Charles Sanders Peirce

Charles Sanders Peirce (10 September 1839 – 19 April 1914) was an American polymath, born in Cambridge, Massachusetts. Although educated as a chemist and employed as a scientist for 30 years, it is for his contributions to logic, mathematics, philosophy, and the theory of signs, or semeiotic, that he is largely appreciated today. The philosopher Paul Weiss, writing in the Dictionary of American Biography for 1934, called Peirce "the most original and versatile of American philosophers and America's greatest logician" (Brent, 1).
The secondary literature on Peirce was scant until after World War II. Much of his huge output is still unpublished. An innovator in fields such as logic, mathematics, philosophy of science, research methodology, semiotics, epistemology, and metaphysics, he considered himself a logician first and foremost. While he made major contributions to the development of formal logic as it is known today, logic for him encompassed much of what is now studied under the philosophies of knowledge, language, and science. Peirce saw logic as the formal branch of the theory of signs, or semiotics, here using formal in the sense of normative or what he called quasi-necessary. In 1886 he saw that logical operations could be carried out by electrical switching circuits, an idea used decades later in the development of electronic computers.
==Life==
Charles Sanders Peirce was the son of Sarah Hunt Mills and Benjamin Peirce, a professor of astronomy and mathematics at Harvard University, perhaps the first serious research mathematician in America. At 12 years of age, Charles read an older brother's copy of Richard Whately's Elements of Logic, then the leading English language text on the subject. Thus began his lifelong fascination with logic and reasoning. He went on to obtain the BA and MA from Harvard, and in 1863 the Lawrence Scientific School awarded him its first M.Sc. in chemistry. This last degree was awarded summa cum laude; otherwise his academic record was undistinguished. At Harvard, he began lifelong friendships with Francis Ellingwood Abbot, Chauncey Wright, and William James. One of his Harvard instructors, Charles William Eliot, formed an unfavorable opinion of Peirce. This opinion proved fateful, because Eliot, while President of Harvard 1869–1909 — a period encompassing nearly all of Peirce's working life — repeatedly vetoed having Harvard employ Peirce in any capacity.
===United States Coast Survey===
Between 1859 and 1891, Charles was intermittently employed in various scientific capacities by the United States Coast Survey, where he enjoyed the protection of his highly influential father until the latter's death in 1880. This employment exempted Charles from having to take part in the Civil War. It would have been very awkward for him to do so, as the Boston Brahmin Peirces sympathized with the Confederacy. At the Survey, he worked mainly in geodesy and in gravimetry, refining the use of pendulums to determine small local variations in the strength of the earth's gravity. The Survey sent him to Europe five times, the first in 1871, as part of a group dispatched to observe a solar eclipse. While in Europe, he sought out Augustus De Morgan, William Stanley Jevons, and William Kingdon Clifford, British mathematicians and logicians whose 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 stars and the shape of the Milky Way. (On Peirce the astronomer, see Lenzen's chapter in Moore and Robin, 1964.) In 1878, he was the first to define the meter as so many wavelengths of light of a certain frequency, the definition employed until 1983 (Taylor, 5).
During the 1880s Peirce's indifference to bureaucratic detail waxed while the quality and timeliness of his Survey work waned. Peirce took years to write reports that he should have completed in mere months. Meanwhile, he wrote hundreds of logic, philosophy, and science entries for the Century Dictionary. In 1885, an investigation by the Allison Commission exonerated Peirce, but led to the dismissal of Superintendent Julius Hilgard and several other Coast Survey employees for misuse of public funds. In 1891, Peirce resigned from the Coast Survey, at the request of Superintendent Thomas Corwin Mendenhall. He never again held regular employment.
===Johns Hopkins University===
In 1879, Peirce was appointed Lecturer in logic at the new Johns Hopkins University. That university was strong in a number of areas that interested Peirce, such as philosophy (Royce and Dewey did their PhDs at Hopkins), psychology (taught by G. Stanley Hall and studied by Joseph Jastrow, who coauthored a landmark empirical study with Peirce), and mathematics (taught by J.J. Sylvester, who came to admire Peirce's work on mathematics and logic). This untenured position proved to be the only academic appointment Peirce ever held. 
Brent 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 American scientist of the day, Simon Newcomb. Peirce's ability to find academic employment may also have been frustrated by a difficult personality.
Peirce's personal life also proved a grave handicap. His first wife, Harriet Melusina Fay, left him in 1875. He soon took up with a woman whose maiden name and nationality remain uncertain to this day (the best guess is that her name was Juliette Froissy and that she was French), but did not marry her until his divorce with Harriet became final in 1883. 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. Just why Peirce's later applications for academic employment at Clark University, University of Wisconsin, University of Michigan, Cornell University, Stanford University, and the University of Chicago were all unsuccessful can no longer be determined. Presumably, his having lived with Juliette for a number of years while still legally married to Harriet led him to be deemed morally unfit for academic employment anywhere in the USA. Peirce had no children by either marriage.
===Arisbe===
In 1887 Peirce spent part of his inheritance from his parents to purchase 2,000 rural acres near Milford, Pennsylvania, land which never yielded an economic return. On that land, he built a large house which he named Arisbe and where he spent the rest of his life, writing prolifically, much of it unpublished to this day. His insistence on living beyond his means soon led to grave financial and legal difficulties. Peirce spent much of the last two decades of his life so destitute that he could not afford heat in winter, and his only food was old bread kindly donated by the local baker. Unable to afford new stationery, he wrote on the verso side of old manuscripts. An outstanding warrant for assault and unpaid debts led to his being a fugitive in New York City for a while. Several people, including his brother James Mills Peirce and his neighbors, relatives of Gifford Pinchot, settled his debts and paid his property taxes and mortgage.
Peirce did some scientific and engineering consulting and wrote a good deal for meager pay, primarily dictionary and encyclopedia entries, and reviews for The Nation (with whose editor, Wendell Phillips Garrison he became friendly). He did translations for the Smithsonian Institution, at the instigation of its director, Samuel Langley. Peirce also did substantial mathematical calculations for Langley's research on powered flight. Hoping to make money, Peirce tried his hand at inventing, and began but did not complete a number of books. In 1888, President Grover Cleveland appointed him to the Assay Commission. From 1890 onwards, he had a friend and admirer in Judge Francis C. Russell of Chicago, who introduced Peirce to Paul Carus and Edward Hegeler, the editor and owner, respectively, of the pioneering American philosophy journal The Monist, which eventually published a number of his articles. He applied to the newly formed Carnegie Institution for a grant to write a book summarizing his life's work. The application was doomed; his nemesis Newcomb served on the Institution's executive committee, and its President had been the President of Johns Hopkins at the time of Peirce's dismissal. 
The one who did the most to help Peirce in these desperate times was his old friend William James, who dedicated his Will to Believe to Peirce, and who arranged for Peirce to be paid to give four series of lectures at or near Harvard. Most important, each year from 1898 until his death in 1910, James would write to his friends in the Boston intelligentsia, asking that they make a financial contribution to help support Peirce. Peirce reciprocated by designating James's eldest son as his heir should Juliette predecease him, and by adding "Santiago", "Saint James" in Spanish, to his full name (Brent, 315–316, 374).
Peirce died destitute in Milford, Pennsylvania, twenty years before his widow.
==Reception==
Bertrand Russell opined, "Beyond doubt... he was one of the most original minds of the later nineteenth century, and certainly the greatest American thinker ever." (Yet his Principia Mathematica does not mention Peirce.) A.N. Whitehead, while reading some of Peirce's unpublished manuscripts soon after arriving at Harvard in 1924, was struck by how Peirce had anticipated his own "process" thinking. (On Peirce and process metaphysics, see the chapter by Lowe in Moore and Robin, 1964.) Karl Popper viewed Peirce as "one of the greatest philosophers of all time". Nevertheless, Peirce's accomplishments were not immediately recognized. His imposing contemporaries William James and Josiah Royce 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, the editor of a 1923 anthology of Peirce's writings titled Chance, Love, and Logic and the author of the first bibliography of Peirce's scattered writings. John Dewey had had Peirce as an instructor at Johns Hopkins, and from 1916 onwards, Dewey's writings repeatedly mention Peirce with deference. His 1938 Logic: The Theory of Inquiry is Peircean through and through. The publication of the first six volumes of the Collected Papers (1931–1935), the most important event to date in Peirce studies and one Cohen made possible by raising the needed funds, did not lead to an immediate outpouring of secondary studies. The editors of those volumes, Charles Hartshorne and Paul Weiss, did not become Peirce specialists. Early landmarks of the secondary literature include the monographs by Buchler (1939), Feibleman (1946), and Goudge (1950), the 1941 Ph.D. thesis by Arthur Burks (who went on to edit volumes 7 and 8 of the Collected Papers), and the edited volume by Wiener and Young (1952). The Charles S. Peirce Society was founded in 1946. Its Transactions, an academic journal specializing in Peirce, pragmatism, and American philosophy, has appeared since 1965.
In 1949, while doing unrelated archival work, the historian of mathematics Carolyn Eisele chanced on an autograph letter by Peirce. Thus began her 40 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 emerged as an authority on Peirce. Fisch (1986) reprints many of the relevant articles, including a wide-ranging survey of the impact of Peirce's thought through 1983.
Peirce has come to enjoy a significant international following. There are university research centers devoted to Peirce studies and pragmatism in Brazil, Finland, Germany, and Spain. For many years the North American philosophy department most devoted to Peirce was the University of Toronto's, thanks in good part to the leadership of Thomas Goudge and David Savan. In recent years, American Peirce scholars have clustered at Indiana University – Purdue University Indianapolis, the home of the Peirce Edition Project, and the Pennsylvania State University.
==Peirce's philosophy==
It is not sufficiently recognized that Peirce's career was that of a scientist, not a philosopher; and that during his lifetime he was known and valued chiefly as a scientist, only secondarily as a logician, and scarcely at all as a philosopher. Even his work in philosophy and logic will not be understood until this fact becomes a standing premise of Peircian studies. (Max Fisch, in (Moore and Robin 1964), p. 486).
Peirce's lifelong work in science bears on a wide array of disciplines, including astronomy, economics, geodesy, linguistics, logic, mathematics, metrology, philosophy, psychology, and the history and philosophy of science. This work has become the subject of renewed interest, inspired not only by his anticipations of subsequent scientific developments but also by his articulation of the ways that philosophy applies to human problems.
Peirce's writings repeatedly refer to a system of three categories, named Firstness, Secondness, and Thirdness, devised early in his career in reaction to his reading of Aristotle, Kant, and Hegel. He later emphasized the principle of logic that he called the maxim of pragmatism, on which his lifelong friend William James based a popular philosophical movement known as pragmatism. Peirce believed that any truth is provisional, and that the truth of any proposition cannot be certain but only probable. The philosophical recognition of this fact he called fallibilism.
===Pragmatism===
: Main articles : Pragmatism, Pragmaticism, Pragmatic maxim
Peirce's recipe for pragmatic thinking, going under the label of pragmatism and also known as pragmaticism, is articulated in several versions of the maxim of logic that he called the pragmatic maxim. Here is one of his more emphatic statements of it:
Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object. (CP 5.438).
William James, among others, regarded two of Peirce's papers, "The Fixation of Belief" (1877) and "How to Make Our Ideas Clear" (1878), as being the origin of pragmatism. Peirce conceived pragmatism to be a method for clarifying the meaning of difficult ideas through the application of the pragmatic maxim.
Peirce's pragmatism may be understood as a method for resolving conceptual confusions by linking the meanings of one's conceptions to one's conceptions of the operational or practical consequences of their use in practice. The concepts of belief, knowledge, and truth were some of the difficult ideas that Peirce sought to clarify by means of the pragmatic maxim. Doing this required a theory of inquiry that moderated the conventional dualism of foundational alternatives, namely:
* Deduction from self-evident truths, or rationalism.
* Induction from experiential phenomena, or empiricism.
His approach is distinguished from either coherentism or foundationalism by virtue of the following three dimensions:
# Active process of theory generation, with no prior assurance of truth.
# Subsequent application of the provisional theory, aimed toward developing its logical and practical consequences.
# Evaluation of the provisional theory's utility for the anticipation of future experience, and that in dual senses of the word: prediction and control.
Peirce's appreciation of these three dimensions serves to flesh out a physiognomy of inquiry far more solid than the flatter image of inductive generalization simpliciter, which is merely the relabeling of phenomenological patterns. Peirce's pragmatism was the first time the scientific method was proposed as an epistemology for philosophical questions.
As advocated by James, John Dewey, Ferdinand Canning Scott Schiller, George Herbert Mead, and others, pragmatism has proved durable and popular. But Peirce did not seize on this fact to enhance his reputation. While it is sometimes stated that James' and other philosophers' use of the word pragmatism so dismayed Peirce that he renamed his own variant pragmaticism, this was not the main reason (Haack, 55). This is revealed by the context in which Peirce introduced the latter term:
But at present, the word [pragmatism] begins to be met with occasionally in the literary journals, where it gets abused in the merciless way that words have to expect when they fall into literary clutches. … So then, the writer, finding his bantling "pragmatism" so promoted, feels that it is time to kiss his child good-by and relinquish it to its higher destiny; while to serve the precise purpose of expressing the original definition, he begs to announce the birth of the word "pragmaticism", which is ugly enough to be safe from kidnappers. (C.S. Peirce, CP 5.414).
===Scholastic realism===
Peirce's at one point describes himself as a "scholastic realist of a somewhat extreme stripe" (CP 5.470). In contrast, some writers call him an idealist, apparently on account of his defining reality as "the object of the final opinion of the scientific community", but this label is based on a peculiar sense of the word idealist and is overall misleading in the case of Peirce, as he consistently and systematically argues that reality is best viewed as independent of mind, at least of minds in particular, if not necessarily of minds in general. The problem of interpretation appears to arise from at least three sources. First, Peirce's use of the word independent needs to be understood in a way that is analogous to its definition in mathematics, where it means orthogonal, or its definition in statistics, where it means uncorrelated. In these senses, independence is a particular kind of relation, not a lack of relation, and certainly not a form of disconnection or exclusion. Second, Peirce did in fact describe himself as being in favor of objective idealism, but what he meant by that is a far cry from ordinary idealism. Third, we need to recognize that scholastic realism is one side of the realist vs. nominalist debate over universals, and not a position in the realist vs. idealist debate about a mind-independent reality. Peirce's scholastic realism in fact supplies essential support for his own thesis of objective idealism regarding the relationship between matter and mind. Two early studies on Peirce's realism and the influence of Duns Scotus thereon, are the chapter by McKeon in Wiener and Young (1952), and that by Moore in Moore and Robin (1964).
In his first remarks on the realist vs. nominalist debate, Peirce sided with nominalism:
Qualities are fictions; for though it is true that roses are red, yet redness is nothing, but a fiction framed for the purpose of philosophizing; yet harmless so long as we remember that the scholastic realism it implies is false. (CE 1, 307, 1865).
Here Peirce is explicitly disparaging a position he is well-known for spending most of his life defending. How might we make sense of this apparent contradiction? The temptation is to simply say Peirce changed his mind. After all, since Peirce asserts nominalism in 1865 and scholastic realism in 1868, Peirce may have gone from denying the reality of universals to asserting it. This explanation is most famously given by Max Fisch in his "Peirce's Progress from Nominalism toward Realism" (1967) and then again in his introduction to Volume 2 of the Chronological Edition of Peirce's writings (1984). More recently this way of understanding Peirce has been indepenently challenged by Rosa Mayorga in her On Universals (2002) and by Robert Lane in his "Peirce's Early Realism" (2004). Both Mayorga and Lane are troubled by several instances where Peirce's self assessment of his own intellectual development contradicts Fisch's account of Peirce development. One of these statemesnts appears in 1893 when Peirce states that "never, during the thirty years in which I have been writing on philosophical questions, have I failed in my allegiance to realistic opinions and to certain Scotistic ideas" (CP 6.605). Remarks like these led Lane to conduct a re-evaluation of Peirce's 1865 declarations for nomianlism, whereupon Lane discovered significant evidence for the same conclusion Mayorga had already reached two years earlier (unbenownst to Lane). Both concluded that the correct way to understand Peirce's shift from outspoken nominalist to outspoken realist is not by reading into Peirce a change in his fundamental philosophical position, but instead to realize that Peirce merely changed his understanding and use of the terms scholastic realism and nominalism. The reason Peirce calls himself a nominalist in 1865 is because he believes realism to only come in the form offered by Plato:
It has been said that these 'abstract names' [blueness, hardness, and loudness] denote qualities and connote nothing. But it seems to me the phrase 'denoted object' is nothing but a roundabout expression for a thing. To say that a quality is denoted is to say it is a thing. [Such terms] were framed at a time when all men were realists in the scholastic sense and consequently things were meant by them, entities which had no quality but that expressed by the word. They, therefore, must denote these things and connote the qualities they relate to. (Peirce, CE 1, 311–312).
When Peirce goes on to call universals 'fictions', he is not condemning their truth; he is simply asserting that they do not exist as particulars. This becomes clearer when in the same paper Peirce argues against psychologism in logic, by establishing the same 'fictional' status for logic and mathematics that he claims for universals. Now by proving logic fictional, Peirce believes he does logic a favor, that is, by saving it from the psychologists. This suggests that Peirce employed fictional in a rather idiosyncratic way. Many things (including universals) covered by Peirce's pre-1868 use of fictional came under his post-1868 use of real. Peirce had been using 'fictional' to refer to things having no physical existence, and not to imply that something was merely the result of human imagination or fancy. By 1868 at least, Peirce had changed his mind about "reality", holding instead that "fictional" should be contrasted with "independent of what we think about it" (real). He no longer deemed existence as a physical object as a prerequisite for being real, so that a lack of physical existence no longer led Peirce to chatacterize universals as "fictional". That something has blueness can be true independent of what anyone thinks of it, and therefore it can be a part of reality despite the fact blueness never has a physical existence anywhere. Blueness is real independent of what anyone thinks, but it does not exist as an entity because it has no secondness.
==Formal perspective==
Peirce did not live or work in a vacuum. No one who appreciates his use of phrases like laws of the symbol in their historical context can fail to hear the echoes of George Boole, nor the undertones of the symbolist movement in mathematics inspired by the writings of George Peacock.
At the outset of his Laws of Thought, Boole tells us how he plans to evade the horns of a dilemma that would otherwise threaten to block his inquiry before he can even begin.
In proceeding to these inquiries, it will not be necessary to enter into the discussion of that famous question of the schools, whether Language is to be regarded as an essential instrument of reasoning, or whether, on the other hand, it is possible for us to reason without its aid. I suppose this question to be beside the design of the present treatise, for the following reason, viz., that it is the business of Science to investigate laws; and that, whether we regard signs as the representatives of things and of their relations, or as the representatives of the conceptions and operations of the human intellect, in studying the laws of signs, we are in effect studying the manifested laws of reasoning. (Boole, Laws of Thought, p. 24)
Boole is saying that the business of science, the investigation of laws, applies itself to the laws of signs at such a level of abstraction that its results are the same no matter whether it finds those laws embodied in objects or in intellects. In short, he does not have to choose one or the other in order to begin. This simple idea is the essence of the formal approach in mathematics, and it is one of the reasons that contemporary mathematicians tend to consider structures that are isomorphic to one another as tantamount to being the same thing. Peirce avails himself of this same depth of perspective for much the same reason. It allows him to investigate the forms of triadic sign relations that exist among objects, signs, and interpretants without being blocked by the impossible task of acquiring knowledge of supposedly unknowable things in themselves, whether outward objects or the contents of other minds. Like Aristotle and Boole before him, Peirce replaces these impossible problems with the practical problem of inquiring into the sign relations that exist among commonly accessible objects and publicly accessible signs.
How often do we think of the thing in algebra? When we use the symbol of multiplication we do not even think out the conception of multiplication, we think merely of the laws of that symbol, which coincide with the laws of the conception, and what is more to the purpose, coincide with the laws of multiplication in the object. Now, I ask, how is it that anything can be done with a symbol, without reflecting upon the conception, much less imagining the object that belongs to it? It is simply because the symbol has acquired a nature, which may be described thus, that when it is brought before the mind certain principles of its use — whether reflected on or not — by association immediately regulate the action of the mind; and these may be regarded as laws of the symbol itself which it cannot as a symbol transgress. ("On the Logic of Science" (1865), CE 1, 173).
The motive themes of the symbolist movement are familiar to anyone who has worked a "story problem" in a mathematics course. One learns to approach the story problem, a roughly realistic representation of a concrete set of circumstances, with the aim of abstracting the appropriate general formula from the mass of concrete details that make up the problem — not all of which data are equally pertinent to the solution and some of which may even be thrown in as distractors. The next step is to derive the logical implications of the abstract formula, generally speaking substituting specific values for some of its variables but just as often leaving other variables unfilled in. The bearing of the formula on the desired answer is obscure at first — that is what makes the problem a problem in the first place. But progressive clarification of the formula leads to an equivalent or implied formula that amounts to an abstract answer or a generic solution to the story problem. Given that, there is nothing more to do but fill in the rest of the concrete data to arrive at the concrete answer or the specific solution to the problem.
The three-phase maneuver for solving a story problem, (1) teasing out, (2) cranking the crank, (3) plugging in, can be articulated in semiotic or sign-relational terms as follows: The first phase passes from the object domain to the sign domain, the second phase passes from the sign domain to the interpretant sign domain, continuing perhaps in a relay of successive passes, and the third phase passes from the last interpretant sign domain back to the object domain.
There are a number of issues that typically arise with the continuing development of a symbolist perspective, in any field of endeavor, over the years of its natural life-cycle. We can see these issues illustrated clearly enough in our story problem paradigm, with its parsing of the problem-solving process into the three phases of abstraction, transformation, and application.
* Once the division of labor among the three phases of the process has been in place for a sufficiently long time, each of the three phases will tend to take on a certain degree of independence, sometimes actual and sometimes merely apparent, from the other two phases.
* As a side-effect of the increasing independence among the various phases of inquiry, there tend to develop specialized disciplines, each devoted to a single aspect of the initially interactive and integral process. A symptom of this stage of development is that references to the 'independence' of the several phases of inquiry may become confused with or even replaced by assertions of their 'autonomy' from one another.
Returning to the formal sciences of logic and mathematics and focusing on the rise of symbolic logic in particular, all of the above issues were clearly recognized and widely discussed among the movers and shakers of the symbolist movement, with especial mention of George Boole, Augustus De Morgan, Benjamin Peirce, and Charles Peirce.
The first symptoms of a crisis typically arise in connection with questions about the status of the abstract symbols that are 'manipulated' in the transformation phase, to express it in sign-relational terms, the sign-to-sign aspect of semiosis.
In the beginning, while it is still evident to everyone concerned that these symbols are mined from the matrix of their usual interpretations, which are generally more diverse than unique, these abstracted symbols are commonly referred to as 'uninterpreted symbols', the sense being that they are transiently detached from their interpretations simply for the sake of extra facility in processing the more general thrust of their meanings, after which intermediary process they will have their concrete meanings restored.
When we start to hear these abstract, general, uninterpreted symbols being described as 'meaningless' symbols, then we can be sure that a certain line in our sand-reckoning has been crossed, and that the crossers thereof have hefted or sublimated 'formalism' to the status of a full-blown Weltanschauung rather than a simple heuristic device.
What we observe here is a familiar form of cyclic process, with the crest of excess followed by the slough of despond. The inflationary boom that raises 'formalism' beyond its formative sphere as one among a host of equally useful heuristic tricks to the status of a totalizing worldview leads perforce to the deflationary bust that makes of 'formalist' a pejorative term.
The point of the foregoing discussion is this, that one of the main difficulties that we have in understanding what the whole complex of words rooted in 'form' meant to Peirce is that we find ourselves, historically speaking, on opposite sides of this cycle of ideas from him.
And so we are required, as so often happens in trying to read a writer of another age, to lift the scales of the years from our eyes, to drop the reticles that have encrusted themselves on our 'reading glasses', our hermeneutic scopes, due to the interpolant philosophical schemata that have managed to enscounce themselves in our unthinking culture over the years that separate us from the writer in question.
===Relationships, relations, relatives===
The reader of Peirce needs to be aware of the distinction between relations and relatives. Succinctly put, relations are objects and relatives are signs. The term "relative" is short for "relative term", and a relative term is a type of sign that forms the main study of the logic of relatives. A relation, on the other hand, is a type of formal object that is treated in the mathematical theory of relations. There is of course an intimate relationship between the two studies, but like most intimate relationships it has its fair share of intricacies.
The following collection of definitions is practically indispensable.
:* A relative, then, may be defined as the equivalent of a word or phrase which, either as it is (when I term it a complete relative), or else when the verb "is" is attached to it (and if it wants such attachment, I term it a nominal relative), becomes a sentence with some number of proper names left blank.
:* A relationship, or fundamentum relationis, is a fact relative to a number of objects, considered apart from those objects, as if, after the statement of the fact, the designations of those objects had been erased.
:* A relation is a relationship considered as something that may be said to be true of one of the objects, the others being separated from the relationship yet kept in view. Thus, for each relationship there are as many relations as there are blanks. For example, corresponding to the relationship which consists in one thing loving another there are two relations, that of loving and that of being loved by. There is a nominal relative for each of these relations, as "lover of ——" and "loved by ——".
:* These nominal relatives belonging to one relationship, are in their relation to one another termed correlatives. In the case of a dyad, the two correlatives, and the corresponding relations are said, each to be the converse of the other.
:* The objects whose designations fill the blanks of a complete relative are called the correlates.
:* The correlate to which a nominal relative is attributed is called the relate.
:* In the statement of a relationship, the designations of the correlates ought to be considered as so many logical subjects and the relative itself as the predicate. The entire set of logical subjects may also be considered as a collective subject, of which the statement of the relationship is predicate.
: (Peirce, CP 3.466–467, "The Logic of Relatives", Monist, 7, 161–217 (1897), CP 3.456–552).
To understand these definitions, as everywhere in Peirce's work, one needs to keep a close watch on the things that are meant as objects of discussion and thought and the things that are meant as signs and thoughts in which discussion and thought take place. Doing this is trickier than it seems at first, since many standard approaches to defining abstract, formal, or hypostatic objects approach their objects by way of formal operations on the corresponding signs.
====Relatives====
: Main article : Logic of relatives
====Relations====
: Main article : Relation theory
A concept of relation that suffices to begin the study of Peirce's logic, mathematics, and semiotics, making use of analogous concepts of relation that have served well enough in other areas of experience to make further experience possible, can be set out as follows.
* Defined in extension, a k-adic relation L is a set of k-tuples.
* A k-tuple x is a sequence of k elements, x1, …, xk, called the components of the k-tuple. The components of the k-tuple x can be indicated by writing either one of the following two forms, the latter form of syntax being the one that Peirce most often used:
: x = (x1, …, xk)
: x = x1 : … : xk
It is critically important to understand that a relation in extension is a set, in other words, an aggregate entity or a collection of things. More to the point, a k-tuple is not a relation, it is only an element of a relation, what Peirce quite naturally called an elementary relation or sometimes an individual relation.
In his time, Peirce found himself forced by the task of understanding the intertwined natures of science and signs to develop the logic of relations from the fairly primitive state in which he found it to a condition of readiness more qualified for the job. There was nothing very cut and dried about trying to do this from scratch, as will be evident in the appropriate Sections below when we sample the fits and starts forward, the culs-de-sac, and the many paths that had to be backtracked in order to arrive at an adequate theory of relations. For the purpose at hand, however, we can rely on the fact that few readers these days will have escaped some encounter with relational databases, and so we can draw on these resources of experience to speed the exposition of relations in general.
Table 1 shows how the k-tuples of a k-adic relation might be conceived in tabular form, with the k-tuple xi = <xi1, …, xik> = xi1 : … : xik supplying the entries for the ith row of the Table.
 Table 1. Relational Database
Domain 1	Domain 2	...	Domain j	...	Domain k
x11 	x12 	...	x1j 	...	x1k
x21 	x22 	...	x2j 	...	x2k
... 	... 	...	... 	...	...
xi1 	xi2 	...	xij 	...	xik
... 	... 	...	... 	...	...
xm1 	xm2 	...	xmj 	...	xmk
For ease of exposition, Table 1 shows the generic form of a discrete k-adic relation, one that contains a countable number of k-tuples, indeed, it shows a finite k-adic relation, one that contains a finite number of k-tuples. Generalizations to relations with an infinite or even a continuous cardinality in respect of their numbers of elementary relations are possible. Indeed, it is possible to conceive of relations with infinite, continuous, or even no fixed numbers of components in their elementary relations, but finite k-adic relations are illustration enough for our immediate aims.
====Dyadic relations====
: Main article : Binary relation
====Triadic relations====
: Main article : Triadic relation
: This completes the classification of dual relatives founded on the difference of the fundamental forms A : A and A : B. Similar considerations applied to triple relatives would give rise to a highly complicated development, inasmuch as here we have no less than five fundamental forms of individuals, namely: 
(A : A) : A	(A : A) : B	(A : B) : A	(B : A) : A	(A : B) : C.
: (Peirce, CP 3.229, "On the Algebra of Logic", American Journal of Mathematics 3, 15–57 (1880), CP 3.154–251).
===Theory of categories===
Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: "Category" from Aristotle and Kant, "Functor" from Carnap (Logische Syntax der Sprache), and "natural transformation" from then current informal parlance.
(Saunders Mac Lane, Categories for the Working Mathematician, 29–30).
Mac Lane did not mention Peirce among the objects of his sincerest flattery, but he might as well have, for his mention of Aristotle and Kant well enough credits his deep indebtedness to the pursers of them all. As Richard Feynman was fond of observing, "the same questions have the same answers", and the problem that a system of categories is aimed to beautify is the same sort of beast whether it's Aristotle, Kant, Peirce, Carnap, or Eilenberg and Mac Lane that bends the bow. What is that problem? To answer that, let's begin again at the source:
Things are equivocally named, when they have the name only in common, the definition (or statement of essence) corresponding with the name being different. For instance, while a man and a portrait can properly both be called animals (ζωον), these are equivocally named. For they have the name only in common, the definitions (or statements of essence) corresponding with the name being different. For if you are asked to define what the being an animal means in the case of the man and the portrait, you give in either case a definition appropriate to that case alone.
Things are univocally named, when not only they bear the same name but the name means the same in each case — has the same definition corresponding. Thus a man and an ox are called animals. The name is the same in both cases; so also the statement of essence. For if you are asked what is meant by their both of them being called animals, you give that particular name in both cases the same definition.
(Aristotle, Categories, 1.1a1–12).
In the logic of Aristotle categories are adjuncts to reasoning that are designed to resolve equivocations and thus to prepare ambiguous signs, that are otherwise recalcitrant to being ruled by logic, for the application of logical laws. An equivocation is a variation in meaning, or a manifold of sign senses, and so Peirce's claim that three categories are sufficient amounts to an assertion that all manifolds of meaning can be unified in just three steps.
The following passage is critical to the understanding of Peirce's Categories:
I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates.
That wonderful operation of hypostatic abstraction by which we seem to create entia rationis that are, nevertheless, sometimes real, furnishes us the means of turning predicates from being signs that we think or think through, into being subjects thought of. We thus think of the thought-sign itself, making it the object of another thought-sign.
Thereupon, we can repeat the operation of hypostatic abstraction, and from these second intentions derive third intentions. Does this series proceed endlessly? I think not. What then are the characters of its different members?
My thoughts on this subject are not yet harvested. I will only say that the subject concerns Logic, but that the divisions so obtained must not be confounded with the different Modes of Being: Actuality, Possibility, Destiny (or Freedom from Destiny).
On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being. Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into Realms for the different Predicaments.
(Peirce, CP 4.549, "Prolegomena to an Apology for Pragmaticism", The Monist 16, 492–546 (1906), CP 4.530–572).
The first thing to extract from this passage is the fact that Peirce's Categories, or "Predicaments", are predicates of predicates. Meaningful predicates have both extension and intension, so predicates of predicates get their meanings from at least two sources of information, namely, the classes of relations and the qualities of qualities to which they refer. Considerations like these tend to generate hierarchies of subject matters, extending through what is traditionally called the logic of second intensions, or what is handled very roughly by second order logic in contemporary parlance, and continuing onward through higher intensions, or higher order logic and type theory.
Peirce arrived at his own system of three categories after a thoroughgoing study of his predecessors, with special reference to the categories of Aristotle, Kant, and Hegel. The names that he used for his own categories varied with context and occasion, but ranged from moderately intuitive terms like quality, reaction, and symbolization to maximally abstract terms like firstness, secondness, and thirdness, respectively. Taken in full generality, k-ness may be understood as referring to those properties that all k-adic relations have in common. Peirce's distinctive claim is that a type hierarchy of three levels is generative of all that we need in logic.
Part of the justification for Peirce's claim that three categories are both necessary and sufficient appears to arise from mathematical facts about the reducibility of k-adic relations. With regard to necessity, triadic relations cannot be completely analyzed in terms or monadic and dyadic predicates. With regard to sufficiency, all higher arity k-adic relations can be analyzed in terms of triadic and lower arity relations.
===Logic as formal semiotic===
On the Definition of Logic. Logic is formal semiotic. A sign is something, A, which brings something, B, its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, C, its object, as that in which itself stands to C. This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time. It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of Weierstrassian severity, and that is perfectly evident. The word "formal" in the definition is also defined. (Peirce, "Carnegie Application", NEM 4, 54).
In 1902 Peirce applied to the newly established Carnegie Institution for aid "in accomplishing certain scientific work", presenting an "explanation of what work is proposed" plus an "appendix containing a fuller statement". These parts of the letter, along with excerpts from earlier drafts, can be found in NEM 4 (Eisele 1976). The appendix is organized as a "List of Proposed Memoirs on Logic", and No. 12 among the 36 proposals is titled "On the Definition of Logic", the earlier draft of which is quoted in full above.
Peirce's major discoveries in formal logic include:
:* Distinguishing (Peirce, 1885) between first-order and second-order quantification.
:* Seeing that Boolean calculations could be carried out by means of electrical switches (W5:421–24), anticipating Claude Shannon by more than 50 years.
:* Devising the existential graphs, a diagrammatic notation for the predicate calculus. These graphs form the basis of the conceptual graphs of John F. Sowa, and of Sun-Joo Shin's diagrammatic reasoning.
A philosophy of logic, grounded in his categories and semeiotic, can be extracted from Peirce's writings. This philosophy, as well as Peirce's logical work more generally, is exposited and defended in , and in Hilary Putnam (1982), the Introduction to Houser et al (1997), and Dipert's chapter in Misak (2004). Jean Van Heijenoort (1967), Jaakko Hintikka in his chapter in Brunning and Forster (1997), and Brady (2000) divide those who study formal (and natural) languages into two camps: the model-theorists / semanticists, and the proof theorists / universalists. Hintikka and Brady view Peirce as a pioneer model theorist. On how the young Bertrand Russell, especially his Principles of Mathematics and Principia Mathematica, did not do Peirce justice, see Anellis (1995).
Peirce's work on formal logic had admirers other than Ernst Schröder:
:* The philosophical algebraist William Kingdon Clifford and the logician William Ernest Johnson, both British;
:* The Polish school of logic and foundational mathematics, including Alfred Tarski;
:* Arthur Prior, whose Formal Logic and chapter in Moore and Robin (1964) praised and studied Peirce's logical work.
==Dynamics of inquiry==
Every mind which passes from doubt to belief must have ideas which follow after one another in time. Every mind which reasons must have ideas which not only follow after others but are caused by them. Every mind which is capable of logical criticism of its inferences, must be aware of this determination of its ideas by previous ideas. (Peirce, "On Time and Thought", CE 3, 68–69).
All through the 1860's, the young but rapidly maturing Peirce was busy establishing a conceptual basecamp and a technical supply line for the intellectual adventures of a lifetime. Taking the longview of this activity and trying to choose the best titles for the story, it all seems to have something to do with the dynamics of inquiry. This broad subject area has a part that is given by nature and a part that is ruled by nurture. On first approach, it is possible to see a question of articulation and a question of explanation:
* What is needed to articulate the workings of the active form of representation that is known as conscious experience?
* What is needed to account for the workings of the reflective discipline of inquiry that is known as science?
The pursuit of answers to these questions finds them to be so entangled with each other that it's ultimately impossible to comprehend them apart from each other, but for the sake of exposition it's convenient to organize our study of Peirce's assault on the summa by following first the trails of thought that led him to develop a theory of signs, one that has come to be known as 'semiotic', and tracking next the ways of thinking that led him to develop a theory of inquiry, one that would be up to the task of saying 'how science works'.
Opportune points of departure for exploring the dynamics of representation, such as led to Peirce's theories of inference and information, inquiry and signs, are those that he took for his own springboards. Perhaps the most significant influences radiate from points on parallel lines of inquiry in Aristotle's work, points where the intellectual forerunner focused on many of the same issues and even came to strikingly similar conclusions, at least about the best ways to begin. Staying within the bounds of what will give us a more solid basis for understanding Peirce, it serves to consider the following loci in Aristotle:
* The basic terminology of psychology, in On the Soul.
* The founding description of sign relations, in On Interpretation;
* The differentiation of the genus of reasoning into three species of inference that are commonly translated into English as abduction, deduction, and induction, in the Prior Analytics.
In addition to the three elements of inference, that Peirce would assay to be irreducible, Aristotle analyzed several types of compound inference, most importantly the type known as 'reasoning by analogy' or 'reasoning from example', employing for the latter description the Greek word 'paradeigma', from which we get our word 'paradigm'.
Inquiry is a form of reasoning process, in effect, a particular way of conducting thought, and thus it can be said to institute a specialized manner, style, or turn of thinking. Philosophers of the school that is commonly called 'pragmatic' hold that all 
thought takes place in signs, where 'sign' is the word they use for the broadest conceivable variety of characters, expressions, formulas, messages, signals, texts, and so on up the line, that might be imagined. Even intellectual concepts and mental ideas are held to be a special class of signs, corresponding to internal states of the thinking agent that both issue in and result from the interpretation of external signs.
The subsumption of inquiry within reasoning in general and the inclusion of thinking within the class of sign processes allows us to approach the subject of inquiry from two different perspectives:
* The syllogistic approach treats inquiry as a species of logical process, and is limited to those of its aspects that can be related to the most basic laws of inference.
* The sign-theoretic approach views inquiry as a genus of semiosis, an activity taking place within the more general setting of sign relations and sign processes.
The distinction between signs denoting and objects denoted is critical to the discussion of Peirce's theory of signs. Wherever needed in the rest of this article, therefore, in order to mark this distinction a little more emphatically than usual, double quotation marks placed around a given sign, for example, a string of zero or more characters, will be used to create a new sign that denotes the given sign as its object.
===Theory of signs, or semiotic===
: Main articles : Semeiotic, Semiotics, Sign relation
Peirce's efforts to articulate the logic of scientific inquiry forced him to examine the role of signs, including cognitive concepts, in the activities by which we inform ourselves about the world and make logical inferences on the basis of that information. The complexities of the possible relationships among signs, objects in the world, and ideas in the mind demanded that he develop a more powerful logic of relations. The hasty map of relations and relatives sketched above makes it possible to venture on the terrain of sign relations, the main subject matter of Peirce's semeiotic, or theory of signs.
Peirce referred to his theory of signs, based on a specific type of triadic relation called a sign relation, as semeiotic or semiotic, either of which terms are currently used in either singular of plural form. He described the corresponding process of semiosis as an "action, or influence, which is, or involves, a cooperation of three subjects, such as a sign, its object, and its interpretant, this tri-relative influence not being in any way resolvable into actions between pairs". (Peirce 1907, in Houser 1998, 411).
====Sign relations====
Here again is Peirce's definition of the triadic sign relation that formed the core of his definition of logic.
Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C. (Peirce 1902, NEM 4, 20–21).
This definition, together with Peirce's definitions of correspondence and determination, is sufficient to derive all of the statements that are necessarily true all sign relations. There is a lot more to the theory of signs than simply proving universal theorems about generic sign relations, however. There is also the task of classifying the different species and subspecies of sign relations. As a practical matter, of course, familiarity with the full range of concrete examples is indispensable to theory and application both.
In Peirce's theory of signs, a sign is something that stands in a well-defined type of relation to two other things, its object and its interpretant sign. Although Peirce's definition of a sign is independent of psychological subject matter and his theory of signs covers more ground than linguistics alone, it is nevertheless the case that many of the more familiar examples and illustrations of sign relations will naturally be drawn from the descriptive sciences of linguistics and psychology, along with our ordinary experience of their subject matters.
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, CP 5.254).
For example, one way to approach the concept of an interpretant is to think of a psycholinguistic process. In this context, an interpretant can be understood as a sign's effect on the mind, or on anything that acts like a mind, what Peirce calls a quasi-mind. An interpretant is what results from a process of interpretation, one of the types of activity that falls under the heading of semiosis. One usually says that a sign stands for an object to an agent, an interpreter. In the upshot, however, it is the sign's effect on the agent that is paramount. This effect is what Peirce called the interpretant sign, or the interpretant for short. The interpretant may be regarded as one of the sign's roughly equivalent meanings, and especial interest attaches to the types of semiosis that proceed from obscure signs to relatively clear interpretants. In logic and mathematics the most clarified and most succinct signs for an object are called canonical forms or normal forms.
Some of the understanding needed by the mind depends on familiarity with the object. In order to know hat a given sign denotes, the mind needs some experience of that sign's object collaterally to that sign or sign system, and in this context Peirce speaks of collateral experience, collateral observation, and so on.
* 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.
* An object (or semiotic object) is a subject matter of a sign and an interpretant. It can be anything discussable or thinkable, a thing, event, relationship, quality, law, argument, etc., and can even be fictional, for instance Hamlet. All of those are special or partial objects. The object most accurately is the universe of discourse to which the partial or special object belongs. For instance, a perturbation of Pluto's orbit is a sign about Pluto but ultimately not only about Pluto. 
* An interpretant (also called an interpretant sign) is the sign's more or less clarified meaning or ramification, a kind of form or idea of the difference which the sign's being true would make. (Peirce's sign theory concerns meaning in the broadest sense, including logical implication, not just the meanings of words as properly clarified by a dictionary.) The interpretant is a sign (a) of the object and (b) of the interpretant's "predecessor" (the interpreted sign) as being a sign of the same object. The interpretant is an interpretation in the sense of a product of an interpretive process or a content in which an interpretive relation culminates, though this product or content may itself be an act or conduct of some kind. Another way to say these things is that the sign stands for the object to the interpretant.
Object, sign, and interpretant are constrained by a relation of informational or logical determination which is more general than the special cases of causal or physical determination. In general terms, any information about one of the items in the sign relation tells you something about the others, although the actual ampont of this information may be nil in some species of sign relations. Moreover, when Peirce says that one thing determines some other things, he means determines in some measure, a measure that is not necessarily completely deterministic.
Peirce made various classifications of his semiotic elements, especially of the sign and the interpretant. Of particular concern in understanding the sign-object-interpretant triad is this: In relation to a sign, its object and its interpretant are either immediate (present in the sign) or mediate.
# Sign, always immediate to itself — that is, in a tautologous sense, present in or at itself, even if it is not immediate to a mind or immediately accomplished without processing or is a general apprehended only in its instances.
# Object
## Immediate object, the object as represented in the sign.
## Dynamic object, the object as it really is, on which the idea which is the immediate object is founded as on bedrock. Also called the dynamoid object, the dynamical object.
# Interpretant
## Immediate interpretant, the quality of the impression which a sign is fit to produce, not any actual reaction, and which the sign carries with it even before there is an interpreter or quasi-interpreter. It is what is ordinarily called the sign's meaning.
## Dynamic interpretant, the actual effect (apart from the feeling) of the sign on a mind or quasi-mind, for instance the agitation of the feeling.
## Final interpretant, the effect which the sign would have on any mind or quasi-mind if circumstances allowed that effect to be fully achieved. It is the sign's end or purpose. The final interpretant of one's inqury about the weather may consist in the effect which the true response would have on one's plans for the day which were the inquiry's purpose. The final interpretant of a line of investigation as such is truth and would be reached sooner or later but still inevitably by investigation adequately prolonged, though the truth remains independent of that which "you or I" or any finite community of investigators believe.
Peirce preferred phrases like dynamic object over real object since the object might be fictive — Hamlet, for instance, to whom one grants a fictive reality, a reality within the universe of discourse of the play Hamlet.
Signhood is a way of being in relation, not a way of being in itself. The role of sign is constituted as one role among three — object, sign, and interpretant sign — where the roles are distinct even when the things that fill them are not. In other words, the question of what a sign is depends on the concept of a sign relation, which depends on the concept of a triadic relation. This, in turn, depends on the concept of a relation itself. There are traditionally two ways of understanding what relations are, corresponding to definition by extension and definition by intension or comprehension. Peirce regards these aspects of relation as necessary but not sufficient, and he adds a third approach, the way of information — including the idea of change of information — in order to integrate the other two approaches into a unified whole.
====Types of signs====
There are three principal ways that a sign can denote its objects. These are usually described as kinds, species, or types of signs, but it is important to recognize that these are not ontological species, that is, they are not mutually exclusive features of description, since the same thing can be a sign in several different ways.
Beginning very roughly, the three main ways of being a sign can be described as follows:
* An icon is a sign that denotes its objects by virtue of a quality that it shares with its objects.
* An index is a sign that denotes its objects by virtue of an existential connection that it has with its objects.
* A symbol is a sign that denotes its objects solely by virtue of the fact that it is interpreted to do so.
One of Peirce's early delineations of the three types of signs is still quite useful as a first approach to understanding their differences and their relationships to each other:
In the first place there are likenesses or copies — such as statues, pictures, emblems, hieroglyphics, and the like. Such representations stand for their objects only so far as they have an actual resemblance to them — that is agree with them in some characters. The peculiarity of such representations is that they do not determine their objects — they stand for anything more or less; for they stand for whatever they resemble and they resemble everything more or less.
The second kind of representations are such as are set up by a convention of men or a decree of God. Such are tallies, proper names, &c. The peculiarity of these conventional signs is that they represent no character of their objects. Likenesses denote nothing in particular; conventional signs connote nothing in particular.
The third and last kind of representations are symbols or general representations. They connote attributes and so connote them as to determine what they denote. To this class belong all words and all conceptions. Most combinations of words are also symbols. A proposition, an argument, even a whole book may be, and should be, a single symbol. (Peirce 1866, "Lowell Lecture 7", CE 1, 467–468).
Many variant descriptions of signs and their types appear throughout Peirce's writings, but one of the more canonical typologies classifies signs into icons, indices, and symbols (CP 2.228, CP 2.229, CP 5.473). A sign may compound these modes, for example, a roadside sign displaying a fork both (1) portrays and (2) indicates a rest stop, and (3) does so in keeping with some pronounced stylistic conventions which add something of a symbolic dimension.
* An icon is a sign that denotes its object by virtue of a quality which is shared by them but which the icon has irrespectively of the object. The icon, for example, a portrait or a diagram, resembles or imitates its object. The icon has, of itself, a certain character or aspect, one which the object also has (or is supposed to have) and which lets the icon be interpreted as a sign even if the object does not exist. The icon signifies essentially on the basis of its ground. Peirce defined the ground as the pure abstraction of a quality, and the sign's ground as the respect in which it resembles its object.
* An index is a sign that denotes its object by virtue of an actual connection to it, one that he also calls a real relation in virtue of its being irrespective of interpretation. If the indexical relation is a resistance or reaction physically or causally connecting an index to its object, then the index is a reagent, for example, smoke coming from a building is a reagent index of fire. Such an index refers to the object because that index is really affected or modified by the object, and is the only kind of index which can be used in order to ascertain facts about its object. Peirce held that an index does not have to be an actual individual fact or thing, but can be a general; a disease symptom is general, its occurrence singular; and he usually considered a designation to be an index, for example, a pronoun, a proper name, or a label on a diagram, actually directing or compelling the mind's attention toward the object just as a reagent does. (In 1903 Peirce said that only an individual is an index (EP 2.274) and called designations "subindices or hyposemes", classing them as symbols; he allowed of a "degenerate index" indicating a non-individual object, as exemplified by an individual thing indicating its own characteristics. But by 1904 he returned to allowing indices to be generals and to classing designations as indices.)
* A symbol is a sign that denotes its object solely by virtue of the fact that it is interpreted to do so. The symbol does not depend on having any resemblance or actual connection to the denoted object but is fundamentally conventional, so that the signifying relationship must be learned and agreed upon, for example, the word cat. A symbol thus denotes, primarily, by virtue of its interpretant. Its sign-action (semeiosis) is ruled by a convention, a more or less systematic set of associations that ensures its interpretation.
These definitions are specific to Peirce's theory of signs and are not exactly equivalent to more general uses of the terms icon, index, or symbol.
===Theory of inquiry===
: Main article : Inquiry
: Upon this first, and in one sense this sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not be satisfied with what you already incline to think, there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy:
 Do not block the way of inquiry.
: Although it is better to be methodical in our investigations, and to consider the economics of research, yet there is no positive sin against logic in trying any theory which may come into our heads, so long as it is adopted in such a sense as to permit the investigation to go on unimpeded and undiscouraged. On the other hand, to set up a philosophy which barricades the road of further advance toward the truth is the one unpardonable offence in reasoning, as it is also the one to which metaphysicians have in all ages shown themselves the most addicted. (Peirce, "F.R.L." (c. 1899), CP 1.135–136).
Peirce extracted the pragmatic model or theory of inquiry from its raw materials in classical logic and refined it in parallel with the early development of symbolic logic to address problems about the nature of scientific reasoning. Borrowing a brace of concepts from Aristotle, Peirce examined three fundamental modes of reasoning that play a role in inquiry, processes that are currently known as abductive, deductive, and inductive inference.
In the roughest terms, abduction is what we use to generate a likely hypothesis or an initial diagnosis in response to a phenomenon of interest or a problem of concern, while deduction is used to clarify, to derive, and to explicate the relevant consequences of the selected hypothesis, and induction is used to test the sum of the predictions against the sum of the data.
These three processes typically operate in a cyclic fashion, systematically operating to reduce the uncertainties and the difficulties that initiated the inquiry in question, and in this way, to the extent that inquiry is successful, leading to an increase in the knowledge or skills, in other words, an augmentation in the competence or performance, of the agent or community engaged in the inquiry.
In the pragmatic way of thinking every thing has a purpose, and the purpose of any thing is the first thing that we should try to note about it. The purpose of inquiry is to reduce doubt and lead to a state of belief, which a person in that state will usually call 'knowledge' or 'certainty'. It needs to be appreciated that the three kinds of inference, insofar as they contribute to the end of inquiry, describe a cycle that can be understood only as a whole, and none of the three makes complete sense in isolation from the others.
For instance, the purpose of abduction is to generate guesses of a kind that deduction can explicate and that induction can evaluate. This places a mild but meaningful constraint on the production of hypotheses, since it is not just any wild guess at explanation that submits itself to reason and bows out when defeated in a match with reality. In a similar fashion, each of the other types of inference realizes its purpose only in accord with its proper role in the whole cycle of inquiry. No matter how much it may be necessary to study these processes in abstraction from each other, the integrity of inquiry places strong limitations on the effective modularity of its principal components.
If we then think to inquire, 'What sort of constraint, exactly, does pragmatic thinking place on our guesses?', we have asked the question that is generally recognized as the problem of 'giving a rule to abduction'. Peirce's way of answering it is given in terms of the so-called 'pragmatic maxim', and this in turn gives us a clue as to the central role of abductive reasoning in Peirce's pragmatic philosophy.
===Logic of information===
: Main article : Logic of information
Peirce was a pioneer in the study of information, lecturing on what he called the laws of information as early as 1865.
Let us now return to the information. The information of a term is the measure of its superfluous comprehension. That is to say that the proper office of the comprehension is to determine the extension of the term. For instance, you and I are men because we possess those attributes — having two legs, being rational, &tc. — which make up the comprehension of man. Every addition to the comprehension of a term lessens its extension up to a certain point, after that further additions increase the information instead. (C.S. Peirce, "The Logic of Science, or, Induction and Hypothesis" (1866), CE 1, 467).
==Logical graphs==
: Main article : Logical graph
==Mathematics==
It may be added that algebra was formerly called Cossic, in English, or the Rule of Cos; and the first algebra published in England was called "The Whetstone of Wit", because the author supposed that the word cos was the Latin word so spelled, which means a whetstone. But in fact, cos was derived from the Italian, cosa, thing, the thing you want to find, the unknown quantity whose value is sought. It is the Latin caussa, a thing aimed at, a cause. (C.S. Peirce, "Elements of Mathematics", MS 165 (c. 1895), NEM 2, 50).
Peirce made a number of discoveries in foundational mathematics, nearly all of which came to be appreciated only long after his death. Among other things, he:
:* Showed how what is now called Boolean algebra could be expressed by means of a single binary operation, either NAND or its dual, NOR. (See also De Morgan's Laws). This discovery anticipated Sheffer by 33 years.
:* In Peirce (1885), set out what can be read as the first (primitive) axiomatic set theory, anticipating Zermelo by about two decades.
:* Discovered the now-classic axiomatization of natural number arithmetic, a few years before Dedekind and Peano did so.
:* Discovered, independently of Dedekind, an important formal definition of an infinite set, namely, as a set that can be put into a one-to-one correspondence with one of its proper subsets. 
Beginning with his first paper on the "Logic of Relatives" (1870), Peirce extended the theory of relations that Augustus De Morgan had recently revived after long years of neglect by logicians. Much of the actual mathematics of relations that is taken for granted today was borrowed from Peirce, not always with all due credit (Anellis 1995). Beginning in 1940, Alfred Tarski and his students rediscovered aspects of Peirce's larger vision of relational logic, developing the perspective of relational algebra. These theoretical resources gradually worked their way into applications, in large part instigated by the work of Edgar F. Codd, who happened to be a doctoral student of the Peirce editor and scholar Arthur W. Burks, on the relational model or the relational paradigm for implementing and using databases.
In the four volume work, The New Elements of Mathematics by Charles S. Peirce (1976), mathematician and Peirce scholar Carolyn Eisele published a large number of Peirce's previously unpublished manuscripts on mathematical subjects, including the drafts for an introductory textbook, titled The New Elements of Mathematics in allusion to Euclid's Elements, that presented mathematics from a novel standpoint.
==Works==
Peirce's reputation is largely based on a number of academic papers published in American scholarly and scientific journals. These papers, along with a selection of Peirce's previously unpublished work and a smattering of his correspondence, fill the eight volumes of the Collected Papers of Charles Sanders Peirce, published between 1931 and 1958. An important recent sampler of Peirce's philosophical writings is the two volume The Essential Peirce (Houser and Kloesel (eds.) 1992, Peirce Edition Project (eds.) 1998).
The only book Peirce published in his lifetime was Photometric Researches (1878), a monograph on the applications of spectrographic methods to astronomy. While at Johns Hopkins, he edited Studies in Logic (1883), containing chapters by himself and his graduate students. He was a frequent book reviewer and contributor to The Nation, work reprinted in Ketner and Cook (1975–87).
Hardwick (2001) published Peirce's entire correspondence with Victoria, Lady Welby. 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 in the Writings.
Harvard University acquired the papers found in Peirce's study soon after his death, but did not microfilm them until 1964. Only after Richard Robin (1967) catalogued this Nachlass did it become clear that Peirce had left approximately 1650 unpublished manuscripts, totalling 80,000 pages. Eisele (1976, 1985) published some of this work, but most of it remains unpublished. For more on the vicissitudes of Peirce's papers, see (Houser 1989).
The limited coverage, and defective editing and organization, of the Collected Papers led Max Fisch and others in the 1970s to found the Peirce Edition Project, whose mission is to prepare a more complete critical chronological edition, known as the Writings. Only 6 out of a planned 31 volumes have appeared to date, but they cover the period from 1859–1890, when Peirce carried out much of his best-known work.
===On a New List of Categories (1867)===
: Main article: On a New List of Categories
===Logic of Relatives (1870)===
: Main article: Logic of Relatives (1870)
By 1870, the drive that Peirce exhibited to understand the character of knowledge, starting with our partly innate and partly inured models of the world and working up to the conduct of our scientific inquiries into it, having led him to inquire into the three-roled relationship of objects, signs, and impressions of the mind, now brought him to the pass of needing more power in a theory of relations than the available logical formalisms were up to providing. His first concerted effort to supply the gap was rolled out in his paper "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic". But the nameplate "LOR of 1870" will do for ease of identification.
===Logic of Relatives (1883)===
: Main article: Logic of Relatives (1883)
===Logic of Relatives (1897)===
===The Simplest Mathematics (1902)===
: Main article: The Simplest Mathematics
===Kaina Stoicheia (1904)===
: Main article: Kaina Stoicheia
==References==
* Abbreviations for frequently cited works:
: CE n, m = Writings of Charles S. Peirce: A Chronological Edition, vol. n, page m.
: CP n.m = Collected Papers of Charles Sanders Peirce, vol. n, paragraph m.
: CTN n, m = Contributions to 'The Nation' , vol. n, page m.
: EP n, m = The Essential Peirce: Selected Philosophical Writings, vol. n, page m.
: NEM n, m = The New Elements of Mathematics by Charles S. Peirce, vol. n, page m.
: SIL m = Studies in Logic by Members of the Johns Hopkins University, page m.
: SS m = Semiotic and Significs … Charles S. Peirce and Lady Welby, page m.
: SW m = Charles S. Peirce, Selected Writings, page m.
===Works by Peirce===
* Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as CP (volume).(paragraph).
* Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–. Cited as CE (volume), (page).
* Peirce, C.S, The New Elements of Mathematics by Charles S. Peirce, 4 volumes in 5, Carolyn Eisele (ed.), Mouton Publishers, The Hague, Netherlands, 1976. Humanities Press, Atlantic Highlands, NJ, 1976. Cited as NEM (volume), (page).
* Peirce, C.S., The Essential Peirce : Selected Philosophical Writings, Volume 1 (1867–1893), Nathan Houser and Christian Kloesel (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1992. Cited as EP 1, (page).
* Peirce, C.S., The Essential Peirce : Selected Philosophical Writings, Volume 2 (1893–1913), Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1998. Cited as EP 2, (page).
* Peirce, C.S. (1865a), "On the Logic of Science", Harvard University Lectures, CE 1, 161–302.
* Peirce, C.S. (1865b), "An Unpsychological View of Logic to which are appended some applications of the theory to Psychology and other subjects", MS 109 (Variant 1), May–Fall 1865, CE 1, 305–309.
* Peirce, C.S. (1865c), "An Unpsychological View of Logic to which are appended some applications of the theory herein contained to Psychology and other subjects", MS 109 (Variant 2), May–Fall 1865, CE 1, 310–321.
* Peirce, C.S. (1866), "The Logic of Science, or, Induction and Hypothesis", Lowell Institute Lectures, CE 1, 357–504.
* Peirce, C.S. (1867), "On a New List of Categories", Proceedings of the American Academy of Arts and Sciences 7 (1868), 287–298. Presented, 14 May 1867. Reprinted (CP 1.545–559), (CE 2, 49–59), (EP 1, 1–10).
* Peirce, C.S. (1877), "The Fixation of Belief", Popular Science Monthly 12, 1–15, 1877. Reprinted, CP 5.358–387. Eprint.
* Peirce, C.S. (1878), "How to Make Our Ideas Clear", Popular Science Monthly 12, 286–302, 1878. Reprinted, CP 5.388–410. Eprint.
* Peirce, C.S. (1899), "F.R.L." [First Rule of Logic], unpaginated manuscript, c. 1899. Reprinted, CP 1.135–140. Eprint.
* Peirce, C.S. (1902), "Application of C.S. Peirce to the Executive Committee of the Carnegie Institution" (1902 July 15). Published, "Parts of Carnegie Application" (L75), pp. 13–73 in The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy, Carolyn Eisele (ed.), Mouton Publishers, The Hague, Netherlands, 1976. Eprint, Joseph Ransdell (ed.).
===Works by others===
* Anellis, I.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. Eprint.
* Aristotle, "The Categories", Harold P. Cooke (trans.), pp. 1–109 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
* Aristotle, "On Interpretation", Harold P. Cooke (trans.), pp. 111–179 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
* Aristotle, "Prior Analytics", Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
* Awbrey, J.L., and Awbrey, S.M. (1995), "Interpretation as Action : The Risk of Inquiry", Inquiry : Critical Thinking Across the Disciplines 15, 40–52. Eprint.
* Boole, George (1854), An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Macmillan, 1854. Reprinted with corrections, Dover Publications, New York, NY, 1958.
* Burch, Robert (2001/2006), "Charles Sanders Peirce", in Edward N. Zalta (ed.), Stanford Encyclopedia of Philosophy (Fall 2006 Edition). Eprint.
* Dewey, John (1910), How We Think, D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.
* Haack, Susan (1998), Manifesto of a Passionate Moderate, The University of Chicago Press, Chicago, IL.
* van Heijenoort, Jean (1967), "Logic as Language and Logic as Calculus", Synthese 17, 324–330.
* Houser, Nathan (1989), "The Fortunes and Misfortunes of the Peirce Papers", Fourth Congress of the International Association for Semiotic Studies, Perpignan, France, 1989. Published, pp. 1259–1268 in Signs of Humanity, vol. 3, Michel Balat and Janice Deledalle–Rhodes (eds.), Gérard Deledalle (gen. ed.), Mouton de Gruyter, Berlin, Germany, 1992. Eprint.
* Liddell, Henry George, and Scott, Robert (1889), An Intermediate Greek–English Lexicon, Oxford University Press, Oxford, UK, 1889. Impression of 1991. Eprint.
* Mac Lane, Saunders (1971), Categories for the Working Mathematician, Springer-Verlag, New York, NY, 1971. Second edition, 1998.
* Robin, Richard S. (1967), Annotated Catalogue of the Papers of Charles S. Peirce, University of Massachusetts Press, Amherst, MA, 1967. Eprint.
* Taylor, Barry N. (ed.), The International System of Units (NIST Special Publication 330), Superintendent of Documents, Washington, DC, 2001. PDF text.
==Bibliography==
See Charles Sanders Peirce (Bibliography).
==See also==
===Abstraction===
* Continuous predicate
* Hypostatic abstraction
* Hypostatic object
* Prescisive abstraction
===Contemporaries===
* John Dewey
* William James
* Ernst Schröder
===Information, inquiry, logic, semiotics===
* Ampheck
* Comprehension
* Entitative graph
* Existential graph
* Inquiry
* Laws of form
* Logic of information
* Logic of relatives
* Logical graph
* Logical matrix
* Logical NAND
* Logical NNOR
* Meaning
* Peirce's law
* Pragmatics
* Rhema, Rheme
* Semeiotic
* Semiosis
* Semiotics
* Semiotic information theory
* Semiotic triangle
* Sign
* Sign relation
* Sole sufficient operator
===Mathematics===
* Dyadic relation
* Kaina Stoicheia
* Quincuncial map
* Relation
* Relation composition
* Relation construction
* Relation reduction
* Theory of relations
* Triadic relation
===Philosophy===
* Pragmatism
* Pragmaticism
* Pragmatic maxim
* Pragmatic theory of truth
==External links==
* Arisbe : The Peirce Gateway, Joseph Ransdell (ed.)
* Charles S. Peirce Society
:* Transactions
* Charles S. Peirce Studies
* Charles Sanders Peirce, MacTutor History of Mathematics, O'Connor and Robertson
* Commens : Virtual Centre for Peirce Studies, University of Helsinki, Bergman and Paavola (eds.)
:* Dictionary of Peirce Terms 
:* Peirce's Writings Online
* Digital Encyclopedia of Charles S. Peirce
* Grupo de Estudios Peirceanos, Jaime Nubiola (ed.)
* His Glassy Essence: Autobiography of Charles S. Peirce, Kenneth Laine Ketner
* Internet Encyclopedia of Philosophy, Fieser & Dowden (eds.)
:* Charles Sanders Peirce (1839–1914), Albert Atkin
:* C.S. Peirce's Architectonic Philosophy, Albert Atkin
:* C.S. Peirce's Pragmatism, Albert Atkin
* Peirce, Charles Sanders (1839–1914), Ralph Lichtensteiger
* Peirce Edition Project
:* Introduction to Essential Peirce, vol. 1, Nathan Houser
:* Introduction to Essential Peirce, vol. 2, Nathan Houser
* Pragmatism Cybrary, John R. Shook (ed.)
:* Cambridge School of Pragmatism
* Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.)
:* Charles Sanders Peirce, Robert Burch
:* Peirce's Logic, Eric Hammer
* Wikipédia, l’encyclopédie libre
:* Philosophie et sémiotique de Peirce, Raymond Robert Tremblay.
==Document history==
Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.
* Charles Sanders Peirce, GetWiki.
 
* Charles Sanders Peirce, Wikinfo.
* Charles Sanders Peirce, Wikipedia.
* Charles S. Peirce, Jaime Nubiola, Nupedia.
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