Charles Sanders Peirce began writing on semiotics, which he also called semeiotics, meaning the philosophical study of signs, in the 1860s, around the time that he devised his system of three categories. During the 20th century, the term "semiotics" was adopted to cover all tendencies of sign researches, including Ferdinand de Saussure's semiology, which began in linguistics as a completely separate tradition.

Peirce adopted the term semiosis (or semeiosis) and defined it to mean an "action, or influence, which is, or involves, a cooperation of three subjects, such as a sign, its object, and its interpretant, this trirelative influence not being in any way resolvable into actions between pairs."1906, EP 2:411 and CP 5.484. Peirce went on to say: "Σημείωσις [Sêmeíôsis] in Greek of the Roman period, as early as Cicero's time, if I remember rightly, meant the action of almost any kind of sign; and my definition confers on anything that so acts the title of a 'sign.'" See Σημείωσις in the Liddell & Scott Ancient Greek Lexicon at the Perseus Digital Library. This specific type of triadic relation is fundamental to Peirce's understanding of logic as formal semiotic. By "logic" he meant philosophical logic. He eventually divided (philosophical) logic, or formal semiotics, into (1) speculative grammar, or stechiology on the elements of semiosis (sign, object, interpretant), how signs can signify and, in relation to that, what kinds of signs, objects, and interpretants there are, how signs combine, and how some signs embody or incorporate others; (2) logical critic, or logic proper, on the modes of inference; and (3) speculative rhetoric, or methodeutic, the philosophical theory of inquiry, including his form of pragmatism.

His speculative grammar, or stechiology, is this article's subject.

Peirce conceives of and discusses things like representations, interpretations, and assertions broadly and in terms of philosophical logic, rather than in terms of psychology, linguistics, or social studies. He places philosophy at a level of generality between mathematics and the special sciences of nature and mind, such that it draws principles from mathematics and supplies principles to special sciences.For Peirce's definitions of philosophy, see for instance "A Syllabus of Certain Topics of Logic", CP 1.183-186, 1903 and "Minute Logic", CP 1.239-241, 1902. See Peirce's definitions of philosophy at CDPT under "Cenoscopy" and "Philosophy". On the one hand, his semiotic theory does not resort to special experiences or special experiments in order to settle its questions. On the other hand, he draws continually on examples from common experience, and his semiotics is not contained in a mathematical or deductive system and does not proceed chiefly by drawing necessary conclusions about purely hypothetical objects or cases. As philosophical logic, it is about the drawing of conclusions deductive, inductive, or hypothetically explanatory. Peirce's semiotics, in its classifications, its critical analysis of kinds of inference, and its theory of inquiry, is philosophical logic studied in terms of signs and their triadic relations as positive phenomena in general.

Peirce's semiotic theory is different from Saussure's conceptualization in the sense that it rejects his dualist view of the Cartesian self. He believed that semiotics is a unifying and synthesizing discipline. More importantly, he included the element of "interpretant" into the fundamental understanding of the sign.

Semiotic elements

Here is Peirce's definition of the triadic sign relation that formed the core of his definition of logic: Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C. (Peirce 1902, NEM 4, 20–21).

This definition, together with Peirce's definitions of correspondence and determination, is sufficient to derive all of the statements that are necessarily true for all sign relations. Yet, there is much more to the theory of signs than simply proving universal theorems about generic sign relations. There is also the task of classifying the various species and subspecies of sign relations. As a practical matter, of course, familiarity with the full range of concrete examples is indispensable to theory and application both.

In Peirce's theory of signs, a sign is something that stands in a well-defined kind of relation to two other things, its object and its interpretant sign. Although Peirce's definition of a sign is independent of psychological subject matter and his theory of signs covers more ground than linguistics alone, it is nevertheless the case that many of the more familiar examples and illustrations of sign relations will naturally be drawn from linguistics and psychology, along with our ordinary experience of their subject matters.

For example, one way to approach the concept of an interpretant is to think of a psycholinguistic process. In this context, an interpretant can be understood as a sign's effect on the mind, or on anything that acts like a mind, what Peirce calls a quasi-mind. An interpretant is a process of interpretation, one of the types of activity that falls under the heading of semiosis. One usually says that a sign stands for an object to an agent, an interpreter. In the upshot, however, it is the sign's effect on the agent that is paramount. This effect is what Peirce called the interpretant sign, or the interpretant for short. An interpretant in its barest form is a sign's meaning, implication, or ramification, and especial interest attaches to the types of semiosis that proceed from obscure signs to relatively clear interpretants. In logic and mathematics the most clarified and most succinct signs for an object are called canonical forms or normal forms. The interpretant, in Peirce's conceptualization, is not the user of the sign but the "proper significate effect" or that mental concept produced by both the sign and by the user's experience of the object.

Peirce argued that logic is the formal study of signs in the broadest sense, not only signs that are artificial, linguistic, or symbolic, but also signs that are semblances or are indexical such as reactions. Peirce held that "all this universe is perfused with signs, if it is not composed exclusively of signs",Peirce, C.S., CP 5.448 footnote, from "The Basis of Pragmaticism" in 1906. along with their representational and inferential relations. He argued that, since all thought takes time, all thought is in signs:

To say, therefore, that thought cannot happen in an instant, but requires a time, is but another way of saying that every thought must be interpreted in another, or that all thought is in signs. (Peirce, 1868"Questions Concerning Certain Faculties Claimed for Man" (Arisbe Eprint), Journal of Speculative Philosophy vol. 2 (1868), pp. 103-114. Reprinted CP 5.213-263, the quote is from para. 253.) Thought is not necessarily connected with a brain. It appears in the work of bees, of crystals, and throughout the purely physical world; and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there. Consistently adhere to that unwarrantable denial, and you will be driven to some form of idealistic nominalism akin to Fichte's. Not only is thought in the organic world, but it develops there. But as there cannot be a General without Instances embodying it, so there cannot be thought without Signs. We must here give "Sign" a very wide sense, no doubt, but not too wide a sense to come within our definition. Admitting that connected Signs must have a Quasi-mind, it may further be declared that there can be no isolated sign. Moreover, signs require at least two Quasi-minds; a Quasi-utterer and a Quasi-interpreter; and although these two are at one (i.e., are one mind) in the sign itself, they must nevertheless be distinct. In the Sign they are, so to say, welded. Accordingly, it is not merely a fact of human Psychology, but a necessity of Logic, that every logical evolution of thought should be dialogic. (Peirce, 1906"Prolegomena To an Apology For Pragmaticism", pp. 492–546, The Monist, vol. XVI, no. 4 (mislabeled "VI"), Oct. 1906, see p. 523. Reprinted CP 4.530–572; see para. 551 Eprint. .)

Sign relation

Signhood is a way of being in relation, not a way of being in itself. Anything is a sign—not as itself, but in some relation to another. The role of sign is constituted as one role among three: object, sign, and interpretant sign. It is an irreducible triadic relation; the roles are distinct even when the things that fill them are not. The roles are but three: a sign of an object leads to interpretants, which, as signs, lead to further interpretants. In various relations, the same thing may be sign or semiotic object. The question of what a sign is depends on the concept of a sign relation, which depends on the concept of a triadic relation. This, in turn, depends on the concept of a relation itself. Peirce depended on mathematical ideas about the reducibility of relations—dyadic, triadic, tetradic, and so forth. According to Peirce's Reduction Thesis,See "The Logic of Relatives", The Monist, Vol. 7, 1897, pp. 161-217, see p. 183 (via Google Books with registration apparently not required). Reprinted in the Collected Papers, vol. 3, paragraphs 456-552, see paragraph 483. (a) triads are necessary because genuinely triadic relations cannot be completely analyzed in terms of monadic and dyadic predicates, and (b) triads are sufficient because there are no genuinely tetradic or larger polyadic relations—all higher-arity n-adic relations can be analyzed in terms of triadic and lower-arity relations and are reducible to them. Peirce and others, notably Robert W. Burch (1991) and Joachim Hereth Correia and Reinhard Pöschel (2006), have offered proofs of the Reduction Thesis.* Burch, Robert (1991), A Peircean Reduction Thesis: The Foundations of Topological Logic, Texas Tech University Press, Lubbock, Texas Anellis, Irving (1993) "Review of A Peircean Reduction Thesis: The Foundations of Topological Logic by Robert Burch" in Modern Logic v. 3, n. 4, 401-406, Project Euclid Open Access PDF 697 KB. Criticism and some suggestions for improvements. Anellis, Irving (1997), "Tarski's Development of Peirce's Logic of Relations" (Google Book Search Eprint) in Houser, Nathan, Roberts, Don D., and Van Evra, James (eds., 1997), Studies in the Logic of Charles Sanders Peirce. Anellis gives an account of a Reduction Thesis proof discussed and presented by Peirce in his letter to William James of August 1905 (L224, 40-76, printed in Peirce, C. S. and Eisele, Carolyn, ed. (1976), The New Elements of Mathematics by Charles S. Peirce, v. 3, 809-835). Hereth Correia, Joachim and Pöschel, Reinhard (2006), "The Teridentity and Peircean Algebraic Logic" in Conceptual Structures: Inspiration and Application (ICCS 2006): 229-246, Springer. Frithjof Dau "the strong version" of proof of Peirce's Reduction Thesis. John F. Sowa in the same discussion that an explanation in terms of conceptual graphs is sufficiently convincing about the Reduction Thesis for those without the time to understand what Peirce was saying. In 1954 W. V. O. Quine claimed to prove the reducibility of larger predicates to dyadic predicates, in Quine, W.V.O., "Reduction to a Dyadic Predicate", Selected Logic Papers. According to Peirce, a genuinely monadic predicate characteristically expresses quality. A genuinely dyadic predicate—reaction or resistance. A genuinely triadic predicate—representation or mediation. Thus Peirce's theory of relations underpins his philosophical theory of three basic categories (see below).