Peirce’s Universal Categories in the S₃ symmetry group: an analysis of the basic color terms

2.1.1 Truth functional values

Unlike Newton’s categorization of 100 % spectral, fully saturated colors, a subset of B&K’s eleven colors considered here are red, orange, yellow, green, blue, purple, which are perceptually based. These colors do not quite process colors at the same level of purity or full saturation as Newton’s spectral colors. Nonetheless, in relative terms, B&K’s perceptual colors are purer and more saturated compared to less saturated shades like pink and brown, or as the fully unsaturated black and white. It is noteworthy that the classification of the eleven colors relies on the simple logic of truth functional values that define basic properties of colors, namely, ±spectral and ±saturated:

Basic properties of color Truth functional values

+Spectral → +Saturated Material Implication

–Spectral ↔ +Saturated Material Biconditional

–Spectral ← –Saturated Contrapositive of the Material Implication

+Spectral ↔ –Saturated Impossible

1. True: Material Implication: Any color that is spectral is also saturated, classifying red, orange, yellow, green, blue, purple

2. True: Material Biconditional: Any color that is not spectral may be saturated or unsaturated or any color that is unsaturated may be spectral or not spectral, classifying pink, brown

3. True: Contrapositive of the Material Implication: Any color that is not saturated cannot be spectral, classifying white, gray, black.

4. False: No color can be spectral and not saturated nor can any color be not saturated spectral.

2.1.2 Sound symbolism in English

We note that sound symbolism, evident in English color names, indicates varying levels of luminance (“brighter than” ↔ “darker than”), as illustrated in Figure 1.^[2]

Figure 1:

Categorizing Berlin and Kay’s eleven colors.

Colors such as white and yellow, which possess brighter luminosity, begin with the glides w/y. In contrast, black and blue, which exhibit darker luminosity, start with voiced labio-lateral blends, bl/bl. Neutral colors like gray and green are based on voiced velar-rhotic blends.

Derived colors exhibit similar patterns. Brown and purple begin with labio-rhotic blends, while pink and orange end with nasal, non-labial blends.

Furthermore, Kay et al. (1991) report that purple and brown typically occur in languages with a grue (the green-blue composite), whereas orange and pink are found in languages that separate grue into distinct green and blue, as noted by Dowman (2007).

Sound symbolism (Hinton et al. 1994) and color symbolism are pervasive in language. English is interesting because it merges sound symbolism with color. Specifically, English names for BCTs reflect an inherent property of color itself: luminosity.

2.3.1 Neurobiological evidence of the unique hues

Rezeanu et al. (2023) have recently disputed these arguments:

In this paper, we propose a neurobiological three-stage color vision model to explain the neural basis of the unique hues without relying on any post hoc adjustments to fit the psychophysical data. By taking into account the wide variability in L:M cone ratios and normalizing second-stage cone-opponent mechanisms to equal-energy white, the model generates accurate color-opponent mechanisms that predict both the position and variability of the spectral unique hues – blue, green, and yellow – without any further adjustments. Furthermore, by taking differential adaptation of the S and the L cones at short and long wavelengths into account, we can explain the non-linearities in the R/G and B/Y color-opponent mechanisms demonstrated by Burns et al. (Greene et al. 2024; Rezeanu et al. 2023)

The eleven B&K categories, outlined in Figure 1, are based on the unique hues – black/white, red/green, and yellow/blue – as presented in Figure 3. If the neurological model proposed by Rezeanu et al. (2023) is accurate, it provides the long-sought neurological foundation for the B&K categories listed in Figure 3. In Figure 3, these same six opponent hues are achieved by excluding the five logically derived colors from Figure 1 orange, purple, pink, brown, and gray, leaving yellow/blue, red/green, and of course black/blue.

Figure 3:

The unique hues.

2.3.2 The transition from six unique hues to a light/warm versus dark/cool color system

Building on the foundational work of Rezeanu et al. (2023) and Greene et al. (2024), the six unique hues can be realized as the three opponent pairs: black ↔ white, blue ↔ yellow, and green ↔ red. In turn, these oppositions can also exhibit properties that enable their grouping into two distinct categories with three members each: light/warm colors (white; red, yellow) and dark/cool colors (black; green, blue).

2.3.3 The evolutionary process based on light/warm, dark/cool colors

Based on the light/warm versus the dark/cool colors, whose source is the unique hues, there is an evolutionary process by which preindustrial languages with a limited number of BCTs gradually acquire additional color terms, eventually reaching the full complement of the 11 generally found in industrialized languages. This process was comprehensively detailed by Kay and Maffi (1999), derived from the 110 languages from the World Color Survey (WCS). Their findings reveal a systematic evolutionary process, as illustrated in Figure 4 (see Kay and Maffi 1999: 751 for a similar description of the process). According to their analysis, 92 % of the WCS languages follow the progression outlined in Figure 4.

Figure 4:

Showing the evolutionary process of going from two to eleven BCSs (after Kay and Moffi 1999).

2.4.1 Computational evidence supporting the BCTs

McCarthy et al. (2019) eschewed using B&K’s celebrated basicness criteria to arrive at the same 11 BCTs. In doing so, they use computational linguistic criteria, using data from 2,491 languages of various quality to test the accuracy of B&K’s 11 basic color terms. Their “‘features’ aggregation correlates strongly with the Berlin and Kay basic/secondary color term partition (γ = 0.96). The aggregation also largely predicts the Berlin and Kay hypothesized universal acquisition sequence, which is in no way directly entailed by the basicness criteria.” In short, this computational methodology eluded the criticisms of the B&K’s basicness methodology (Lucy 1997; Roberson et al. 2005; Saunders and van Brakel 1988, 1997, and others). McCarthy et al.’s methodology did not rely on Berlin and Kay’s approach, but by using computational linguistics, they came up with largely the same results.

3.3.1 Cunén K’iche’ and Tz’utujil Mayan as an example of the shift from the fifth to the sixth-stage of development

Unlike Himba and Bernmo, Cunén K’iche’ (my data) is an example of a normal prototype with five colors, which includes ‘grue’ (green/blue). On the other hand, Tz’utujil (Mendoza and Mendosa 2001) innovated by splitting the historic ‘grue’ into the two categories, green and blue. The Tz’utujil term asuul was borrowed from Spanish, as shown in Figure 6. What is unusual about the fifth categorial division in Himba, is that it did not have the fixed category ‘grue’.

Figure 6:

Comparing Cunén K’iche’ Maya with Tz’utujil Maya.

3.3.2 Possible reason for lack of universal prototypes in the Himba language

I would note here that Himba’s Burou is a recent borrowing: “This term is more recently borrowed from Afrikaans (blau) via Herero.” Because of its recentness, not “all observers spontaneously used the term ‘Burou’ in the naming task, but those who did not were able to select chips from the appropriate range when asked to indicate a ‘Burou’ stimulus.” It is interesting that for one of the tests, “[d]ata was excluded for two older participants who failed to use the term Burou to name any of the stimuli in the sets.” Apparently, older speakers had not fully absorbed Burou as a part of their speaking lexicon (quotes from Roberson et al. 2005).

These data suggest a language in the throes of change. This may explain why “[o]nly 35 % of Himba observers chose a black chip as best example [of zoozu]; other choices included light blue (10B 6/10), medium green (5G 5/10) and best example blue (5B 4/10). 2.6 % of observers used the term only for the black chip.” One wonders if zoozu might have historically been the word for grue before burou was borrowed, because, “light blue (6/10), medium green, and the best example blue, whereas only 35 % chose black”. The newly acquired burou is used “mostly in the blue/green/purple range. This corresponds to a grue term in Berlin and Kay’s (1969) stage theory.”

Could the newly borrowed burou’s increasing acceptance into the Himba language be the reason for the confusion regarding the universal prototypes, in particular grue as the referent? One suspects that for the rising generation, burou might be making its way to be the new word for grue presaging prototypical changes. Perhaps burou is adjusting the language to become the standard word for grue. Perhaps Roberson et al. captured Himba in an incomplete process of change.

4.1.1 Peirce’s Universal Categories

In Figure 7 Peirce’s Universal Categories, refer to the term “experience.” Please associate experience with the notion of perceptual awareness – that is, experiencing perceptions.

Firstness represents the immediate, qualitative aspects of experience – the sensations, feelings, and possibilities that arise in the present moment. Secondness encapsulates the realm of brute actuality – the resistances and reactions we encounter in the external world. Thirdness bridges the gap between the subjective and the objective, mediating between possibilities and necessities through signs, habits, and reasoned interpretation.

1. Firstness stands for “a monadic quality [that] is a mere potentiality, without existence” (CP 1.328). In other words, think of “ monadic experiences , or simples , [as] being elements [as] though there were nothing else in all experience.”

2. Secondness stands for a dyadic experience, which is an “experience of an opposing pair of objects.”

3. Thirdness stands for “ triadic experiences , or comprehensions , each a direct experience, which connects other possible experiences” (CP 1.328).

Peirce’s six Universal Categories – Firstness of Firstness, Firstness of Secondness, Secondness of Secondness, Firstness of Thirdness, Secondness of Thirdness, Thirdness of Thirdness – extend beyond their abstract philosophical foundations, providing a multifaceted framework for analyzing real-world experiences. Peirce sees his categories as representing or standing for different types of perceptual experiences. As symbolic representations, the six categories offer a lens through which to understand potentials, objects, and systems, rendering complex data transparent and accessible. As shown below, the categories provide the foundational framework for this proposed explanation for the widespread existence of the BCTs.

4.1.2 Structure and organization of S₃ symmetry

Figure 8 shows significant isomorphisms (one-to-one relationships) comparing the Universal Categories with the S₃ transformations: Firstness ≃ Identity, Secondness ≃ Rotation, and Thirdness ≃ Reflection.

These isomorphisms establish a thoroughgoing connection between Firstness, Secondness, and Thirdness. They suggest a nonrandom relationship between these concepts, further supported by six correlations. Compare the following:

1. Firstness of Firstness: The firstness of Identity represents the category on which the transformations of Rotation and Reflection must operate. Its mode of being is, therefore, a potential or a “firstness,” the subject to all transformations of the S₃ symmetry group, including Rotation and Reflection.

2. Firstness of Secondness: a first Rotation, representing the permutation of 120 degrees from Identity.

3. Secondness of Secondness: a second Rotation, representing the permutation of 240 degrees from Identity.

4. Firstness of Thirdness: a first Reflection represents swapping Identity’s 1 and 2.

5. Secondness of Thirdness: a second Reflection represents swapping Identity’s 2 and 3.

6. Thirdness of Thirdness: a third Reflection represents the swapping Identity’s 3 and 1.

Peirce’s Universal Categories offer a fundamental framework aligned with the conceptual structure of the symmetry and permutations associated with the S3 symmetry group. As Peirce noted, “the universal Categories … belong to every phenomenon, one being perhaps more prominent in one aspect of that phenomenon than another but all of them belonging to every phenomenon” (CP 5.43).^[3]

The S₃ group’s structure embodies the intuitive logic of the Universal Categories, underscoring the fact that these concepts and their corresponding isomorphisms are not merely abstract notions but are rather an intrinsic part of the fabric of reality. After all, Peirce defines Firstness, Secondness, and Thirdness in terms of perceptual “experiences” (CP 1.328). This group serves as a telling example of how abstract philosophical concepts can find tangible expression in the world around us, illustrating the connection between philosophical metaphysics and the actual physics of the material world.

Plainly put, “Geometric algebra is useful to a physicist because it automatically incorporates the structure of the world we inhabit, and accordingly provides a natural language for physics” (Gull et al. 1993).

As shown below, the formal relationship of Peirce’s Universal Categories and the algebra of the S₃ symmetry transformations lays out the possibility of an abstract diagram produces each of the eleven BCTs. Peirce describes such possibilities in this way:

The purpose of a Diagram is to represent certain relations in such form that it can be transformed into another form, representing other relations involved in the first represented, and this transformed icon can be interpreted in the symbolic statement … No other kind of sign can make a Truth evident for the evidence is that which is presented in an image, leaving for the work of the understanding merely the interpretation of the image in a Symbol. (MS Am 1632 [339], 1906; emphasis added)

4.2.1 The relationship of the spectral and saturated colors

To understand the implicational values that lead to the generation of the BCTs, we must first explain the logic of Spectral and Saturated Colors relative to their relationship of the truth values mentioned above. For simplicity numbers will represent the structure of the three categories, Implication, Contrapositive, and Biconditional, represented in Figure 10.

Figure 10:

The truth functional value of ±Spectral, ±Saturated.

In Figure 10, 2x represents Spectral colors, and 2y represents Saturated colors. 1x represents non-Spectral colors and 1y represents non-saturated colors.

A spectral color is monochromatic light composed of a single wavelength. Saturation is a measure of how vivid a color appears to the viewer. It describes the colors’ intensity, and purity. A highly saturated color is perceived as rich and vibrant, while a low saturation appears closer to a neutral shade of white, gray, black.

4.2.2 The contrapositive

The Contrapositive belongs to the category Firstness ≃ Identity ≃ Contrapositive. Please note: The Contrapositive has the truth value of –spectral ← –saturation of Figures 9 and 10. In both tables, the leftward arrow (1x ← 1y) represents the relationship “not Spectral ← not Saturated,” which is to be understood as “If any color is not Saturated, then it cannot be Spectral.” The Identity triangle of Figure 9 is the instantiation of the Contrapositive of the Material Implication: not Spectral ← not Saturated. The BCT colors Black, Gray, and White are neither Saturated nor Spectral.

The Identity triangle is a key focus of this paper because it lays the groundwork for the application of the Rotation and Reflexive transformations. The function of Identity is nothing more than to passively undergo transformations, to be transformed, i.e., to be Rotated, and then to undergo Reflection.

In Figure 11, Identity Triangle, red is in the first position, and as potential it plays no part in what is in second position, namely, white, and in third position, , black. The combination of white + black yields gray, the color of the triangle.

Figure 11:

Identity triangle.

The color of the Contrapositive, white + black = gray, highlights a fundamental principle: a color that is not saturated cannot have a specific wavelength, making it fully non-spectral.

4.2.3 The material biconditional: rotation

In Figure 10, the left-right arrow (1x ↔ 1y) represents a biconditional reading: “If not Spectral, then Saturated” and “If not Saturated, then Spectral.”

It is the role of the Material Biconditional to determine the quality of colors that are Spectral (Red) and not Saturated, in this case, White. It is the rotation of 120⁰ that brings black to the hidden, first position, red to the second position, and white to the third position. As a consequence, red + white results in the pink and brown triangles, as shown in Figures 12 and 13.

Figure 12:

Transformation from gray to pink.

Figure 13:

Transformation from gray to brown.

The second rotation at 240⁰ brings white to the potential position, black the second position, and red to the third position, which results in black + red coloring the triangle to a brown color. It is the Material Biconditional that also determines the quality of colors that are not Saturated (Black) and Saturated (Red) = Brown, in that order.

In the second rotation at 240⁰, white moves to the potential position, black occupies the second position, and red takes the third position. The Material Biconditional also includes non-Saturated (Black) combined with Saturated (Red) yielding Brown, in that specific order.

Figure 14 summarizes.

Figure 14:

Summary of rotation.

Figure 15:

First reflection from gray to purple.

4.2.4 The material implication: reflection

In every instance of transforming from the Contrapositive (white, gray, black) to the Material Implication (yellow, green, blue), we observe that white is substituted with yellow, gray with green, and black with blue. These substitutions arise from the transformation of the Contrapositive (–spectral ← –saturated; white, gray, black) to the Material Implication (+spectral → +saturated; yellow, green, blue). Notably, red remains unchanged throughout all transformations, because it begins as potential, absent from position 2 or 3, in the Contrapositive (see Figure 11).

In Figure 15, the Identity transformation undergoes the first Reflective transformation: Swap red and yellow, placing yellow in the first position, with red in second position, with blue remaining in third position, such that red + blue = purple.

In Figure 16, the Identity transformation undergoes the second Reflective transformation: swap red and yellow, which yields orange. The swap places blue in the second position and yellow in the third position with red remaining in the first position, where blue + yellow = green.

Figure 16:

Second transformation from gray to green.

In Figure 17, the Identity transformation undergoes the third Reflective transformation: swap red and blue, which yields places red in third position and yellow remaining in second position with blue remaining the first position, where red + yellow = orange.

Figure 17:

Third transformation from gray to orange.

Figure 18 summarizes.

Figure 18:

Summary of reflection.

In summary:

1. The Identity transformation i.e., –Spectral ← –Saturation, yields red, white + black = gray.

2. The Material Biconditional i.e., –Spectral ↔ +Saturation, yields black, white + red = pink, and white, red + black = brown.

3. The Material Implication i.e., +Spectral → +Saturation, yellow, red + blue = purple, red, blue + yellow = green, and blue, yellow + red = orange.