Charles Peirce

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Western Philosophy
19th/20th century philosophy
Charles Sanders Peirce theb3558.jpg
Name: Charles Sanders Peirce
Birth: September 10, 1839 (Cambridge, Massachusetts)
Death: April 19, 1914 (Milford, Pennsylvania)
School/tradition: Pragmaticism (Pragmatism)
Main interests
Metaphysics, Logic, Epistemology, Mathematics, Science
Notable ideas

Charles Sanders Peirce (pronounced purse), (September 10, 1839 – April 19, 1914) was an American polymath, born in Cambridge, Massachusetts. Although educated as a chemist and employed as a scientist for 30 years, it is for his contributions to logic, mathematics, philosophy, and the theory of signs, or semeiotic, that he is largely appreciated today. The philosopher Paul Weiss, writing in the Dictionary of American Biography for 1934, called Peirce "the most original and versatile of American philosophers and America's greatest logician"[1]

Peirce was largely ignored during his lifetime, and secondary literature on his works was scant until after World War II. Much of his huge output is still unpublished. An innovator in fields such as mathematics, research methodology, the philosophy of science, epistemology, and metaphysics, he considered himself a logician first and foremost. While he made major contributions to formal logic, "logic" for him encompassed much of what is now called the philosophy of science and epistemology. He, in turn, saw logic as a branch of semiotics, of which he is a founder. In 1886, he saw that logical operations could be carried out by electrical switching circuits, an idea used decades later to produce digital computers.


The only Peirce biography in English is Joseph Brent's Charles Peirce, A Life, (1998). Charles Sanders Peirce was born September 10, 1839, the son of Sarah Hunt Mills and Benjamin Peirce, a professor of astronomy and mathematics at Harvard University, who was perhaps the first serious research mathematician in America. When he was 12 years old, Charles read an older brother's copy of Richard Whately's Elements of Logic, then the leading English language text on the subject, and acquired a lifelong fascination with logic and reasoning. He went on to obtain a BA and MA from Harvard, and in 1863 Harvard's Lawrence Scientific School awarded him its first M.Sc. in chemistry. This last degree was awarded summa cum laude; otherwise his academic record was undistinguished. At Harvard, he began lifelong friendships with Francis Ellingwood Abbot, Chauncey Wright, and William James. One of his Harvard instructors, Charles William Eliot, formed an unfavorable opinion of Peirce. This opinion proved fateful, because Eliot, while President of Harvard 1869-1909—a period encompassing nearly all of Peirce's working life—repeatedly vetoed having Peirce employed in any capacity at Harvard.

United States Coast Survey

Between 1859 and 1891, Charles was intermittently employed in various scientific capacities by the United States Coast Survey, where he enjoyed the protection of his highly influential father until the latter's death in 1880. This employment exempted Charles from having to take part in the Civil War. It would have been very awkward for him to do so, as the wealthy Boston Peirce family sympathized with the Confederacy. At the Survey, he worked mainly in geodesy and in gravimetry, refining the use of pendulums to determine small local variations in the strength of the earth's gravity. The Survey sent him to Europe five times, the first in 1871, as part of a group dispatched to observe a solar eclipse. While in Europe, he sought out Augustus De Morgan, William Stanley Jevons, and William Kingdon Clifford, British mathematicians and logicians whose interests resembled his own. From 1869 to 1872, he was employed as an Assistant in Harvard's astronomical observatory, doing important work on determining the brightness of stars and the shape of the Milky Way.[2] In 1878, he was the first to define the meter as so many wavelengths of light of a certain frequency, the definition employed until 1983[3].

During the 1880s, Peirce's became increasingly indifferent to bureaucratic detail, and the quality and timeliness of his Survey work suffered. Peirce took years to write reports that he should have completed in a few months. Meanwhile, he wrote hundreds of logic, philosophy, and science entries for the Century Dictionary. In 1885, an investigation by the Allison Commission exonerated Peirce, but led to the dismissal of Superintendent Julius Hilgard and several other Coast Survey employees for misuse of public funds. In 1891, Peirce resigned from the Coast Survey, at the request of Superintendent Thomas Corwin Mendenhall. He never again held regular employment.

Johns Hopkins University

In 1879, Peirce was appointed Lecturer in logic at the new Johns Hopkins University in Baltimore. That university was strong in a number of areas that interested him, such as philosophy; (Royce and Dewey earned their PhDs at Hopkins), psychology (taught by G. Stanley Hall and studied by Joseph Jastrow, who coauthored a landmark empirical study with Peirce), and mathematics (taught by J. J. Sylvester, who came to admire Peirce's work on mathematics and logic). This untenured position proved to be the only academic appointment Peirce ever held.

Brent, his biographer, documents something Peirce never suspected; his efforts to obtain academic employment, grants, and scientific respectability were repeatedly frustrated by the covert opposition of a major American scientist of the day, Simon Newcomb. A difficult personality may have contributed to Peirce's difficulty in finding academic employment. Brent conjectures that Peirce may have been manic-depressive, claiming that Peirce experienced eight nervous breakdowns between 1876 and 1911. Brent also believes that Peirce tried to alleviate his symptoms with ether, morphine, and cocaine.

Peirce's personal life also proved a grave handicap. His first wife, Harriet Melusina Fay, from the prominent Cambridge family of Reverend Charles Fay, whom he married in October 1863, had left him in 1875. Peirce soon began living openly with a woman whose maiden name and nationality remain uncertain to this day (the best guess is that her name was Juliette Froissy Pourtalès and that she was French), but did not marry her until his divorce with Harriet became final in 1883. That year, Simon Newcomb pointed out to a Johns Hopkins trustee that Peirce, while a Hopkins employee, had lived and traveled with a woman to whom he was not married. The ensuing scandal led to his dismissal. Just why Peirce's later applications for academic employment at Clark University, University of Wisconsin, University of Michigan, Cornell University], Stanford University, and the University of Chicago were all unsuccessful can no longer be determined. Presumably, his having lived with Juliette for a number of years while still legally married to Harriet led him to be deemed morally unfit for academic employment anywhere in the United States. Peirce had no children by either marriage.


In 1887, Peirce spent part of his inheritance from his parents to purchase 2000 rural acres near Milford, Pennsylvania, land which never yielded an economic return. On that land, he built a large house which he named "Arisbe," where he spent the rest of his life, writing prolifically. Much of his writing remains unpublished to this day. His insistence on living beyond his means soon led to serious financial and legal difficulties. Peirce spent much of the last two decades of his life so destitute that he could not afford heat in winter, and his only food was old bread kindly donated by the local baker. Unable to afford new stationery, he wrote on the verso side of old manuscripts. An outstanding warrant for assault and unpaid debts led to his being a fugitive in New York City for a while. Several people, including his brother James Mills Peirce and his neighbors, relatives of Gifford Pinchot, settled his debts and paid his property taxes and mortgage.

Peirce did some scientific and engineering consulting and wrote a good deal for meager pay, primarily dictionary and encyclopedia entries, and reviews for The Nation (with whose editor, Wendell Phillips Garrison he became friendly). He did translations for the Smithsonian Institution, at the instigation of its director, Samuel Langley. Peirce also did substantial mathematical calculations for Langley's research on powered flight. Hoping to make money, Peirce tried his hand at inventing, and began but did not complete a number of books. In 1888, President Grover Cleveland appointed him to the Assay Commission. From 1890 onwards, he had a friend and admirer in Judge Francis C. Russell of Chicago, who introduced Peirce to Paul Carus and Edward Hegeler, the editor and owner, respectively, of the pioneering American philosophy journal The Monist, which eventually published a number of his articles. He applied to the newly formed Carnegie Institution for a grant to write a book summarizing his life's work. This application was doomed; his nemesis Newcomb served on the Institution's executive committee, and its President had been the President of Johns Hopkins at the time of Peirce's dismissal.

The one who did the most to help Peirce in these desperate times was his old friend William James, who dedicated his book of essays The Will to Believe (1896) to Peirce, and who arranged for Peirce to be paid to give four series of lectures at or near Harvard. Most important, each year from 1898 until his death in 1910, James would write to his friends in the Boston academic circles, asking that they make a financial contribution to help support Peirce. Peirce reciprocated by designating James's eldest son as his heir should Juliette predecease him, and by adding "Santiago," "Saint James" in Spanish, to his full name[4].

Peirce died destitute in Milford, Pennsylvania, on April 19, 1914, 20 years before his widow.


Bertrand Russell once said about Peirce, "Beyond doubt… he was one of the most original minds of the later nineteenth century, and certainly the greatest American thinker ever." (Yet his Principia Mathematica does not mention Peirce.) A. N. Whitehead, while reading some of Peirce's unpublished manuscripts soon after arriving at Harvard in 1924, was struck by how Peirce had anticipated his own "process" thinking. (On Peirce and process metaphysics, see the chapter by Lowe in Moore and Robin, 1964.) Karl Popper viewed Peirce as "one of the greatest philosophers of all times." Nevertheless, Peirce's accomplishments were not immediately recognized. His imposing contemporaries William James and Josiah Royce admired him, and Cassius Jackson Keyser at Columbia University and C. K. Ogden wrote about Peirce with respect, but he received little pubic recognition.

The first scholar to give Peirce his considered professional attention was Royce's student Morris Raphael Cohen, the editor of a 1923 anthology of Peirce's writings titled Chance, Love, and Logic, and the author of the first bibliography of Peirce's scattered writings. John Dewey had had Peirce as an instructor at Johns Hopkins, and from 1916 onwards, Dewey's writings repeatedly mention Peirce with deference. His 1938 Logic: The Theory of Inquiry reflects Peirce’s theories. The publication of the first six volumes of the Collected Papers (1931-1935), the most important event to date in Peirce studies and one Cohen made possible by raising the needed funds, did not lead to an immediate outpouring of secondary studies. The editors of those volumes, Charles Hartshorne and Paul Weiss, did not become Peirce specialists. Early landmarks of the secondary literature include the monographs by Buchler (1939), Feibleman (1946), and Goudge (1950), the 1941 Ph.D. thesis by Arthur Burks (who went on to edit volumes 7 and 8 of the Collected Papers), and the edited volume Wiener and Young (1952). The Charles S. Peirce Society was founded in 1946; its Transactions, an academic journal specializing in Peirce, pragmatism, and American philosophy, has appeared since 1965.

In 1949, while doing unrelated archival work, the historian of mathematics Carolyn Eisele (1902-2000) chanced on an autograph letter by Peirce. She began forty years of research on Peirce as a mathematician and scientist, culminating in Eisele (1976, 1979, 1985). Beginning around 1960, the philosopher and historian of ideas Max Fisch (1900-1995) emerged as an authority on Peirce; Fisch (1986) reprinted many of the relevant articles, including a wide-ranging survey (Fisch 1986: 422-448) of the impact of Peirce's thought through 1983.

Peirce has come to enjoy a significant international following. There are university research centers devoted to Peirce studies and pragmatism in Brazil, Finland, Germany, and Spain. Since 1950, there have been French, Italian, and British Peirceans of note. For many years, the North American philosophy department most devoted to Peirce was the University of Toronto's, thanks in good part to the leadership of Thomas Goudge and David Savan. In recent years, American Peirce scholars have clustered at Indiana University - Purdue University Indianapolis, the home of the Peirce Edition Project, and the Pennsylvania State University.

Robert Burch has commented on Peirce's current influence as follows:

Currently, considerable interest is being taken in Peirce's ideas from outside the arena of academic philosophy. The interest comes from industry, business, technology, and the military; and it has resulted in the existence of a number of agencies, institutes, and laboratories in which ongoing research into and development of Peircean concepts is being undertaken.[5].


Peirce's reputation is largely based on a number of academic papers published in American scholarly and scientific journals. These papers, along with a selection of Peirce's previously unpublished work and a smattering of his correspondence, fill the eight volumes of the Collected Papers of Charles Sanders Peirce, published between 1931 and 1958[6]. An important recent sampler of Peirce's philosophical writings is the two volume The Essential Peirce[7] .

The only book Peirce published in his lifetime was Photometric Researches (1878), a monograph on the applications of spectrographic methods to astronomy. While at Johns Hopkins, he edited Studies in Logic (1883), containing chapters by himself and his graduate students. He was a frequent book reviewer and contributor to The Nation,reprinted in Ketner and Cook (1975-1987).

In 2001 Peirce's entire correspondence with Lady Victoria Welby was published.[8] Peirce's other published correspondence is largely limited to 14 letters included in volume 8 of the Collected Papers, and about 20 pre-1890 items included in the Writings.

Harvard University acquired the papers found in Peirce's study soon after his death, but did not microfilm them until 1964. Only after Richard Robin (1967) catalogued this Nachlass did it become clear that Peirce had left approximately 1,650 unpublished manuscripts, totalling 80,000 pages. Carolyn Eisele[9] published some of this work, but most of it remains unpublished.[10].

The limited coverage, and defective editing and organization, of the Collected Papers led Max Fisch and others in the 1970s to found the Peirce Edition Project, whose mission is to prepare a more complete critical chronological edition, known as the Writings. Only six out of a planned 31 volumes have appeared to date, but they cover the period from 1859-1890, when Peirce carried out much of his best-known work.

On a New List of Categories (1867)

On May 14, 1867, Peirce presented a paper entitled "On a New List of Categories" to the American Academy of Arts and Sciences, which published it the following year. Among other things, this paper outlined a theory of three universal categories that Peirce continued to apply throughout philosophy and elsewhere for the rest of his life. Peirce scholars generally regard the "New List" as his blueprint for a pragmatic philosophy.

Logic of Relatives (1870)

By 1870, the drive that Peirce exhibited to understand the character of knowledge, starting with our partly innate and partly inured models of the world and working up to the conduct of our scientific inquiries into it, which had led him to inquire into the three-roled relationship of objects, signs, and impressions of the mind, now brought him to a point where he needed a theory of relations more powerful than that provided by the available logical formalisms. His first concerted effort to supply this gap was rolled out in his 60-page paper "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic,"[11]published in Memoirs of the American Academy of Arts and Sciences in 1870 and also separately as an extraction. The logic of relatives, short for the logic of relative terms, is the study of relations in their logical, philosophical, or semiotic aspects, as distinguished from—though closely coordinated with—their more properly formal, mathematical, or objective aspects. The consideration of relative terms has its roots in antiquity, but it entered a radically new phase of development with Peirce's 1870 paper, which is one of the wellsprings of contemporary systems of logic.

Illustrations of the Logic of Science (1877-1878)

Published in Popular Science Monthly Vols. 12-13 (see entries at the Charles Sanders Peirce bibliography, this series of articles is foundational for Peirce's pragmatism as a method of inquiry, especially "The Fixation of Belief" (1877) and "How to Make Our Ideas Clear" (1878).

Logic of Relatives (1883)

"Logic of Relatives (1883)," more precisely, "Note B. The Logic of Relatives," is the title of a 17-page addendum to the chapter entitled "A Theory of Probable Inference" that C.S. Peirce contributed to the volume Studies in Logic by Members of the Johns Hopkins University, 1883[12]. This volume, edited by Peirce, collected works of his students at Johns Hopkins. As a body, these works broke new ground in several different directions of logical exploration at once.

Logic of Relatives (1897)

Published in The Monist vol. VII, (2): 161-217.

The Simplest Mathematics (1902)

"The Simplest Mathematics" is the title of a paper by Peirce, intended as Chapter 3 of his unfinished magnum opus The Minute Logic. The paper is dated January–February 1902 but was not published until the appearance of his Collected Papers, Volume 4. in 1933. Peirce introduces the subject of the paper as "certain extremely simple branches of mathematics which, owing to their utility in logic, have to be treated in considerable detail, although to the mathematician they are hardly worth consideration"[13].

"Kaina Stoicheia" (1904)

"Kaina Stoicheia" (Καινα στοιχεια) or "New Elements" is the title of several manuscript drafts of a document that Peirce wrote circa 1904, intended as a preface to a book on the foundations of mathematics. It presents a consummate integration of his ideas on the interrelations of logic, mathematics, and semeiotic, or the theory of signs[14].


In order to understand Peirce’s philosophical work, it is important to remember that Peirce was a working scientist for 30 years, and was a professional philosopher only during the five years he lectured at Johns Hopkins. He learned philosophy mainly by reading a few pages of Kant's Critique of Pure Reason, in the original German, every day while a Harvard undergraduate. His writings bear on a wide array of disciplines, including astronomy, metrology, geodesy, mathematics, logic, philosophy, the history and philosophy of science, linguistics, economics, and psychology. This work has become the subject of renewed interest and approval, resulting in a revival inspired not only by his anticipations of recent scientific developments, but also by his demonstration of how philosophy can be applied effectively to human problems.

Peirce's writings repeatedly refer to a system of three categories, named "Firstness", "Secondness", and "Thirdness", devised early in his career in reaction to his reading of Aristotle, Kant, and Hegel. He later initiated the philosophical tendency known as pragmatism, a variant of which his life-long friend William James made popular. Peirce believed that any truth is provisional, and that the truth of any proposition cannot be certain but only probable. The name he gave to this state of affairs was "fallibilism." This fallibilism and pragmatism may be seen as taking roles in his work similar to those of skepticism and positivism, respectively, in the work of others.

Upon this first, and in one sense this sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not be satisfied with what you already incline to think, there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy:

Do not block the way of inquiry.
Although it is better to be methodical in our investigations, and to consider the economics of research, yet there is no positive sin against logic in trying any theory which may come into our heads, so long as it is adopted in such a sense as to permit the investigation to go on unimpeded and undiscouraged. On the other hand, to set up a philosophy which barricades the road of further advance toward the truth is the one unpardonable offence in reasoning, as it is also the one to which metaphysicians have in all ages shown themselves the most addicted.[15]


Peirce's recipe for pragmatic thinking, labeled pragmatism and also known as pragmaticism, is recapitulated in several versions of the so-called pragmatic maxim. Here is one of his more emphatic statements of it:

Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object.[16]

William James, among others, regarded two of Peirce's papers, "The Fixation of Belief" (1877) and "How to Make Our Ideas Clear" (1878) as being the origin of pragmatism. Peirce conceived pragmatism to be a method for clarifying the meaning of difficult ideas through the application of the pragmatic maxim. He differed from William James and the early John Dewey, in some of their tangential enthusiasms, by being decidedly more rationalistic and realistic.

Peirce's pragmatism may be understood as a method of sorting out conceptual confusions by linking the meaning of concepts to their operational or practical consequences. This understanding of pragmatism bears no resemblance to "vulgar" pragmatism, in which the search for truth is driven by a ruthless and Machiavellian regard for mercenary or political advantage. Rather, Peirce sought an objective method of verification to test the truth of putative knowledge. His pragmatism was a method of experimentational mental reflection, arriving at conceptions in terms of conceivable confirmatory and disconfirmatory circumstances, a method which allowed the generation of explanatory hypotheses, and which was conducive to the employment and improvement of verification. This pragmatism went beyond the usual foundational alternatives or rationalism (deduction from self-evident truths), and empiricism (inductive reasoning|induction]] from experiential phenomena).

His approach is often confused with empiricism, but is distinguished from it by the following three dimensions:

  • An active process of theory generation, with no prior assurance of truth;
  • Subsequent application of the contingent theory, aimed toward developing its logical and practical consequences;
  • Evaluation of the provisional theory's utility for the anticipation of future experience, in the senses of prediction and control.

Peirce's pragmatism was the first time the scientific method was proposed as an epistemology for philosophical questions. A theory that proves itself more successful in predicting and controlling our world than its rivals, is said to be nearer the truth. This is an operational notion of truth employed by scientists. Unlike the other pragmatists, Peirce never explicitly advanced a theory of truth. His scattered comments about truth proved influential to several epistemic truth theorists, and served as a useful foil for deflationary and correspondence theories of truth.

Pragmatism is regarded as a distinctively American philosophy. As advocated by James, John Dewey, Ferdinand Canning Scott Schiller, George Herbert Mead, and others, it has proved durable and popular. But Peirce did not seize on this fact to enhance his reputation. While it is sometimes stated that James' and other philosophers' use of the word pragmatism so dismayed Peirce that he renamed his own variant pragmaticism, this was not the main reason (Haack, 55). This is revealed by the context in which Peirce introduced the latter term:

But at present, the word [pragmatism] begins to be met with occasionally in the literary journals, where it gets abused in the merciless way that words have to expect when they fall into literary clutches. … So then, the writer, finding his bantling "pragmatism" so promoted, feels that it is time to kiss his child good-by and relinquish it to its higher destiny; while to serve the precise purpose of expressing the original definition, he begs to announce the birth of the word "pragmaticism," which is ugly enough to be safe from kidnappers.[17].

In a 1908 article [39] he expressed areas of agreement and disagreement with his fellow pragmatists. Peirce remained joined with them about:

the reality of generals and habits, to be understood, as are hypostatic abstractions, in terms of potential concrete effects even if unactualized;
the falsity of necessitarianism;
the character of consciousness as only "visceral or other external sensation."

and differed with their:

"angry hatred of strict logic";
view that "truth is mutable";
view that infinity is unreal; and
"confusion of active willing (willing to control thought, to doubt, and to weigh reasons) with willing not to exert the will (willing to believe)."

Peirce's pragmatism, in its core senses as a method and theory of definitions and the clearness of ideas, is a department within his theory of method of inquiry[18], which he variously called Methodeutic and Philosophical or Speculative Rhetoric. He applied his pragmatism as a method throughout his work.

Formal Logic

Peirce was very conscious of the limitations of language, and of the attempt to define human thought in terms of logical steps. He acknowledged that the intuitive mind understands reality in ways that have not yet been defined, and sought to harness intuitive thought so that it could be applied scientifically to yield new insights in research and investigation.

How often do we think of the thing in algebra? When we use the symbol of multiplication we do not even think out the conception of multiplication, we think merely of the laws of that symbol, which coincide with the laws of the conception, and what is more to the purpose, coincide with the laws of multiplication in the object. Now, I ask, how is it that anything can be done with a symbol, without reflecting upon the conception, much less imagining the object that belongs to it? It is simply because the symbol has acquired a nature, which may be described thus, that when it is brought before the mind certain principles of its use—whether reflected on or not—by association immediately regulate the action of the mind; and these may be regarded as laws of the symbol itself which it cannot as a symbol transgress.[19]

Logic as Formal Semiotic

On the Definition of Logic. 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.[20]

Though Frege is credited with being the first to invent “formal logic,” Hilary Putnam points out that Peirce and his students discovered it in the effective sense that they developed it independently and made it widely known. The main evidence for Putnam's claims is Peirce (1885), published in the premier American mathematical journal of the day. Giuseppe Peano, Ernst Schröder, among others, cited this article. Peirce was apparently ignorant of Frege's work, despite their rival achievements in logic, philosophy of language, and the foundations of mathematics. [21][22] [23]

Peirce's other major discoveries in formal logic include:

  • Distinguishing (Peirce, 1885) between first-order and second-order quantification.
  • Seeing that Boolean calculations could be carried out by means of electrical switches (W5:421-24), anticipating Claude Shannon by more than 50 years.
  • Devising the existential graphs, a diagrammatic notation for the predicate calculus. These graphs form the basis of the conceptual graphs of John F. Sowa, and of Sun-Joo Shin's diagrammatic reasoning.

A philosophy of logic, grounded in his categories and semeiotic, can be extracted from Peirce's writings. This philosophy, as well as Peirce's logical work more generally, is exposited and defended in [24] [25] , and [26] Jean Van Heijenoort (1967)[27], Jaakko Hintikka [28]in his chapter in Brunning and Forster (1997), and Geraldine Brady (2000)[29] divide those who study formal (and natural) languages into two camps: the model-theorists / semanticists, and the proof theorists / universalists. Hintikka and Brady view Peirce as a pioneer model theorist. On how the young Bertrand Russell, especially his Principles of Mathematics and Principia Mathematica, did not do Peirce justice, see Anellis (1995).

Peirce's work on formal logic had admirers other than Ernst Schröder; the philosophical algebraist William Kingdon Clifford and the logician William Ernest Johnson, both British; the Polish school of logic and foundational mathematics, including Alfred Tarski; and Arthur Prior, whose Formal Logic and chapter in Moore and Robin (1964) praised and studied Peirce's logical work.


It may be added that algebra was formerly called Cossic, in English, or the Rule of Cos; and the first algebra published in England was called "The Whetstone of Wit," because the author supposed that the word cos was the Latin word so spelled, which means a whetstone. But in fact, cos was derived from the Italian cosa, thing, the thing you want to find, the unknown quantity whose value is sought. It is the Latin caussa, a thing aimed at, a cause. ("Elements of Mathematics," MS 165 (c. 1895), NEM 2, 50).

Peirce made a number of striking discoveries in foundational mathematics, nearly all of which came to be appreciated only long after his death. He:

  • Showed how what is now called Boolean algebra could be expressed by means of a single binary operation, either NAND or its dual, NOR. (See also De Morgan's Laws). This discovery anticipated Sheffer by 33 years.
  • In Peirce (1885), set out what can be read as the first (primitive) axiomatic set theory, anticipating Zermelo by about two decades.
  • Discovered the now-classic axiomatization of natural number arithmetic, a few years before Dedekind and Peano did so.
  • Discovered, independently of Dedekind, an important formal definition of an infinite set, namely, as a set that can be put into a one-to-one correspondence with one of its proper subsets.

Beginning with his first paper on the "Logic of Relatives" (1870), Peirce extended the theory of relations that Augustus De Morgan had just recently revived. Much of the actual mathematics of relations that is taken for granted today was "borrowed" from Peirce, not always with all due credit (Anellis 1995). Beginning in 1940, Alfred Tarski and his students rediscovered aspects of Peirce's larger vision of relational logic, developing the perspective of relational algebra. These theoretical resources gradually worked their way into applications, in large part instigated by the work of Edgar F. Codd, who happened to be a doctoral student of the Peirce editor and scholar Arthur W. Burks, on the relational model or the relational paradigm for implementing and using databases.

In the four-volume work, The New Elements of Mathematics by Charles S. Peirce (1976), mathematician and Peirce scholar Carolyn Eisele published a large number of Peirce's previously unpublished manuscripts on mathematical subjects, including the drafts for an introductory textbook, allusively titled The New Elements of Mathematics, that presented mathematics from a decidedly novel, if not revolutionary, standpoint.

Theory of Signs, or Semiotic

Peirce referred to his general study of signs, based on the concept of a triadic sign relation, as semiotic or semeiotic; both terms are currently used in either singular of plural form. Peirce began writing on semeiotic in the 1860s, around the time that he devised his system of three categories. He eventually defined semiosis as an "action, or influence, which is, or involves, a cooperation of three subjects, such as a sign, its object, and its interpretant, this tri-relative influence not being in any way resolvable into actions between pairs." (Houser 1998: 411, written 1907). This triadic relation grounds the semeiotic.

The meaning of the concept sign must be understood in the context of its role in a triadic relation. The role of a sign is constituted as one among three roles which are distinct, even when the things that fill them are not. In order to understand what a triadic relation is we need to understand what a relation is, and here there To the two traditional ways of understanding relation, the way of extension and the way of intension. Peirce added a third way, the way of information, which integrated the other two approaches in a unified whole.

Semiotic Elements

Peirce held there are exactly three basic elements in semiosis (sign action): 1. A sign (or representamen) represents something, in the broadest possible sense of "represents." It conveys information about something. It is not necessarily symbolic, linguistic, or artificial. 2. An object (or semiotic object) is the subject matter of a sign and an interpretant. It can be anything discussable or thinkable, a thing, event, relationship, quality, law, argument, etc., and can even be fictional, for instance Hamlet[30]. All of those are special or partial objects. The object most accurately is the universe of discourse to which the partial or special object belongs[31]. For instance, a perturbation of Pluto's orbit is a sign about Pluto but ultimately not only about Pluto. 3. An interpretant (or interpretant sign) is the sign's more or less clarified meaning or ramification. (Peirce's sign theory concerns meaning in the broadest sense, including logical implication, not just the meanings of words as properly clarified by a dictionary.) The interpretant is a sign (a) of the object and (b) of the interpretant's "predecessor" (the interpreted sign) as being a sign of the same object. The interpretant is an interpretation in the sense of a product of an interpretive process or a content in which an interpretive relation culminates, though this product or content may itself be an act, a state of agitation, or a conduct. Such is what is meant in saying that the sign stands for the object to the interpretant. Some of the mind’s understanding of a sign depends on familiarity with the object. In order to know what a given sign denotes, the mind needs some experience of that sign's object collaterally to that sign or sign system, and in that context, Peirce speaks of collateral experience, collateral observation, and collateral acquaintance, all in much the same terms.[32] The relationship between an object and a sign determines another sign—the interpretant – which is related to the object in the same way as the sign is related to the object. The interpretant, fulfilling its function as a sign of the object, determines a further interpretant sign. The process is logically structured to perpetuate itself.

Types of signs

Peirce proposes several typologies and definitions of the signs. More that 76 definitions of what a sign is have been collected throughout Peirce's work.[33] Some canonical typologies can nonetheless be observed, one crucial one being the distinction between "icons," "indices" and "symbols." This typology emphasizes the different ways in which the representamen (or its ground) addresses or refers to its "object," through a particular mobilization of an "interpretant."


The sign is perceived as resembling or imitating the object it refers to (such as a fork on a sign by the road indicating a rest stop). An icon thus "resembles" to its object. It shares a character or an aspect with it, which allows for it to be interpreted as a sign even if the object does not exist. It signifies essentially on the basis of its "ground."


For an index to signify, its relation to the object is crucial. The representamen is directly connected in some way (physically or casually) to the object it denotes (smoke coming from a building is an index of fire). Hence, an index refers to the object because it is really affected or modified by it, and thus may stand as a trace of the existence of the object.


The representamen does not resemble the object signified but is fundamentally conventional, so that the signifying relationship must be learned and agreed upon (such as the word “cat”). A symbol thus denotes, primarily, by virtue of its interpretant. Its action (semeiosis) is ruled by a convention, a more or less systematic set of associations that guarantees its interpretation, independently of any resemblance or any material relation with its object.

Theory of inquiry

Peirce extracted the pragmatic model or theory of inquiry from its raw materials in classical logic and refined it in parallel with the early development of symbolic logic to address problems about the nature of scientific reasoning. Borrowing a brace of concepts from Aristotle, Peirce examined three fundamental modes of reasoning that play a role in inquiry, processes that are currently known as abductive, deductive, and inductive inference.

Abduction is what we use to generate a likely hypothesis or an initial diagnosis in response to a phenomenon of interest or a problem of concern, while deduction is used to clarify, to derive, and to explicate the relevant consequences of the selected hypothesis, and induction is used to test the sum of the predictions against the sum of the data. Abuction, deduction, and induction typically operate in a cyclic fashion, systematically operating to reduce the uncertainties and the difficulties that initiated the inquiry in question, and in this way, to the extent that inquiry is successful, leading to an increase in the knowledge or skills, in other words, an augmentation in the competence or performance, of the agent or community engaged in the inquiry.

In the pragmatic way of thinking in terms of conceivable consequences, every thing has a purpose, and that purpose is the first thing that we should try to note about it. The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call 'knowledge' or 'certainty.' It needs to be appreciated that the three kinds of inference contribute and collaborate toward the end of inquiry, and none of the three makes complete sense in isolation from the others. For instance, the purpose of abduction is to generate guesses of a kind that deduction can explicate and that induction can evaluate. This places a mild but meaningful constraint on the production of hypotheses, since a hypothesis cannot be just any wild guess, but must be able to be subjected to the process of verification. In a similar way, each of the other types of inference realizes its purpose only as part of the whole cycle of inquiry. If we then think to inquire, 'What sort of constraint, exactly, does pragmatic thinking place on our guesses?', we have asked the question that is generally recognized as the problem of 'giving a rule to abduction'. Peirce's way of answering it is given in terms of the so-called 'pragmatic maxim.' In 1903 Peirce called the question of pragmatism "the question of the logic of abduction"[34]. Peirce characterized the scientific method as follows[35]:

1. Abduction (or retroduction). Generation of explanatory hypothesis. From abduction, Peirce distinguishes induction as inferring, on the basis of tests, the proportion of truth in the hypothesis. Every inquiry, whether into ideas, brute facts, or norms and laws, arises as a result of surprising observations in the given realm or realms, and the pondering of the phenomenon in all its aspects in the attempt to resolve the wonder. All explanatory content of theories is reached by way of abduction, the most insecure among modes of inference. Induction as a process is far too slow, so economy of research demands abduction, whose success depends on intuition and previous knowledge. Abduction has general inductive justification in that it works often enough and that nothing else works as quickly. Since abduction depends on mental processes that are not necessarily conscious and deliberate, its explanatory hypotheses should be optimally simple and should have consequences with a conceivable practical bearing that allow at least mental tests, and, in science, lend themselves to scientific testing.
2. Deduction is the analysis of hypothesis and deduction of its consequences in order to test the hypothesis, and has two stages: explication (logical analysis of the hypothesis in order to render it as distinct as possible); and demonstration (or deductive argumentation, the deduction of hypothesis's consequence).
3. Induction. The long-run validity of the rule of induction is deducible from the principle (presuppositional to reasoning in general) that the real "is only the object of the final opinion to which sufficient investigation would lead".[36] In other words, if there were something to which an inductive process involving ongoing tests or observations would never lead, then that thing would not be real. Induction has three stages: classification (classing objects of experience under general ideas); probation (direct inductive argumentation), the enumeration of instances, the arrival at new estimates of the proportion of truth in the hypothesis after each test, including the application of statistical analysis; and sentential induction. "…which, by Inductive reasonings, appraises the different Probations singly, then their combinations, then makes self-appraisal of these very appraisals themselves, and passes final judgment on the whole result"[37].

See also


  • Continuous predicate
  • Hypostatic abstraction
  • Hypostatic object
  • Prescisive abstraction



  • Dyadic relation
  • Kaina Stoicheia
  • Quincuncial map

  • Relation
  • Relation composition
  • Relation construction

  • Relation reduction
  • Theory of relations
  • Triadic relation


  • Pragmatism
  • Pragmaticism
  • Pragmatic maxim
  • Pragmatic theory of truth


  1. Joseph Brent (1998). Charles Sanders Peirce: A Life, Revised and enlarged ed. (Bloomington, IN: Indiana University Press. ISBN 0253333504), 1).
  2. On Peirce the astronomer, see Lenzen's chapter in Edward C. Moore and Richard S. Robin. Studies in the Philosophy of Charles Sanders Peirce. (Amherst, MA: University of Massachusetts Press, 1964)
  3. Barry N. Taylor, (ed.) (2001) The International System of Units, NIST Special Publication 330. (Washington DC: Superintendent of Documents.), 5. Retrieved December 16, 2007.
  4. Brent, 1998, 315-316, 374
  5. Robert Burch, Charles Sanders Peirce, in Edward N. Zalta, ed. Stanford Encyclopedia of Philosophy, Revised, Winter 2005. Retrieved December 16, 2007.
  6. Charles Sanders Pierce. Collected Papers of Charles Sanders Peirce, Vols I - VI, edited by Charles Hartshorne and Paul Weiss. (Cambridge, MA: Harvard University Press/Belnap Press, ISBN 978-0674138001)
  7. Charles Peirce. The Essential Peirce. (2 vols) (Houser and Kloesel (eds.) 1992, (Peirce Edition Project) (eds.) 1998. )
  8. Hartwick 2001
  9. Carolyn Eisele, (ed.) (1976). The New Elements of Mathematics, by Charles S. Peirce. (Four volumes in five books.) (The Hague: Mouton Publishers.)
  10. For more on the vicissitudes of Peirce's papers, see Nathan Houser, (1989) "The Fortunes and Misfortunes of the Peirce Papers," (Fourth Congress of the International Association for Semiotic Studies. Perpignan, France: 1989.) Published, 1259-1268 in Signs of Humanity / L'homme et ses signes. vol. 3, Michel Balat and Janice Deledalle-Rhodes, (eds.); Gérard Deledalle, gen. ed. (Berlin: Mouton de Gruyter, 1992. ISBN 3110116758)
  11. Charles Peirce, Description of a notation for the logic of Retrieved January 21, 2009.
  12. Studies in Logic by Members of the Johns Hopkins University,(Boston, MA: Little, Brown, and Company, 1883)
  13. (CP 4.227)
  14. (MS 517. NEM 4, 235–263. Cf. "New Elements," EP 2, 300–324). Arisbe
  15. (Peirce, "F.R.L." (c. 1899), CP 1.135-136).
  16. (CP 5.438).
  17. (C.S. Peirce, CP 5.414)
  18. See Joseph Ransdell's comments and his tabular list of titles of Peirce's proposed list of memoirs in 1902 for his Carnegie application,
  19. ("On the Logic of Science" (1865), CE 1, 173).
  20. (Peirce, "Carnegie Application," NEM 4, 54).
  21. In 1902 Peirce applied to the newly established Carnegie Institution for aid "in accomplishing certain scientific work," presenting an "explanation of what work is proposed" plus an "appendix containing a fuller statement." These parts of the letter, along with excerpts from earlier drafts, can be found in NEM 4 (Eisele 1976). The appendix is organized as a "List of Proposed Memoirs on Logic," and No. 12 among the 36 proposals is titled "On the Definition of Logic,"
  22. While, to my knowledge, no one except Gottlob Frege ever published a single paper in Frege's notation, many famous logicians adopted Peirce-Schröder notation, and famous results and systems were published in it. Löwenheim stated and proved the Löwenheim-Skolem theorem … in Peirce's notation. In fact, there is no reference in Löwenheim's paper to any logic other than Peirce's. To cite another example, Zermelo presented his axioms for set theory in Peirce–Schröder notation, and not, as one might have expected, in Russell–Whitehead notation.
  23. One can sum up these simple facts (which anyone can quickly verify) as follows: Frege certainly discovered the quantifier first (four years before O. H. Mitchell did so, going by publication dates, which are all we have as far as I know). But Leif Ericson probably discovered America 'first' (forgive me for not counting the native Americans, who of course really discovered it 'first'). If the effective discoverer, from a European point of view, is Christopher Columbus, that is because he discovered it so that it stayed discovered (by Europeans, that is), so that the discovery became known (by Europeans). Frege did 'discover' the quantifier in the sense of having the rightful claim to priority; but Peirce and his students discovered it in the effective sense. The fact is that until Russell appreciated what he had done, Frege was relatively obscure, and it was Peirce who seems to have been known to the entire world logical community. How many of the people who think that 'Frege invented [formal] logic' are aware of these facts?
  24. Hilary Putnam (1982) "Pierce the Logician" Retrieved January 22, 2009. excerpt from Hilary Putnam. Realism with a Human Face. (Harvard University Press, 1990), 252-260.
  25. the Introduction to Houser et al (1997)
  26. Dipert's chapter in Cheryl Misak, (ed.) (2004). The Cambridge Companion to Peirce. (Cambridge University Press, ISBN 0521579104)
  27. Jean Van Heijenoort (1967), "Logic as Language and Logic as Calculus," Synthese 17: 324-330.
  28. Jaakko Hintikka Brunning and Paul Forster, (1997) [1] Retrieved January 22, 2009.
  29. Geraldine Brady. (2000). From Peirce to Skolem: A Neglected Chapter in the History of Logic. (Amsterdam, Netherlands: North-Holland/Elsevier Science BV.) (2000)
  30. Peirce (1909), A Letter to William James, in The Essential Pierce, 2, 498,
  31. Peirce (1909), A Letter to William James, in The Essential Pierce, 2, 492,
  32. "Pragmatism," 404-409 in The Essential Peirce.
  33. Robert Marty's 76 Definitions of what a Sign is. Retrieved January 22, 2009.
  34. C. S. Peirce, (1903), "Pragmatism—The Logic of Abduction," CP 5.195-205, especially para. 196.
  35. C. S. Peirce, (1908), "A Neglected Argument for the Reality of God," Hibbert Journal 7 (1909): 90-112. Reprinted (CP 6.452-485); (Selected Writings, 358-379); (Essential Pierce 2, 434-450); online, /"A Neglected Argument for the Reality of God,". wikisource. Retrieved January 22, 2009.
  36. "That the rule of induction will hold good in the long run may be deduced from the principle that reality is only the object of the final opinion to which sufficient investigation would lead," in C. S. Peirce, (April 1878), "The Probability of Induction," in Popular Science Monthly 12: 705-718. 718.; Reprinted in C. S. Peirce. Chance, Love, and Logic: Philosophical Essays. (International Library of Philosophy) Routledge, 2000. ISBN 0415225418), 82-105.
  37. C. S. Peirce, (1908), "A Neglected Argument for the Reality of God," Hibbert Journal 7: 90-112; Reprinted (CP 6.452-485); (Selected Writings, 358-379); (EP 2, 434-450)

ISBN links support NWE through referral fees

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External links

All links retrieved December 4, 2023.

Encyclopedia Entries

  • Internet Encyclopedia of Philosophy
  • Stanford Encyclopedia of Philosophy

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