is not assumed to be presupposed, is .38. It is not clear that this relatively small difference in mean certainty rating (.07) motivates analyzing the CC of announce as presupposed in contrast to that of demonstrate.

Schlenker’s (2021) analysis may be able to predict that projection is sensitive to interpreters’ prior beliefs. For instance, the CC of (6) might be presupposed if i) (6) contextually entails that Julian dances salsa (which may very well be the case in a context in which Julian is Cuban, and Cole is a reliable source of information about Julian), and ii) if a generic agent typically believes that Julian dances salsa prior to interpreting (6).

If, on the other hand, (6) is uttered in a context in which the CC is not contextually entailed (e.g., if Julian is German or Cole is an unreliable source), the CC might not be presupposed in (6). Thus, if condition i) allows for prior beliefs to be considered in determining contextual entailment, Schlenker’s (2021) analysis might be able to predict that projection is sensitive to interpreters’ prior beliefs.

(II) The relation between at-issueness and lexical meaning in modulating projection inferences. It is not clear how at-issueness in the form of sensitivity to the QUD would modulate projection in the analysis proposed in Schlenker (2021). It is conceivable, however, to enrich the analysis by incorporating Abrusán’s (2011) constraint that entailments are not presupposed if they are at-issue with respect to the QUD addressed by the utterance.

In sum, Schlenker’s (2021) analysis may be able to predict that the CCs of some nonfactive predicates project and it may also be able to predict that projection is sensitive to prior beliefs. The analysis does not, however, predict that the CCs of many predicates which he takes to be nonpresupposed are projective when compared to main clause content and that, for many predicates, the CCs are more projective when they are more not-at-issue.

The projection analyses in Qing et al. (2016), Warstadt (2022) and Scontras and Tonhauser (2025) are formulated in the Rational Speech Act (RSA) framework (for an introduction see Degen 2023). These analyses were developed for the pre-state content of stop (e.g., that Sam stopped smoking implies that Sam smoked), for the genus content of species predications (e.g., that Tom has a Green Card implies that Tom is not a US citizen), and for the CC of know, respectively.16 As such, none of these analysis capture the variable contributions of the lexical meanings of clause-embedding predicates to projection inferences. We can nevertheless consider how these analyses capture the relation between (I) prior beliefs and lexical meaning, and (II) at-issueness and lexical meaning. We focus on the analysis in Qing et al. (2016) but the same considerations apply to that in Warstadt (2022) and Scontras and Tonhauser 2025.

Qing et al.’s (2016) analysis follows the basic RSA model in assuming a literal listener who interprets utterances according to their semantics and who forms the basis of the reasoning process, a speaker who reasons about the literal listener in choosing their utterances, and a pragmatic listener who infers a probability distribution over the speaker’s intended meaning. In Qing et al.’s (2016) analysis, the literal listener observes an utterance (e.g., Sam didn’t stop smoking) and interprets its literal meaning. There are four possible world states against which utterances are interpreted; these differ in whether Sam smoked in the past, and in whether Sam smokes now. The literal meaning of the positive utterance Sam stopped smoking is compatible only with the world state in which Sam smoked in the past and does not smoke now, which means that the literal meaning of the negated variant is compatible with the other three world states. This means that the projection inference, which is the inference to the world state in which Sam smoked in the past and does not smoke now, is not coded in the lexical meaning of stop and not achieved by the literal listener.

The literal listener in Qing et al.’s (2016) analysis does not simply return a distribution over the world states in which the utterance is literally true, but it rather returns a distribution over answers to the possible QUDs. Following Kao et al. (2014), QUDs are modeled as partitions of the set of world states. For instance, the QUD of whether Sam smokes now partitions the set of world states into the set of two world states in which he does not smoke now, and the set of two in which he smokes now. What is not at-issue with respect to this QUD is whether Sam smoked in the past. For instance, when the literal listener interprets the utterance Sam didn’t stop smoking relative to the QUD of whether Sam smokes now and against a world state in which Sam did not smoke in the past but smokes now, the literal listener returns a distribution in which the positive answer to the QUD has a very high probability (approaching 1) and the negative answer has a very low one (approaching 0).

A modification to the basic RSA model introduced in Qing et al. (2016) is that interpretation, including that by the literal listener, is relative to an assumed common ground, which is a non-empty element of the power set of the four possible world states. Given that interpretation is relative to an assumed common ground, this means that the literal listener considers only those world states that are compatible with the literal meaning of the utterance and that, additionally, are compatible with the common ground assumed by the speaker.

The speaker in Qing et al.’s (2016) analysis wants to convey a particular world state given a common ground and a QUD. The speaker evaluates all possible utterance alternatives with respect to how likely the literal listener is to infer the world state that the speaker wants to communicate, given that common ground and QUD. For instance, if the common ground is the set of world states (that is, nothing is known yet), the QUD is whether Sam smokes now, and the speaker wants to communicate the world state in which Sam smoked in the past and does not smoke now, the utterance Sam stopped smoking has much more utility than the utterance Sam didn’t stop smoking, because the literal listener assigns a higher probability to the intended answer to the QUD (namely that Sam does not smoke now) given the former utterance than given the latter utterance.

The pragmatic listener, finally, observes an utterance, and reasons about the world state and the common ground intended by the speaker. As shown in Qing et al. (2016), the projection inference is sensitive to the QUD: When the pragmatic listener observes Sam didn’t stop smoking and the QUD is whether Sam smokes now, the most likely world state is the state in which Sam smoked in the past and does not smoke now (the projection inference), whereas it is the state that Sam didn’t smoke in the past or now, if the QUD is whether Sam smoked in the past (no projection inference).

(I) The relation between prior beliefs and lexical meaning in modulating projection inferences. Qing et al. (2016) assumed a uniform prior over world states, which means (for instance) that Sam is just as likely to have smoked in the past as he is to not have smoked in the past. Warstadt (2022) and Scontras and Tonhauser 2025 also assumed a uniform prior for the literal listener, but a non-uniform one for the pragmatic listener. This allows one to specify that, if Sam is known to be a very health-conscious person, it is more likely that Sam didn’t smoke in the past than that he did, by assigning a higher prior probability to two world states in which Sam didn’t smoke in the past than to the two world states in which he did smoke in the past. These examples illustrate that prior beliefs can be straightforwardly integrated into projection analyses formulated in the RSA framework, by modifying the distribution over world states (see also, (e.g., Degen, 2023; Goodman & Stuhlmüller, 2013). Scontras and Tonhauser (2025) predicted that the CC of know is more projective from under negation than that of think, and that the projection of both CCs is sensitive to prior beliefs. Future research needs to investigate how these analyses can be extended to the broader set of clause-embedding predicates investigated in this paper.

(II) The relation between at-issueness and lexical meaning in modulating projection inferences. Qing et al. (2016), Warstadt (2022) and Scontras and Tonhauser (2025) showed that the investigated projection inferences are QUD-sensitive: Content is more likely to project when it is not at-issue with respect to the QUD than when it is at-issue with respect to the QUD. These examples illustrate that analyses of projection formulated in the RSA framework can, in principle, integrate the effect of the QUD, and therefore of at-issueness. Qing et al. (2016) and Warstadt (2022) were not developed for the projection inferences contributed by utterances of sentences with clause-embedding predicates, and Scontras and Tonhauser (2025) did not investigate a differential effect of at-issueness on the projection of the CCs of think and know. It is therefore an open question how such analyses can capture the observed by-predicate variation observed in our experiments.

In sum, the RSA analyses in Qing et al. (2016), Warstadt (2022) and Scontras and Tonhauser (2025) are able to predict projection without assuming that the projective content is lexically specified as presupposed. These analyses also show that the effects of prior beliefs and at-issueness can be captured in analyses formulated within the RSA framework. An exciting task for future research is to develop an analysis of the projection of the CCs of clause-embedding predicates in the RSA framework that is able to predict the observed by-predicate projection variation and the observed by-predicate variation in the effect of at-issueness on projection.

A predictive analysis of projection inferences must be able to predict that there is by-predicate projection variation, that prior beliefs modulate projection across clause-embedding predicates, and that there is by-predicate variation in the effect of at-issueness on projection. As illustrated in this section, there is no analysis on the market that can capture all of these observations. One of the largest challenges to developing such an analysis is that lexical meaning modulates projection in more fine-grained ways than previously assumed and that the CCs of nonfactive predicates may also project. The lack of a principled account of how lexical meaning modulates projection also means that there is not yet a satisfactory analysis of the by-predicate variation observed in how at-issueness modulates projection. We hypothesize that (yet to be identified) components of lexical meaning constrain whether the projection of the CC is modulated by the QUD. For instance, as suggested above, an analysis according to which the lexical meaning of pretend entails that that the CC is false may predict that this content does not project even when it is not at-issue. The question of which components of lexical meaning play a role in modulating projection inferences and its interaction with at-issueness is an important question for future research.