The arbitrariness principle can be applied not only to the sign, but to the whole sign-system. The fundamental arbitrariness of language is apparent from the observation that each language involves different distinctions between one signifier and another (e.g. 'tree' and 'free') and between one signified and another (e.g. 'tree' and 'bush'). The signified is clearly arbitrary if reality is perceived as a seamless continuum (which is how Saussure sees the initially undifferentiated realms of both thought and sound): where, for example, does a 'corner' end? Commonsense suggests that the existence of things in the world preceded our apparently simple application of 'labels' to them (a 'nomenclaturist' notion which Saussure rejected and to which we will return in due course). Saussure noted that 'if words had the job of representing concepts fixed in advance, one would be able to find exact equivalents for them as between one language and another. But this is not the case' (Saussure 1983, 114-115; Saussure 1974, 116). Reality is divided up into arbitrary categories by every language and the conceptual world with which each of us is familiar could have been divided up very differently. Indeed, no two languages categorize reality in the same way. As John Passmore puts it, 'Languages differ by differentiating differently' (cited in Sturrock 1986, 17). Linguistic categories are not simply a consequence of some predefined structure in the world. There are no 'natural' concepts or categories which are simply 'reflected' in language. Language plays a crucial role in 'constructing reality'.
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The principle of arbitrariness does not mean that the form of a word is accidental or random, of course. Whilst the sign is not determined extralinguistically it is subject to intralinguistic determination. For instance, signifiers must constitute well-formed combinations of sounds which conform with existing patterns within the language in question. Furthermore, we can recognize that a compound noun such as 'screwdriver' is not wholly arbitrary since it is a meaningful combination of two existing signs.
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The arbitrariness principle does not, of course mean that an individual can arbitrarily choose any signifier for a given signified. The relation between a signifier and its signified is not a matter of individual choice; if it were then communication would become impossible. 'The individual has no power to alter a sign in any respect once it has become established in the linguistic community' (Saussure 1983, 68; Saussure 1974, 69). From the point-of-view of individual language-users, language is a 'given' - we don't create the system for ourselves. Saussure refers to the language system as a non-negotiable 'contract' into which one is born... The ontological arbitrariness which it involves becomes invisible to us as we learn to accept it as 'natural'.
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Saussure added that 'any means of expression accepted in a society rests in principle upon a collective habit, or on convention - which comes to the same thing'

Everything that has been said up to this point boils down to this: in language there are only differences. Even more important: a difference generally implies positive terms between which the difference is set up; but in language there are only differences without positive terms. Whether we take the signified or the signifier, language has neither ideas nor sounds that existed before the linguistic system, but only conceptual and phonic difference that have issued from the system. The idea or phonic substance that a sign contains is of less importance than the other signs that surround it. Proof of this is that the value of a term may be modified without either its meaning or its sound being affected, solely because a neighboring term has been modified (see p. 115).

But the statement that everything in language is negative is true only if the signified and the signifier are considered separately; when we consider the sign in its totality, we have something that is positive in its own class. A linguistic system is a series of differences of sound combined with a series of differences of ideas; but the pairing of a certain number of acoustical signs with as many cuts made from the mass thought engenders a system of values; and this system serves as the effective link between the phonic and psychological elements within each sign. Although both the signified and the signifier are purely differential and negative when considered separately, their combination is a positive fact; it is even the sole type of facts that language has, for maintaining the parallelism between the two classes of differences is the distinctive function of the linguistic institution. — link

Durkheim conceived of sociology as the scientific study of a reality sui generis, a clearly defined group of phenomena different from those studied by all other sciences, biology and psychology included. It was for these phenomena that Durkheim reserved the term social facts, i.e., "a category of facts which present very special characteristics: they consist of manners of acting, thinking, and feeling external to the individual, which are invested with a coercive power by virtue of which they exercise control over him." Since these facts consisted of actions, thoughts, and feelings, they could not be confused with biological phenomena; but neither were they the province of psychology, for they existed outside the individual conscience. It was to define the proper method for their study that Durkheim wrote The Rules of Sociological Method (1895).

Durkheim was particularly concerned to distinguish social facts, which he sometimes described as "states of the collective mind," from the forms these states assumed when manifested through private, individual minds. This distinction is most obvious in cases like those treated in The Division of Labor -- e.g., customs, moral and legal rules, religious beliefs, etc. -- which indeed appear to have an existence independent of the various actions they determine. It is considerably less obvious, however, where the social fact in question is among those more elusive "currents of opinion" reflected in lower or higher birth, migration, or suicide rates; and for the isolation of these from their individual manifestations, Durkheim recommended the use of statistics, which "cancel out" the influence of individual conditions by subsuming all individual cases in the statistical aggregate.2 Durkheim did not deny, of course, that such individual manifestations were in some sense "social," for they were indeed manifestations of states of the collective mind; but precisely because they also depended in part on the psychological and biological constitution of the individual, as well as his particular circumstances, Durkheim reserved for them the term "socio-psychical," suggesting that they might remain of interest to the sociologist without constituting the immediate subject matter of sociology.3

It might still be argued, of course, that the external, coercive power of social facts is derived from their being held in common by most of the individual members of a society; and that, in this sense, the characteristics of the whole are the product of the characteristics of the parts. But there was no proposition to which Durkheim was more opposed. The obligatory, coercive nature of social facts, he argued, is repeatedly manifested in individuals because it is imposed upon them, particularly through education; the parts are thus derived from the whole rather than the whole from the parts. — link

I'd like to add the perhaps already implicit notion of the equivalence class to Saussure's thinking. Here's a concrete example. There are trillions (an infinity?) of ways to pronounce the word "bumblebee." First we can consider the billions of different human voices on this planet, and second we can consider all the different ways that each individual could pronounce the word. I think it's absurd to say that there's a right or ideal way to pronounce the word. All that matters is that each sounding of the word is recognized as equivalent to the others. So pronunciations of 'bumblebee' form an equivalence class without a privileged representative. — j0e

A category, such as a species of animal, is a population of unique members who vay from one another, with no fingerprint at their core. The category can be described at the group level only in abstract, statistical terms. Just as no American family consists of 3.13 people, no instance of anger must include an average anger pattern (should we be able to identify one). Nor will any instance necessarily resemble the elusive fingerprint of anger. What we have been calling a fingerprint might just be a stereotype.
Once I adopted a mindset of population thinking, my whole landscape shifted, scientifically speaking. I began to see variation not as error but as normal and even desirable. — Lisa Feldman Barrett, ‘How Emotions Are Made’

A category, such as a species of animal, is a population of unique members who vay from one another, with no fingerprint at their core. — Lisa Feldman Barrett, ‘How Emotions Are Made’

It is even more curious when we consider that the practical consequences have been widely drawn, indeed had been drawn thousands of years before Saussure, for it is only through the concept of a difference between form and substance that we can explain the possibility of speech and writing existing at the same time as expressions of one and the same language. If either of these two substances, the stream of air or the stream of ink, were an integral part of the language itself, it would not be possible to go from one to the other without changing the language. — Uldall

It is well-known that Saussure distinguishes between the “sound-image” and the objective sound. He thus gives himself the right to “reduce,” in the phenomenological sense, the sciences of acoustics and physiology at the moment that he institutes the science of language. The sound-image is the structure of the appearing of the sound [l'apparaître du son] which is anything but the sound appearing [le son apparaissant]. It is the sound-image that he calls signifier, reserving the name signified not for the thing, to be sure (it is reduced by the act and the very ideality of language), but for the “concept,” undoubtedly an unhappy notion here; let us say for the ideality of the sense... The sound-image is what is heard; not the sound heard but the being-beard of the sound. Being-heard is structurally phenomenal and belongs to an order radically dissimilar to that of the real sound in the world. — Derrida

The conceptual side of value is made up solely of relations and differences with respect to the other terms of language, and the same can be said of its material side. The important thing in the word is not the sound alone but the phonic differences that make it possible to distinguish this word from all others, for differences carry signification.

This may seem surprising, but how indeed could the reverse be possible? Since one vocal image is no better suited than the next for what it is commissioned to express, it is evident, even a priori, that a segment of language can never in the final analysis be based on anything except its noncoincidence with the rest. Arbitrary and differential are two correlative qualities.

In addition, it is impossible for sound alone, a material element, to belong to language. It is only a secondary thing, substance to be put to use. All our conventional values have the characteristic of not being confused with the tangible element which supports them. For instance, it is not the metal in a piece of money that fixes its value. A coin nominally worth five francs may contain less than half its worth of silver. Its value will vary according to the amount stamped upon it and according to its use inside or outside a political boundary. This is even more true of the linguistic signifier, which is not phonic but incorporeal—constituted not by its material substance but by the differences that separate its sound-image from all others.
— Sausure

The background and foil for this article was the position, dominant at the time, that axiomatic set theory provides the foundation for modern mathematics, including allowing us to identify all mathematical objects with sets. For example, the natural numbers 0, 1, 2, … can be identified with the finite von Neumann ordinals (starting with ∅ for 0 and using the successor function f:x→x∪{x}); similarly, the real numbers can be identified with Dedekind cuts constructed set-theoretically. Arithmetic truths are then truths about these set-theoretic objects; and this generalizes to other mathematical theories, all of whose objects are taken to be sets as well.

According to Benacerraf, such a set-theoretic foundationalist position misrepresents the structuralist character of arithmetic in particular and of mathematics more generally. To begin with, instead of working with the finite von Neumann ordinals, we can equally well work with the finite Zermelo ordinals (starting again with ∅ for 0 but using the alternative successor function f:x→{x}); and there are infinitely many other choices that are equivalent. Similarly, instead of working with a set-theoretic construction of the real numbers in terms of Dedekind cuts, we can work with a construction based on equivalence classes of Cauchy sequences on the rational numbers as suggested by Cantor and others. This basic observation is hard to deny, and even set-theorist foundationalists can agree with it (more on that below). But Benacerraf draws some further, more controversial conclusions from this basic observation.

Benacerraf argues, in particular, that the natural numbers should not be identified with any set-theoretic objects; in fact, they should not be taken to be objects at all. Instead, numbers should be treated as “positions in structures”, e.g., in “the natural number structure”, “the real number structure”, etc. All that matters about such positions are their structural properties, i.e., those “stem[ming] from the relations they bear to one another in virtue of being arranged in a progression” (1965: 70), as opposed to further set-theoretic properties of the von Neumann ordinals, Dedekind cuts, etc. What we study and try to characterize in modern mathematics, along such lines, are the corresponding “abstract structures”. — link

To prove that language is only a system of pure values, it is enough to consider the two elements involved in its functioning: ideas and sounds.

Psychologically our thought—apart from its expression in words—is only a shapeless and indistinct mass. Philosophers and linguists have always agreed in recognizing that without the help of signs we would be unable to make a clear-cut, consistent distinction between two ideas. Without language, thought is a vague uncharted nebula. There are no pre-existing ideas, and nothing is distinct before the appearance of language.

Against the floating realm of thought, would sounds by them selves yield predelimited entities? No more so than ideas. Phonic substance is neither more fixed nor more rigid than thought; it is not a mold into which thought must of necessity fit but a plastic substance divided in turn into distinct parts to furnish the signifiers needed by thought. The linguistic fact can therefore be pictured in its totality—i.e. language—as a series of contiguous subdivisions marked off on both the indefinite plane of jumbled ideas (A) and the equally vague plane of sounds (B). The following diagram gives a idea of it:

[NICE IMAGE. FOLLOW LINK TO SEE]

The characteristic role of language with respect to thought is not to create a material phonic means for expressing ideas but to serve as a link between thought and sound, under conditions that of necessity bring about the reciprocal delimitations of units. Thought, chaotic by nature, has to become ordered in the process of its decomposition. Neither are thoughts given material form nor are sounds transformed into mental entities;the somewhat mysterious fact is rather that “thought-sound" implies division, and that language works out its units while taking shape between two shapeless masses. Visualize the air in contact with a sheet of water; if the atmospheric pressure changes, the surface of the water will be broken up into a series of divisions, waves; the waves resemble the union or coupling of thought with phonic substance.

Language might be called the domain of articulations, using the word as it was defined earlier (see p. 10). Each linguistic term is a member, an articulus in which an idea is fixed in a sound and a sound becomes the sign of an idea.

Language can also be compared with a sheet of paper: thought is the front and the sound the back; one cannot cut the front with out cutting the back at the same time; likewise in language, one can neither divide sound from thought nor thought from sound; the division could be accomplished only abstractedly, and the result would be either pure psychology or pure phonology.

Linguistics then works in the borderland where the elements of sound and thought combine; their combination produces a form, not a substance.
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Everything that has been said up to this point boils down to this: in language there are only differences. Even more important: a difference generally implies positive terms between which the difference is set up; but in language there are only differences without positive terms. Whether we take the signified or the signifier, language has neither ideas nor sounds that existed before the linguistic system, but only conceptual and phonic difference that have issued from the system. The idea or phonic substance that a sign contains is of less importance than the other signs that surround it....

But the statement that everything in language is negative is true only if the signified and the signifier are considered separately; when we consider the sign in its totality, we have something that is positive in its own class. A linguistic system is a series of differences of sound combined with a series of differences of ideas; but the pairing of a certain number of acoustical signs with as many cuts made from the mass thought engenders a system of values; and this system serves as the effective link between the phonic and psychological elements within each sign. Although both the signified and the signifier are purely differential and negative when considered separately, their combination is a positive fact; it is even the sole type of facts that language has, for maintaining the parallelism between the two classes of differences is the distinctive function of the linguistic institution.
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When we compare signs—positive terms—with each other, we can no longer speak of difference; the expression would not be fitting, for it applies only to the comparing of two sound-images, e.g. father and mother, or two ideas, e.g. the idea “father” and the idea “mother”; two signs, each having a signified and signifier, are not different but only distinct. Between them there is only opposition. The entire mechanism of language, with which we shall be concerned later, is based on oppositions of this kind and on the phonic and conceptual differences that they imply.
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What is true of value is true also of the unit (see pp. 110 ff.). A unit is a segment of the spoken chain that corresponds to a certain concept; both are by nature purely differential.

Applied to units, the principle of differentiation can be stated in this way: the characteristics of the unit blend with the unit itself. In language, as in any semiological system, whatever distinguishes one sign from the others constitutes it. Difference makes character just as it makes value and the unit.
Putting it another way, language is a form and not a substance (see p. 113). This truth could not be overstressed, for all the mistakes in our terminology, all our incorrect ways of naming things that pertain to language, stem from the involuntary supposition that the linguistic phenomenon must have substance. — S

Let us begin with speech. It is subject to measurement, but only in so far as it is sound; it is not a quantity in its essential nature, which nature is that it be significant, as noun and verb are significant. The air is its Matter, as it is Matter to verb and noun, the components of speech.

To be more precise, we may define speech as an impact [made upon the outer air by the breath], though it is not so much the impact as the impression which the impact produces and which, as it were, imposes Form [upon the air]. Speech, thus, is rather an action than a quantity- an action with a significance. Though perhaps it would be truer to say that while this motion, this impact, is an action, the counter-motion is an experience [or Passion]; or each may be from different points of view either an action or an experience: or we may think of speech as action upon a substrate [air] and experience within that substrate.

If however voice is not characteristically impact, but is simply air, two categories will be involved: voice is significant, and the one category will not be sufficient to account for this significance without associating with a second. — Plotinus

Value is the sign as it is determined by the other signs in a semiotic system. For linguist Ferdinand de Saussure, for example, the content of a sign in linguistics is ultimately determined and delimited not by its internal content, but by what surrounds it: the synonyms redouter ("to dread"), craindre ("to fear"), and avoir peur ("to be afraid") have their particular values because they exist in opposition to one another. If two of the terms disappeared, then the remaining sign would take on their roles, become vaguer, less articulate, and lose its "extra something" because it would have nothing to distinguish itself from.

For de Saussure, this suggests that thought is a chaotic nebula until linguistic structure dissects it and holds its divisions in equilibriums. This is akin to the philosophy of Sir William Hamilton, who indirectly influenced Saussure and believed that the mind could only grasp an idea through distinguishing it from something that it is not. He reasoned that the two objects would otherwise collapse together for the mind and become indistinguishable from one another. — link

In certain ways Herder’s philosophical texts are easier to read than others from the period. For example, he avoids technical jargon, writes in a way that is lively and rich in examples rather than dry and abstract, and has no large, complex system for the reader to keep track of.
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Herder—especially in the Ideas—laid the foundations for the discipline of linguistics with the following five principles:

Thought is essentially dependent on and bounded by language—i.e., one can only think if one has a language, and one can only think what one can express linguistically.
Meanings or concepts consist (not in referents, Platonic forms, or the subjective mental “ideas” favored by the British Empiricists, but instead) in word-usages.
Humankind exhibits profound differences in modes of thought, concepts, and language, especially between different historical periods and cultures.
Because of principles (1) and (2), investigating the characters of peoples’ languages and their differences from each other is a primary and dependable means for discovering the character of, and the differences between, their modes of thought and concepts. For principles (1) and (2) entail that their languages constitute an empirically accessible and reliable window onto the nature of their modes of thought and concepts.
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First, Hegel developed his conception of the transition from nature to the (human) mind through a reflection on, and revision of, a position of Herder’s. In the Ideas Herder had argued that human beings (a) exhibit a strong continuity with the “realm of animals” (he even calls the animals their “older brothers”) but (b) are distinguished from it by their possession of spirit, constituting a special “spiritual realm”
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For another thing, Herder’s rejection in On the Cognition and Sensation of hard and fast divisions between mental faculties, for example, between cognition and sensation or between cognition and volition, led to a similar rejection of such hard and fast divisions in Hegel.
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Whereas the early Herder of the Treatise on the Origin (like the Enlightenment before him) had implied that languages were mere aggregates of particular words/concepts, in the Ideas he came to emphasize that grammatical structure [Bau] plays a fundamental role in languages.
(1) As we have seen, Herder had already in early works such as On Diligence and the Fragments championed the doctrine that thought is essentially dependent on and bounded by language. In chapters of the Phenomenology of Spirit such as “Sense-certainty”, “Phrenology”, and “The Artificer”, as well as in other works from the Jena period, Hegel assumes the same doctrine (in addition taking over Herder’s further conception that oral language is more fundamental than written). (2) Especially in On the Cognition and Sensation, Herder had championed the theory that language, and therefore also thought, depends on a linguistic community. This position too plays a central role in Hegel’s Phenomenology of Spirit as well as in other works of his. (3) In his Treatise on the Origin Herder had argued that the whole human mind is dependent on thought and hence language, intimately involving them both. Similarly, Hegel in the Phenomenology of Spirit characterizes language as the very “existence [Dasein]” of the subject, or the mind, and he continues to champion this Herderian conception strongly in as late a text as the preface to the second edition of his Science of Logic (1832). (4) In keeping with that position, Herder had championed the thesis in On the Cognition and Sensation that not only language and therefore thought but also the subject depends on a linguistic community.
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Fourth, Herder exercised a strong influence on Hegel’s philosophy of history. Several components of this influence can be distinguished: (1) Herder had argued in This Too a Philosophy of History that history has a necessary course that works to fulfill a final purpose, then in the Ideas he had identified the purpose in question as the realization of (humanity or) reason. In the Phenomenology of Spirit and his lectures on the philosophy of history Hegel takes over this position and develops it further. (2) Especially in This Too a Philosophy of History Herder had emphasized that the human spirit, or mind, changes in profound ways over the course of history. Hegel adopts this position too, especially in the Phenomenology of Spirit and other works concerned with history. (3) In Attempt at a History of Lyric Poetry (1764), the Fragments (1767/8), and This Too a Philosophy of History (1774) Herder had developed a “genetic” method that undertakes to explain modern spiritual or mental phenomena in terms of their gradual development out of earlier historical origins and antecedents. Hegel subsequently took over this method, especially in his Phenomenology of Spirit (where he discusses his version of it in the preface). (4) Finally, in This Too a Philosophy of History Herder had extended the application of the concept of formation [Bildung] from its then normal use in connection with individuals and their education to humankind and its historical self-development as a whole as well. Hegel takes over this new broader application of the concept of formation in the Phenomenology of Spirit. (This list of Hegel’s debts to Herder is by no means exhaustive.) — link