Since so many of the posts on this blog refer to Peirce’s theory of signs––either explicitly or ex silentio––perhaps a synoptic view of what he called semeiotic would be of use to readers. What follows has been adapted from the seminal work of the late dean of Peirce studies, Max H. Fisch, Peirce, Semeiotic, and Pragmatism (Bloomington: Indiana University Press, 1986, pp. 324-326). (The key to the abbreviations of the volumes of Peirce’s writings can be found at the end of this post.)

The first published sketch of Peirce’s semeiotic was in a paper “On a New List of Categories,” which he presented to the American Academy of Arts and Sciences on May 14, 1867. Forty years later he described this paper as the outcome of “the hardest two years’ mental work that I have ever done in my life” (CP 1.561). He first establishes, in place of Aristotle’s ten categories and Kant’s twelve, a new list of three: Quality, Relation, Representation. He then uses these categories to distinguish: (i) three kinds of representations [i.e., SIGNS]––likenesses (which he will later call icons), indices, and symbols; (2) a trivium of conceivable sciences—formal grammar, logic, and formal rhetoric; (3) a general division of symbols, common to all three of these sciences—terms, propositions, and arguments; and (4) three kinds of argument, distinguished by their three relations between premisses and conclusion—deduction (symbol), hypothesis (likeness), induction (index) (W 2:491-59; CP 1.545-59).
Peirce is a logician, and he concerns himself with semeiotic only so far as is necessary to place logic within the larger framework of that one of the three most general kinds of science that Locke, following the ancient Greeks, had distinguished. To that objection, however, it may fairly be replied that at no time of his life did Peirce set any limit to the intensity of cultivation of the larger field of semeiotic that would be advantageous for purposes of logic, even if the cultivating had to be done by logicians themselves because, for the time being, they were the only semeioticians.
In any case, it was not enough in Peirce’s eyes for semeiotic to provide a pigeonhole for logic in the classification of the sciences. This became fully apparent in 1868-69 in a series of three articles in the Journal of Speculative Philosophy: “Questions Concerning Certain Faculties Claimed for Man,” “Some Consequences of Four Incapacities,” and “Grounds of Validity of the Laws of Logic: Further Consequences of Four Incapacities” (W 2:193-272; CP 5.213-357).
The first two papers are there for the sake of the third. The upshot of the series is a theory of the validity of the laws of logic, including those of’ the logic of science (that is, of hypothesis and induction) as well as those of the logic of mathematics (that is, of deduction). Yet the first paper is in the form of a medieval quaestio, a disputed question, and the second begins with a four-point statement of “the spirit of Cartesianism,” followed by an opposed four-point statement of the spirit of the scholasticism that it displaced. In respect of these four antitheses, “modern science and modern logic” are closer to the spirit of scholasticism. The first paper was “written in this spirit of opposition to Cartesianism.” It was meant to illustrate as well as to commend the “multiform argumentation of the Middle Ages.” It resulted in four denials:
1. We have no power of Introspection, but all knowledge of the internal world is derived by hypothetical reasoning from our knowledge of external facts.
2. We have no power of Intuition, but every cognition is determined logically by previous cognitions.
3. We have no power of thinking without signs.
4. We have no conception of the absolutely incognizable. (CP 5.265)
These propositions cannot be regarded as certain, Peirce says; and the second paper puts them to the further test of tracing out some of their consequences. The third paper then constructs a theory of the validity of the laws of logic in the form of “further consequences” of these “four incapacities.”
The central positive doctrine of the whole series is that “all thought is in signs” (5.253). Every thought continues another and is continued by still another. There are no uninferred premisses and no inference-terminating conclusions. Inferring is the sole act of cognitive mind. No cognition is adequately or accurately described as a two-term or dyadic relation between a knowing mind and an object known, whether that be an intuited first principle or a sense-datum, a “first impression of sense” (5.291). Cognition is a minimally three-termed or triadic relation (5.283). The sign-theory of cognition thus entails rejection not only of Cartesian rationalism but also of British empiricism.
The sign-theory of cognition leads into a semeiotic theory of the human self, “the
man-sign” (5.313), and thence into a social theory of logic. “When we think, then, we ourselves, as we are at that moment, appear as a sign” (5.383); “the word or sign which man uses is the man himself” (5.314). “Finally, no present actual thought (which is a mere feeling) has any meaning, any intellectual value; for this lies not in what is actually thought, but in what this thought may be connected with in representation by subsequent thoughts; so that the meaning of a thought is altogether something virtual” (5.289). “Accordingly, just as we say that a body is in motion, and not that motion is in a body, we ought to say that we are in thought and not that thoughts are in us” (5.289n1).
“The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of me and you. Thus, the very origin of the conception of reality shows that this conception essentially involves the notion of a COMMUNITY, without definite limits, and capable of an indefinite increase of knowledge” (5.311). “So the social principle is rooted intrinsically in logic” (5.354).
Along the way, with the help of his three categories, Peirce’s doctrine of signs is worked out in greater detail in these three papers, and especially in the second of them.
The semeiotic thus founded was semeiotic as viewed from the standpoint of logic and studied for the purposes of logic, and more particularly for those of the logic of science rather than for those of the logic of mathematics. But it was a semeiotic that included logic.

PEIRCE, Charles Sanders.
1931-1958. Collected Papers of Charles Sanders Peirce, vols. 1-8, ed. Charles Hartshorne and Paul Weiss (vols. 1-6) and Paul Weiss (vols. 7-8). (Cambridge, Mass: Harvard University Press). Abbreviated CP [references by volume and paragraph number]
1982-2009. Writings of Charles S. Peirce: A Chronological Edition, ed. Max H. Fisch et al., vols. 1-6, 8. (Bloomington: Indiana University Press). Abbreviated W [references by volume and page number]
1992-1998. The Essential Peirce, vols. 1-2., ed Nathan Houser and Christian Kloesel (vol. 1) and Peirce Edition Project (vol. 2). (Bloomington: Indiana University Press). Abbreviated EP [references by volume and page number]