Charles Peirce: Encyclopedia - Charles Peirce

Charles Peirce

Charles Sanders Santiago Peirce (pronounced purse), (September 10, 1839, Cambridge, Massachusetts – April 19, 1914, Milford, Pennsylvania) was an American polymath. Although educated as a chemist and employed as a scientist for 30 years, he is now mostly seen as a philosopher. He is the greatest American builder of architectonic systems, and his admirers deem him the most important systematizer since Kant and Hegel, who were major influences.

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 Much of his huge output is still unpublished.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

Charles Peirce - Life

Right from the beginning, the relations of America as New England with Europe were, from the philosophical point of view, ambiguous, when they were not simply difficult and, in the end, impossible. Peirce is in himself the ‘’resumé’’ of this story… from the rejection of European philosophical paradigms to the creation of new paradigms which are not only Peirce’s but America’s, and slowly but inevitably [those] of the global world of tomorrow. (Deledalle 2000: 3).

Brent (1998) is the only Peirce biography in English. 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 devoured an older brother's copy of Richard Whately's Elements of Logic, then the leading English language text of its kind. Thus began his lifelong fascination with logic and reasoning. He went on to obtain the BA and MA from Harvard, and in 1863 was awarded the Lawrence Scientific School's first B.Sc. in chemistry. This last degree was awarded summa cum laude; his academic record was otherwise 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 him; they clashed on later occasions. This was unfortunate, because Eliot was President of Harvard 1869-1909, a period encompassing nearly all of Peirce's working life, during which he repeatedly vetoed having Harvard employ Peirce in any capacity.

Charles was employed as a scientist by the United States Coast Survey (1859–1891), 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, sparing him a very awkward situation, as his Boston Brahmin family sympathized with the Confederacy. At the Survey, he worked mainly in geodesy and in gravimetry, refining the use of pendulums to determine small local variations in the strength of the earth's gravity. The Survey sent him to Europe five times, the first in 1871, as part of a group dispatched to observe a solar eclipse. While in Europe, he sought out Augustus De Morgan, William Stanley Jevons, and William Kingdon Clifford, British mathematicians and logicians whose turn of mind resembled his own. During 1869-72, 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 today.

Over 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 required 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, he resigned from the Coast Survey, at the request of Superintendent Thomas Corwin Mendenhall. He never again held regular employment.

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 John Dewey were students), 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 was ever held. It is a fact that Clark, Wisconsin, Michigan, Cornell, Stanford, and Chicago all declined to hire him, although the precise reasons for their so doing can no longer be determined. 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 (1835-1909). Peirce's ability to find academic employment may also have been frustrated by a difficult personality. Brent conjectures that Peirce may have been manic-depressive, further claiming that Peirce experienced 8 nervous breakdowns between 1876 and 1911. Brent also believes that Peirce tried to alleviate his symptoms with ether, morphine, and cocaine.

Peirce's personal life also proved a grave handicap. His first wife, Harriet Melusina Fay, 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), marrying her immediately upon divorcing Harriet 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, and to his being deemed morally unfit for academic employment anywhere in the USA. Peirce had no children by either marriage.

In 1887, Peirce used an 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. He insisted on living well beyond his means, which 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. His only food was bread donated by the local baker, and he wrote on the verso side of old manuscripts because he could not afford new stationery. For a while an outstanding warrant for assault and debt led to his becoming a fugitive in New York. A variety of people including his brother James Mills Peirce and his neighbors, relatives of Gifford Pinchot, paid his property taxes and mortgage, and settled other debts.

During this long final twilit phase of Peirce’s life, he 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, William Lloyd 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. Peirce tried his hand at inventing, and began but did not complete a number of books, all in the hope of making money. 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 this his hour of desperate need was his old friend William James, who helped arrange four series of lectures at or near Harvard, and dedicated his Will to Believe to Peirce. 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 showed his gratitude for these remarkable gestures of friendship 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 1998: 315-16, 374).

Peirce died destitute, as did his widow 20 years later. How she managed to survive that long is a puzzle.

Charles Peirce - 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 fails to mention Peirce.) While reading some of Peirce's unpublished manuscripts soon after arriving at Harvard in 1924, Alfred North Whitehead was struck by the extent to which Peirce had anticipated his own "process" thinking. (On Peirce and process metaphysics, see the chapter by Lowe in Moore and Robin, 1964.) Karl Popper viewed Peirce as "one of the greatest philosophers of all times". Nevertheless, Peirce's accomplishments were not immediately recognized. His imposing contemporaries William James and Josiah Royce admired him, and Cassius 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 Peirce bibliography. From 1916 until his death, John Dewey's writings repeatedly mention Peirce with deference, and his 1938 Logic: The Theory of Inquiry is Peircian through and through. The publication of the first six volumes of the Collected Papers (1931-35), 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 Buchler (1939), Feibleman (1946), and Goudge (1950), the 1941 Ph.D. thesis by Arthur Burks (who went on to edit vols. 7 and 8 of the Collected Papers), and the edited volume Wiener and Young (1952). The Charles S. Peirce Society was founded in 1946; its Transactions, an academic journal specializing in the history of American philosophy, including pragmatism, has appeared since 1965.

In 1949, while doing unrelated archival work, the historian of mathematics Carolyn Eisele (1902-2000) chanced on an autograph letter by Peirce. 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 (1900-1995) emerged as an authority on Peirce; Fisch (1986) reprints many of the relevant articles, including (pp. 422-48) a wide-ranging survey of the impact of Peirce's thought through 1983.

Peirce has come to enjoy an international following. University research centers devoted to Peirce Studies and pragmatism can be found in Brazil, Finland, Germany, and Spain. There have been French and Italian Peirceans of note since 1950. For many years, the University of Toronto housed the North American philosophy department most devoted to Peirce. In recent years, Peirce scholars have clustered at Indiana University - Purdue University Indianapolis, the home of the Peirce Edition Project, and the Pennsylvania State University.

Charles Peirce - Works

Peirce's reputation is based in large part on a number of academic papers published in American scholarly and scientific journals. These papers fill most of the eight volumes of the Collected Papers of Charles Sanders Peirce, published between 1931 and 1958. Perhaps the best introduction to Peirce's writings is the two volumes titled The Essential Peirce (Houser 1992, 1998).

In Peirce's day, one made a name in philosophy by publishing monographs on the subject, which he never did. Nor did he ever lay out systematically his thoughts on mathematics and logic. His only book was Photometric Researches, a little-noted 1878 short technical monograph on astronomy. While at Johns Hopkins, he edited Studies in Logic (Peirce 1883), containing chapters by himself and his graduate students. He was also a frequent book reviewer and contributor to the Nation; this journalism is reprinted in Peirce (1975-), edited by Ketner and Cook.

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 vol. 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 published his catalog of these manuscripts in 1967, did it become clear that Peirce had left approximately 1650 unpublished manuscripts, totalling 80,000 pages. A number of these manuscripts were published in Eisele (1976, 1985), but most are still unpublished. For more on the vicissitudes of Peirce's manuscripts, see [1].

Because the Collected Papers include very little of this vast trove of manuscripts and are flawed in other ways, Max Fisch set up in the 1970s, then headed for some years, the Peirce Edition Project whose mission is to prepare a more complete critical edition, known as the Writings and organized chronologically. Only six of a planned 31 volumes of the Writings have appeared to date, but those six cover the period 1859-1890, when Peirce published most of the work upon which his fame rests. Each volume of the Writings begins with one or more introductory essays, all worthy additions to the secondary literature. Houser (1998) is a good selection of what Peirce wrote after 1890.

All citations without a name are to Peirce's writings. The standard collections of those writings are referenced using a system of abbreviations described in the Bibliography below.

Charles Peirce - 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 secondly 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: 486).

Upon this first, and in one sense 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 written upon every wall of the city of philosophy: Do not block the way of inquiry. (CP 1.135)

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 a few pages of Kant's Critique of Pure Reason in the original German, every day while a Harvard undergraduate. His writings bear on an wide array of disciplines, including astronomy, metrology, geodesy, mathematics, logic, philosophy, the history and philosophy of science, linguistics, economics, and psychology. This work has become the subject of renewed interest and approval, resulting in a revival inspired not only by his anticipations of recent scientific developments but also by his demonstration of how philosophy can be applied effectively to human problems.

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

Charles Peirce - Pragmatism

τα δε μοι παθήματα μαθήματα γέγονε.
My sufferings have been my lessons.
(Herodotus, in Liddell & Scott).

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

Pragmaticism was originally enounced in the form of a maxim, as follows: 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, "Issues of Pragmaticism", Monist, 15, 481-499 (1905), CP 5.438-463).

William James, among others, deemed two of Peirce's papers "The Fixation of Belief" (1877) and "How to Make Our Ideas Clear" (1878) as being the origin of pragmatism. Unlike James and some later pragmatists, e.g., John Dewey, Peirce conceived of pragmatism primarily as a method for the clarification of the meaning of ideas, by applying the scientific method to philosophical issues.

Peirce's pragmatism may be understood as a method of sorting out conceptual confusions by linking the meaning of concepts to their operational or practical consequences. This pragmatism bears no resemblance to "vulgar" pragmatism, which misleadingly connotes a ruthless and Machiavellian search for mercenary or political advantage. Rather, Peirce sought an objectively verifiable method to test the truth of putative knowledge on a way that goes beyond the usual duo of foundational alternatives, namely:

• Deduction from self-evident truths, or rationalism;
• Induction from experiential phenomena, or empiricism.

His approach is often confused with the latter form of foundationalism, but is distinct from it by virtue of the following three dimensions:

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

A theory that proves itself more successful in predicting and controlling our world than its rivals is said to be nearer the truth. This is an operational notion of truth employed by scientists. Unlike the other pragmatists, Peirce never explicitly advanced a theory of truth. But his scattered comments about truth have proved influential to several epistemic truth theorists, and as a useful foil for deflationary and correspondence theories of truth.

Pragmatism is regarded as a distinctively American philosophy. As advocated by James, John Dewey, Ferdinand Canning Scott Schiller, George Herbert Mead, and others, it has proved durable and popular. But Peirce did not seize on this fact to enhance his reputation. Instead, what James and others called "pragmatism" so dismayed Peirce that he renamed his own variant pragmaticism, joking that it was "ugly enough to be safe from kidnappers" (CP 5.414), i.e., no one would ever appropriate a neologism so ugly.

Charles Peirce - Scholastic realism

Peirce’s confession to being a “scholastic realist of a somewhat extreme stripe” (CP 5.470) is well known and baffles some. He has been described by careless writers as an idealist (“reality” = “the object of the final opinion of the scientific community”), but this description is inaccurate, since he believed that reality was best described 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 (where Peirce sided with the idealists). 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, May-Fall 1865)

Here Peirce—-in no uncertain terms—-disowns scholastic realism. So how does this square with his later confession to be a scholastic realist? The temptation is to explain this discontinuity by the dates of the statements in question: since Peirce asserted nominalism in 1865, but was clearly a scholastic realist in 1867, Peirce must have changed his mind over that period. This standard explanation is that of an essay Fisch published in 1967, reprinted in Fisch (1986: 184-200), and his Introduction to Vol. 2 of the Writings. Fisch portrays Peirce as having begun (at age 15!) as a nominalist, but then coming to adopt stronger and stronger versions of realism as his philosophy developed. Eventually, Fisch tells us, Peirce became the “extreme” scholastic realist of his famed 1905 confession.

Robert Lane (2004) is one of many Peirce scholars to challenge this received understanding of Peirce's evolution, pointing to two instances where Peirce’s self-assessment of his own intellectual development contradicts Fisch's more orthodox understanding. The clearest contradiction is Peirce's 1893 remark 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) This remark leads Lane to re-evaluate Peirce’s 1865 statements and find that a better way to explain Peirce’s change from outspoken nominalist to outspoken realist is not by reading into Peirce a change in his philosophical position, but instead to see that between 1865 and 1868, Peirce changed his understanding of “Scholastic realism”. Before 1868, Peirce took a universal as existing out in the world; as an object that could be set next to a clock and a desk and counted, a position he would later call “platonic nominalism”. Here is Peirce explaining scholastic realism in 1865:

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. (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, i.e., 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; it has no secondness).

Charles Peirce - Formal perspective

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. (George Boole, Laws of Thought, p. 24)

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).

No matter what you hear, 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 could fail to hear the echoes of Boole, nor indeed the background stirrings of the contemporary Zeitgeist in mathematics that went under the name of the 'symbolist movement', the clarion call to which is commonly attributed to George Peacock (1791-1858). If Peirce appears at times to march out of step, it is because he hears the beat of many different drummers, not just one.

The main themes of the symbolist movement, though they may have presented novelties to the general understanding of mathematics in the 19th Century, are nowadays familiar to anyone who has had a brush with the 'art of the story problem' in an elementary algebra course. There one learns to approach the story problem, a roughly realistic representation of a concrete circumstance, with the aim to abstract or to 'tease out' a general formula from the concrete data that specify the situation. Next one proceeds to 'crank the formula', starting from a form that is true but problematically obscure in its implications, and, circumstances warranting, continuing until a logically equivalent or a lower implied form is reached, but one that is maximally clarified in its implications. That most clear result one dubs the 'abstract answer' or the 'general solution' to the story problem, leaving nothing more to do but 'plug in' the concrete data that came with the story problem to arrive at the 'concrete answer' or the 'specific solution'.

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.

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. ("Application to the Carnegie Institution", L75 (1902), NEM 4, 54).

Peirce, in his 1902 "application for aid from the Carnegie Institution" (L75, published with excerpts from earlier drafts in NEM 4, 13-73), includes a "List of Proposed Memoirs on Logic". Numbered among the 36 proposals is No. 12. On the Definition of Logic, the final version of which is quoted in full above.

On Peirce and his contemporaries Ernst Schröder and Frege, Hilary Putnam (1982) wrote:

When I started to trace the later development of logic, the first thing I did was to look at Schröder's Vorlesungen über die Algebra der Logik. This book … has a third volume on the logic of relations (Algebra und Logik der Relative, 1895). [These] three volumes were the best-known logic text in the world among advanced students, and they can safely be taken to represent what any mathematician interested in the study of logic would have had to know, or at least become acquainted with in the 1890s.

While, to my knowledge, no one except Frege ever published a single paper in Frege's notation, many famous logicians adopted Peirce-Schröder notation, and famous results and systems were published in it. Löwenheim stated and proved the Löwenheim-Skolem theorem ... in Peircian notation. In fact, there is no reference in Löwenheim's paper to any logic other than Peirce's. To cite another example, Zermelo presented his axioms for set theory in Peirce-Schröder notation, and not, as one might have expected, in Russell-Whitehead notation.

One can sum up these simple facts (which anyone can quickly verify) as follows: Frege certainly discovered the quantifier first (four years before O. H. Mitchell did so, going by publication dates, which are all we have as far as I know). But Leif Ericson probably discovered America 'first' (forgive me for not counting the native Americans, who of course really discovered it 'first'). If the effective discoverer, from a European point of view, is Christopher Columbus, that is because he discovered it so that it stayed discovered (by Europeans, that is), so that the discovery became known (by Europeans). Frege did 'discover' the quantifier in the sense of having the rightful claim to priority; but Peirce and his students discovered it in the effective sense. The fact is that until Russell appreciated what he had done, Frege was relatively obscure, and it was Peirce who seems to have been known to the entire world logical community. How many of the people who think that 'Frege invented [formal] logic' are aware of these facts?

The main evidence for Putnam's claims is Peirce (1885), published in the premier American mathematical journal of the day. Peano, Ernst Schröder, among others, cited this article. Peirce was apparently ignorant of Frege's work, despite their rival achievements in logic, philosophy of language, and the foundations of mathematics.

Peirce's other major discoveries in formal logic include:

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

A philosophy of logic, grounded in his categories and semeiotic, can be extracted from Peirce's writings. This philosophy, as well as Peirce's logical work more generally, is exposited and defended in , 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.

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.

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.

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, …, x_k, 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:

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-uple x_i = <x_i1, …, x_ik> = x_i1 : … : x_ik supplying the entries for the i^th row of the Table.

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.

(Text in progress, 06 January 2006)

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, pp. 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, 1.1^a1-12).

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. (CP 4.549, "Prolegomena to an Apology for Pragmaticism", Monist, 16, 492-546 (1906), CP 4.530-572).

The first thing that we need to extract from this text is the fact that Categories are predicates of predicates, in effect, types of relations.

Main article: Logical graphs

(Text in preparation, 08 January 2005)

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

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

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

Peirce much extended the relational algebra begun by Augustus DeMorgan, and wrote about its philosophical power. Principia Mathematica owed much to Peirce's work on relations, but never mentioned his name. Beginning in 1940, Alfred Tarski and his students rediscovered and extended the relational algebra, which has been applied of late to database theory.

Peirce admired Georg Cantor and Albert Bray Kempe, and wrote a dismissive review of Bertrand Russell's Principles of Mathematics.

Eisele (1976) published many mathematical manuscripts Peirce did publish in his lifetime, many of them drafts of textbooks which found no publisher. These writings reveal that Peirce preferred discursiveness and motivation to rigor and technique.

Charles Peirce - Dynamics of representation

All through the 1860's, the young but rapidly maturing Charles Peirce — our focus now being his coming of age in the sphere of intellect — 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 representation, divided into the portion that we are given by nature and the portion that we are given to nurture. In this quest we may discern a question of articulation and a question of explanation:

• How best to articulate the workings of that wary form of representation that we know as 'conscious experience'?

• How best to account for the workings of that discipline of inquiry that we mark out for recognition 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.

Nota Bene. 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.

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

In order to understand what a sign is we need to understand what a sign relation is, for signhood is a way of being in relation, not a way of being in itself. In order to understand what a sign relation is we need to understand what a triadic relation is, for the role of a sign is constituted as one among three, where roles in general are distinct even when the things that fill them are not. In order to understand what a triadic relation is we need to understand what a relation is, and here there are traditionally two ways of understanding what a relation is, both of which are necessary if not sufficient to complete understanding, namely, the way of extension and the way of intension. To these traditional approximations, Peirce adds a third way, the way of information, that integrates the other two approaches in a unified whole.

Main article: Sign relations

With that hasty map of relations and relatives sketched above (§ 4.3.2), we may now trek into the terrain of sign relations, the main subject matter of Peirce's semiotic, or theory of signs.

(Text in preparation, 08 January 2006)

Main article: 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. (["On Time and Thought"], MS 215 (1873), CE 3, 68-69).

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.

For our present purposes, the first feature to note in distinguishing the three principal modes of reasoning from each other is whether each of them is exact or approximate in character. In this light, deduction is the only one the three types of reasoning that can be made exact, in essence, always deriving true conclusions from true premisses, while abduction and induction are unavoidably approximate in their modes of operation, involving elements of fallible judgment in practice and inescapable error in their application.

The reason for this is that deduction, in the ideal limit, can be rendered a purely internal process of the reasoning agent, while the other two modes of reasoning essentially demand a constant interaction with the outside world, a source of phenomena and problems that will no doubt continue to exceed the capacities of any finite resource, human or machine, to master. Situated in this larger reality, approximations can be judged appropriate only in relation to their context of use and can be judged fitting only with regard to a purpose in view.

A parallel distinction that is often made in this connection is to call deduction a demonstrative form of inference, while abduction and induction are classed as non-demonstrative forms of reasoning. Strictly speaking, the latter two modes of reasoning are not properly called inferences at all. They are more like controlled associations of words or ideas that just happen to be successful often enough to be preserved as useful heuristic strategies in the repertoire of the agent. But non-demonstrative ways of thinking are inherently subject to error, and must be constantly checked out and corrected as needed in practice.

In classical terminology, forms of judgment that require attention to the context and the purpose of the judgment are said to involve an element of 'art', in a sense that is judged to distinguish them from 'science', and in their renderings as expressive judgments to implicate arbiters in styles of rhetoric, as contrasted with logic.

In a figurative sense, this means that only deductive logic can be reduced to an exact theoretical science, while the practice of any empirical science will always remain to some degree an art.

Many aspects of inquiry can be recognized and usefully studied in very basic logical settings, even simpler than the level of syllogism, for example, in the realm of reasoning that is variously known as boolean algebra, propositional calculus, sentential calculus, or zeroth order logic. By way of approaching the learning curve on the gentlest availing slope, we may well begin at the level of zeroth order inquiry, in effect, taking the syllogistic approach to inquiry only so far as the propositional or sentential aspects of the associated reasoning processes are concerned. One of the bonuses of doing this in the context of Peirce's logical work is that it provides us with doubly instructive exercises in the use of his logical graphs, taken at the level of his so-called 'alpha graphs'.

In the case of propositional calculus or sentential logic, deduction comes down to applications of the transitive law for conditional implications and the approximate forms of inference hang on the properties that derive from these. In describing the various types of inference I will employ a few old 'terms of art' from classical logic that are still of use in treating these kinds of simple problems in reasoning.

For ease of reference, Figure 1 and the Legend beneath it summarize the classical terminology for the three types of inference and the relationships among them.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` Z ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |\` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | \ ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `\` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` \ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` `\` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` \ ` R U L E ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` F ` | ` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` A ` | ` ` ` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` o Y ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` C ` | ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` T ` | ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` / ` C A S E ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Deduction takes a Case of the form X => Y,` ` ` |
| matches it with a Rule of the form Y => Z,` ` ` |
| then adverts to a Fact of the form X => Z.` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Induction takes a Case of the form X => Y,` ` ` |
| matches it with a Fact of the form X => Z,` ` ` |
| then adverts to a Rule of the form Y => Z.` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Abduction takes a Fact of the form X => Z,` ` ` |
| matches it with a Rule of the form Y => Z,` ` ` |
| then adverts to a Case of the form X => Y.` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Even more succinctly: ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` Abduction `Deduction` Induction ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Premiss:` ` `Fact ` ` ` Rule` ` ` `Case ` ` ` ` |
| Premiss:` ` `Rule ` ` ` Case` ` ` `Fact ` ` ` ` |
| Outcome:` ` `Case ` ` ` Fact` ` ` `Rule ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 1.  Elementary Structure and Terminology

In its original usage a statement of Fact has to do with a deed done or a record made, that is, a type of event that is openly observable and not riddled with speculation as to its very occurrence. In contrast, a statement of Case may refer to a hidden or a hypothetical cause, that is, a type of event that is not immediately observable to all concerned. Obviously, the distinction is a rough one and the question of which mode applies can depend on the points of view that different observers adopt over time. Finally, a statement of a Rule is called that because it states a regularity or a regulation that governs a whole class of situations, and not because of its syntactic form. So far in this discussion, all three types of constraint are expressed in the form of conditional propositions, but this is not a fixed requirement. In practice, these modes of statement are distinguished by the roles that they play within an argument, not by their style of expression. When the time comes to branch out from the syllogistic framework, we will find that propositional constraints can be discovered and represented in arbitrary syntactic forms.

The three kinds of inference that Peirce would come to refer to as 'abductive', 'deductive', and 'inductive' inference he gives his earliest systematic treatment in two series of lectures on the logic of science: the Harvard University Lectures of 1865 and the Lowell Institute Lectures of 1866. There he sums up the characters of the three kinds of reasoning in the following terms:

• We have then three different kinds of inference:

Early in the first series of lectures Peirce gives a very revealing illustration of how he then thinks of the natures, operations, and relationships of this trio of inference types:

• If I reason that certain conduct is wise

• If I think it is wise because it once turned out

• But if I think it is wise because a wise man does it,

We may begin the analysis of Peirce's example by making the following assignments of letters to the qualitative attributes mentioned in it:

• A = 'Wisdom',
• B = 'a certain character',
• C = 'a certain conduct',
• D = 'done by a wise man',
• E = 'a certain occasion'.

Recognizing that a little more concreteness will serve as an aid to the understanding, let's augment the Spartan features of Peirce's illustration in the following way:

• B = 'Benevolence', a certain character,
• C = 'Contributes to Charity', a certain conduct,
• E = 'Earlier today', a certain occasion.

The converging operation of all three reasonings is shown in Figure 2.

o---------------------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `D ("done by a wise man") ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `\* ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` \ * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `\` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` \ ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` * A ("a wise act")` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/| * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / | ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\`/` | ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` . ` | ` ` ` o B ("benevolence", a certain character)` |
| ` ` ` ` ` ` `/ \` | ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` / ` \ | ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `/` ` `\| * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` / ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `/` ` ` * C ("contributes to charity", a certain conduct) ` |
| ` ` ` ` / ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `/` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` / ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `/` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` / * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `/* ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `E ("earlier today", a certain occasion)` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------------------------------------------------------------------o
Figure 2.  A Thrice Wise Act

One of the styles of syntax that Aristotle uses for syllogistic propositions suggests the composite symbols that geometers have long used for labeling line intervals in a geometric figure, and it comports quite nicely with the Figure that we have just drawn. Specifically, the proposition that predicates X of the subject Y is represented by the digram 'XY' and associated with the line interval XY that descends from the point X to the point Y in the corresponding lattice diagram. In this wise we make the following observations:

The common proposition that concludes each argument is AC. Introducing the symbol '=>' to denote the relation of logical implication, the proposition AC can be written as C => A, and read as 'C implies A'. Adopting the parenthetical form of Peirce's alpha graphs, in their 'existential interpretation', AC can be written as (C (A)), and most easily comprehended as 'not C without A'. In the context of the present example, all of these forms are equally good ways of expressing the same concrete proposition, namely, 'contributing to charity is wise'.

• Deduction could have obtained the Fact AC from the Rule AB, 'benevolence is wisdom', along with the Case BC, 'contributing to charity is benevolent'.

• Induction could have gathered the Rule AC, after a manner of saying that 'contributing to charity is exemplary of wisdom', from the Fact AE, 'the act of earlier today is wise', along with the Case CE, 'the act of earlier today was an instance of contributing to charity'.

• Abduction could have guessed the Case AC, in a style of expression stating that 'contributing to charity is explained by wisdom', from the Fact DC, 'contributing to charity is done by this wise man', and the Rule DA, 'everything that is wise is done by this wise man'. Thus, a wise man, who happens to do all of the wise things that there are to do, may nevertheless contribute to charity for no good reason, and even be known to be charitable to a fault. But all of this notwithstanding, on seeing the wise man contribute to charity we may find it natural to conjecture, in effect, to consider it as a possibility worth examining further, that charity is indeed a mark of his wisdom, and not just the accidental trait or the immaterial peculiarity of his character — in essence, that wisdom is the 'cause' of his contribution or the 'reason' for his charity.

As a general rule, and despite many obvious exceptions, an English word that ends in '-ion' denotes equivocally either a process or its result. In our present application, this means that each of the words 'abduction', 'deduction', 'induction' can be used to denote either the process of inference or the product of that inference, that is, the proposition to which the inference in question leads.

One of the morals of Peirce's illustration can now be drawn. It demonstrates in a very graphic fashion that the three kinds of inference are three kinds of process and not three kinds of proposition, not if one takes the word 'kind' in its literal sense as denoting a genus of being, essence, or substance. Said another way, it means that being an abductive Case, a deductive Fact, or an inductive Rule is a category of relation, indeed, one that involves at the very least a triadic relation among propositions, and not a category of essence or substance, that is, not a property that inheres in the proposition alone.

This category distinction between the absolute, essential, or monadic predicates and the more properly relative predicates constitutes a very important theme in Peirce's architectonic. There is of course a parallel application of it in the theory of sign relations, or semiotics, where the distinctions among the sign relational roles of Object, Sign, and Interpretant are distinct ways of relating to other things, modes of relation that may vary from moment to moment in the extended trajectory of a sign process, and not distinctions that mark some fixed and eternal essence of the thing in itself.

In the normal course of inquiry, the elementary types of inference proceed in the order: Abduction, Deduction, Induction. However, the same building blocks can be assembled in other ways to yield different types of complex inferences. Of particular importance, reasoning by analogy can be analyzed as a combination of induction and deduction, in other words, as the abstraction and the application of a rule. Because a complicated pattern of analogical inference will be used in our example of a complete inquiry, it will help to prepare the ground if we first stop to consider an example of analogy in its simplest form.

The locus classicus for the study of abductive reasoning is lodged in Aristotle's Prior Analytics, Book 2, Chapt. 25. It begins this way:

We have Reduction (απαγωγη, abduction):

1. When it is obvious that the first term applies to the middle, but that the middle applies to the last term is not obvious, yet is nevertheless more probable or not less probable than the conclusion;
2. Or if there are not many intermediate terms between the last and the middle;

By way of explanation, Aristotle supplies two very instructive examples, one for each of the two varieties of abductive inference steps that he has just described in the abstract:

1. For example, let A stand for "that which can be taught", B for "knowledge", and C for "morality". Then that knowledge can be taught is evident; but whether virtue is knowledge is not clear. Then if BC is not less probable or is more probable than AC, we have reduction; for we are nearer to knowledge for having introduced an additional term, whereas before we had no knowledge that AC is true.
2. Or again we have reduction if there are not many intermediate terms between B and C; for in this case too we are brought nearer to knowledge. For example, suppose that D is "to square", E "rectilinear figure", and F "circle". Assuming that between E and F there is only one intermediate term — that the circle becomes equal to a rectilinear figure by means of lunules — we should approximate to knowledge.

Aristotle's latter variety of abductive reasoning, though it will take some explaining in the sequel, is well worth our contemplation, since it hints already at streams of inquiry that course well beyond the syllogistic source from which they spring, and into regions that Peirce will explore more broadly and deeply.

Much of Peirce's work deals with the scientific and logical questions of knowledge and truth, questions grounded in his experience as a working logician and experimental scientist, one who was a member of the international community of scientists and thinkers of his day. He made important contributions to deductive logic (see below), but was primarily interested in the logic of science and specifically in what he called abduction or "hypothesis," as opposed to deduction and induction. Abduction is the process whereby a hypothesis is generated, so that surprising facts may be explained. "There is a more familiar name for it than abduction," Peirce wrote, "for it is neither more nor less than guessing." Indeed, Peirce considered abduction to be at the heart not only of scientific research but of all ordinary human activities as well.

In his "Illustrations of the Logic of Science" (CE 3: 325-26), Peirce gave the following classic example of how abduction nests with classical deductive and inductive reasoning. Peirce begins by positing the following three statements:

• Rule: "All the beans from this bag are white."

• Case: "These beans are from this bag."

• Result: "These beans are white."

Now let any two of these statements be Givens (their order not mattering), and let the remaining statement be the Conclusion. The result is an argument, of which three kinds are possible:

Deduction encompasses, of course, the classical syllogism.

(Text In Preparation, 24 December 2005)

(Text in preparation, 24 December 2005)

The classic description of analogy in the syllogistic frame comes from Aristotle, who called this form of inference by the name 'paradeigma', that is, reasoning by way of example or through the parallel comparison of cases.

We have an Example [παραδειγμα, analogy] when the major extreme is shown to be applicable to the middle term by means of a term similar to the third. It must be known both that the middle applies to the third term and that the first applies to the term similar to the third. (Aristotle, "Prior Analytics", 2.24).

Aristotle illustrates this pattern of argument with the following sample of reasoning. The setting is a discussion, taking place in Athens, on the issue of going to war with Thebes. It is apparently accepted that a war between Thebes and Phocis is or was a bad thing, perhaps from the objectivity lent by non-involvement or perhaps as a lesson of history.

For example, let A be 'bad', B 'to make war on neighbors', C 'Athens against Thebes', and D 'Thebes against Phocis'. Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad. Evidence of this can be drawn from similar examples, for example, that war by Thebes against Phocis is bad. Then since war against neighbors is bad, and war against Thebes is war against neighbors, it is evident that war against Thebes is bad.(Aristotle, "Prior Analytics", 2.24, with minor alterations).

Aristotle's sample of argument from analogy may be analyzed in the following way:

First, a Rule is induced from the consideration of a similar Case and a relevant Fact:

• Case: D => B, Thebes vs Phocis is war against neighbors.
• Fact: D => A, Thebes vs Phocis is bad.
• Rule: B => A, War against neighbors is bad.

Next, the Fact to be proved is deduced from the application of the previously induced Rule to the present Case:

• Case: C => B, Athens vs Thebes is war against neighbors.
• Rule: B => A, War against neighbors is bad.
• Fact: C => A, Athens vs Thebes is bad.

In practice, of course, it would probably take a mass of comparable cases to establish a rule. As far as the logical structure goes, however, this quantitative confirmation only amounts to 'gilding the lily'. Perfectly valid rules can be guessed on the first try, abstracted from a single experience or adopted vicariously with no personal experience. Numerical factors only modify the degree of confidence and the strength of habit that govern the application of previously learned rules.

Figure 3 gives a graphical illustration of Aristotle's example of 'Example', that is, the form of reasoning that proceeds by Analogy or according to a Paradigm.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` A ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `/*\` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` / * \ ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `/` * `\` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` / ` * ` \ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `/` ` * ` `\` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` / ` ` * ` ` \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/` `R u l e` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` * ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` * ` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` ` ` ` * ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` ` ` ` * ` ` ` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `F a c t` ` ` ` o ` ` ` `F a c t` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `/` ` ` ` ` * B * ` ` ` ` `\` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` / ` ` ` ` * ` ` ` * ` ` ` ` \ ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `/` ` ` ` * ` ` ` ` ` * ` ` ` `\` ` ` ` ` ` ` |
| ` ` ` ` ` ` / ` ` ` * ` ` ` ` ` ` ` * ` ` ` \ ` ` ` ` ` ` |
| ` ` ` ` ` `/` `C a s e` ` ` ` ` ` C a s e ` `\` ` ` ` ` ` |
| ` ` ` ` ` / ` ` * ` ` ` ` ` ` ` ` ` ` ` * ` ` \ ` ` ` ` ` |
| ` ` ` ` `/` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` * ` `\` ` ` ` ` |
| ` ` ` ` / ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * ` \ ` ` ` ` |
| ` ` ` `/` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * `\` ` ` ` |
| ` ` ` / * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * \ ` ` ` |
| ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` |
| ` ` C ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` D ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A `=` Atrocious, Adverse to All, A bad thing` ` ` ` ` ` ` |
| B `=` Belligerent Battle Between Brethren ` ` ` ` ` ` ` ` |
| C `=` Contest of Athens against Thebes` ` ` ` ` ` ` ` ` ` |
| D `=` Debacle of Thebes against Phocis` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A is a major term ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| B is a middle term` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| C is a minor term ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| D is a minor term, similar to C ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 3.  Aristotle's 'War Against Neighbors' Example

In this analysis of reasoning by Analogy, it is a complex or a mixed form of inference that can be seen as taking place in two steps:

• The first step is an Induction that abstracts a Rule from a Case and a Fact.

• The final step is a Deduction that applies this Rule to a Case to arrive at a Fact.

As we see, Aristotle analyzed analogical reasoning into a phase of inductive reasoning followed by a phase of deductive reasoning. Peirce would pick up the story at this juncture and eventually parse analogy in a couple of different ways, both of them involving all three types of inference: abductive, deductive, and inductive.

Examples of inquiry, that illustrate the full cycle of its abductive, deductive, and inductive phases, and yet are both concrete and simple enough to be suitable for a first (or zeroth) exposition, are somewhat rare in Peirce's writings, and so let us draw one from the work of fellow pragmatician John Dewey, analyzing it according to the model of zeroth order inquiry that we developed above.

A man is walking on a warm day. The sky was clear the last time he observed it; but presently he notes, while occupied primarily with other things, that the air is cooler. It occurs to him that it is probably going to rain; looking up, he sees a dark cloud between him and the sun, and he then quickens his steps. What, if anything, in such a situation can be called thought? Neither the act of walking nor the noting of the cold is a thought. Walking is one direction of activity; looking and noting are other modes of activity. The likelihood that it will rain is, however, something suggested. The pedestrian feels the cold; he thinks of clouds and a coming shower. (John Dewey, How We Think, pp. 6-7).

Let's first give Dewey's elegant example of inquiry in everyday life the quick once over, hitting just the high points of its analysis into Peirce's three kinds of reasoning.

In Dewey's 'Rainy Day' or 'Sign of Rain' story, we find our peripatetic hero presented with a surprising Fact:

• Fact: C => A, In the Current situation the Air is cool.

Responding to an intellectual reflex of puzzlement about the situation, his resource of common knowledge about the world is impelled to seize on an approximate Rule:

• Rule: B => A, Just Before it rains, the Air is cool.

This Rule can be recognized as having a potential relevance to the situation because it matches the surprising Fact, C => A, in its consequential feature A.

All of this suggests that the present Case may be one in which it is just about to rain:

• Case: C => B, The Current situation is just Before it rains.

The whole mental performance, however automatic and semi-conscious it may be, that leads up from a problematic Fact and a previously settled knowledge base of Rules to the plausible suggestion of a Case description, is what we are calling an abductive inference.

The next phase of inquiry uses deductive inference to expand the implied consequences of the abductive hypothesis, with the aim of testing its truth. For this purpose, the inquirer needs to think of other things that would follow from the consequence of his precipitate explanation. Thus, he now reflects on the Case just assumed:

• Case: C => B, The Current situation is just Before it rains.

He looks up to scan the sky, perhaps in a random search for further information, but since the sky is a logical place to look for details of an imminent rainstorm, symbolized in our story by the letter B, we may safely suppose that our reasoner has already detached the consequence of the abduced Case, C => B, and has begun to expand on its further implications. So let us imagine that our up-looker has a more deliberate purpose in mind, and that his search for additional data is driven by the new-found, determinate Rule:

• Rule: B => D, Just Before it rains, Dark clouds appear.

Contemplating the assumed Case in combination with this new Rule leads him by an immediate deduction to predict an additional Fact:

• Fact: C => D, In the Current situation Dark clouds appear.

The reconstructed picture of reasoning assembled in this second phase of inquiry is true to the pattern of deductive inference.

Whatever the case, our subject observes a Dark cloud, just as he would expect on the basis of the new hypothesis. The explanation of imminent rain removes the discrepancy between observations and expectations and thereby reduces the shock of surprise that made this process of inquiry necessary.

Figure 4 gives a graphical illustration of Dewey's example of inquiry, isolating for the purposes of the present analysis the first two steps in the more extended proceedings that go to make up the whole inquiry.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` A ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` D ` ` |
| ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` |
| ` ` ` \ * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * / ` ` ` |
| ` ` ` `\` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * `/` ` ` ` |
| ` ` ` ` \ ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * ` / ` ` ` ` |
| ` ` ` ` `\` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` * ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` * ` ` ` ` ` ` ` ` ` ` ` * ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` `R u l e` ` ` ` ` ` `R u l e` `/` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` * ` ` ` ` ` ` ` * ` ` ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` * ` ` ` ` ` * ` ` ` `/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` * ` ` ` * ` ` ` ` / ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` * B * ` ` ` ` `/` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `F a c t` ` ` ` o ` ` ` `F a c t` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` * ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` * ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` ` * ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` ` * ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `C a s e` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ ` ` * ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\` ` * ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` * ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` * `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ * / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\*/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` C ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A `=` the Air is cool ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| B `=` just Before it rains` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| C `=` the Current situation ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| D `=` a Dark cloud appears` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A is a major term ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| B is a middle term` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| C is a minor term ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| D is a major term, associated with A` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 4.  Dewey's 'Rainy Day' Inquiry

In this analysis of the first steps of Inquiry, we have a complex or a mixed form of inference that can be seen as taking place in two steps:

• The first step is an Abduction that abstracts a Case from the consideration of a Fact and a Rule.

• The final step is a Deduction that admits this Case to another Rule and so arrives at a novel Fact.

This is nowhere near a complete analysis of the Rainy Day inquiry, even insofar as it might be carried out within the constraints of the syllogistic framework, and it covers only the first two steps of the relevant inquiry process, but maybe it will do for a start.

One other thing needs to be noticed here, the formal duality between this expansion phase of inquiry and the argument from analogy. This can be seen most clearly in the propositional lattice diagrams shown in Figures 3 and 4, where analogy exhibits a rough "A" shape and the first two steps of inquiry exhibit a rough "V" shape, respectively. Since we find ourselves repeatedly referring to this expansion phase of inquiry as a unit, let's give it a name that suggests its duality with analogy — 'catalogy' will do for the moment. This usage is apt enough if one thinks of a catalogue entry for an item as a text that lists its salient features. Notice that analogy has to do with the examples of a given quality, while catalogy has to do with the qualities of a given example. Peirce noted similar forms of duality in many of his early writings, leading to the consummate treatment in his 1867 paper "On a New List of Categories" (CP 1.545-559, CE 2, 49-59).

In order to comprehend the bearing of inductive reasoning on the closing phases of inquiry there are a couple of observations that we need to make:

• First, we need to recognize that smaller inquiries are typically woven into larger inquiries, whether we view the whole pattern of inquiry as carried on by a single agent or by a complex community.

• Further, we need to consider the different ways in which the particular instances of inquiry can be related to ongoing inquiries at larger scales. Three modes of inductive interaction between the micro-inquiries and the macro-inquiries that are salient here can be described under the headings of the 'Learning', the 'Transfer', and the 'Testing' of rules.

Throughout inquiry the reasoner makes use of rules that have to be transported across intervals of experience, from the masses of experience where they are learned to the moments of experience where they are applied. Inductive reasoning is involved in the learning and the transfer of these rules, both in accumulating a knowledge base and in carrying it through the times between acquisition and application.

• Learning. The principal way that induction contributes to an ongoing inquiry is through the learning of rules, that is, by creating each of the rules that goes into the knowledge base, or ever gets used along the way.

• Transfer. The continuing way that induction contributes to an ongoing inquiry is through the exploit of analogy, a two-step combination of induction and deduction that serves to transfer rules from one context to another.

• Testing. Finally, every inquiry that makes use of a knowledge base constitutes a 'field test' of its accumulated contents. If the knowledge base fails to serve any live inquiry in a satisfactory manner, then there is a prima facie reason to reconsider and possibly to amend some of its rules.

Let's now consider how these principles of learning, transfer, and testing apply to John Dewey's 'Sign of Rain' example.

Learning

Rules in a knowledge base, as far as their effective content goes, can be obtained by any mode of inference.

For example, a rule like:

• Rule: B => A, Just Before it rains, the Air is cool,

is usually induced from a consideration of many past events, in a manner that can be rationally reconstructed as follows:

• Case: C => B, In Certain events, it is just Before it rains,
• Fact: C => A, In Certain events, the Air is cool,

• Rule: B => A, Just Before it rains, the Air is cool.

However, the very same proposition could also be abduced as an explanation of a singular occurrence or deduced as a conclusion of a presumptive theory.

Transfer

What is it that gives a distinctively inductive character to the acquisition of a knowledge base? It is evidently the 'analogy of experience' that underlies its useful application. Whenever we find ourselves prefacing an argument with the phrase 'If past experience is any guide …' then we can be sure that this principle has come into play. We are invoking an analogy between past experience, considered as a totality, and present experience, considered as a point of application. What we mean in practice is this: 'If past experience is a fair sample of possible experience, then the knowledge gained in it applies to present experience'. This is the mechanism that allows a knowledge base to be carried across gulfs of experience that are indifferent to the effective contents of its rules.

Here are the details of how this notion of transfer works out in the case of the 'Sign of Rain' example:

Let K(pres) be a portion of the reasoner's knowledge base that is logically equivalent to the conjunction of two rules, as follows:

• K(pres) = (B => A) and (B => D).

K(pres) is the present knowledge base, expressed in the form of a logical constraint on the present universe of discourse.

It is convenient to have the option of expressing all logical statements in terms of their logical models, that is, in terms of the primitive circumstances or the elements of experience over which they hold true.

• Let E(past) be the chosen set of experiences, or the circumstances that we have in mind when we refer to 'past experience'.

• Let E(poss) be the collective set of experiences, or the projective total of possible circumstances.

• Let E(pres) be the present experience, or the circumstances that are present to the reasoner at the current moment.

If we think of the knowledge base K(pres) as referring to the 'regime of experience' over which it is valid, then all of these sets of models can be compared by the simple relations of set inclusion or logical implication.

Figure 5 schematizes this way of viewing the 'analogy of experience'.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `K(pres)` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `/|\` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` / | \ ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `/` | `\` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` / ` | ` \ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `/` Rule` `\` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` / ` ` | ` ` \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/` ` ` | ` ` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` | ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` `E(poss)` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `Fact / ` ` ` ` o ` ` ` ` \ Fact` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` ` ` * ` * ` ` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` ` * ` ` ` * ` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `/` ` ` * ` ` ` ` ` * ` ` `\` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` / ` ` * ` ` ` ` ` ` ` * ` ` \ ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `/` ` * ` ` ` ` ` ` ` ` ` * ` `\` ` ` ` ` ` ` |
| ` ` ` ` ` ` / ` * `Case ` ` ` ` ` Case `* ` \ ` ` ` ` ` ` |
| ` ` ` ` ` `/` * ` ` ` ` ` ` ` ` ` ` ` ` ` * `\` ` ` ` ` ` |
| ` ` ` ` ` / * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * \ ` ` ` ` ` |
| ` ` ` ` `/* ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` *\` ` ` ` ` |
| ` ` ` ` o<<<---------------<<<---------------<<<o ` ` ` ` |
| ` ` `E(past)` ` ` ` Analogy Morphism` ` ` ` `E(pres)` ` ` |
| ` `More Known ` ` ` ` ` ` ` ` ` ` ` ` ` ` `Less Known ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 5.  Analogy of Experience

In these terms, the 'analogy of experience' proceeds by inducing a Rule about the validity of a current knowledge base and then deducing a Fact, its applicability to a current experience, as in the following sequence:

Inductive Phase:

• Given Case: E(past) => E(poss), Chosen events fairly sample Collective events.
• Given Fact: E(past) => K(pres), Chosen events support the Knowledge regime.

• Induce Rule: E(poss) => K(pres), Collective events support the Knowledge regime.

Deductive Phase:

• Given Case: E(pres) => E(poss), Current events fairly sample Collective events.
• Given Rule: E(poss) => K(pres), Collective events support the Knowledge regime.

• Deduce Fact: E(pres) => K(pres), Current events support the Knowledge regime.

Testing

If the observer looks up and does not see dark clouds, or if he runs for shelter but it does not rain, then there is fresh occasion to question the utility or the validity of his knowledge base. But we must leave our foulweather friend for now and defer the logical analysis of this testing phase to another occasion.

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).

(Text in preparation, 02 January 2006)

Charles Peirce - Normative sciences

(Text in preparation, 04 January 2006)

Charles Peirce - Parallels with Leibniz

Peirce was aware that the breadth and depth of his ideas resembled those of the 17th century German polymath Gottfried Wilhelm Leibniz; (see Fisch 1986: 249-60). But the parallels between Peirce and Leibniz were even more striking than he knew. One should keep in mind that the scope of Leibniz's achievement was not as well appreciated in Peirce's day as in ours. Both men were mathematicians, logicians, historians, philosophers of language and mind, and metaphysicians. Neither was by any means primarily educated in philosophy (Leibniz's degree was in law). Both were passionate about natural science and contributed thereto, dabbled in inventions, and worked on engineering projects. Both were fascinated by semiotics and mathematical notation, and the interplay between philosophy and mathematics. Both were surprisingly friendly to some parts of scholastic metaphysics as well as logic; e.g., Peirce frequently invoked the Scotistic notion of haecceity. Both published few books, many articles, and died leaving a vast amount in manuscript. The ideas of both men underwent oversimplification in the hands of others, and were little appreciated for some time after their deaths. The critical editions of the works of both men are far from complete. The secondary literature on both men mostly dates from the end of WWII. Leibniz differs from Peirce in his greater range, vast correspondence, freedom from financial difficulties, and his passionate Christianity.

Charles Peirce - Bibliography

Charles Peirce - Primary literature

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.
• 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.
• SS m = Semiotic and Significs: the Correspondence between C.S. Peirce and Victoria Lady Welby, page m.

Ketner, K.L., et al, 1986. A Comprehensive Bibliography of the Works of C. S. Peirce. Bowling Green OH: Philosophy Documentation Center.

• Peirce, C.S., 1885, "On the Algebra of Logic: A Contribution to the Philosophy of Notation", American Journal of Mathematics, 7.2, 180-202. Also CP 3.359-403 and CE 5, 162-190.
• Peirce, C.S., 1931-1935, 1958, 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. Thoemmes reprint. Vol. 1 online. All 8 vols. on CD-ROM.
• Peirce, C.S., 1975-. C.S. Peirce: Contributions to the Nation, 4 Vols. Compiled and annotated by K.L. Ketner and J.E. Cook. Texas Tech Press.
• Peirce, C.S., 1982-. Writings of Charles S. Peirce: A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN. This edition in progress meets the standards of the Modern Language Association for critical editions. Available on CD-ROM.
• Peirce, C.S., ed., 1983 (1883). Studies in Logic by Members of the Johns Hopkins University (Foundations of Semiotics). Introduction by M.H. Fisch, Preface by A. Eschbach. John Benjamins.
• Peirce, C.S., 1992. Reasoning and the Logic of Things. Edited, with Introduction, commentary, and notes, by K. L. Ketner and H. Putnam. Harvard Uni. Press. Being the text of 1898 lectures William James invited Peirce to give in Cambridge MA.
• Eisele, Carolyn, ed., 1976. The New Elements of Mathematics, 4 vols. The Hague: Mouton.
• Eisele, Carolyn, ed., 1985. Historical Perspectives on Peirce's Logic of Science: A History of Science, 2 vols. Berlin: DeGruyter.
• Hardwick, C. S. ed., 2001 (1977). Semiotic and Significs: the Correspondence between C. S. Peirce and Victoria Lady Welby. Texas Tech Press.
• Houser, N. et al eds., 1992, 1998. The Essential Peirce, 2 vols. Indiana Uni. Press. Intro to Vol. 1 Intro to Vol. 2.
• Peirce writings available on-line.

Charles Peirce - Secondary literature

For a bibliography of the secondary literature prior to 1964, see Moore and Robin (1964: 486-514).

• Anellis, Irving, 1995, "Peirce Rustled, Russell Pierced: How Peirce and 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. Online
• Awbrey, Jon, and Awbrey, Susan, 1995, "Interpretation as Action: The Risk of Inquiry", Inquiry: Critical Thinking Across the Disciplines, vol. 15, pp. 40-52. Online
• Brady, Geraldine, 2000. From Peirce to Skolem. North Holland.
• Brent, Joseph, 1998. Charles Sanders Peirce: A Life. Revised and enlarged edition. Indiana Uni. Press.
• Brunning, J., and Forster, P., eds., 1997. The Rule of Reason. Uni. of Toronto Press.
• Deledalle, Gérard, 2000. C.S. Peirce's Philosophy of Signs. Indiana Uni. Press.
• Eisele, Carolyn, 1979. Studies in the Scientific and Mathematical Philosophy of C.S. Peirce. Edited by Richard Milton Martin. The Hague: Mouton.
• Esposito, Joseph, 1980. Evolutionary Metaphysics: The Development of Peirce's Theory of Categories. Ohio Uni. Press.
• Fisch, Max, 1986. Peirce, Semeiotic, and Pragmatism. Indiana Uni. Press.
• Hookway, Christopher, 1985. Peirce. London: Routledge and Kegan Paul.
• Houser, Nathan, Roberts, D.D., and Van Evra, J., eds., 1997. Studies in the Logic of C.S. Peirce. Indiana Uni. Press.
• Lane, Robert, 2004. "On Peirce's Early Realism", Transactions of the C.S. Peirce Society 40: 575-605.
• Liszka, J.J., 1996. A General Introduction to the Semeiotic of C.S. Peirce. Indiana Uni. Press.
• Menand, Louis, 2001. The Metaphysical Club: A Story of Ideas in America. Farrar, Straus and Giroux. ISBN 0374199639, (paperback ISBN 0374528497). Biography of Pierce, Oliver Wendell Holmes, Jr., William James, and John Dewey.
• Misak, Cheryl, ed., 2004. Cambridge Companion to C. S. Peirce. Cambridge Uni. Press.
• Moore, Edward, and Robin, Richard, 1964. Studies in the Philosophy of C. S. Peirce, Second Series. Uni. of Massachusetts Press.
• Murphey, Murray, 1993 (1961). The Development of Peirce's Thought. Indianapolis: Hackett.
• Walker Percy, 2000. Signposts in a Strange Land. Samway, P, ed. Saint Martin's Press: 271-91.
• Hilary Putnam, 1982. 'Peirce the Logician'. Historia Mathematica 9: 290-301. Reprinted in Putnam, H., 1990. Realism with a Human Face. Harvard Uni. Press: 252-60. Online fragment
• Robin, Richard, 1967. Annotated Catalog of the Papers of C. S. Peirce. University of Massachussetts Press.
• Josiah Royce and Kernan, W. F., 1916, "Charles Sanders Peirce,"Journal of Philosophy, Psychology, and Scientific Method 13: 701-09. Online.
• Wiener, Philip, and Young, Frederick, eds., 1952. Studies in the Philosophy of C.S. Peirce. Harvard Uni. Press.
• Writings about Peirce available on-line.

• Chiasson, Phyllis, 2001. Peirce's Pragmatism. The Design for Thinking. Amsterdam: Rodopi.
• Debrock, Guy, 1992. "Peirce, a Philosopher for the 21st Century. Introduction." Transactions of the C. S. Peirce Society 28: 1-18.
• Hintikka Jakko, 1980, "Peirce's first real discovery," The Monist 63: 257-304.
• Kirkham, Richard, 1995. Theories of Truth. MIT Press.
• Parker, Kelly, A., 1998. The Continuity of Peirce's Thought. Vanderbilt Uni. Press.
• Skagestad, Peter, 1981. The Road of Inquiry, Charles Peirce's Pragmatic Realism. Columbia Uni. Press.
• Walther, Elizabeth, 1989. C.S. Peirce: Leben und Werk. Baden-Baden Germany: Agis-Verlag.
• Wennerberg, Hjalmar, 1962. The Pragmatism of C.S. Peirce: An Analytical Study. Lund, Sweden: Gleerup.

See also

• Abstraction

• Hypostatic abstraction
• Prescisive abstraction

• Inquiry
• Logic

• Logical graph
• Logical nand — The Sheffer stroke Peirce anticipated in 1880.
• Peirce's law — An axiom in Peirce (1885).
• Laws of form — System of logical graphs given in the book Laws of Form (George Spencer Brown, 1969) that revived and extended certain aspects of Peirce's logical graphs.
• Ernst Schröder — Popularized Peirce's work on relations and quantification.

• Pragmatism

• Pragmaticism
• Pragmatic maxim — Seven ways of looking at it.

• Semiotics

• Sign relation — The basic construct of Peircean semiotics.

Adapted from the Wikipedia article "Charles Peirce", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki