3.1. Surprisal effects and explaining time-series data

As mentioned, the IPLP is quite sensitive to surprisal effects, and as a result, special care must be exercised in its design. In the familiarisation phase of our experiment, bicoloured balls were paired with conjunctive sentences (the same bicoloured ball appeared on each side of the screen), and unicoloured balls were paired with disjunctive sentences (for a sentence such as look, the ball is blue or yellow, a blue ball appeared on one side of the screen, and a yellow one on the other); see Figure 2 for examples of the graphics used in the experiment. In the experimental phase, children would see a bicoloured ball on one side of the screen and a unicoloured ball on the other side in each trial, and thus, the surprisal effect would be expected to be triggered by the unicoloured ball in the conjunction condition and by the bicoloured ball in the disjunction condition, as these were not the pairings children had observed during the familiarisation phase. This is not quite what we obtained in our results.

One reason for this is the fact that our task was not a purely perceptual task, where surprisal effects are strongest, but a language comprehension task with a perceptual component. This is par for the course when using the IPLP with young children, where the general expectation is that language comprehension processes ought to mitigate the strength of the surprisal effect, and this is what was observed in our data. Before we discuss this point, though, it is important to note that children did not manifest a preference for a particular image to begin with. In each experimental trial, we presented the images side by side for 1000 ms before the sentence started – as stressed, a bicoloured ball on one side of the screen and a unicoloured one on the other, varying the position – and the eye-tracking record shows that toddlers did not direct their fixations to any particular image in the absence of linguistic input during this time window – no effects were identify by our GAMMs in the first 1000 ms. We had included this 1000-ms segment as a baseline in order to control for any possible biases, and none arose.

The situation is as follows, then. In the conjunction condition, children would be expected to be drawn to the novel image (the unicoloured ball) as the result of a surprisal effect, an image they would then reject as an interpretation for conjunction, eventually fixating on the bicoloured ball instead. Toddlers may indeed have been drawn to the competitor image, as indicated by the switches shown in Figure 5 within the 1000–3000 time window, but fixations to the novel image were not consistent, plausibly as a result of comprehension processes inhibiting the surprisal effect – that is, children might have considered the unicoloured ball, but did not accept it as a reading of conjunction. Nevertheless, children did not then fixate on the target image (the bicoloured ball) especially for the remainder of the trial, despite the apparent increase in fixations to the bicoloured balls from the 3000-ms mark onwards, when the conjunction and the two colour terms were being presented; in this case, there were no significant differences between the target and competitor in the conjunction condition in any time window, according to our models.Footnote ⁷

In the disjunction condition, the bicoloured ball constitutes the novel image, and it did draw children’s attention to it; but, crucially in this condition, this image is compatible with the meaning of inclusive disjunction – if children of course regard the true–true reading represented by bicoloured balls as a possible reading of disjunctive sentences in language. And as it transpired, children stayed focused on this image for the second half of the trial onwards, thereby demonstrating that they accept language’s disjunction as an inclusive kind of disjunction. In this case too, then, the surprisal effect is operative, establishing which image children first fixate on at around 1000–2000 ms, as shown in the bottom panel of Figure 5, a time window in which the sentence has just started. And in fact, and this is key, fixations to the target (the bicoloured ball) greatly increased in the 3000–4400 period (or 2000–3200 ms into the sentences), the time window in which the disjunction and the colour terms appeared; and given, moreover, that children accepted this reading, they did not deviate from it. Indeed, and this is telling, children do not switch from the true–true interpretation for the remainder of the trial, as they do not seem to have favoured exclusive readings. This datum does not necessarily mean that children are (mistakenly) treating disjunction as if it were conjunction, as in the data from Singh et al. (Reference Singh, Wexler, Astle-Rahim, Kamawar and Fox2016), given the differing behaviour between the two conditions in our study; more likely, though we do not want to state this conclusively, children this young cannot compute an exclusive reading from inclusive disjunction (the pragmatic enrichment process mentioned earlier on), and in any case, the semantic and intonational cues of our materials did not signal an exclusivity reading (we’ll come back to this).Footnote ⁸

The overall argument may be a bit tortuous, so let us make it as explicit as possible. We are postulating that a surprisal effect is at play in the data and that this effect applies differently in each condition, as toddlers fixate on the bicoloured ball in the disjunction condition on account of the surprisal effect, whilst they do not seem to go to unicoloured balls in the conjunction condition in quite the same way. We thus propose the following account: there is a surprisal effect in the conjunction condition, but the novel image is not compatible with the meaning of conjunction, so children quickly reject this interpretation; the novelty is not long-lasting, and thus, the surprisal effect is inhibited. In the disjunction condition, on the other hand, the novel image is compatible with the meaning of (inclusive) disjunction, and thus, toddlers do not look away once the novelty has decreased, as they accept the image as a match for disjunction. In sum, we are arguing that the toddlers in our experiment did indeed process the connectives appropriately and their comprehension of conjunction and disjunction is partly responsible for the patterns we obtained, though there was certainly a trade-off between the mental resources involved in dealing with the surprisal effect and those that language comprehension exercises, as explained.

According to an alternative account, suggested by a reviewer, children might not have processed the connective in the disjunction condition and instead fixated on the most salient elements of the auditory and visual information, disregarding the logical relations that disjunction signals. If so, children would have simply tracked whether the balls and colours mentioned in the sentences matched the graphics, thus fixating on bicoloured balls and abstracting away from the disjunction mediating the two colour predicates in this case. This kind of worry is a common one, in fact, going back at least to Paris (Reference Paris1973), who reported a number of experiments involving the logical connectives with 7-year-olds and older participants, and where this kind of explanation was also considered when analysing the results with disjunctive sentences. We are rather sceptical of the general import of the alternative account to begin with, though, and for well-known reasons. As Fodor (Reference Fodor1983) argued some time ago now, and has been reargued once again more recently by Ferreira and Nye (Reference Ferreira, Nye, De Almeida and Gleitman2018), language comprehension is a compulsory mental process – given a linguistic input, the language processing system cannot but be activated – and this is magnified in well-controlled experimental settings, where conditions do not allow for abstracting away from the linguistic material that is presented to participants – the contrary view would have to be properly substantiated, and it is unclear how this would proceed.

This is not to say that there might be experimental conditions in which children might become confused and perhaps not attend as well as they are expected to; we mentioned earlier that the experiments in Singh et al. (Reference Singh, Wexler, Astle-Rahim, Kamawar and Fox2016) may have introduced significant pragmatic uncertainty for the children, but this was not the case in our own experiment in any way – pragmatic considerations were minimised, the IPLP is designed to constantly draw children’s attention to the screen and the audio, and moreover, we had applied a slight stressed intonation to the connectives, further drawing children’s attention to (this part of) the audio. Nevertheless, a reviewer worries that in the disjunction task, it may not be entirely clear how to interpret the sentences given the visual graphics and the overall setting, even for adults. We disagree as speakers of Italian, but in order to rule out this possibility we collected online survey data with the PsyToolkit software (Stoet, Reference Stoet2010, Reference Stoet2017) from adult Italian speakers on a minimally modified version of each task (adults were asked to select the graphic that matched the sentences instead, and no eye tracking was involved). We collected data from 32 respondents in the conjunction task, and 37 in the disjunction task. We expected 100% accuracy in the conjunction condition, and this was indeed the case: adults selected bicoloured balls in this condition for every single item (as in the original tasks, there were four experimental items). In the disjunction condition, we expected adult speakers to favour an exclusive reading and thus choose unicoloured balls, as per the expectation, discussed earlier, that adult speakers tend to compute an exclusivity implicature for disjunctive sentences. In this case, there was a mean accuracy of 79% (SD 0.37), and this shows that there was a clear favoured interpretation in this condition. It is important to recall that disjunctive sentences are ambiguous between inclusive and exclusive readings, and 79% is a pretty high response value in such circumstances; in fact, six participants had an accuracy of 0%, which means they selected the inclusive reading each time, one had an accuracy of 50% (two out of two items answered correctly), four had an accuracy of 75% (three out of four answered correctly), and the remainder 26 out of 37 participants had an accuracy of 100%. In sum, respondents were not confused at all by the design, graphics, or audio (or a combination thereof). A Welch two sample t-test shows that there is a significant difference in mean accuracy between conditions (t = 3.3152, df = 36, p-value = 0.002098), with a large size effect (Cohen’s d = 0.8), whilst a Frith logistic regression (conducted in order to handle perfect separation correctly) confirmed that the factor condition (with two levels: conjunction and disjunction) was the main predictor and age (mean age 30.4, SD 9.0) and sex (18 male, 52 female), the other two predictors we collected in the survey, had no detectable effect.Footnote ⁹

More substantially, perhaps, there are two further reasons to favour our own account to the alternative. Firstly, the alternative take does not strike us as a fully principled explanation, for it does not provide a unified account for the data from the two conditions; instead, the alternative account applies to the disjunction condition only, with the data from the conjunction condition to be explained in rather different terms, and this is hard to justify – why would children not attend to the disjunction but do so for a conjunction, for instance? What an experiment typically requires is an integrated explanation covering the data from all experimental conditions, and in this case one that centres on the relevant factors at play: the surprisal effect, the processing of the linguistic material, and the interaction between these two factors and the experimental design – in the case of the latter, the progression from the familiarisation to the testing phase is a particularly important issue, as we have discussed.Footnote ¹⁰

Secondly, GAMMs provide a nuanced account of time-series data, allowing us to identify the time windows in which the postulated effects are significant (though not, as stressed, the actual point when an effect starts). As Figure 4 shows, this plot representing fixations modelled as empirical logits to the target and competitor, per condition, fixations to bicoloured balls (the target in each condition) increased for both conditions at around the 3000-ms mark, and thus, 2000 ms into the sentences, with the greatest amplitudes noticeable after the 4000-ms mark, where the sentences have been fully, or mostly fully, presented (conjunctive sentences ended at 4000 ms, and disjunctive ones at 4200 ms). Here, the difference between the target and competitor proved to be significant only in the disjunction condition, as shown in Figure 5, but the “difference curves” from these plots confirm that the greatest difference between the target and competitor is identifiable after 4000 ms.

In terms of our explanation, then, the increase in looks to bicoloured balls in the disjunction condition, starting at around 3000 ms, would have been partly due to the surprisal effect, as disjunctive sentences were paired with unicolouring balls only in the familiarisation phase, but the continuation of these fixations, along with the increasing difference between target and competitor, peaking after the 4000-ms mark into the trials, precisely around when the sentence has finished, as marked in Figure 5 (thus, 3200 ms into disjunctive sentence, which had a length of 3200 ms), would have eventually been the result of processing and comprehending the sentences – indeed, of accepting an inclusive reading for disjunction. As anticipated early on in this paper, the full interpretation of compound sentences is only possible once the sentences have been fully presented and processed; the mere presence of the connective, for instance, cannot establish the meaning of the sentence in the absence of the second (colour) predicate. In other words, and to restate our position, given that all children were exposed to both conjunctive and disjunctive sentences in the familiarisation phase, with conjunction matched with bicoloured balls and disjunction with unicoloured balls in order to create a contrast between the two conditions, the more parsimonious result would have been for toddlers in the conjunction group to have fixated on bicoloured balls once the sentences had been fully presented and for children in the disjunction group to have fixated on unicoloured balls once they had been exposed to the full sentences, but this is not what transpired in either case.

Putting it all together, then, the results show, first of all, discrimination between conjunction and disjunction by 28-month-old toddlers, thus showcasing the very early comprehension of both inclusive disjunction and conjunction (though, indirectly so, in the case of the latter). And secondly, the surprisal effect typical of the IPLP is modulated and effectively inhibited by what interpretations are entertainable by children for conjunctive and disjunctive sentences, with this going in one direction in the case of conjunction, but in another direction for disjunction, as explained.

There is no indication that children consider the exclusive readings of disjunction in this experiment, though this per se does not mean that children at this age do not have access to exclusive interpretations. As argued earlier, however, this age group is not expected to be able to derive exclusivity from (inclusive) disjunction, as per the consensus in the literature that such derivations require pragmatic processes that are not be available to toddlers. Nevertheless, the possibility that children may have access to exclusive disjunction independently of inclusive disjunction cannot be discounted, perhaps in the form of a connective of an underlying conceptual representational system (recall that in formal logic inclusive and exclusive disjunction are two separate connectives). This brings us back full circle to the issue of the semantics of the connectives as evidenced by our experimental data, to which we now turn. We shall also offer some pointers on the primitives of the logical vocabulary in both language and thought, and that will bring the paper to an end.

3.2. Semantics and the logical vocabulary in thought

A running thread of the whole discussion has been the close connection between formal logic and corresponding properties of language. Natural languages, however, are not formal languages stricto sensu, and some features pertain to natural language only. One such feature is the fact that the clauses of compound sentences need not always be spelled out in full but can appear in reduced form in language – that is, John ate an apple and a banana instead of John ate an apple and John ate a banana. This is not the case in logic, where everything is spelled out in a formal language. The worry in such cases is that the connectives might not compute the meaning contributed by each clause equally or even return illicit interpretations (recall that logic’s connectives are boolean operators that take the true or false values of each clause and return a true or false value for the entire sentence). This can be the case with reduced clauses in the subject position; a common example in the literature usually involves a sentence such as John and Mary are married, which may be regarded as a reduced version of John is married to Mary and Mary is married to John, the latter one of the possible meanings of the sentence, in fact (John and Mary may well not be married to each other!). In such cases, these sentences would not be tracking logical relationships very closely. This does not seem to apply to the object position as much; John ate an apple and a banana does not introduce similar issues, as there is no apparent ambiguity as to who did the eating, and likewise for John ate an apple or a banana, though in the latter case there is a hint of a proposition that expresses a doubt or a possibility, more a case of (as far as I know) John ate an apple or a banana than an affirmation (this is part of the story of how disjunction is usually processed and interpreted, as we shall see). The question here is whether our materials suffer the same sort of shortcomings.

Consider Winter (Reference Winter2001), who discusses the kind of materials we employed here. Thus, after noting that sentences such as the flag is green and white do not appear to be boolean in nature – the sentence does not imply that the flag is green and that the flag is white – Winter argues that this is an epiphenomenon of independently known facts to do with the special semantics of colour predicates, as they can also function as nominals (p. 70). That is, colour terms can be used as both adjectives, as in our materials, and nouns (e.g. in blue is my favourite colour), and this gives rise to a contrast between adjectives that have nominal uses and adjectives that do not, which in turn results in a contrast between whether colour predicates behave in boolean or non-boolean ways. In particular, adjectives that do not have a nominal use behave in a straightforward, and expected, boolean fashion – e.g. the house is comfortable and small does imply that the house is comfortable and that the house is small – and such facts, as Winter puts it, “speak against any general non-boolean analysis of and in singular predicate conjunctions” (p. 71). We defer to Winter (Reference Winter2001) for the actual details; here we simply note that conjunctive sentences involving colour predicates are boolean in nature and no troubling ambiguity seems to arise. The case of disjunction is slightly different, however.

Jasbi et al. (Reference Jasbi, Jaggi, Clark and Frank2024), discussed earlier, showed that a significant amount of adults’ use of disjunctive sentences, especially when directed at children, are questions, and naturally enough these typically pose an option between two or more possibilities, all within the realm of what is possible. This is particularly relevant here because the disjunctive sentences in our study were of the kind look, where is the ball that is blue or yellow? As Jasbi et al. demonstrated, exclusive interpretations of disjunction usually come with intonational and semantic cues – a rise–fall prosody and the presence of two incompatible clauses (i.e. both cannot be true) – and without these, the interpretation is inclusive instead. Our materials were of the relevant kind to elicit an inclusive interpretation, which children appear to have accepted, as the intonation was not rise–fall and the clauses were compatible given the graphics on display (if anything, the disjunction connective was slightly stressed, as discussed earlier, and this ought to have signalled an exclusive interpretation, but in this case did not). In terms of the meaning of disjunction, our data accord well with a non-boolean account of disjunction, mentioned in Introduction: namely, that of Zimmerman (Reference Zimmerman2000), according to which linguistic disjunctions are conjunctive lists of epistemic possibilities (though we leave open the possibility that a boolean meaning of disjunction may also be part of natural language; perhaps language’s or is ambiguous between a boolean and non-boolean operator).

Based on an analysis of free choice disjunctive sentences such as Mr X may take a bus or a taxi, which imply both that Mr X may take a bus and that Mr X may take a taxi, Zimmerman proposes a non-boolean, modal account of disjunction, where disjunctions of the kind $ {S}_1 $ $ \left[ or\right] $ $ {S}_2, $ $ \dots $ $ or $ $ {S}_n $ express a list of propositions such as $ {S}_1,\dots {S}_n $ , all of which are compatible with the person uttering a disjunctive sentence. Under this view, a disjunctive sentence amounts to a “list answer” to the question of “what might be the case?”, with the response effectively being $ {S}_1 $ $ \left[ and\right] $ $ {S}_2, $ $ \dots $ $ \left[ and\right] $ $ {S}_n $ , where the brackets indicate that each addition of a further proposition is but a possibility entailed by the disjunction. Similar to the account in Jasbi et al. (Reference Jasbi, Jaggi, Clark and Frank2024), Zimmerman’s proposal also includes a role for prosody in the interpretation of disjunctive sentences, and along the same lines; in particular, in Zimmerman’s account a high phrase-final tone indicates that all possibilities in a disjunction can apply (the inclusive reading), whereas a low phrase-final tone signals that this is not the case (a low phrase-final tone is equivalent to the rise–fall intonation of Jasbi et al. and of course exemplifies an exclusive reading).

Zimmerman’s account is especially attractive here because it suggests a possible explanation for the observation that our toddlers preferred disjunctive sentences to conjunctive sentences for the true–true reading. Namely, the task may have created a situation where children indeed considered what might be the case given the images on display and the sentences played to them (actual questions), with the design of the materials furthermore supporting the application of a list of possibilities as a possible answer to the question – a neutral tone, compatible clauses, a picture–sentence matching task which involved some novelty and uncertainty, etc, and indeed children did look at both graphics to begin with, but focused on bicoloured balls early on. In such a setting, children may well have favoured an inclusive reading of disjunction over the much stronger, and less open to uncertainty, option of conjunction. An additional contrast is to be found in the adult data reported in Lobina et al. (Reference Lobina, Demestre, García-Albea and Guasch2023), where participants undertook a visual world experiment similar in kind to ours, as noted in Introduction. In this study, disjunctive sentences such as the triangle is yellow or the square is blue were played to participants as they inspected a screen that was divided into quadrants, with each quadrant representing each one of the possible four truth values of disjunction. The true–true reading received the most fixations overall in this experiment – that is, participants fixated the most on a quadrant that exhibited, for the example sentence, a yellow triangle and a blue square. The set-up is not entirely equivalent to our own, but these data show nonetheless that disjunctive sentences that are presented with the relevant intonation and exhibiting two compatible clauses yield an inclusive interpretation.

Thus stated, then, we take our conjunctive sentences to be underlain by a boolean conjunction operator, following Winter (Reference Winter2001), whilst the comprehension data in our disjunctive condition appear to be compatible with Zimmerman’s (Reference Zimmerman2000) non-boolean account of disjunction. Moreover, we have adult data converging on the true–true, online reading of (inclusive) disjunction that our toddlers manifested with disjunctive sentences.

A final point we wish to broach, but which we shall not discuss extensively, is the issue of whether acquisitional data of this kind can inform on what sort of primitives there might be in language and thought, especially in terms of the logical vocabulary (in this case, the logical connectives). As an example, in a study on how young children reason with the disjunctive syllogism (viz., p or q and not-q, therefore p), Mody and Carey (Reference Mody and Carey2016) report an experiment in which children below the age of 3 do not appear to employ the syllogism, and this, these authors suggest, may be related to the apparent fact that English or is not used properly until about age 3, in line with the literature we reviewed earlier. How so? In order to employ the syllogism, one would presumably have to have the concepts OR and NOT (we use capitals for concepts, as per the norm), and one way to figure out whether these concepts are available at a certain age, according to Mody & Carey, is to establish how early in ontogeny the corresponding lexical items are acquired. If there is a large temporal gap between a given word entering the lexicon and it being given the appropriate meaning, we could then conclude that there was no corresponding concept in thought before the acquisition of the relevant lexical item (if this is the case, language would have a rather central role in both bringing about the relevant meanings and assisting the development of logical capacities).

This argumentative line must be regarded as very tentative. The mapping between lexical items and their meanings is a rather intricate affair (Bloom, Reference Bloom2000), and in many cases, the mapping will not be swift or instantaneous, especially when the semantics of a given lexical item is not a straightforward matter, as is the case with disjunction. It is also doubtful that a given concept will always be available for use in a given experimental task, even if it is already part of the conceptual system at the time, as myriad factors may conspire to block its employment (misidentification of the task, cognitive maturation, etc.). In the case of Mody and Carey (Reference Mody and Carey2016), these scholars employed an act-out task, which requires children to plan a set of actions and then execute them, and this is often rather taxing for young children (see Crain and Thornton, Reference Crain and Thornton1998, for a description of act-out tasks and the like, and Brandt-Kobele and Höhle, Reference Brandt-Kobele and Höhle2010 for a discussion on how different techniques yield different kinds of data).

Cesana-Arlotti et al. (Reference Cesana-Arlotti, Martín, Téglás, Vorobyova, Cetnarski and Bonatti2018) employed a more neutral method to probe whether toddlers can use the disjunctive syllogism to reason with alternatives and argued that their results showed that 12- and 19-month-olds demonstrated the requisite abilities. This study measured the eye movements and pupil dilations of very young children during the presentation of scenes that were congruent and incongruent in relation to the application of the disjunctive syllogism, and under these conditions, the children did in fact behave as if following the disjunctive syllogism. Though we do not discount alternative explanations, these skills would be presumably in place before the acquisition of or, suggesting that the concepts OR and NOT (along with a way to put them together) may be available at such an early age, contrary to what Mody and Carey (Reference Mody and Carey2016) had suggested (see, however, the discussion in Feiman et al., Reference Feiman, Mody and Carey2022).

Viger (Reference Viger2005) runs a similar argument in terms of the kinds of learning involved with different kinds of vocabularies (logical and natural kinds), arguing that the earlier on a particular kind of vocabulary is acquired in language acquisition, the more likely it is that this vocabulary will be part of a language of thought or similar (i.e. as part of an innate conceptual system; the locus classicus here is Fodor, Reference Fodor1975). And in Viger (Reference Viger2005) too, we find the observation that the logical vocabulary is acquired relatively late (the data he cites are now certainly out of date, but the evidence also circled around the age of 3 then) and thus the claim that conjunction, disjunction, and the like would not be part of thought ab initio. We will not claim to settle this issue here (we hope to do some further work on this), but we do wish to suggest that our data show that children have conjunction and and inclusive disjunction or relatively early in language acquisition, not late.