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Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher,
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
ian, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for thirty years, Peirce made major contributions to
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, a subject that, for him, encompassed much of what is now called
epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...
and the philosophy of science. He saw logic as the formal branch of
semiotics Semiotics (also called semiotic studies) is the systematic study of sign processes (semiosis) and meaning making. Semiosis is any activity, conduct, or process that involves signs, where a sign is defined as anything that communicates something, ...
, of which he is a founder, which foreshadowed the debate among logical positivists and proponents of philosophy of language that dominated 20th-century Western philosophy. Additionally, he defined the concept of
abductive reasoning Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th centu ...
, as well as rigorously formulated mathematical induction and
deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...
. As early as 1886, he saw that logical operations could be carried out by electrical switching circuits. The same idea was used decades later to produce digital computers. See Also In 1934, the philosopher Paul Weiss called Peirce "the most original and versatile of American philosophers and America's greatest logician".

# Life

Peirce was born at 3 Phillips Place in
Cambridge, Massachusetts Cambridge ( ) is a city in Middlesex County, Massachusetts, United States. As part of the Boston metropolitan area, the cities population of the 2020 U.S. census was 118,403, making it the fourth most populous city in the state, behind Boston ...
. He was the son of Sarah Hunt Mills and Benjamin Peirce, himself a professor of
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxies ...
and mathematics at
Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher ...
. At age 12, Charles read his older brother's copy of Richard Whately's ''Elements of Logic'', then the leading English-language text on the subject. So began his lifelong fascination with logic and reasoning. He went on to earn a Bachelor of Arts degree and a Master of Arts degree (1862) from Harvard. In 1863 the Lawrence Scientific School awarded him a Bachelor of Science degree, Harvard's first ''summa cum laude''
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties, ...
degree. His academic record was otherwise undistinguished. At Harvard, he began lifelong friendships with Francis Ellingwood Abbot, Chauncey Wright, and
William James William James (January 11, 1842 – August 26, 1910) was an American philosopher, historian, and psychologist, and the first educator to offer a psychology course in the United States. James is considered to be a leading thinker of the lat ...
. One of his Harvard instructors, Charles William Eliot, formed an unfavorable opinion of Peirce. This proved fateful, because Eliot, while President of Harvard (1869–1909—a period encompassing nearly all of Peirce's working life), repeatedly vetoed Peirce's employment at the university. Peirce suffered from his late teens onward from a nervous condition then known as "facial neuralgia", which would today be diagnosed as
trigeminal neuralgia Trigeminal neuralgia (TN or TGN), also called Fothergill disease, tic douloureux, or trifacial neuralgia is a long-term pain disorder that affects the trigeminal nerve, the nerve responsible for sensation in the face and motor functions such as ...
. His biographer, Joseph Brent, says that when in the throes of its pain "he was, at first, almost stupefied, and then aloof, cold, depressed, extremely suspicious, impatient of the slightest crossing, and subject to violent outbursts of temper". Its consequences may have led to the social isolation of his later life.

## Early employment

Between 1859 and 1891, Peirce was intermittently employed in various scientific capacities by the
United States Coast Survey United may refer to: Places * United, Pennsylvania, an unincorporated community * United, West Virginia, an unincorporated community Arts and entertainment Films * ''United'' (2003 film), a Norwegian film * ''United'' (2011 film), a BBC Two f ...
and its successor, the
United States Coast and Geodetic Survey The United States Coast and Geodetic Survey (abbreviated USC&GS), known from 1807 to 1836 as the Survey of the Coast and from 1836 until 1878 as the United States Coast Survey, was the first scientific agency of the United States Government. It ...
,Burch, Robert (2001, 2010),
Charles Sanders Peirce
, ''Stanford Encyclopedia of Philosophy''
where he enjoyed his highly influential father's protection until the latter's death in 1880. That employment exempted Peirce from having to take part in the
American Civil War The American Civil War (April 12, 1861 – May 26, 1865; also known by other names) was a civil war in the United States. It was fought between the Union ("the North") and the Confederacy ("the South"), the latter formed by states ...
; it would have been very awkward for him to do so, as the Boston Brahmin Peirces sympathized with the Confederacy. At the Survey, he worked mainly in
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
and
gravimetry Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest. Units of measurement Gr ...
, refining the use of pendulums to determine small local variations in the Earth's gravity. He was elected a resident fellow of the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...
in January 1867. The Survey sent him to Europe five times, first in 1871 as part of a group sent to observe a
solar eclipse A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby obscuring the view of the Sun from a small part of the Earth, totally or partially. Such an alignment occurs during an eclipse season, approximately every six mont ...
. There, he sought out Augustus De Morgan,
William Stanley Jevons William Stanley Jevons (; 1 September 183513 August 1882) was an English economist and logician. Irving Fisher described Jevons's book ''A General Mathematical Theory of Political Economy'' (1862) as the start of the mathematical method in ec ...
, and
William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in h ...
, British mathematicians and logicians whose turn of mind 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
star A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ...
s and the shape of the
Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye. ...
.Moore, Edward C., and Robin, Richard S., eds., (1964), ''Studies in the Philosophy of Charles Sanders Peirce, Second Series'', Amherst: U. of Massachusetts Press. On Peirce the astronomer, see Lenzen's chapter. On April 20, 1877, he was elected a member of the
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...
. Also in 1877, he proposed measuring the meter as so many wavelengths of light of a certain frequency, the kind of definition employed from 1960 to 1983. During the 1880s, Peirce's indifference to bureaucratic detail waxed while his Survey work's quality and timeliness waned. Peirce took years to write reports that he should have completed in months. Meanwhile, he wrote entries, ultimately thousands, during 1883–1909 on philosophy, logic, science, and other subjects for the encyclopedic '' 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 Superintendent Thomas Corwin Mendenhall's request.

## Johns Hopkins University

In 1879, Peirce was appointed lecturer in logic at
Johns Hopkins University Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hem ...
, which had strong departments in areas that interested him, such as philosophy (
Royce Markus Bennett is an American hip-hop recording artist. People Surname * Ed Royce (born 1951), American politician * Elwyn E. Royce (1868–1960), American politician * Henry Royce (1863–1933), cofounder of the Rolls-Royce automobile company ...
and
Dewey Dewey may refer to: Places In the United States * Dewey, Arizona, a former unincorporated town, now part of the town of Dewey-Humboldt * Wasco, California, formerly Dewey, a city * Dewey, Illinois, an unincorporated community * Dewey, Indiana, an ...
completed their Ph.D.s 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). His '' Studies in Logic by Members of the Johns Hopkins University'' (1883) contained works by himself and Allan Marquand, Christine Ladd, Benjamin Ives Gilman, and Oscar Howard Mitchell,Randall R. Diper
(1994) The Life and Logical Contributions of O. H. Mitchell, Peirce's Gifted Student
/ref> several of whom were his graduate students.Houser, Nathan (1989),

, ''Writings of Charles S. Peirce'', 4:xxxviii, find "Eighty-nine".
Peirce's nontenured position at Hopkins was the only academic appointment he ever held. Brent documents something Peirce never suspected, namely that his efforts to obtain academic employment, grants, and scientific respectability were repeatedly frustrated by the covert opposition of a major Canadian-American scientist of the day,
Simon Newcomb Simon Newcomb (March 12, 1835 – July 11, 1909) was a Canadian– American astronomer, applied mathematician, and autodidactic polymath. He served as Professor of Mathematics in the United States Navy and at Johns Hopkins University. Born in ...
. Peirce's efforts may also have been hampered by what Brent characterizes as "his difficult personality". In contrast, Keith Devlin believes that Peirce's work was too far ahead of his time to be appreciated by the academic establishment of the day and that this played a large role in his inability to obtain a tenured position. Peirce's personal life undoubtedly worked against his professional success. After his first wife, Harriet Melusina Fay ("Zina"), left him in 1875, Peirce, while still legally married, became involved with Juliette, whose last name, given variously as Froissy and Pourtalai, and nationality (she spoke French) remains uncertain. When his divorce from Zina became final in 1883, he married Juliette. That year, 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 in January 1884. Over the years Peirce sought academic employment at various universities without success. He had no children by either marriage.

## Poverty

In 1887, Peirce spent part of his inheritance from his parents to buy of rural land near Milford, Pennsylvania, which never yielded an economic return. There he had an 1854 farmhouse remodeled to his design. The Peirces named the property " Arisbe". There they lived with few interruptions for the rest of their lives, Charles writing prolifically, much of it unpublished to this day (see Works). Living beyond their means soon led to grave financial and legal difficulties. He spent much of his last two decades unable to afford heat in winter and subsisting on old bread donated by the local baker. Unable to afford new stationery, he wrote on the
verso ' is the "right" or "front" side and ''verso'' is the "left" or "back" side when text is written or printed on a leaf of paper () in a bound item such as a codex, book, broadsheet, or pamphlet. Etymology The terms are shortened from Latin ...
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 Gifford Pinchot (August 11, 1865October 4, 1946) was an American forester and politician. He served as the fourth chief of the U.S. Division of Forestry, as the first head of the United States Forest Service, and as the 28th governor of Pennsy ...
, settled his debts and paid his property taxes and mortgage. Peirce did some scientific and engineering consulting and wrote much for meager pay, mainly encyclopedic dictionary entries, and reviews for ''
The Nation ''The Nation'' is an American liberal biweekly magazine that covers political and cultural news, opinion, and analysis. It was founded on July 6, 1865, as a successor to William Lloyd Garrison's '' The Liberator'', an abolitionist newspaper th ...
'' (with whose editor, Wendell Phillips Garrison, he became friendly). He did translations for the
Smithsonian Institution The Smithsonian Institution ( ), or simply the Smithsonian, is a group of museums and education and research centers, the largest such complex in the world, created by the U.S. government "for the increase and diffusion of knowledge". Founded ...
, at its director Samuel Langley's instigation. Peirce also did substantial mathematical calculations for Langley's research on powered flight. Hoping to make money, Peirce tried inventing. He began but did not complete several books. In 1888, President Grover Cleveland appointed him to the Assay Commission. From 1890 on, he had a friend and admirer in Judge Francis C. Russell of Chicago, who introduced Peirce to editor Paul Carus and owner Edward C. Hegeler of the pioneering American philosophy journal '' The Monist'', which eventually published at least 14 articles by Peirce. He wrote many texts in James Mark Baldwin's '' Dictionary of Philosophy and Psychology'' (1901–1905); half of those credited to him appear to have been written actually by Christine Ladd-Franklin under his supervision. He applied in 1902 to the newly formed Carnegie Institution for a grant to write a systematic book describing his life's work. The application was doomed; his nemesis, Newcomb, served on the Carnegie Institution executive committee, and its president had been 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 William James (January 11, 1842 – August 26, 1910) was an American philosopher, historian, and psychologist, and the first educator to offer a psychology course in the United States. James is considered to be a leading thinker of the lat ...
, dedicating his ''Will to Believe'' (1897) to Peirce, and arranging for Peirce to be paid to give two series of lectures at or near Harvard (1898 and 1903). Most important, each year from 1907 until James's death in 1910, James wrote to his friends in the Boston intelligentsia to request financial aid for Peirce; the fund continued even after James died. Peirce reciprocated by designating James's eldest son as his heir should Juliette predecease him. It has been believed that this was also why Peirce used "Santiago" ("St. James" in English) as a middle name, but he appeared in print as early as 1890 as Charles Santiago Peirce. (See
Charles Santiago Sanders Peirce Charles Santiago Sanders Peirce was the adopted name of Charles Sanders Peirce (September 10, 1839 – April 19, 1914), an American philosopher, logician, mathematician, and scientist. Peirce's name appeared in print as "Charles Santiago Pei ...
for discussion and references). Peirce died destitute in Milford, Pennsylvania, twenty years before his widow. Juliette Peirce kept the urn with Peirce's ashes at Arisbe. In 1934, Pennsylvania Governor
Gifford Pinchot Gifford Pinchot (August 11, 1865October 4, 1946) was an American forester and politician. He served as the fourth chief of the U.S. Division of Forestry, as the first head of the United States Forest Service, and as the 28th governor of Pennsy ...
arranged for Juliette's burial in Milford Cemetery. The urn with Peirce's ashes was interred with Juliette.

## Slavery, the American Civil War, and racism

Peirce grew up in a home where white supremacy was taken for granted, and Southern slavery was considered natural. Until the outbreak of the Civil War, his father described himself as a
secessionist Secession is the withdrawal of a group from a larger entity, especially a political entity, but also from any organization, union or military alliance. Some of the most famous and significant secessions have been: the former Soviet republics l ...
, but after the outbreak of the war, this stopped and he became a Union partisan, providing donations to the Sanitary Commission, the leading Northern war charity. No members of the Peirce family volunteered or enlisted. Peirce shared his father's views and liked to use the following
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...
to illustrate the unreliability of traditional forms of logic, if one doesn't keep the meaning of the words, phrases, and sentences consistent throughout an argument:
All Men are equal in their political rights. Negroes are Men. Therefore, negroes are equal in political rights to whites.

# Reception

Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
(1959) wrote "Beyond doubt ..he was one of the most original minds of the later nineteenth century and certainly the greatest American thinker ever". Russell and Whitehead's ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'', published from 1910 to 1913, does not mention Peirce (Peirce's work was not widely known until later).Anellis, Irving H. (1995), "Peirce Rustled, Russell Pierced: How Charles Peirce and Bertrand Russell Viewed Each Other's Work in Logic, and an Assessment of Russell's Accuracy and Role in the Historiography of Logic", ''Modern Logic'' 5, 270–328. ''Arisbe'
Eprint
/ref> 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 Lowe 1964.)
Karl Popper Sir Karl Raimund Popper (28 July 1902 – 17 September 1994) was an Austrian-British philosopher, academic and social commentator. One of the 20th century's most influential philosophers of science, Popper is known for his rejection of the c ...
viewed Peirce as "one of the greatest philosophers of all times". Yet Peirce's achievements were not immediately recognized. His imposing contemporaries
William James William James (January 11, 1842 – August 26, 1910) was an American philosopher, historian, and psychologist, and the first educator to offer a psychology course in the United States. James is considered to be a leading thinker of the lat ...
and
Josiah Royce Josiah Royce (; November 20, 1855 – September 14, 1916) was an American objective idealist philosopher and the founder of American idealism. His philosophical ideas included his version of personalism, defense of absolutism, idealism and his ...
admired him and Cassius Jackson Keyser, at Columbia and C. K. Ogden, wrote about Peirce with respect but to no immediate effect. The first scholar to give Peirce his considered professional attention was Royce's student
Morris Raphael Cohen Morris Raphael Cohen ( be, Мо́рыс Рафаэ́ль Ко́эн; July 25, 1880 – January 28, 1947) was an American philosopher, lawyer, and legal scholar who united pragmatism with logical positivism and linguistic analysis. This union co ...
, the editor of an anthology of Peirce's writings entitled '' Chance, Love, and Logic'' (1923), and the author of the first bibliography of Peirce's scattered writings.
John Dewey John Dewey (; October 20, 1859 – June 1, 1952) was an American philosopher, psychologist, and educational reformer whose ideas have been influential in education and social reform. He was one of the most prominent American scholars in the fi ...
studied under Peirce at Johns Hopkins. From 1916 onward, Dewey's writings repeatedly mention Peirce with deference. His 1938 ''Logic: The Theory of Inquiry'' is much influenced by Peirce. The publication of the first six volumes of ''Collected Papers'' (1931–1935), the most important event to date in Peirce studies and one that Cohen made possible by raising the needed funds, did not prompt an 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 PhD thesis by Arthur W. Burks (who went on to edit volumes 7 and 8), and the studies edited by Wiener and Young (1952). The Charles S. Peirce Society was founded in 1946. Its ''Transactions'', an academic quarterly specializing in Peirce's pragmatism and American philosophy has appeared since 1965. (See Phillips 2014, 62 for discussion of Peirce and Dewey relative to transactionalism.) By 1943 such was Peirce's reputation, in the US at least, that ''Webster's Biographical Dictionary'' said that Peirce was "now regarded as the most original thinker and greatest logician of his time". In 1949, while doing unrelated archival work, the historian of mathematics Carolyn Eisele (1902–2000) chanced on an autograph letter by Peirce. So began her forty years of research on Peirce, “the 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).Fisch, Max (1986), ''Peirce, Semeiotic, and Pragmatism'', Kenneth Laine Ketner and Christian J. W. Kloesel, eds., Bloomington, Indiana: Indiana U. Press. He includes many of his relevant articles in a survey (Fisch 1986: 422–48) of the impact of Peirce's thought through 1983. Peirce has gained an international following, marked by university research centers devoted to Peirce studies and pragmatism in Brazil ( CeneP/CIEP), Finland ( HPRC and ), Germany ( Wirth's group, Hoffman's and Otte's group, and Deuser's and Härle's group), France ( L'I.R.S.C.E.), Spain ( GEP), and Italy ( CSP). His writings have been translated into several languages, including German, French, Finnish, Spanish, and Swedish. Since 1950, there have been French, Italian, Spanish, British, and Brazilian Peirce scholars of note. For many years, the North American philosophy department most devoted to Peirce was the
University of Toronto The University of Toronto (UToronto or U of T) is a public university, public research university in Toronto, Ontario, Canada, located on the grounds that surround Queen's Park (Toronto), Queen's Park. It was founded by royal charter in 1827 ...
, thanks in part to the leadership of Thomas Goudge and David Savan. In recent years, U.S. Peirce scholars have clustered at Indiana University – Purdue University Indianapolis, home of the Peirce Edition Project (PEP) –, and Pennsylvania State University. In recent years, Peirce's trichotomy of signs is exploited by a growing number of practitioners for marketing and design tasks. John Deely writes that Peirce was the last of the "moderns" and "first of the postmoderns". He lauds Peirce's doctrine of signs as a contribution to the dawn of the Postmodern epoch. Dewey additionally comments that "Peirce stands...in a position analogous to the position occupied by
Augustine Augustine of Hippo ( , ; la, Aurelius Augustinus Hipponensis; 13 November 354 – 28 August 430), also known as Saint Augustine, was a theologian and philosopher of Berber origin and the bishop of Hippo Regius in Numidia, Roman North Af ...
as last of the Western Fathers and first of the medievals".

# Works

Peirce's reputation rests largely on academic papers published in American scientific and scholarly journals such as ''Proceedings of the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...
'', the ''Journal of Speculative Philosophy'', '' The Monist'', ''
Popular Science ''Popular Science'' (also known as ''PopSci'') is an American digital magazine carrying popular science content, which refers to articles for the general reader on science and technology subjects. ''Popular Science'' has won over 58 awards, incl ...
Monthly'', the '' American Journal of Mathematics'', ''Memoirs of the
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...
'', ''
The Nation ''The Nation'' is an American liberal biweekly magazine that covers political and cultural news, opinion, and analysis. It was founded on July 6, 1865, as a successor to William Lloyd Garrison's '' The Liberator'', an abolitionist newspaper th ...
'', and others. See Articles by Peirce, published in his lifetime for an extensive list with links to them online. The only full-length book (neither extract nor pamphlet) that Peirce authored and saw published in his lifetime was '' Photometric Researches'' (1878), a 181-page 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 Postgraduate or graduate education refers to academic or professional degrees, certificates, diplomas, or other qualifications pursued by post-secondary students who have earned an undergraduate ( bachelor's) degree. The organization and ...
. Besides lectures during his years (1879–1884) as lecturer in Logic at Johns Hopkins, he gave at least nine series of lectures, many now published; see Lectures by Peirce. After Peirce's death,
Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher ...
obtained from Peirce's widow the papers found in his study, 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, totaling over 100,000 pages, mostly still unpublished except on microfilm. On the vicissitudes of Peirce's papers, see Houser (1989). Reportedly the papers remain in unsatisfactory condition. The first published anthology of Peirce's articles was the one-volume '' Chance, Love and Logic: Philosophical Essays'', edited by
Morris Raphael Cohen Morris Raphael Cohen ( be, Мо́рыс Рафаэ́ль Ко́эн; July 25, 1880 – January 28, 1947) was an American philosopher, lawyer, and legal scholar who united pragmatism with logical positivism and linguistic analysis. This union co ...
, 1923, still in print. Other one-volume anthologies were published in 1940, 1957, 1958, 1972, 1994, and 2009, most still in print. The main posthumous editions of Peirce's works in their long trek to light, often multi-volume, and some still in print, have included: 1931–1958: '' Collected Papers of Charles Sanders Peirce'' (CP), 8 volumes, includes many published works, along with a selection of previously unpublished work and a smattering of his correspondence. This long-time standard edition drawn from Peirce's work from the 1860s to 1913 remains the most comprehensive survey of his prolific output from 1893 to 1913. It is organized thematically, but texts (including lecture series) are often split up across volumes, while texts from various stages in Peirce's development are often combined, requiring frequent visits to editors' notes. Edited (1–6) by Charles Hartshorne and Paul Weiss and (7–8) by
Arthur Burks Arthur Walter Burks (October 13, 1915 – May 14, 2008) was an American mathematician who worked in the 1940s as a senior engineer on the project that contributed to the design of the ENIAC, the first general-purpose electronic digital computer. ...
, in print and online. 1975–1987: ''Charles Sanders Peirce: Contributions to'' The Nation, 4 volumes, includes Peirce's more than 300 reviews and articles published 1869–1908 in ''
The Nation ''The Nation'' is an American liberal biweekly magazine that covers political and cultural news, opinion, and analysis. It was founded on July 6, 1865, as a successor to William Lloyd Garrison's '' The Liberator'', an abolitionist newspaper th ...
''. Edited by Kenneth Laine Ketner and James Edward Cook, online. 1976: '' The New Elements of Mathematics by Charles S. Peirce'', 4 volumes in 5, included many previously unpublished Peirce manuscripts on mathematical subjects, along with Peirce's important published mathematical articles. Edited by Carolyn Eisele, back in print. 1977: '' Semiotic and Significs: The Correspondence between C. S. Peirce and Victoria Lady Welby'' (2nd edition 2001), included Peirce's entire correspondence (1903–1912) with
Victoria, Lady Welby Victoria, Lady Welby (27 April 1837 – 29 March 1912), more correctly Lady Welby-Gregory, was a self-educated British philosopher of language, musician and watercolourist. Life Welby was born to the Hon. Charles Stuart-Wortley-Mackenzie ...
. Peirce's other published correspondence is largely limited to the 14 letters included in volume 8 of the ''Collected Papers'', and the 20-odd pre-1890 items included so far in the ''Writings''. Edited by Charles S. Hardwick with James Cook, out of print. 1982–now: '' Writings of Charles S. Peirce, A Chronological Edition'' (W), Volumes 1–6 & 8, of a projected 30. The limited coverage, and defective editing and organization, of the ''Collected Papers'' led Max Fisch and others in the 1970s to found th
Peirce Edition Project
(PEP), whose mission is to prepare a more complete critical chronological edition. Only seven volumes have appeared to date, but they cover the period from 1859 to 1892, when Peirce carried out much of his best-known work. ''Writings of Charles S. Peirce'', 8 was published in November 2010; and work continues on ''Writings of Charles S. Peirce'', 7, 9, and 11. In print and online. 1985: '' Historical Perspectives on Peirce's Logic of Science: A History of Science'', 2 volumes. Auspitz has said, "The extent of Peirce's immersion in the science of his day is evident in his reviews in the ''Nation'' ..and in his papers, grant applications, and publishers' prospectuses in the history and practice of science", referring latterly to ''Historical Perspectives''. Edited by Carolyn Eisele, back in print. 1992: '' Reasoning and the Logic of Things'' collects in one place Peirce's 1898 series of lectures invited by William James. Edited by Kenneth Laine Ketner, with commentary by
Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, and computer scientist, and a major figure in analytic philosophy in the second half of the 20th century. He made significant contributions ...
, in print. 1992–1998: '' The Essential Peirce'' (EP), 2 volumes, is an important recent sampler of Peirce's philosophical writings. Edited (1) by Nathan Hauser and Christian Kloesel and (2) by ''Peirce Edition Project'' editors, in print. 1997: '' Pragmatism as a Principle and Method of Right Thinking'' collects Peirce's 1903 Harvard "Lectures on Pragmatism" in a study edition, including drafts, of Peirce's lecture manuscripts, which had been previously published in abridged form; the lectures now also appear in ''The Essential Peirce'', 2. Edited by Patricia Ann Turisi, in print. 2010: '' Philosophy of Mathematics: Selected Writings'' collects important writings by Peirce on the subject, many not previously in print. Edited by Matthew E. Moore, in print.

# Mathematics

Peirce's most important work in pure mathematics was in logical and foundational areas. He also worked on
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
,
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, various geometries,
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
and Listing numbers,
Bell number In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of epon ...
s,
graph Graph may refer to: Mathematics * Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of disc ...
s, the four-color problem, and the nature of continuity. He worked on applied mathematics in economics, engineering, and map projections (such as the Peirce quincuncial projection), and was especially active in probability and statistics. Burks, Arthur W., "Review: Charles S. Peirce, ''The new elements of mathematics''", ''Bulletin of the American Mathematical Society'' v. 84, n. 5 (1978)
pp. 913–18 (PDF)
;Discoveries Peirce made a number of striking discoveries in formal logic and foundational mathematics, nearly all of which came to be appreciated only long after he died: In 1860 he suggested a cardinal arithmetic for infinite numbers, years before any work by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his libe ...
's 1851 (posthumous) ''Paradoxien des Unendlichen''.
The Peirce arrow,
symbol for "(neither) ... nor ...", also called the ''Quine dagger''
In 1880–1881 he showed how
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
could be done via a repeated sufficient single binary operation ( logical NOR), anticipating Henry M. Sheffer by 33 years. (See also De Morgan's Laws.) In 1881 he set out the axiomatization of natural number arithmetic, a few years before
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
and
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The stand ...
. In the same paper Peirce gave, years before Dedekind, the first purely cardinal definition of a finite set in the sense now known as "
Dedekind-finite In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there exists a bijective function from ''A'' onto s ...
", and implied by the same stroke an important formal definition of an
infinite set In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set t ...
(Dedekind-infinite), as a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
that can be put into a
one-to-one correspondence In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
with one of its proper
subsets In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset ...
. In 1885Peirce (1885), "On the Algebra of Logic: A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7, two parts, first part published 1885, pp
180–202
(see Houser i

in "Introduction" in ''Writings of Charles S. Peirce'', 4). Presented, National Academy of Sciences, Newport, RI, October 14–17, 1884 (see ''The Essential Peirce'', 1

. 1885 is the year usually given for this work. Reprinted ''Collected Papers of Charles Sanders Peirce'', 3.359–403, ''Writings of Charles S. Peirce'', 5:162–90, ''The Essential Peirce'', 1:225–28, in part.
he distinguished between first-order and second-order quantification. In the same paper he set out what can be read as the first (primitive)
axiomatic set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, anticipating
Zermelo Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...
by about two decades (Brady 2000,Brady, Geraldine (2000), ''From Peirce to Skolem: A Neglected Chapter in the History of Logic'', North-Holland/Elsevier Science BV, Amsterdam, Netherlands. pp. 132–33). In 1886, he saw that Boolean calculations could be carried out via electrical switches, anticipating
Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, and cryptographer known as a "father of information theory". As a 21-year-old master's degree student at the Massachusetts Instit ...
by more than 50 years. By the later 1890s he was devising existential graphs, a diagrammatic notation for the
predicate calculus Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
. Based on them are John F. Sowa's
conceptual graph A conceptual graph (CG) is a formalism for knowledge representation. In the first published paper on CGs, John F. Sowa used them to represent the conceptual schemas used in database systems. The first book on CGs applied them to a wide range o ...
s and Sun-Joo Shin's diagrammatic reasoning. ;''The New Elements of Mathematics'' Peirce wrote drafts for an introductory textbook, with the working title ''The New Elements of Mathematics'', that presented mathematics from an original standpoint. Those drafts and many other of his previously unpublished mathematical manuscripts finally appeared in ''The New Elements of Mathematics by Charles S. Peirce'' (1976), edited by mathematician Carolyn Eisele. ;Nature of mathematics Peirce agreed with Auguste Comte in regarding mathematics as more basic than philosophy and the special sciences (of nature and mind). Peirce classified mathematics into three subareas: (1) mathematics of logic, (2) discrete series, and (3) pseudo-continua (as he called them, including the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
) and continua. Influenced by his father
Benjamin Benjamin ( he, ''Bīnyāmīn''; "Son of (the) right") blue letter bible: https://www.blueletterbible.org/lexicon/h3225/kjv/wlc/0-1/ H3225 - yāmîn - Strong's Hebrew Lexicon (kjv) was the last of the two sons of Jacob and Rachel (Jacob's thir ...
, Peirce argued that mathematics studies purely hypothetical objects and is not just the science of quantity but is more broadly the science which draws necessary conclusions; that mathematics aids logic, not vice versa; and that logic itself is part of philosophy and is the science ''about'' drawing conclusions necessary and otherwise.

## Mathematics of logic

Mathematical logic and foundations, some noted articles * "On an Improvement in Boole's Calculus of Logic" (1867) * "Description of a Notation for the Logic of Relatives" (1870) * "On the Algebra of Logic" (1880) * "A Boolean Algebra with One Constant" (1880 MS) * "On the Logic of Number" (1881) * "Note B: The Logic of Relatives" (1883) * "On the Algebra of Logic: A Contribution to the Philosophy of Notation" (1884/1885) * "The Logic of Relatives" (1897) * "The Simplest Mathematics" (1902 MS) * "Prolegomena to an Apology for Pragmaticism" (1906, on existential graphs)
Beginning with his first paper on the "Logic of Relatives" (1870), Peirce extended the
theory of relations In mathematics, a finitary relation over sets is a subset of the Cartesian product ; that is, it is a set of ''n''-tuples consisting of elements ''x'i'' in ''X'i''. Typically, the relation describes a possible connection between the elemen ...
that Augustus De Morgan had just recently awakened from its Cinderella slumbers. Much of the mathematics of relations now taken for granted was "borrowed" from Peirce, not always with all due credit; on that and on how the young
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
, especially his ''Principles of Mathematics'' and ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'', did not do Peirce justice, see Anellis (1995). In 1918 the logician C. I. Lewis wrote, "The contributions of C.S. Peirce to symbolic logic are more numerous and varied than those of any other writer—at least in the nineteenth century." Beginning in 1940,
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
and his students rediscovered aspects of Peirce's larger vision of relational logic, developing the perspective of relation algebra. Relational logic gained applications. In mathematics, it influenced the abstract analysis of E. H. Moore and the
lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper boun ...
of Garrett Birkhoff. In computer science, the
relational model The relational model (RM) is an approach to managing data using a structure and language consistent with first-order predicate logic, first described in 1969 by English computer scientist Edgar F. Codd, where all data is represented in terms of tu ...
for databases was developed with Peircean ideas in work of Edgar F. Codd, who was a doctoral student of Arthur W. Burks, a Peirce scholar. In economics, relational logic was used by Frank P. Ramsey,
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cov ...
, and Paul Samuelson to study preferences and utility and by Kenneth J. Arrow in ''
Social Choice and Individual Values Kenneth Arrow's monograph ''Social Choice and Individual Values'' (1951, 2nd ed., 1963, 3rd ed., 2012) and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor ...
'', following Arrow's association with Tarski at City College of New York. On Peirce and his contemporaries Ernst Schröder and
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phi ...
,
Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, and computer scientist, and a major figure in analytic philosophy in the second half of the 20th century. He made significant contributions ...
(1982)Putnam, Hilary (1982), "Peirce the Logician", ''Historia Mathematica'' 9, 290–301. Reprinted, pp. 252–60 in Putnam (1990), ''Realism with a Human Face'', Harvard
Excerpt with article's last five pages
documented that Frege's work on the logic of quantifiers had little influence on his contemporaries, although it was published four years before the work of Peirce and his student Oscar Howard Mitchell. Putnam found that mathematicians and logicians learned about the logic of quantifiers through the independent work of Peirce and Mitchell, particularly through Peirce's "On the Algebra of Logic: A Contribution to the Philosophy of Notation" (1885), published in the premier American mathematical journal of the day, and cited by
Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
and Schröder, among others, who ignored Frege. They also adopted and modified Peirce's notations, typographical variants of those now used. Peirce apparently was ignorant of Frege's work, despite their overlapping achievements in logic, philosophy of language, and the
foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathem ...
. Peirce's work on formal logic had admirers besides Ernst Schröder: * Philosophical algebraist
William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in h ...
and logician
William Ernest Johnson William Ernest Johnson, FBA (23 June 1858 – 14 January 1931), usually cited as W. E. Johnson, was a British philosopher, logician and economic theorist.Zabell, S.L. (2008"Johnson, William Ernest (1858–1931)"In: Durlauf S.N., Blume L.E. ( ...
, both British; * The Polish school of logic and foundational mathematics, including
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
; * Arthur Prior, who praised and studied Peirce's logical work in a 1964 paper and in ''Formal Logic'' (saying on page 4 that Peirce "perhaps had a keener eye for essentials than any other logician before or since"). A philosophy of logic, grounded in his categories and semiotic, can be extracted from Peirce's writings and, along with Peirce's logical work more generally, is exposited and defended in Hilary Putnam (1982); the Introduction in Nathan Houser ''et al.'' (1997); and Randall Dipert's chapter in Cheryl Misak (2004).

## Continua

Continuity and synechism are central in Peirce's philosophy: "I did not at first suppose that it was, as I gradually came to find it, the master-Key of philosophy". From a mathematical point of view, he embraced infinitesimals and worked long on the mathematics of continua. He long held that the real numbers constitute a pseudo-continuum; that a true continuum is the real subject matter of ''analysis situs'' (
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
); and that a true continuum of instants exceeds—and within any lapse of time has room for—any
Aleph number In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are name ...
(any infinite ''multitude'' as he called it) of instants. In 1908 Peirce wrote that he found that a true continuum might have or lack such room. Jérôme Havenel (2008): "It is on 26 May 1908, that Peirce finally gave up his idea that in every continuum there is room for whatever collection of any multitude. From now on, there are different kinds of continua, which have different properties."

## Probability and statistics

Peirce held that science achieves statistical probabilities, not certainties, and that spontaneity (absolute chance) is real (see Tychism on his view). Most of his statistical writings promote the frequency interpretation of probability (objective ratios of cases), and many of his writings express skepticism about (and criticize the use of) probability when such models are not based on objective
randomization Randomization is the process of making something random. Randomization is not haphazard; instead, a random process is a sequence of random variables describing a process whose outcomes do not follow a deterministic pattern, but follow an evolution d ...
. Though Peirce was largely a frequentist, his
possible world semantics A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their m ...
introduced the "propensity" theory of probability before
Karl Popper Sir Karl Raimund Popper (28 July 1902 – 17 September 1994) was an Austrian-British philosopher, academic and social commentator. One of the 20th century's most influential philosophers of science, Popper is known for his rejection of the c ...
. Peirce (sometimes with Joseph Jastrow) investigated the probability judgments of experimental subjects, "perhaps the very first" elicitation and estimation of subjective probabilities in
experimental psychology Experimental psychology refers to work done by those who apply experimental methods to psychological study and the underlying processes. Experimental psychologists employ human participants and animal subjects to study a great many topics, in ...
and (what came to be called)
Bayesian statistics Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about the event, ...
. Peirce was one of the
founders of statistics Statistics is the theory and application of mathematics to the scientific method including hypothesis generation, experimental design, sampling, data collection, data summarization, estimation, prediction and inference from those results to the ...
. He formulated modern statistics in " Illustrations of the Logic of Science" (1877–1878) and " A Theory of Probable Inference" (1883). With a repeated measures design, Charles Sanders Peirce and Joseph Jastrow introduced blinded, controlled randomized experiments in 1884 (Hacking 1990:205) (before Ronald A. Fisher). He invented optimal design for experiments on gravity, in which he " corrected the means". He used
correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
and
smoothing In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena. In smoothing, the d ...
. Peirce extended the work on
outliers In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter ar ...
by Benjamin Peirce, his father. He introduced terms " confidence" and "
likelihood The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood funct ...
" (before Jerzy Neyman and Fisher). (See
Stephen Stigler Stephen Mack Stigler (born August 10, 1941) is Ernest DeWitt Burton Distinguished Service Professor at the Department of Statistics of the University of Chicago. He has authored several books on the history of statistics; he is the son of the ...
's historical books and
Ian Hacking Ian MacDougall Hacking (born February 18, 1936) is a Canadian philosopher specializing in the philosophy of science. Throughout his career, he has won numerous awards, such as the Killam Prize for the Humanities and the Balzan Prize, and been ...
1990.)

# Philosophy

Peirce was a working scientist for 30 years, and arguably was a professional philosopher only during the five years he lectured at Johns Hopkins. He learned philosophy mainly by reading, each day, a few pages of
Immanuel Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and aest ...
's '' Critique of Pure Reason'', in the original German, while a Harvard undergraduate. His writings bear on a wide array of disciplines, including mathematics,
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, philosophy, statistics,
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxies ...
,
metrology Metrology is the scientific study of measurement. It establishes a common understanding of units, crucial in linking human activities. Modern metrology has its roots in the French Revolution's political motivation to standardise units in Fra ...
,
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
,
experimental psychology Experimental psychology refers to work done by those who apply experimental methods to psychological study and the underlying processes. Experimental psychologists employ human participants and animal subjects to study a great many topics, in ...
, economics, linguistics, and the history and philosophy of science. This work has enjoyed renewed interest and approval, 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 philosophy includes (see below in related sections) a pervasive three-category system: belief that truth is immutable and is both independent from actual opinion ( fallibilism) and discoverable (no radical skepticism), logic as formal semiotic on signs, on arguments, and on inquiry's ways—including philosophical pragmatism (which he founded), critical common-sensism, and
scientific method The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific m ...
—and, in metaphysics: Scholastic realism, e.g. John Duns Scotus, belief in God, freedom, and at least an attenuated immortality,
objective idealism Objective idealism is a form of metaphysical idealism that accepts Naïve realism (the view that empirical objects exist objectively) but rejects epiphenomenalist materialism (according to which the mind and spiritual values have emerged due to ...
, and belief in the reality of continuity and of absolute chance, mechanical necessity, and creative love. In his work, fallibilism and pragmatism may seem to work somewhat like
skepticism Skepticism, also spelled scepticism, is a questioning attitude or doubt toward knowledge claims that are seen as mere belief or dogma. For example, if a person is skeptical about claims made by their government about an ongoing war then the pe ...
and
positivism Positivism is an empiricist philosophical theory that holds that all genuine knowledge is either true by definition or positive—meaning ''a posteriori'' facts derived by reason and logic from sensory experience.John J. Macionis, Linda M. G ...
, respectively, in others' work. However, for Peirce, fallibilism is balanced by an anti-skepticism and is a basis for belief in the reality of absolute chance and of continuity,Peirce (1897) "Fallibilism, Continuity, and Evolution", ''Collected Papers of Charles Sanders Peirce'', 1.141–75
Eprint
, placed by the ''Collected Papers of Charles Sanders Peirce'', editors directly after "F.R.L." (1899, ''Collected Papers of Charles Sanders Peirce'', 1.135–40).
and pragmatism commits one to anti- nominalist belief in the reality of the general (CP 5.453–57). For Peirce, First Philosophy, which he also called cenoscopy, is less basic than mathematics and more basic than the special sciences (of nature and mind). It studies positive phenomena in general, phenomena available to any person at any waking moment, and does not settle questions by resorting to special experiences.Peirce (1903), ''Collected Papers of Charles Sanders Peirce'', 1.180–202 and (1906) "The Basis of Pragmaticism", ''The Essential Peirce'', 2:372–73, see
Philosophy
at ''Commens Digital Companion to C.S. Peirce''.
He
divided Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication. At an elementary level the division of two natural numbers ...
such philosophy into (1) phenomenology (which he also called phaneroscopy or categorics), (2) normative sciences (esthetics, ethics, and logic), and (3) metaphysics; his views on them are discussed in order below.

## Theory of categories

On May 14, 1867, the 27-year-old Peirce presented a paper entitled "On a New List of Categories" to the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...
, which published it the following year. The paper outlined a theory of predication, involving three universal categories that Peirce developed in response to reading
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phi ...
,
Immanuel Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and aest ...
, and G. W. F. Hegel, categories that Peirce applied throughout his work for the rest of his life. Peirce scholars generally regard the "New List" as foundational or breaking the ground for Peirce's "architectonic", his blueprint for a pragmatic philosophy. In the categories one will discern, concentrated, the pattern that one finds formed by the three grades of clearness in " How To Make Our Ideas Clear" (1878 paper foundational to pragmatism), and in numerous other trichotomies in his work. "On a New List of Categories" is cast as a Kantian deduction; it is short but dense and difficult to summarize. The following table is compiled from that and later works. In 1893, Peirce restated most of it for a less advanced audience.

## Aesthetics and ethics

Peirce did not write extensively in aesthetics and ethics, but came by 1902 to hold that aesthetics, ethics, and logic, in that order, comprise the normative sciences. He characterized aesthetics as the study of the good (grasped as the admirable), and thus of the ends governing all conduct and thought.

# Philosophy: logic, or semiotic

## Logic as philosophical

Peirce regarded logic ''per se'' as a division of philosophy, as a normative science based on esthetics and ethics, as more basic than metaphysics,Peirce (1899 MS), "F.R.L." irst Rule of Logic ''Collected Papers of Charles Sanders Peirce'', 1.135–40
Eprint
/ref> and as "the art of devising methods of research".Peirce (1882), "Introductory Lecture on the Study of Logic" delivered September 1882, ''Johns Hopkins University Circulars'', v. 2, n. 19, pp
11–12
(via Google), November 1882. Reprinted (''The Essential Peirce'', 1:210–14; ''Writings of Charles S. Peirce'', 4:378–82; ''Collected Papers of Charles Sanders Peirce'', 7.59–76). The definition of logic quoted by Peirce is by
Peter of Spain __NOTOC__ Peter of Hispania ( la, Petrus Hispanus; Portuguese and es, Pedro Hispano; century) was the author of the ', later known as the ', an important medieval university textbook on Aristotelian logic. As the Latin ''Hispania'' was consider ...
.
More generally, as inference, "logic is rooted in the social principle", since inference depends on a standpoint that, in a sense, is unlimited. Peirce called (with no sense of deprecation) "mathematics of logic" much of the kind of thing which, in current research and applications, is called simply "logic". He was productive in both (philosophical) logic and logic's mathematics, which were connected deeply in his work and thought. Peirce argued that logic is formal semiotic: 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", along with their representational and inferential relations. He argued that, since all thought takes time, all thought is in signsPeirce, (1868), "Questions concerning certain Faculties claimed for Man", ''Journal of Speculative Philosophy'' v. 2, n. 2
pp. 103–14
On thought in signs, see p. 112. Reprinted ''Collected Papers of Charles Sanders Peirce'', 5.213–63 (on thought in signs, see 253), ''Writings of Charles S. Peirce'', 2:193–211, ''The Essential Peirce'', 2:11–27. ''Arisbe'

and sign processes ("semiosis") such as the inquiry process. He
divided Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication. At an elementary level the division of two natural numbers ...
logic into: (1) speculative grammar, or stechiology, on how signs can be meaningful and, in relation to that, what kinds of signs there are, how they combine, and how some embody or incorporate others; (2) logical critic, or logic proper, on the modes of inference; and (3) speculative or universal rhetoric, or methodeutic, the philosophical theory of inquiry, including pragmatism.

### Presuppositions of logic

In his "F.R.L." irst Rule of Logic(1899), Peirce states that the first, and "in one sense, the sole", rule of reason is that, ''to learn, one needs to desire to learn'' and desire it without resting satisfied with that which one is inclined to think. So, the first rule is, ''to wonder''. Peirce proceeds to a critical theme in research practices and the shaping of theories:
...there follows one
corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
which itself deserves to be inscribed upon every wall of the city of philosophy: ::Do not block the way of inquiry.
Peirce adds, that method and economy are best in research but no outright sin inheres in trying any theory in the sense that the investigation via its trial adoption can proceed unimpeded and undiscouraged, and that "the one unpardonable offence" is a philosophical barricade against truth's advance, an offense to which "metaphysicians in all ages have shown themselves the most addicted". Peirce in many writings holds that logic precedes metaphysics (ontological, religious, and physical). Peirce goes on to list four common barriers to inquiry: (1) Assertion of absolute certainty; (2) maintaining that something is absolutely unknowable; (3) maintaining that something is absolutely inexplicable because absolutely basic or ultimate; (4) holding that perfect exactitude is possible, especially such as to quite preclude unusual and anomalous phenomena. To refuse absolute theoretical certainty is the heart of ''fallibilism'', which Peirce unfolds into refusals to set up any of the listed barriers. Peirce elsewhere argues (1897) that logic's presupposition of fallibilism leads at length to the view that chance and continuity are very real ( tychism and synechism). The First Rule of Logic pertains to the mind's presuppositions in undertaking reason and logic; presuppositions, for instance, that truth and the real do not depend on yours or my opinion of them but do depend on representational relation and consist in the destined end in investigation taken far enough ( see below). He describes such ideas as, collectively, hopes which, in particular cases, one is unable seriously to doubt.

### Four incapacities

The ''Journal of Speculative Philosophy'' series (1868–1869), including * Questions concerning certain Faculties claimed for Man (1868) * Some Consequences of Four Incapacities (1868) * Grounds of Validity of the Laws of Logic:
Further Consequences of Four Incapacities (1869)
In three articles in 1868–1869,Peirce (1868), "Some Consequences of Four Incapacities", ''Journal of Speculative Philosophy'' v. 2, n. 3
pp. 140–57
Reprinted ''Collected Papers of Charles Sanders Peirce'', 5.264–317, ''Writings of Charles S. Peirce'', 2:211–42, ''The Essential Peirce'', 1:28–55. ''Arisbe'

Peirce, "Grounds of Validity of the Laws of Logic: Further Consequences of Four Incapacities", ''Journal of Speculative Philosophy'' v. II, n. 4
pp. 193–208
Reprinted ''Collected Papers of Charles Sanders Peirce'', 5.318–57, ''Writings of Charles S. Peirce'', 2:242–72 (''Peirce Edition Project''

, ''The Essential Peirce'', 1:56–82.
Peirce rejected mere verbal or
hyperbolic doubt Cartesian doubt is a form of methodological skepticism associated with the writings and methodology of René Descartes (March 31, 1596Feb 11, 1650). Scruton, R.''Modern Philosophy: An Introduction and Survey''(London: Penguin Books, 1994). Leiber, ...
and first or ultimate principles, and argued that we have (as he numbered them): # No power of Introspection. All knowledge of the internal world comes by hypothetical reasoning from known external facts. # No power of Intuition (cognition without logical determination by previous cognitions). No cognitive stage is absolutely first in a process. All mental action has the form of inference. # No power of thinking without signs. A cognition must be interpreted in a subsequent cognition in order to be a cognition at all. # No conception of the absolutely incognizable. (The above sense of the term "intuition" is almost Kant's, said Peirce. It differs from the current looser sense that encompasses instinctive or anyway half-conscious inference.) Peirce argued that those incapacities imply the reality of the general and of the continuous, the validity of the modes of reasoning, and the falsity of philosophical
Cartesianism Cartesianism is the philosophical and scientific system of René Descartes and its subsequent development by other seventeenth century thinkers, most notably François Poullain de la Barre, Nicolas Malebranche and Baruch Spinoza. Descartes ...
( see below). Peirce rejected the conception (usually ascribed to Kant) of the unknowable thing-in-itself and later said that to "dismiss make-believes" is a prerequisite for pragmatism.

### Logic as formal semiotic

Peirce sought, through his wide-ranging studies through the decades, formal philosophical ways to articulate thought's processes, and also to explain the workings of science. These inextricably entangled questions of a dynamics of inquiry rooted in nature and nurture led him to develop his semiotic with very broadened conceptions of signs and inference, and, as its culmination, a theory of inquiry for the task of saying 'how science works' and devising research methods. This would be logic by the medieval definition taught for centuries: art of arts, science of sciences, having the way to the principles of all methods. Influences radiate from points on parallel lines of inquiry in
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phi ...
's work, in such ''loci'' as: the basic terminology of psychology in ''
On the Soul ''On the Soul'' ( Greek: , ''Peri Psychēs''; Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Lat ...
''; the founding description of sign relations in '' On Interpretation''; and the differentiation of inference into three modes that are commonly translated into English as '' abduction'', '' deduction'', and '' induction'', in the ''
Prior Analytics The ''Prior Analytics'' ( grc-gre, Ἀναλυτικὰ Πρότερα; la, Analytica Priora) is a work by Aristotle on reasoning, known as his syllogistic, composed around 350 BCE. Being one of the six extant Aristotelian writings on logic ...
'', as well as inference by
analogy Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject (th ...
(called ''paradeigma'' by Aristotle), which Peirce regarded as involving the other three modes. Peirce began writing on semiotic in the 1860s, around the time when he devised his system of three categories. He called it both ''
semiotic Semiotics (also called semiotic studies) is the systematic study of sign processes ( semiosis) and meaning making. Semiosis is any activity, conduct, or process that involves signs, where a sign is defined as anything that communicates something ...
'' and ''semeiotic''. Both are current in singular and plural. He based it on the conception of a triadic sign relation, and defined ''
semiosis Semiosis (, ), or sign process, is any form of activity, conduct, or process that involves signs, including the production of meaning. A sign is anything that communicates a meaning, that is not the sign itself, to the interpreter of the sign. ...
'' as "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". As to signs in thought, Peirce emphasized the reverse: "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 held that all thought is in signs, issuing in and from interpretation, where ''sign'' is the word for the broadest variety of conceivable semblances, diagrams, metaphors, symptoms, signals, designations, symbols, texts, even mental concepts and ideas, all as determinations of a mind or ''quasi-mind'', that which at least functions like a mind, as in the work of crystals or bees—the focus is on sign action in general rather than on psychology, linguistics, or social studies (fields which he also pursued). Inquiry is a kind of inference process, a manner of thinking and semiosis. Global divisions of ways for phenomena to stand as signs, and the subsumption of inquiry and thinking within inference as a sign process, enable the study of inquiry on semiotics' three levels: # Conditions for meaningfulness. Study of significatory elements and combinations, their grammar. # Validity, conditions for true representation. Critique of arguments in their various separate modes. # Conditions for determining interpretations. Methodology of inquiry in its mutually interacting modes. Peirce uses examples often from common experience, but defines and discusses such things as assertion and interpretation in terms of philosophical logic. In a formal vein, Peirce said:

## Signs

### Sign relation

Peirce's theory of signs is known to be one of the most complex semiotic theories due to its generalistic claim. Anything is a sign—not absolutely as itself, but instead in some relation or other. The '' sign relation'' is the key. It defines three roles encompassing (1) the sign, (2) the sign's subject matter, called its ''object'', and (3) the sign's meaning or ramification as formed into a kind of effect called its ''interpretant'' (a further sign, for example a translation). It is an irreducible '' triadic relation'', according to Peirce. The roles are distinct even when the things that fill those roles are not. The roles are but three; a sign of an object leads to one or more interpretants, and, as signs, they lead to further interpretants. ''Extension × intension = information.'' Two traditional approaches to sign relation, necessary though insufficient, are the way of '' extension'' (a sign's objects, also called breadth, denotation, or application) and the way of ''
intension In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an intension is any property or quality connoted by a word, phrase, or an ...
'' (the objects' characteristics, qualities, attributes referenced by the sign, also called depth, comprehension, significance, or connotation). Peirce adds a third, the way of ''
information Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random, ...
'', including change of information, to integrate the other two approaches into a unified whole. For example, because of the equation above, if a term's total amount of information stays the same, then the more that the term 'intends' or signifies about objects, the fewer are the objects to which the term 'extends' or applies. ''Determination.'' A sign depends on its object in such a way as to represent its object—the object enables and, in a sense, determines the sign. A physically causal sense of this stands out when a sign consists in an indicative reaction. The interpretant depends likewise on both the sign and the object—an object determines a sign to determine an interpretant. But this determination is not a succession of dyadic events, like a row of toppling dominoes; sign determination is triadic. For example, an interpretant does not merely represent something which represented an object; instead an interpretant represents something ''as'' a sign representing the object. The object (be it a quality or fact or law or even fictional) determines the sign to an interpretant through one's collateral experienceSee pp. 404–09 in "Pragmatism" in ''The Essential Peirce'', 2. Ten quotes on collateral experience from Peirce provided by Joseph Ransdell can be viewe
here
at peirce-l's Lyris archive. Note: Ransdell's quotes from ''Collected Papers of Charles Sanders Peirce'', 8.178–79 are also in ''The Essential Peirce'', 2:493–94, which gives their date as 1909; and his quote from ''Collected Papers of Charles Sanders Peirce'', 8.183 is also in ''The Essential Peirce'', 2:495–96, which gives its date as 1909.
with the object, in which the object is found or from which it is recalled, as when a sign consists in a chance semblance of an absent object. Peirce used the word "determine" not in a strictly deterministic sense, but in a sense of "specializes", ''bestimmt'',Peirce, letter to William James, dated 1909, see ''The Essential Peirce'', 2:492. involving variable amount, like an influence.See

, collected by Robert Marty (U. of Perpignan, France).
Peirce came to define representation and interpretation in terms of (triadic) determination. The object determines the sign to determine another sign—the interpretant—to be related to the object ''as the sign is related to the object'', hence the interpretant, fulfilling its function as sign of the object, determines a further interpretant sign. The process is logically structured to perpetuate itself, and is definitive of sign, object, and interpretant in general.

### Semiotic elements

Peirce held there are exactly three basic elements in semiosis (sign action): # A ''sign'' (or ''representamen'') represents, in the broadest possible sense of "represents". It is something interpretable as saying something about something. It is not necessarily symbolic, linguistic, or artificial—a cloud might be a sign of rain for instance, or ruins the sign of ancient civilization. As Peirce sometimes put it (he defined ''sign'' at least 76 times), the sign stands ''for'' the object ''to'' the interpretant. A sign represents its object in some respect, which respect is the sign's ''ground''. # An ''object'' (or ''semiotic object'') is a subject matter of a sign and an interpretant. It can be anything thinkable, a quality, an occurrence, a rule, etc., even fictional, such as
Prince Hamlet A prince is a male ruler (ranked below a king, grand prince, and grand duke) or a male member of a monarch's or former monarch's family. ''Prince'' is also a title of nobility (often highest), often hereditary, in some European states. The ...
.Peirce (1909), A Letter to William James, ''The Essential Peirce'', 2:492–502. Fictional object, 498. Object as universe of discourse, 492. See
Dynamical Object
at ''Commens Digital Companion to C.S. Peirce''.
All of those are special or partial objects. The object most accurately is the
universe of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The doma ...
to which the partial or special object belongs. For instance, a perturbation of Pluto's orbit is a sign about Pluto but ultimately not only about Pluto. An object either (i) is ''immediate'' to a sign and is the object as represented in the sign or (ii) is a ''dynamic'' object, the object as it really is, on which the immediate object is founded "as on bedrock". # An ''
interpretant Interpretant is a subject (philosophy) / sign (semiotics) that refers to the same object (philosophy) as another sign (semiotics), transitively. History The concept of "interpretant" is part of Charles Sanders Peirce's "triadic" theory of ...
'' (or ''interpretant sign'') is a sign's meaning or ramification as formed into a kind of idea or effect, an interpretation, human or otherwise. An interpretant is a sign (a) of the object and (b) of the interpretant's "predecessor" (the interpreted sign) as a sign of the same object. An interpretant either (i) is ''immediate'' to a sign and is a kind of quality or possibility such as a word's usual meaning, or (ii) is a ''dynamic'' interpretant, such as a state of agitation, or (iii) is a ''final'' or ''normal'' interpretant, a sum of the lessons which a sufficiently considered sign ''would'' have as effects on practice, and with which an actual interpretant may at most coincide. Some of the understanding needed by the mind depends on familiarity with the object. To know what a given sign denotes, the mind needs some experience of that sign's object, experience outside of, and collateral to, that sign or sign system. In that context Peirce speaks of collateral experience, collateral observation, collateral acquaintance, all in much the same terms.

### Classes of signs

Among Peirce's many sign typologies, three stand out, interlocked. The first typology depends on the sign itself, the second on how the sign stands for its denoted object, and the third on how the sign stands for its object to its interpretant. Also, each of the three typologies is a three-way division, a trichotomy, via Peirce's three phenomenological categories: (1) quality of feeling, (2) reaction, resistance, and (3) representation, mediation. I. ''Qualisign, sinsign, legisign'' (also called'' tone, token, type,'' and also called ''potisign, actisign, famisign''):On the varying terminology, look up i
''Commens Digital Companion to C.S. Peirce''
This typology classifies every sign according to the sign's own phenomenological category—the qualisign is a quality, a possibility, a "First"; the sinsign is a reaction or resistance, a singular object, an actual event or fact, a "Second"; and the legisign is a habit, a rule, a representational relation, a "Third". II. ''Icon, index, symbol'': This typology, the best known one, classifies every sign according to the category of the sign's way of denoting its object—the icon (also called semblance or likeness) by a quality of its own, the index by factual connection to its object, and the symbol by a habit or rule for its interpretant. III. ''Rheme, dicisign, argument'' (also called ''sumisign, dicisign, suadisign,'' also ''seme, pheme, delome,'' and regarded as very broadened versions of the traditional ''term, proposition, argument''): This typology classifies every sign according to the category which the interpretant attributes to the sign's way of denoting its object—the rheme, for example a term, is a sign interpreted to represent its object in respect of quality; the dicisign, for example a proposition, is a sign interpreted to represent its object in respect of fact; and the argument is a sign interpreted to represent its object in respect of habit or law. This is the culminating typology of the three, where the sign is understood as a structural element of inference. Every sign belongs to one class or another within (I) ''and'' within (II) ''and'' within (III). Thus each of the three typologies is a three-valued parameter for every sign. The three parameters are not independent of each other; many co-classifications are absent, for reasons pertaining to the lack of either habit-taking or singular reaction in a quality, and the lack of habit-taking in a singular reaction. The result is not 27 but instead ten classes of signs fully specified at this level of analysis.

## Modes of inference

Borrowing a brace of concepts from
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phi ...
, Peirce examined three basic modes of inference—'' abduction'', '' deduction'', and '' induction''—in his "critique of arguments" or "logic proper". Peirce also called abduction "retroduction", "presumption", and, earliest of all, "hypothesis". He characterized it as guessing and as inference to an explanatory hypothesis. He sometimes expounded the modes of inference by transformations of the categorical syllogism Barbara (AAA), for example in "Deduction, Induction, and Hypothesis" (1878). He does this by rearranging the ''rule'' (Barbara's major premise), the ''case'' (Barbara's minor premise), and the ''result'' (Barbara's conclusion): Deduction. ''Rule:'' All the beans from this bag are white.
''Case:'' These beans are beans from this bag.
$\therefore$ ''Result:'' These beans are white. Induction. ''Case:'' These beans are [randomly selected] from this bag.
''Result:'' These beans are white.
$\therefore$ ''Rule:'' All the beans from this bag are white. Hypothesis (Abduction). ''Rule:'' All the beans from this bag are white.
''Result:'' These beans ddlyare white.
$\therefore$ ''Case:'' These beans are from this bag. Peirce 1883 in "A Theory of Probable Inference" ('' Studies in Logic'') equated hypothetical inference with the induction of characters of objects (as he had done in effect before). Eventually dissatisfied, by 1900 he distinguished them once and for all and also wrote that he now took the syllogistic forms and the doctrine of logical extension and comprehension as being less basic than he had thought. In 1903 he presented the following logical form for abductive inference: The logical form does not also cover induction, since induction neither depends on surprise nor proposes a new idea for its conclusion. Induction seeks facts to test a hypothesis; abduction seeks a hypothesis to account for facts. "Deduction proves that something ''must'' be; Induction shows that something ''actually is'' operative; Abduction merely suggests that something ''may be''." Peirce did not remain quite convinced that one logical form covers all abduction. In his methodeutic or theory of inquiry (see below), he portrayed abduction as an economic initiative to further inference and study, and portrayed all three modes as clarified by their coordination in essential roles in inquiry: hypothetical explanation, deductive prediction, inductive testing.

## Pragmatism

Some noted articles and lectures * Illustrations of the Logic of Science (1877–1878):
inquiry, pragmatism, statistics, inference # The Fixation of Belief (1877) # How to Make Our Ideas Clear (1878) # The Doctrine of Chances (1878) # The Probability of Induction (1878) # The Order of Nature (1878) # Deduction, Induction, and Hypothesis (1878) * The Harvard lectures on pragmatism (1903) * What Pragmatism Is (1905) * Issues of Pragmaticism (1905) * Pragmatism (1907 MS in ''The Essential Peirce'', 2)
Peirce's recipe for pragmatic thinking, which he called '' pragmatism'' and, later, ''
pragmaticism "Pragmaticism" is a term used by Charles Sanders Peirce for his pragmatic philosophy starting in 1905, in order to distance himself and it from pragmatism, the original name, which had been used in a manner he did not approve of in the "literary ...
'', is recapitulated in several versions of the so-called ''
pragmatic maxim {{C. S. Peirce articles, abbreviations=no The pragmatic maxim, also known as the maxim of pragmatism or the maxim of pragmaticism, is a maxim of logic Logic is the study of correct reasoning. It includes both formal and informal logic. F ...
''. Here is one of his more emphatic reiterations of it: As a movement, pragmatism began in the early 1870s in discussions among Peirce,
William James William James (January 11, 1842 – August 26, 1910) was an American philosopher, historian, and psychologist, and the first educator to offer a psychology course in the United States. James is considered to be a leading thinker of the lat ...
, and others in
the Metaphysical Club The Metaphysical Club was a name attributed by the philosopher Charles Sanders Peirce, in an unpublished paper over thirty years after its foundation, to a conversational philosophical club that Peirce, the future Supreme Court Justice Oliver Wen ...
. James among others regarded some articles by Peirce such as " The Fixation of Belief" (1877) and especially " How to Make Our Ideas Clear" (1878) as foundational to pragmatism. Peirce (CP 5.11–12), like James ('' Pragmatism: A New Name for Some Old Ways of Thinking'', 1907), saw pragmatism as embodying familiar attitudes, in philosophy and elsewhere, elaborated into a new deliberate method for fruitful thinking about problems. Peirce differed from James and the early
John Dewey John Dewey (; October 20, 1859 – June 1, 1952) was an American philosopher, psychologist, and educational reformer whose ideas have been influential in education and social reform. He was one of the most prominent American scholars in the fi ...
, in some of their tangential enthusiasms, in being decidedly more rationalistic and realistic, in several senses of those terms, throughout the preponderance of his own philosophical moods. In 1905 Peirce coined the new name
pragmaticism "Pragmaticism" is a term used by Charles Sanders Peirce for his pragmatic philosophy starting in 1905, in order to distance himself and it from pragmatism, the original name, which had been used in a manner he did not approve of in the "literary ...
"for the precise purpose of expressing the original definition", saying that "all went happily" with James's and F.C.S. Schiller's variant uses of the old name "pragmatism" and that he coined the new name because of the old name's growing use in "literary journals, where it gets abused". Yet he cited as causes, in a 1906 manuscript, his differences with James and Schiller and, in a 1908 publication, his differences with James as well as literary author
Giovanni Papini Giovanni Papini (9 January 18818 July 1956) was an Italian journalist, essayist, novelist, short story writer, poet, literary critic, and philosopher. A controversial literary figure of the early and mid-twentieth century, he was the earliest and ...
's declaration of pragmatism's indefinability. Peirce in any case regarded his views that truth is immutable and infinity is real, as being opposed by the other pragmatists, but he remained allied with them on other issues. Pragmatism begins with the idea that belief is that on which one is prepared to act. Peirce's pragmatism is a method of clarification of conceptions of objects. It equates any conception of an object to a conception of that object's effects to a general extent of the effects' conceivable implications for informed practice. It is a method of sorting out conceptual confusions occasioned, for example, by distinctions that make (sometimes needed) formal yet not practical differences. He formulated both pragmatism and statistical principles as aspects of scientific logic, in his "Illustrations of the Logic of Science" series of articles. In the second one, " How to Make Our Ideas Clear", Peirce discussed three grades of clearness of conception: # Clearness of a conception familiar and readily used, even if unanalyzed and undeveloped. # Clearness of a conception in virtue of clearness of its parts, in virtue of which logicians called an idea "distinct", that is, clarified by analysis of just what makes it applicable. Elsewhere, echoing Kant, Peirce called a likewise distinct definition "nominal" (CP 5.553). # Clearness in virtue of clearness of conceivable practical implications of the object's conceived effects, such that fosters fruitful reasoning, especially on difficult problems. Here he introduced that which he later called the
pragmatic maxim {{C. S. Peirce articles, abbreviations=no The pragmatic maxim, also known as the maxim of pragmatism or the maxim of pragmaticism, is a maxim of logic Logic is the study of correct reasoning. It includes both formal and informal logic. F ...
. By way of example of how to clarify conceptions, he addressed conceptions about truth and the real as questions of the presuppositions of reasoning in general. In clearness's second grade (the "nominal" grade), he defined truth as a sign's correspondence to its object, and the real as the object of such correspondence, such that truth and the real are independent of that which you or I or any actual, definite community of inquirers think. After that needful but confined step, next in clearness's third grade (the pragmatic, practice-oriented grade) he defined truth as that opinion which ''would'' be reached, sooner or later but still inevitably, by research taken far enough, such that the real does depend on that ideal final opinion—a dependence to which he appeals in theoretical arguments elsewhere, for instance for the long-run validity of the rule of induction."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 Peirce (1878 April), "The Probability of Induction", p
718
(via ''Internet Archive'' ) in ''Popular Science Monthly'', v. 12, pp. 705–18. Reprinted in ''Collected Papers of Charles Sanders Peirce'', 2.669–93, ''Writings of Charles S. Peirce'', 3:290–305, ''The Essential Peirce'', 1:155–69, elsewhere.
Peirce argued that even to argue against the independence and discoverability of truth and the real is to presuppose that there is, about that very question under argument, a truth with just such independence and discoverability. Peirce said that a conception's meaning consists in " all general modes of rational conduct" implied by "acceptance" of the conception—that is, if one were to accept, first of all, the conception as true, then what could one conceive to be consequent general modes of rational conduct by all who accept the conception as true?—the whole of such consequent general modes is the whole meaning. His pragmatism does not equate a conception's meaning, its intellectual purport, with the conceived benefit or cost of the conception itself, like a meme (or, say, propaganda), outside the perspective of its being true, nor, since a conception is general, is its meaning equated with any definite set of actual consequences or upshots corroborating or undermining the conception or its worth. His pragmatism also bears no resemblance to "vulgar" pragmatism, which misleadingly connotes a ruthless and Machiavellian search for mercenary or political advantage. Instead the pragmatic maxim is the heart of his pragmatism as a method of experimentational mental reflection arriving at conceptions in terms of conceivable confirmatory and disconfirmatory circumstances—a method hospitable to the formation of explanatory hypotheses, and conducive to the use and improvement of verification. Peirce's pragmatism, as method and theory of definitions and conceptual clearness, is part of his theory of inquiry, which he variously called speculative, general, formal or universal rhetoric or simply methodeutic.Se
rhetoric definitions
at ''Commens Digital Companion to C.S. Peirce''.
He applied his pragmatism as a method throughout his work.

### Theory of inquiry

In
The Fixation of Belief
(1877), Peirce gives his take on the psychological origin and aim of inquiry. On his view, individuals are motivated to inquiry by desire to escape the feelings of anxiety and unease which Peirce takes to be characteristic of the state of doubt. Doubt is described by Peirce as an "uneasy and dissatisfied state from which we struggle to free ourselves and pass into the state of belief." Peirce uses words like “irritation” to describe the experience of being in doubt and to explain why he thinks we find such experiences to be motivating. The irritating feeling of doubt is appeased, Peirce says, through our efforts to achieve a settled state of satisfaction with what we land on as our answer to the question which led to that doubt in the first place. This settled state, namely, belief, is described by Peirce as “a calm and satisfactory state which we do not wish to avoid.” Our efforts to achieve the satisfaction of belief, by whichever methods we may pursue, are what Peirce calls “inquiry”. Four methods which Peirce describes as having been actually pursued throughout the history of thought are summarized below in the section after next.

### =Critical common-sensism

= Critical common-sensism, treated by Peirce as a consequence of his pragmatism, is his combination of Thomas Reid's common-sense philosophy with a fallibilism that recognizes that propositions of our more or less vague common sense now indubitable may later come into question, for example because of transformations of our world through science. It includes efforts to work up in tests genuine doubts for a core group of common indubitables that vary slowly if at all.

### =Rival methods of inquiry

= In " The Fixation of Belief" (1877), Peirce described inquiry in general not as the pursuit of truth ''per se'' but as the struggle to move from irritating, inhibitory doubt born of surprise, disagreement, and the like, and to reach a secure belief, belief being that on which one is prepared to act. That let Peirce frame scientific inquiry as part of a broader spectrum and as spurred, like inquiry generally, by actual doubt, not mere verbal, quarrelsome, or
hyperbolic doubt Cartesian doubt is a form of methodological skepticism associated with the writings and methodology of René Descartes (March 31, 1596Feb 11, 1650). Scruton, R.''Modern Philosophy: An Introduction and Survey''(London: Penguin Books, 1994). Leiber, ...
, which he held to be fruitless. Peirce sketched four methods of settling opinion, ordered from least to most successful: # The method of (policy of sticking to initial belief) – which brings comforts and decisiveness but leads to trying to ignore contrary information and others' views as if truth were intrinsically private, not public. The method goes against the social impulse and easily falters since one may well notice when another's opinion seems as good as one's own initial opinion. Its successes can be brilliant but tend to be transitory. # The method of – which overcomes disagreements but sometimes brutally. Its successes can be majestic and long-lasting, but it cannot regulate people thoroughly enough to withstand doubts indefinitely, especially when people learn about other societies present and past. # The method of the – which promotes conformity less brutally but fosters opinions as something like tastes, arising in conversation and comparisons of perspectives in terms of "what is agreeable to reason". Thereby it depends on fashion in
paradigm In science and philosophy, a paradigm () is a distinct set of concepts or thought patterns, including theories, research methods, postulates, and standards for what constitute legitimate contributions to a field. Etymology ''Paradigm'' comes ...
s and goes in circles over time. It is more intellectual and respectable but, like the first two methods, sustains accidental and capricious beliefs, destining some minds to doubt it. # The method of – wherein inquiry supposes that the real is discoverable but independent of particular opinion, such that, unlike in the other methods, inquiry can, by its own account, go wrong ( fallibilism), not only right, and thus purposely tests itself and criticizes, corrects, and improves itself. Peirce held that, in practical affairs, slow and stumbling ratiocination is often dangerously inferior to instinct and traditional sentiment, and that the scientific method is best suited to theoretical research, which in turn should not be trammeled by the other methods and practical ends; reason's "first rule" is that, in order to learn, one must desire to learn and, as a corollary, must not block the way of inquiry.
Scientific method The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific m ...
excels over the others finally by being deliberately designed to arrive—eventually—at the most secure beliefs, upon which the most successful practices can be based. Starting from the idea that people seek not truth ''per se'' but instead to subdue irritating, inhibitory doubt, Peirce showed how, through the struggle, some can come to submit to truth for the sake of belief's integrity, seek as truth the guidance of potential conduct correctly to its given goal, and wed themselves to the scientific method.

### =Scientific method

= Insofar as clarification by pragmatic reflection suits explanatory hypotheses and fosters predictions and testing, pragmatism points beyond the usual duo of foundational alternatives: deduction from self-evident truths, or '' rationalism''; and induction from experiential phenomena, or ''
empiricism In philosophy, empiricism is an epistemological theory that holds that knowledge or justification comes only or primarily from sensory experience. It is one of several views within epistemology, along with rationalism and skepticism. Empiri ...
''. Based on his critique of three modes of argument and different from either
foundationalism Foundationalism concerns philosophical theories of knowledge resting upon non-inferential justified belief, or some secure foundation of certainty such as a conclusion inferred from a basis of sound premises.Simon Blackburn, ''The Oxford Dictio ...
or coherentism, Peirce's approach seeks to justify claims by a three-phase dynamic of inquiry: # Active,
abductive Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th centur ...
genesis of theory, with no prior assurance of truth; # Deductive application of the contingent theory so as to clarify its practical implications; # Inductive testing and evaluation of the utility of the provisional theory in anticipation of future experience, in both senses: '' prediction'' and ''control''. Thereby, Peirce devised an approach to inquiry far more solid than the flatter image of inductive generalization ''simpliciter'', which is a mere re-labeling of phenomenological patterns. Peirce's pragmatism was the first time the
scientific method The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific m ...
was proposed as an
epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...
for philosophical questions. A theory that succeeds better than its rivals in predicting and controlling our world is said to be nearer the truth. This is an operational notion of truth used by scientists. Peirce extracted the pragmatic
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
or
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...
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. Abduction, deduction, and induction make incomplete sense in isolation from one another but comprise a cycle understandable as a whole insofar as they collaborate toward the common end of inquiry. In the pragmatic way of thinking about conceivable practical implications, every thing has a purpose, and, as possible, its purpose should first be denoted. Abduction hypothesizes an explanation for deduction to clarify into implications to be tested so that induction can evaluate the hypothesis, in the struggle to move from troublesome uncertainty to more secure belief. No matter how traditional and needful it is to study the modes of inference in abstraction from one another, the integrity of inquiry strongly limits the effective
modularity Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
of its principal components. Peirce's outline of the scientific method in §III–IV of "A Neglected Argument"Peirce (1908), "
A Neglected Argument for the Reality of God A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes' ...
", published in large part, ''Hibbert Journal'' v. 7, 90–112. Reprinted with an unpublished part, ''Collected Papers of Charles Sanders Peirce'', 6.452–85, ''Selected Writings'' pp. 358–79, ''The Essential Peirce'', 2:434–50, ''Peirce on Signs'' 260–78.
is summarized below (except as otherwise noted). There he also reviewed plausibility and inductive precision (issues of critique of arguments). 1. (or retroductive) phase. Guessing, inference to explanatory hypotheses for selection of those best worth trying. 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 from surprising observations in one or more of those realms (and for example at any stage of an inquiry already underway). All explanatory content of theories comes from abduction, which guesses a new or outside idea so as to account in a simple, economical way for a surprising or complicated phenomenon. The modicum of success in our guesses far exceeds that of random luck, and seems born of attunement to nature by developed or inherent instincts, especially insofar as best guesses are optimally plausible and simple in the sense of the "facile and natural", as by
Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
's natural light of reason and as distinct from "logical simplicity". Abduction is the most fertile but least secure mode of inference. Its general rationale is inductive: it succeeds often enough and it has no substitute in expediting us toward new truths. In 1903, Peirce called pragmatism "the logic of abduction". Coordinative method leads from abducting a plausible hypothesis to judging it for its testability and for how its trial would economize inquiry itself.See MS L75.329–30, from Draft D o
Memoir 27
of Peirce's application to the Carnegie Institution:
The hypothesis, being insecure, needs to have practical implications leading at least to mental tests and, in science, lending themselves to scientific tests. A simple but unlikely guess, if not costly to test for falsity, may belong first in line for testing. A guess is intrinsically worth testing if it has plausibility or reasoned objective probability, while subjective likelihood, though reasoned, can be misleadingly seductive. Guesses can be selected for trial strategically, for their caution (for which Peirce gave as example the game of Twenty Questions), breadth, or incomplexity. One can discover only that which would be revealed through their sufficient experience anyway, and so the point is to expedite it; economy of research demands the leap, so to speak, of abduction and governs its art. 2. phase. Two stages: :i. Explication. Not clearly premised, but a deductive analysis of the hypothesis so as to render its parts as clear as possible. :ii. Demonstration: Deductive Argumentation,
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
ean in procedure. Explicit deduction of consequences of the hypothesis as predictions about evidence to be found. Corollarial or, if needed, Theorematic. 3. phase. Evaluation of the hypothesis, inferring from observational or experimental tests of its deduced consequences. 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"; in other words, anything excluding such a process would never be real. Induction involving the ongoing accumulation of evidence follows "a method which, sufficiently persisted in", will "diminish the error below any predesignate degree". Three stages: :i. Classification. Not clearly premised, but an inductive classing of objects of experience under general ideas. :ii. Probation: direct Inductive Argumentation. Crude or Gradual in procedure. Crude Induction, founded on experience in one mass (CP 2.759), presumes that future experience on a question will not differ utterly from all past experience (CP 2.756). Gradual Induction makes a new estimate of the proportion of truth in the hypothesis after each test, and is Qualitative or Quantitative. Qualitative Gradual Induction depends on estimating the relative evident weights of the various qualities of the subject class under investigation (CP 2.759; see also ''Collected Papers of Charles Sanders Peirce'', 7.114–20). Quantitative Gradual Induction depends on how often, in a fair sample of instances of ''S'', ''S'' is found actually accompanied by ''P'' that was predicted for ''S'' (CP 2.758). It depends on measurements, or statistics, or counting. :iii. 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".

### =Against Cartesianism

= Peirce drew on the methodological implications of the four incapacities—no genuine introspection, no intuition in the sense of non-inferential cognition, no thought but in signs, and no conception of the absolutely incognizable—to attack philosophical
Cartesianism Cartesianism is the philosophical and scientific system of René Descartes and its subsequent development by other seventeenth century thinkers, most notably François Poullain de la Barre, Nicolas Malebranche and Baruch Spinoza. Descartes ...

# Philosophy: metaphysics

Some noted articles * The ''Monist'' Metaphysical Series (1891–1893) ** The Architecture of Theories (1891) ** The Doctrine of Necessity Examined (1892) ** The Law of Mind (1892) ** Man's Glassy Essence (1892) ** Evolutionary Love (1893) * Immortality in the Light of Synechism (1893 MS)
Peirce
divided Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication. At an elementary level the division of two natural numbers ...
metaphysics into (1) ontology or general metaphysics, (2) psychical or religious metaphysics, and (3) physical metaphysics.

## Ontology

Peirce was a scholastic realist, declaring for the reality of
generals A general officer is an officer of high rank in the armies, and in some nations' air forces, space forces, and marines or naval infantry. In some usages the term "general officer" refers to a rank above colonel."general, adj. and n.". OED ...
as early as 1868. Regarding modalities (possibility, necessity, etc.), he came in later years to regard himself as having wavered earlier as to just how positively real the modalities are. In his 1897 "The Logic of Relatives" he wrote: Peirce retained, as useful for some purposes, the definitions in terms of information states, but insisted that the pragmaticist is committed to a strong modal realism by conceiving of objects in terms of predictive general conditional propositions about how they ''would'' behave under certain circumstances.

## Psychical or religious metaphysics

Peirce believed in God, and characterized such belief as founded in an instinct explorable in musing over the worlds of ideas, brute facts, and evolving habits—and it is a belief in God not as an ''actual'' or ''existent'' being (in Peirce's sense of those words), but all the same as a ''real'' being.Peirce in his 1906 "Answers to Questions concerning my Belief in God", ''Collected Papers of Charles Sanders Peirce'', 6.495
Eprint
, reprinted in part as "The Concept of God" in ''Philosophical Writings of Peirce'', J. Buchler, ed., 1940, pp. 375–78:
In "
A Neglected Argument for the Reality of God A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes' ...
" (1908), Peirce sketches, for God's reality, an argument to a hypothesis of God as the Necessary Being, a hypothesis which he describes in terms of how it would tend to develop and become compelling in musement and inquiry by a normal person who is led, by the hypothesis, to consider as being purposed the features of the worlds of ideas, brute facts, and evolving habits (for example scientific progress), such that the thought of such purposefulness will "stand or fall with the hypothesis"; meanwhile, according to Peirce, the hypothesis, in supposing an "infinitely incomprehensible" being, starts off at odds with its own nature as a purportively true conception, and so, no matter how much the hypothesis grows, it both (A) inevitably regards itself as partly true, partly vague, and as continuing to define itself without limit, and (B) inevitably has God appearing likewise vague but growing, though God as the Necessary Being is not vague or growing; but the hypothesis will hold it to be ''more'' false to say the opposite, that God is purposeless. Peirce also argued that the will is free and (see Synechism) that there is at least an attenuated kind of immortality.

## Physical metaphysics

Peirce held the view, which he called
objective idealism Objective idealism is a form of metaphysical idealism that accepts Naïve realism (the view that empirical objects exist objectively) but rejects epiphenomenalist materialism (according to which the mind and spiritual values have emerged due to ...
, that "matter is effete mind, inveterate habits becoming physical laws". Peirce asserted the reality of (1) absolute chance (his tychist view), (2) mechanical necessity ( anancist view), and (3) that which he called the law of love ( agapist view), echoing his categories Firstness, Secondness, and Thirdness, respectively. He held that fortuitous variation (which he also called "sporting"), mechanical necessity, and creative love are the three modes of evolution (modes called "tychasm", "anancasm", and "agapasm") of the cosmos and its parts. He found his conception of agapasm embodied in Lamarckian evolution; the overall idea in any case is that of evolution tending toward an end or goal, and it could also be the evolution of a mind or a society; it is the kind of evolution which manifests workings of mind in some general sense. He said that overall he was a synechist, holding with reality of continuity, especially of space, time, and law.See p. 115 in '' Reasoning and the Logic of Things'' (Peirce's 1898 lectures).

# Science of review

Peirce outlined two fields, "Cenoscopy" and "Science of Review", both of which he called philosophy. Both included philosophy about science. In 1903 he arranged them, from more to less theoretically basic, thus: # Science of Discovery. ## Mathematics. ## Cenoscopy (philosophy as discussed earlier in this article – categorial, normative, metaphysical), as First Philosophy, concerns positive phenomena in general, does not rely on findings from special sciences, and includes the ''general'' study of inquiry and scientific method. ## Idioscopy, or the Special Sciences (of nature and mind). # Science of Review, as Ultimate Philosophy, arranges "... the results of discovery, beginning with digests, and going on to endeavor to form a philosophy of science". His examples included Humboldt's ''
Cosmos The cosmos (, ) is another name for the Universe. Using the word ''cosmos'' implies viewing the universe as a complex and orderly system or entity. The cosmos, and understandings of the reasons for its existence and significance, are studied in ...
'', Comte's '' Philosophie positive'', and Spencer's ''Synthetic Philosophy''. # Practical Science, or the Arts. Peirce placed, within Science of Review, the work and theory of classifying the sciences (including mathematics and philosophy). His classifications, on which he worked for many years, draw on argument and wide knowledge, and are of interest both as a map for navigating his philosophy and as an accomplished polymath's survey of research in his time.

* Charles Sanders Peirce's type–token distinction * Continuous predicate * Entitative graph * Howland will forgery trial * Hypostatic abstraction * * ''
Laws of Form ''Laws of Form'' (hereinafter ''LoF'') is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy. ''LoF'' describes three distinct logical systems: * The "primary arithmetic" (described in C ...
'' * List of American philosophers *
Logic of information The logic of information, or the logical theory of information, considers the information content of logical signs and expressions along the lines initially developed by Charles Sanders Peirce. In this line of work, the concept of information serve ...
* Logical machine * Logical matrix *
Mathematical psychology Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus character ...
* * Peirce triangle * Peircean realism *
Phaneron The phaneron (Greek φανερός 'phaneros''"visible, manifest"Note that φανερόν is the neuter nominative form (see e.g. Wiktionary: φανερός)) is the subject matter of phenomenology, or of what Charles Sanders Peirce Charl ...
* Pragmatics * * * Relation algebra *
Truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...

## Contemporaries associated with Peirce

* Oliver Wendell Holmes Jr. *
George Herbert Mead George Herbert Mead (February 27, 1863 – April 26, 1931) was an American philosopher, sociologist, and psychologist, primarily affiliated with the University of Chicago, where he was one of several distinguished pragmatists. He is regarded a ...

# References

Arisbe: The Peirce Gateway
Joseph Ransdell, ed. Over 100 online writings by Peirce as of November 24, 2010, with annotations. Hundreds of online papers on Peirce. The peirce-l e-forum. Much else.
Center for Applied Semiotics (CAS)
(1998–2003), Donald Cunningham & Jean Umiker-Sebeok, Indiana U. * and previously et al., Pontifical Catholic U. of (PUC-SP), Brazil. In Portuguese, some English.
Commens Digital Companion to C.S. Peirce
Mats Bergman, Sami Paavola, & , formerl

Includes Commens Dictionary of Peirce's Terms with Peirce's definitions, often many per term across the decades, and the Digital Encyclopedia of Charles S. Peirce ( old edition still at old website).
Peirce
Carlo Sini, Rossella Fabbrichesi, et al., U. of Milan, Italy. In Italian and English. Part o
Pragma

Charles S. Peirce Foundation
Co-sponsoring the 2014 Peirce International Centennial Congress (100th anniversary of Peirce's death).
Charles S. Peirce Society
br>

'. Quarterly journal of Peirce studies since spring 1965

of all issues.
Charles S. Peirce Studies
Brian Kariger, ed. *
Collegium for the Advanced Study of Picture Act and Embodiment
The Peirce Archive. Humboldt U, Berlin, Germany. Cataloguing Peirce's innumerable drawings & graphic materials
(Prof. Aud Sissel Hoel).
Digital Encyclopedia of Charles S. Peirce

now at UFJF
& Ricardo Gudwin
at Unicamp
, eds., U. of , Brazil, in English. 84 authors listed, 51 papers online & more listed, as of January 31, 2009. Newer edition now at ''Commens Digital Companion to C.S. Peirce''.
Existential Graphs
Jay Zeman, ed., U. of Florida. Has 4 Peirce texts. * , ed., U. of Navarra, Spain. Big study site, Peirce & others in Spanish & English, bibliography, more.
Helsinki Peirce Research Center
(HPRC), Ahti-Veikko Pietarinen et al., U. of Helsinki.
His Glassy Essence
Autobiographical Peirce. Kenneth Laine Ketner.
Institute for Studies in Pragmaticism
Kenneth Laine Ketner, Clyde Hendrick, et al., Texas Tech U. Peirce's life and works.
International Research Group on Abductive Inference
et al., eds., U., Frankfurt, Germany. Uses frames. Click on link at bottom of its home page for English. Moved to U. of Gießen, Germany
not in English but see Artikel section there.

(1974–2003) – , U. of , France.
Minute Semeiotic
, U. of , Brazil. English, Portuguese.
Peirce
at ''Signo: Theoretical Semiotics on the Web'', Louis Hébert, director, supported by U. of Québec. Theory, application, exercises of Peirce'
Semiotics
an
Esthetics
English, French.
Peirce Edition Project (PEP)
Indiana U.–Purdue U. Indianapolis (IUPUI). André De Tienne, Nathan Houser, et al. Editors of the ''Writings of Charles S. Peirce'' (W) and ''The Essential Peirce'' (EP) v. 2. Many study aids such as the Robin Catalog of Peirce's manuscripts & letters and:
Biographical introductions t

an

b

Working on ''Writings of Charles S. Peirce'', 7: Peirce's work on the ''Century Dictionary''
Definition of the week

Peirce's Existential Graphs
Frithjof Dau, Germany
Peirce Research Group
Department of Philosophy "Piero Martinetti" – University of Milan, Italy.
Pragmatism Cybrary
David Hildebrand & John Shook.

(late 1990s), Germany). See ''Peirce Project Newsletter'' v. 3, n. 1
p. 13

wit

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