Stanford Encyclopedia of Philosophy 7. Probability, Verisimilitude, and Plausibility Given Peirce's tychism and his view that statistical information is often the best information we can have about phenomena, it should not be surprising that Peirce devoted close attention to the analysis of situations in which perfect exactness and perfect certitude were unattainable. It is only to be expected that he would devote a great deal of attention, for example, to probability theory. Indeed, Peirce did so from the dates of even his earliest thinking. Not only, for example, did he extensively employ the concept of probability, but also he offered a pragmaticistic account of the notion of probability itself. Yet it would be a huge mistake to think that Peirce's philosophizing about situations of imperfection of exactness and certitude were confined merely to the theory of probability. Rather, from the outset of his thinking about the matters, in about 1863, his attention was directed to the broadest sorts of issues connected with statistical inference. And, as his thinking progressed, Peirce came ever more clearly to see that there are three distinct and mutually incommensurable measures of imperfection of certitude. Only one was probability. The other two he called “verisimilitude” (or “likelihood”) and “plausibility”. Each of the three measures was associated with one of his types of argument. Probability he associated with deduction. Verisimilitude he associated with induction. And plausibility he associated with abduction. Let us look more closely at each of these three distinct measures of uncertainty. By the time Peirce wrote on probability, the concept and its calculus were well over two hundred years old. The probability calculus itself had become more or less standardized by Peirce's time, and indeed Peirce's own axioms for the calculus are more or less the same that Kolmogoroff gives for his “ elementary theory of probability.” By contrast with the calculus, the philosophical theory of the meaning of probability was hotly disputed. Two sides to the dispute existed. There were the subjectivists, or “conceptualists,” as Peirce designated them. These believed that probability was a measure of the strength of belief actually accorded to a proposition or a measure of the degree of rational belief that ought to be accorded to a proposition. Among the defenders of this sort of view, Augustus de Morgan and Adolphe Quetelet were major figures. And there were the objectivists, or “ materialists,” as Peirce designated them. These believed that probability was a measure of the relative frequency with which an event of some specific sort repeatedly happened. John Venn was a major defender of this sort of view. Pierre Simon Laplace had spoken sometimes in a subjectivist way, sometimes in an objectivist way; but his arguments basically depended on a subjectivist interpretation of probability. Peirce vigorously attacked the subjectivist view of de Morgan and others to the effect that probability is merely a measure of our level of confidence or strength of belief. He allowed that the logarithm of the odds of an event might be used to assess the degree of rational belief that should be accorded to an event, but only provided that the odds of the event were determined by the objective relative frequency of the event. In other words, Peirce suggested that any justifiable use of subjectivism in connection with probability theory must ultimately rest on quantities obtained by means of an objectivist understanding of probability. Rather than holding that probability is a measure of degree of confidence or belief, then, Peirce adopted an objectivist notion of probability that he explicitly likened to the doctrine of John Venn. Indeed, he even held that probability is actually a notion with clear empirical content and that there are clear empirical procedures for ascertaining that content. First, he held, that that to which a probability is assigned, insofar as the notion of probability is used scientifically, is not a proposition or an event or a state; nor is it a type of event or state. Rather, what is assigned a probability is an argument, an argument having premisses (Peirce insisted on this spelling rather than the spelling “premises.”) and a conclusion. Peirce's view in this regard is virtually indistinguishable from the view of Kolmogoroff that all probabilities are conditional probabilities. Second, Peirce held that, in order to ascertain the probability of a particular argument, the observer notes all occasions on which all of its premisses are true, case by case, just as they come under observation. For each of these occasions the observer notes whether the conclusion is true or not. The observer keeps a running tally, the ongoing ratio whose numerator is the number of occasions so far observed on which the conclusion as well as the premisses are true and whose denominator is the number of occasions so far observed on which the premisses are true irrespective of whether or not the conclusion is also true. At each observation the observer computes this ratio, which obviously encompasses all the observer's past observations of occasions on which the premisses are true. The probability of the argument in question is defined by Peirce to be the limit of the crucial ratio as the number of observations tends to grow infinitely large (if this limit exists). Peirce's earliest account of the meaning of probability, then, is a version of what is called the “long run relative frequency view” of probability. Late in his philosophical career, about 1910, Peirce found fault with his earliest views on account of their failure to make clear just how the occasions of observation are to be chosen. He also emphasized that probability judgments are judgments about what he called “would-be's.” For this reason Peirce is often considered to be the originator of the sort of “propensity view” of probability that is associated with Karl Popper. One should not, however, think that viewing Peirce as a propensity theorist is in conflict with viewing him as some sort of long term relative frequency theorist. Rather, Peirce's view seems to be that the propensity in question, when its sense is spelled out in accord with the pragmatic (or: pragmaticistic) theory of meaning, is a dispositional property that manifests itself in the set of concrete facts that amount to a certain long term relative frequency tending toward a certain limit as the number of appropriate occasions of observation 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 ultimately logical. 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, Peirce also attacked the use of what came to be called the method of “inverse probabilities” as a way of solving the problem of induction. In the process, he also excoriated the theoretical work, in this connection, of de Morgan and Adolphe Quetelet (the Belgian criminologist and early user of statistical analysis in sociology). Induction, as we have seen, Peirce counted as an inference from sample to population. The method of inverse probabilities offers itself as a way of calculating the (conditional) probability that a population has a trait in a certain proportion given that a sample drawn from that population has the trait in that proportion. It proposes to calculate this conditional probability by applying the so-called “Bayes's Theorem” in order to express it in terms of the (inverse conditional) probability that the sample has the trait in the crucial proportion given that the population has the trait in the crucial proportion. In the expression of the first conditional probability in terms of the second conditional probability, however, there occur certain quantities, known as the “Bayesian prior probabilities.” What Peirce pointed out is that there is no way to assign any quantities in a rational fashion to the requisite Bayesian prior probabilities. The appearance that one does have a reason for assigning particular quantities results only from an illicit substitution of subjective probabilities for the needed objectivist probabilities. What the user of the method of inverse probabilities does is to equate complete lack of information about something with the claim that all possibilities must have equal probabilities. This equation was called “the principle of insufficient reason” in the nineteenth century; John Maynard Keynes later named it “the principle of indifference.” This principle is, however, completely irrational without a dependence on a subjectivist account of probability. What we need, however, is objective probabilities, and so we have no reason for assigning any particular values to the Bayesian prior probabilities. Only “if universes were as plenty as blackberries,” wrote Peirce, would the 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 to inductive arguments. Instead of probability, a different measure of imperfection of certitude must be assigned to inductive arguments: verisimilitude or likelihood. In explaining this notion Peirce offered an account of hypothesis-testing that is equivalent to standard statistical hypothesis-testing. In effect we get an account of confidence intervals and choices of statistical significance for rejecting null hypotheses. Such ideas became standard only in the twentieth century as a result of the work of R. A. Fisher, Jerzy Neyman, and others. But already by 1878, in his paper “The Probabilitiy of Induction,” Peirce had worked out the whole matter. (This topic has been discussed expertly by Deborah Mayo, who also has shown that the error-correction implicit in statistical hypothesis-testing is intimately affiliated with Peirce's notion of science as self-correcting and convergent to “the truth.”) Peirce's accounts of his third type of deviance from perfect certitude, namely plausibility, are much sketchier than his accounts of probability and verisimilitude. Unlike the other two forms of uncertainty, which can be spelled out mathematically with great precision, plausibility seems to be capable of only a qualitative account, even though plausibility does seem to comes in greater and lesser degree. The question of the plausibility of a claim arises, apparently, only in contexts in which one is seeking to adduce an explanatory hypothesis for some actual fact that is surprising. The key point is that the hypothesis must be plausible in order to taken seriously. If we were, for example, to come upon a lump of ice in the middle of a desert, we might plausibly say that perhaps someone put it there, or perhaps a freak storm had left a great hailstone. But we would not plausibly say that it had been thrown off a flying saucer that previously had swooped through. It should be obvious that the notion of plausibility is a difficult one, which strongly invites further analysis but which is not easy to analyze in technical detail. 8. Psycho-physical Monism and Anti-nominalism Peirce held that science suggests that the universe has evolved from a condition of maximum freedom and spontaneity into its present condition, in which it has taken on a number of habits, sometimes more entrenched habits and sometimes less entrenched ones. With pure freedom and spontaneity Peirce tended to associate mind, and with firmly entrenched habits he tended to associate matter (or, more generally, the physical). Matter he tended to regard as “congealed” mind, and mind he tended to regard as “effete” matter. Thus he tended to see the universe as the end-product-so-far of a process in which mind has acquired habits and has “congealed” (this is the very word Peirce used) into matter. This notion of all things as being evolved psycho-physical unities of some sort places Peirce well within the sphere of what might be called “the grand old-fashioned metaphysicians,” along with such thinkers as Plato, Aristotle, Aquinas, Spinoza, Leibniz, Hegel, Schopenhauer, Whitehead, et al. Some contemporary philosophers might be inclined to reject Peirce out of hand upon discovering this fact. Others might find his notion of psycho-physical unities not so very offputting or indeed even attractive. What is crucial is that Peirce argued that mind pervades all of nature in varying degrees: it is not found merely in the most advanced animal species. This pan-psychistic view, combined with his synechism, meant for Peirce that mind is extended in some sort of continuum throughout the universe. Peirce tended to think of ideas as existing in mind in somewhat the same way as physical forms exist in physically extended things. He even spoke of ideas as “spreading” out through the same continuum in which mind is extended. This set of conceptions is part of what Peirce regarded as (his own version of) Scotistic realism, which he sharply contrasted with nominalism. He tended to blame what he regarded as the errors of much of the philosophy of his contemporaries as owing to its nominalistic disregard for the objective existence of form. 9. Triadism and the Universal Categories Merely to say that Peirce was extremely fond of placing things into groups of three, of trichotomies, and of triadic relations, would fail miserably to do justice to the overwhelming obtrusiveness in his philosophy of the number three. Indeed, he made the most fundamental categories of all “things” of any sort whatsoever the categories 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 he called a second, and necessity he called a third. Again: quality was a first, fact was a second, and habit (or rule or law) was a third. Again: entity was a first, relation was a second, and representation was a third. Again: rheme (by which Peirce meant a relation of arbitrary adicity or arity) was a first, proposition was a second, and argument was a third. The list goes on and on. Let us refer to Peirce's penchant for describing things in terms of trichotomies and triadic relations as Peirce's “triadism.” If Peirce had a general technical rationale for his triadism, Peirce scholars have not yet made it abundantly clear what this rationale might be. He seemed to base his triadism on what he called “phaneroscopy,” by which word he meant the mere observation of phenomenal appearances. He regularly commented that the phenomena in the phaneron just do fall into three groups and that they just do display irreducibly triadic relations. He seemed to regard this matter as simply open for verification by direct inspection. Although there are many examples of phenomena that do seem more or less naturally to divide into three groups, Peirce seems to have been driven by something more than mere examples in his insistence on applying his categories to almost everything imaginable. Perhaps it was the influence of Kant, whose twelve categories divide into four groups of three each. Perhaps it was the triadic structure of the stages of thought as described by Hegel. Perhaps it was even the triune commitments of orthodox Christianity (to which Peirce, at least in some contexts and during some swings of mood, seemed to subscribe). Certainly involved was Peirce's commitment to the ineliminability of mind in nature, for Peirce closely associated the activities of mind with the aforementioned triadic relation that he called the “sign” relation. (More on this topic appears below.) Also involved was Peirce's so-called “ reduction thesis” in logic (on which more will given below), to which Peirce had concluded as early as 1870. It is difficult to imagine even the most fervently devout of the passionate admirers of Peirce, of which there are many, saying that his account (or, more accurately, his various accounts) of the three universal categories is (or are) absolutely clear and compelling. Yet, in almost everything Peirce wrote from the time the categories were first introduced, Peirce's firsts, seconds, and thirds found a place. Giving their exact and general analysis and providing an exact and general account of their rationale, if there be such, constitute chief problems in Peirce scholarship. 10. Mind and Semeiotic Connected with Peirce's insistence on the ubiquity of mind in the cosmos is the importance he attached to what he called “semeiotic,” the theory of signs in the most general sense. Although a few points concerning this subject were made earlier in this article, some further discussion is in order. What Peircean meant by “semeiotic” is almost totally different from what has come to be called “semiotics,” and which hails not so much from Peirce as from Ferdinand de Saussure and Charles W. Morris. Even though Peircean semeiotic and semiotics are often confused, it is important not to do so. Peircean semeiotic derives ultimately from the theory of signs of Duns Scotus and its later development by John of St. Thomas (John Poinsot). In Peirce's theory the sign relation is a triadic relation that is a special species of the genus: the representing relation. Whenever the representing relation 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 third thing (the “interpretant.”) Moreover, the object is represented by the representamen in such a way that the interpretant is thereby “ determined” to be also a representamen of the object to yet another interpretant. That is to say, the interpretant stands in the representing relation to the same object represented by the original representamen, and thus the interpretant represents the object (either again or further) to yet another interpretant. Obviously, Peirce's complicated definition entails that we have an infinite sequence of representamens of an object whenever we have any one representamen of it. The sign relation is the special species of the representing relation that obtains whenever the first interpretant (and consequently each member of the whole infinite sequence of interpretants) has a status that is mental, i.e. (roughly) is a cognition of a mind. In any instance of the sign relation an object is signified by a sign to a mind. One of Peirce's central tasks was that of analyzing all possible kinds of signs. For this purpose he introduced various distinction among signs, and discussed various ways of classifying them. One set of distinctions among signs was introduced by Peirce in the early stages of his analysis. The distinctions in this set turn on whether the particular instance of the sign relation is “degenerate” or “non-degenerate.” The notion of “degeneracy” here is the standard mathematical notion, and as applied to sign theory non-degeneracy means simply that the triadic relation cannot be analyzed as a logical conjunction of any combination of dyadic relations and monadic relations. More exactly, a particular instance of the obtaining of the sign relation is degenerate if and only if the fact that a sign s means an object o to an interpretant i can be analyzed into a conjunction of facts of the form P(s) & Q(o) & R(i) & T(s,o) & U(o,i) & W(i,s) (where not all the conjuncts have to be present). Either an obtaining of the sign relation is non-degenerate, in which case it falls into one class; or it is degenerate in various possible ways (depending on which of the conjuncts are omitted and which retained), in which cases it falls into various other classes. Other distinctions regarding signs were introduced later by Peirce. Some of them will be discussed very briefly in the following section of this article. 11. Semeiotic and Logic Peirce's settled opinion was that logic in the broadest sense is to be equated with semeiotic (the general theory of signs), and that logic in a much narrower sense (which he typically called “logical critic”) is one of three major divisions or parts of semeiotic. Thus, in his later writings, he divided semeiotic into speculative grammar, logical critic, and speculative rhetoric (also called “methodeutic”). Peirce's word “speculative” is his Latinate version of the Greek-derived word “theoretical,” and should be understood to mean exactly what the word “theoretical” means. Peirce's tripartite division of semeiotic is not to be confused with Charles W. Morris's division: syntax, semantics, and pragmatics (although there may be some commonalities in the two trichotomies). By speculative grammar Peirce understood the analysis of the kinds of signs there are and the ways that they can be combined significantly. For example, under this heading he introduced three trichotomies of signs and argued for the real possibility of only certain 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 (also called “semes”), or symbols (sometimes called “tokens”), accordingly as they derive their significance from resemblance to their objects, a real relation (for example, of causation) with their objects, or are connected only by convention to their objects, respectively. Signs are rhematic signs (also called “sumisigns” and “rhemes”), dicisigns (also called “quasi-propositions”), or arguments (also called “suadisigns”), accordingly as they are predicational/relational in character, propositional in character, or argumentative in character. Because the three trichotomies are independent of each other, together they yield the abstract possibility that there are 27 distinct kinds of signs. Peirce argued, however, that 17 of these are logically impossible, so that finally only 10 kinds of signs are genuinely possible. In terms of these 10 kinds of signs, Peirce endeavored to construct a theory of all possible natural and conventional signs, whether simple or complex. What Peirce meant by “logical critic” is pretty much logic in the ordinary, accepted sense of “logic” from Aristotle's logic to present-day mathematical logic. As might be expected, a crucial concern of logical critic is to characterize the difference between correct and incorrect reasoning. Peirce achieved extraordinarily extensive and deep results in this area, and a few of his accomplishments in this area will be discussed below. By “speculative rhetoric” or “methodeutic” Peirce understood all inquiry into the principles of the effective use of signs for producing valuable courses of research and giving valuable expositions. Methodeutic studies the methods that researchers should use in investigating, giving expositions of, and creating applications of the truth. Peirce also understood, under the heading of speculative rhetoric, the analysis of communicational interactions and strategies, and their bearing on the evaluation of inferences. Peirce's important topic of the economy of research is closely affiliated with his idea of speculative rhetoric. The idea of methodeutic may overlap to some small extent with Morris's notion of “pragmatics,” but the spirit of Peirce's notion is much more extensive than that of Morris's notion. Moreover, Peirce handled the notion of indexical reference under the heading of speculative grammar and not under the heading of speculative rhetoric, whereas the topic 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 considerable philosophical attention in recent years, especially among Finnish Peirce scholars centering about the University of Helsinki. These have noted that there are extensive affiliations between Peirce's discussions of the communicational and dialogical aspects of semeiotic, on the one hand, and the many and varied “ game-theoretical” approaches to logic that have been for some time of interest to Finnish philosophers (as well as many others), on the other hand. Various proposals for game-theoretic semantical approaches to logic have been developed and applied to Peirce's logic, as well as being used to understand Peircean points. 12. The Classification of the Sciences Peirce maintained a considerable interest in the topic of classification or taxonomy in general, and he considered biology and geology the foremost sciences to have made progress in developing genuinely useful systems of classification for things. In his own theory of classification, he seemed to regard some sort of cluster analysis as holding the key to creating really useful classifications. He regularly strove to create a classification of all the sciences that would be as useful to logic as the taxonomies of the biologists and geologists were to these scientists. Of special interest in this regard is the fact that he considered the relation of similarity to be a triadic relation, rather than a dyadic relation. Thus, for Peirce taxonomies and taxonomic trees are only one sort of classificatory system, albeit the most highly-developed one. He would not be in the least surprised to find that the topic of constructing “ontologies” is in vogue among computer scientists, and he would applaud endeavors to construct ontologies. He would not find in the least alien many contemporary analytic discussions of the notion of similarity; he would be right at home among them. As with many of Peirce's classificatory divisions, his classification of the sciences is a taxonomy whose tree is trinary. For example he classifies all the sciences into those of discovery, review, and practicality. Sciences of discovery he divides into mathematics, philosophy, and what he calls “ idioscopy” (by which he seems to mean the class of all the particular or special sciences like physics, psychology, and so forth). Mathematics he divides into mathematics of logic, of discrete series, and of continua and pseudo-continua. Philosophy he divides into phenomenology, normative science, and metaphysics. Normative science he divides into aesthetics, ethics, and logic. And so on and on. Very occasionally there is found a binary division: for example, he divides idioscopy into the physical sciences and the psychical (or human) sciences. But, hardly surprisingly given his penchant for triads, most of his divisions are into threes. Peirce scholars have found the topic of Peirce's classification of the sciences a fertile ground for assertions about what is most basic in all thinking, in Peirce's view. Whether or not such assertions run afoul of Peirce's anti-foundationalism is itself a topic for further study. 13. Logic In the extensiveness and originality of his contributions to mathematical logic, Peirce is almost without equal. His writings and original ideas are so numerous that there is no way to do them justice in a small article such as the present one. Accordingly, only a few of his numerous achievements will be mentioned here. Peirce's special strength lay not so much in theorem-proving as rather in the invention and developmental elaboration of novel systems of logical syntax and fundamental logical concepts. He invented dozens of different systems of logical syntax, including a syntax based on a generalization of de Morgan's relative product operation, an algebraic syntax that mirrored Boolean algebra to some extent, a quantifier-and-variable syntax that (except for its specific symbols) is identical to the much later Russell-Whitehead syntax. He even invented two systems of graphical two-dimensional syntax. The first, the so-called “entitative graphs,” is based on disjunction and negation. A version of the entitative graphs later appeared in G. Spencer Brown's Laws of Form, without anything remotely like proper citation of Peirce. A second, and better, system of graphical two-dimensional syntax followed: the so-called “existential graphs.” This system is based on conjunction and negation. Even though the syntax is two dimensional, the surface it actually requires in its most general form is a torus of finite genus. So, the system of existential graphs actually requires three dimensions for its representations, although the third dimension in which the torus is embedded can usually be represented in two dimensions by the use of pictorial devices that Peirce called “fornices” or “tunnel-bridges” and by the use of identificational devices that Peirce called “selectives.” The existential graphs are essentially a syntax for logic that uses the whole mathematical apparatus of topological graph theory. There are three parts of it: alpha (for propositional logic), beta (for quantificational logic with identity but without functions), and gamma (for modal logic and meta-logic). In 1870 Peirce published a long paper “Description of a Notation for the Logic of Relatives” in which he introduced for the first time in history, two years before Frege's Begriffschrift a complete syntax for the logic of relations of arbitrary adicity (or: arity). In this paper the notion of the variable (though not under the name “variable”) was invented, and Peirce provided devices for negating, for combining relations (basically by building upon de Morgan's relative product and relative sum), and for quantifying existentially and universally. By 1883, along with his student O. H. Mitchell, Peirce had developed a full syntax for quantificational logic that was only a very little different (as was mentioned just above) from the standard Russell-Whitehead syntax, which did not appear until 1910 (with no adequate citations of Peirce). Peirce introduced the material-conditional operator into logic, developed the Sheffer stroke and dagger operators 40 years before Sheffer, and developed a full logical system based only on the stroke function. As Garret Birkhoff notes in his Lattice Theory it was in fact Peirce who invented the concept of a lattice (around 1883). (Quite possibly, it is Peirce's lattice theory that holds the key to his technical theory of infinitesimals and the continuum.) During his years teaching at Johns Hopkins University, Peirce began to research the four-color map conjecture, to work on the graphical mathematics of de Morgan's associate A. B. Kempe, and to develop extensive connections between logic. algebra, and topology, especially topological graph theory. Ultimately these researches bore fruit in his existential graphs, but his writings in this area also contain a considerable number of other valuable ideas and results. He hinted that he had made great progress in the theory of provability and unprovability by exploring the connections between logic and topology. 14. Peirce's Reduction Thesis Peirce's so-called “Reduction Thesis” is the thesis that all relations, relations of arbitrary adicity, may be constructed from triadic relations alone, whereas monadic and dyadic relations alone are 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 Reduction Thesis lay in his 1870 paper “Description of a Notation for the Logic of Relatives,” the Thesis was for over a century doubted by many, especially after the publication of a proof by Willard Van Orman Quine that all relations could be constructed exclusively from dyadic ones. As it turns out, both Peirce and Quine were correct: the issue entirely depends on exactly what constructive resources are to be allowed to be used in building relations out of other relations. (Obviously, the more extensive and powerful are the constructive resources, the more likely it is that all relations can be constructed from dyadic ones alone by using them.) An exact exposition and proof of Peirce's Reduction Thesis was finally accomplished in 1988 (Burch 1991), and it makes clear that Peirce's constructive resources are to be understood to include only negation, a generalization of de Morgan's relative product operation, and the use of a particular triadic relation that Peirce called “the teridentity relation” and that we might today write as x = y = z. Peirce felt that the teridentity relation was in some way more primitive logically and thus more fundamental than the usual dyadic identity relation x = y, which he derived from two instances of the triadic identity relation by two applications of the relative product operation of de Morgan. Peirce also felt that de Morgan's relative product operation was logically a more primitive and fundamental operation than, say, the Boolean product or the Boolean sum. The full philosophical import of his Reduction Thesis, and the philosophical importance of his triadism insofar as this triadism rests on his Reduction Thesis, cannot be ascertained without a prior understanding of his non-typical theory of identity and his special view 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 by researchers wholly outside the arena of academic philosophy. The interest comes from industry, business, technology, intelligence organizations, and the military; and it has resulted in the existence of a substantial number of agencies, institutes, businesses, and laboratories in which ongoing research into and development of Peircean concepts are being vigorously undertaken. This interest arose, originally, in two ways. First, some thirty years ago in the former Soviet Union interest in Peirce and Karl Popper had led logicians and computer scientists like Victor Konstantinovich Finn and Dmitri Pospelov to try to find ways in which computer programs could generate Peircean hypotheses (Popperian “conjectures”) in “semeiotic” contexts (non-numerical or qualitative contexts). Under the guide in particular of Finn's intelligent systems laboratory in VINITI-RAN (the All-Russian Institute of Scientific and Technical Information of the Russian Academy of Sciences), elaborate techniques for automatic generation of hypotheses were found and were extensively utilized for many practical purposes. Finn called his approach to hypothesis generation the “JSM Method of Automatic Hypothesis Generation ” (so named for similarities to John Stuart Mill's methods for identifying causes). Among the purposes for which the JSM Method has proved fruitful are sociological prediction, pharmacological discovery, and the analysis of processes of industrial production. Interest in Finn's work, and through it in the practical application of Peirce's philosophy, has spread to France, Germany, Denmark, Finnland, and ultimately the United States. Second, as the limits of expert systems and production rule programming in the area of artificial intelligence became increasingly clear to computer scientists, they began to search for methods beyond those that depended merely on imitating experts. One promising line of research has been to automate phases of (Peirce's concept of) the scientific method, complete with techniques for hypothesis generation and making assessments of the costs and benefits of exploring hypotheses. In some areas of research added impetus has been provided by the similarity of Peircean techniques to techniques that have already proven useful. For example, in the field of automated multi-track radar, the similarity of Peircean scientific method to the so-called “ Kalman filter” has been noted by many systems analysts. Again, those interested in military command-and-control often note the similarity of Peircean scientific method to the classic OODA loop (“observe, orient, decide, act”) of command-and-control-theory. The aerospace industry, especially in France and the United States, is currently investigating Peircean ideas in connection with avionics systems that monitor aircraft “health.” Almost simultaneously with Finn's development of Automatic Generation of Hypotheses, German mathematicians Rudolf Wille and Bernhardt Ganter were developing an aspect of Galois Theory and lattice theory (the latter being, as was said, Peirce's invention) that came to be known as “Formal Concept Analysis.” Interestingly enough, even though the two groups of researchers initially were working completely independently of each other, the mathematical apparatus of Finn's Automatic Generation of Hypotheses is at its core the very same apparatus as that of Wille's and Ganter's Formal Concept Analysis. For obvious reasons, then, there has now grown up an extensive cooperation between the German researchers and the Russian researchers, principally through the writings and intermediary work of Sergei Kuznetsov, who has been working both with the German group and with the Russian group. The heart of both sets of ideas is the notion of clustering items by similarity. The algorithms for clustering into formal concepts are the same as the algorithms for preliminary groupings by similarity for the purpose of automatically generating hypotheses. As it turns out, and as Kuznetsov has shown, these algorithms are equivalent in their effect to algorithms for finding the maximal complete subgraphs of arbitrary graphs. This fact has proved extremely useful in recent years, since the latter algorithms are the core of what has come to be known as “Social Network Analysis.” And Social Network Analysis has become a major intellectual tool in the world's battles against criminal organizations and terrorist networks. So all three sets of ideas have become matters of crucial practical importance and even urgency in contemporary affairs. Such practical applications of Peircean ideas may seem surprising to many philosophers whose minds are rooted strictly in the academic world. The applications, however, most certainly would not have surprised Peirce in the least. Indeed, given his lifelong ideas and goals as a scientist-philosopher, he probably would have found the current practical importance of his ideas entirely to be expected. 16. Significant Students of Peirce During the time of Peirce's teaching logic at Johns Hopkins University, that is: during the years from 1879 through 1884, Peirce had a number of students in logic who then went on to establish significant reputations in their own right. Often mentioned in this connection are John Dewey, Allan Marquand, Christine Ladd-Franklin, Oscar Howard Mitchell, Benjamin I. Gilman, Fabian Franklin, and Thorstein Veblen. Here we provide brief descriptions of three of these students, Dewey, Ladd-Franklin, and Mitchell. Of necessity the accounts given here of the work of these students will be extremely brief. It is obvious that full-length accounts of each of them 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's Hopkins during the years 1882 through 1883. Along with Peirce, Dewey understood the subject of logic in an extremely broad way, so that the subject in his mind, as well as in Peirce's mind, comprised the entire topic 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 book Logic: the Theory of Inquiry (1938), is a close approximation of the structure of the scientific method as Peirce understands it. Recall that for Peirce inquiry begins with an anomalous situation, in which a particular puzzle or set of puzzles is elicited from an indeterminate background. Then, by an ongoing, and in fact ultimately endless, process, hypotheses are formulated (abduction) and tested (deduction and induction). so that at each iteration of the methodological "loop," indeterminacy as to the original anomalous situation is successively, though never totally, eliminated. Dewey's own account of inquiry, especially insofar as Dewey considers it to be the successive elimination of indeterminacy, is remarkably akin to Peirce's account of scientific methodology in action. Of course there are differences. Dewey often writes as though at each stage of development of the method of logic (of science) the next stage is more or less already specified; for example, at any stage of indeterminacy, Dewey writes as if the relevant hypothesis or hypotheses to test at the next stage are more or less already determined. By contrast, Peirce (although he sometimes speaks this way) more often emphasizes the creative and non-determined aspect of eliciting/developing hypotheses. Nevertheless, even despite the differences in emphasis from Peirce to Dewey, the similarities between their positions are unmistakeable. Unlike Dewey, both Christine Ladd and Oscar Howard Mitchell concentrated on formal deductive logic, i.e. mathematical logic, rather than on the informal methodology of the sciences. Also unlike Dewey, Ladd and Mitchell both published articles in the 1883 volume Studies in Logic, which was edited by Peirce. Again, in his own “Preface” to this volume Peirce singles out both 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 to Fabian Franklin in 1882) known as Christine Ladd-Franklin. Although she is almost unknown today, she was very well-known and highly-regarded as a mathematician, logician, and psychologist from the 1870's until her death on March 5, 1930 at the age of 82. Her earliest published work was in mathematics, in the Educational Times of England. Here it attracted the attention of the great mathematician J. J. Sylvester, who also published in the Educational Times and who in 1876 assumed the position of the first professor of mathematics at the new Johns Hopkins University. (It was Sylvester who hired Peirce in 1879.) On the strength of her already-published work, as known by Sylvester, Ladd applied to Johns Hopkins in 1888 to become a graduate student in mathematics. Although, because she was a woman, she could not become a regular graduate student at Johns Hopkins, still at the ardent and insistent urgings of Sylvester, she was admitted as a special-status student, and she was even awarded a fellowship to study mathematics. She held this fellowship from 1879 until 1882, when she completed all the requirements for the Ph.D. degree. It was during this period that she became attracted to mathematical logic, and to the teachings on logic of Peirce, with whom she studied carefully. Because of her gender, her status as a student of mathematics was recorded only in notes rather than on the usual student lists. For the same reason, she could not actually be given the Ph.D. degree in 1882, although she was ultimately (but not until 1926), awarded the Ph.D. degree in mathematics. (Meanwhile, Vassar College awarded her an LL.D. degree in 1887.) On August 24, 1882, upon the completion of her mathematics fellowship, she married a member of the Johns Hopkins department of mathematics and a fellow-student of Peirce's, Fabian Franklin (b. 1853, d. 1939). Both she and Fabian Franklin seem to have stayed closely connected to Peirce until his death in 1914. As can be gathered from the foregoing, it was studying with Peirce that focussed Ladd's attention from mathematics in general to mathematical logic (also called symbolic logic) in particular. Ladd's best-known, and most-celebrated work was her paper "On the Algebra of Logic," published in Studies in Logic by the members of the Johns Hopkins University (C. S. Peirce, editor), Little, Brown, 1883, pp. 17–71. In it, she is (or at least was once) widely regarded as having achieved for the first time in history a completely general and unified account of the Aristotelian syllogism, including a general account of the differences between valid and invalid syllogisms. (Josiah Royce, for example, held this view of Ladd-Franklin's work in logic.) It is thus worth looking in some detail at how she achieves this result through the algebraizing of the syllogism, but here such detail will not be entered into. Basically, however, we can say that her basic idea is to associate with each possible syllogism a certain “triad”. The syllogism is valid if and only if the “triad” is an “antilogism,” that is to say, is an inconsistent “triad”. Ladd-Franlin wrote on the antilogism as late as 1928 in volume 37 of Mind, pp. 532–534. Whether or not her work is really the first work to unify the theory of the syllogism, and whether or not she ultimately merits the exalted status as a logician that Royce assigns to her must remain a task for future exploration to determine. In addition to her major work on the syllogism, equivalently on the notion of the antilogism, Ladd-Franklin as well wrote some of the entries 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.” In fact she was an associate editor of Baldwin's Dictionary. She also wrote at least one related paper in the Journal of Philosophy, Psycholgy, and Scientific Method, later named simply the Journal of Philosophy.Her specialty in psychology was the theory of vision, and she early espoused the theory that all color vision both developed historically and was based on three primary colors, hence three distinct processes of chemical reaction. Her writings on the psychology of vision continued to pour firth voluminously, and these continued from the early 1890's through at least 1926, when she published a paper on the mysterious visual phenomenon known as “blue arcs” in the Proceedings of the National Academy of Sciences. Peirce praised O. H. Mitchell as being his very best student in logic. Mitchell was trained as a mathematician even befoe coming to Johns Hopkins, and he was especially noted by all those who knew him as being extraordinarily meticulous and attentive to detail. He was so attentive to exactitude that he was even noted for slowness and ponderousness in speech. But what was there in Mitchell's 1883 paper that stimulated Peirce's high praise? It seems to be closely connected, perhaps identical, with what Peirce praises Mitchell for accomplishing in Peirce's entry in Baldwin's Dictionary of Philosophy and Psychology, particularly in his article there for the word “ dimension.” In the article Peirce credits Mitchell with having introduced “the concept of a multidimensional logical universe” into exact logic, and Peirce claims that it is one of the “fecund contributions” that Mitchell made to exact logic. There seem to be two components to the idea of a multidimensional logical universe. The first one is simply the idea of multiple quantification in connection with polyadic predicates, so that (in modern terminology) we can have propositions with quantifiers occurring within the scope of other quantifiers. That Mitchell employs a formalism suggestive of this idea is undeniable, but whether he successfully distinguishes (for all x)(there exists a y) from (there exists a y)(for all x) is very much less than obvious. Moreover, that Mitchell was the very first logician to have conceived of multiple quantification and polyadic predicates seems a bit dubious, especially since by 1870 Peirce himself was already making use of notions for which multiple quantification and polyadic predicates are involved. The second component of the idea of a multidimensional logical universe is (again, using modern terminology) the idea that the universe of discourse relevant to one variable might not be the same as the universe of discourse relevant to another variable. Moreover, since the universe of discourse relevant to one variable might be all the times there are, all the places there are, all possible situations, or even all possible worlds, it is not difficult (though it is perhaps a bit far-fetched) to connect Mitchell's idea of a multidimensional logical universe to the idea of tense logic or even modal logic. Whether a careful study of Mitchell's logical contribution really supports such a reading of Mitchell's contributions to logic is not as yet clearly determined. Clearly, close study of the logical accomplishments of Peirce's notable students is in order.
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