The dataset	We used the annotated dataset reported by degen2015investigating, a dataset of the utterances from the Switchboard corpus of telephone dialogues  that contain the word some. The dataset consists of 1,362 unique utterances with a noun phrase containing some (some-NP). For each example with a some-NP, degen2015investigating collected inference strength ratings from at least 10 participants recruited on Amazon's Mechanical Turk. Participants saw both the target utterance and ten utterances from the preceding discourse context. They then rated the similarity between the original utterance like (UNKREF8) and an utterance in which some was replaced with some, but not all like (UNKREF9), on a 7-point Likert scale with endpoints labeled “very different meaning” (1) and “same meaning” (7). Low similarity ratings thus indicate low inference strength, and high similarity ratings indicate high inference strength.   I like, I like to read some of the philosophy stuff. I like, I like to read some, but not all, of the philosophy stuff.

The new, no-context ratings correlated with the original ratings ($r=0.68$, see Appendix SECREF11) but were overall more concentrated towards the center of the scale, suggesting that in many cases, participants who lacked information about the conversational context were unsure about the strength of the scalar inference. Since the original dataset exhibited more of a bi-modal distribution with fewer ratings at the center of the scale, this suggests that the broader conversational context contains important cues to scalar inferences. For our model, these results suggest that the representation of the conversational context is inadequate, which highlights the need for more sophisticated representations of linguistic contexts beyond the target utterance. We further find that the model trained on the original dataset is worse at predicting the no-context ratings ($r=0.66$) than the original ratings ($r=0.78$), which is not surprising considering the imperfect correlation between ratings across experiments, but also provides additional evidence that participants indeed behaved differently in the two experiments.

While the weak/strong distinction has been notoriously hard to pin down , degen2015investigating used strength norms elicited independently for each item, which exploited the presuppositional nature of strong some: removing some (of) from utterances with weak some leads to higher ratings on a 7-point Likert scale from `different meaning' to `same meaning' than removing it from utterances with strong some. Items with stronger some – e.g., (UNKREF19a), strength 3.3 – yielded stronger inference ratings than items with weaker some – e.g., (UNKREF19b), strength 6.7.   And some people don't vote. (5.2) Well, we could use some rain up here. (2.1)  The quantitative findings from degen2015investigating are summarized in Figure FIGREF27, which shows in blue the regression coefficients for all predictors she considered (see the original paper for more detailed descriptions). For our experiments, we randomly split the dataset into a 70% training and 30% test set, resulting in 954 training items and 408 test items.

Experiments ::: Tuning results	We find that the attention layer improves predictions; that contextual word embeddings lead to better results than the static GloVe embeddings; and that including the conversational context does not improve predictions (see Appendix SECREF8, for learning curves of all models, and Section SECREF6, for a discussion of the role of conversational context). Otherwise, the model is quite insensitive to hyperparameter settings: neither the dimension of the hidden LSTM states nor the dropout rate had considerable effects on the prediction accuracy. We do find, however, that there are differences depending on the BERT and ELMo layer that we use as word representations. We find that higher layers work better than lower layers, suggesting that word representations that are influenced by other utterance tokens are helpful for this task. Based on these optimization runs, we chose the model with attention that uses the BERT-large embeddings but no conversational context for the subsequent experiments and analyses.

Context, revisited	In the tuning experiments above, we found that including the preceding conversational context in the input to the model did not improve or lowered prediction accuracy. At the same time, we found that the model is capable of making accurate predictions in most cases without taking the preceding context into account. Taken together, these results suggest either that the conversational context is not necessary and one can draw inferences from the target utterance alone, or that the model does not make adequate use of the preceding context.  did not systematically investigate whether the preceding conversational context was used by participants judging inference strength. To assess the extent to which the preceding context in this dataset affects inference strength, we re-ran her experiment, but without presenting participants with the preceding conversational context. If the context is irrelevant for drawing inferences, then mean inference strength ratings should be very similar across the two experiments, suggesting the model may have rightly learned to not utilize the context. If the presence of context affects inference strength, ratings should differ across experiments, suggesting that the model's method of integrating context is ill-suited to the task.

Conclusion and future work	We showed that neural network-based sentence encoders are capable of harnessing the linguistic signal to learn to predict human inference strength ratings from some to not all with high accuracy. Further, several model behavior analyses provided consistent evidence that the model learned associations between previously established linguistic features and the strength of scalar inferences. Taken together, these results suggest that it is possible to learn associations between linguistic features and scalar inferences from statistical input consisting of a relatively small set of utterances. In an analysis of the contribution of the conversational context, we found that humans make use of the preceding context whereas the models we considered failed to do so adequately. Considering the importance of context in drawing both scalar and other inferences in communication , , , , , , , the development of appropriate representations of larger context is an exciting avenue for future research. One further interesting line of research would be to extend this work to other pragmatic inferences. Recent experimental work has shown that inference strength is variable across scale and inference type , . We treated some as a case study in this work, but none of our modeling decisions are specific to some. It would be straightforward to train similar models for other types of inferences.

Harnessing the richness of the linguistic signal in predicting pragmatic inferences	The strength of pragmatic inferences systematically depends on linguistic and contextual cues. For example, the presence of a partitive construction increases the strength of a so-called scalar inference: humans perceive the inference that Chris did not eat all of the cookies to be stronger after hearing "Chris ate some of the cookies" than after hearing the same utterance without a partitive, "Chris ate some cookies". In this work, we explore to what extent it is possible to learn associations between linguistic cues and inference strength ratings without direct supervision. We show that an LSTM-based sentence encoder with an attention mechanism trained on a dataset of human inference strength ratings is able to predict ratings with high accuracy (r=0.78). We probe the model's behavior in multiple analyses using corpus data and manually constructed minimal pairs and find that the model learns associations between linguistic cues and scalar inferences, suggesting that these associations are inferable from statistical input.

Lastly, the fact that the attention weights provided insights into the model's decisions suggests possibilities for using neural network models for developing more precise theories of pragmatic language use. Our goal here was to investigate whether neural networks can learn associations for already established linguistic cues but it would be equally interesting to investigate whether such models could be used to discover new cues, which could then be verified in experimental and corpus work, potentially providing a novel model-driven approach to experimental and formal pragmatics.

Using this corpus, degen2015investigating found that several linguistic and contextual factors influenced inference strength ratings, including the partitive form of, subjecthood, the previous mention of the embedded NP referent, determiner strength, and modification of the head noun. Partitive: (UNKREF10a-b) are example utterances from the corpus with and without partitive some-NPs, respectively. Values in parentheses indicate the mean inference strength rating for that item. On average, utterances with partitives yielded stronger inference ratings than ones without.   I've seen some of them on repeat. (5.8) You sound like you have some small ones in the background. (1.5)  Subjecthood: Utterances in which the some-NP appears in subject position, as in (UNKREF13a), yielded stronger inference ratings than utterances in which the some-NP appears in a different grammatical position, e.g., as a direct object as in (UNKREF13b).   Some kids are really having it. (5.9) That would take some planning. (1.4)  Previous mention: Discourse properties also have an effect on inference strength.

The full list of sentences can be found in Appendix SECREF10.  Some (of the) (organic) farmers (in the mountains) milked the brown goats who graze on the meadows. The organic farmers in the mountains milked some (of the) (brown) goats (who graze on the meadows). The brown goats who graze on the meadows were milked by some (of the) (organic) farmers (in the mountains). Some (of the) (brown) goats (who graze on the meadows) were milked by the organic farmers in the mountains. Figure FIGREF41 shows the model predictions for the manually constructed sentences grouped by the presence of a partitive construction, the grammatical function of the some-NP, and the presence of a modifier. As in the natural dataset from , sentences with a partitive received higher predicted ratings than sentences without a partitive; sentences with subject some-NPs received higher predicted ratings than sentences with non-subject some-NPs; and sentences with a modified head noun in the some-NP received lower predictions than sentences with an unmodified some-NP. All these results provide additional evidence that the model learned the correct associations.

Experiments ::: Training	We used 5-fold cross-validation on the training data to optimize the following hyperparameters. Word embedding model: GloVe, ELMo, BERT-base, BERT-large. Output layer of word embedding models: $[1,3]$ for ELMo, $[1,12]$ for BERT-base, and $[1,24]$ for BERT-large. Number of training epochs: $[1,800]$. Dimension of LSTM hidden states: $\lbrace 100,200,400,800\rbrace $. Dropout rate in LSTM: $\lbrace 0.1,0.2,0.3,0.4\rbrace $. We first optimized the output layer of the word embedding model for each embedding model while keeping all other parameters fixed. We then optimized the other parameters for each embedding model by computing the average correlation between the model predictions and the human ratings across the five cross-validation folds. Architectural variants. We also evaluated all combinations of two architectural variants: First, we evaluated models in which we included the attention layer (LSTM+Attention) or simply used the final hidden state of the LSTM (LSTM) as a sentence representation.

Model	The objective of the model is to predict mean inference strength rating $i$ given an utterance (a sequence of words) $U = \lbrace w_1,w_2,...,w_N\rbrace $. While the original participant ratings were on a Likert scale from 1 to 7, we rescale these values to the interval $[0,1]$. Figure FIGREF22 shows the overall model architecture. The model is a sentence classification model akin to the model proposed by . The model first embeds the utterance tokens using pre-trained embedding models, and then forms a sentence representation by passing the embedded tokens through a 2-layer bidirectional LSTM network (biLSTM)  with dropout  followed by a self-attention mechanisms that provides a weighted average of the hidden states of the top-most biLSTM hidden states. This sentence representation is then passed through a transformation layer with a sigmoid activation function, which outputs the predicted score in the interval $[0,1]$. We rescale this predicted value to fall in the original interval $[1,7]$.

The attention weights are generally higher for tokens early in the utterance, but the attention weights of utterances with a subject some-NP are on average higher for tokens early in the utterance compared to utterances with the some-NP in non-subject positions. Both of these findings provide evidence that the model assigns high weight to the tokens within and surrounding the some-NP. In a more targeted analysis to assess whether the model learned to use the partitive cue, we examined whether the model assigned higher attention to the preposition of in partitive some-NPs compared to when of occurred elsewhere. As utterance length was again a potential confound, we conducted the analysis separately on the full set of utterances with raw attention weights and on a subset that included only utterances with at least two instances of of (128 utterances), in which we renormalized the weights of of-tokens to sum to 1. Results are shown in the right panel of Figure FIGREF30. The attention weights were higher for of tokens in partitive some-NPs, suggesting that the model learned an association between partitive of in some-NPs and inference strength.

Model behavior analyses ::: Minimal pair analysis.	As a final analysis, we constructed artificial minimal pairs that differed along several factors of interest and compared the model predictions. Such methods have been recently used to probe what kind of syntactic dependencies different types of recurrent neural network language models are capable of encoding (e.g., , , , , , ), and also allow us to probe whether the model is sensitive to controlled changes in the input. We constructed a set of 25 initial sentences with some-NPs. For each sentence, we created 32 variants that differed in the following four properties of the some-NP: subjecthood, partitive, pre-nominal modification, and post-nominal modification. For the latter three features, we either included or excluded of the or the modifier, respectively. To manipulate subjecthood of the some-NP, we created variants in which some was either the determiner in the subject NP as in (UNKREF36a) or in the object-NP as in (UNKREF36b). We also created passive versions of each of these variants (UNKREF36c-d). Each set of sentences included a unique main verb, a unique pair of NPs, and unique modifiers.

Second, since participants providing inference strength ratings also had access to the preceding conversational context, we also compared models that make predictions based only the target utterance with the some-NP and models that make predictions based on target utterances and the preceding conversational context. For the models using GloVe and ELMo, we prepended the conversational context to the target utterance to obtain a joint context and sentence embedding. For models using BERT, we made use of the fact that BERT had been trained to jointly embed two sentences or documents, and we obtained embeddings for the tokens in the target utterance by feeding the target utterance as the first document and the preceding context as the second document into the BERT encoder. For these models, we discarded the hidden states of the preceding context and only used the output of the BERT encoder for the tokens in the target utterance. Implementation details. We implemented the model in PyTorch . We trained the model using the Adam optimizer  with default parameters and a learning rate of 0.001, minimizing the mean squared error of the predicted ratings.

A some-NP with a previously mentioned embedded NP referent yields stronger inferences than a some-NP whose embedded NP referent has not been previously mentioned. For example, (UNKREF16a) contains a some-NP in which them refers to previously mentioned Mission Impossible tape recordings, whereas planning in the some-NP in (UNKREF16b) has not been previously mentioned.   I've seen some of them on repeats. (5.8) That would take some planning. (1.4)  Modification:  also found a small effect of whether or not the head noun of the some-NP was modified such that some-NP with unmodified head nouns yielded slightly stronger inferences than those with modified head nouns. Determiner strength: Finally, the strength of the determiner some has traditionally been analyzed as having a weak, indefinite, non-presuppositional reading as well as a strong, quantificational, presuppositional reading , .

These results suggest that the NN model indeed learned associations between linguistic features and inference strength rather than only explaining variance caused by individual items. This is particularly true for the grammatical and lexical features; we find that the NN predictor explains most of the variance originally explained by the partitive, subjecthood, and modification predictors. More surprisingly, the NN predictions also explain a lot of the variance originally explained by the determiner strength predictor, which was empirically determined by probing human interpretation and is not encoded explicitly in the surface form utterance. One potential explanation for this is that strong and weak some have different context distributions. For instance, weak some occurs in existential there constructions and with individual-level predicates, whereas strong some tends not to , , . Since pre-trained word embedding models capture a lot of distributional information, the NN model is presumably able to learn this association.

We evaluated three word embedding models: GloVe, a static pre-trained word embedding matrix , and pre-trained contextual word embedding models in the form of English ELMo ,  and English BERT ,  models. We used the 100d GloVe embeddings, and we evaluated the 768d uncased BERT-base and 1024d BERT-large models.

Model behavior analyses ::: Attention weight analysis.	As a second type of analysis, we analyzed the attention weights that the model used for combining the token embeddings to a sentence embedding. Attention weight analyses have been successfully used for inspecting and debugging model decisions (e.g., , , , ; but see , and , for critical discussions of this approach). Based on these results, we expected the model to attend more to tokens that are relevant for making predictions. Given that many of the hand-mined features that predict inference strength occur within or in the vicinity of the some-NP, we should therefore expect the model to attend most to the some-NP. To test this, we first explored whether the model attended on average more to some than to other tokens in the same position. Further, we exploited the fact that subjects generally occur at the beginning of a sentence. If the model attends to the vicinity of the some-NP, the average attention weights should be higher on early positions for utterances with a subject some-NP compared to utterances with a non-subject some-NP, and conversely for late utterance positions.

Introduction	An important property of human communication is that listeners can infer information beyond the literal meaning of an utterance. One well-studied type of inference is scalar inference , , whereby a listener who hears an utterance with a scalar item like some infers the negation of a stronger alternative with all:  Chris ate some of the cookies. $\rightsquigarrow $ Chris ate some, but not all, of the cookies. Early accounts of scalar inferences (e.g., , , ) considered them to arise by default unless specifically cancelled by the context. However, in a recent corpus study, degen2015investigating showed that there is much more variability in the strength of scalar inferences from some to not all than previously assumed. degen2015investigating further showed that this variability is not random and that several lexical, syntactic, and semantic/pragmatic features of context explain much of the variance in inference strength. Recent Bayesian game-theoretic models of pragmatic reasoning , , which are capable of integrating multiple linguistic cues with world knowledge, are able to correctly predict listeners' pragmatic inferences in many cases (e.g., , ).

Hyperparameter tuning	Figure FIGREF44 shows the learning curves averaged over the 5 cross-validation tuning runs for models using different word embeddings. As these plots show, the attention layer improves predictions; contextual word embeddings lead to better results than the static GloVe embeddings; and including the conversational context does not improve predictions and in some cases even lowers prediction accuracy.

Model behavior analyses	Regression analysis. As a first analysis, we investigated whether the neural network model predictions explain (some of) the variance explained by the linguistic factors that modulate inference strength. To this end, we used a slightly simplified Bayesian implementation of the linear mixed-effects model by degen2015investigating using the brms  and STAN  packages and compared this original model to an extended model that included the output of the above NN model as a predictor. For this comparison, we investigated whether the magnitude of a predictor in the original model significantly decreased in the extended model that included the NN predictions, based on the reasoning that if the NN predictions already explain the variance previously explained by these manually coded predictors, then the original predictor should explain no or less additional variance. We approximated the probability that the magnitude of the coefficient in the extended model including the NN predictor is smaller than the coefficient in the original model, $P(|\beta _i^{extended}| < |\beta _i^{original}|)$, by sampling values for each coefficient from the distributions of the original and the extended models and comparing the magnitude of the sampled coefficients. We repeated this process 1,000,000 times and treated the simulated proportions as approximate probabilities.

We thus compared the average attention weights for each position in the utterance across utterances with subject versus non-subject some-NPs. To make sure that any effects were not only driven by the attention weight of the some-tokens, we set the attention weights of the token corresponding to some to 0 and re-normalized the attention weights for this analysis. Further, since the attention weights are dependent on the number of tokens in the utterance, it is crucial that the average utterance length across the two compared groups be matched. We addressed this by removing outliers and limiting our analysis to utterances up to length 30 (1,028 utterances), which incidentally equalized the number of tokens across the two groups. (While these exclusions resulted in tiny quantitative difference in the average attention weights, the qualitative patterns are not affected.) The left panel of Figure FIGREF30 shows the average attention weight by position for some versus other tokens. The model assigns much higher weight to some. The center panel of Figure FIGREF30 shows the average attention weight by position for subject vs. non-subject some-NP utterances.

In the no-context experiments, we truncated target utterances longer than 30 tokens, and in the experiments with context, we truncated the beginning of the preceding context such that the number of tokens did not exceed 150. Evaluation. We evaluated the model predictions in terms of their correlation $r$ with the human inference strength ratings. For the optimization of hyperparameters and architectural variants, we evaluated the model using 5-fold cross-validation. We then took the best set of parameters and trained a model on all the available training data and evaluated that model on the held-out data.

Experiments ::: Test results	Figure FIGREF26 shows the correlation between the best model according to the tuning runs (now trained on all training data) and the empirical ratings on the 408 held-out test items. As this plot shows, the model predictions fall within a close range of the empirical ratings for most of the items ($r=0.78$). Further, similarly as in the empirical data, there seem to be two clusters in the model predictions: one that includes lower ratings and one that includes higher ratings, corresponding to strong and weak scalar inferences, respectively. The only systematic deviation appears to be that the model does not predict any extreme ratings – almost all predictions are greater than 2 or less than 6, whereas the empirical ratings include some cases outside of this range. Overall, these results suggest that the model can learn to closely predict the strength of scalar inferences. However, this result by itself does not provide evidence that the model learned associations between linguistic cues and inference strength, since it could also be that, given the large number of parameters, the model learned spurious correlations independent of the empirically established cue-strength associations. To rule out the latter explanation, we probed the model's behavior in multiple ways, which we discuss next.

These experimental and modeling results suggest that listeners integrate multiple linguistic and contextual cues in utterance interpretation, raising the question how listeners are able to draw these pragmatic inferences so quickly in interaction. This is an especially pressing problem considering that inference in Bayesian models of pragmatics is intractable when scaling up beyond toy domains to make predictions about arbitrary utterances. One possibility is that language users learn to use shortcuts to the inference (or lack thereof) by learning associations between the speaker's intention and surface-level cues present in the linguistic signal across many instances of encountering a scalar expression like some. In this work, we investigate whether it is possible to learn such associations between cues in the linguistic signal and speaker intentions by training neural network models to predict empirically elicited inference strength ratings from the linguistic input. In this enterprise we follow the recent successes of neural network models in predicting a range of linguistic phenomena such as long distance syntactic dependencies (e.g., , , , , ), semantic entailments (e.g., , ), acceptability judgements , factuality , and, to some extent, speaker commitment . In particular, we ask:  How well can a neural network sentence encoder learn to predict human inference strength judgments for utterances with some?

Results from no-context experiment	Figure FIGREF48 shows the correlation between the mean inference strength ratings for each item in the experiment from  and the mean strength ratings from the new no-context experiment, discussed in Section SECREF6.

List of manually constructed sentences	Tables TABREF46 and TABREF47 show the 25 manually created sentences for the analyses described in the minimal pairs analysis in Section SECREF5. As described in the main text, we created 16 variants of the sentence with the some-NP in subject position (sentences in the left column), and 16 variants of the sentence with the some-NP in object position (sentences in the right column), yielding in total 800 examples.

An issue with this analysis is that estimating the regression model only on the items in the held-out test set yields very wide credible intervals for some of the predictors–in particular for some of the interactions–since the model infers these values from very little data. We therefore performed this (and all subsequent) analyses on the entire data, and obtained the NN predictions through 6-fold cross-validation, so that the NN model always made predictions on data that it had not seen during training. This did yield the same qualitative results as the analyses only performed on the held-out test items (see Appendix SECREF9) but it also provided us with narrower credible intervals that highlight the differences between the coefficient estimates of the two models. Figure FIGREF27 shows the estimates of the coefficients in the original model and the extended model. We find that the NN predictions explain some or all of the variance originally explained by many of the manually coded linguistic features: the estimated magnitude of the predictors for partitive, determiner strength, linguistic mention, subjecthood, modification, utterance length, and two of the interaction terms decreased in the extended model.

This is particularly remarkable considering the train-test mismatch: the model was trained on noisy transcripts of spoken language that contained many disfluencies and repairs, and was subsequently tested on clean written sentences.

Regression analysis on held-out test data	Figure FIGREF45 shows the estimates of the predictors in the original and extended Bayesian mixed-effects models estimated only on the held-out test data. We find the same qualitative effects as in Figure FIGREF27, but since these models were estimated on much less data (only 408 items), there is a lot of uncertainty in the estimates and therefore quantitative comparisons between the coefficients of the different models are less informative.

To what extent does such a model capture the qualitative effects of hand-mined contextual features previously identified as influencing inference strength? To address these questions, we first compare the performance of neural models that differ in the underlying word embedding model (GloVe, ELMo, or BERT) and in the sentence embedding model (LSTM, LSTM$+$attention). We then probe the best model's behavior through a regression analysis, an analysis of attention weights, and an analysis of predictions on manually constructed minimal sentence pairs.

Question: Which dataset do they use?
Answer: the annotated dataset reported by degen2015investigating, a dataset of the utterances from the Switchboard corpus of telephone dialogues  that contain the word some