1.1. Theoretical background

One issue that is actively discussed in contemporary phonological studies is whether or not phonological systems can count only up to two. The predominant view in the generative literature has been that linguistic systems, including phonological systems, may count up to two but not more (e.g., Goldsmith, 1976; Hewitt & Prince, 1989; Ito & Mester, 2003; McCarthy & Prince, 1986; Myers, 1997, among many others). This view is succinctly summarized by the following quote from McCarthy & Prince (1986: 1, quoted from the 1996 version):

Consider first the role of counting in grammar. How long may a count run? General considerations of locality, now the common currency in all areas of linguistic thought, suggest that the answer is probably ‘up to two’: a rule may fix on one specified element and examine a structurally adjacent element and no other.

McCarthy and Prince (1986) claim, for example, that no reduplicative patterns copy three segments—i.e., [bad-badupi] versus [bla-bladupi] versus [adu-adupi]: They argue that this is a pattern that is predicted to arise if phonological systems can refer to three segments.

A similar view was reiterated by Ito and Mester (2003)—one of the main inspirations of the current study—who proposed to capture dissimilation effects in terms of local self-conjunction of markedness constraints (Smolensky, 1995, 1997; see also Alderete, 1997, and Blust, 2012, for related proposals). In their view, a dissimilation force against two instances of the same structure [A] is modeled as resulting from a self-conjoined version of the markedness constraint prohibiting [A] within a particular domain, i.e., *[A]&*[A]_domain. Since Ito and Mester (2003) take local conjunction to be a recursive operation, they raise the concern that the theory might predict a constraint prohibiting three instances of a particular structure. They doubt that this actually happens in the phonology of natural languages, stating that:

With local conjunction as a recursive operation, ternary (and higher) conjunction such as (No-φ&_δNo-φ)&_δNo-φ = No-φ²&_δNo-φ = No-φ³_δ are formally derivable. In the example given, the third violation of No-φ would be the fatal one. No convincing evidence has been found so far that No-φ³ is ever linguistically operative separate from No-φ², which tends to support the old idea in generative linguistics (cf. syntactic movement theory) that the genuine contrast in grammars is not “1 vs. 2 vs. 3 vs. 4 vs….”, but “1 vs. greater than 1 (p. 265).” [note by SK and GK: φ is a variable representing a phonological structure and δ is a variable representing a domain]

In other words, Ito and Mester (2003) argue that constraints that prohibit the co-occurrence of two tokens of the same segment/feature (=“No-φ²”) are omnipresent in natural languages, but that constraints that prohibit the presence of three tokens (=“No-φ³”) are unattested.

The hypothesis that phonology only counts to two, however, was recently challenged by Paster (2019) in an article titled “Phonology counts.” Paster (2019) argues, for example, that H-tones can spread twice (ternary H spreading), and likewise, H-tones can be displaced two moras to the right (ternary H displacement). In addition to these show-case examples, Paster (2019) adduces several other cases in which the phonological system apparently counts beyond two.

This question regarding whether phonological systems can count is also recently addressed in the context of counting cumulativity (Jäger, 2007; Jäger & Rosenbach, 2006), in which the numbers of constraint violations appear to additively affect phonological patterns. Some recent studies, in particular Hayes (2022), have proposed to take a linguistic scale—e.g., propensity to undergo vowel harmony in Hungarian—as a scale with actual numeric values and use these values to model various probabilistic phonological patterns (see also Breiss, 2020; Kawahara, 2020; McPherson & Hayes, 2016; Smith & Pater, 2020; Zuraw & Hayes, 2017, for related proposals). In this view, linguistic systems can literally count the numbers of constraint violations and link those constraint violations to the predicted probabilities of the relevant output candidates. One widely used model to achieve this link is MaxEnt Harmonic Grammar (Goldwater & Johnson, 2003; Hayes & Wilson, 2008; Smolensky, 1986), in which the numbers of weighted constraint violations are summed up to calculate the predicted probabilities of output candidates.¹

Inspired by this debate, the current study addressed this general question about the (in)capability of counting by studying Rendaku and Lyman’s Law in Japanese. Rendaku is a process in which the morpheme-initial voiceless obstruent of a second member of a compound becomes voiced. Lyman’s Law reduces the applicability of Rendaku by prohibiting morphemes with two voiced obstruents (Ito & Mester, 1986, 2003; Lyman, 1894). Two experiments were conducted in order to explore whether Lyman’s Law is able to count beyond two or not.

1.2. Background on Rendaku and Lyman’s Law

The two experiments reported below make use of Rendaku and Lyman’s Law to address the general question regarding the possibility of counting in phonological systems. In this subsection, we briefly review some background information on Rendaku and Lyman’s Law. Rendaku is a morphophonological process in Japanese, in which the morpheme-initial obstruent of the second element (henceforth, E2) in a compound undergoes voicing, as in (1).² Rendaku is blocked when E2 already contains a voiced obstruent, as in (2). The second generalization is known as Lyman’s Law after Lyman (1894).

Patterns of Rendaku are not as simple as the examples in (1) and (2) would appear to suggest, since various factors, both linguistic and idiosyncratic, affect the applicability of Rendaku (e.g., Kawahara, 2015; Rosen, 2016; Vance, 2014, 2016, and especially Vance 2022). For example, for some items, the application of Rendaku is optional; e.g., both [kaɾa+seki] (without Rendaku) and [kaɾa+zeki] (with Rendaku) ‘dry cough’ are attested forms, and there is some non-negligible degree of inter-speaker variability as well (see especially Vance, 2022: Section 7.7 and references cited therein). There are lexical items like [kemuɾi] ‘smoke’ and [saki] ‘point,’ which never undergo Rendaku, despite the fact that there are no (known) reasons for them not to undergo Rendaku. Thus, there is a lot to be said about idiosyncratic properties of Rendaku.

However, one important aspect of Rendaku that we would like to highlight at this point is that there are also good reasons to consider it to be a (semi-productive) phonological pattern (Kawahara, 2015). For example, it interacts with a phonological restriction such as OCP (labial), a constraint that prohibits two labial constraints in proximity, as well as with OCP (voice) (i.e., Lyman’s Law), which prohibits two voiced obstruents within the same morpheme. See also Kobayashi, Sugioka, and Ito (2014) for evidence based on ERP patterns that Rendaku is a ruled-governed process.

Another aspect of Rendaku that we would like to make clear at this point is that when we run nonce word experiments on Rendaku, the results invariably show that Rendaku is semi-productive and that there is a rather large between-speaker variability (Kawahara, 2012; Kawahara & Sano, 2014a; Vance, 1979, 1980). Even when we use nonce words that do not contain any factor that would block Rendaku, not all speakers apply Rendaku 100% of the time, which is likely due to the fact that Rendaku is not fully productive in the contemporary Japanese, as we reviewed above. For instance, Kawahara and Sano (2014a) found that nonce words that do not violate Lyman’s Law undergo Rendaku about 60% of the time on average. How often native speakers apply Rendaku to nonce words show some variation, and the source of such inter-speaker variability remains a mystery to date. This variation does not mean, however, that Rendaku is a random, unpredictable process: The influences of phonological factors—such as the effects of Lyman’s Law and the avoidance of identical segments/moras—become clearly visible in nonce word experimentation, suggesting that Rendaku shows systematicity.

1.3. Direct motivations of the current study

Now moving on to the more direct motivations of the current study, in addition to the concern raised by Ito and Mester (2003), which is quoted above in Section 1.1, another study which directly motivated our current study is the recent claim about Rendaku and Lyman’s Law, namely that two nasal consonants seem to block Rendaku. Kim (2020) has argued, based on the analysis of the Corpus of Spontaneous Japanese (Maekawa, 2004), that no forms that contain two nasals (e.g., [hanami] ‘cherry watching’) undergo Rendaku. After excluding those forms whose Rendaku would be blocked for independent reasons, Kim (2020) found that there were 1,586 tokens and that about 40% of them showed Rendaku in that corpus. On the other hand, in the same corpus, there were 55 tokens which contain two nasals, and none of them underwent Rendaku.

Kumagai (2017) reports a nonce-word judgment study, which shows that nonce words which contain two nasals (e.g., [hanama]) were less likely to undergo Rendaku than those which contain just one nasal (e.g., [çimasa]). In that experiment, 133 native speakers of Japanese judged whether each nonce word should undergo Rendaku or not. The control condition showed about 60% of Rendaku application, whereas the target condition, in which /h/ was followed by two nasal consonants, showed only 43% of Rendaku application. These observations, if correct, imply that Japanese phonology disfavors a configuration in which a voiced obstruent is followed by two nasals, a statement which seems to require counting three segments (i.e., *[D… N… N]). Kim (2020) proposes a mechanism within a MaxEnt Harmonic Grammar in which the numbers of violations of Lyman’s Law are scaled to account for the blocking of Rendaku by two nasals, assuming that nasals contribute to the violations of Lyman’s Law. In short, this observation implies that Lyman’s Law can count three segments. We thus aimed to examine this general possibility that Lyman’s Law can count beyond two in further detail via experimentation.

2.1. Introduction

Since whether nasals contribute to the violations of Lyman’s Law is at best a controversial assumption (e.g., Ito & Mester, 1986; Mester & Ito, 1989; Rice, 1993), Experiment 1 more directly addressed the possibility that a constraint can count three segments by testing whether Lyman’s Law distinguishes words containing three voiced obstruents ([D… D… D]) from those containing two voiced obstruents ([D… D]), where “D” stands for a voiced obstruent. While Lyman’s Law more-or-less categorically blocks Rendaku in real Japanese words (Vance, 2015), the blockage of Rendaku by Lyman’s Law is only probabilistic in nonce words (Vance, 1979, 1980). Experiment 1 took advantage of this nature of Lyman’s Law to address the question of counting in phonological systems.

To preview the results, we did not obtain strong evidence that Japanese speakers distinguish words containing three voiced obstruents ([D… D… D]) from those containing two voiced obstruents ([D… D]). In light of this result, Experiment 2 re-examined the claim that two nasals reduce the applicability of Rendaku (Kim, 2020; Kumagai, 2017).

2.3. Results

Figure 1 shows the Rendaku application rate for each condition in the form of violin plots, whose width represent normalized probability distributions of the responses. Each panel shows a different segment type. Within each panel, each violin shows the three conditions with different numbers of voiced obstruents (0, 1, 2 from left to right). Transparent blue circles represent average response values from each speaker (horizontally jittered). Solid red diamonds represent the average of each condition. Abstracting away from segmental differences, the three voicing conditions resulted in the following Rendaku application rates: (1) 57.8%, (2) 30.8%, (3) 33.0%.⁸

Figure 1

The results of Experiment 1.

We observe that the first condition (no violations of Lyman’s Law) differs from the second and the third conditions (violations of Lyman’s Law). This overall result is in line with previous experimental studies of Rendaku and Lyman’s Law, providing further support for the psychological reality of Lyman’s Law (Kawahara, 2012; Kawahara & Sano, 2014a; Vance, 1979, 1980). On the other hand, no apparent differences were observed between the second and the third conditions—Rendaku was no less likely to be observed if it resulted in three voiced obstruents compared to when it resulted in two voiced obstruents. If anything, the third condition overall showed higher Rendaku rate than the second condition. This result is compatible with the formulation of Lyman’s Law by Ito and Mester (2003), but not with a general idea advanced by Kim (2020) (though we hasten to add at this point that Kim’s claim is not based on the number of voiced obstruents).

The model summary of the Bayesian mixed effects logistic regression analysis appears in Table 2. For the sound type (=the coefficients in (b)), /h/ serves as the baseline. All of the relevant 95% CrIs for the coefficients in (b) include 0, suggesting that differences among the four segment types were not very meaningful, although /t/ and /k/ were slightly more likely to undergo Rendaku compared to /h/. None of the interaction terms (=the coefficients in (d)) appear to be meaningful either, suggesting that the effects of voiced obstruents do not differ substantially among different consonant types, though the first interaction term shows that the effects of Lyman’s Law were slightly less pronounced for /k/ than for /h/.

More relevant to the main aim of the experiment are the effects of voiced obstruents (=the coefficients in (c)). The difference between the no voiced obstruent condition and the one voiced obstruent condition is highly meaningful, suggesting that Lyman’s Law reduced Rendaku applicability. In fact, all the posterior samples for this β-coefficient were positive (p(β > 0) = 1). The difference between the one voiced obstruent and the two voiced obstruent condition does not seem credible, however. For this comparison, we examined what proportion of posterior samples were negative, because if anything, we expected that Rendaku might be less likely to apply when it resulted in three voiced obstruents (à la Kim, 2020 and Kumagai, 2017). Only 49% of the posterior samples of this β-coefficient were negative (p(β < 0) = 0.49).

Since the difference between the one voiced obstruent condition and the two voiced obstruent condition was not apparent, we took advantage of a Bayesian analysis to explore to what extent we can believe in “the null effect” for this difference. To do so, we deployed an analysis using ROPE (Region of Practical Equivalence: Kruschke & Liddell, 2018; Makowski, Ben-Shachar, Chen, & Lüdecke, 2019). The basic idea is that we define a range that is equivalent to a point estimate—here β = 0—and examine how many posterior samples are contained in that region, a region that can be considered to be equivalent to 0 for practical purposes. Following Makowski et al. (2019), we take 0.1—a negligible effect size according to Cohen (1988)—of a standardized parameter to define that ROPE. In logistic regression models, this ROPE ranges from [–0.18, 0.18]. We used bayestestR (Makowski, Lüdecke, Ben-Shachar, Wilson, Bürkner, Mahr, Singmann, Gronau, & Crawley, 2020) to calculate how many posterior samples are contained in this ROPE. This analysis shows that 55.8% of the posterior samples within the 95% Credible Intervals were contained in this ROPE. In other words, we can be about 56% certain that there are no differences between the two conditions.

Finally, the associate editor pointed out that there seems to be some substantial between-speaker variability in Figure 1. Such inter-speaker variability was to be expected given that Rendaku shows some between-speaker variability in real words, and that such variation has been observed in previous nonce-word experiments, starting with the seminal experimental work by Vance (1979, 1980) (see Section 1.2). To further explore the patterns of inter-speaker variation in the current experiment, Figure 2 compares the first condition and the second condition (the left panel) and the second condition and the third condition (the right panel) for each speaker. Each dot represents the Rendaku applicability rate for each speaker. In the left panel, we observe that most if not all dots are below the diagonal axis, suggesting that most speakers applied Rendaku more often when it does not violate Lyman’s Law, again supporting the psychological reality of Lyman’s Law. The right panel shows that there is no clear systematicity with respect to whether Rendaku is more likely to apply when it results in two voiced obstruents or three voiced obstruents.

Figure 2

Comparing the Rendaku application rate of the different conditions for each speaker in Experiment 1. Each dot represents each speaker, which are slightly jittered for the sake of illustration.

2.4. Discussion

The specific question we addressed in Experiment 1 is whether or not Lyman’s Law counts the number of voiced obstruents, i.e., whether it distinguishes forms with two voiced obstruents from those with three voiced obstruents. A short answer is that it apparently does not. While we were unable to prove “the null effect,” no convincing evidence was obtained that Lyman’s Law counts beyond two. The results are compatible with the remark by Ito and Mester (2003), which we quoted in Section 1.1, as well as the general view reviewed in that section that phonological systems do not count beyond two (Goldsmith, 1976; Hewitt & Prince, 1989; Ito & Mester, 2003; McCarthy & Prince, 1986; Myers, 1997).

From the perspective of Optimality Theory (Prince & Smolensky, 1993/2004), we can interpret the current results as suggesting that, regardless of whether a morpheme contains two voiced obstruents or three voiced obstruents, the constraint behind Lyman’s Law is violated to an equal degree. For example, this constraint can assign a violation mark for every morpheme that contains more than one voiced obstruent, rather than assigning a violation mark for each pair of voiced obstruents. The latter formulation is assumed by Kim (2020) and Ito and Mester (2003), the latter of whom state that “[f]or C₁&_δC₁, the special case of self-conjunction with C₁ = C₂, this implies that a candidate receives a violation mark for each pair of violation marks (*C₁, *C₁) it has accrued for C₁ in domain δ” (p. 23, emphasis ours). The current experiment seems to suggest that instead, it is a domain (i.e., morpheme) that receive a violation mark in this case. This is compatible with the definition of local conjunction that Moreton and Smolensky (2002) give: “The local conjunction of C₁ and C₂ in D, is a constraint which is violated whenever there is a domain of type D in which both C₁ and C₂ are violated” (p. 306, emphasis in the original).

At this point, we note that our study is specifically about how Lyman’s Law behaves with respect to the number of voiced obstruents—it may as well be the case that Lyman’s Law counts only up to two, but that other phonological systems are able to count beyond two (Paster, 2019). We will come back to this general issue in the conclusion section.

A question that arises given the current results is how we should reconcile the current results with one of the direct motivations of the current study—the observation that two nasals seem to block Rendaku (Kim, 2020; Kumagai, 2017). One possibility is that this observation was actually epiphenomenal. Inspection of the actual examples used by Kim (2020) shows that many of the E2s are actually compounds.⁹ For example, [hanami] “cherry watching” consists of [hana] “flower/cherry” and [mi] “watching.” Other examples of this kind include [kami-no-ke] ‘(lit.) head’s hair’ and [tate-mono] ‘(lit.) built things.’ Since it is independently known that Rendaku applies only to the elements on right branches of compounds (Ito & Mester, 1986; Otsu, 1980), these examples may be explained away in terms of this independently motivated restriction. Other examples include those complex E2s whose left member already contain a voiced obstruent (e.g., [tabe-mono] ‘food’ and [hidaɾi-mimi] ‘left ear’), and Rendaku in such examples should be blocked by that voiced obstruent, not necessarily by the two nasals. Some other items included in Kim’s (2020) data are actually those that can undergo Rendaku (e.g., [konomi] ‘favorite’ vs. [joɾi-gonomi] ‘pick and choose’ and [tanomi] ‘plea’ vs. [kami-danomi] ‘plea to a god’), although non-Rendaku forms may have appeared in the corpus.

These alternative explanations, however, do not provide an explanation for the experimental finding by Kumagai (2017), because that experiment made use of monomorphemic nonce words as E2s. One issue that can be raised about the experiment by Kumagai (2017), however, is that it had only three items for each condition, and thus the generalizability of his findings can be questioned. In light of the results of Experiment 1, we feel that it is necessary to reexamine Kumagai’s (2017) experimental finding by expanding the number of items tested. Experiment 2 takes up on this task.

3.1. Introduction

Experiment 1 found that two voiced obstruents and three voiced obstruents are treated alike for the calculation of Lyman’s Law, which means Lyman’s Law seems to count only up to two. Given this result, the next experiment was designed to re-examine the claim that two nasal consonants may trigger Lyman’s Law and inhibit Rendaku (Kim, 2020; Kumagai, 2017). Recall that many examples used by Kim (2020) can potentially be explained away in terms of other independently motivated restrictions on Rendaku, and that Kumagai’s (2017) experiment had only three items for each condition.

There are independent reasons to test—more robustly than Kumagai (2017) did—the possibility that two nasals can block Rendaku in Japanese. Specifically, the [voice] specifications of sonorant consonants in Japanese have been known to be ambivalent. On the one hand, the standard view about the role of sonorants in triggering Lyman’s Law is that they do not, and there have been several attempts to model this observation. The inertness of sonorant voicing with respect to Lyman’s Law has been modeled by using the underspecification theory (Ito & Mester, 1986), by positing a privative [voice] feature that is specific to obstruents (Mester & Ito, 1989), or by positing different [voice] features for sonorants and obstruents (Rice, 1993). See Kawahara and Zamma (2016) for a review of these proposals.

On the other hand, there is some evidence that sonorants, especially nasals, are specified for [voice] in Japanese phonology. The clearest evidence comes from the fact that nasals trigger voicing of following voiceless consonants, as observed in the past tense formation (e.g., /kam-ta/ → [kan-da] ‘bite + PAST’), which seems to suggest that moraic nasals in Japanese are specified for [+voice] (Ito, Mester, & Padgett, 1995; Rice, 1993).¹⁰ An analysis of half rhymes in Japanese rap lyrics likewise shows that sonorant consonants are more likely to rhyme with voiced obstruents than with voiceless obstruents (Kawahara, 2007), and the same generalization holds in the pairing patterns of imperfect puns (Kawahara & Shinohara, 2009), although these studies argue that these pairing patterns are based on perceptual similarity rather than phonological similarity. In short, there are some ways in which nasals—and perhaps sonorants in general—could be interpreted as being specified as [+voice] in Japanese, and it would be interesting to test whether this feature can trigger Lyman’s Law, especially when there are two instances of nasals/sonorants.

3.3. Results

Figure 3 shows the Rendaku application rate for each condition in the form of violin plots, whose width represent normalized probability distributions. Transparent blue circles represent average response values from each speaker (horizontally jittered). Solid red diamonds represent the average in each condition. The seven phonological conditions resulted in the following Rendaku application rates: (a) [h-vls-vls] = 43.6%; (b) [h-nas-vls] = 43.8%; (c) [h-nas-nas] = 40.2%; (d) [h-ɾ-vls] = 45.0%; (e) [h-ɾ-ɾ] = 44.9%; (f) [h-App-vls] = 43.5%; (g) [h-App-App] = 38.0%.

Figure 3

The results of Experiment 2.

Overall, the effects of phonological compositions of the stimuli were not very apparent.¹² The critical condition, which contained two nasal consonants, showed 3.4% reduction in Rendaku responses compared to the baseline condition. The conditions which contained one sonorant, whether it was a nasal, [ɾ], or an approximant, did not show any substantial reduction in Rendaku responses. The clearest case was the stimuli with two approximants, which showed the reduction in Rendaku responses by 5.6% compared to the baseline condition.

The model summary of a Bayesian mixed effects model is shown in Table 4. As observed in the table, the condition with two approximants is the only condition whose 95% CrI does not include 0. Since we did observe some reduction in Rendaku applicability for the condition with two nasals, we calculated the proportions of posterior samples that are negative for this β-coefficient, and found that 90.6% of them were negative. If we take the conservative measure and assume that the lower edge of the ROPE (i.e., –0.18) should define the critical region, then only 64.9% of the posterior samples are below –0.18. This result suggests that we can only be 65% confident that two nasals lower Rendaku responses to a non-negligible degree. We conclude that the evidence for the probabilistic blocking of Rendaku by two nasals is weak or at best moderate.¹³ We can also conclude that there is no strong evidence that [+voice] feature of nasals, to the extent that Japanese nasal consonants are specified as such, trigger Lyman’s Law either, regardless of whether nasals occur once or twice.

As with Experiment 1, we observe some inter-speaker variability, and thus Figure 4 compares the control condition (forms with two voiceless obstruents) and the two nasal condition. There does not seem to be systematic patterns which suggest a clear difference between the two conditions.

Figure 4

The comparison between the control condition and the two nasal condition by each speaker.

3.4. Discussion

This experiment was set out to re-examine the previous claim that two nasals may block Rendaku. The results show, however, that the evidence for this blockage effect was weak or at best moderate. Comparing the current results with those of Kumagai (2017), the crucial items used in the latter experiment were [hanama], [çinama], and [ϕunama], which all end with [nama]. The current stimuli contained [hanama], and therefore, as a post-hoc comparison, we compared [hanama] and other items. Indeed, [hanama] showed slightly lower Rendaku responses than other items in the same condition: 38.5% vs. 40.6%. The blockage of Rendaku may have something to do with that specific [nama] sequence, but does not seem to generalize to other items containing two nasals.

On the other hand, the condition with two approximants showed reduction in Rendaku rates to a degree that can be considered credible. We find this result puzzling. We know of no good reason why approximants, to the exclusion of nasals or [ɾ]s, interact with a voiced obstruent in the calculation of Lyman’s Law in Japanese phonology. If anything, the [voice] specification is more clearly motivated for nasals than for approximants, as the former arguably triggers post-nasal voicing in Japanese (Ito et al., 1995, though see Hayashi & Iverson, 1998, and Vance, 1991).