Contrafactives, learnability, and production

David Strohmaier & Simon Wimmer*

Abstract. No natural language has contrafactive attitude verbs. Because factives are
universal across natural languages, this means that there is a major asymmetry between
contrafactives and factives. We previously hypothesised that this asymmetry arises
partly because the meaning of contrafactives is significantly harder to learn than that
of factives. Here we test this hypothesis by using a production-oriented computational
experiment that overcomes two limitations of our previous experiments. We find that
our results do not support our previous hypothesis.
Keywords. Contrafactives; Factives; Semantic universals; Transformers; Learnability

1. Introduction. No natural language appears to have what Holton (2017) calls a ‘contrafactive’, i.e.
a morphologically atomic attitude verb that would mirror know and other factives.1,2 Like know, a
contrafactive would entail a belief in the content of its declarative complement. But unlike know,
it would presuppose the falsity (not truth) of that complement.3 For instance, if English had a
contrafactive contra, Ayesha contras that Beatrice is cool would entail that Ayesha believes that
Beatrice is cool, but presuppose that it is false that Beatrice is cool.

That no natural language appears to have a contrafactive is one part of an asymmetry between
contrafactives and factives.4 The second part is that know, and so at least one factive, appears to
have counterparts in all natural languages (Goddard 2010, Hannon 2015). Given that a contrafac-
tive would mirror a factive, this two-part asymmetry raises the question: why do contrafactives and
factives differ so greatly in their frequency?

There have been several attempts to explain this asymmetry. Holton (2017) suggests that con-
trafactives are ruled out because there are no entities suitable to serve as the semantic values of their
declarative complements.5 Roberts & Özyildiz (2023) suggest that they are ruled out because they
do not satisfy a necessary condition on presupposition triggering: that asserted content depends on

*This paper reports on research supported by Cambridge University Press and Assessment, University of Cam-
bridge. We thank the NVIDIA Corporation for the donation of the Titan X Pascal GPU used in this research. We
are grateful to Giulia Martina as well as audiences at and anonymous reviewers for ELM3 and the MECORE Clos-
ing Workshop, especially Kajsa Djärv, Natasha Korotkova, Todor Koev, Tom Roberts, and Wataru Uegaki. David
Strohmaier designed and ran the computational experiment, Simon Wimmer brought philosophical and linguistic dis-
cussions to bear on design and interpretation. Authors: David Strohmaier, Department of Computer Science and
Technology, ALTA Institute, University of Cambridge (ds858@cam.ac.uk) & Simon Wimmer, Department of Philos-
ophy, Heinrich-Heine-University Duesseldorf (simon.wimmer@hhu.de).

1For discussion of relevant cross-linguistic evidence see Rosenberg (1975), Kierstead (2015), Krifka (2016), Holton
(2017), Hsiao (2017), Anvari et al. (2019), Sander (2020), Hoeksema (2021), Bochnak & Hanink (2022), Bossi (2022),
Roberts & Özyildiz (2023), Strohmaier & Wimmer (2022, 2023), McGregor (2024).

2Given the atomicity condition, Anvari et al. (2019)’s creerse is no contrafactive, despite their use of the label.
3The presupposition condition entails that adjectives like falsely or erroneously cannot give rise to a contrafactive,

since falsely believe entails, but does not presuppose, the falsity of its declarative complement.
4Potential candidates not yet discussed in sufficient detail to assess whether they should count as contrafactives

include Pitta-Pitta’s widi and other examples by McGregor (2024) as well as German’s wähnen by Sander (2020). But
as we previously noted, our target asymmetry would remain even if some contrafactives were found.

5For critical discussion see Hyman (2017), Wimmer (2019), Roberts & Özyildiz (2023).

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presupposed content. Finally, in earlier work we hypothesized that contrafactives are less common
than factives partly because their meaning is harder to learn than that of factives (Strohmaier &
Wimmer 2022, 2023).6,7 Our aim here is to further explore our hypothesis.

We initially motivated our hypothesis via two mismatches generated by contrafactive, but not
factive, attitude ascriptions (Strohmaier & Wimmer 2022; 300-1, 2023; 71-3): first, a mismatch
between their matrix subject’s commitment to the truth, and their speaker’s commitment to the fal-
sity, of the contrafactive’s declarative complement; second, a mismatch between the primary use of
a contrafactive’s declarative complement to make assertions (and thereby commit to its truth) and
the commitment to the declarative complement’s falsity incurred by using a contrafactive attitude
ascription. We expected these mismatches to make it harder to learn the meaning of a contrafac-
tive than that of a factive.8 To test our expectation, we ran two experiments with artificial neural
networks designed to capture those mismatches. The results from these experiments supported our
hypothesis: in both cases, the loss dropped more slowly for contrafactives than for factives.

But as section 2 explains, our previous experiments had two key limitations. They did not dis-
tinguish falsity from presupposition failure. And they only tested comprehension, not production.
We therefore conducted a third experiment designed to overcome those limitations. Yet as section
3 shows, the results from this experiment do not support our previous hypothesis. We find no
consistent difference between contrafactives and factives. For reasons we sketch in our conclusion
(section 4), however, we are not yet sure whether this undermines our previous hypothesis.

2. Previous Experiments. In Strohmaier & Wimmer 2022, 2023 we used Transformer encoders
with Binary Cross Entropy Loss from the pyTorch library. We trained them to predict the truth
value of factive, non-factive, and contrafactive attitude ascriptions based on three inputs: an attitude
ascription; a mind representation to tell the model what the matrix subject of the ascription believes
about the world; and a world representation to tell the model what the world is like.9

We encoded the mind and world representations differently across the two experiments. In
‘Experiment 1’, reported in Strohmaier & Wimmer 2022, they consisted of sequences of tokens
that we glossed as representing a 3x3 grid filled with varying geometrical objects of varying colours
(e.g. ‘blue-square empty empty ...’);10 in ‘Experiment 2’, reported in Strohmaier & Wimmer 2023,
the representations consisted of a single token representing a proposition (e.g. ‘r p110’). Given
this difference, we also encoded the attitude ascriptions we fed into the model differently across
Experiments 1 and 2. To illustrate, in Experiment 1 contrafactive attitude ascriptions looked like
contra blue square above red triangle; in Experiment 2 like contra p110.

Our Transformers produced the same kind of output in both experiments: a value within [0,1].
This value encodes an estimate of the probability that an input attitude ascription is true, given

6And, as Steinert-Threlkeld & Szymanik (2019; 4) note, natural languages intuitively tend to use compositional,
non-atomic means to express meanings that are harder to learn.

7Maldonado et al. (2022) show that humans can learn contrafactives. Partly for this reason, we did not intend our
hypothesis to fully explain the asymmetry between contrafactives and factives.

8Our emphasis on the first mismatch was inspired by literature on theory of mind, especially Phillips & Norby
(2021), our emphasis on the second by work on pragmatic-syntactic bootstrapping, especially Hacquard & Lidz (2022).

9Our paradigm for a non-factive attitude ascription is a belief ascription. Like know and a contrafactive, believe
entails a belief in the content of its declarative complement. However, it presupposes neither truth nor falsity.

10We did not train the model on visual information. Our gloss is merely intended to ease human interpretation.

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the input mind and world representations. A value of 1 encodes an estimate of ‘definitely true’; 0
encodes one of ‘definitely not true.’ In evaluating our models’ learning performance, we considered
the distance between their estimates, given the input world and mind representations, and the
correct value (1 or 0), given those representations.

In both experiments, the rolling mean of the loss fell more slowly for contrafactives than
for factives. In Experiment 1, for instance, the mean loss after 100,000 training examples for
contrafactives was 0.54, for factives 0.39. Our Transformers approached the correct values of
contrafactive attitude ascriptions more slowly than that of their factive counterparts.

2.1. LIMITATIONS. We previously noted several limitations of our experiments. Some hold for
one experiment, but not the other. For instance, our Transformer in Experiment 1 may have made
some mistakes partly due to difficulties Transformers have with word order (compare Pham et al.
2021). This was one key reason why Experiment 2 used simpler inputs.

A more general limitation is the question of whether Transformers approximate human lan-
guage learning closely enough for us to draw conclusions about humans from results about Trans-
formers. We previously referred to several encouraging results (e.g. Caucheteux & King 2022,
Merkx & Frank 2021, Schrimpf et al. 2021) that show correlations between Transformer and hu-
man performance. For instance, Schrimpf et al. argue that Transformers (especially GPT-2) explain
a high proportion of all explainable variance from brain measurements (fMRI and ECoG) during
sentence processing tasks, such as reading and listening tasks.

We can add to this list of encouraging results here. Kallini et al. (2024) argue that Transform-
ers learn English more easily than languages that humans cannot learn. Paape (2023) suggests that
Transformers approximate human performance for depth charge illusions and their non-illusory
counterparts. There are also encouraging results for attitude verbs specifically. As Ziembicki et al.
(2023) explain, Transformers (BERT in particular) approximate the factivity of Polish attitude
verbs as judged by Polish-speaking expert linguists more closely than Polish-speaking non-expert
linguists. Similarly, Ross & Pavlick (2019) suggest that Transformers (again BERT) closely ap-
proximate human judgments about the veridicality of English attitude verbs.

Admittedly, Transformers do not match human performance perfectly. Still, the overall trend
is clear: Transformers closely approximate human performance. Given this, we continue to tenta-
tively draw conclusions about humans from results about Transformers.

2.2. MOTIVATION FOR CURRENT EXPERIMENT. Our latest experiment is motivated by, and ad-
dresses, two further limitations of both of our previous experiments.

The first is that an output value of 0 only encodes an estimate that the input ascription is
definitely not true. For instance, if in Experiment 2 our Transformer outputs 0 for contra p110, we
take our Transformer to estimate that contra p110 is definitely not true. But this estimate does not
distinguish whether the ascription is false or a presupposition failure.

In fact, our Transformer cannot distinguish these cases. To see this, contrast two cases. In case
1, our Transformer gets r p110 as mind and world representation. This world representation veri-
fies the declarative complement p110. This makes the ascription a presupposition failure. In case
2, our Transformer gets input mind and world representations distinct from r p110. A non-r p110
world representation falsifies the declarative complement and a non-r p110 mind representation
means the matrix subject does not believe p110. Contra p110’s falsity presupposition is satisfied,

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but its belief entailment is not. This makes the ascription false. The problem is that our model
treats both cases alike: in both cases, we train it to output 0.

That our Transformer should distinguish these cases does not depend on our view of presup-
positions. Of course, if presupposition failure yields undefined or a third truth value, rather than
falsity, our Transformers must learn to distinguish false from undefined or third value ascriptions to
approximate human performance. But even if presupposition failure yields falsity, our Transform-
ers should treat ‘mere’ falsity and presupposition failure differently. For that is just what human
language users do: the ‘hey! wait a minute’ diagnostic (e.g. von Fintel 2004) suggests as much.

The second key limitation of both of our previous experiments is that they only consider how
easily our Transformers learn to comprehend the attitude ascriptions we feed into them. (They
successfully learn to comprehend them iff their output values for those ascriptions, given some
input mind and world representations, match the correct values for those ascriptions, given those
representations.)11 But this means that the results of our experiments do not speak to the relative
ease of learning how to produce contrafactive and factive attitude ascriptions. Our Transformers
got attitude ascriptions as inputs; they did not produce them as outputs.

This is problematic because Transformers may learn how to comprehend and to produce con-
trafactive attitude ascriptions at different speeds. Ideally, they also learn how to produce contrafac-
tive attitude ascriptions more slowly than to produce factives ones. This would strengthen our
hypothesis. But pessimistically, Transformers may also more quickly learn how to produce con-
trafactive attitude ascriptions than factive ones. This would raise questions about our hypothesis.
If the meaning of a contrafactive is easier to learn than that of a factive in a production-oriented
setting, but harder to learn in a comprehension-oriented setting, can a difference in learnability
explain any of the asymmetry in frequency between contrafactives and factives?

To address the two key limitations just noted, we ran a third computational experiment. Now,
our Transformer learns how to produce the expressions whose meaning it has to learn: factive,
non-factive, and contrafactive attitude ascriptions. This immediately addresses the second key
limitation and yields a paradigm that, as far as we know, has not been used yet in the computational
literature on learnability and semantic universals (compare Steinert-Threlkeld & Szymanik 2019,
2020, Steinert-Threlkeld 2020).12

Our production paradigm also allows us to address the first key limitation. To do this, we
enrich the input to our Transformer. In addition to a mind and world representation, the input also
tells the model what we want it to produce. Put roughly, we either ask the network to produce the
most informative true ascription, or ask for the most informative merely false ascription, or ask for
the most informative ascription that is a presupposition failure.13

11This does not model the kind of comprehension one has if one grasps the truth conditions of an utterance, but
lacks information that allows one to decide whether an ascription is true, or has such information but is somehow
blocked from exploiting it. Still, we take it to give us at least some access to how well one grasps the truth conditions
of an utterance, notwithstanding concerns about the relation between competence and performance (compare Dupre
2021). Although our latest experiment tests production, similar points apply here too.

12Johnson et al. (2021) investigated morphological universals for affixes using a production-based experiment with
LSTMs as well as human subjects. Outside the computational literature, Maldonado & Culbertson (2019) investigated
semantic universals for person systems using a production-based experiment with human subjects.

13We thus test for presupposition failure without requiring our Transformer to learn how to comprehend sentences
used in the family of sentences diagnostic (attitude ascriptions embedded under negation, possibility, and question).

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Whether we ask for a truth, falsehood, or presupposition failure, we always ask for the most
informative ascription. This ‘informativity demand’ highlights that the inputs to our Transformer
encode some pragmatic principles. Since we effectively ask our Transformer to maximize the
informativity of asserted and presupposed content, our Transformer learns how to satisfy Grice
(1989)’s maxim of quantity and Heim (1991)’s maximize presupposition.14 This is key to getting
our Transformer to produce contrafactive and factive attitude ascriptions. Because both types of
ascription entail their non-factive counterpart, the model could maximize its chances of produc-
ing true ascriptions by producing non-factive attitude ascriptions only. Our informativity demand
forbids this behavior, since contrafactive and factive attitude ascriptions are more informative than
their non-factive counterparts. Given this key role of pragmatic principles in our new inputs, we
label the inputs we use ‘semantic-pragmatic conditions.’

Our new paradigm addresses both key limitations at once. This is more economical than
addressing them separately. But it also raises a concern. We build in presupposition failure by
sometimes asking our model to produce presupposition failures. But human language learners are
hardly ever, if at all, asked to produce presupposition failures. So, our new paradigm is not as
naturalistic as we would want. Although we will weight our data to partially address this concern,
we feel its force. To address it, we plan to do follow-up experiments that test for presupposition
failure without asking for the production of presupposition failures.

3. Current Experiment. Let’s turn to the details of our latest experiment.15

3.1. ARCHITECTURE. Our Transformer closely follows Vaswani et al. (2017)’s models using the
pyTorch implementation. Thus, our model consists of a Transformer encoder and decoder. We pre-
viously used an encoder-only approach. But this is only appropriate for testing comprehension.16

To initialise the weights, we use the Xavier uniform distribution. We 0-initialise biases and
use sinusoidal position embeddings. But we use position-specific linear layers to constrain the
output for each position in an output sequence to a proper subset of the words our model learns.17

In effect, our model learns an artificial language with a fixed word order and can only produce
sentences with that word order. This means we can rule out the word order confound in Experiment
1, whilst letting our model learn a language that is more naturalistic (due to its larger vocabulary)
than in Experiment 2. Minimizing the role of syntax also allows our model to learn our artificial
language more easily, rendering the training more economical.18

14Our Transformer also learns how to satisfy Grice’s maxim of quality because it learns to produce a true ascription
when we ask for one. Of course, it also learns to produce falsehoods or presupposition failures when we ask for them.
But this does not mean that it violates the maxim of quality because our demand that it produce something other than
a true ascription suspends that maxim.

15The code for running our experiments as well as the output of our experiments can be found at https://
github.com/dstrohmaier/productive_contrafactives.

16The third and as yet unexplored option is a decoder-only approach, as popularised by the GPT-family of mod-
els (Radford et al. 2018).

17One may worry that we are not testing production, as this constraint makes our task akin to a multiple classification
task. However, even without that constraint, our task would be akin to a multiple classification task. The difference
would merely be that, in classification, at a given position all words, and not just a proper subset of them, would have
to be considered. More generally, there is no strict division between neural language modelling and classification.

18One may worry about minimizing the role of syntax. For cross-linguistically the meaning of attitude ascriptions

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3.2. INPUT AND OUTPUT DATA. We train our Transformer on a sequence-to-sequence task. It
takes sequences of tokens and outputs sequences of tokens. The input sequences provide our
semantic-pragmatic conditions: more specifically, what we call the ‘main value’, ‘sub value’,
‘mind-world relation’, and ‘attitude content’. The output sequences can be glossed as attitude
ascriptions in an artificial language, consisting of an attitude verb and an embedded clause. The
function from input to output can be written as:19

main value × sub value × mind-world relation × attitude content → attitude verb × embedded clause

The first two inputs tell the model what we want it to produce. The main value is the value
we want for the attitude ascription the model produces: True, False, or P-Failure. The sub value is
the value we want for the embedded clause the model uses in its attitude ascription: True, False,
or Unknown.20 The second two inputs give the model the information it needs to decide how to
produce what we want it to produce. The mind-world relation has three possible values: mind and
world match (=), are incompatible (!=), or the world state is unknown (?). Finally, the attitude
content exhaustively tells the model what the matrix subject believes.

Table 5 in the appendix lists all possible combinations of semantic-pragmatic conditions
(in particular, mind-world relation, main value, and sub value) alongside their required attitude
verbs.21 But let’s walk through three examples. Say we have some attitude content, mind-world
relation =, main value true, and sub value true. Here (row 1), our model is required to produce a
factive. Or, say we have some attitude content, mind-world relation =!, main value true, and sub
value false. Now (row 11), our model is required to produce a contrafactive. Or, say we have some
attitude content with mind-world relation ?, main value true, and sub value unknown. In this case
(row 21), our model is required to produce a non-factive.

We let the sub value vary independently of main value and mind-world relation to permit the
model to produce a wider range of attitude ascriptions. As mentioned earlier, a non-factive attitude
ascription is less informative than its contrafactive and factive counterparts. So our model would

often depends on syntactic properties. For instance, Ozyildiz (2017) shows that whether Turkish bil- ‘know’ has a
truth presupposition depends on whether it embeds a nominalized clause or a tensed clause headed by diye. Similar
‘factivity alternations’ are attested across many languages (Bondarenko 2019, Lee 2019, Grano & Park 2022). Ex-
amples of dependence on syntactic properties can also be found in English (Karttunen 1971). Whether forget has a
truth presupposition depends on whether it embeds a finite that-clause or a to-infinitive. A similar ‘factive-implicative’
alternation is also attested for remember.

These examples are not problematic for our architecture because the artificial language our model learns is highly
constrained. Any language with factive attitude ascriptions has a type of embedded clause that gives us a truth presup-
position. And one way to understand our artificial language is as containing just that type of embedded clause. This
means that insofar as our Transformers do not learn the factivity alternation, or the factive-implicative alternation, they
do not learn the full meaning of predicates like bil- and forget. But we do not intend our Transformers to do so. We
focus on two dimensions along which contrafactives and factives are at least sometimes the same (belief entailment)
and different (truth/falsity presupposition), and set aside potential alternations in which they participate.

19We use ‘function’ loosely here, as some inputs permit more than one kind of output. See below.
20Although some tokens the model uses in the embedded clause can be glossed in English as presupposition triggers,

e.g. ‘rory’, we set aside the possibility that the embedded clause is a presupposition failure. In effect, our Transformer
learns a language whose only presupposition triggers are contrafactives and factives.

21Our data does not contain any of the rows marked as impossible.

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not be permitted to produce a non-factive if we just asked for the main value false and gave it some
mind-world relation. But by also asking for the sub value unknown, we can require the model to
produce a false non-factive attitude ascription, even given our informativity demand.

An input sequence generally requires exactly one output sequence. Sometimes, though, the
model has more than one option. Table 5, row 27 gives an example: if we have some attitude
content, mind-world relation ?, main value presupposition failure, and sub value unknown, both a
contrafactive and a factive are permitted.

We use slightly different languages for the input attitude content and the output embedded
clause. Table 1 lists the vocabulary used for the attitude content. This vocabulary gets us contents
like ‘eat rory tomato basil soup lunch tomorrow’ or ‘buy ahab carrot oregano pie dinner yesterday’.
The function from input attitude content to output embedded clause can then be written as:

verb × agent × ingredient × spice × dish × meal × day → × agent × verb (with tense) × main
ingredient + spice × preposition × dish × meal × day

To make the task more naturalistic our mapping from input to output language is slightly indirect.
The ‘+’ indicates that the ingredient and spice combine to a single output token. And the output
verbs’ tense must be inferred from the input day indexical, e.g. the ‘tomorrow’ token.

Category Lexical Items

Verb eat, cook, order, buy
Subject rory, lorelai, lane, paris, timon, ahab
Ingredient tomato, pumpkin, mushroom, carrot, potato
Spice basil, oregano, pepper, chili, coconut
Dish soup, pie, rice, stew, curry
Meal lunch, dinner, breakfast
Day day-before-yesterday, yesterday, now, today, tomorrow, day-after-tomorrow

Table 1: Attitude content vocabulary. Content has one token of each category.

An attitude content and embedded clause match iff applying the attitude content to the func-
tion from attitude content to embedded clause results in the embedded clause. For instance, ‘eat
rory tomato basil soup lunch tomorrow’ matches ‘rory will-eat tomato-basil soup for lunch tomor-
row’, but not ‘ahab bought carrot-oregano pie for dinner yesterday’. Some semantic-pragmatic
conditions require attitude content and embedded clause to match, some require them not to. To
illustrate, the only way to produce a merely false non-factive attitude ascription is to use a non-
matching embedded clause (see Table 5, rows 6, 15, and 24). This is because the input attitude
content exhaustively describes what the matrix subject believes. So, any non-factive attitude as-
cription with a matching embedded clause is true, and any with a non-matching one false.

3.3. DATA GENERATION. We generated input and output sequences automatically. Because not
all types of sequence would occur equally frequently if we generated all possible combinations in
the same number (factives, in particular, would be over-represented amongst the output sequences),
we subsampled to get a more balanced dataset.

We also weighted the main semantic value, while ensuring that the sub value remained bal-
anced. As noted earlier, human audiences rarely ask one to produce a sentence that is merely false

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or a presupposition failure. To partially address this, whilst still allowing our model to learn how
to produce such outputs, the number of instances where the main value is true (and we thereby ask
for a true ascription) equals the sum of the number of instances where the main value is false and
the number of instances where the main value is presupposition failure.

The balanced data set contained 194,400 instances. For each attitude verb, 70,200 instances
permit that verb. 16,200 instances permit both factives and contrafactives.22 113,400 instances
require matching embedded clauses; 81,000 non-matching ones. We randomly split the balanced
data into 90% training and 10% test data.

3.4. TRAINING. We split the training phase into two parts. The first was a hyperparameter search
to identify suitable settings for hyperparameters, such as learning rate, size of training batches etc.
The second was the training of the selected settings on all training data.

We used the AdamW algorithm (Loshchilov & Hutter 2018) with standard settings for the
pyTorch library (except for the learning rate, which is explored as a hyperparameter) and an adapted
version of pyTorch’s binary cross entropy (BCE) loss. Given semantic-pragmatic conditions that
require a matching embedded clause, we set all output tokens (attitude verbs plus embedded clause
constituents) required by those conditions to 1, the rest to 0. For conditions that permit any non-
matching clause, but not the matching one, we instead set the tokens that occur in the impermissible
clause to 0.5. This reduces the likelihood that the model produces all these tokens together, while
still allowing the model to produce individual ones. We then calculated the loss as the divergence
from these values using the standard pyTorch formula for BCE. This resembles standard language
modelling practice (e.g. the use of cross entropy loss in Kaplan et al. 2020), but permits more than
one output sequence per combination of semantic-pragmatic conditions.

The hyperparameter search explored 41 different settings using a randomised search and 5-
fold cross-validation. Our model learned how to produce our target expressions in 2 of these
settings.23 Table 2 details the search space and successful settings, which we call settings 1 and 2.

Name Space Setting 1 Setting 2

Dim. Embedding {80, 100, 120, 140, 160, 180, 200} 180 200
Dim. Hidden {160, 180, 200, 220, 240, 260, 280, 300, 320, 340} 240 260
# Attention Heads {2, 5, 10, 20} 10 20
# Encoder Layers {5, 10, 15, 20, 25} 5 5
# Decoder Layers {5, 10, 15, 20, 25} 15 20
Epochs {3, 5, 7} 3 5
Batch Size {120, 240, 360, 480, 600} 120 240
Learning Rate {1 ·10−3, 5 ·10−4, 1 ·10−4, 5 ·10−5, 1 ·10−5, 5 ·10−6, 1 ·10−6} 1 · 10−4 1 · 10−4

Dropout {0.1, 0.2, 0.3} 0.1 0.3

Table 2: Hyperparameter space and two successful settings.

22One might worry that this gives a slight advantage to learning non-factives, as they never have to compete. We
find, however, that our Transformer learns non-factives most slowly. See below.

23The high failure rate in the search may raise concerns about the robustness of our results. However, neural
networks regularly fail to learn throughout much of the hyperparameter space. Our Transformer is also relatively
small, which leads us to expect that, unlike larger models, it only learns given relatively specific hyperparameters.

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3.5. EVALUATION. To track the learning process, we evaluated settings 1 and 2 on the complete
test data every 20 training batches.24 This allowed us to compare how well our Transformer had
learned our target expressions at different stages of the training process. We also varied the original
random seed for each setting four times to get 10 evaluations overall. By varying the random seed,
we checked whether our results are robust to a random change in our network’s initial conditions.
Our evaluation metric was the correctness of the output sequence given the semantic-pragmatic
conditions. If a condition permits contrafactive and factives, the output is correct with either verb.

3.6. RESULTS AND DISCUSSION. Figure 1 and Table 3 show that we find no consistent difference
between contrafactives and factives. While there are some differences, these reverse over the
course of training. For instance, at setting 1 batch #200, the model performs better on factives
(factives: 40.9%, contrafactives: 36.2%), but by batch #600 the order has reversed.

Figure 1: Performance over the course of training.

batch factive contrafactive non-factive

0 39.0 (±14.5) 26.5 (±17.4) 0.6 (±1.4)
200 40.9 (±5.9) 36.2 (±6.3) 12.8 (±8.4)
400 44.4 (±8.7) 41.8 (±9.0) 29.2 (±5.0)
600 65.5 (±3.1) 71.9 (±2.9) 56.1 (±4.9)
800 90.7 (±0.6) 94.2 (±1.8) 94.1 (±2.9)
1000 98.6 (±1.1) 99.5 (±0.3) 99.9 (±0.1)
1200 99.9 (±0.1) 99.9 (±0.1) 100.0 (±0.0)
1400 100.0 (±0.0) 100.0 (±0.0) 100.0 (±0.0)

(a) Setting 1.

factive contrafactive non-factive

0 25.1 (±18.3) 29.8 (±21.9) 4.8 (±9.2)
400 48.5 (±5.1) 25.5 (±6.3) 0.5 (±0.6)
800 37.0 (±6.4) 41.0 (±8.8) 17.4 (±7.5)
1200 55.4 (±2.2) 52.7 (±6.6) 27.1 (±2.7)
1600 64.8 (±7.4) 65.1 (±6.7) 40.8 (±12.3)
2000 83.8 (±13.3) 83.9 (±13.8) 74.2 (±23.8)
2400 98.8 (±2.2) 98.9 (±2.1) 98.8 (±2.3)
2800 100.0 (±0.0) 100.0 (±0.0) 100.0 (±0.0)

(b) Setting 2.

Table 3: Percentage of correct output sequences given permitted attitude verb. Numbers in paren-
theses give standard deviation across random seeds.

24Settings 1 and 2 differ in batch size (see Table 2). Nonetheless, comparison is relatively straightforward, as the
batch size of setting 2 is exactly twice that of setting 1.

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(a) Matching. (b) Non-matching.

Figure 2: Performance over the course of training split by whether model had to produce matching
or non-matching embedded clauses. Shaded area gives standard deviation between random seeds.

NON-FACTIVES. In early training, our model performs worst on non-factives. For example, at
setting 2 batch #1,200, the model only treats correctly 17.4% of instances that require a non-
factive, but other attitude verbs perform above 50%. This parallels results from Experiment 1. But
the reason why we get this result is specific to our current experiment.

Some semantic-pragmatic conditions that require non-factives require non-matching embed-
ded clauses. These are the conditions where non-factives perform worst. Figure 2 shows that with
matching embedded clauses non-factives perform no worse than contrafactives and factives. For
setting 1 non-factives even perform better.

To see why non-factives underperform with non-matching embedded clauses, consider non-
matching embedded clauses more generally. Figure 2 shows that semantic-pragmatic conditions
that require non-matching embedded clauses yield higher variability in early training and account
for much of the variability in early training in Figure 1. Given the model’s freedom to choose which
embedded clause to produce if a non-matching one is required, this may seem surprising. But, in
fact, the dynamics are what we expect from a Transformer. Because only one embedded clause is
matching, but 129,599 are non-matching, the model initially fares better with non-matching em-
bedded clauses. But, as the model learns to produce matching clauses, it overadapts and produces
them also in inappropriate cases. This leads to a correction in the other direction, giving rise to
variability. In effect, the model is pushed every which way by the instances it learns from.

The model’s overadaptation to matching clauses produces an even worse outcome for non-
factives than for contrafactives and factives. Only one out of the four semantic-pragmatic condi-
tions that require non-factives requires a matching embedded clause: Table 5, row 21. But because
of how we balanced the data (weighting main value true and balancing the verbs) this condition
occurs much more frequently than conditions that require a non-factive with a non-matching em-
bedded clause: 54,000 instances require a non-factive with a matching embedded clause, 16,200 a
non-factive with a non-matching one.We do not get this imbalance with contrafactives and factives,
for which 32,400 instances require matching embedded clauses and 37,800 require non-matching
ones. This difference between non-factives on one side and contrafactives and factives on the other
means that the model sees more conditions that require non-factives with matching embedded
clauses than conditions that require contrafactives or factives with matching embedded clauses.

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And this leads the model to overadapt to matching clauses more strongly for non-factives. Hence,
in our setup, non-factives underperform with non-matching embedded clauses.

In effect, non-factives underperform because of the proportion of matching to non-matching
embedded clauses that our model has to produce with non-factives. This proportion in turn results
from our efforts to weight and balance semantic-pragmatic conditions and attitude verbs. We
would, therefore, not be surprised if non-factives performed better given appropriate changes to
how we weight and balance our inputs and outputs.

SELECTION PREFERENCES. Some semantic-pragmatic conditions permit both contrafactives and
factives. Does our Transformer prefer one verb over the other in these conditions? If so, this could
indicate that the preferred verb is easier to learn.

To answer this question, we look at selection preferences after the model has stabilised (batch
>3,000). For every 20 batches after this threshold and every random seed, we consider the propor-
tion of contrafactives and factives produced in semantic-pragmatic conditions that permit both. We
focus on what happens after the model has stabilised because we expect that if one verb is easier
to learn, the model would settle on producing that verb as a local optimum. And that would mean
that it would consistently prefer that verb over the other once it stabilizes. Table 4 and Figure 3
show that the selection preference depends on the hyperparameters, not the attitude verb. Our
Transformer does not prefer one verb over the other.

setting verb count mean std min 25% 50% 75% max

1 contrafactive 345 0.76 0.13 0.28 0.68 0.77 0.85 0.99
factive 345 0.24 0.13 0.01 0.15 0.23 0.32 0.72

2 contrafactive 165 0.37 0.15 0.05 0.25 0.37 0.48 0.76
factive 165 0.63 0.15 0.24 0.52 0.63 0.75 0.95

Table 4: Statistics for selection (with batch >3,000). Count (number of evaluations) differs be-
tween settings as batch sizes differ. Numbers given are for the mean selected proportion, the
standard deviation, and the quantiles for each setting and verb.

Figure 3: Selection preference when both contrafactives and factives are permitted.

4. Conclusion. Natural languages appear to universally have factives, but lack contrafactives. We
previously hypothesised that this asymmetry arises partly because the meaning of a contrafactive is
harder to learn than that of a factive. Our previous comprehension-oriented experiments supported
this hypothesis, but the results from the production-oriented experiment reported here do not.

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To fully interpret our results, more work is needed. Given our results, the meaning of con-
trafactives may be harder to learn than that of factives in comprehension-oriented settings only.
This would raise the question whether a difference in learnability in one setting is enough to ex-
plain any of the difference in frequency between contrafactives and factives. Alternatively, it may
be that our latest experiment has limitations that undermine the validity of its results.

In fact, we highlighted one potential limitation earlier on: We built in presupposition failure
by sometimes asking our model to produce presupposition failures. But human language learners
are hardly ever, if at all, asked to produce presupposition failures. So, although we did ask for
twice as many true ascriptions as presupposition failures, our latest experiment may not allow us
to draw conclusions about human language learners.25

Another potential limitation, compared to Experiment 2 in particular, is that our current set-up
does not capture how human infants seem to learn attitude verb meanings. On the pragmatic syn-
tactic bootstrapping model (Hacquard & Lidz 2022), they infer the meaning of non-factive think,
for instance, partly from the parallel between the use of non-factive attitude ascriptions like Ayesha
thinks that Rebecca swims as indirect assertions and the primary use of their declarative comple-
ments as direct assertions. Experiment 2 captured this developmental priority of complements over
attitude verbs. We pre-trained our model to learn the meanings of embedded clauses; only once it
learnt those meanings, did we train it on our attitude verbs. Our current experiment, however, does
not capture this developmental priority. This may be (part of) why our current results differ from
those of Experiment 2, and may mean that our current results do not give us as much insight into
human language learners as we would want.

In sum, given the results and (potential) limitations of our three experiments to date, more
work remains to be done to assess whether a difference in learnability can contribute to an expla-
nation of the difference in frequency between contrafactives and factives.

25One may also worry that our current experiment treats the difference between contrafactives and factives dif-
ferently than our previous experiments. Factive attitude ascriptions are true only if mind and world representations
correspond, contrafactive attitude ascriptions only if they do not. Our previous experiments treated these conditions
asymmetrically. While there was only one world representation that corresponded to the mind representation, there
were many world representations that failed to correspond. This asymmetry, the worry goes, may have made it harder
to learn the meaning of contrafactives than that of factives. Yet our current experiment treats the conditions sym-
metrically. Both correspond to exactly one mind-world relation: = and !=. So, by making symmetrical what was
asymmetrical, we may have removed a significant explanatory factor. And insofar as this factor is naturalistically
motivated, our current experiment may run up against another limitation.

In reply, note that we also levelled out an asymmetry in the other direction. Factive attitude ascriptions are not true
if mind and world representations do not correspond, whilst contrafactive attitude ascriptions are not true if mind and
world representations do correspond. Our previous experiments treated these conditions asymmetrically: only one
world representation corresponded to the mind representation, but many world representations failed to correspond.
By contrast, in our current experiment, both conditions correspond to exactly one mind-world relation: != and =.

Now, if, in our previous experiments, the first asymmetry made it harder to learn the meaning of a contrafactive
than that of a factive, the second asymmetry may have made it harder to learn the meaning of a factive than that
of a contrafactive. These asymmetries would then have balanced out, given that true and not true ascriptions were
balanced in training and test data. In effect, our current experiment treats contrafactives and factives symmetrically
from the start, whilst our previous experiments treat them asymmetrical twice and in opposite, and therefore balancing,
directions. Ultimately, we expect no significant difference between these treatments.

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Appendix

Mind-World Main Value Sub Value Attitude Verb Embedded Clause

1 = True True Factive Matching
2 = True False IMPOSSIBLE —
3 = True Unknown IMPOSSIBLE —
4 = False True Factive Non-Matching
5 = False False Contrafactive Non-Matching
6 = False Unknown Non-Factive Non-Matching
7 = P-Failure True Contrafactive Matching
8 = P-Failure False Factive Non-Matching
9 = P-Failure Unknown Factive or Contrafactive Non-Matching

10 != True True IMPOSSIBLE —
11 != True False Contrafactive Matching
12 != True Unknown IMPOSSIBLE —
13 != False True Factive Non-Matching
14 != False False Contrafactive Non-Matching
15 != False Unknown Non-Factive Non-Matching
16 != P-Failure True Contrafactive Non-Matching
17 != P-Failure False Factive Matching
18 != P-Failure Unknown Factive or Contrafactive Non-Matching

19 ? True True IMPOSSIBLE —
20 ? True False IMPOSSIBLE —
21 ? True Unknown Non-Factive Matching
22 ? False True Factive Non-Matching
23 ? False False Contrafactive Non-Matching
24 ? False Unknown Non-Factive Non-Matching
25 ? P-Failure True Contrafactive Non-Matching
26 ? P-Failure False Factive Non-Matching
27 ? P-Failure Unknown Factive or Contrafactive Matching

Table 5: Possible combinations of semantic-pragmatic conditions and output tokens.

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