For Peirce, the classification of various "sign relations" has the function of comprehensively and precisely characterizing all the possible modes in which we recognize and represent the world around us.

While Peirce often and strongly modified this classification in various stages of his philosophical development, the basic structure has remained the same. In the following, we shall first present Peirce¹s general definition of sign and continue by sketching his classification of possible sign relations as it was conclusive to him about the year 1903.
Basic for Peirce¹s concept of sign are the following three definitions:

1. Signs are bound into a triadic structure whose relata are determined in their relation to one another as follows:

• "A sign, or representamen, is something which stands to somebody for something in some respect or capacity. It addresses somebody, that is, creates in the mind of that person an equivalent sign, or perhaps a more developed sign. That sign which it creates I call the interpretant of the first sign. The sign stands for something, its object. It stands for that object, not in all respects, but in reference to a sort of idea, which I have sometimes called the ground of the representamen." (CP 2.228, 1897)
• As Liszka put it, each of this three elements of a sign relation "is mediated through the others: the ability of the sign to represent also requires, inherently, its power to be interpreted as a sign of that object in some respect; the ability of the sign to be interpreted can only work if it is interpreted as representing an object in some respect; and it can only be understood as representing an object in some respect if it is interpreted as representing an object as such." (p.19)

2. This relation must be read, so-to-say, in two directions, firstly as determination, and secondly as representation: the object "determines" the interpretant, mediated by the sign, and both the sign and the interpretant "represent" the object. As Parmentier says, these are "two opposed yet interlocking vectors involved in semiosis." (p.4)

Any interpretation of a sign represents a certain perspective on an object.

Objects can be represented by different signs, and any sign can be interpreted quite differently. Epistemologically, the triadic relation, with its three relata of object, sign, and interpretant is thus irreducible.

• "A Sign is anything which is related to a Second thing, its Object, in respect to a Quality, in such a way as to bring a Third thing, its Interpretant, into relation to the same Object, and that in such a way as to bring a Fourth into relation to that Object in the same form, ad infinitum." (CP 2.92, 1902)
• A sign¹s meaning is constituted in a sequence of interpretants which is in principle infinite.

As we have seen from the above sign triad, signs, for Peirce, are always integrated into triadic relations, i.e. they mediate between object and interpretant. The interpretation developed below now does not concern "signs" proper, but rather possible triadic relations in which the sign itself  is only one relatum besides object and interpretant.

The differentiation of sign relations proposed by Peirce now is based on that each of the three "branches" in the above figure can be given in three modes. This means: in each possible sign relation, the object relation (OR) can be given in three different modes, and this is true for the interpretant relation  (IR) and the sign itself (S) as well. One could designate OR, IR and S the "dimensions" of every sign relation, and the three modes of appearance of each sign dimension as "sign aspects:"

My view is that there are three modes of being. I hold that we can directly observe them in elements of whatever is at any time before the mind in any way. They are the being of positive qualitative possibility, the being of actual fact, and the being of law that will govern facts in the future.

Let us begin with considering actuality, and try to make out just what it consists in. If I ask you what the actuality of an event consists in, you will tell me that it consists in its happening then and there. The specifications then and there involve all its relations to other existents. The actuality of the event seems to lie in its relations to the universe of existents. A court may issue injunctions and judgments against me and I not care a snap of my finger for them. I may think them idle vapor. But when I feel the sheriff¹s hand on my shoulder, I shall begin to have a sense of actuality. Actuality is something brute. There is no reason in it. I instance putting your shoulder against a door and trying to force it open against an unseen, silent, and unknown resistance. We have a two-sided consciousness of effort and resistance, which seems to me to come tolerably near to a pure sense of actuality. On the whole, I think we have here a mode of being of one thing which consists in how a second object is. I call that Secondness.

Besides this, there are two modes of being that I call Firstness and Thirdness. Firstness is the mode of being which consists in its subject¹s being positively such as it is regardless of aught else. That can only be a possibility. For as long as things do not act upon one another there is no sense or meaning in saying that they have any being, unless it be that they are such in themselves that they may perhaps come into relation with others. The mode of being a redness, before anything in the universe was yet red, was nevertheless a positive qualitative possibility. And redness in itself, even if it be embodied, is something positive and sui generis. That I call Firstness. We naturally attribute Firstness to outward objects, that is we suppose they have capacities in themselves which may or may not be already actualized, which may or may not ever be actualized, although we can know nothing of such possibilities [except] so far as they are actualized.

Now for Thirdness. Five minutes of our waking life will hardly pass without our making some kind of prediction; and in the majority of cases these predictions are fulfilled in the event. Yet a prediction is essentially of a general nature, and cannot ever be completely fulfilled. To say that a prediction has a decided tendency to be fulfilled, is to say that the future events are in a measure really governed by a law. If a pair of dice turns up sixes five times running, that is a mere uniformity. The dice might happen fortuitously to turn up sixes a thousand times running. But that would not afford the slightest security for a prediction that they would turn up sixes the next time. If the prediction has a tendency to be fulfilled, it must be that future events have a tendency to conform to a general rule. "Oh," but say the nominalists, "this general rule is nothing but a mere word or couple of words!" I reply, "Nobody ever dreamed of denying that what is general is of the nature of a general sign; but the question is whether future events will conform to it or not. If they will, your adjective Œmere¹ seems to be ill-placed." A rule to which future events have a tendency to conform is ipso facto an important thing, an important element in the happening of those events. This mode of being which consists, mind my word if you please, the mode of being which consists in the fact that future facts of Secondness will take on a determinate general character, I call a Thirdness. (Peirce CP 1.23-26, 1903)

Peirce¹s theory of categories forms the basis for his differentiation shown above of three sign aspects, respectively, for each sign dimension: each of the three sign dimensions can be differentiated into a "first," a "second," and a "third" sign aspect. The triadic sign relations proper (cf. the first figure above) are formed by various combinations of sign aspects, one of each of the three "branches" mentioned above, i.e. one of OR, of S, and of IR. As can be easily seen from the second figure above, 3³, i.e. 27 combinations are mathematically possible. (Cf. CP 2.2264, 1903) This variety of possibilities assure a wealth of possible sign relations that really permits to characterize a multitude of modes in which we recognize and represent the world around us in a comprehensive and precise manner.

Among the mathematically possible 27 triadic combinations only 10, however, can really occur, as Peirce says. The reason for that results from combining the following points: (Cf. Lieb 1977  <1953>, p.161; Liszka, p.45)

1. If you consider the sign relation under the aspect of determination, this means that a sign will generally be determined, as we have seen, by an object, while itself determines its interpretant. This is to say that an interpretant will always be determined firstly by the kind of  sign itself (S) and, secondly, by the kind of the object relation (OR).
2. Peirce assumes that a First can only determine a First, and that a Third can only be determined by a Third. This means that if the sign itself (S) is a First, the object relation (OR) represented and the interpretant determined can only be a First as well. If the sign (S), against that, is a Second or a Third, there is a corresponding increase in possibilities of categorically determined sign aspects with regard to the other relata.

According to Peirce CP 2.254 ff, 1903; for a description of each sign relation cf. Liszka, pp.48-52.

It should be noted that the concern here is solely to separate analytically diverse classes of sign relations, whereas, in daily life, we are always confronted with combinations of different sign relations. Thus, in concrete or "singular" signs (SR_2-4) we are always simultaneously in contact with qualities and hence with "qualisigns" (SR₁)‹given in the materiality of the sign,‹and lawlike signs (SR_6-10), again, can only be present to us by their "replicas" or embodiments in "singular" signs (more on this below).

To present the order of these ten possible sign relation classes even more illustratively with regard to their categorical determination, a three-dimensional representation might be suggestive‹"three-dimensional" to be understood firstly in the sense of "spatial," and secondly in the sense of distinguishing between the three so-called "sign dimensions":

Every cube in this drawing represents a sign relation class (SR_1-10), i.e. a triad according to the first figure above. The three sign dimensions, resp. the three subrelations of the triads, are presented as the three spatial dimensions of each cube; depth referring to the dimension of the sign for itself (S_1-3), breadth referring to the relations between sign and object (OR_1-3), and height referring to the relations between sign and interpretant (IR_1-3). A similar three-dimensional model of the ten sign relations was developed by Farias and Queiroz (2000).

To reflect on this at first glance certainly rather abstract classification of sign relations, I should like to point out some particularly important points:

• The classification has not been obtained, as can already be seen from its schematic structure, simply from "observing" given signs, but it is constructed, the most important means of construction being, as has been shown, the theory of the three fundamental categories. Applying these will force, at every level of differentiation, ever new tripartitions‹the Peirce of later years develops even far more differentiated classifications going well beyond what has been presented here. (Cf. Lieb 1977  <1953>,  Müller 1994)
• One of the consequences of this constructivist moment is that the definition of the sign itself (S_1-3) within the sign triad is formulated in a way that is hardly accessible to our common use of language: the sign itself can be either "qualitative," "singular," or "lawlike," as is evident from the above figure. If it is "qualitative," we have to do with a sign relation (SR₁) that is a First in all its aspects. This, however, means that such a sign can never be given as a "sign" in the sense of a perceptible object, it is pure "qualitative possibility." The same is true, in another way, for the "lawlike" signs as well, for the character of this "lawlikeness" itself cannot be "seen." What is seen in the presence of a symbol like  or "H₂O" as a lawlike sign (SR₈), for instance, is no more than a "replica" or token of the lawlike type, and hence a "singular" sign (SR₃ or SR₄). Singular signs alone are perceptible as concrete and individual objects by virtue of their secondness. Nevertheless, Peirce consciously speaks of First or Third signs which he needs to mention the qualitative side of perceptible signs and the side or their meaning:

 
The difference between a legisign and a qualisign, neither of which is an individual thing, is that a legisign has a definite identity, though usually admitting a great variety of appearances. Thus, &, and, and the sound are all one word. The qualisign, on the other hand, has no identity. It is the mere quality of an appearance and is not exactly the same throughout a second. Instead of identity, it has great similarity, and cannot differ much without being called quite another qualisign.  (Peirce CP 8.334, 1904)

• From this follows in particular that Peirce¹s semiotics is not concerned with an "empirical study of signs," but rather with a concept of sign which may serve as a foundation for a general epistemology that is not only interested in perceptible singular things, but also in qualities, possibilities, laws, rules, and conventions.

The Object Relation: Icon, Index, and Symbol

1. When a sign "partakes in the characters of the object, I call the sign an Icon." (CP 4.531) In this sense, iconic signs exclusively represent similarities. Iconic signs are important mainly if the concern is to depict relations. This is true for photos, cartoons, or footprints, which are iconic with regard to their similarity with what is being depicted, but also for diagrams which map relations or data structures.
2. When the sign is "really and in its individual existence connected with the individual object, ... I call the sign an Index." (CP 4.531) If smoke is understood to be a sign of fire, then this sign is an indexical sign, for "the index ... forces the attention to the particular object intended without describing it." (CP 1.369) There is an immediate dyadic relation between smoke and fire which would exist even if there were nobody to interpret smoke as a sign of fire.

As far as the interpretation must be given only potentially in case of the indexical sign‹as opposed to the symbol‹it is true that the indexical sign has no "meaning" in the sense proper; its only "content" being the characterization of a dyadic relation between an object and a sign; it "forces" attention, as Peirce says in the above quote, into a certain direction, but does not describe anything.
3. When the sign "will be interpreted by more or less approximate certainty as denoting the object, in consequence of a habit (which term I use as including a natural disposition, ...I call the sign a Symbol." (CP 4.531) The only signs which can represent meaning according to Peirce, are symbolic. Wherever we have to do with meaning, we have to do with symbols. It is constitutive for the symbolic sign that there would be no relation between it and the object it designates if there were no interpretant which establishes this relation in the first place. In contrast to the indexical, dyadic relation between sign and object, this here is no "factual" relation between the two, but a relation mediated by the interpretant. A symbol requires interpretation. Thus, only the symbol represents a genuinely triadic relation.

The symbol will designate something only if there is a general convention, a specific use, and this means: if there is an "interpretant" which establishes a relation between sign and object. The sign , for instance, is a symbol in the context of geometry: it can fulfill its function only if there is an interpretant who is able to mediate the relation between this sign and the object it designates. Or, vice versa:  will function only if it determines an interpretant precisely so as to make this interpretant mediate the symbol with what it symbolizes. On the basis of his / her own habits, a person must be able to see a as a symbol.
The essential thing about the symbol is not, however, that it is established by convention, but that there is a "habit" which, as interpretant, constitutes a relation between symbol and object. As Peirce intends to extend the concept of habit to include "natural dispositions," also an instinctive reaction could be an interpretant. In short: only that sign is symbolic that fulfills its function of designating an object on the basis of a general law. Thus, it is clear that symbols, for Peirce, can only be lawlike signs or "Legisigns" (see below). This contradicts or usual use of language, for if we say this  drawn here is a symbol, Peirce would say what has been drawn here is merely a replica of a symbol, but not this symbol itself. For Peirce, the symbol is exclusively what determines the meaning of the  drawn, making possible to understand it. If it has no meaning, or if it is not understood, it is no symbol.

Considering the Sign Itself: Qualisign, Sinsign, and Legisign
According to the distinction between the three categories, the sign in itself, i.e. the relatum in the above triadic sign relation called "sign," can either be "a mere quality, ... an actual existent, or ... a general law." (CP 2.243) Accordingly, Peirce speaks of "Qualisign," "Sinsign" (from SINgular), and "Legisign" (from Latin LEX):

A Qualisign is a quality which is a Sign. It cannot actually act as a sign until it is embodied; but the embodiment has nothing to do with its character as a sign.

A Sinsign (where the syllable sin is taken as meaning "being only once," as in single, simple, Latin semel, etc.) is an actual existent thing or event which is a sign. It can only be so through its qualities; so that it involves a qualisign, or rather, several qualisigns. But these qualisigns are of a peculiar kind and only form a sign through being actually embodied.

A Legisign is a law that is a Sign. This law is usually established by men. Every conventional sign is a legisign [but not conversely]. It is not a single object, but a general type which, it has been agreed, shall be significant. Every legisign signifies through an instance of its application, which may be termed a Replica of it. Thus, the word "the" will usually occur from fifteen to twenty-five times on a page. It is in all these occurrences one and the same word, the same legisign. Each single instance of it is a Replica. The Replica is a Sinsign. Thus, every Legisign requires Sinsigns. But these are not ordinary Sinsigns, such as are peculiar occurrences that are regarded as significant. Nor would the Replica be significant if it were not for the law which renders it so. (CP 2.244-246)

Peirce confines the term "association"

to the process whereby one idea acquires the power to attract another from the depths of memory to the surface of consciousness, and to the habit resulting from this process, (Peirce MS 318, CSP 45 ff.)

the action by which, an association having once been established, that act by which, in accordance with it, one idea calls up another. (ibid.)

The significance of this distinction lies in that it may serve to describe processes of learning: if a child is induced to associate six markers or six apples or six circles with the idea of "six," then every time six objects suggest to the child the idea of "six," we are not in the presence of a random connection, but rather of one determined by law.

Semiotically, this means that the interpretation of six objects by the idea of "six," or vice versa the interpretation of the idea of "six" by the idea of six objects, is mediated by a Legisign. As opposed to that, using the Arabic number "6," or the Roman cipher "VI," or the word "six" to designate six objects would be using a Sinsign that relates to the Legisign as a token to a type. It is only the "expression" of a lawlike connection, but not this law itself.

The Interpretant Relation: Rhema, Dicisign and Argument
The third sign dimension concerns three ways how an interpretant "represents" a sign. With regard to that, the classification developed in 1903 by Peirce has clearly been inspired by language, "rhema" being "word" in Greek, "dici" alluding to  judgment or proposition, and "argument" to an argumentative chain of propositions. This will perhaps be understood best if one distinguishes whether something is represented, at the side of the interpretant, as something singular‹as a substantive, a predicate, or even as a singular feeling‹or as a dyadic connection as in a simple proposition, or as a triadic connection mediated by a law. The latter would not only include "arguments" as logical relations, but also phenomena like causality. Thus, Peirce¹s concept of "argument" is applicable to processes of nature as well (cf. NEM IV, p.254, 1904; for some ontological implications of Peirce¹s semiotics cf. Parker 1994). It should be noted at this point that what Peirce called "argument" here is a Third in all of its sign aspects, and thus the most "abstract" and lawlike sign relation (as it is also indicated by its position at the "top" of our cube-model of sign dimensions).
After the revision of his own sign classification which he began about 1904 (cf. Pape 1990, p.46-52; Müller 1994), Peirce, in particular, differentiated his concept of the interpretant anew. Instead of reducing it to linguistic matters alone, the interpretant is now defined more generally as "the proper significant outcome of a sign," or as the "proper significate effects of signs." (CP 5.475, 5.473; 1907) The distinction between different "significate effects" now corresponds to the distinction between different interpretants: on the one hand, Peirce distinguishes in his later texts between the "emotional," the "energetic," and the "logical" interpretants, and on the other between the "immediate," the "dynamical," and the "final" interpretants. T.L. Short has argued extensively that these two distinctions (which evidently have been formulated again in accordance with the basic distinction into three categories) are two irreducible trichotomies. A note in Peirce¹s "Logic" notebook (MS 339) shows that both the immediate and the dynamical interpretant can be emotional, energetic, and logical, respectively, while the final interpretant obviously can only be grasped as a logical interpretant.

Peirce defines the Immediate Interpretant here as

...the immediate pertinent possible effect in its unanalyzed primitive entirety. ... It may be a quality of feeling, more or less vague, or an idea of an effort or experience awaked by the air of previous experience or it may be the idea of a form or anything of a general type.

The Dynamical Interpretant is the actual effect produced upon a given interpreter on a given occasion in a given stage of his consideration of the sign. This again may be 1^st a feeling merely, or 2^nd an action, or 3^rd a habit. ...

The Final Interpretant is the ultimate effect of the sign, so far as it is intended or destined, from the character of the sign, being more or less of a habitual and formal nature." (MS 339, 1906 Oct. 23, p.288r, 289r = SEM III, p.224 f.).

As becomes clear already from the concept of "significate effect," the basic idea of Peirce¹s pragmatism consists in that a sign¹s meaning cannot be separated from its effects. But a distinction has to be made between three classes of effects: the concrete significate effect as a Second, that is singular emotions, actions, or habits, as they are summarized by the concept of "dynamical interpretant;" the qualitative possibility of such effects; and the general significate effect as a Third, that is the final-logical interpretant. According to Peirce,

pragmatism says ... the meaning of any sign for anybody consists in the way he reacts to the sign. When the captain of infantry gives the word "Ground arms!" the dynamic Interpretant is in the thump of the muskets on the ground, or rather it is the Act of their Minds. ... The Final Interpretant does not consist in the way in which any mind does act but in the way in which every mind would act. That is, it consists in a truth which might be expressed in a conditional proposition of this type: "If so and so were to happen to any mind this sign would determine that mind to such and such conduct." (Peirce CP 8.315, 1909)