Charles Sanders Peirce (1839–1914) was the founder of Americanpragmatism (after about 1905 called by Peirce“pragmaticism” in order to differentiate his views fromthose of William James, John Dewey, and others, which were beinglabelled “pragmatism”), a theorist of logic, language,communication, and the general theory of signs (which was often calledby Peirce “semeiotic”), an extraordinarily prolificlogician (mathematical and general), and a developer of anevolutionary, psycho-physically monistic metaphysicalsystem. Practicing geodesy and chemistry in order to earn a living, henevertheless considered scientific philosophy, and especially logic,to be his true calling, his real vocation. In the course of hispolymathic researches, he wrote voluminously on an exceedingly widerange of topics, ranging from mathematics, mathematical logic,physics, geodesy, spectroscopy, and astronomy, on the one hand (thatof mathematics and the physical sciences), to psychology,anthropology, history, and economics, on the other (that of thehumanities and the social sciences).

• 1. Brief Biography
• 2. Difficulty of Access to Peirce’s Writings
• 3. Deduction, Induction, and Abduction
• 4. Pragmatism, Pragmaticism, and the Scientific Method
• 5. Anti-determinism, Tychism, and Evolutionism
• 6. Synechism, the Continuum, Infinites, and Infinitesimals
• 7. Probability, Verisimilitude, and Plausibility
• 8. Psycho-physical Monism and Anti-nominalism
• 9. Triadism and the Universal Categories
• 10. Mind and Semeiotic
• 11. Semeiotic and Logic
• 12. The Classification of the Sciences
• 13. Logic
• 14. Peirce’s Reduction Thesis
• 15. Contemporary Practical Application of Peirce’s Ideas
• 16. Significant Students of Peirce
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Brief Biography

Charles Sanders Peirce was born on September 10, 1839 in Cambridge,Massachusetts, and he died on April 19, 1914 in Milford,Pennsylvania. His writings extend from about 1857 until near hisdeath, a period of approximately 57 years. His published works run toabout 12,000 printed pages and his known unpublished manuscripts runto about 80,000 handwritten pages. The topics on which he wrote havean immense range, from mathematics and the physical sciences at oneextreme, to economics, psychology, and other social sciences at the otherextreme.

Peirce’s father Benjamin Peirce was Professor of Mathematics atHarvard University and was one of the founders of, and for a while a director of, the U. S. Coast andGeodetic Survey as well as one of the founders of the SmithsonianInstitution. The department of mathematics at Harvard was essentiallybuilt by Benjamin. From his father, Charles Sanders Peirce receivedmost of the substance of his early education as well as a good deal ofintellectual encouragement and stimulation. Benjamin’s didactictechnique mostly took the form of setting interesting problems for hisson and checking Charles’s solutions to them. In this challenginginstructional atmosphere Charles acquired his lifelong habit ofthinking through philosophical and scientific problems entirely on hisown. To this habit, perhaps, is to be attributed Charles Peirce’sconsiderable originality.

Peirce graduated from Harvard in 1859 and received the bachelor ofscience degree in chemistry in 1863, graduatingsumma cumlaude. Except for his remarkable marks in chemistry Peirce was apoor student, typically in the bottom third of his class. Obviously,the standard curriculum bored him, so that he mostly avoided doingseriously its required work. For thirty-two years, from 1859 until thelast day of 1891, he was employed by the U. S. Coast and GeodeticSurvey, mainly surveying and carrying out geodeticinvestigations. Some of this work Peirce undertook simply to financehis diurnal existence (and that of his first wife Melusina (Zina)Fay), while he devoted the main force of his thinking to abstractlogic. Nevertheless, the geodetic tasks involved making carefulmeasurements of the intensity of the earth’s gravitational field bymeans of using swinging pendulums. The pendulums that Peirce used wereoften of his own design. For over thirty years, then, Peirce wasinvolved in practical and theoretical problems associated with makingvery accurate scientific measurements. This practical involvement inphysical science was crucial in his ultimately coming to rejectscientific determinism, as we shall see.

From 1879 until 1884, Peirce maintained a second job teachinglogic in the Department of Mathematics at Johns HopkinsUniversity. During that period the Department of Mathematics washeaded by the famous mathematician J. J. Sylvester, whom Peirce hadmet earlier through his father Benjamin. This teaching period also wascharacterized by Peirce’s having several students who made names forthemselves in their own right. Among these were Oscar Howard Mitchell,Allan Marquand, Benjamin Ives Gilman, Joseph Jastrow, Fabian Franklin,Christine Ladd (later, after having married Fabian Franklin, ChristineLadd-Franklin), Thorstein Veblen, and John Dewey. Brief commentarywill be offered at the end of this essay on three of these figures:John Dewey, Oscar Howard Mitchell, and Christine Ladd. It issometimes said that William James was also one of Peirce’s students,but this claim is erroneous: it conflates the fact of James’s being anold and a close friend of Peirce, as well as being a fellow-memberwith Peirce in the so-called “Metaphysical Club” inCambridge, Massachusetts, with the non-fact of James’s being a studentof Peirce at Johns Hopkins University along with John Dewey andothers.

Peirce’s teaching job at Johns Hopkins was suddenly terminated forreasons that are apparently connected with the fact that Peirce’ssecond wife (Juliette Annette Froissy, a.k.a. Juliette AnnettePourtalai) was Romani, moreover a Roma with whom Peirce had more orless openly cohabited before marriage and before his divorce from hisfirst wife Zina. (In fact Peirce obtained his divorce from Zina onlytwo days before marrying Juliette.) The Johns Hopkins position wasPeirce’s only academic employment, and after losing it Peirce workedthereafter only for the U. S. Coast and Geodetic Survey (andconstructing entries for theCentury Dictionary) and writingbook reviews for theNation. The government employment cameto an end the last day of 1891, ultimately because of fundingobjections to pure research (and perhaps also to Peirce’s extravagantspending and to his procrastination in finishing his required reports)that were generated in an ever-practical-minded Congress. Thereafter,Peirce often lived on the edge of penury, eking out a living doingintellectual odd-jobs (such as translating or writing occasionalpieces) and carrying out consulting work (mainly in chemicalengineering and analysis). For the remainder of his life, except formoney inherited from his mother and aunt, Peirce was often in direfinancial straits; sometimes he managed to survive only because of theovert or covert charity of relatives or friends, for example that ofhis old friend William James.

In his youth Peirce was amazingly precocious, and he began to studylogic seriously at an extraordinarily early age. According to notedPeirce scholar Max Fisch in his“Introduction” to Volume 1ofThe Writings of Charles S. Peirce: A ChronologicalEdition, p. xviii, Peirce’s introduction to and first immersionin the study of logic came in 1851 within a week or two of his turning12 years of age. Remembering the occasion in 1910, in his “Noteon the Doctrine of Chances,” inCollected Papers of CharlesS. Peirce, Volume II, Section 408 (hereinafter suchCollectedPapers references will be cited asCP, 2.408), Peircehimself remembered the crucial event as having occurred in 1852, whenhe was 13 years old. Regardless of his exact age, at the time of theevent Charles encountered and then over a period of at most a few daysstudied and absorbed a standard textbook of the time on logic byBishop Richard Whately. Having become fascinated by logic, he began tothink of all issues as problems in logic. During his freshman year atcollege (Harvard), in 1855, when he was 16 years old, he and a friendbegan private study of philosophy in general, starting with Schiller’sLetters on the Aesthetic Education of Man and continuing withKant’sCritique of Pure Reason. Schiller’s distinction amongthe three basic human drives ofStofftrieb,Formtrieb, andSpieltrieb Peirce never forgot orrenounced, and it became the basis for Peirce’s distinction betweenthe man of practical affairs, the man of scientific activity, and theman of aesthetic practice. By Kant Peirce was initially more or lessrepelled. After three years of intense study of Kant, Peirce concludedthat Kant’s system was vitiated by what he called its “puerilelogic,” and about the age of 19 he formed the fixed intention ofdevoting his life to the study of and to research in logic. It was,however, impossible at that time, as indeed Peirce’s father Benjamininformed him, to earn a living as a research logician; and Peircedescribed himself at the time of his graduation from Harvard in 1859,just short of his 20th birthday, as wondering “what I would doin life.” Within two years, however, he had more or lessresolved his problem. During those two years he had worked as an Aidon the Coast Survey, in Maine and Louisiana, then had returned toCambridge and had studied natural history and natural philosophy atHarvard. He said of himself that in 1861 he “No longer wonderedwhat I would do in life but defined my object.” It is evidentthat his adoption of the profession of chemistry and his practice ofgeodesy allowed Charles both to support himself (and before long alsohis first wife Zina) and to continue to engage in researches onlogic. From the early 1860s until his death in 1914 his output inlogic was voluminous and varied. One of his logical systems becamethe basis for Ernst Schroeder’s great three-volume treatise on logic,theVorlesungen ueber die Algebra der Logik.

Peirce, then, had early and deep disagreements with Kant’s positionabout logic, and he never altered his view that Kant’s view of logicwas superficial: “… he [i.e. Kant] never touches thislast doctrine [i.e. logic] without betraying marks of hasty,superficial study” (Collected Papers of Charles SandersPeirce, Volume 2, Section 3; hereafter suchCollectedPapers references will be cited as asCP, 2.3). Evenworse, Peirce held, was theLogik of Hegel: Kant’s fault“… is a hundredfold more true of Hegel’sLogik… . That work cannot justly be regarded as anything morethan a sketch” (CP, 2.32).

Nevertheless, Peirce continued to respect and read the firstCritique throughout his life. For a fuller discussionof Peirce’s own views about how his work related to that ofKant, Hegel, and Schelling, see the supplementary document:

Peirce’s View of the Relationship Between His Own Work and German Idealism

2. Difficulty of Access to Peirce’s Writings

Peirce’s extensive publications are scattered among variouspublication media, and have been difficult to collect. Shortly afterhis death in 1914, his widow Juliette sold his unpublished manuscriptsto the Department of Philosophy at Harvard University. Initially theywere under the care of Josiah Royce, but after Royce’s death in 1916,and especially after the end of the First World War, the papers werepoorly cared for. Many of them were misplaced, lost, given away,scrambled, and the like. Carolyn Eisele, one of several genuine heroesin the great effort to locate and assemble Peirce’s writings,discovered a lost trunk full of Peirce’s papers and manuscripts onlyin the mid-1950s; the trunk had been secreted, apparently for decades,in an unlit, obscure part of the basement in Harvard’s WidenerLibrary.

In the 1930s volumes ofThe Collected Papers of Charles SandersPeirce began to appear, with Charles Hartshorne, Paul Weiss, andArthur Burks as their editors. For almost three decades these volumes,and various collections of entries culled from them were the onlygenerally available source for Peirce’s thoughts. Unfortunately, manyof the entries in theCollected Papers are not integralpieces of Peirce’s own design, but rather stretches of writing thatwere cobbled together by the editors at their own discretion(sometimes one might almost say “whim”) from different Peirceansources. Often a single entry will consist of patches of writing fromvery different periods of Peirce’s intellectual life, and thesepatches might even be in tension or outright contradiction with eachother. Such entries in theCollected Papers make verydifficult reading if one tries to regard them as consistent, sustainedpassages of argument. They also tend to give the reader a falsepicture of Peirce as unsystematic, desultory, and unable to complete atrain of thought. In general, even though Peirce is often obscure and even at his best is seldomeasy to read, theCollected Papers make Peirce’s thinking look much more obscure than it really is.

The only sensible and intelligent way to publish the works of someonelike Peirce, who wrote voluminously and over such a long period oftime (57 years), is to arrange the publication chronologically and toemploy extremely careful editing. In such a fashion, the entire set ofPeircean works can be presented, as Peirce conceived them and in theirnatural temporal setting and order. Finally, beginning in 1976 withthe organizational conception of Max H. Fisch and the help of EdwardMoore, the Peirce Edition Project (PEP) was created at IndianaUniversity-Purdue University at Indianapolis (UIPUI). Then, under thePEP, in the 1980s, there began to appear a meticulously editedchronological edition of carefully selected works of Peirce: this istheWritings of Charles S. Peirce: a Chronological Edition,edited by The Peirce Edition Project of the Indiana University-PurdueUniversity at Indianapolis. Although theChronologicalEdition has been fettered from time to time by lack of properfunding, theChronological Edition has succeeded in coveringextremely well in its first seven published volumes the major writingsfrom 1857 to 1892. (At the present time, October 2014, Volume 7 isstill awaiting publication, even though Volume 8, covering writingsfrom 1890 to 1892 already has been published. Volume 7 is to be anedition of Peirce’s definitions for theCenturyDictionary. It is to be edited by the Peirce Edition Project inconjunction with the University of Quebec at Montreal (UQAM)), underthe supervision of Professor Francois Latraverse.) The impressiveachievement of the PEP is finally making it possible to assess thereal Peirce, instead of the chopped-up and then re-pasted-togetherpicture of Peirce previously available. In particular theChronological Edition has made it possible to see thedevelopment of Peirce’s thinking from its earliest stages to its laterdevelopments. Questions long vexed in Peirce scholarship are finallybeginning to be debated usefully by Peirce scholars: whether there isgenuine systematic unity in Peirce’s thought, whether his ideaschanged or remained the same over time, in what particulars histhought did change and why, when exactly certain notions were firstconceived by Peirce, whether there were definite “periods”in Peirce’s intellectual development, and what exactly Peirce meant bysome of his more obscure notions such as his universal categories (onwhich see below). Continued funding for the Peirce Edition Project isobviously a crucial priority in the ongoing effort to bring to publiclight the thoughts of this extremely important Americanphilosopher.

In addition to theChronological Edition of the PeirceEdition Project, other venues for editing and publishing Peirce’s workare regularly found, and there are several excellent editions ofparticular lectures, lecture-series, chains of correspondence, and thelike. Just four such editions will be mentioned here. First, there isthe edition of Peirce’s Cambridge Conferences Lectures of 1898, editedby Kenneth Laine Ketner and with an introduction on the consequencesof mathematics by Kenneth Laine Ketner and Hilary Putnam and commentson the lectures by Hilary Putnam, entitledReasoning and the Logicof Things. Second, there is the edition of Peirce’s HarvardLectures on Pragmatism of 1903, edited by Patricia Ann Turrisi,entitledPragmatism as a Principle and Method of RightThinking. Third, there is the four-volume edition of Peirce’smathematical writings edited by Carolyn Eisele, entitledThe NewElements of Mathematics by Charles S. Peirce. Fourth, there isthe two-part edition of Peirce’s writings on the history and logic ofscience edited by Carolyn Eisele, entitledHistorical Perspectiveson Peirce’s Logic of Science: A History of Science.

3. Deduction, Induction, and Abduction

Prior to about 1865, thinkers on logic commonly had divided argumentsinto two subclasses: the class of deductive arguments (a.k.a.necessary inferences) and the class of inductive arguments (a.k.a.probable inferences). About this time, Peirce began to hold that therewere two utterly distinct classes of probable inferences, which hereferred to as inductive inferences and abductive inferences (which healso called hypotheses and retroductive inferences). Peirce reachedthis conclusion by entertaining what would happen if one were tointerchange propositions in the syllogism AAA-1 (Barbara): AllMs arePs; allSs areMs;therefore, allSs arePs. This valid syllogismPeirce accepted as representative of deduction. But he also seemedtypically to regard it in connection with a problem of drawingconclusions on the basis of taking samples. For let us regard beinganM as being a member of a population of some sort, saybeing a ball of the population of balls in some particular urn. Let usregardP as being some property a member of this populationcan have, say being red. And, finally, let us regard beinganS as being a member of a random sample taken from thispopulation. Then our syllogism in Barbara becomes: All balls in thisurn are red; all balls in this particular random sample are taken fromthis urn; therefore, all balls in this particular random sample arered. Peirce regarded the major premise here as being the Rule, theminor premise as being the particular Case, and the conclusion asbeing the Result of the argument. The argument is a piece ofdeduction. In this example the argument is also an argument frompopulation to random sample that is also a necessary inference.

But now let us see what happens if we form a new argument byinterchanging the conclusion (the Result) with the major premise (theRule). The resultant argument becomes: AllSs arePs(Result); allSs areMs (Case); therefore, allMs arePs (Rule). This is the invalid syllogismAAA-3. But let us now construe it as pertaining to drawing conclusionson the basis of taking samples. The argument then becomes: All ballsin this particular random sample are red; all balls in this particularrandom sample are taken from this urn; therefore, all balls in thisurn are red. What we have here is an argument from sample topopulation. This sort of argument is what Peirce understood to be thecore meaning of induction. That is to say, for Peirce, induction inthe most basic sense is argument from random sample to population. It should be clear that inductive inference is not necessary inference: it might well turn out that the claims stated in the premises are true even though the claim made in the conclusion is false.

Let us now go further and see what happens if, from the deductionAAA-1, we form a new argument by interchanging the conclusion (theResult) with the minor premise (the Case). The resultant argumentbecomes: AllMs arePs (Rule); allSs arePs (Result); therefore, allSs areMs(Case). This is the invalid syllogism AAA-2. But let us now regard itas pertaining to drawing conclusions on the basis of takingsamples. The argument then becomes: All balls in this urn are red; allballs in this particular random sample are red; therefore, all ballsin this particular random sample are taken from this urn. What we havehere is nothing at all like an argument from population to sample oran argument from sample to population: rather, it is a form of probableargument entirely different from both deduction and induction. It hasthe air of conjecture or “educated guess” about it. This new type ofargument Peirce called hypothesis (also, retroduction, and also, abduction). It should be clear that abduction is never necessary inference

There is no need to consider the variant of AAA-1 that is obtained byinterchanging the Rule and the Case in AAA-1. The resultant argumentis of the form AAA-4, which is exactly the same argument as AAA-1 withinterchanged premises. So it is simply deduction over again.

Peirce’s thinking about deduction, induction, and abduction can be seen also from examples he gives of arguments that are similar to the syllogisms he discusses, but retain the universal affirmative judgment only for the Case, using a definite percentage between 0% and 100% for both the Rule and the Result.

Corresponding to AAA-1 (deduction) we have the followingargument:X% ofMs arePs (Rule);allSs areMs (Case); therefore,X%ofSs arePs (Result). Construing this argument, aswe did before, as applying to drawing balls from urns, the argumentbecomes:X% of the balls in this urn are red; all the ballsin this random sample are taken from this urn; therefore,X%of the balls in this random sample are red. Peirce still regards thisargument as being a deduction, even though it is not—as theargument AAA-1 is—a necesary inference. He calls such anargument a “statistical deduction” or a“probabilistic deduction proper.”

Corresponding to AAA-3 (induction) we have the followingargument:X% ofSs arePs (Result);allSs areMs (Case); therefore,X%ofMs arePs (Rule). Construing this argument asapplying to drawing balls from urns, the argument becomes:X%of the balls in this random sample are red; all the balls in thisrandom sample are taken from this urn; therefore,X% of theballs in this urn are red. Here we still have an argument whoseessence is the logical transition from a random sample to thepopulation from which the sample is taken. The inference is made byvirtue of what Hans Reichenbach called “the straightrule”: the proportion of a trait found in the sample isattributed also to the population.

Corresponding to AAA-2 (abduction) we have the followingargument:X% ofMs arePs(Rule);X% ofSs arePs (Result);therefore, allSs areMs (Case). Construing thisargument as applying to drawing balls from urns, the argument becomes:X% of the balls in this urn are red;X% of the ballsin this random sample are red; therefore, all the balls in this randomsample are taken from this urn. Again here we have the character of aneducated guess or inference to a plausible explanation.

Over many years Peirce modified his views on the three types ofarguments, sometimes changing his views but mostly extending them byexpanding his commentary upon the original trichotomy. Occasionallyhe swerved between one view and another concerning which larger classof arguments a particular instance or sub-type of argument belongedto. For example, he seemed to have some hesitation about whetherarguments from analogy should be construed as inductions (argumentsfrom a sample of the properties of things to a population of theproperties of things) or abductions (conjectures made on the basis ofsufficient similarity, which notion might not easily be analyzed interms of sets of properties).

The most important extension Peirce made of his earliest views onwhat deduction, induction, and abduction involved was to integrate thethree argument forms into his view of the systematic procedure forseeking truth that he called the “scientific method.” Asso integrated, deduction, induction, and abduction are not simplyargument forms any more: they are three phases of the methodology ofscience, as Peirce conceived this methodology. In fact, in Peirce’smost mature philosophy he virtually (perhaps totally and literally)equates the trichotomy with the three phases he discerns in thescientific method. Scientific method begins with abduction orhypothesis: because of some perhaps surprising or puzzling phenomenon,a conjecture or hypothesis is made about what actually is goingon. This hypothesis should be such as to explain the surprisingphenomenon, such as to render the phenomenon more or less a matter ofcourse if the hypothesis should be true. Scientific method thenproceeds to the stage of deduction: by means of necessary inferences,conclusions are drawn from the provisionally-adopted hypothesis aboutthe obtaining of phenomena other than the surprising one thatoriginally gave rise to the hypothesis. Conclusions are reached, thatis to say, about other phenomena that must obtain if the hypothesisshould actually be true. These other phenomena must be such thatexperimental tests can be performed whose results tell us whether thefurther phenomena do obtain or do not obtain. Finally, scientificmethod proceeds to the stage of induction: experiments are actuallycarried out in order to test the provisionally-adopted hypothesis byascertaining whether the deduced results do or do not obtain. At thispoint scientific method enters one or the other of two “feedbackloops.” If the deduced consequences do obtain, then we loop backto the deduction stage, deducing still further consequences of ourhypothesis and experimentally testing for them again. But, if thededuced consequences do not obtain, then we loop back to the abductionstage and come up with some new hypothesis that explains both ouroriginal surprising phenomenon and any new phenomena we have uncoveredin the course of testing our first, and now failed, hypothesis. Thenwe pass on to the deduction stage, as before. The entire procedure ofhypothesis-testing, and not merely that part of it that consists ofarguing from sample to population, is called induction in Peirce’slater philosophy.

An important part of Peirce’s full conception of scientific method iswhat he called the “economics (or: economy) of research.”The idea is that, because research is difficult, research labor-timeis valuable and should not be wasted. Both in the creation ofhypotheses to be tested and in the experiments chosen to test thesehypotheses, we should act so as to get the very most cognitive bangfor the buck, so to say. The object is to proceed at every stage so asto maximize the reduction in indeterminacy of our beliefs. Peirce hadan elaborate, mathematical theory of some aspects of the economy ofresearch, and he published several complex papers on this topic. Thefollowing section of the present article contains further informationon Peirce’s notion of the economy of research.

4. Pragmatism, Pragmaticism, and the Scientific Method

Probably Peirce’s best-known works are the first two articles in aseries of six that originally were collectively entitledIllustrations of the Logic of Science and published inPopular Science Monthly from November 1877 through August1878. The first is entitled “The Fixation of Belief” andthe second is entitled “How to Make Our Ideas Clear.” Inthe first of these papers Peirce defended, in a manner consistent withnot accepting naive realism, the superiority of the scientific methodover other methods of overcoming doubt and “fixingbelief.” In the second of these papers Peirce defended a“pragmatic” notion of clear concepts.

Perhaps the single most important fact to keep in mind in trying tounderstand Peirce’s philosophy concerning clarity and the propermethod of fixing belief is that all his life Peirce was a practicingphysical scientist: already mentioned is the fact that he worked as aphysical scientist for 32 years in his job with the United StatesCoast and Geodetic Survey. As Peirce understood the topics ofphilosophy and logic, philosophy and logic were themselves alsosciences, although not physical sciences. Moreover, he understoodphilosophy to be the philosophy of science, and he understood logic tobe the logic of science (where the word “science” has asense that is best captured by the GermanwordWissenschaft).

It is in this light that his specifications of the nature ofpragmatism are to be understood. It is also in this light that hislater calling of his views “pragmaticism,” in order todistinguish his own scientific philosophy from other conceptions andtheories that were trafficked under the title“pragmatism,” is to be understood. When he said that thewhole meaning of a (clear) conception consists in the entire set ofits practical consequences, he had in mind that a meaningfulconception must have some sort of experiential “cashvalue,” must somehow be capable of being related to some sort ofcollection of possible empirical observations under specifiableconditions. Peirce insisted that the entire meaning of a meaningfulconception consisted in the totality of such specifications ofpossible observations. For example, Peirce tended to spell out themeaning of dispositional properties such as “hard” or“heavy” by using the same sort of counterfactualconstructions as, say, Karl Hempel would use. Peirce was not a simpleoperationalist in his philosophy of science; nor was he a simpleverificationist in his epistemology: he believed in the reality ofabstractions, and in many ways his thinking about universals resemblesthat of the medieval realists in metaphysics. Nevertheless, despitehis metaphysical leanings, Peirce’s views bear a strong familyresemblance to operationalism and verificationism. In regard tophysical concepts in particular, his views are quite close to thoseof, say, Einstein, who held that the whole meaning of a physicalconcept is determined by an exact method of measuring it.

The previous point must be tempered with the fact that Peirceincreasingly became a philosopher with broad and deep sympathies forboth transcendental idealism and absolute idealism. His Kantianaffinities are simpler and easier to understand than his Hegelianleanings. Having rejected a great deal in Kant, Peirce neverthelessshared with Charles Renouvier the view that Kant’s (quasi-)concept oftheDing an sich can play no role whatsoever in philosophy orin science other than the role that Kant ultimately assigned to it,viz. the role of aGrenzbegriff: a boundary-concept, or,perhaps a bit more accurately, a limiting concept. A supposed“reality” that is “outside” of every logicalpossibility of empirical or logical interaction with “it”can play no direct role in the sciences. Science can deal only withphenomena, that is to say, only with what can “appear”somehow in experience. All scientific concepts must somehow betraceable back to phenomenological roots. Thus, even when Peirce callshimself a “realist” or is called by others a“realist,” it must be kept in mind that Peirce was alwaysa realist of the Kantian “empirical” sort and not aKantian “transcendental realist.” His realism is similarto what Hilary Putnam has called “internal realism.” (Aswas said, Peirce was also a realist in quite another sense of he word:he was a realist or an anti-nominalist in the medieval sense.)

Peirce’s Hegelianism, to which he increasingly admitted as heapproached his most mature philosophy, is more difficult to understandthan his Kantianism, partly because it is everywhere intimately tiedto his entire late theory of signs (semeiotic) and sign use(semeiosis), as well as to his evolutionism and to his rather puzzlingdoctrine of mind. There are at least four major components of hisHegelian idealism. First, for Peirce the world of appearances, whichhe calls “the phaneron,” is a world consisting entirely ofsigns. Signs are qualities, relations, features, items, events,states, regularities, habits, laws, and so on that have meanings,significances, or interpretations. Second, a sign is one term in athreesome of terms that are indissolubly connected with each other bya crucial triadic relation that Peirce calls “the signrelation.” The sign itself (also called the representamen) isthe term in the sign relation that is ordinarily said to represent ormean something. The other two terms in this relation are called theobject and the interpretant. The object is what would ordinarilywould be said to be the “thing” meant or signified orrepresented by the sign, what the sign is a signof. The interpretant of a sign is said by Peirce to be thatto which the sign represents the object. What exactly Peircemeans by the interpretant is difficult to pin down. It is somethinglike a mind, a mental act, a mental state, or a feature or quality ofmind; at all events the interpretant is something ineliminablymental. Third, the interpretant of a sign, by virtue of the verydefinition Peirce gives of the sign-relation, must itself be a sign,and a sign moreover of the very same object that is (or: was)represented by the (original) sign. In effect, then, the interpretantis a second signifier of the object, only one that now has an overtlymental status. But, merely in being a sign of the original object,this second sign must itself have (Peirce uses the word“determine”) an interpretant, which then in turn is a new,third sign of the object, and again is one with an overtly mentalstatus. And so on. Thus, if there is any sign at all of any object,then there is an infinite sequence of signs of that same object. So,everything in the phaneron, because it is a sign, begins an infinitesequence of mental interpretants of an object.

But now, there is a fourth component of Peirce’s idealism: Peircemakes everything in the phaneron evolutionary. The whole systemevolves. Three figures from the history of culture loomed exceedinglylarge in the intellectual development of Peirce and in the culturalatmosphere of the period in which Peirce was most active: Hegel inphilosophy, Lyell in geology, and Darwin (along with Alfred RusselWallace) in biology. These thinkers, of course, all have a singletheme in common: evolution. Hegel described an evolution of ideas,Lyell an evolution of geological structures, and Darwin an evolutionof biological species and varieties. Peirce absorbed it all. Peirce’sentire thinking, early on and later, is permeated with theevolutionary idea, which he extended generally, that is to say, beyondthe confines of any particular subject matter. For Peirce, the entireuniverse and everything in it is an evolutionary product. Indeed, heconceived that even the most firmly entrenched of nature’s habits (forexample, even those habits that are typically called “naturallaws”) have themselves evolved, and accordingly can and shouldbe subjects of philosophical and scientific inquiry. One can sensiblyseek, in Peirce’s view, evolutionary explanations of the existence ofparticular natural laws. For Peirce, then, the entire phaneron (theworld of appearances), as well as all the ongoing processes of itsinterpretation through mental significations, has evolved and isevolving.

Now, no one familiar with Hegel can escape the obvious comparison: wehave in Peirce an essentially idealist theory that is similar to theidealism that Hegel puts forward in thePhaenomenologie desGeistes. Furthermore, both Hegel and Peirce make the wholeevolutionary interpretation of the evolving phaneron to be a processthat is said to be logical, the “action” of logicitself. Of course there are differences between the twophilosophers. For example, what exactly Hegel’s logic is has beenshrouded in mystery for every Hegelian after Hegel himself (and somephilosophers, for example Popper, would say for every Hegelianincluding Hegel). By contrast Peirce’s logic is reasonably clear, andhe takes great pains to work it out in intricate detail; basicallyPeirce’s logic is the whole logical apparatus of the physical andsocial sciences.

One implication of the unending nature of the interpretation ofappearances through infinite sequences of signs is that Peirce cannotbe any type of epistemological foundationalist or believer in absoluteor apodeictic knowledge. He must be, and is, an anti-foundationalistand a fallibilist. From his earliest to his latest writings Peirceopposed and attacked all forms of epistemological foundationalism andin particular all forms of Cartesianism andapriorism. Philosophy must begin wherever it happens to be at themoment, he thought, and not at some supposed ideal foundation,especially not in some world of “private references.” Theonly important thing in thinking scientifically to apply thescientific method itself. This method he held to be essentially publicand reproducible in its activities, as well as self-correcting in thefollowing sense: No matter where different researchers may begin, aslong as they follow the scientific method, their results willeventually converge toward the same outcome. (The pragmatic, orpragmaticistic, conception of meaning implies that two theories withexactly the same empirical content must have, despite superficialappearances, the same meaning.) This ideal point of convergence iswhat Peirce means by “the truth,” and“reality” is simply what is meant by “thetruth.” That these Peircean notions of reality and truth areinherently idealist rather than naively realist in character shouldrequire no special pleading.

Connected with Peirce’s anti-foundationalism is his insistence on thefallibility of particular achievements in science. Although thescientific method will eventually converge to something as a limit,nevertheless at any temporal point in the process of scientificinquiry we are only at a provisional stage of it and cannot ascertainhow far off we may be from the limit to which we are somehowconverging. This insistence on the fallibilism of human inquiry isconnected with several other important themes of Peirce’sphilosophy. His evolutionism has already been discussed: fallibilismis obviously connected with the fact that science is not shooting at afixed target but rather one that is always moving. What Peirce callshis “tychism,” which is his anti-deterministic insistencethat there is objective chance in the world, is also intimatelyconnected to his fallibilism. (Tychism will be discussed below.)Despite Peirce’s insistence on fallibilism, he is far from being anepistemological pessimist or sceptic: indeed, he is quite theopposite. He tends to hold that every genuine question (that is, everyquestion whose possible answers have empirical content) can beanswered in principle, or at least should not be assumed to beunanswerable. For this reason, one his most important dicta, which hecalled his first principle of reason, is “Do not block the wayof inquiry!”

For Peirce, as we saw, the scientific method involves three phases orstages: abduction (making conjectures or creating hypotheses),deduction (inferring what should be the case if the hypotheses are thecase), and induction (the testing of hypotheses). The process of goingthrough the stages should also be carried out with concern for theeconomy of research. Peirce’s understanding of scientific method,then, is not very different from today’s idea of scientificmethod which, seems to have been derived historically both fromWilliam Whewell and Peirce (see Niiniluoto 1984, 18; Ambrosio 2016;and Wagenmakers, Dutilh, & Sarafoglou 2018, 423). Today’sunderstanding sees the scientific method as the method of constructinghypotheses, deriving consequences from these hypotheses, and thenexperimentally testing these hypotheses (guided always by theeconomics of research). Also, as was said above, Peirce increasinglycame to understand his three types of logical inference as beingphases or stages of the scientific method. For example, as Peirce cameto extend and generalize his notion of abduction, abduction becamedefined as inference to and provisional acceptance of an explanatoryhypothesis for the purpose of testing it. Abduction is not alwaysinference to the best explanation, but it is always inference to someexplanation or at least to something that clarifies or makes routinesome information that has previously been “surprising,” inthe sense that we would not have routinely expected it, given ourthen-current state of knowledge. Deduction came to mean for Peirce thedrawing of conclusions as to what observable phenomena should beexpected if the hypothesis is correct. Induction came for him to meanthe entire process of experimentation and interpretation performed inthe service of hypothesis testing.

A few further comments are perhaps in order in connection withPeirce’s idea of the economy (or: the economics) of research. Concernfor the economy of research is a crucial and ineliminable part ofPeirce’s idea of the scientific method. He understood that science isessentially a human and social enterprise and that it always operatesin some given historical, social, and economic context. In such acontext some problems are crucial and paramount and must beattended-to immediately, while other problems are trivial or frivolousor at least can be put off until later. He understood that in the realcontext of science some experiments may be vitally important whileothers may be insignificant. Peirce also understood that the economicresources of the scientist (time, money, ability to exert effort,etc.) are always scarce, even though all the while the “greatocean of truth,” which lies undiscovered before us, isinfinite. All resources for carrying out research, such as personnel,person-hours, and apparatus, are quite costly; accordingly, it iswasteful, indeed irrational, to squander them. Peirce proposed,therefore, that careful consideration be paid to the problem of how toobtain the biggest epistemological “bang for the buck.” Ineffect, the economics of research is a cost/benefit analysis inconnection with states of knowledge. Although this idea has beeninsufficiently explored by Peirce scholars, Peirce himself regarded itas central to the scientific method and to the idea of rationalbehavior. It is connected with what he called “speculativerhetoric” or “methodeutic” (which will be discussedbelow).

5. Anti-determinism, Tychism, and Evolutionism

Against powerful currents of determinism that derived from theEnlightenment philosophy of the eighteenth century, Peirce urged thatthere was not the slightest scientific evidence for determinism andthat in fact there was considerable scientific evidence againstit. Always by the words “science” and“scientific” Peirce understood reference to actualpractice by scientists in the laboratory and the field, and notreference to entries in scientific textbooks. In attackingdeterminism, therefore, Peirce appealed to the evidence of the actualphenomena in laboratories and fields. Here, what is obtained as theactual observations (e.g. measurements) does not fit neatly into someone point or simple function. If we take, for example, a thousandmeasurements of some physical quantity, even a simple one such aslength or thickness, no matter how carefully we may do so, we will notobtain the same result a thousand times. Rather, what we get is adistribution (often, but not always and certainly not necessarily,something akin to a normal or Gaussian distribution) of hundreds ofdifferent results. Again, if we measure the value of some variablethat we assume to depend on some given parameter, and if we let theparameter vary while we take successive measurements, the result ingeneral will not be a smooth function (for example, a straight line oran ellipse); rather, it will typically be a “jagged”result, to which we can at bestfit a smooth function byusing some clever method (for example, fitting a regression line bythe method of least-squares). Naively, we might imagine that thevariation and relative inexactness of our measurements will becomeless pronounced and obtrusive the more refined and microscopic are ourmeasurement tools and procedures. Peirce, the practicing scientist,knew better. What actually happens, if anything, is that ourvariations get relatively greater the finer is our instrumentation andthe more delicate our procedures. (Obviously, Peirce would not havebeen the least surprised by the results obtained from measurements atthe quantum level.)

What the directly measured facts of scientific practice seem to tellus, then, is that, although the universe displays varying degrees ofhabit (that is to say, of partial, varying, approximate, andstatistical regularity), the universe does not display deterministiclaw. It does not directly show anything like total, exact,non-statistical regularity. Moreover, the habits that nature doesdisplay always appear in varying degrees of entrenchment or“congealing.” At one end of the spectrum, we have the nearly law-likebehavior of larger physical objects like boulders and planets; but atthe other end of the spectrum, we see in human processes ofimagination and thought an almost pure freedom and spontaneity; and inthe quantum world of the very small we see the results of almost purechance.

The immediate, “raw” result, then, of scientificobservation through measurement is that not everything is exactlyfixed by exact law (even if everything should be constrained to somedegree by habit). In his earliest thinking about the significance ofthis fact, Peirce opined that natural law pervaded the world but thatcertain facets of reality were just outside the reach or grasp oflaw. In his later thinking, however, Peirce came to understand thisfact as meaning that reality in its entirety was lawless and that purespontaneity had an objective status in the phaneron. Peirce called hisdoctrine that chance has an objective status in the universe“tychism,” a word taken from the Greek word for“chance” or “luck” or “what the godshappen to choose to lay on one.” Tychism is a fundamentaldoctrinal part of Peirce’s mature view, and reference to his tychismprovides an added reason for Peirce’s insisting on the irreduciblefallibilism of inquiry. For nature is not a static world of unswervinglaw but rather a dynamic and dicey world of evolved and continuallyevolving habits that directly exhibit considerablespontaneity. (Peirce would have embraced quantum indeterminacy.)

One possible path along which nature evolves and acquires its habitswas explored by Peirce using statistical analysis in situations of experimental trials in which the probabilities of outcomes in later trials are not independent of actual outcomes in earlier trials, situations of so-called “non-Bernoullian trials.” Peirce showed that, if we posit a certain primal habitin nature, viz. the tendency however slight to take on habits howevertiny, then the result in the long run is often a high degree ofregularity and great macroscopic exactness. For this reason, Peircesuggested that in the remote past nature was considerably morespontaneous than it has now become, and that in general and as a wholeall the habits that nature has come to exhibit have evolved. Just as ideas, geological formations, and biological species have evolved, natural habit has evolved.

In this evolutionary notion of nature and natural law we have an addedsupport of Peirce’s insistence on the inherent fallibilism ofscientific inquiry. Nature may simply change, even in its mostentrenched fundamentals. Thus, even if scientists were at one point intime to have conceptions and hypotheses about nature that survivedevery attempt to falsify them, this fact alone would not ensure thatat some later point in time these same conceptions and hypotheseswould remain accurate or even pertinent.

An especially intriguing and curious twist in Peirce’s evolutionism isthat in Peirce’s view evolution involves what he calls its“agapeism.” Peirce speaks of evolutionary love. Accordingto Peirce, the most fundamental engine of the evolutionary process isnot struggle, strife, greed, or competition. Rather it is nurturinglove, in which an entity is prepared to sacrifice its own perfectionfor the sake of the wellbeing of its neighbor. This doctrine had asocial significance for Peirce, who apparently had the intention ofarguing against the morally repugnant but extremely popularsocio-economic Darwinism of the late nineteenth century. The doctrinealso had for Peirce a cosmic significance, which Peirce associatedwith the doctrine of the Gospel of John and with the mystical ideas ofSwedenborg and Henry James. In Part IV of the third of Peirce’s sixpapers inPopular Science Monthly, entitled “TheDoctrine of Chances,” Peirce even argued that simply beinglogical presupposes the ethics of self-sacrifice: “He who wouldnot sacrifice his own soul to save the whole world, is, as it seems tome, illogical in all his inferences, collectively.” To socialDarwinism, and to the related sort of thinking that constituted forHerbert Spencer and others a supposed justification for the morerapacious practices of unbridled capitalism, Peirce referred indisgust as “The Gospel of Greed.”

6. Synechism, the Continuum, Infinites, and Infinitesimals

Along with Richard Dedekind and Georg Cantor, Peirce was one of the first scientific thinkers to argue in favor of the existence of actually infinite collections, and to maintain that the paradoxes that Bernard Bolzano had associated with the idea of infinite collections were not really contradictions at all. His criterion of the difference between finite and infinite collections was that the so-called “syllogism of transposed quantity,” which had been introduced by Augustus de Morgan, constituted a deductively validargument only when applied to finite sets; as applied to infinite setsit was invalid. The syllogism of transposed quantity runs asfollows. We have a binary relationR defined on a setS, such that the following two premises are true of the relation (where the quantifications are taken over the setS). First, for allx there is ay such thatRxy. Second, for allx,y,z,Rxz andRyz implies thatx = y. Theconclusion (of the syllogism of transposed quantity) is that for allx there exists ay such thatRyx. One ofPeirce’s favorite examples helps elucidate the idea, even if itperhaps be not perfectly politically correct: Every Texan kills someTexan; no Texan is killed by more than one Texan; therefore everyTexan is killed by some Texan. The argument’s conclusion followsvalidly only if the set of Texans is finite.

If byRxy in the syllogism of transposed quantity we takef(x) =y, where the function is defined onand has values in the setS, then the second premise of the syllogism of transposed quantity says thatf is a one-one function. The conclusion says that every member ofS is the image underf of some member ofS. Thus the syllogism of transposed quantity says that no one-one function can map the setS to a proper subset of itself. This assertion holds, of course, only ifS is a finite set. So, as it turns out, Peirce’s definition of the difference between finite and infinite sets is virtually equivalent to the standard one, which is found in Section 5 of Richard Dedekind’sWas Sind und Was Sollen die Zahlen?, to the effect that an infinite set is one that can be placed into a one-to-one correspondence with a proper subset of itself. Peirce claimed on various occasions to have reached his definition of the difference between finite and infinite collections at least six years before Dedekind reached his own definition.

Peirce held that the continuity of space, time, ideation, feeling, andperception is an irreducible deliverance of science, and that anadequate conception of such continua is an extremely important part ofall the sciences. The doctrine of the continuity of nature he called“synechism,” a word deriving from the Greek prepositionthat means “(together) with.” In mid-1892, somewhat underthe influence of reading Cantor’s works, Peirce defined a (linear)continuum to be a linearly-ordered infinite setC such that(1) for any two distinct members ofC there exists a third member ofC that is strictlybetween these; and (2) every countably infinite subset ofCthat has an upper (lower) bound inC has a least upper bound(greatest lower bound) inC. The first property he called“Kanticity” and the second “Aristotelicity.”(Today we would likely call these properties “density” and“closedness,” respectively.) The second condition has thecorollary that a continuum contains all its limit points, andsometimes Peirce used this property in conjunction with“Kanticity” to define a continuum.

Toward the end of the nineteenth century, however, Peirce began tohold that Kanticity and Aristotelicity, even when conjoined, wereinsufficient to define adequately the notion of a continuum. Hemaintained that he had framed an updated conception of continua bysomewhat loosening his attachment to Cantor’s ideas. He began to writein ways that, at least at first glance, seem close to falling intoCantor’s Paradox; Peirce, however, tried to avoid outrightcontradiction by means of embracing some sort of non-standard ideaabout the identity of points on a line. For example, in Lecture 3 ofhisCambridge Conferences Lectures of 1898, publishedasReasoning and the Logic of Things, Peirce says that if aline is cut into two portions, the point at which the cut takes placeactually becomes two points. What Peirce’s new approach is, inmathematical detail, and whether or not it contains hidden but realcontradictions, is a problem that has not yet been solved byresearchers into Peirce’s logic and mathematics.

Connected with his new conception of the continuum is Peirce’sincreasingly frequent and sometimes pugnacious defenses of thedoctrine of the reality of infinitesimal quantities. The doctrine wasnot newly taken up by Peirce late in the nineteenth century; indeed,he had held the doctrine for some time, and it had been the doctrineof his father Benjamin. He considered it superior to the newerdoctrine of limits for providing a foundation for the differential andintegral calculus. What was new was that Peirce began to see thedoctrine of infinitesimals as the key to his updated doctrine of thecontinuum. Thus, adding to his long-standing defense of infinitelylarge magnitudes (Peirce often used the word“multitudes.”), Peirce began vigorously to defendinfinitely small magnitudes, infinitesimal magnitudes. Many examplesof such defenses can be found. Carolyn Eisele collected a number ofsuch examples in her edited workThe New Elements of Mathematicsby Charles S. Peirce. See, for example, Volume 2, pages169–170, where Peirce says “My personal opinion is thatthere is positive evidence of the real existence of infinitesimals;and that the admission of them would considerably simplify theintroduction to the calculus.” See also Volume 3, Part 1, pages121–124, 125–127, 128–131, and 742–750. By theend of the nineteenth century Peirce’s view about infinitesimals wasso rare and remarkable that Josiah Royce remarked, in a footnote ofhis “Supplementary Essay” forThe World and theIndividual, First Series, that outside of Italy Peirce wasvirtually the only mathematical philosopher who believed ininfinitesimals. (See footnote 2, page 562 of this work by Royce.)

Not only did Peirce defend infinitesimals. He furthermore claimed that hehad proved the consistency of introducing infinitesimals into thesystem of real numbers in such a way as to form a new system in whichthere were infinitely many entities that were not equal to zero andyet were all smaller than any real numberr that is not equalto zero, no matter how smallr might be. To use modernterminology, Peirce was claiming to have shown the existence ofordered fields that were non-Archimedean. It was these non-Archimedeanfields that Peirce now wanted to call genuine continua. Additionally,Peirce wanted to use his notion infinitesimal quantities and hisrevised concept of the continuum in order to justify the traditionalpre-Gaussian definitions and underpinnings of the differentialcalculus.

Peirce also made a number of remarks that suggest, in connection withthe foregoing enterprise, that he had a novel conception of thetopology of points in a continuum. All these remarks he connected withhis previous defenses of infinite sets. For these reasons some Peircescholars, and in particular the great Peirce scholar Carolyn Eisele,have suggested that his ideas were an anticipation of AbrahamRobinson’s non-standard analysis of 1964. Whether this actually be soor not, however, is at the present time far from clear. Peircecertainly says many things that are quite suggestive of theconstruction of non-standard models of the theory of ordered fields bymeans of using equivalence classes of countably infinite CartesianProducts of the standard real numbers and then applying Loś’sTheorem. However, no commentator up to now has provided anything evenremotely resembling a careful and detailed exposition of Peirce’sthinking in this area. Unfortunately, most of Peirce’s publishedwriting and public talks on this topic were designed for audiencesthat were extremely unsophisticated mathematically (a fact that helamented). For that reason most of what Peirce said on the topic ispicturesque and intriguing, but extremely obscure. The entire analysisof Peirce’s notion of an infinitesimal, as well as the exact bearingthis notion has on his concept of a real continuum and on his idea ofthe topology of the points of a continuum, still awaits meticulousmathematical discussion.

7. Probability, Verisimilitude, and Plausibility

Given Peirce’s tychism and his view that statistical information isoften the best information we can have about phenomena, it should notbe surprising that Peirce devoted close attention to the analysis ofsituations in which perfect exactness and perfect certitude wereunattainable. It is only to be expected that he would devote a greatdeal of attention, for example, to probability theory. Indeed, Peircedid so from the dates of even his earliest thinking. Not only, forexample, did he extensively employ the concept of probability, butalso he offered a pragmaticistic account of the notion of probabilityitself. Yet it would be a huge mistake to think that Peirce’sphilosophizing about situations of imperfection of exactness andcertitude were confined merely to the theory of probability.

Rather, from the outset of his thinking about the matters, in about1863, his attention was directed to the broadest sorts of issuesconnected with statistical inference. And, as his thinking progressed,Peirce came ever more clearly to see that there are three distinct andmutually incommensurable measures of imperfection of certitude. Onlyone was probability. The other two he called“verisimilitude” (or “likelihood”) and“plausibility”. Each of the three measures was associatedwith one of his types of argument. Probability he associated withdeduction. Verisimilitude he associated with induction. Andplausibility he associated with abduction. Let us look more closely ateach of these three distinct measures of uncertainty.

By the time Peirce wrote on probability, the concept and its calculuswere well over two hundred years old. The probability calculus itselfhad become more or less standardized by Peirce’s time, and indeedPeirce’s own axioms for the calculus are more or less the same thatKolmogoroff gives for his “elementary theory ofprobability.” By contrast with the calculus, the philosophicaltheory of the meaning of probability was hotly disputed. Two sides tothe dispute existed. There were the subjectivists, or“conceptualists,” as Peirce designated them. Thesebelieved that probability was a measure of the strength of beliefactually accorded to a proposition or a measure of the degree ofrational belief that ought to be accorded to a proposition. Among thedefenders of this sort of view, Augustus de Morgan and AdolpheQuetelet were major figures. And there were the objectivists, or“materialists,” as Peirce designated them. These believedthat probability was a measure of the relative frequency with which anevent of some specific sort repeatedly happened. John Venn was amajor defender of this sort of view. Pierre Simon Laplace had spokensometimes in a subjectivist way, sometimes in an objectivist way; buthis arguments basically depended on a subjectivist interpretation ofprobability.

Peirce vigorously attacked the subjectivist view of de Morgan andothers to the effect that probability is merely a measure of our levelof confidence or strength of belief. He allowed that the logarithm ofthe odds of an event might be used to assess the degree of rationalbelief that should be accorded to an event, but only provided that theodds of the event were determined by the objective relative frequencyof the event. In other words, Peirce suggested that any justifiableuse of subjectivism in connection with probability theory mustultimately rest on quantities obtained by means of an objectivistunderstanding of probability.

Rather than holding that probability is a measure of degree ofconfidence or belief, then, Peirce adopted an objectivist notion ofprobability that he explicitly likened to the doctrine of JohnVenn. Indeed, he even held that probability is actually a notion withclear empirical content and that there are clear empirical proceduresfor ascertaining that content. First, he held, that that to which aprobability is assigned, insofar as the notion of probability is usedscientifically, is not a proposition or an event or a state; nor is ita type of event or state. Rather, what is assigned a probability isanargument, an argument having premisses (Peirce insisted onthis spelling rather than the spelling “premises.”) and aconclusion. Peirce’s view in this regard is virtuallyindistinguishable from the view of Kolmogoroff that all probabilitiesare conditional probabilities. Second, Peirce held that, in order toascertain the probability of a particular argument, the observer notesall occasions on which all of its premisses are true, case by case,just as they come under observation. For each of these occasions theobserver notes whether the conclusion is true or not. The observerkeeps a running tally, the ongoing ratio whose numerator is the numberof occasions so far observed on which the conclusion as well as thepremisses are true and whose denominator is the number of occasions sofar observed on which the premisses are true irrespective of whetheror not the conclusion is also true. At each observation the observercomputes this ratio, which obviously encompasses all the observer’spast observations of occasions on which the premisses are true. Theprobability of the argument in question is defined by Peirce to be thelimit of the crucial ratio as the number of observations tends to growinfinitely large (if this limit exists).

Peirce’s earliest account of the meaning of probability, then, is aversion of what is called the “long run relative frequencyview” of probability. Late in his philosophical career, about1910, Peirce found fault with his earliest views on account of theirfailure to make clear just how the occasions of observation are to bechosen. He also emphasized that probability judgments are judgmentsabout what he called “would-be’s.” For this reason Peirceis often considered to be the originator of the sort of“propensity view” of probability that is associated withKarl Popper. One should not, however, think that viewing Peirce as apropensity theorist is in conflict with viewing him as some sort oflong term relative frequency theorist. Rather, Peirce’s view seems tobe that the propensity in question, when its sense is spelled out inaccord with the pragmatic (or: pragmaticistic) theory of meaning, is adispositional property that manifests itself in the set of concretefacts that amount to a certain long term relative frequency tendingtoward a certain limit as the number of appropriate occasions ofobservation increases indefinitely.

There is an interesting connection between Peirce’s tychism, his view that there is objective spontaneity in the universe, and the foregoing account of probability. For Peirce understood the universe of appearances as a logical process, somewhat in the same manner that Hegel understood the universe of appearances as the phenomenology of spirit. He tended to consider a given state of the universe as being a given set of premisses, so to say, of a possible inference. Then a subsequent state of the universe could be seen as being the conclusion of an actual inference. Thus Peirce tended to see the universe of appearances as bringing itself into being by a process that is ultimatelylogical. The world, as it were, evolves by abducing,deducing, and inducing itself. It is in some sense Hegel’s “Thought thinking thought.”

Along with his attack on a subjectivist account of probability, Peircealso attacked the use of what came to be called the method of“inverse probabilities” as a way of solving the problem ofinduction. In the process, he also excoriated the theoretical work, inthis connection, of de Morgan and Adolphe Quetelet (the Belgiancriminologist and early user of statistical analysis insociology). Induction, as we have seen, Peirce counted as an inferencefrom sample to population. The method of inverse probabilities offersitself as a way of calculating the (conditional) probability that apopulation has a trait in a certain proportion given that a sampledrawn from that population has the trait in that proportion. Itproposes to calculate this conditional probability by applying theso-called “Bayes’s Theorem” in order to express it interms of the (inverse conditional) probability that the sample has thetrait in the crucial proportion given that the population has thetrait in the crucial proportion. In the expression of the firstconditional probability in terms of the second conditionalprobability, however, there occur certain quantities, known as the“Bayesian prior probabilities.” What Peirce pointed out isthat there is no way to assign any quantities in a rational fashion tothe requisite Bayesian prior probabilities. The appearance that onedoes have a reason for assigning particular quantities results onlyfrom an illicit substitution of subjective probabilities for theneeded objectivist probabilities. What the user of the method ofinverse probabilities does is to equate complete lack of informationabout something with the claim that all possibilities must have equalprobabilities. This equation was called “the principle ofinsufficient reason” in the nineteenth century; John MaynardKeynes later named it “the principle of indifference.”This principle is, however, completely irrational without a dependenceon a subjectivist account of probability. What we need, however, isobjective probabilities, and so we have no reason for assigning anyparticular values to the Bayesian prior probabilities. Only “ifuniverses were as plenty as blackberries,” wrote Peirce, wouldthe analysis of de Morgan and Quetelet make any sense.

In rejecting Bayesianism and the method of inverse probabilities,Peirce argued that in fact no probability at all can be assigned toinductive arguments. Instead of probability, a different measure ofimperfection of certitude must be assigned to inductive arguments:verisimilitude or likelihood. In explaining this notion Peirce offeredan account of hypothesis-testing that is equivalent to standardstatistical hypothesis-testing. In effect we get an account ofconfidence intervals and choices of statistical significance forrejecting null hypotheses. Such ideas became standard only in thetwentieth century as a result of the work of R. A. Fisher, JerzyNeyman, and others. But already by 1878, in his paper “TheProbabilitiy of Induction,” Peirce had worked out the wholematter. (This topic has been discussed expertly by Deborah Mayo, whoalso has shown that the error-correction implicit in statisticalhypothesis-testing is intimately affiliated with Peirce’s notion ofscience as self-correcting and convergent to “thetruth.”)

Peirce’s accounts of his third type of deviance from perfectcertitude, namely plausibility, are much sketchier than his accountsof probability and verisimilitude. Unlike the other two forms ofuncertainty, which can be spelled out mathematically with greatprecision, plausibility seems to be capable of only a qualitativeaccount, even though plausibility does seem to comes in greater andlesser degree. The question of the plausibility of a claim arises,apparently, only in contexts in which one is seeking to adduce anexplanatory hypothesis for some actual fact that is surprising. Thekey point is that the hypothesis must be plausible in order to takenseriously. If we were, for example, to come upon a lump of ice in themiddle of a desert, we might plausibly say that perhaps someone put itthere, or perhaps a freak storm had left a great hailstone. But wewould not plausibly say that it had been thrown off a flying saucerthat previously had swooped through. It should be obvious that thenotion of plausibility is a difficult one, which strongly invitesfurther analysis but which is not easy to analyze in technicaldetail.

8. Psycho-physical Monism and Anti-nominalism

Peirce held that science suggests that the universe has evolved from acondition of maximum freedom and spontaneity into its presentcondition, in which it has taken on a number of habits, sometimes moreentrenched habits and sometimes less entrenched ones. With purefreedom and spontaneity Peirce tended to associate mind, and withfirmly entrenched habits he tended to associate matter (or, moregenerally, the physical). Matter he tended to regard as“congealed” or “effete” mind (1891; CP, 6.25;EP, 1.293). Thus he tended to see the universe as theend-product-so-far of a process in which mind has acquired habits andhas “congealed” (this is the very word Peirce used) intomatter.

This notion of all things as being evolved psycho-physical unities ofsome sort places Peirce well within the sphere of what might be called“the grand old-fashioned metaphysicians,” along with suchthinkers as Plato, Aristotle, Aquinas, Spinoza, Leibniz, Hegel,Schopenhauer, Whitehead,et al. Some contemporaryphilosophers might be inclined to reject Peirce out of hand upondiscovering this fact. Others might find his notion of psycho-physicalunities not so very offputting or indeed even attractive. What iscrucial is that Peirce argued that mind pervades all of nature invarying degrees: it is not found merely in the most advanced animalspecies.

This pan-psychistic view, combined with his synechism, meant for Peircethat mind is extended in some sort of continuum throughout theuniverse. Peirce tended to think of ideas as existing in mind insomewhat the same way as physical forms exist in physically extendedthings. He even spoke of ideas as “spreading” out through the samecontinuum in which mind is extended. This set of conceptions is partof what Peirce regarded as (his own version of) Scotistic realism,which he sharply contrasted with nominalism. He tended to blame whathe regarded as the errors of much of the philosophy of hiscontemporaries as owing to its nominalistic disregard for theobjective existence of form.

9. Triadism and the Universal Categories

Merely to say that Peirce was extremely fond of placing things intogroups of three, of trichotomies, and of triadic relations, would failmiserably to do justice to the overwhelming obtrusiveness in hisphilosophy of the number three. Indeed, he made the most fundamentalcategories of all “things” of any sort whatsoever thecategories of “Firstness,” “Secondness,” and“Thirdness,” and he often described “things”as being “firsts” or “seconds” or“thirds.” For example, with regard to the trichotomy“possibility,” “actuality,” and“necessity,” possibility he called a first, actuality hecalled a second, and necessity he called a third. Again: quality was afirst, fact was a second, and habit (or rule or law) was athird. Again: entity was a first, relation was a second, andrepresentation was a third. Again: rheme (by which Peirce meant arelation of arbitrary adicity or arity) was a first, proposition was asecond, and argument was a third. The list goes on and on. Let usrefer to Peirce’s penchant for describing things in terms oftrichotomies and triadic relations as Peirce’s“triadism.”

If Peirce had a general technical rationale for his triadism, Peircescholars have not yet made it abundantly clear what this rationalemight be. He seemed to base his triadism on what he called“phaneroscopy,” by which word he meant the mereobservation of phenomenal appearances. He regularly commented that thephenomena in the phaneron justdo fall into three groups andthat they justdo display irreducibly triadic relations. Heseemed to regard this matter as simply open for verification by directinspection.

Although there are many examples of phenomena that do seem more orless naturally to divide into three groups, Peirce seems to have beendriven by something more than mere examples in his insistence onapplying his categories to almost everything imaginable. Perhaps itwas the influence of Kant, whose twelve categories divide into fourgroups of three each. Perhaps it was the triadic structure of thestages of thought as described by Hegel. Perhaps it was even thetriune commitments of orthodox Christianity (to which Peirce, at leastin some contexts and during some swings of mood, seemed tosubscribe). Certainly involved was Peirce’s commitment to theineliminability of mind in nature, for Peirce closely associated theactivities of mind with the aforementioned triadic relation that he calledthe “sign” relation. (More on this topic appears below.)Also involved was Peirce’s so-called “reduction thesis” inlogic (on which more will given below), to which Peirce had concludedas early as 1870.

It is difficult to imagine even the most fervently devout of thepassionate admirers of Peirce, of which there are many, saying thathis account (or, more accurately, his various accounts) of the threeuniversal categories is (or are) absolutely clear and compelling. Yet,in almost everything Peirce wrote from the time the categories werefirst introduced, Peirce’s firsts, seconds, and thirds found aplace. Giving their exact and general analysis and providing an exactand general account of their rationale, if there be such, constitutechief problems in Peirce scholarship.

10. Mind and Semeiotic

Connected with Peirce’s insistence on the ubiquity of mind in thecosmos is the importance he attached to what he called“semeiotic,” the theory of signs in the most generalsense. Although a few points concerning this subject were made earlierin this article, some further discussion is in order. What Peirceanmeant by “semeiotic” is almost totally different from whathas come to be called “semiotics,” and which hails not somuch from Peirce as from Ferdinand de Saussure and CharlesW. Morris. Even though Peircean semeiotic and semiotics are oftenconfused, it is important not to do so. Peircean semeiotic derivesultimately from the theory of signs of Duns Scotus and its laterdevelopment by John of St. Thomas (John Poinsot). In Peirce’s theorythe sign relation is a triadic relation that is a special species ofthe genus: the representing relation. Whenever the representingrelation has an instance, we find one thing (the “object”)being represented by (or: in) another thing (the“representamen”) and being represented to (or: in) a thirdthing (the “interpretant.”) Moreover, the object isrepresented by the representamen in such a way that the interpretantis thereby “determined” to be also a representamen of theobject to yet another interpretant. That is to say, the interpretantstands in the representing relation to the same object represented bythe original representamen, and thus the interpretant represents theobject (either again or further) to yet anotherinterpretant. Obviously, Peirce’s complicated definition entails thatwe have an infinite sequence of representamens of an object wheneverwe have any one representamen of it.

The sign relation is the special species of the representing relationthat obtains whenever the first interpretant (and consequently eachmember of the whole infinite sequence of interpretants) has a statusthat ismental, i.e. (roughly) is a cognition of a mind. Inany instance of the sign relation an object is signified by a sign toa mind. One of Peirce’s central tasks was that of analyzing allpossible kinds of signs. For this purpose he introduced variousdistinction among signs, and discussed various ways of classifyingthem.

One set of distinctions among signs was introduced by Peirce in theearly stages of his analysis. The distinctions in this set turn onwhether the particular instance of the sign relation is“degenerate” or “non-degenerate.” The notionof “degeneracy” here is the standard mathematical notion,and as applied to sign theory non-degeneracy means simply that thetriadic relation cannot be analyzed as a logical conjunction of anycombination of dyadic relations and monadic relations. More exactly, aparticular instance of the obtaining of the sign relation isdegenerate if and only if the fact that a signs means an objecto to an interpretantican be analyzed into a conjunction of facts of the formP(s) &Q(o) &R(i) &T(s,o) &U(o,i) &W(i,s) (where not all the conjuncts have tobe present). Either an obtaining of the sign relation isnon-degenerate, in which case it falls into one class; or it isdegenerate in various possible ways (depending on which of theconjuncts are omitted and which retained), in which cases it fallsinto various other classes. Other distinctions regarding signs wereintroduced later by Peirce. Some of them will be discussed verybriefly in the following section of this article.

11. Semeiotic and Logic

Peirce’s settled opinion was that logic in the broadest sense is to beequated with semeiotic (the general theory of signs), and that logicin a much narrower sense (which he typically called “logicalcritic”) is one of three major divisions or parts ofsemeiotic. Thus, in his later writings, he divided semeiotic intospeculative grammar, logical critic, and speculative rhetoric (alsocalled “methodeutic”). Peirce’s word“speculative” is his Latinate version of the Greek-derivedword “theoretical,” and should be understood to meanexactly what the word “theoretical” means. Peirce’stripartite division of semeiotic is not to be confused with CharlesW. Morris’s division: syntax, semantics, and pragmatics (althoughthere may be some commonalities in the two trichotomies).

By speculative grammar Peirce understood the analysis of the kinds ofsigns there are and the ways that they can be combinedsignificantly. For example, under this heading he introduced threetrichotomies of signs and argued for the real possibility of onlycertain kinds of signs. Signs are qualisigns, sinsigns, or legisigns,accordingly as they are mere qualities, individual events and states,or habits (or laws), respectively. Signs are icons, indices (alsocalled “semes”), or symbols (sometimes called“tokens”), accordingly as they derive their significancefrom resemblance to their objects, a real relation (for example, ofcausation) with their objects, or are connected only by convention totheir objects, respectively. Signs are rhematic signs (also called“sumisigns” and “rhemes”), dicisigns (alsocalled “quasi-propositions”), or arguments (also called“suadisigns”), accordingly as they arepredicational/relational in character, propositional in character, orargumentative in character. Because the three trichotomies areindependent of each other, together they yield the abstract possibilitythat there are 27 distinct kinds of signs. Peirce argued, however,that 17 of these are logically impossible, so that finally only 10kinds of signs are genuinely possible. In terms of these 10 kinds ofsigns, Peirce endeavored to construct a theory of all possible naturaland conventional signs, whether simple or complex.

What Peirce meant by “logical critic” is pretty much logicin the ordinary, accepted sense of “logic” fromAristotle’s logic to present-day mathematical logic. As might beexpected, a crucial concern of logical critic is to characterize thedifference between correct and incorrect reasoning. Peirce achievedextraordinarily extensive and deep results in this area, and a few ofhis accomplishments in this area will be discussed below.

By “speculative rhetoric” or “methodeutic”Peirce understood all inquiry into the principles of the effective useof signs for producing valuable courses of research and givingvaluable expositions. Methodeutic studies the methods thatresearchers should use in investigating, giving expositions of, andcreating applications of the truth. Peirce also understood, under theheading of speculative rhetoric, the analysis of communicationalinteractions and strategies, and their bearing on the evaluation ofinferences. Peirce’s important topic of the economy of research isclosely affiliated with his idea of speculative rhetoric. The idea ofmethodeutic may overlap to some small extent with Morris’s notion of“pragmatics,” but the spirit of Peirce’s notion is muchmore extensive than that of Morris’s notion. Moreover, Peirce handledthe notion of indexical reference under the heading of speculativegrammar and not under the heading of speculative rhetoric, whereas thetopic certainly belongs to Morris’s pragmatics. There clearly exist connections between Peirce’s speculative rhetoric, on the one hand, and the attention paid by twentieth-century philosophers such as Ludwig Wittgenstein and J. L. Austin to matters having to do with language as a set of various social practices. Unfortunately, however, little attention has been paid by Peirce scholars to the relations between Peirce’s thinking and familiar twentieth-century notions such as Wittgenstein’s language-games and Austin’s speech-acts.

Speculative rhetoric, however, has attracted considerablephilosophical attention in recent years, especially among FinnishPeirce scholars centering about the University of Helsinki. These havenoted that there are extensive affiliations between Peirce’sdiscussions of the communicational and dialogical aspects ofsemeiotic, on the one hand, and the many and varied“game-theoretical” approaches to logic that have been forsome time of interest to Finnish philosophers (as well as manyothers), on the other hand. Various proposals for game-theoreticsemantical approaches to logic have been developed and applied toPeirce’s logic, as well as being used to understand Peirceanpoints.

12. The Classification of the Sciences

Peirce maintained a considerable interest in the topic ofclassification or taxonomy in general, and he considered biology andgeology the foremost sciences to have made progress in developinggenuinely useful systems of classification for things. In his owntheory of classification, he seemed to regard some sort of clusteranalysis as holding the key to creating really usefulclassifications. He regularly strove to create a classification of allthe sciences that would be as useful to logic as the taxonomies of thebiologists and geologists were to these scientists. Of specialinterest in this regard is the fact that he considered the relation ofsimilarity to be a triadic relation, rather than a dyadicrelation. Thus, for Peirce taxonomies and taxonomic trees are only onesort of classificatory system, albeit the most highly-developedone. He would not be in the least surprised to find that the topic ofconstructing “ontologies” is in vogue among computerscientists, and he would applaud endeavors to construct ontologies. Hewould not find in the least alien many contemporary analyticdiscussions of the notion of similarity; he would be right at homeamong them.

As with many of Peirce’s classificatory divisions, his classificationof the sciences is a taxonomy whose tree is trinary. For example heclassifies all the sciences into those of discovery, review, andpracticality. Sciences of discovery he divides into mathematics,philosophy, and what he calls “idioscopy” (by which heseems to mean the class of all the particular or special sciences likephysics, psychology, and so forth). Mathematics he divides intomathematics of logic, of discrete series, and of continua andpseudo-continua. Philosophy he divides into phenomenology, normativescience, and metaphysics. Normative science he divides intoaesthetics, ethics, and logic. And so on and on. Very occasionallythere is found a binary division: for example, he divides idioscopyinto the physical sciences and the psychical (or human) sciences. But,hardly surprisingly given his penchant for triads, most of hisdivisions are into threes.

Peirce scholars have found the topic of Peirce’s classification of thesciences a fertile ground for assertions about what is most basic inall thinking, in Peirce’s view. Whether or not such assertions runafoul of Peirce’s anti-foundationalism is itself a topic for furtherstudy.

13. Logic

In the extensiveness and originality of his contributions tomathematical logic, Peirce is almost without equal. His writings andoriginal ideas are so numerous that there is no way to do them justicein a small article such as the present one. Accordingly, only a few ofhis numerous achievements will be mentioned here.

Peirce’s special strength lay not so much in theorem-proving as ratherin the invention and developmental elaboration of novel systems oflogical syntax and fundamental logical concepts. He invented dozens ofdifferent systems of logical syntax, including a syntax based on ageneralization of de Morgan’s relative product operation, an algebraicsyntax that mirrored Boolean algebra to some extent, aquantifier-and-variable syntax that (except for its specific symbols)is identical to the much later Russell-Whitehead syntax. He eveninvented two systems of graphical two-dimensional syntax. The first,the so-called “entitative graphs,” is based on disjunctionand negation. A version of the entitative graphs later appeared inG. Spencer Brown’sLaws of Form, without anything remotelylike proper citation of Peirce. A second, and better, system ofgraphical two-dimensional syntax followed: the so-called“existential graphs.” This system is based on conjunctionand negation. Even though the syntax is two dimensional, the surfaceit actually requires in its most general form is a torus of finitegenus. So, the system of existential graphs actually requires threedimensions for its representations, although the third dimension inwhich the torus is embedded can usually be represented in twodimensions by the use of pictorial devices that Peirce called“fornices” or “tunnel-bridges” and by the useof identificational devices that Peirce called“selectives.” The existential graphs are essentially asyntax for logic that uses the whole mathematical apparatus oftopological graph theory. There are three parts of it: alpha (forpropositional logic), beta (for quantificational logic with identitybut without functions), and gamma (for modal logic andmeta-logic).

In 1873 Peirce published a long paper (completed in 1870)“Description of a Notation for the Logic of Relatives” inwhich he introduced (six years before Frege’sBegriffsschrift was published) a complete syntax for thelogic of relations of arbitrary adicity (or: arity). In this paper thenotion of the variable (though not under the name“variable”) was invented, and Peirce provided devices fornegating, for combining relations (basically by building upon deMorgan’s relative product and relative sum), and for quantifyingexistentially and universally. By 1883, along with his studentO. H. Mitchell, Peirce had developed a full syntax forquantificational logic that was only a very little different (as wasmentioned just above) from the standard Russell-Whitehead syntax,which did not appear until 1910 (with no adequate citations ofPeirce).

Peirce introduced the material-conditional operator into logic,developed the Sheffer stroke and dagger operators 40 years beforeSheffer, and developed a full logical system based only on the strokefunction. As Garret Birkhoff notes in hisLattice Theory itwas in fact Peirce who invented the concept of a lattice (around1883). (Quite possibly, it is Peirce’s lattice theory that holds thekey to his technical theory of infinitesimals and the continuum.)

During his years teaching at Johns Hopkins University, Peirce began toresearch the four-color map conjecture, to work on the graphicalmathematics of de Morgan’s associate A. B. Kempe, and to developextensive connections between logic, algebra, and topology, especiallytopological graph theory. Ultimately these researches bore fruit inhis existential graphs, but his writings in this area also contain aconsiderable number of other valuable ideas and results. He hintedthat he had made great progress in the theory of provability andunprovability by exploring the connections between logic andtopology.

14. Peirce’s Reduction Thesis

Peirce’s so-called “Reduction Thesis” is the thesis thatall relations, relations of arbitrary adicity, may be constructed fromtriadic relations alone, whereas monadic and dyadic relations aloneare not sufficient to allow the construction of even a single“non-degenerate” (that is: non-Cartesian-factorable)triadic relation. Although the germ of his argument for the ReductionThesis lay in his 1873 [1870] paper “Description of a Notation for theLogic of Relatives,” the Thesis was for over a century doubtedby many, especially after the publication of a proof by Willard VanOrman Quine that all relations could be constructed exclusively fromdyadic ones. As it turns out, both Peirce and Quine were correct: theissue entirely depends on exactly what constructive resources are tobe allowed to be used in building relations out of otherrelations. (Obviously, the more extensive and powerful are theconstructive resources, the more likely it is that all relations canbe constructed from dyadic ones alone by using them.) An exactexposition and proof of Peirce’s Reduction Thesis was finallyaccomplished in 1988 (Burch 1991), and it makes clear that Peirce’sconstructive resources are to be understood to include only negation,a generalization of de Morgan’s relative product operation, and theuse of a particular triadic relation that Peirce called “theteridentity relation” and that we might today write asx = y= z.

Peirce felt that the teridentity relation was in some way moreprimitive logically and thus more fundamental than the usual dyadicidentity relationx = y, which he derived from two instancesof the triadic identity relation by two applications of the relativeproduct operation of de Morgan. Peirce also felt that de Morgan’srelative product operation was logically a more primitive andfundamental operation than, say, the Boolean product or the Booleansum. The full philosophical import of his Reduction Thesis, and thephilosophical importance of his triadism insofar as this triadismrests on his Reduction Thesis, cannot be ascertained without a priorunderstanding of his non-typical theory of identity and his specialview of the fundamental nature of the relative product operation.

15. Contemporary Practical Applications of Peirce’s Ideas

Currently, considerable interest is being taken in Peirce’s ideas byresearchers wholly outside the arena of academic philosophy. Theinterest comes from industry, business, technology, intelligenceorganizations, and the military; and it has resulted in the existenceof a substantial number of agencies, institutes, businesses, andlaboratories in which ongoing research into and development ofPeircean concepts are being vigorously undertaken.

This interest arose, originally, in two ways. First, some thirty yearsago in the former Soviet Union interest in Peirce and Karl Popper hadled logicians and computer scientists like Victor Konstantinovich Finnand Dmitri Pospelov to try to find ways in which computer programscould generate Peircean hypotheses (Popperian“conjectures”) in “semeiotic” contexts(non-numerical or qualitative contexts). Under the guide in particularof Finn’s intelligent systems laboratory in VINITI-RAN (theAll-Russian Institute of Scientific and Technical Information of theRussian Academy of Sciences), elaborate techniques for automaticgeneration of hypotheses were found and were extensively utilized formany practical purposes. Finn called his approach to hypothesisgeneration the “JSM Method of Automatic HypothesisGeneration” (so named for similarities to John Stuart Mill’smethods for identifying causes). Among the purposes for which the JSMMethod has proved fruitful are sociological prediction,pharmacological discovery, and the analysis of processes of industrialproduction. Interest in Finn’s work, and through it in the practicalapplication of Peirce’s philosophy, has spread to France, Germany,Denmark, Finland, and ultimately the United States.

Second, as the limits of expert systems and production ruleprogramming in the area of artificial intelligence became increasinglyclear to computer scientists, they began to search for methods beyondthose that depended merely on imitating experts. One promising line ofresearch has been to automate phases of (Peirce’s concept of) thescientific method, complete with techniques for hypothesis generationand making assessments of the costs and benefits of exploringhypotheses. In some areas of research added impetus has been providedby the similarity of Peircean techniques to techniques that havealready proven useful. For example, in the field of automatedmulti-track radar, the similarity of Peircean scientific method to theso-called “Kalman filter” has been noted by many systemsanalysts. Again, those interested in military command-and-controloften note the similarity of Peircean scientific method to the classicOODA loop (“observe, orient, decide, act”) ofcommand-and-control-theory. The aerospace industry, especially inFrance and the United States, is currently investigating Peirceanideas in connection with avionics systems that monitor aircraft“health.”

Almost simultaneously with Finn’s development of Automatic Generationof Hypotheses, German mathematicians Rudolf Wille and Bernhardt Ganterwere developing an aspect of Galois Theory and lattice theory (thelatter being, as was said, Peirce’s invention) that came to be knownas “Formal Concept Analysis.” Interestingly enough, even though thetwo groups of researchers initially were working completelyindependently of each other, the mathematical apparatus of Finn’sAutomatic Generation of Hypotheses is at its core the very sameapparatus as that of Wille’s and Ganter’s Formal Concept Analysis.For obvious reasons, then, there has now grown up an extensivecooperation between the German researchers and the Russianresearchers, principally through the writings and intermediary work ofSergei Kuznetsov, who has been working both with the German group andwith the Russian group.

The heart of both sets of ideas is the notion of clustering items bysimilarity. The algorithms for clustering into formal concepts are thesame as the algorithms for preliminary groupings by similarity for thepurpose of automatically generating hypotheses. As it turns out, andas Kuznetsov has shown, these algorithms are equivalent in theireffect to algorithms for finding the maximal complete subgraphs ofarbitrary graphs. This fact has proved extremely useful in recentyears, since the latter algorithms are the core of what has come to beknown as “Social Network Analysis.” And Social Network Analysis hasbecome a major intellectual tool in the world’s battles againstcriminal organizations and terrorist networks. So all three sets ofideas have become matters of crucial practical importance and evenurgency in contemporary affairs.

Such practical applications of Peircean ideas may seem surprising tomany philosophers whose minds are rooted strictly in the academicworld. The applications, however, most certainly would not havesurprised Peirce in the least. Indeed, given his lifelong ideas andgoals as a scientist-philosopher, he probably would have found thecurrent practical importance of his ideas entirely to be expected.

16. Significant Students of Peirce

During the time of Peirce’s teaching logic at Johns HopkinsUniversity, that is: during the years from 1879 through 1884, Peircehad a number of students in logic who then went on to establishsignificant reputations in their own right. Often mentioned in thisconnection are John Dewey, Allan Marquand, Christine Ladd-Franklin,Oscar Howard Mitchell, Benjamin I. Gilman, Fabian Franklin, and Thorstein Veblen. Here we provide brief descriptions ofthree of these students, Dewey, Ladd-Franklin, and Mitchell. Ofnecessity the accounts given here of the work of these students willbe extremely brief. It is obvious that full-length accounts of each ofthem can be given, and in the case of one of them, John Dewey,full-length accounts have, indeed, often been given.

Without a doubt, the best known of all Peirce’s students, even including Thorstein Veblen, is John Dewey (1859–1952). Dewey attended Peirce’s logic class at John’sHopkins during the years 1882 through 1883. Along with Peirce, Deweyunderstood the subject of logic in an extremely broad way, so that thesubject in his mind, as well as in Peirce’s mind, comprised the entiretopic of the methodology of the exact sciences. It is not surprising,then, that the structure of logic in Dewey’s own works in logic,viz. most notably in his bookLogic: the Theory of Inquiry(1938), is a close approximation of the structure of the scientificmethod as Peirce understands it. Recall that for Peirce inquiry beginswith an anomalous situation, in which a particular puzzle or set ofpuzzles is elicited from an indeterminate background. Then, by anongoing, and in fact ultimately endless, process, hypotheses areformulated (abduction) and tested (deduction and induction). so thatat each iteration of the methodological “loop,” indeterminacy as tothe original anomalous situation is successively, though nevertotally, eliminated. Dewey’s own account of inquiry, especiallyinsofar as Dewey considers it to be the successive elimination ofindeterminacy, is remarkably akin to Peirce’s account of scientificmethodology in action. Of course there are differences. Dewey oftenwrites as though at each stage of development of the method of logic(of science) the next stage is more or less already specified; forexample, at any stage of indeterminacy, Dewey writes as if therelevant hypothesis or hypotheses to test at the next stage are moreor less already determined. By contrast, Peirce (although he sometimesspeaks this way) more often emphasizes the creative and non-determinedaspect of eliciting/developing hypotheses. Nevertheless, even despitethe differences in emphasis from Peirce to Dewey, the similaritiesbetween their positions are unmistakeable.

Unlike Dewey, both Christine Ladd and Oscar Howard Mitchellconcentrated on formal deductive logic, i.e. mathematical logic,rather than on the informal methodology of the sciences. Also unlikeDewey, Ladd and Mitchell both published articles in the 1883volumeStudies in Logic, which was edited by Peirce. Again,in his own “Preface” to this volume Peirce singles outboth Ladd and Mitchell for commentary and overt praise(p.v). To these two students of Peirce we now turn.

Christine Ladd, born December 1, 1847, was later (from her marriage toFabian Franklin in 1882) known as Christine Ladd-Franklin. Althoughshe is almost unknown today, she was very well-known andhighly-regarded as a mathematician, logician, and psychologist fromthe 1870s until her death on March 5, 1930 at the age of 82. Herearliest published work was in mathematics, in theEducationalTimes of England. Here it attracted the attention of the greatmathematician J. J. Sylvester, who also published in theEducational Times and who in 1876 assumed the position of thefirst professor of mathematics at the new Johns HopkinsUniversity. (It was Sylvester who hired Peirce in 1879.) On thestrength of her already-published work, as known by Sylvester, Laddapplied to Johns Hopkins in 1888 to become a graduate student inmathematics. Although, because she was a woman, she could not becomea regular graduate student at Johns Hopkins, still at the ardent andinsistent urgings of Sylvester, she was admitted as a special-statusstudent, and she was even awarded a fellowship to studymathematics. She held this fellowship from 1879 until 1882, when shecompleted all the requirements for the Ph.D. degree. It was duringthis period that she became attracted to mathematical logic, and tothe teachings on logic of Peirce, with whom she studiedcarefully. Because of her gender, her status as a student ofmathematics was recorded only in notes rather than on the usualstudent lists. For the same reason, she could not actually be giventhe Ph.D. degree in 1882, although she was ultimately (but not until1926), awarded the Ph.D. degree in mathematics. (Meanwhile, VassarCollege awarded her an LL.D. degree in 1887.) On August 24, 1882, uponthe completion of her mathematics fellowship, she married a member ofthe Johns Hopkins department of mathematics and a fellow-student ofPeirce’s, Fabian Franklin (b. 1853, d. 1939). Both she and FabianFranklin seem to have stayed closely connected to Peirce until hisdeath in 1914.

As can be gathered from the foregoing, it was studying with Peircethat focussed Ladd’s attention from mathematics in general tomathematical logic (also called symbolic logic) in particular. Ladd’sbest-known, and most-celebrated work was her paper “On the Algebra ofLogic,” published inStudies in Logic by the members of the JohnsHopkins University (C. S. Peirce, editor), Little, Brown, 1883,pp. 17–71. In it, she is (or at least was once) widely regardedas having achieved for the first time in history a completely generaland unified account of the Aristotelian syllogism, including a generalaccount of the differences between valid and invalidsyllogisms. (Josiah Royce, for example, held this view ofLadd-Franklin’s work in logic.) It is thus worth looking in somedetail at how she achieves this result through the algebraizing of thesyllogism, but here such detail will not be entered into. Basically,however, we can say that her basic idea is to associate with eachpossible syllogism a certain “triad”. The syllogism isvalid if and only if the “triad” is an“antilogism,” that is to say, is an inconsistent“triad”. Ladd-Franklin wrote on the antilogism as late as1928 in volume 37 ofMind, pp. 532–534. Whether or nother work is really the first work to unify the theory of thesyllogism, and whether or not she ultimately merits the exalted statusas a logician that Royce assigns to her must remain a task for futureexploration to determine.

In addition to her major work on the syllogism, equivalently on thenotion of the antilogism, Ladd-Franklin as well wrote some of theentries in the three-volume work of James Mark Baldwin(1861–1934),Dictionary of Philosophy and Psychology,1901–1905, for example the entries “syllogism,”“symbolic logic,” and “algebra of logic.” Infact she was an associate editor of Baldwin’sDictionary. Shealso wrote at least one related paper in theJournal ofPhilosophy, Psychology, and Scientific Method, later named simplytheJournal of Philosophy.

Peirce praised O. H. Mitchell as being his very best student inlogic. Mitchell was trained as a mathematician even befoe coming toJohns Hopkins, and he was especially noted by all those who knew himas being extraordinarily meticulous and attentive to detail. He was soattentive to exactitude that he was even noted for slowness andponderousness in speech. But what was there in Mitchell’s 1883 paperthat stimulated Peirce’s high praise? It seems to be closelyconnected, perhaps identical, with what Peirce praises Mitchell foraccomplishing in Peirce’s entry in Baldwin’sDictionary ofPhilosophy and Psychology, particularly in his article there forthe word “dimension.” In the article Peirce creditsMitchell with having introduced “the concept of amultidimensional logical universe” into exact logic, and Peirceclaims that it is one of the “fecund contributions” thatMitchell made to exact logic. There seem to be two components to theidea of a multidimensional logical universe. The first one is simplythe idea of multiple quantification in connection with polyadicpredicates, so that (in modern terminology) we can have propositionswith quantifiers occurring within the scope of other quantifiers. ThatMitchell employs a formalism suggestive of this idea is undeniable,but whether he successfully distinguishes (for allx)(thereexists ay) from (there exists ay)(forallx) is very much less than obvious. Moreover, thatMitchell was the very first logician to have conceived of multiplequantification and polyadic predicates seems a bit dubious, especiallysince by 1870 Peirce himself was already making use of notions forwhich multiple quantification and polyadic predicates areinvolved. The second component of the idea of a multidimensionallogical universe is (again, using modern terminology) the idea thatthe universe of discourse relevant to one variable might not be thesame as the universe of discourse relevant to anothervariable. Moreover, since the universe of discourse relevant to onevariable might be all the times there are, all the places there are,all possible situations, or even all possible worlds, it is notdifficult (though it is perhaps a bit far-fetched) to connectMitchell’s idea of a multidimensional logical universe to the idea oftense logic or even modal logic. Whether a careful study of Mitchell’slogical contribution really supports such a reading of Mitchell’scontributions to logic is not as yet clearly determined.

Clearly, close study of the logical accomplishments of Peirce’snotable students is in order.

Bibliography

Primary Sources

Peirce’s Writings

• 1873 [1870], “Description of a Notation for the Logic ofRelatives, Resulting from an Amplification of the Conceptions ofBoole's Calculus of Logic,”Memoirs of the American Academyof Arts and Sciences (New Series), 9(2): 317–378.[Peirce 1873 [1870] available online]
• Collected Papers of Charles SandersPeirce (CP), 8 volumes, edited by CharlesHartshorne, Paul Weiss, and Arthur W. Burks (Harvard University Press,Cambridge, Massachusetts, 1931–1958; volumes 1–6 edited byCharles Harteshorne and Paul Weiss, 1931–1935; vols. 7–8edited by Arthur W. Burks, 1958).
• The Essential Peirce (EP), 2 volumes,edited by Nathan Houser, Christian Kloesel, and the Peirce EditionProject (Indiana University Press, Bloomington, Indiana, 1992,1998).
• The New Elements of Mathematics by Charles S. Peirce,Volume I Arithmetic, Volume II Algebra and Geometry, Volume III/1 andIII/2 Mathematical Miscellanea, Volume IV Mathematical Philosophy.Edited by Carolyn Eisele (Mouton Publishers, The Hague, 1976).
• Historical Perspectives on Peirce’s Logic of Science: A History of Science, 2 Parts. Edited by Carolyn Eisele (de Gruyter & Co., Berlin, 1985).
• Pragmatism as a Principle and Method of Right Thinking: the 1903 Harvard Lectures on Pragmatism by Charles Sanders Peirce. Edited, Introduced, and with a Commentary by Patricia Ann Turrisi (State University of New York Press, Albany, New York, 1997).
• Reason and the Logic of Things: The Cambridge ConferencesLectures of 1898. Edited by Kenneth Laine Ketner (HarvardUniversity Press, Cambridge, Massachusetts, 1992).
• Semiotic and Significs: The Correspondence between Charles S. Peirce and Victoria Lady Welby. Edited by Charles S. Hardwick (Bloomington: Indiana University Press, 1977).
• Writings of Charles S. Peirce: a Chronological Edition,Volume I 1857–1866, Volume II 1867–1871, Volume III1872–1878, Volume IV 1879–1884, Volume V1884–1886, Volume VI 1886–1890, Volume VIII 1890–1892. Edited by the Peirce Edition Project (Indiana University Press, Bloomington, Indiana, 1982, 1984, 1986, 1989,1993, 2000, 2010).
• Peirce, Charles S. (Editor), 1883, 1983,Studies inLogic, Introduction by Max Fisch, Preface by Achim Eschenbach,Amsterdam/Philadelphia: John Benjamins Publishing Company.
• “The Architecture of Theories,”, 1891,The Monist, 1(2), pp. 161–176.

Work by Others

• Baldwin, James Mark, 1901–1905,Dictionary of Philosophyand Psychology, 3 volumes, New York: The Macmillan Company.
• Dewey, John, 1916,Essays in Experimental Logic, Chicago,Ill.: University of Chicago Press.
• Dewey, John, 1938,Logic: The Theory of Inquiry, NewYork: H. Holt.
• Hegel, Georg Wilhelm Friedrich,The Phenomenology of Mind, tr., intro, and notes by J. B. Baillie, New York: The Macmillan Company, 1910.
• Kant, Immanuel,Critique of Pure Reason, tr. by Norman Kemp Smith, New York: The Macmillan Company, 2003.
• Schiller, Friedrich,On the Aesthetic Education of Man, in aSeries of Letters, tr. and intro. by Reginald Snell, New York:F. Unger Publishing Company, 1965.
• Schroeder, Ernst,Vorlesungen über die Algebra der Logik(exakte Logik), 3 volumes, Bronx, N. Y.: Chelsea PublishingCompany, 1966.

Secondary Sources

• Ambrosio, C., 2016. “The Historicity of Peirce’sClassification of the Sciences,”European Journal ofPragmatism and American Philosophy, 8: 9–43.
• Burch, Robert, 1991,A Peircean Reduction Thesis: Foundationsof Toplogical Logic, Lubbock, TX: Texas Tech UniversityPress.
• Dipert, Randall R., 1994, “The Life and LogicalContributions of O. H. Mitchell, Peirce’s GiftedStudent,”Transactions of the Charles S. PeirceSociety, 30 (3): 515–542.
• Eisele, Carolyn, 1979,Studies in the Scientific and Mathematical Philosophy of Charles S. Peirce, The Hague: Mouton.
• Hacking, Ian, 1990,The Taming of Chance, Cambridge: Cambridge University Press, especially Chapter 23.
• Houser, Nathan, Roberts, Don D., and Van Evra, James, (eds.),1997,Studies in the Logic of Charles Sanders Peirce, Bloomington and Indianapolis: Indiana University Press.
• Houser, Nathan, 2005, “The Scent ofTruth,”Semiotica, 153 (1/4): 455–466.
• Hudry, Jean-Louis, 2004, “Peirce’s Potential Continuity andPure Geometry,”Transactions of the Charles S. PeirceSociety, 40 (2): 229–243.
• Hurvich, Dorothea Jameson, 1971, “Ladd-Franklin, Christine(Dec. 1, 1847–Mar. 5, 1930,” inNotable AmericanWomen: 1607–1950, Edward T. James, Janet Wilson James, andPaul S.Boyer (Eds.), Radcliffe College.
• Ketner, Kenneth Laine, 1998,His Glassy Essence: AnAutobiography of Charles S. Peirce, Volume 1, Nashville, TN:University of Vanderbilt Press.
• Ketner, Kenneth Laine, (ed.), 1995,Peirce and Contemporary Thought: Philosophical Inquiries, New York: Fordham University Press.
• Martin, R. M., (ed.), 1979,Studies in the Scientific andMathematical Philosophy of Charles S. Peirce: Essays by CarolynEisele, The Hague: Mouton.
• Marty, Robert, 1990,L’algèbra des signes, Amsterdam: John Benjamins.
• Marty, Robert, 1982, “C. S. Peirce’s Phaneroscopy and Semiotics,”Semiotica, 41 (1/4): 169–181.
• Mayo, Deborah G., 1996,Error and the Growth of ExperimentalKnowledge, Chicago: University of Chicago Press.
• Mayo, Deborah G., 2005, “Peircean Induction and theError-Correcting Thesis,”Transactions of the Charles S. PeirceSociety, 41 (2): 299–319.
• McLaughlin, Thomas G., 2004, “C. S. Peirce’s Proof of Frobenius’Theorem on Finite-Dimensional Real Associative Division Algebras,”Transactions of the Charles S. Peirce Society, 40 (2):701–710.
• Niiniluoto, I., 1984, “Notes on Popper as Follower ofWhewell and Peirce,” in I. Niiniluoto,Is ScienceProgressive?, Dordrecht: D. Reidel.
• Paavola, Sami, 2004, “Abduction Through Grammar, Critic, andMethodeutic,”Transactions of the Charles S. Peirce Society,40 (2): 245–270.
• Pietarinen, Ahti-Veikko, 2005,Signs of Logic: Peircean Themeson the Philosophy of Language, Games, and Communication, Dordrecht: Springer.
• Pollard, Stephen, 2005, “Some Mathematical Facts about Peirce’sGames,”Transactions of the Charles S. Peirce Society, 41 (1):189–201.
• Quieroz, João and Merrell, Floyd, 2005, “Abduction: BetweenSubjectivity and Objectivity,”Semiotica, 153 (1/4): 1–7.
• Royce, Josiah, 1959,The World and the Individual, First Series, New York: Dover.
• Russinoff, I. Susan, 1999, “The Syllogism’s FinalSolution,”The Bulletin of Symbolic Logic, 5 (4):451–469.
• Thayer, H. S., 1952,The Logic of Pragmatism: an Examinationof John Dewey’s Logic. New York: Humanities Press.
• Thayer, H. S., 1973,Meaning and Action: A Critical Expositionof American Pragmatism, Indianapolis: Indiana UniversityPress.
• Wagenmakers, E.-J., Dutilh, G., and Sarafoglou, A., 2018, “TheCreativity-Verification Cycle in Psychological Science: New Methods toCombat Old Idols,”Perspectives on Psychological Science, 13:418–427. doi:10.1177/1745691618771357

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