I had been reading Peirce for a decade or more before I found a math-strength definition of signs and sign relations. A lot of the literature on semiotics takes almost any aperçu Peirce penned about signs as a “definition” but barely a handful of those descriptions are consequential enough to support significant theory. For my part, the definition of a sign relation coming closest to the mark is one Peirce gave in the process of defining logic itself. Two variants of that definition are linked and copied below.
Selections from C.S. Peirce, “Carnegie Application” (1902)
No. 12. On the Definition of Logic
Logic will here be defined as formal semiotic. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. 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. It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized. (NEM 4, 20–21).
No. 12. On the Definition of Logic [Earlier Draft]
Logic is formal semiotic. A sign is something, A, which brings something, B, its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, C, its object, as that in which itself stands to C. This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time. It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of Weierstrassian severity, and that is perfectly evident. The word “formal” in the definition is also defined. (NEM 4, 54).
- Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75), published in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73. Online.