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Logical Connectives

Abstract and Keywords

This chapter reviews results from recent experimental studies on the acquisition of logical connectives. Developmental psychologists have long been interested in the development of logic in children, and recent research in this field has made great advancement by incorporating insights from theoretical linguistics. There are two important theoretical grounds that were crucial to such advancement. One is dissociation between pragmatic implicature and lexical semantics. The other is a model of semantic interaction between a logical connective and another logical word in the same sentence. Experimental results from recent studies that incorporated these insights strongly suggest that preschool children have sophisticated semantic knowledge of logical connectives, even though their behavior may sometimes deviate from adults’ behavior.

Keywords: logical connective, disjunction, conjunction, negation, pragmatic implicature, scope, focus, positive polarity

Developmental psychologists have long been interested in the development of logical concepts and logical thinking in children (e.g. Inhelder and Piaget 1958, 1964; Piaget 1967, 1968). Studies on children’s comprehension of logical connectives in natural languages, such as English conjunction and and disjunction or, have an equally long history. In this chapter, rather than trying to spell out the whole history of research, I will focus on presenting findings from more recent studies on the acquisition of logical connectives, and discuss some implications of the findings. This is because the recent research in this field has made great advances, which I believe is a fundamental leap forward. The key feature of this “paradigm shift” in the field is the incorporation of insights from theoretical linguistics. Over the last few decades, theoretical linguists have elaborated on the descriptions of logical connectives in natural language, proposing formal analyses that encompass syntax, semantics, and pragmatics. These formal models help elucidate the nature of the knowledge of logical connectives in natural language, thereby providing grounds for designing appropriate experiments to assess the knowledge in children. In what follows, I review some of the experimental investigations of the acquisition of logical connectives that are built on the results of theoretical linguistics. The experimental findings point in the same direction: preschool children have sophisticated knowledge of logical connectives, even though their behavior may sometimes deviate from adult behavior. I begin by describing how semantics and pragmatics affect interpretations of natural language disjunction.

23.1 Disjunction: Semantics and Pragmatics

In English, sentences containing a disjunctive phrase “A or B” often invoke the so-called exclusive interpretation of disjunction. For example, upon hearing the sentence in (1), (p. 548) the hearer would typically infer that it is not the case that John speaks both French and Spanish.

  1. (1) Logical Connectives

It is a widely accepted view that this exclusive interpretation of disjunction arises from a scalar implicature (e.g. Horn 1972, 1989; Gazdar 1979). Building on the ideas of Grice (1975), it is assumed that in cooperative conversations the speaker is expected to provide the maximally specific information within the relevant context to the hearer. In the current case, using the conjunction and instead of or yields a more restricted interpretation, hence it provides more specific information. Therefore, if the speaker knows that John speaks both French and Spanish, he should have uttered “John speaks French and Spanish,” rather than using the form in (1). This allows the hearer to infer that the speaker does not believe John speaks both French and Spanish. Assuming that the speaker is cooperative and reliable, the hearer then concludes that John in fact does not speak both languages, giving rise to the exclusive interpretation of disjunction.

In what follows, I will follow the pragmatic account of the exclusive interpretation of disjunction. I will assume English or corresponds to inclusive disjunction in classical logic (∨), and a derived exclusive implicature is computed and added onto the basic lexical meaning, due to the pragmatic principles that govern interpretations of sentences in conversation. Under this assumption, making judgments about the truth or falsity of sentences like (1) involves more than just knowing the meanings of each word and the rules of semantic composition: language users must also be aware of pragmatic principles that give rise to the relevant scalar implicature. This compound nature of the interpretation of disjunction or becomes particularly relevant when we examine children’s acquisition of the connective. In order to manifest the full range of adultlike behavior with or, a child must have acquired (a) knowledge of the lexical semantics of the word, and (b) the ability to compute scalar implicatures. Therefore, a study of the acquisition of the logical connective must appropriately distinguish the contributions of these two different components to children’s behavior.

It has been widely observed that young children do not reliably compute pragmatic implicature in general (i.e. they fail to reject test sentences on the basis of pragmatic implicature) in experimental setups (e.g. Noveck 2001; Papafragou and Musolino 2003; Guasti et al. 2005). For example, using the word some in a descriptive sentence invokes a scalar implicature “not every.” However, in one of Papafragou and Musolino’s (2003) experiments, 5-year-old Greek-speaking children almost consistently accepted the Greek version of the test sentence Some of the horses jumped over the fence under a situation where all of the horses jumped over the fence, while adults overwhelmingly rejected the same sentence in that situation. Given this background, the observation that young children often fail to assign the exclusive interpretation to disjunction or (e.g. Paris 1973; Chierchia et al. 2001) would most naturally be interpreted as reflecting their general difficulty in computing pragmatic implicatures. Put differently, children’s non-adult behavior with respect to the exclusive interpretation of or does not endorse the conclusion (p. 549) that their knowledge of the meaning of the lexical item is flawed. An appropriate test for children’s lexical semantics of or requires an experimental design that abstracts away from the exclusive interpretation. I will discuss such a design in the next section.

Another factor that can affect children’s behavior in experimental setups is the felicity conditions for using a disjunctive statement such as (1). In normal conversation, a speaker would avoid using the sentence in (1) if he knows exactly what language John speaks; he would instead say something like (2) or (3).

  1. (2) Logical Connectives

  2. (3) Logical Connectives

In the situations where (2) or (3) is appropriate, a disjunctive statement such as (1) is infelicitous. In other words, the use of or in a descriptive statement is pragmatically felicitous only when the speaker is not in a position to provide more specific information. Some examples of utterances where or is felicitously used are as follows. The disjunctive statement in (4) expresses the speaker’s uncertainty, and the sentence in (5) expresses John’s minimum requirement:

  1. (4) Logical Connectives

  2. (5) Logical Connectives

In earlier studies on the acquisition of or (e.g. Suppes and Feldman 1969; Paris 1973; Beilin and Lust 1975; Braine and Rumain 1981), little attention was paid to the felicity requirements. Test sentences that involve or were often presented without contextual supports that would justify the use of a disjunctive statement. However, the emerging consensus in more recent development of experimental studies on language acquisition is that failing to meet the felicity requirements for test sentences can cause young (especially, pre-school) children to behave differently from adults (see Crain and Thornton 1998: ch. 35; Gualmini 2003). Therefore, the results from earlier studies, which involved younger children’s non-adult-like performance with the disjunction or and gradual development of adult-like behavior, must be reevaluated with the data from recent experiments that controlled for the felicity of test sentences.

23.2 Boolean Disjunction in Preschool English

Since early 2000s, Stephen Crain and his colleagues have developed a research paradigm that draws upon the Boolean property of the disjunction or (e.g. Crain et al. 2002; Gualmini and Crain 2002, 2004, 2005; Goro et al. 2005). This approach focuses on the semantic interaction between or and another logical word (in particular, negation) in the same sentence. When the disjunction or appears within the scope of negation, it (p. 550) allows an inference that closely resembles one of De Morgan’s laws of Boolean logic. In (6), to illustrate, the truth conditions of the sentence that contains a negated disjunction can be recast with the conjunction and presiding over both of the disjuncts.

  1. (6) Logical Connectives

We call this interpretation the “conjunctive” interpretation of disjunction because it is logically equivalent to the conjunction of two negated expressions (e.g. Higginbotham 1991):

  1. (7) Logical Connectives

Therefore in normal contexts, the sentence in (6) is judged to be false if John speaks either Spanish or French.

The following pair of sentences from Crain et al. (2002) illustrates the interpretive contrast between the conjunctive interpretation of disjunction and the ordinary disjunctive interpretation:

  1. (8) Logical Connectives

  2. (9) Logical Connectives

Both of the sentences involve the disjunction operator or and negation. In addition, in both of the sentences negation precedes disjunction. However, the structural relations between negation and disjunction are different in the two sentences. In (8), the negation not appears in the matrix clause, and is structurally higher than the disjunction or. In contrast, the negation n’t in (9) is embedded within a relative clause, and hence is not structurally higher than the disjunction or. This difference in the structural relations affects the interpretations of the sentences. In (8), or is interpreted under the scope of negation, yielding a conjunctive truth condition: the girl will not get a dime AND will not get a jewel. In (9), or is interpreted outside the scope of negation, and the sentence is not associated with conjunctive truth conditions. Therefore, under a situation where the girl ends up getting a jewel but not a dime, (8) is judged to be false, whereas (9) is judged to be true.

Notice that the exclusive interpretation of disjunction, which involves computation of scalar implicatures, is irrelevant in making these judgments. First, the conjunctively interpreted disjunction in (8) does not invoke a scalar implicature, presumably because it yields the most restrictive truth condition (see the truth table in (7)). Second, regardless of whether the disjunction or in (9) is interpreted exclusively or inclusively, the sentence should be judged true in the above situation. On the other hand, assigning distinct truth conditions to (8) and (9) requires sophisticated semantic knowledge. In order to (p. 551) derive the conjunctive truth condition for (8), or must be understood as the Boolean inclusive disjunction. In addition, the knowledge of semantic compositional principles is necessary to correctly determine scope relations between the disjunction and negation. Thus, the interpretive contrast between (8) and (9) provides an excellent testing ground for children’s semantic knowledge of the disjunction or, abstracting away from the problem of pragmatic implicature computation.

Crain et al. (2002) investigated children’s interpretation of sentences such as (8) and (9), using a Truth Value Judgment (TVJ) task (Crain and McKee 1985; Crain and Thornton 1998). In one of the experimental trials, children were told a story about two girls who had both lost a tooth, and were looking forward to getting a reward from the Tooth Fairy in exchange for their lost tooth. At the moment the Tooth Fairy arrived in the story, the puppet (manipulated by one experimenter) interrupted the story and presented his prediction about what would happen in the reminder of the story. The “prediction mode” was employed in order to satisfy the felicity conditions for using a disjunctive statement. Since the puppet was making a prediction about what would happen, rather than presenting a description about what had happened, or can be felicitously used to express his uncertainty. One group of children were presented with (8), while the other group heard (9). Then the story resumed, and the Tooth Fairy gave a jewel, but no dime, to the girl who had stayed up late. Following the completion of the story, the child participant was asked to judge whether the puppet’s prediction was right or wrong.

Under the adult interpretations of the test sentences, (8) is false in the situation, because of the conjunctive truth conditions associated with the sentence (i.e. the girl will not get a dime AND will not get a jewel). In contrast, (9) is true in the situation, because the disjunction or is interpreted outside the scope of the negation. Crain et al. (2002) found that English-speaking children at age 4–5 rejected (i.e. said the puppet was “wrong”) sentences like (8) 92 percent of the time, while they accepted (i.e. said the puppet was “right”) sentences like (9) 87 percent of the time. This highly robust adultlike performance by preschool children has been replicated in other studies that employed similar design (e.g. Gualmini and Crain 2002, 2004, 2005; Goro et al. 2005). Therefore, the available evidence strongly suggests that preschool children as young as age 4 can correctly compute the semantic interaction between the disjunction or and other logical words, for example, negation. Given the data, we conclude that the semantics of the disjunction or is fully developed at age 4, although children at this age may have problems in computing pragmatic implicatures, or in accommodating themselves to test sentences that fail to meet the relevant felicity conditions.

23.3 Cross-linguistic Variation and Universality

As we have seen in the previous section, English disjunction or yields the “conjunctive” interpretation within single-clause negative sentences such as (6). In contrast, the (p. 552) Japanese counterpart of (6) appears to lack the conjunctive interpretation. As illustrated in (10), a Japanese simple negative sentence that involves the disjunction ka is most naturally paraphrased by the disjunction of two negated expressions:

  1. (10) Logical Connectives

Thus, the sentence in (11) can be truthfully uttered in a situation where, for example, John speaks Spanish but not French. In order to convey the “neither” meaning of (6), Japanese speakers use the conjunction … mo … mo:

  1. (11) Logical Connectives

This cross-linguistic contrast was first pointed out by Szabolcsi (2002), on the basis of the observation that Hungarian disjunction vagy lacks the “conjunctive” interpretation in simple negative sentences.

  1. (12) Logical Connectives

However, when the disjunction and negation are separated by a clause boundary, the interpretive contrast between English and Hungarian evaporates. For example, (13) is interpreted as “I don’t think we closed the door AND don’t think we closed the window,” just as in English.

  1. (13) Logical Connectives

A parallel observation can be made in Japanese. The following examples illustrate that identical conjunctive interpretations of ka and or emerge when they appear in a sentential complement ((14)), and in a relative clause ((15)), embedded under a matrix negation.

  1. (14) Logical Connectives

  2. (p. 553) (15) Logical Connectives

These data suggest that both Hungarian vagy and Japanese ka are Boolean disjunction, just like English or. However, vagy and ka are subject to a language-specific constraint that blocks them from being interpreted under the scope of local negation. Due to the effect of the constraint, they take scope over negation in single clause sentences such as (10) and (12), yielding the “disjunctive” truth conditions.

Szabolcsi (2002) and Goro (2004a, 2007) argue that the relevant constraint on scope interpretation is lexical in nature. Specifically, they propose that disjunctions in natural language are divided into two classes. In one class, which includes Hungarian vagy and Japanese ka, disjunctions are Positive Polarity Items (PPI). In the other class, which includes English or, disjunctions are not PPIs. The defining property of PPIs is that they cannot take scope under local negation, while they can be interpreted under the scope of extraclausal negation. English existential quantifier some, which is assumed to be a typical PPI (e.g. Baker 1979; Progovac 1994; Szabolsci 2004), shows this locality effect in scope interpretation. As illustrated in (16), the scope behavior of some parallels that of vagy and ka:

  1. (16) Logical Connectives1

Given this analysis, children learning a language with a PPI-disjunction (i.e. Hungarian, Japanese, etc.) must learn that the relevant lexical item is a Boolean inclusive disjunction, and also that the item is a PPI. However, an equally plausible hypothesis seems to be available for children: the hypothesis that the relevant disjunction operator is not a (p. 554) Boolean connective, and therefore it does not semantically interact with negation (or with any other logical words). As we have seen in (10) and (12) above, PPI disjunctions do not yield the conjunctive interpretation in simple negative sentences. Consequently, adults chose to use an alternative form (e.g. the conjunction … mo … mo as in (11)) to express the “neither” meaning. As a result, in the vast majority of input data to children, PPI disjunctions fail to manifest their Boolean nature. The direct evidence for the Boolean nature presumably involves sentential embedding, such as in (13), (14), and (15), which may not be included in the primary linguistic data used by children (e.g. Lightfoot 1999). Given these considerations, cross-linguistic research on the acquisition of disjunction bears on the issue of how experience affects language acquisition. If children are to learn the Boolean semantics of disjunction from experience, then it is expected that English-speaking children and Japanese-speaking children would go through different developmental courses, because of the difference in the availability of crucial evidence in input.

Goro and Akiba (2004) examined Japanese children’s interpretation of disjunction in simple negative sentences. The experiment used the TVJ task. One experimenter acted out a short story about an “eating-game.” In the game, there are 12 animals who are each asked to eat vegetables that they don’t like: a carrot and a green pepper. Each animal gets a different prize depending on how well they did. First, if an animal eats not only cake, but also the vegetables, then it receives a shining gold medal. Second, if an animal eats cake but only one of the vegetables, then it receives a blue medal. Finally, if an animal only eats cake and does not eat any vegetables, then it gets a black cross (a symbol of failure in Japanese culture). The story phase continued until all twelve animals finished their trials and were presented with their rewards. After the story, the puppet started to guess how well each animal did in the game, starting with the first animal. First, the puppet said that he didn’t remember exactly what each animal ate, then he started to make guesses about this, based on the color of the prizes the animals had been presented as awards. The crucial test cases are the puppet’s guess about those animals with a blue medal, that is, those who ate only one of the vegetables. For example, the puppet uttered the test sentence (17) for the pig, who had a blue medal as his reward:

  1. (17) Logical Connectives

Recall that a blue medal was awarded only to those animals who had eaten just one of the vegetables—it did not indicate which vegetable the animals had actually eaten. Therefore, the color of prizes provides only incomplete information with respect to what the pig actually ate. Given this incompleteness of information, all that the puppet could reasonably guess was something like “he didn’t eat the pepper or he didn’t eat the carrot,” which corresponds to the adult-Japanese interpretation of the target sentence. In this situation, the adult control group accepted the test sentences with disjunction, as in (17), (p. 555) 100 percent of the time. This is just as expected. In contrast to the pattern of results from adults, however, 30 Japanese-speaking children (age: 3;7–6;3, mean: 5;3) only accepted the crucial test sentences under the same situation 25 percent of the time. Among the 30 children, only four were adultlike in consistently accepting the test sentences. The remainder of the children rejected the test sentences 87 percent of the time. When these children were asked to explain the reason for their negative judgments, they said, for example either “because the pig did eat one of the vegetables” or “because it is only one of the vegetables that the pig didn’t eat.”

The negative judgments from the vast majority of children, combined with their explanation for their negative judgments, suggest that Japanese children were assigning the conjunctive interpretation to ka in simple negative sentences. That is, with respect to the interpretation of negated disjunction, Japanese children are more like English-speaking children/adults than like Japanese adults. This result has been replicated in other languages with a PPI-disjunction: Mandarin Chinese2 (Jing et al. 2005) and Russian (Verbuk 2006a). Thus, the conjunctive interpretation of negated disjunction appears to be the “default” interpretation for children. This in turn suggests that children uniformly assign Boolean semantics to disjunction, regardless of whether the input data provide evidence that supports the hypothesis.

Summarizing, available evidence suggests a lack of cross-linguistic variation in child language: children uniformly assign Boolean semantics to disjunction, and interpret negated disjunction conjunctively, even if the interpretation is not available in their target language. In other words, the cross-linguistic variation with respect to the interpretation of negated disjunction in adult language fails to affect children’s semantic knowledge of disjunction: experience does not seem to play a significant role in the acquisition of semantics of disjunction. If children are to construct the meaning of disjunction from linguistic experience, then the difference between adult English and Japanese is expected to affect the outcome: children learning English should encounter a sizable amount of evidence that shows the disjunction or is Boolean (i.e. not … or expressing “neither”), but Japanese children are virtually deprived of such evidence.

These results from cross-linguistic studies on the acquisition of disjunction have led researchers to propose that the semantics of disjunction is innately determined (Goro 2007; Crain and Khlentzos 2008, 2010; among others). On this view (“logical nativism” by Crain and Khlentzos 2010), humans have an innate logical faculty that structures thought, providing universal logical concepts that all natural languages draw on. The Boolean inclusive semantics of disjunction is specified in the innate endowment, and children learn an appropriate linguistic label for the innate concept. It is then predicted that all human languages associate the same truth conditions to an expression of disjunction. Crain and Khlentzos (2008, 2010) argue that this is indeed the case: the semantics of disjunction is uniform across languages.

Further support for the logical nativism account would come from a study of younger children. According to this view, the innate logical concepts including disjunction (p. 556) should be present even at the earliest stages of development. Therefore, with an appropriate task design, it should be possible to find evidence that younger children, possibly infants, engage in logical reasoning using such concepts. Studies on the development of logical capacity in children have found that, with tasks that do not require explicit conscious reasoning, children as young as 2.5 years of age show some success on making what looks to be logical inference (e.g. Pea 1982; Fabricius et al. 1987; Scholnick and Wing 1995; Watson et al. 2001). For example, Halberda (2006) showed a familiar object (e.g. a cup) and a novel object (e.g. a vacuum tube-looking object) to 3-year-old children, providing a novel label (“watch the dax”). Eye-tracking analysis revealed that upon hearing the novel label, children systematically switched their gaze back to the familiar object, rejected it, and then returned their gaze to the novel target. Halberda argues that this behavior reflects the computational structure of Disjunctive Syllogism (i.e. A or B, not A, therefore B) that the children worked through to motivate the mapping of the novel label to the novel object. Thus, 3-year-olds engage in logical inference that involves disjunction, suggesting that the concept is present and operative in the mind of children at that age.

Summarizing, we have reviewed studies on the acquisition of disjunction from a cross-linguistic perspective. Due to the existence of language-specific constraints on scope interpretation, simple negative sentences involving disjunction are assigned contrastive truth conditions in different languages. However, the cross-linguistic contrast is absent in child language. Children across different languages uniformly assign the Boolean, “De Morgan” interpretation to negated disjunction. This universality in child language motivates logical nativism: the idea that logical concepts including disjunction are specified as part of the innate endowment of the species.

23.4 Acquisition of Scope

In Goro and Akiba’s (2004) experiment, Japanese children at around age 5 assigned the conjunctive interpretation to the disjunction ka in test sentences like (17). This means that they interpreted ka under the scope of local negation, even though the interpretation is not possible in adult Japanese. Given this observation, it is important to ask how Japanese children learn the scope constraint on ka. Since the scope constraint on disjunction is not universal, first language learners must somehow learn it from experience. In this section, we will discuss how the acquisition of the scope constraint is possible.

First, it must be pointed out that the non-adult scope interpretation of ka by Japanese children cannot be attributed to a general inability to access “inverse-scope” interpretations. The behavior of Japanese children, at first glance, is somewhat reminiscent of English-speaking children’s non-adult performance observed by Musolino et al. (2000). In their TVJ task experiments, Musolino et al. found that young children often failed to assign “inverse scope” readings to test sentences like (18), resulting in a failure to accept the sentence in a situation where the detective found two of his friends but missed one.

(p. 557)

  1. (18) Logical Connectives

Note that in adult English some is a PPI, and resists taking scope under local negation. However, young children adhered to the narrow-scope interpretation of the existential quantifier (i.e. “the detective didn’t find anyone”), ignoring the constraint on scope interpretation.

Given this background, Goro and Akiba (2004) carried out a control experiment to make sure that Japanese children at the relevant age (i.e. children at around age 5) can access inverse-scope interpretations. The control condition was established by replacing the disjunctive phrase in the target sentences, such as (17), with another quantificational expression nanika. Nanika is an indefinite existential corresponding to English something:

  1. (19) Logical Connectives

In the control experiment, Japanese children (age: 3;7–6;3, mean: 5;4) were presented the target sentences such as (19) as the puppet’s guess about those animals with a blue medal, that is, those who ate only one of the vegetables. In contrast to the experiment with disjunction ka, children did not show the same non-adult performance: they correctly accepted the crucial test sentences 88 percent of the time. This result suggests that Japanese children in Goro and Akiba’s experiments did not experience general problems in accessing inverse-scope interpretations of object QPs. Given this result, Goro and Akiba concluded that the non-adult performance in interpreting sentences with ka must have to do specifically with the lexical item.

In Goro (2004a, 2007), I proposed that innate linguistic knowledge restricts a learner’s possible hypotheses about the scope of natural language disjunctions, arguing that (20) is a part of innate linguistic knowledge:

  1. (20) Logical Connectives

Under the analysis, Hungarian vagy and Japanese ka are disjunctions with the value [+PPI], and English or has the value [–PPI ]. The first language learner’s task is then to determine which value of the parameter a particular disjunction in the target language has. Based on a learnability consideration, Goro (2004a, 2007) argued that the lexical parameter has a default value. Assuming that direct negative evidence does not play a crucial role in language acquisition, the learnability argument goes as follows. Suppose that a language L has the Boolean disjunction OR. In order to determine whether this item is +PPI or –PPI, the crucial data is a single clause negative sentence, in which the form A OR B appears in the potential scope domain of sentential negation (e.g. the object position in transitive sentences). With such a form as input, there are two different output truth conditions for the connective, corresponding to whether the item is +PPI or –PPI:

  1. (p. 558) (21) Logical Connectives

Notice that the two truth conditions stand in a subset/superset relation: the situations in which ¬A ∧ ¬B is true are a subset of the situations in which ¬A ∨ ¬B is true. Thus, in every logical situation where ¬A ∧ ¬B is true, ¬A ∨ ¬B is also true. Given this, the learnability argument contends that an incorrect hypothesis that yields the superset truth conditions can never be falsified by positive input data. Therefore, the relevant parameters must have a default value so that children always start with a hypothesis that yields the subset truth condition. Such a default value for disjunction is [–PPI], hence the conjunctive interpretation of negated disjunction is the default interpretation across children learning different languages.

The experimental data from young children are compatible with the predicted default values for the parameter. However, it must be pointed out that the conceptual underpinnings of the learnability argument has been challenged. Gualmini and Schwarz (2007), for example, argued that the semantic entailment problem could be gotten around either by taking pragmatic implicature into account, or by considering cases in which the relevant forms are embedded under a downward-entailing operator (see Gualmini and Schwarz 2007 for details). Furthermore, Jing (2008) pointed out that a –PPI disjunction (e.g. English or) may take scope over local negation, and showed that in appropriate experimental settings both adults and children can access the wide-scope interpretation of disjunction. Thus, the truth conditions associated with the two parameter values may not stand in a proper subset/superset relation, with [–PPI] value yielding both ¬A ∧ ¬B and ¬A ∨ ¬B. But even if the conceptual argument has to be lifted, the question of whether or not the relevant parameter has a default value remains as a valid empirical issue for future research.

23.5 Using Disjunction as a Probe into Children’s Semantic Capacity

As we have reviewed in the previous sections, there are strong empirical grounds to support the conclusion that English-speaking children, at least by roughly age 4, have adultlike knowledge about the Boolean property of the disjunction or. This conclusion invites the research strategy of using children’s interpretation of or as a diagnostic test of their knowledge of the semantics of different operators: if children assign correct adultlike interpretations to or within the immediate scope of a certain operator, that would suggest they identify the semantics of the operator correctly. This strategy can bear a particularly insightful result when it is applied to an operator with which children have been reported to show non-adult performance. If children are able to compute the (p. 559) semantic interaction between the operator and or, that would strongly suggest that they have adultlike lexical semantics of the operator. This result would, in turn, narrow down the range of possible sources of children’s non-adult behavior in interpreting sentences with the operator.

One such case concerns the universal quantifier every. Various studies have demonstrated that when presented with a sentence like “Is every boy riding a pony?” with a picture showing three boys on a pony with an extra rider-less pony, 3- to 5-year-olds often respond “no,” pointing to the extra pony (e.g. Philip 1995; cf. Inhelder and Piaget 1964). This non-adult behavior is often referred to as symmetrical response. To account for the non-adult behavior, Philip (1995) argues that children’s grammar may allow the determiner every in sentences like every boy is riding a pony to quantify over events, rather than over individuals. Under that event quantification interpretation, the sentence means something like “For every event e that involves a boy or a pony, a boy is riding a pony.” Thus, for the sentence to be true on that interpretation, it must be the case that every pony is ridden by a boy, in addition to that every boy is riding a pony.

Here, studies on children’s interpretation of sentences that involve every and or becomes relevant. Within sentences containing every in the subject position, the interpretation of or varies systematically according to its position. Specifically, when or appears within the noun phrase that every combines with (“restrictor”), it yields the conjunctive entailment. In contrast, when or appears within the predicate phrase (“nuclear scope”), it continues to provide the ordinary “disjunctive” reading:

  1. (22) Logical Connectives

  2. (23) Logical Connectives

Experimental studies (Boster and Crain 1993; Guamlini et al. 2003) observed that children assigned different interpretations to or in sentences like (22) and (23). Specifically, when or appears within the restrictor of every, children assigned the conjunctive interpretation to or, and consistently rejected the test sentence in (22) when those trolls who ordered onion rings did not get any mustard. In contrast, children interpreted or in the nuclear scope of every disjunctively, and consistently accepted sentences such as (23) even when there are some ghostbusters who didn’t choose a pig. These asymmetrical interpretations of or are unexpected if children interpret every as taking sentential scope, quantifying over events. This in turn suggests that children’s semantic composition of sentences with every is essentially adultlike, and the source of the non-adult, symmetrical response by children must reside somewhere else.

Another case involves the focus operator only. Previous research found that preschool children often assigned non-adult interpretations to sentences involving only. For example, Crain et al. (1994) observed that children often associate pre-subject only (p. 560) with VP, and interpret, for example, Only John speaks Spanish as meaning that John only speaks Spanish, and the exact nature of the non-adult interpretations is under ongoing debate (e.g. Paterson et al. 2003). Goro et al. (2005) sought to bring about a new perspective on the issue, by investigating children’s interpretation of sentences with only and or. They pointed out that the meaning of sentences involving only and or, such as (24), is decomposed into two parts, as in (25) (e.g. Horn 1969):

  1. (24) Logical Connectives

  2. (25) Logical Connectives

Notice that the two different meaning components affect the disjunction or differently. Within the first component of the decomposed propositions, or is interpreted disjunctively with respect to what Bunny Rabbit ate. Thus, for the sentence to be true, it must be the case that Bunny Rabbit ate a carrot or a pepper (but not necessarily both). In contrast, within the second component, or receives the conjunctive interpretation with respect to what everyone other than Bunny Rabbit did not eat. Hence, for the sentence to be true, it must be the case that everyone other than Bunny Rabbit did not eat a carrot and everyone other than Bunny Rabbit did not eat a pepper. In other words, the disjunction or in (24) is “two-faced”: it is effectively interpreted twice, giving rise to distinct truth conditions.

In their experimental study, Goro et al. (2005) found that children at around age 4 correctly interpreted the two-faced or. In one of their experimental stories, Bunny Rabbit and another two characters were introduced, and the puppet made a prediction about what would happen next in the story: for example, “I think only Bunny Rabbit will eat a carrot or a pepper.” In one condition, Bunny Rabbit proceeded to eat a carrot, but neither of the other characters ate anything. In this condition, children consistently accepted the test sentence, showing that they interpreted or disjunctively with respect to what Bunny Rabbit ate. In the second condition, Bunny Rabbit proceeded to eat a carrot, and another character ate a pepper. In this condition, children consistently rejected the test sentence, showing that they interpreted or conjunctively with respect to what everyone else failed to eat. Taken together, the results suggest that adultlike semantic composition of the meaning of sentences containing or and only is fully operative at around age 4. This implies that the relevant semantic computation is not subject to a performance problem that is caused by some limitation on the processing capacity of children, although the computation appears to be quite complex. In interpreting the crucial test sentences, children must have created representations for two separate propositions, and computed the truth conditions for each of the propositions individually, in order to derive the distinct truth conditions with the “two-faced” or. Children’s adultlike behavior with the “two-faced” or suggests that the processing capacity of children at around age 4 is capable of carrying out the computation.

(p. 561) 23.6 Conjunction

In most of the earlier studies on the acquisition of logical connectives (e.g. Suppes and Feldman 1969; Paris 1973; Beilin and Lust 1975; Roberge 1975), the conjunction and was found to be easier than the disjunction or for young children. With preschool children, and almost always elicited a higher rate of adultlike responses than or did. In the 2000s, as we have reviewed 23.2 and 23.3 above, research on disjunction shifted its focus on the acquisition of semantic interaction between disjunction operator and another logical expression in the same sentence. This research strategy has proven to be fruitful, revealing that preschool children have adultlike semantic knowledge of the disjunction or, even though they may have problems in the domain of pragmatics.

To date, however, there are only a handful of studies on conjunction that focus on the acquisition of the semantic interaction between conjunction and another logical element. One reason behind this situation is that the Boolean character of natural language conjunction is not as stable as that of disjunction. For example, Szabolcsi and Haddican (2004) point out that the Hungarian conjunction és conjoining two definite NPs systematically resists the Boolean “not both” interpretation under the scope of negation.

  1. (26) Logical Connectives

  2. (27) Logical Connectives

As illustrated in (27), the non-Boolean, “neither” interpretation of és persists even under extra-clausal negation. This behavior of és contrasts with Hungarian disjunction vagy, which is a PPI and yields the Boolean interpretation when there is a clause boundary between negation and the disjunction (e.g. (13)). Szabolcsi and Haddican argue that definite conjunctions with és are given the same denotation as definite plurals and, therefore, yield a non-Boolean interpretation under negation. Szabolcsi and Haddican further argue that the same definite plural interpretation is available for English definite conjunctions with and. In fact, the Boolean “not both” interpretation of and in simple negative sentences seems to be a rather marked reading for English-speaking adults, sometimes requiring focal stress on and. In an experiment by Goro et al. (2006), adult English speakers resisted assigning the “not both” interpretation to the conjunction and (without focal stress) in sentences like (28) for over two-thirds of the time.

(p. 562)

  1. (28) Logical Connectives

Japanese conjunction … mo … mo, according to Goro (2004a, 2007), displays another pattern. While it lacks the “not both” interpretation in simple negative sentences (e.g. (11)), the Boolean interpretation becomes systematically available in embedded contexts, suggesting that the lexical item is a Boolean PPI conjunction:

  1. (29) Logical Connectives

In short, the semantics of conjunctions in natural language shows greater variation than that of disjunction, and the exact nature of the variation is not yet fully understood. A problem for language acquisition research is that it is not always possible to exploit the Boolean character of conjunction in order to assess children’s knowledge of the lexical item. English conjunction and, for example, seems to be ambiguous between Boolean and non-Boolean interpretations, and therefore most experimental designs with the disjunction or cannot be straightforwardly carried over to the study of the acquisition of conjunction.

To resolve the problem, Goro et al. (2006) adopted different experimental materials, namely sentences with the focus operator only. In these sentences, the Boolean interpretation of the conjunction and systematically emerges even when conjunction does not receive focal stress. For example, imagine that the sentence in (29) describes the result of a “jumping contest” with the Smurf and other two characters:

  1. (29) Logical Connectives

Goro et al. (2006) found that English-speaking adults systematically accept the sentence when the Smurf jumped over both the tree and the pond, and the other characters jumped over either the tree or the pond (but not both). This suggests that the sentence is interpreted as “everyone other than the Smurf didn’t jump over both of the obstacles,” with the Boolean “not both” interpretation of the conjunction and. Goro et al. further observed that English-speaking 4-year-olds showed exactly the same behavior: children consistently accepted the Boolean “not both” interpretations of and within sentences containing only.

Goro (2004a, 2007) employed this experimental material to investigate Japanese children’s knowledge of the conjunction … mo … mo. As we have just discussed, … mo … mo is a PPI, and is therefore forced to scope over local negation, failing to instantiate the “not both” interpretation in simple negative sentences. This property of … mo … mo parallels that of the disjunction ka in Japanese. However, children’s behavior with … mo … mo contrasts with their behavior with ka. Remember that Japanese children, at around age 5, assigned non-adult Boolean (i.e. “neither”) interpretations to ka in simple (p. 563) negative sentences. The same children, however, systematically assigned adultlike interpretations to … mo … mo in sentences such as (30) (Goro and Akiba 2004):

  1. (30) Logical Connectives

Children consistently rejected the sentence when the pig ate, for example, the carrot but not the pepper. This suggests that children interpreted the negated … mo … mo as ‘neither.’ This can be because Japanese children at this age have already learned that … mo … mo is a PPI. Another possibility is that they were assigning non-Boolean semantics to the conjunction. To tease these possibilities apart, Goro (2004a, 2007) carried out an experiment using test sentences with … mo … mo and dake ‘only.’ In one condition, the child participant was introduced to three Pokemon characters (Pikachu, Zenigame, and Hitokage) who were going to attempt to open boxes with their PSI power. There was a blue box and a black box. Pikachu successfully opened both of the boxes, and the other two opened either one of the boxes. Then the puppet presented the test sentence (31) as his description of what happened:

  1. (31) Logical Connectives

Under the adult interpretation, the conjunction … mo … mo in (31) receives the non-Boolean “not both” interpretation with respect to what everyone other than Pikachu opened. Hence the sentence is true in this condition because nobody other than Pikachu succeeded in opening both boxes. The finding was that Japanese children (at age 5) systematically accepted the test sentence, just like adult controls. This result suggests that Japanese children identify … mo … mo as a Boolean conjunction, and demonstrates that neither children’s lexical semantics nor the scope property of … mo … mo deviate from adult knowledge. It remains mysterious, however, how exactly Japanese children had avoided the hypothesis that … mo … mo is a non-Boolean conjunction. Since natural languages do allow non-Boolean conjunctions (e.g. Hungarian és), the hypothesis cannot be ruled out by turning to some innate restriction on possible semantics of conjunction. This is an open issue for future research.

23.7 Conclusion

In this chapter, I have tried to sketch out the results of recent studies on the acquisition of logical connectives in natural language. My main goal is to show how developmental psycholinguistics informed by theoretical linguistics has uncovered preschool children’s knowledge of logical connectives. There are two important theoretical grounds that (p. 564) greatly advanced developmental research. One is the dissociation of pragmatic implicature from the semantics of disjunction, and the other is the model of semantic interaction between a logical connective and another logical word in the same sentence. Studies that incorporated these perspectives have revealed that preschool children’s knowledge of logical connectives, especially of disjunction, had been underestimated, and children in fact have quite sophisticated semantic knowledge of the expression. Furthermore, theoretical studies on cross-linguistic variations of logical connectives motivated cross-linguistic studies on the acquisition of connectives, which resulted in finding otherwise unexpected universality in child language. These results illustrate the importance of combining experimental studies on children’s knowledge with formal theories of adult language. I hope that our understanding of child development and adult knowledge of logical connectives advance in tandem in the future.

Notes:

(1) Although the literature often claims that the narrow scope interpretation of some is impossible, there seem to be nontrivial numbers of native speakers who find the reading just fine. This may possibly be due to a lexical variation among speakers: for those speakers who allow the narrow scope interpretation of some in simple negative sentences, some is not a PPI.

(2) See Jing (2008) for an alternative view on Mandarin disjunction.