Symmetry in category systems across languages

Domain: physics
Published: 2026-01-14
Editorial outcome: minor_revision

Abstract
Language reflects how people organize experience into categories, and cross-linguistic comparison can help to identify general principles that shape categorization. Here we argue that symmetry is one such principle, and present a symmetry-based theory that predicts whether category systems for a given domain tend to include an even or an odd number of categories. We test the theory against cross-linguistic data previously compiled for a range of domains and find that deictic day-naming and tense-marking systems tend to have an odd number of categories, but that systems for domains including seasons, phases of the moon, kinship, and cardinal directions tend to have an even number of categories. Our results therefore provide evidence of the widespread influence of symmetry on categorization across languages and domains.

Introduction
Anyone familiar with two or more languages will know that terms for kintypes, colors, seasons, animals, and many other domains often do not translate perfectly across languages. Despite this variation, scholars have identified universal principles that appear to shape categorization across cultures. Named categories tend to be useful for communication1,2 and relatively easy to learn and remember3,4, and properties that support these goals include perceptual salience5, convexity6, compositionality7 and hierarchical organization3. We propose that symmetry is another such property, and show that the principle of symmetry supports predictions about whether category systems across domains tend to have an even or an odd number of categories.

Symmetry is evident in day-naming systems that treat the future (e.g., tomorrow) similarly to the past (yesterday), in season-naming systems that give balanced treatment to the hottest (summer) and coldest (winter) periods of the year, and in directional systems that label both ends of an axis (e.g., east and west). Many such symmetries reflect symmetries in the world, such as predictable seasonal cycles, but they also suggest a cognitive inclination towards symmetry. Symmetric category systems are compressible8, which makes them relatively simple2 and thus relatively easy to learn and remember. Previous researchers have proposed that category systems are shaped by evolutionary pressures towards simplicity2,8,9, and our proposed symmetry preference can be seen as a special case of this more general proposal.

Our work is related to previous investigations of the role of symmetry in perception and cognition10–13, and in particular to previous approaches that characterize the simplicity or goodness of a category in terms of structural properties such as symmetry14. Most work in this tradition focuses on visual objects such as patterns of dots, but ideas about symmetry and invariance have also been used to characterize the goodness of categories defined in terms of abstract and binary features15. We go beyond previous psychological work on symmetry and categorization by considering the symmetry of entire systems of categories, and by using symmetry to make predictions about categorization across languages.

Our work also builds on and extends the structuralist approach in anthropology, which has been applied to domains including food, mythology, and kinship. Levi-Strauss16(p2) describes structuralism as the “quest for the invariant, or for the invariant elements among superficial differences,” and invariance is closely related to symmetry. A standard idea in group theory, the mathematical theory of symmetry, is that symmetry can be defined as invariance with respect to a set of transformations, and our theory makes use of this definition. Our approach is therefore highly compatible with structuralism, but we go beyond prior structuralist approaches by evaluating our theory against cross-linguistic data from a diverse set of domains. The structuralist tradition has been criticized for placing more emphasis on theory than data17, but our work highlights the fact that structuralist approaches can make testable empirical predictions.

Each domain we consider is represented by a structure, and four examples are shown in Fig. 1. A structure includes a space along with a set of transformations defined over the space. We introduce our theory using the linear structure as an example, and the remaining three structures are introduced in subsequent sections.

The linear structure includes a one-dimensional space and is appropriate for systems of temperature terms (e.g., English cold, cool, warm and hot), where the underlying dimension is temperature18. Each structure may be relevant to multiple semantic domains, and the linear structure is also appropriate for systems of life-stage terms (e.g., English infant, toddler, child, teenager, adult), where the underlying dimension is age19.

A category system organizes the points in the space into categories, and Fig. 1 shows examples that partition the one-dimensional space into two, three, and four categories. All of these categories are connected, and we will assume that any category within a connected space must also be connected. For example, we do not allow a category that includes both ends of the one-dimensional space but excludes the central region.

All of the example systems in Fig. 1 are also symmetric, where symmetry is defined in terms of the transformations shown. For the linear structure in Fig. 1, there is a single transformation that inverts the one-dimensional space so that the left endpoint maps to the right endpoint. A category system is symmetric if it is invariant under all transformations associated with a structure. For example, inverting the first example system shown for the linear structure affects the category labels (‘purple’ is now on the left rather than the right), but the resulting system is isomorphic to the original. A more formal definition of symmetry is provided in the Methods.

We will say that a structure is odd if all symmetric category systems defined over that structure have an odd number of categories. Similarly, the structure is even if all symmetric category systems (except the system that assigns all points to the same category) have an even number of categories. If the symmetric systems for a structure include both odd and even examples (as is the case for the linear structure in Fig. 1), we will say that the structure has both parities.

Our core theoretical proposal is that human languages show a preference for symmetric category systems. If so, then systems for domains with an odd structure should tend to have an odd number of categories, and attested systems for even domains should tend to have an even number of categories. If a structure has both parities (such as the linear structure in Fig. 1), then our theory suggests that both odd and even systems are expected, and by default should have relatively similar frequencies. We test these predictions here using a diverse collection of cross-linguistic datasets that were mostly compiled by previous scholars and are archived at Zenodo20. We initially focus on domains for which our theory predicts a tendency toward either even or odd systems, and test the prediction that domains with a bisected linear structure show a preference for odd-numbered systems, and that domains with bisected circular or binary feature structures show a preference for even-numbered systems. A subsequent section considers two control domains that are expected to show no preference toward either even or odd systems.

Results
A bisected linear structure includes a one-dimensional space bisected by a privileged central point that must be assigned to a category. Figure 1 shows that the single transformation associated with the structure inverts the dimension about its central point. A bisected structure is similar to a linear structure, and the key difference between the two is whether the one-dimensional space has a privileged central point. We consider two temporal domains that are instances of the bisected linear structure. In both cases, the dimension is time, and the privileged point represents the present, or the point that separates past from future. The first domain is deictic day-naming: for example, English includes the terms “yesterday,” “today,” and “tomorrow,” and many languages also have terms for the day before yesterday and the day after tomorrow. The second domain is tense marking: for example, English distinguishes between past, present and future using inflectional morphology (e.g., “she walked”, “she walks”) and periphrasis ("she will walk”).

In both of these temporal domains, symmetric systems must have an odd number of categories. Intuitively, a symmetric system will have one category that includes the present, and each category for the past will be mirrored by a corresponding category for the future, which leads to an odd number of categories overall.

Deictic day names
We analyzed a set of 157 day-naming systems compiled by Tent21. Distributions of these systems across areas and language families are reported in Tables S1, S2 of the Supplementary Methods. The distribution over the sizes of these systems is shown in Fig. 2a, and counts for odd and even systems are shown in the same panel. The most common system has a size of 5 and picks out two days on either side of ‘today’, and systems of size 7 are also relatively common. The sawtooth shape of the size distribution (high for odd sizes and lower for intervening even sizes) reveals a strong preference for odd-numbered systems across languages.

Tent’s day-naming data are not balanced across language families, and the size distribution in Fig. 2a does not take genetic relatedness into account. To estimate the prevalence of odd-numbered systems while controlling for genetic relatedness, we implemented an intercept-only phylogenetic regression model22 with system parity as the dependent variable. For this and other regression models reported in the paper, 95% credible intervals on the intercepts are reported in Fig. 3, and analogous results showing frequentist confidence intervals are reported in Fig. S5. We take a credible interval that excludes and lies to the left of 0.5 as strong evidence that a domain favors odd systems, and an interval that excludes and lies to the right of 0.5 as strong evidence that a domain favors even systems. In Fig. 3, the credible interval for day-naming lies to the left of 0.5, indicating that this domain shows a strong preference for odd systems.

Tense
Next, we analyzed a database of 233 tense systems compiled by Velupillai23. The size distribution in Fig. 2b shows that the most common size is three, and the credible interval in Fig. 3 excludes 0.5, suggesting a reliable preference for odd-numbered systems. The size distribution in Fig. 2b, however, is relatively smooth and does not have the sawtooth shape that provides the strongest evidence for a parity preference. Further reason for caution is provided by systems with remoteness distinctions that divide either the past or future or both into two or more categories. Remoteness distinctions are more likely to subdivide the past than the future23,24, and Velupillai23 notes that symmetric remoteness distinctions are relatively rare. Overall, then, tense-marking does show a preference for odd-numbered systems, but the role of symmetry is less apparent for tense marking than for deictic day-naming.

Seasons
Consider now the bisected circular structure shown in Fig. 1. The space for this structure is a circle that is bisected by a privileged axis, leading to two privileged points. Unlike bisected linear structures, bisected circular structures are predicted to favor systems with an even number of categories. We consider three domains with bisected circular structures: season naming, names for phases of the moon, and names for cardinal directions.

Consider a simple seasonal cycle in which a single variable (temperature, say) varies from low to high and back to low over the course of a year. The bisected circular structure in Fig. 1 can capture this seasonal progression, and the two privileged points could be the summer and winter solstices, or the two equinoxes (see Supplementary Discussion for additional discussion). The two transformations in Fig. 1 correspond to reversing the temporal order of the cycle and shifting the cycle by half a year.

The Western season system divides the circle into four arcs corresponding to winter, spring, summer, and autumn, but other systems of season categories are attested across cultures. We analyzed 53 season systems compiled by Kemp et al.25, and the size distribution for these systems (Fig. 2c) shows a sawtooth pattern that favors even systems. The credible interval for seasons in Fig. 3 excludes 0.5, indicating a robust preference for even systems. A second database of season systems was compiled by Orlove26, and 23 out of the 28 systems in this database are even. Orlove notes that odd-numbered systems are rare, but did not provide an explanation of this result. Our approach suggests that this result is a special case of a more general preference for symmetric category systems.

Moon phases
Consider now a lunar cycle that begins with a new moon, which waxes to become a full moon, then wanes to return to the start of the cycle. The two privileged points in this cycle may correspond to the new moon and the full moon, and the transformations in Fig. 1 correspond to reversing the temporal order of the cycle and shifting the cycle by half a lunar month.

English has labels for four major phases of the lunar cycle (new moon, first quarter, full moon and third quarter), but the number of labeled phases varies across cultures. We collected a set of 52 moon-phase naming systems from the Human Relations Area Files27 and a range of other sources. The distribution of system sizes is shown in Fig. 2d and reveals a preference for even systems driven primarily by a preference for systems of size four. The credible interval for moon phases, however, includes 0.5, indicating that the predicted preference for even systems does not have strong support. Some systems are odd because they use the same term for the first and third quarters, and others because they include two terms for the start of the lunar cycle (one for the new moon and another for the darkness preceding the new moon), but only one term for the midpoint of the cycle (full moon).

The lunar cycle leads to a monthly tidal cycle that includes spring tides near the new and full moons and neap tides near the first and third quarters. Many languages name parts of this monthly cycle, and our approach predicts that these monthly tide-naming systems will tend to include an even number of terms. Languages also have names for parts of the diurnal tidal cycle that moves between high and low tides every lunar day, and again, our theory predicts that these diurnal tide-naming systems will tend to include an even number of terms.

Cardinal directions
Names for seasons, moon phases, and tides label processes that unfold through time, and the bisected circular structure in Fig. 1 seems appropriate for these and other temporal cycles. The structure, however, is not limited to temporal phenomena and can also be used to capture points of the compass. In this case, the privileged points may correspond to the directions of sunrise (east) and sunset (west), and the two transformations correspond to a reflection in the east-west axis and a rotation of 180∘.

English labels four main cardinal directions (north, south, east and west), but this system is not universal. We analyzed a set of 90 cardinal directional systems from a database of Australian directional systems compiled by Hoffmann et al.28. The distribution over the sizes of these systems is shown in Fig. 2e, and reveals a strong preference for systems of size 4. There is also the hint of a sawtooth pattern, because there are more systems of sizes 2 and 4 than systems of size 3.

Kinship
All of the structures considered so far incorporate continuous spaces, but our framework can also accommodate discrete feature spaces. Feature representations have been considered for domains including personal pronouns29, indefinite pronouns30, spatial demonstratives31, and spatial adpositions32, and we consider several examples drawn from the domain of kinship.

The binary feature space in Fig. 1 represents a set of four sibling kin types defined in terms of two binary features: sex (male vs female) and relative age (older vs younger). The English kinship system ignores the relative age feature and organizes the kin types into two categories (brother and sister), but many other languages categorize these kin types differently. The two transformations in Fig. 1 exchange values of the two binary features: the first exchanges male and female, and the second exchanges older and younger. Any category system that is symmetric with respect to these transformations must have an even number of categories.

Figure 2f shows the size distribution over 1172 sibling systems derived from Kinbank33. Some of these systems include overlapping categories (e.g., younger brother and brother), and some are defined using features that go beyond the example in Fig. 1, such as a relative sex feature that indicates whether the referent of a sibling term has the same sex as the speaker34. As a result, a number of systems have sizes exceeding four. Figure 2f shows that even systems are preferred across languages, and that systems of size 3 are less common than both systems of size 2 and systems of size 4.

Figure 2 includes analogous results for five other subdomains: parents’ siblings, grandparents, grandchildren, nieces and nephews, and cousins. The features needed to represent these subdomains vary: for example, the subdomain of grandparents requires a “sex of connecting relative” feature (paternal vs maternal) that is not needed for the subdomain of siblings. All of these subdomains, however, can be represented using a collection of binary features, which mean that even systems are expected for all of these subdomains. Figure 3 indicates that all 6 kinship subdomains show a robust preference for even systems, and sawtooth patterns are evident for siblings, parents’ siblings, grandparents, and cousins. The data are therefore consistent with the prediction that even systems are typical in the domain of kinship.

Figure 1 focuses on domains with especially simple structures, but we now demonstrate that our theory has implications that go beyond these four simplest structures.

Australian Aboriginal social classification
In communities with classificatory kinship systems, rules of marriage and descent are formulated in terms of some number of social categories (also called skin names, sections, or marriage classes)35. For example, in a four-category Kariera system, individuals in category A may only marry individuals in category B. Any children produced by such a marriage will belong to category C, and can only marry individuals belonging to category D. Capturing the structure of these systems was one of the early successes of the structuralist approach to anthropology36, and Levi-Strauss37 includes an appendix demonstrating that the structures of certain classificatory systems are isomorphic to mathematical groups.

Here, we consider one simple property of these systems. Suppose that the set of possible marriage relationships is invariant under a transformation that changes the sex of each individual from male to female or vice versa. If so, then the system must organize individuals into pairs of marriageable categories (e.g., members of categories A and B may marry each other), which mean that the total number of categories must be even. Figure 2l shows the size distribution over 331 systems classified by AustKin35,38,39 as having sections, sub-sections, or moieties. All of them are even, and all have sizes that are powers of two.

Non-cardinal directional systems
We previously considered cardinal directional systems, but many languages have other kinds of directional systems, including systems based on local rivers (upstream vs downstream), local coasts (coastal vs inland), wind direction (upwind vs downwind), and topographic landmarks (uphill vs downhill). Although these locational systems do not have the bisected circular structure used in our analysis of cardinal directional systems, they can often be represented using pairs of opposing concepts, such as upstream and downstream. If these systems are symmetric with respect to transformations that exchange these pairs, then both concepts need to be named, which mean that symmetric locational systems will have an even number of categories.

Figure 2m shows the size distribution over 141 non-cardinal systems drawn from the database of Australian directional systems mentioned earlier28. We included systems based on rivers, coasts, wind direction, path of the sun, and tides, but excluded topographic systems because it was difficult to establish the sizes of these systems. Some languages include multiple non-cardinal systems, and these systems are included as separate entries in our data set. Figure 2m shows that the most common system size is 2, but overall, there is not a strong preference for even-numbered systems. Several systems label only one member of an opposition, perhaps because the other member is taken as a default. For example, Kuku Yalanji has a form ngubar for ‘otherside of the river’, but appears to have no form for ‘same side of the river’40.

Control domains
We now turn to two domains for which a strong preference for even or odd systems is not expected. The first is color. Color can be roughly modeled using a three-dimensional solid in which the axes correspond to red-green, blue-yellow, and light-dark. Psychophysical work suggests that color space has no true symmetries, but transformations that invert the three axes (e.g., that map red to green) are approximate symmetries of color space41. Even if these approximate symmetries are allowed, the Supplementary Methods show that symmetric color systems can be either even or odd, which mean that no strong parity preference is expected. Figure 2n shows the size distribution over 110 color-naming systems drawn from the World Color Survey42. The distribution is relatively smooth and is almost perfectly balanced between odd and even systems.

The second control domain is folk biology, and we consider systems of life-form terms such as bird, fish and snake. The space of living creatures has a branching structure induced by evolution, but appears to have no symmetries that should favor either odd-numbered or even-numbered category systems. Figure 2o shows the size distribution over 144 systems of life form categories compiled by Brown43. As for color, the distribution is relatively smooth and shows no preference toward either odd or even systems.

Additional control domains could be considered, including domains with the linear structure in Fig. 1 (e.g., temperature terms18 and life-stage terms19), and domains with no apparent symmetries (e.g., smell terms44). In all of these cases, we expect a relatively even balance between odd-numbered and even-numbered systems.

Methods
We begin by providing a formal definition of symmetry, then provide detailed information about the datasets analyzed for each domain.

Parity and domain structure
Let S be a space and T be a set of transformations acting on that space. Space S may be connected (as for the first three examples in Fig. 1) or not (as for the binary feature space in Fig. 1). Each transformation t in T is a bijection t: S → S from S to itself.

A category system C is a partition of the space, and can be represented as a binary relation R_C: S × S → {0,1} that indicates whether or not two items belong to the same category. If S is connected, then we assume that all categories in C are also connected. A category system is symmetric if it is left unchanged by applying each transformation in T. More formally, a symmetric system C satisfies

R_C(x,y) = R_C(t(x), t(y)) (1)

for each transformation t in T.

The approach just described can be characterized in terms of a mathematical group acting on space S. For the linear and bisected linear structures in Fig. 1, the transformation group is D_2, the dihedral group of order 2, which is equivalent to the cyclic group Z_2 of order 2. For the bisected circular and binary feature structures, the transformation group is D_4, the dihedral group of order 4.

Although we assumed that each category system C is a partition of S, the same approach applies to category systems with “holes,” or items that are not assigned to any category. Our approach can also be generalized to systems with overlapping categories, including hierarchical category systems.

Domains
For each domain, we aimed to count the number of named categories per language. Some categories have multiple names, and it is sometimes hard to tell whether two names should be treated as labels for the same category or labels for different categories. For example, the database of Hoffmann et al.28 lists Jaru terms glossed as ‘south,’ ‘to south,’ ‘south side’ and ‘south across water,’ and we treated all of these terms as labels for the single category ‘south’.

Some of our sources (especially older sources) are likely to be influenced by Western categories. For example, an author may elicit terms that correspond to the Western concepts of new moon, first quarter, full moon, and third quarter but fail to record additional terms for lunar phases. Our impression is that this issue arises more frequently for lunar phases and cardinal directional systems than for the other domains we consider, and may contribute to the finding that four is the most common size for both lunar-phase and cardinal systems.

Most of our regression analyses incorporated a phylogeny over language families, and all exceptions are noted below. Phylogenies for all domains were derived from Glottolog55, and we used Glottolog data included in version 1.1.20 of the lingtypology package in R56. Glottolog uses language identifiers called glottocodes, and we used these glottocodes to determine the language family for each language in our statistical analyses. Some of our data sets already included glottocodes, others included ISO language codes, which were mapped to glottocodes using the lingtypology package, and all remaining data sets were manually annotated with glottocodes.

Deictic day names. Tent’s database21 includes 157 day-naming systems. A glottocode could not be determined for one language (Natleba), leaving 156 for the regression analysis. Tent reports that some of these systems show dual symmetry and use the same forms for days in the past and future. For example, Komba (Papua New Guinea) uses mara-nan for ‘3 days before today’ and ‘3 days after today’, and these meanings are distinguished using tense marking. Tent21 gives Komba, Hindi, and Capanahua (Peru) as examples of systems with dual symmetry, but his dataset does not specify the full set of systems with this property. As a result, our analyses do not allow for dual symmetry, and we treat Komba as a 7-term system (i.e., a system that picks out meanings ranging from ‘3 days before today’ to ‘3 days after today’) instead of a 4-term system. Although our theory assumes that categories are connected, dual symmetry provides one example of how disconnected categories can arise for principled reasons.

Tense. Velupillai’s tense database includes 318 languages. 77 of those languages have no grammatical tense, and an additional 8 make remoteness distinctions, but the number of these distinctions was not specified, meaning that the sizes of these systems could not be determined. We excluded these 85 languages, leaving 233 for analysis. Velupillai’s classification allows for zero marking, and a category without overt expression contributes to the size of a tense system as long as it stands in paradigmatic opposition to categories that are marked.

Seasons. The seasonal cycle for a climate variable such as temperature can be captured by a function such as f: time → temperature with the form of a sine wave (Fig. 4a). We suggested earlier that the first transformation of the bisected circular structure in Fig. 1 corresponds to time reversal, but this transformation also corresponds to inverting the temperature axis.

The database of season systems compiled by Kemp et al.25 includes 54 systems, and we removed one language (Grand Valley Dani) that is reported to have no terms for seasons. Two glottocodes appear twice each in the data, and in both cases, the two systems for those glottocodes are both even. We included only one representative of each glottocode in the regression analysis, yielding 51 systems in total for that analysis.

Moon phases. The monthly lunar cycle can be captured by a function {f: time → phase} that traces out a helix over time (Fig. 4b). The phase variable ranges over a circular configuration of states, including new moon (bottom of Fig. 4b), first quarter, full moon, and third quarter. Comparing the two panels in Fig. 4 reveal that the domains of seasons and moon phases are not isomorphic. For seasons, the temperature variable corresponds to an interval, not a circle, and a circular structure only arises if we keep track of whether the temperature is rising or falling. For moon phases, the phase variable is intrinsically circular. Although the domains are not isomorphic, both can be represented by the bisected circular structure in Fig. 1.

We know of no existing database of terms for moon phases and therefore compiled our own set of 52 systems. 32 were drawn from the Human Relations Area Files27 by searching for paragraphs that were classified with the subjects “ordering of time” or “ethnometeorology”, and that included either “phase” or “phases” and “lunar” or “moon.” The remaining systems were collected by searching Google, Google Scholar and Google Books for combinations of terms and phrases such as “terms for”, “phases of” and “moon.” Some cultures name every night in a lunar cycle: for example, Kelley and Milone57 give sequences of 30 Tahitian names and 32 Maori names for nights of the moon. We did not include these cases, and the largest system in our database has 14 terms.

One culture was removed from the analysis because a glottocode could not be reliably identified. An additional two cultures appear twice in our data because different authors give different numbers of terms for these cultures. All of these counts were even, and we included each culture only once in the regression analysis, giving 49 systems overall for that analysis.

Cardinal directions. For cardinal systems, we previously noted that the two transformations in Fig. 1 correspond to a reflection in the east-west axis and a rotation of 180∘. The rotation can also be replaced by a reflection in the north-south axis, because both alternatives lead to the same transformation group.

Three glottocodes appeared twice in our set of 90 cardinal systems, and we kept one representative of each for the regression analysis. The two systems listed for one of these glottocodes (waja1257) have different parities, and we kept the odd-numbered system to be conservative with respect to our hypothesis. Roughly 80% of the 87 languages included in the regression analysis belonged to the same language family (Pama-Nyungyan), and only two other families were represented more than twice in the data set. We therefore ran a standard logistic regression rather than a phylogenetic regression for cardinal directional systems.

Kinship. We defined the six subdomains of kinship using the core kin types in Table S2 of Passmore et al.33. For siblings, we included 12 kin types: mB and mZ (which specify the brother and sister of a male speaker), meB and meZ (which specify the elder brother and sister of a male speaker), myB and myZ (which specify the younger brother and sister of a male speaker), and the six analogous kin types for a female speaker. We deliberately dropped mG and fG (sibling of a male and sibling of a female) because we are interested in proper subsets of each subdomain. Lists of the kin types included for the remaining five subdomains are provided in Table S3.

Given a list of kin types, we identified all terms in a language that refer to at least one of these kin types. Kinbank includes many dialectal and orthographic variants, and we wanted to group all terms with the same extensions. We therefore counted all terms that could be used to refer to the same kin types as a single category. For example, three different terms that can each be used to refer to both meB and feB would be counted as a single category corresponding to “elder brother.”

Some languages appear several times in Kinbank with entries attributed to different sources. We kept all of these systems and included a random effect for language in the phylogenetic regression analysis. The number of systems for each subdomain was 1172 (siblings), 1106 (parents’ siblings), 869 (grandparents), 449 (grandchildren), 795 (nieces and nephews) and 880 (cousins). The number of languages represented by these systems was 1081 (siblings), 1018 (parents’ siblings), 808 (grandparents), 419 (grandchildren), 724 (nieces and nephews) and 803 (cousins).

As a robustness check, we computed distributions over system sizes for kinship systems derived from a second dataset compiled by Murdock58. The results are reported in Fig. S1, and are similar to the kinship results shown in Fig. 2.

Australian Aboriginal social classification. Data for Australian social classification were derived from the AustKin website38, and our work is consistent with the conditions of use at http://www.austkin.net/index.php?loc=disclaimer. AustKin includes information about ten types of social systems, and we included all of these types except for phratries and totemic groups. The eight types included were sections, subsections, underspecified sections, matri-moieties, patri-moieties, generational moieties, matri-semi-moieties, and patri-semi-moieties. Multiple systems are specified for some languages: for example, the Alyawarr data includes systems of type sections based on four different sources, and an additional system of subsections. Figure 2l is based on a set of 331 systems that includes only the largest system for each language. Because all systems were even, we did not run a regression analysis for this domain.

White59 demonstrates that any society that allows sister exchange or that allows patrilateral cross-cousins to marry must have an even number of categories. Systems with odd-numbered sizes, however, are theoretically possible, and it is reported that the Purum of India have a five-category system60.

Non-cardinal directional systems. The database of Hoffmann et al.28 distinguishes between multiple types of non-cardinal systems, and we included those with the following types: sun only, wind only, wind and climate, climate/weather, coast and wind, coast only, sun coast and river, river and sun, river only, tide, spiritual/cultural places of significance, or other. We excluded systems based purely on topographic landmarks (topographic elevation only) or vertical elevation (vertical elevation only), along with mixed systems that incorporated topographic landmarks or vertical elevation as a component (e.g., vertical and river and topographic elevation and river). When attempting to determine the sizes of these excluded systems, we often found it difficult to tell whether two terms should be treated as labels for the same concept. For example, the vertical/topographic system of Yadhayekenu includes akinta ‘up’, anpaantu ‘upwards’ and anpaa/anpima ‘above’, and we were unsure whether to treat these terms as labels for one, two or three categories. Because the data set includes multiple non-cardinal systems for some languages, the regression analysis for this domain included a random effect for language. As for our analysis of cardinal systems, however, we used a standard regression rather than a phylogenetic regression because the data are dominated by languages from a single family. The analysis included 141 systems drawn from 82 different languages.

Color. We used the version of the World Color Survey released in the wcsR package61. The database includes labels of color chips provided by individual participants, and for each language, we identified all terms that were provided by at least 60% of participants for at least one chip. The number of such terms was taken as the size of the color-naming system for that language.

Folk biology. For each language appearing in Appendix B of Brown43, we counted the number of entries under the heading “life-form terms.” Brown’s comments on each language sometimes suggest that one or more of the terms listed is not a true life-form term. For example, he lists three terms for Iwaidja (Australia), but suggests that they are labels for “incipient life-form classes” rather than full-fledged life-form terms. We ignored these distinctions for the purposes of our analysis.

Brown’s work has been criticized for relying on a notion of life-form that does not apply across languages, and that leaves out “a large number of highly inclusive categories of intellectual, communicative, ecological, and economic significance”(62,63, p344). These concerns threaten some of the conclusions that Brown draws from his data, but seem less relevant to our finding that life-form systems show no systematic tendency toward even or odd sizes.

Regression models
The phylogenetic regressions used phylogenies derived from Glottolog55 and were implemented using the brms package in R64. All priors were set to the default used by the brms package, which is a half student-t prior with 3 degrees of freedom, a location of 0 and a scale of 2.5.

For each domain, we started by running four Markov chains for 3000 iterations each, including 1500 burn-in iterations. The target average acceptance probability (adapt_delta) was set to 0.95. If brms returned any warnings (e.g., warnings about divergent transitions), we increased the length of each chain and/or the acceptance probability until no errors were returned. Our final runs used acceptance probabilities of 0.98 for deictic day names and 0.99 for parents’ siblings, grandparents, and grandchildren, and the spatial demonstratives analysis (orientation symmetry) described in the Supplementary Methods. The number of iterations was 6000 for deictic day names and 10,000 for tense, the six kinship subdomains and spatial demonstratives (orientation symmetry). In all remaining cases, the default settings (3000 iterations and acceptance probability of 0.95) produced no warnings.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Review: Even or odd? Symmetry and the structure of category systems across languages\nThis paper presents a theory of symmetry as it applies to the categorisation of different linguistic domains. The paper\nproposes that amongst linear 1-dimensional domains, like temperature or life cycles, there is no preference for even or odd\ncategories because there is no focal point to divide a consistent system. Comparably, when a linear structure is bisected\nwith a focal point, there is a tendency to create odd category systems because this symmetry preference asks for the same\nnumber of categories before and after the focal point, imposing an odd number of categories. For a similar reason, bisected\ncircular structures, like the circular notion of seasons repeating each year, are more likely to be even. Applied to non-\ntemporal systems, like cardinal directions, also show this symmetry. Finally, a binary feature system will also show a\npreference for symmetry because each feature imposes an opposition that will result in an even number of relatives. For\nexample, sibling systems are often the combination of binary rules, which result in an even number of categories. In each\nscenario, this paper argues that the idea of symmetry dictates a preference for odd or even categories, whether that be the\ndivision of a particular structure into even categories (resulting in even categories), or the division of sections on either side\nof a bifurcation (resulting in an odd number of categories).\nOverall, I found the breadth of data used to support the symmetry argument convincing. The simplicity and logical nature of a\npreference for symmetry seems well supported, but I wonder whether the result needs to be more cautiously interpreted, as it\nis shown throughout the paper, that these results are statistical, and there are always examples to the contrary. In my review,\nI will tend to focus on the binary features selection, and kinship components of the paper, as that is my speciality, but some\nother comments interspersed.\nTo first address bisected circular structures. Although I don’t have any complaints about the support for even systems in\nthese structures, the paper seems to overlook why we observe symmetry in these systems. Unlike the bisecting linear\nsystem, where it seems logical to have the same number of categories before and after the bisecting point, the bisecting\ncircular structure could feasibly have three bisects and still have symmetry, at least in the mathematical sense, but perhaps\nthere is a different mode of symmetry being invoked here? For example, a circle with three evenly spaced privileged points\ncan be symmetrically divided by drawing a line straight through one point and dividing the other two. Perhaps there is more\nto be said about the relationship between symmetry and privileged points. The privileged points being of different types (i.e.\na cold vs a hot point if we consider seasons), seems like a possible solution the author could explore.\nA secondary point on bisected circular structures is that Figure three seems to imply that the symmetry arises from the\nunderlying environmental features (seasons, moon phases). I would like to see the author justify why these support a\ncognitive argument for the preference of symmetry (which the author seems to be implying is the case across domains on\nline 28). To me, these particular domains just seem to reflect the symmetry that we observe in the natural setting. In general,\nit would be nice to have a cognitive theory for why symmetry is observed, which the author seems to be implying is the\ncause, but to me isn’t fully developed.\nTurning to binary feature selection, and kinship. Although I think the result is sound, the analysis could be more grounded, I\nthink. As I am sure any specialist would say about any of the domains studied, kinship is complicated!\nMy first clarification is asking how each category is defined. A typical division of each kinship category could be found in\nJones (2010). My summary would be:\n- Siblings: 16 total categories; Older vs Younger, Male vs Female, Relative Sex\n- Aunts+Uncles: 16 categories; Older / Younger (of connecting relative), Male vs Female, Lineage (mother of father’s line),\nRelative Sex. Note: these terms should be more specifically Parent’s siblings – since the English category Aunts +\nUncles includes relatives by marriage which I don’t think are included here.\n- Grandparents: 8 categories; Male vs Female, Distinguish lineage\n- Niblings (nieces and nephews): 16 categories: Relative gender, Relative Age, Gender, Lineage\nIn Figure 1, however, we can see that the maximum number of words for each subset seems to be higher than the maximum\nnumber of categories. I think this is likely because the analysis hasn’t considered that some languages in Kinbank have\nmultiple sources and therefore have more words than categories (because either sources disagree or offer different\northographies). For example: Latvian is listed as having 10 grandparent terms. If I do this from the raw data, I actually get 12\nterms, but as is fairly obvious from the data, most of the words are just variants of kinship terms – like in English where we\nhave Granny, Nanna, Grandmother, Grandma, but we might still say we have one kinship term for Grandmother. Ideally, this\nwouldn’t be the case for Kinbank, but most sources are ambiguous about the terms they record. An approach that might work\nbetter with Kinbank data, and would probably result in the same outcome, is using the kinship structure of each language\n(See https://osf.io/8wfmu). You can see in Figure 2A of this paper, cross-referenced with Nerlove and Romney (1967), that\nthe most frequent sibling structures have an even number of categories, and odd-numbered systems are rare. The exception\nis type 5 which has an elder brother term, an elder sister term, and a younger sibling term, which is the second most common\nstructure in Kinbank, but the most common, third, and fourth systems are all even. The fifth most common is to have a single\nkinterm for siblings.\nA final minor point is that these counts appear to assume that each language is independent of all other languages. It could\nbe the case that languages are from a biased sample of languages that contain even numbered categories, and therefore\nthe result is a sampling problem, rather than an actual feature of language. I think given the scale of the datasets used\n(maybe aside from the Australian-specific datasets), this problem could be explained away by justifying the likelihood of\nseeing the predicted preference across such a large sample of languages, oversampling some areas and undersampling\nothers should be balanced out. I would worry that people reading the paper who think that is important will dismiss the result\nif this point is not addressed at least in brief.\nJones, D. (2010). Human kinship, from conceptual structure to grammar. Behavioral and Brain Sciences, 33(5), 367–381.\nhttps://doi.org/10.1017/S0140525X10000890", "reviewer_2": "review of Even or odd? Symmetry and the structure of category systems across languages\nThis paper uses a number of test cases from various domains to argue that category systems in language tend to be symmetric given the underlying structure of the domain. This leads to the prediction that some types of category systems\nshould show a preference for a odd number of categories and others an even number of categories. There is some support for the\nhypothesis provided, though not all test cases provide clear evidence, and in some cases there appears to be a\npreference for just one type of system. In general, I found the paper intriguing, but not necessarily super convincing. I also\nhave some general questions about how symmetry relates to other cognitive preferences that shape language and about\nhow privilege points can be determined.\n(1) A good number of the plots are about kinship, and this is the only binary feature system tested. It wasn't clear to me\nwhether there was a case to be made for separating out each dimension of kinship in this way. And why not look at other\nbinary feature systems? There are typological databased available for the others mentioned. Relatedly, I don't really see\nwhat the Aboriginal social classification data get us. There is only a single system (type?) and these languages are all\nareally and genetically related, no?\n(2) There were several points in the manuscript where I wondered whether it was really so simple to determine what the\nstructure of the underlying system should be, i.e., linear vs. bisected-linear. Take seasons for example, where the prediction\nis odd, but some languages have a bipartition. But this is described in the SI as having a different underlying structure. The\nsame issue came up for me in the description of moon phases. I suppose my question is, how do we separate statistical\nnoise (i.e., the bias is soft, and not all language will conform to the symmetry preference), from differences in category\nstructure, from differences in privileged points? Could it be that some languages differ in whether there IS a privileged point\nat all in a given domain? It feels to me that more evidence is needed to show that there are\nuniversal(ish) category structures that in fact underlie these domains. That kind of work has\nbeen done in the colour domain, and a few other places (some initial work on person systems in https://escholarship.org/uc/item/69d6v383), but without that, I'm not so sure it makes sense\nto assign a given domain a given structure in the way that this work does.\n(3) Perhaps related to the above comment, I also wondered about the relationship between defaults (or unmarked forms) and\nprivileged points. For example in the non-cardinal systems section, there is brief mention of a language that has a term for\n\"other side of the river\" but not \"same side of the river\". Similar question came up in the tense section, where perhaps\npresent tense is the default. What role if any do defaults play here?\n(4) A big question I had while reading this was how symmetry relates to other concepts like monotonicity, connectedness,\nstructural iconicity, and isomorphism. These have all been argued to shape semantic systems, and they all feel related to,\nbut not necessarily the same as symmetry. Perhaps structural iconicity and isomorphism are most closely related to\nsymmetry? I also wondered whether some instances of symmetry are explain by compositionality, i.e., when languages\ncreate a new form for use with one end of a scale, it is then available for use at the other end, So why not? If you see what I\nmean. I would appreciate more discussion of this, and indeed I think this would make the paper more interesting, though\nmaybe more for specialists (see my next comment).\n(5) My final comment is about the abstract/introduction and the audience. First, for me the abstract was not a great summary\nof the paper. I would not understand what the paper is claiming, or showing based on what it says. What is \"the parity of\ncategory systems\"? What is the relation between symmetry and odd and even? What is \"the principle of symmetry\"\nand why do the results \"therefore provide evidence\"? I don't see how the two halves of this sentence connect. I also found the\nintroduction to the paper tough going, particularly section 1. The very beginning gives not much sense for what is at stake, or\nhow symmetry is related to simplicity (or other concepts, see above). The discussion around Figure 1 is really not very clear,\nonly the linear structure is really discussed in the text. There are a few critical sentences that need more scaffolding, \"All of\nthese categories are connected...(line 63)\", \"...if it is left unchanged...(line 68)\". I'm wondering whether it makes sense to use\nsome of the examples in the text that follows to help the reader here. The rest of the paper is more clearly written. My\ngeneral feeling is that paper is in some sense presenting some initial findings, that would benefit from more nuanced discussion, and\ntherefore might be better in a more specialised journal.", "reviewer_3": "The paper is highly interesting and based on a very clear idea, that human systems are symmetrical, something that can be\nexplained by their functionality and salience. Systems are symmetrical in different ways, either even or odd, and that there is\na tendency for systems globally to be either of the one or the other type. In the paper, these questions are investigated by\nmass data and Bayesian methodologies. I think the topic is highly interesting and presented in an appropriate way, even if I\nbelieve that there are several shortcomings in the study, that should be accounted for before publication.\n1. The sources: the data stems from various sources, of which some are global, some more restricted. This is mentioned in\nthe study. However, of the global sources, it is not clear how the distributions of data is in relation to areas and language\nfamilies (at least not in the material as it is presented). I guess that it would be possible to extract this information from\nthe raw data sets, but the supplementary material should have all available statistics on the data sets used for the study.\nAs it is now, data sets stem from a number of different sources, and it is fully unclear in which way the results are representative if\nthe data sets are not compared to each other. Therefore, the data needs a significant review and presentation by the author.\nIf the information is there somewhere, the paper needs to point out where the reader can find this information.\n2. The model: Results of global statistics involving language systems must account for possible autocorrelation or other\nskewing of results related to an area- or family-bias. This can be accounted for in different ways by the model, most\nappropriately by a model that builds in family or area. A more advanced model would build in a phylogeny, but I assume that\nthis is not possible in the cases presented here. In any case, it is uncertain whether - and to what extent - the data represents\nsimilar systems because of this underlying factor (area- or family-bias, as described in previous paragraph).\nFurther, I think that the graphs are very uninformative. All different systems are presented in one graph (Figure 2) and it is\nentirely unclear what represents what etc. This can only be concluded by reading the text carefully, which is no advantage.\nFigure captions should carefully describe what they represent and how they reflect the results of the study."}, "author_response": "RESPONSE TO REVIEWERS\nI thank all reviewers for their careful consideration of my manuscript. I have attempted to\naddress all points raised and believe that the manuscript is stronger as a result.\nReviewer #1 (Remarks to the Author):\nReview: Even or odd? Symmetry and the structure of category systems across languages\nThis paper presents a theory of symmetry as it applies to the categorisation of different\nlinguistic domains. The paper proposes that amongst linear 1-dimensional domains, like\ntemperature or life cycles, there is no preference for even or odd categories because there is\nno focal point to divide a consistent system. Comparably, when a linear structure is bisected\nwith a focal point, there is a tendency to create odd category systems because this symmetry\npreference asks for the same number of categories before and after the focal point, imposing\nan odd number of categories. For a similar reason, bisected circular structures, like the circular\nnotion of seasons repeating each year, are more likely to be even. Applied to non-temporal\nsystems, like cardinal directions, also show this symmetry. Finally, a binary feature system will\nalso show a preference for symmetry because each feature imposes an opposition that will\nresult in an even number of relatives. For example, sibling systems are often the combination\nof binary rules, which result in an even number of categories. In each scenario, this paper\nargues that the idea of symmetry dictates a preference for odd or even categories, whether\nthat be the division of a particular structure into even categories (resulting in even categories),\nor the division of sections on either side of a bifurcation (resulting in an odd number of\ncategories).\nOverall, I found the breadth of data used to support the symmetry argument convincing. The\nsimplicity and logical nature of a preference for symmetry seems well supported, but I wonder\nwhether the result needs to be more cautiously interpreted, as it is shown throughout the\npaper, that these results are statistical, and there are always examples to the contrary. In my\nreview, I will tend to focus on the binary features selection, and kinship components of the\npaper, as that is my speciality, but some other comments interspersed.\nThanks for these positive comments. I have attempted to acknowledge the statistical\nnature of the predictions throughout – for example the abstract uses the phrase “tend to”\nthree times, and the opening paragraph of the discussion explicitly acknowledges that the\npredictions are “statistical tendencies”\nTo first address bisected circular structures. Although I don’t have any complaints about the\nsupport for even systems in these structures, the paper seems to overlook why we observe\nsymmetry in these systems. Unlike the bisecting linear system, where it seems logical to have\nthe same number of categories before and after the bisecting point, the bisecting circular\nstructure could feasibly have three bisects and still have symmetry, at least in the\nmathematical sense, but perhaps there is a different mode of symmetry being invoked here?\nFor example, a circle with three evenly spaced privileged points can be symmetrically divided\nby drawing a line straight through one point and dividing the other two. Perhaps there is more\nto be said about the relationship between symmetry and privileged points. The privileged\npoints being of different types (i.e. a cold vs a hot point if we consider seasons), seems like a\npossible solution the author could explore.\nThanks for suggesting that more discussion of privileged points is needed. The supporting\nmaterial contains an extended discussion of privileged points and the role that they play in\nthe theory. I assume that privileged points form part of the space (second row of Figure 1)\nover which categories are defined – in other words, the privileged points form part of\npeople’s mental representation of a given domain. From this perspective, a circular structure\nwith three privileged points is possible in principle, but I propose that people do not\nrepresent the relevant domains (seasons, lunar phases, cardinal directions) in this way.\nAs suggested in the supplementary material, for seasons the privileged points could\ncorrespond to the hottest and coldest time of the year, or the two equinoxes. Either\npossibility produces the prediction that season systems tend to have an even number of\ncategories.\nA secondary point on bisected circular structures is that Figure three seems to imply that the\nsymmetry arises from the underlying environmental features (seasons, moon phases). I would\nlike to see the author justify why these support a cognitive argument for the preference of\nsymmetry (which the author seems to be implying is the case across domains on line 28). To\nme, these particular domains just seem to reflect the symmetry that we observe in the natural\nsetting. In general, it would be nice to have a cognitive theory for why symmetry is observed,\nwhich the author seems to be implying is the cause, but to me isn’t fully developed.\nThanks for raising this point. I have added a paragraph to the discussion (line 297) that\nconsiders whether the symmetries arise from the world or the mind. This paragraph\nacknowledges that the results do not resolve this question, and suggests that lab studies\ncould be used to see whether people create category systems with greater degrees of\nsymmetry than are present in the environment.\nTurning to binary feature selection, and kinship. Although I think the result is sound, the\nanalysis could be more grounded, I think. As I am sure any specialist would say about any of\nthe domains studied, kinship is complicated!\nMy first clarification is asking how each category is defined. A typical division of each kinship\ncategory could be found in Jones (2010). My summary would be:\n- Siblings: 16 total categories; Older vs Younger, Male vs Female, Relative Sex\n- Aunts+Uncles: 16 categories; Older / Younger (of connecting relative), Male vs Female,\nLineage (mother of father’s line), Relative Sex. Note: these terms should be more specifically\nParent’s siblings – since the English category Aunts + Uncles includes relatives by marriage\nwhich I don’t think are included here.\n- Grandparents: 8 categories; Male vs Female, Distinguish lineage\n- Niblings (nieces and nephews): 16 categories: Relative gender, Relative Age, Gender, Lineage\nIn Figure 1, however, we can see that the maximum number of words for each subset seems to\nbe higher than the maximum number of categories. I think this is likely because the analysis\nhasn’t considered that some languages in Kinbank have multiple sources and therefore have\nmore words than categories (because either sources disagree or offer different\northographies). For example: Latvian is listed as having 10 grandparent terms. If I do this from\nthe raw data, I actually get 12 terms, but as is fairly obvious from the data, most of the words\nare just variants of kinship terms – like in English where we have Granny, Nanna, Grandmother,\nGrandma, but we might still say we have one kinship term for Grandmother. Ideally, this\nwouldn’t be the case for Kinbank, but most sources are ambiguous about the terms they\nrecord. An approach that might work better with Kinbank data, and would probably result in the\nsame outcome, is using the kinship structure of each language (See https://osf.io/8wfmu).\nYou can see in Figure 2A of this paper, cross-referenced with Nerlove and Romney (1967), that\nthe most frequent sibling structures have an even number of categories, and odd-numbered\nsystems are rare. The exception is type 5 which has an elder brother term, an elder sister term,\nand a younger sibling term, which is the second most common structure in Kinbank, but the\nmost common, third, and fourth systems are all even. The fifth most common is to have a\nsingle kinterm for siblings.\nThanks for the general suggestion here and for the pointer to Passmore (2023). The\nMethods (line 443) and Supplementary Material (p 1) now describe how the subdomains of\nkinship are defined, and I have followed your suggestion and changed “aunts+uncles” to\n“parents’ siblings”.\nI now acknowledge around line 451 that some categories can have multiple synonymous\nlabels, and my approach aims to count categories rather than labels. I have changed my\ncode, however, to better achieve this goal – for example, the code is no longer sensitive to\ncapitalization, and I identified and fixed a couple of other bugs. Currently the 12 Latvian\nterms for grandparents are organized into seven categories:\n1) vectētin̩š, vectēvs, vectētin̩š : mFF, mMF, fFF, fMF\n2) vecmāmin̩a, vecmāmul̩a : mFM, mMM, fFM, fMM\n3) tēva tēvs: mFF\n4) vecmate, vecāmāte: mFM, mMM\n5) vacaistevs, vecaistēvs: mFF, mMF\n6) mātes māte: mMM\n7) Vecmāmiņa: mFM\nIdeally the count here would be six: vecmāmin̩a and Vecmāmiņa are distinguished here\nbecause the two have different diacritics on the n. But I think that a general policy of\nignoring diacritics would be a mistake because they may carry important distinctions for\nsome languages. The real problem here is an issue of missing data: Vecmāmiņa would\nhave been merged with category 2 if Vecmāmiņa had been paired with MM, fFM, fMM, but\nin Kinbank it is only paired with mFM.\nTo ensure that missing data has not distorted the results, I added a second kinship analysis\nusing a different data set compiled by Murdock. The results (Figure S1) are compatible with\nthe kinship results, suggesting that the conclusions about kinship are robust. I did not run\nphylogenetic regressions based on the Murdock data because this data set does not\ninclude glottocodes.\nInstead of using the Murdock data set, I considered implementing the “structural vector”\napproach described by Passmore (2023) in https://osf.io/8wfmu . I found the code released\nalong with this paper helpful, and incorporated some of it into my own scripts. But I ended\nup using the Murdock data for two reasons. First, analysing a second data set seemed a\nuseful way to demonstrate robustness. Second, if I understand correctly, the “structural\nvector” approach takes the first term listed in Kinbank for each kin type, and this does not\nseem ideal. As far as I can tell there is no guarantee that the first term listed in Kinbank is\nactually the “best” or “default” term for that kin type.\nA final minor point is that these counts appear to assume that each language is independent\nof all other languages. It could be the case that languages are from a biased sample of\nlanguages that contain even numbered categories, and therefore the result is a sampling\nproblem, rather than an actual feature of language. I think given the scale of the datasets used\n(maybe aside from the Australian-specific datasets), this problem could be explained away by\njustifying the likelihood of seeing the predicted preference across such a large sample of\nlanguages, oversampling some areas and undersampling others should be balanced out. I\nwould worry that people reading the paper who think that is important will dismiss the result if\nthis point is not addressed at least in brief.\nThanks for raising this point. I attempt to address it by reporting estimates derived from\nphylogenetic regression models that control for language relatedness (Figure 3). I have also\nadjusted the text (line 115) to explain that the regression models are included for this purpose.\nJones, D. (2010). Human kinship, from conceptual structure to grammar. Behavioral and Brain\nSciences, 33(5), 367–381. https://doi.org/10.1017/S0140525X10000890\nI now cite this paper on line 202.\nReviewer #2 (Remarks to the Author):\nreview of Even or odd? Symmetry and the structure of category systems across languages\nThis paper uses a number of test cases from various domains to argue that category systems\nin language tend to be symmetric given the underlying structure of the domain. This leads to\nthe prediction that some types of category systems should show a preference for a odd number of categories and others an even number of categories. There is some support for the\nhypothesis provided, though not all test cases provide clear evidence, and in some cases there\nappears to be a preference for just one type of system. In general, I found the paper intriguing,\nbut not necessarily super convincing. I also have some general questions about how\nsymmetry relates to other cognitive preferences that shape language and about how privilege\npoints can be determined.\n(1) A good number of the plots are about kinship, and this is the only binary feature system\ntested. It wasn't clear to me whether there was a case to be made for separating out each\ndimension of kinship in this way.\nConsidering subsystems (e.g. siblings, grandparents, grandchildren, etc) separately is relatively\ncommon in work on kinship – for example, the supplementary material now includes results\nbased on a dataset compiled by Murdock (1970) that is organized in terms of subsystems.\nFollowing this practice here has the advantage of allowing my results to be compared with\nprevious work that focuses on subsystems (as R1 did above when comparing with previous\nwork by Passmore and Nerlove and Romney on siblings).\nAnd why not look at other binary feature systems? There are typological databased available\nfor the others mentioned.\nThanks for this suggestion. I have added an analysis of spatial demonstratives to the\nsupplementary material. I considered the other domains listed but all of them present various\ncomplications. For example, the representation of personal pronoun systems used by Zaslavsky\net al (2021) includes 11 elements and cannot be characterized as a simple product of a person\nfeature (speaker, addressee, and other) and a number feature (one, two, more than two),\nbecause this product would generate only 9 elements. Similarly, Haspelmath’s (1997)\nfeature-based representation of indefinite pronoun systems includes one feature (scale reversal)\nthat depends on another (scalar endpoint), which makes these representations complicated to\nwork with. Given the relatively large number of domains already considered I feel that analyzing\ntwo feature-based domains (ie kinship and spatial demonstratives) is sufficient.\nRelatedly, I don't really see what the Aboriginal social classification data get us. There is only a\nsingle system (type?) and these languages are all areally and genetically related, no?\nThese languages are all related, but there is more than one system – Figure 2l shows that the\ndata set includes systems of several different sizes (ie some make finer-grained distinctions\nthan others). In my mind the social classification data are valuable for two main reasons. First,\nthe absence of odd-numbered systems in this domain provides a counterpoint to other domains\nfor which symmetry is a statistical tendency. Second, the group-theoretic analysis of social\nclassification included in Levi-Strauss (1949) is probably the most influential prior attempt to\nanalyze category systems using symmetry principles. Including social classification as a domain\nprovides a way to connect with that work, which illustrates that symmetry has implications that\ngo beyond parity alone.\n(2) There were several points in the manuscript where I wondered whether it was really so\nsimple to determine what the structure of the underlying system should be, i.e., linear vs.\nbisected-linear. Take seasons for example, where the prediction is odd, but some languages\nhave a bipartition. But this is described in the SI as having a different underlying structure. The\nsame issue came up for me in the description of moon phases. I suppose my question is, how do we separate statistical\nnoise (i.e., the bias is soft, and not all language will conform to the symmetry preference), from differences in category\nstructure, from differences in privileged points? Could it be that some languages differ in whether there IS a privileged point\nat all in a given domain? It feels to me that more evidence is needed to show that there are\nuniversal(ish) category structures that in fact underlie these domains. That kind of work has\nbeen done in the colour domain, and a few other places (some initial work on person systems\nin https://escholarship.org/uc/item/69d6v383), but without that, I'm not so sure it makes\nsense to assign a given domain a given structure in the way that this work does.\nI agree that the structures proposed in the current paper should be viewed as working\nhypotheses rather than well-established universals, and have added a paragraph to the\nDiscussion that makes this point (line 288). I see the current paper as analogous to some of the\nearly work on efficient communication and colour. Initial studies assumed that need probabilities\n(ie the frequency with which people talk about different hues) were constant across cultures, but\nsubsequent work has relaxed that assumption.\n(3) Perhaps related to the above comment, I also wondered about the relationship between\ndefaults (or unmarked forms) and privileged points. For example in the non-cardinal systems\nsection, there is brief mention of a language that has a term for \"other side of the river\" but not\n\"same side of the river\". Similar question came up in the tense section, where perhaps\npresent tense is the default. What role if any do defaults play here?\nThanks for raising this. Of the domains considered, I agree that tense and non-cardinal\ndirectional systems are the two that raise questions about zero marking. I now say that\nVelupillai’s tense data incorporates categories without overt expression (line 399), and that\ndefaults may explain why only one member of a spatial opposition is named (line 247).\n(4) A big question I had while reading this was how symmetry relates to other concepts like\nmonotonicity, connectedness, structural iconicity, and isomorphism. These have all been\nargued to shape semantic systems, and they all feel related to, but not necessarily the same as\nsymmetry. Perhaps structural iconicity and isomorphism are most closely related to\nsymmetry? I also wondered whether some instances of symmetry are explain by\ncompositionality, i.e., when languages create a new form for use with one end of a scale, it is\nthen available for use at the other end, So why not? If you see what I mean. I would appreciate\nmore discussion of this, and indeed I think this would make the paper more interesting, though\nmaybe more for specialists (see my next comment).\nThanks for this question. I have added two new paragraphs to the discussion that consider\nsymmetry of linguistic form and mention both iconicity and compositionality (line 307).\nCompositionality is also now mentioned in the following paragraph (line 330).\nI have not discussed monotonicity and connectedness because do not see clear links between\nsymmetry and these principles. In the case of monotonicity, I’m not sure how to apply this\nprinciple to structures other than the linear or bisected linear structures, and for these structures\nI think that monotonicity may not be related to symmetry. Monotonicity predicts, I think, that the\nstructure will be divided into two pieces by making a single cut somewhere along the dimension\n– that way one of the resulting categories will be upward monotonic and the other will be\ndownward monotonic. But monotonicity does not require that the single cut occurs in the middle\nof the dimension, which means that monotonicity does not imply symmetry.\nFor connectedness, I think that symmetry does not imply connectedness (consider a symmetric\nsystem that uses one term for both ends of a dimension and a second term for the middle),\nand\nthat connectedness does not imply symmetry. So I’m not sure what to say about connectedness\nother than the fact that for connected domains, my predictions about parity rely on the\nassumption that categories are connected (line 65).\n(5) My final comment is about the abstract/introduction and the audience. First, for me the\nabstract was not a great summary of the paper. I would not understand what the paper is\nclaiming, or showing based on what it says. What is \"the parity of category systems\"? What is\nthe relation between symmetry and odd and even? What is \"the principle of symmetry\" and\nwhy do the results \"therefore provide evidence\"? I don't see how the two halves of this\nsentence connect. I also found the introduction to the paper tough going, particularly section\n1. The very beginning gives not much sense for what is at stake, or how symmetry is related to\nsimplicity (or other concepts, see above). The discussion around Figure 1 is really not very\nclear, only the linear structure is really discussed in the text. There are a few critical sentences\nthat need more scaffolding, \"All of these categories are connected...(line 63)\", \"...if it is left\nunchanged...(line 68)\". I'm wondering whether it makes sense to use some of the examples in\nthe text that follows to help the reader here. The rest of the paper is more clearly written. My\ngeneral feeling is that paper is in some sense presenting some initial findings, that would\nbenefit from more nuanced discussion, and therefore might be better in a more specialised\njournal.\nThanks for pointing out that the abstract and introduction could be improved. I have attempted\nto address as follows:\n(i) I have rewritten both the abstract and the introduction in an attempt to be clearer – for\nexample, the term “parity” is no longer mentioned in the abstract, and I try to highlight the key\nidea that symmetry shapes categorization across cultures.\n(ii) When Figure 1 is first introduced, I now say explicitly say that the initial discussion will focus\non the linear structure, and that the other structures will be introduced in subsequent sections\n(line 56).\n(iii) I have adjusted the critical sentences (formerly lines 63 and 68) and added concrete\nexamples that illustrate what it means for a category to be connected and for a category system\nto be symmetric (see lines 67 and 73)\nReviewer #3 (Remarks to the Author):\nThe paper is highly interesting and based on a very clear idea, that human systems are\nsymmetrical, something that can be explained by their functionality and salience. Systems are\nsymmetrical in different ways, either even or odd, and that there is a tendency for systems\nglobally to be either of the one or the other type. In the paper, these questions are investigated\nby mass data and Bayesian methodologies. I think the topic is highly interesting and presented\nin an appropriate way, even if I believe that there are several shortcomings in the study, that\nshould be accounted for before publication.\nThanks for these positive comments!\n1. The sources: the data stems from various sources, of which some are global, some more\nrestricted. This is mentioned in the study. However, of the global sources, it is not clear how\nthe distributions of data is in relation to areas and language families (at least not in the\nmaterial as it is presented). I guess that it would be possible to extract this information from\nthe raw data sets, but the supplementary material should have all available statistics on the\ndata sets used for the study. As it is now, data sets stem from a number of different sources,\nand it is fully unclear in which way the results are representative if the data sets are not\ncompared to each other. Therefore, the data needs a significant review and presentation by the\nauthor. If the information is there somewhere, the paper needs to point out where the reader\ncan find this information.\nThanks for raising this. The supplementary material now opens with tables documenting\nthe distribution of languages across areas and families, and the main text refers to these\ntables on line 108.\n2. The model: Results of global statistics involving language systems must account for\npossible autocorrelation or other skewing of results related to an area- or family-bias. This can\nbe accounted for in different ways by the model, most appropriately by a model that builds in\nfamily or area. A more advanced model would build in a phylogeny, but I assume that this is\nnot possible in the cases presented here. In any case, it is uncertain whether - and to what\nextent - the data represents similar systems because of this underlying factor (area- or\nfamily-bias, as described in previous paragraph).\nTo account for genetic relatedness, the previous version used mixed effects regression\nmodels with a random intercept for language family. Following your suggestion, I have now\nreplaced these with phylogenetic regression models.\nFurther, I think that the graphs are very uninformative. All different systems are presented in\none graph (Figure 2) and it is entirely unclear what represents what etc. This can only be\nconcluded by reading the text carefully, which is no advantage. Figure captions should\ncarefully describe what they represent and how they reflect the results of the study.\nI have now broken Figure 2 into two separate figures (Figs 2 and 3), and the current Figure 2\nplaces the size distribution and parity distribution for a domain side-by-side in a single panel\ninstead of separating them. I have also adjusted and expanded the captions, and I hope that\nthese changes have made the figures more informative."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Thank-you for addressing and clarifying my comments. I think this is a highly interesting paper, and I commend the\nunification of domains under a single, simple idea, as well as the supporting quantitative analysis, which utilises the breadth\nof datasets that are increasingly available. I look forward to seeing the reception of the paper more broadly once it is\npublished.\n(Remarks on code availability)\nI didn't review the code the first time, but I have looked at it this time and found it, at least qualitatively, well described and\ndocumented. I had various problems with running the BRMS models, but that is due to my own technological limitations (and\nthe limitations imposed on my by Apple!). I've listed some minor notes here that might help others reproduce the analysis,\nbut these shouldn't delay production.\n- The package wcs is a custom package. It is worth putting a comment showing how this package can be installed\n(remotes::install_github(\"jvosten/wcs\")) - particularly since all other packages are on CRAN.\n- There are some utility functions found throughout the code. It might make sense to move these into the regression functions\nchunk and call it \"functions\".\n- Given it is unlikely that this code will work forever, it might also benefit from some more descriptive comments throughout.\nAnd a final comment unrelated to the paper, but that might be helpful in the future: There is a note that the regression models\ntake some time to run. I assume this is the brms MCMC process. A non-MCMC Bayesian alternative is INLA, which has a lot\nof speed improvements (although 2 hours isn't all that long either I suppose!).", "reviewer_2": "I thank the author for engaging thoroughly with my comments. I'm satisfied with the revised manuscript.", "reviewer_3": "My comments/concerns have been met in this new version of the paper."}, "author_response": "RESPONSE TO REVIEWERS\nI thank all reviewers for engaging with the revision of my manuscript. All reviewers seemed\nsatisfied with the paper itself, and Reviewer 1 made some suggestions about the code.\nI didn't review the code the first time, but I have looked at it this time and found it, at least\nqualitatively, well described and documented. I had various problems with running the BRMS\nmodels, but that is due to my own technological limitations (and the limitations imposed on my by\nApple!). I've listed some minor notes here that might help others reproduce the analysis, but these\nshouldn't delay production.\n- The package wcs is a custom package. It is worth putting a comment showing how this package\ncan be installed (remotes::install_github(\"jvosten/wcs\")) - particularly since all other packages are\non CRAN.\nThe code uses the R package renv which should allow all packages to be easily installed using\nrenv::restore(). I’ve now documented this in the top level README. That README also now\nspecifically mentions the wcs package and explains how it can be installed without renv if\nneeded.\n- There are some utility functions found throughout the code. It might make sense to move these\ninto the regression functions chunk and call it \"functions\".\nThanks for the suggestion. I’ve added a new chunk for utility functions after the regression\nfunctions chunk.\n- Given it is unlikely that this code will work forever, it might also benefit from some more\ndescriptive comments throughout.\nI have added some comments in an attempt to make things clearer.\nAnd a final comment unrelated to the paper, but that might be helpful in the future: There is a note\nthat the regression models take some time to run. I assume this is the brms MCMC process. A\nnon-MCMC Bayesian alternative is INLA, which has a lot of speed improvements (although 2 hours\nisn't all that long either I suppose!).\nThank you for the pointer to INLA – I will keep it in mind for future projects!\nIn addition to the changes just mentioned I I also updated all R packages to their current\nversions and made some minor changes to avoid functions that are now deprecated. As\nmentioned in the manuscript, all code is now publicly available on GitHub (\nhttps://github.com/cskemp/parity ) and archived under a DOI on Zenodo (\nhttps://doi.org/10.5281/zenodo.17429647 ).\nThank you for the opportunity to improve the code,\nCharles Kemp"}]

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