In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place.
Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian product A × B of some sets A and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product A × B × C of three sets A, B and C.
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." 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.