Peirce’s Philosophy of Logic1

J. Jay Zeman

The roughly two and a half millennia over which we can trace the development of mathematics as a discipline have seen ups and downs in its study; the "ups" have involved varying emphases and interests depending on the problems and the temper of the time. The 19th Century may be characterized as a period of development of rigor and attention to the axiomatic method in mathematics. This focus on the deductive process in mathematics was accompanied by the application of mathematics in the study of the deductive process itself. It is safe, I think, to say that the best-known and most influential of the lines of development of the resulting mathematical logic and philosophy of mathematics was that finding its formal expression in the Principia Mathematica of Whitehead and Russell. But there were other paths—less well-known, perhaps, but worthy of our attention; one of the most important was that of Charles Sanders Peirce.

The work of Peirce is remarkable in its size and scope and in its vision; his ongoing study of mathematics, logic, and their interrelationship is no exception. Peirce developed independently of the Frege Peano-Russell (FPR) tradition all of the key formal logical results of that tradition. He did this first in an algebraic format similar to that employed later in Principia Mathematica and then, for philosophical reasons founded in the theory of signs, he became dissatisfied with algebraic notation for logic; this dissatisfaction resulted in his development of a successful graphical logical notation; in his work in this notation (his "Existential Graphs") he anticipated philosophically important extensions of basic mathematical logic which eluded independent rediscovery till after the middle of this century.

For Peirce, logic is the study of mind, the theory of mind—of mind, that is, in its functioning as revealed in our use of signs. Peirce held that all thought is in signs (see 4.551,2 e.g.); indeed, I believe that it is fair to say that all experience is a matter of signs. The semiotic may thus be considered a theory of experience, or, from the intellectual perspective on experience, a theory of mind. So, in the most general sense of logic, logic is semiotic, the general theory of signs. Mathematical logic as we understand it, although an important area for Peirce, is just a part of the theory of deduction, which is itself part of the theory of inquiry (and whether viewed from the perspective of semiotic—the theory of signs—or that of the theory of inquiry, Peircean logic is far more extensive than deductive logic). Recognizing this philosophical context for Peirce’s mathematical logic, I shall first set down and discuss some of the important basic mathematical formalisms employed by Peirce in his study of logic; I shall then examine the role of this logic in Peirce’s philosophy, and especially the relationship that Peirce saw between logic, mathematics, and reasoning. This seems appropriate and significant in the light of the well-known connection between logic and mathematics in the philosophically influential FPR tradition. Peirce’s thought offers alternatives which can enrich the background of many areas of inquiry significant in human process.

The Basic Mathematical Work

Peirce’s work in algebraic logic goes back at least to 1865 (Robin, ms. 341ff.), when he was in his mid-twenties, and less than twenty years after the seminal work of Boole and De Morgan. This early interest in algebra as applied to logic was just a starting point, however. Peirce was to use the work of Boole and De Morgan as a point of departure for his own developments. From 1865 to about 1885, Peirce worked on the algebra of logic, following the lead of his predecessors, but from the start adding his own touch. I shall give a summary of this early work of Peirce, and then shall concentrate my mathematical discussion principally on the period beginning about 1885, including both algebraic and graphical work.

Peirce’s early logical investigations (see 3.45ff.) involved an effort to improve the work of De Morgan on the logic of relatives, which, although acknowledged by Peirce to be valuable

still leaves something to be desired. Moreover, Boole’s logical algebra has such singular beauty, so far as it goes, that it is interesting to inquire whether it cannot be extended over the whole realm of formal logic, instead of being restricted to that simplest and least useful part of the subject, the logic of absolute terms, which, when he wrote, was the only formal logic known. (3.45)

In the 1870 paper from which this quotation is drawn, Peirce employs the "crow’s foot" (—< ) sign as what we today could call a partial ordering. He makes it clear that, while it carries the general sense of class inclusion (3.66), it also may be used to express hypotheticals3 — "if-then" statements (3.139,140); the sense of "if-then’, expressed by this sign at this point in Peirce’s work is more like that expressed by the metalinguistic concepts of entailment and deducibility in today’s treatments of symbolic logic than it is like a conditional function. Thus, he employs it in this paper generally like what we would think of algebraically as a relation — which forms sentences from terms—rather than a function, such as logical sum, which takes classes, pairs, etc. into classes, pairs, etc. His work in logical algebra shows some development on this point. By 1880 (3.154), he is using this sign much as we do the conditional function, although I feel that even in 1880 there is some ambiguity about the linguistic level of his usage.

In the 1870 paper earlier quoted from, he also deals with the question of quantification (as he must for the logic of relatives). He has not at this point developed the quantifier as a separate notation, although there is a hint of it in his treatment of individuals and relations (3.97). Interestingly, he tends in this paper to deal with quantification as implicit in assertions involving the signs of inclusion and identity (e.g., 3.66); we shall see this concept (i.e., implicit quantification) emerge again—metamorphosized—a quarter of a century later in his most successful logical systems, the existential graphs.

As I have noted, by 1880, Peirce has begun employing his "inclusion sign ‘—< ‘ much as we do the conditional function, with multiple occurrences of it in a single formula. In a paper (3.l54ff.) whose introductory paragraphs show strong philosophical connections with the 1877 Popular Science Monthly series that included "The Fixation of Belief’ (5.358ff.), he remarks that, "From the identity of the relation expressed by the copula [i.e., inclusion] with that of illation [the conditional] springs an algebra" (3.182). This algebra is one whose basic units are propositions rather than general terms pr relations, as previously; Peirce’s symbolic logic is at this point looking more and more like that of today. We also find further use Of symbols which will eventually emerge as true quantifiers (3.223).

As Peirce’s logical work progresses, his treatment not only of propositional logic, but also of the logic of relatives grows simpler, clearer, and more powerful. In 1882 he comments that

A dual relative term, such as "lover," "benefactor," "servant," is a common name signifying a pair of objects. . . . (3.328) A general relative may be conceived of as a logical aggregate of a number of such individual relatives. (3.329)

Peirce at this point begins to move to more compact and succinct statements of relative logic; in this paper he is aided by his use of explicit universal and existential quantifiers; employing these and continuing his use of the conditional function, he lays out and discusses a large variety of statements and arguments involving relations and various compounds thereof.

Peirce was a genuine innovator in symbolic logic. The roughly twenty years culminating in his 1885 paper (subtitled "A Contribution to the Philosophy of Notation") (3.359ff.) saw a development in his treatment of logic which, as we have remarked, included his use of he conditional function and of the quantifiers and which also saw a movement to greater and greater elegance in his presentations of the subject. The 1885 paper marks a watershed in Peirce’s work on the mathematics of logic; he comes here to a complete presentation of logic in the algebraic format. After setting down some of the philosophico-semiotic background for this work, he tells us that

In this paper I purpose to develop an algebra adequate to the treatment of all problems of deductive logic, showing as I proceed what kind of signs have necessarily to be employed at each stage of the development. I shall thus attain three objects. The first is the extension of the power of logical algebra over the whole of its proper realm. The second is the illustration of principles which underlie all algebraic notation. The third is the enumeration of the essentially different kinds of necessary inference; for when the notation which suffices for exhibiting one inference is found inadequate for explaining another, it is clear that the latter involves an inferential element not present in the former. (3.364)

The third of Peirce’s aims as stated above is carried out in his division of the matter of this paper into "Non-relative logic" (3.365ff.), "First intentional logic of relatives" (3.392ff.), and "Second-intentional logic" (3.398ff.). The first of these topics is what we sometimes call the "propositional calculus"; the second is basic quantification theory involving relations whose relate are "ordinary individuals" (what counts as ordinary individuals or "primary substances" for Peirce is intimately involved with his theory of abstraction, and is relative to context; see Zeman 1982; we shall sketch out this theory later in this paper). The third, second-intentional logic, effectively involves quantification over predicates and relations; this is a branch of logic which Peirce continued to work on and never quite finished with.

The second of Peirce’s aims involves, I think, his contention that the very formulas of algebra are themselves icons—signs which represent by resemblance, which are mappings of that which they represent (see, for example, 3.363); we note that he calls the formulas in which he presents the principles of logic in this paper icons.

And his first aim includes a claim of completeness: "[it] is the extension of the power of logical algebra over the whole of its proper domain" (3.364, emphasis mine). This is interesting, since Peirce does not have access to what we recognize as completeness proofs; nevertheless, his claim is not groundless; note the "icons" which Peirce presents us for non-relative logic (I employ Polish notation for typographical reasons [in the HTML version, I include (within the "[ ]") ordinary algebraic versions of the formulas4]):

(1) Cpp [p p] (3.376)

(2) CCpCqrCqCpr [(p (q r)) ((q (p r))] (3.377)

(3) CCpqCCqrCpr [(p q) ((q r) (p r))] (3.378)

(4) Cop [o p], o = absurd (3.381)

(5) CCCpqpp [((p q) p) p] (3.384)

These "icons," taken as axioms and supplemented by the standard rules of inference of detachment and substitution for atomic formulas! are a complete base for the classical PC (see Prior 1958); indeed, the first of them is redundant; it is remarkable that, even though he lacked the kind of metatheoretical techniques that we now have, Peirce was able to present us with an elegant base for this first stage of symbolic logic.

In this he paralleled his remarkable contemporary, Frege, who also at about the same time (in the Begriffschrift, 1879) developed a complete base for the same system. It is generally known that Frege also there developed a complete base for quantification theory; I once thought that Peirce had not developed a complete quantification theory till his graphical logics some ten years later (we will look at this in some detail), but more recent investigation of the 1885 paper we are examining tells me that this paper has a complete quantification theory implicit in its work on the logic of relatives.

I note first of all that Peirce credits O. H. Mitchell with the invention of the quantifier as it is employed in the "Philosophy of Notation" article; I think that it is clear from Peirce’s writing both preceding and following this article that his work on the quantifier is in no way derivative, although Mitchell may well have employed the quantifier with "indexes" in published form first.

For my examination of Peirce’s treatment of quantification in the 1885 article, I shall again employ the Polish notation for propositional connectives, lower case a, b, c, . . . for relations, x, y, z for quantified variables (Peirce’s "indexes" in this context), ‘E’ for the existential quantifier, and ‘A’ for the universal quantifier. The existential quantifier may be taken as = NAN; this is implicit in the discussion of 3.393 which treats the universal and existential quantifiers in terms of products and sums respectively. Peirce presents the material on quantifiers in a more scattered and intuitive way than he does the non-relative logic; we will, however, find asserted formulas which we recognize as a sufficient basis for complete quantification theory; first of all, in 3.393 he points out the connection between universal quantification and conjunction and existential quantification and disjunction; this connection is basic to contemporary treatments of quantification such as that by semantic tableaux or consistency trees, and is, if properly construed, enough to give us the full theory. At the very least, we can see in it an implicit statement of "universal instantiation,"

(6) CPxbxbx    ["xBx Bx]

There is a more explicit statement of this in formulas at the end of 3.403E. At the end of 3.403F occur a number of formulas including

(7) EPxApbxApPxbx  ["x(p Bx) (p "xBx)]

where x is not free in p; taking p as Np [~p] in (7), this yields by the methods of non-relative logic

(8) CPxCpbxCpPxbx ["x(p Bx) (p "xBx)]

where, again, x is not free in p.

In 3.403I, Peirce lists as provable the formula

(9) PxIxx ["x(x = x)]

From the formulas in 3.403A, it is clear that

( 10) EvIxx [v (x = x)]

holds, where v = "truth"; since v is equivalent to anything provable here, formula (9) above along with (10) gives us, implicitly,

(11) If proven p, then proven Pxp [If proven p, then proven "xp]

This universal generalization, plus (6) and (8) added to the classical propositional calculus (non-relative logic) give us a complete quantification theory. We could go even further: the first formula under "Second-intentional Logic" in this article (3.398) is effectively an assertion of the substitutivity of identity; we will have, then, at least full quantification theory with identity in this article.

We saw Peirce claiming (in 3.364) that he intended in the "Philosophy of Notation" article to "extend the power of logical algebra over the whole of its proper domain"; making very reasonable assumptions about what he was doing there, we see that—in terms of the classical logic of today—his claim is well-founded.

I shall mention one other algebraic achievement of Peirce which is not too well-known, but for which he should receive due credit. In an 1880 paper titled by Peirce’s editors "A Boolean Algebra with One Constant" (4.12ff.), Peirce shows that "neither-nor" is a sufficient sole connective for the classical propositional logic; this is thirty-three years before Sheffer’s showing and being acclaimed for showing that one such connective can suffice.

As it turns out, Peirce’s formal mathematical work in logic hardly stops with his work on algebraic logic. He grows dissatisfied not with the mathematics involved, but with the notation, with the way that the mathematical forms are represented in the algebraic notations he has been employing for more than twenty years. Beginning in about 1889 he worked on a system of logical notation he called the "Entitative Graphs." This system still was not what he wanted; he saw it as lacking in iconicity (4.434).

In the 1890's Peirce settled on a system of logical notation which he felt came closer to doing the job he wanted such a system to do. This was the system he called his ‘`Existential Graphs" and which he declared to be his "chef d’oeuvre" (Peirce 1931, v. 4, p. 191). Unfortunately, typographical considerations make it impractical to go into the mathematics of the graphs in the present paper. Relevant expositions and results are available, however, in (Zeman 1964, 1967, 1974); the basic source material on the graphs is in 4.372-4.584. In brief, Peirce felt that these systems, which lay out the logical notation in an ingenious quasi-topological fashion (which is multi-dimensional) are more iconic than the algebraic notations to which he had previously devoted so much attention. The iconicity of the Existential Graphs will figure in my later discussion of Peirce’s philosophy of math and logic; although we will not go into detail about the mathematics of the graphs, I will mention one of the notations of the system Peirce called "Beta." This notation, Peirce’s "line of identity," (passim in v. 4 of Peirce 1931) will have a role in the philosophical discussion to follow. The line of identity is the basic quantificational mechanism of the existential graphs; lines of identity are implicitly quantified variables (Zeman 1964, 1967) which fill blanks in predicates and connect those blanks which should be identified; thus

isredround.GIF (1294 bytes)

means SxK(x is red)(x is round) [$x(x is red x is round)], or "Something is both red and round." Peirce also gives an alternative quantificational notation in his selectives (e.g., 4.460ff.), which are letter variables like those of ordinary quantification theory, although they also bear an implicit quantification;

X is red

X is round

means the same as the configuration above. As I have mentioned, we shall refer to these notations in discussion a bit later.

As mathematical systems, the graphs were quite successful: the system Peirce calls "Alpha" is a complete Classical propositional calculus, and "Beta" is complete quantification theory with identity (Zeman 1964).

The remarkable work of Peirce in these unusual graphical systems extends to the systems he called "Gamma." He was unable to bring gamma to the sort of completion he had achieved with alpha and beta, but he did anticipate in a remarkable way some philosophically important developments. For example, in 4.510ff., he discusses a graphical system which will have not just one, but a book of sheets of assertion; each of these will represent a different possible universe of discourse; although he fell just short of the contemporary notion of an accessibility relation for these "possible worlds" (see 4.518), the anticipation of this fruitful area of logic by Peirce is clear; his "tinctured existential graphs" (4.530ff.) are another effort at possible-worlds semantics. Peirce’s discussions of the gamma graphs also involve quantification over predicates and metatheoretical and other notations; once again, we will be unable to go into the mathematical details of these notations.

Mathematics, Logic, and Mind

Peirce’s contributions to the mathematics of mathematical logic are, as we can see, very considerable. Even in cases where, as with possible-worlds semantics, he was not able to present us with a neatly tied-together package, his insights and the directions he probed show a remarkable feel for the application of novel mathematical thought and technique to the problems of philosophy. Peirce had a well-developed notion of how mathematics, logic, and philosophy are related. His opinions on this, which differ considerably from the FPR tradition, continue, I believe, as a worthwhile guide to those interested in these disciplines today. This applies if your interest is in research in these areas, of course, but it is at least as important for the teaching of logic, mathematics, and the related philosophy. Peirce’s awareness of the process involved in deductive reasoning was remarkable, and is most instructive pedagogically.

The FPR tradition, culminating as it does in Principia Mathematica, has, of course, been tremendously influential. Taken as a philosophy of mathematics, it is logicism, the view that the concepts of mathematics are, at root, concepts of logic—complex and developed concepts of logic in most cases of interest, but concepts of logic nonetheless. And, according to this view, the proofs of mathematics are shorthand paraphrases of proofs of logic (these, of course, are views that most mathematicians would not subscribe to). This tradition also exercised great influence on the early Wittgenstein, and through him on logical atomism; this early effort in analytic philosophy sought to extend the reductionism of Principia Mathematica to language and the objects of language at large.

In contrast to the views of the FPR tradition, Peirce saw mathematics and logic as having essentially different thrusts, although they still bear intimately upon each other. While Frege, typical of that tradition, held that, if his view were correct,

arithmetic would be only a further developed logic, every arithmetic theorem a logical law, albeit a more developed one. (Frege, 107)

Peirce, on the other hand, contended—surprisingly enough—that

An application [of] logical theory is only required by way of exception in reasoning, and not at all in mathematical deduction. (Eisele 4, 272)

This is a theme which emerges in a significant number of locations in Peirce; in a Baldwin’s Dictionary article on symbolic logic, for example, we find him saying that "the purpose and end of a system of logical symbols" is

simply and solely the investigation of the theory of logic, and not at all the construction of a calculus to aid the drawing of inferences. These two purposes are incompatible, for the reason that the system devised for the investigation of logic should be as analytical as possible, breaking up inferences into the greatest possible number of steps, and exhibiting them under the most general categories possible, while a calculus would aim, on the contrary, to reduce the number of processes as much as possible, and to specialize the symbols so as to adapt them to special kinds of inference. (4.373)

A "calculus" in the sense used by Peirce in this passage is any mathematical system used to carry out necessary inference; indeed,

all mathematical reasoning is diagrammatic, and all necessary reasoning is mathematical reasoning, no matter how simple it may be. (Eisele 4,47)

So from a Peircean point of view, not only does the FPR view miss the point, but even the commonest view of the relationship between logic and reasoning seems misguided. For Peirce, logic in its most general sense is semiotic, the theory of signs (2.227); thus, it "rests upon observations of real fact about mental products’, (Eisele 4, 267); it is, then, an empirical science in a sense that mathematics is not (however, we shall indicate a sense in which mathematics too is, for Peirce, an observational science).

The field within which a Peircean logician works is thought; figure against this background is the reasoning process observed as it operates, as it manipulates signs; logic is not, however, merely descriptive of this process; it is, Peirce tells us, normative, along with and dependent on the other normative sciences, esthetics and ethics (1.573, Eisele 4, 197, and passim). Peirce held that the reasoning employed in inquiry may be viewed from three different perspectives, those of deduction, induction, and retroduction (or abduction) (1.65 and passim). His use of the first two of these terms is, of course, not to be confused with the hack logician’s distinction, supposedly based on "arguing from generals to particulars" for deduction, and vice-versa for induction. The broadest sketch of the process of inquiry in Peirce’s terms begins with abductive reasoning, which is the educated hypothesis-formation which proposes initial organizations of figure in the problematic field. Deduction enters in a mediating way, drawing out the consequences of the abductive hypotheses. And induction consists in the return to experience which aims at confirming or refuting those hypotheses by seeing whether the deduced consequences hold or not (2.269). I note in passing that Peirce saw this process as applying in mathematical reasoning as well as in that about "the world out there"; more on this anon.

A prime focus of this paper is on logic as it studies deductive reasoning. Indeed, we may narrow this down even further, for Peirce tells us that

Deductions are either Necessary or Probable. Necessary Deductions are those which have nothing to do with any ratio of frequency, but profess (or their interpretants profess for them) that from true premisses they must invariably produce true conclusions. (2.267)

It is in the area of necessary deductive reasoning that Peirce’s mathematical logic falls, or better, it is this area that it studies. Since, as we have noted, Peirce saw all necessary reasoning as mathematical reasoning, we can see the import for him of mathematical methods in a study of deductive logic.

We can go even further along these lines, moreover. Peirce remarked that not only is all necessary reasoning mathematical, but all mathematical reasoning is diagrammatic. This has important consequences for his views on mathematical logic and appropriate notations therefor, and helps explain the movement from algebraic to graphical notations that we have noted in the development of the formal elements of his theory.

This is a feature of Peirce’s work in logic that cannot but be remarkable to anyone who has examined it in detail. After a near lifetime of most fruitful work which includes seminal efforts in algebraic logic, he gives his attention to a system which turns out to be his most successful mathematical logic, but which is also, as we have suggested, strangely different. He introduces concepts, such as the line of identity, which do their assigned tasks beautifully, but which were not seen before Peirce, and have only recently been recognized for what they are. In spite of everything else he had done, Peirce called the graphs "my chef d’oeuvre"; this emphatic appellation reflects that, as we have noted, logic is, for Peirce, a study of the reasoning process; as such, it should provide us with a diagrammatic representation of the process; this representation should be iconic—indeed, it cannot avoid being so. As an icon, it will represent by resemblance: it provides a mapping of the process it studies (see 4.512, 513). In 1906, Peirce incisively criticizes an element he had presented as part of the beta graphs—his "selectives," which, although implicitly quantified, are individual letters like the variables of algebraic logic; they are contrasted with the line of identity (which is the fundamental quantified variable of the graphs); his criticism applies a fortiori to the standard quantificational notation of algebraic logic:

The essential error … of the Selectives, and their inevitable error… lies in their putting forth [misleading representations] in a system which aims at giving, in its visible forms, a diagram of the logical structure of assertions....

[The] purpose of the System of Existential Graphs … [is] to afford a method (1) as simple as possible (that is to say, with as small a number of arbitrary conventions as possible), for representing propositions (2) as ironically, or diagrammatically and (3) as analytically as possible. … Those three essential aims of the system are, every one of them, missed by Selectives. (4.561n)

He goes on to expand in detail on this; the aniconicity of the selectives is, I believe, his chief objection to them—and an objection which applies to the signs of the algebra of logic, Peirce felt, no matter how successful such an algebra might be. Even in discussing the analyticalness of the selectives Peirce cannot avoid reference to iconicity:

The first respect in which Selectives are not as analytical as they might be, and therefore ought to be, is in representing identity. The identity of [two occurrences of a bound letter variable] is only symbolically expressed.... Iconically, they appear to be merely coexistent; but by the special convention they are interpreted as identical, although identity is not a matter of interpretation.... But the line of identity which may be substituted for the selectees [letter variables] very explicitly represents Identity to belong to the genus Continuity and to the species Linear Continuity. (ibid.)

Thus does the very notation which the mature Peirce prefers for the mathematics of his logic tell much about his view of the nature of logic.

Peirce’s symbolic logic is to be a tool for the study of necessary reasoning; in fact, for Peirce, this is its basic reason for existence. It is quite appropriate, then, that we examine Peirce’s views on the nature of necessary, which is to say mathematical, which is to say diagrammatic reasoning (Eisele 4, 49 and passim).

First of all, let us look further at what Peirce saw mathematics itself to be. Pointing out the essential inadequacy of old definitions of mathematics which held it to be "the science of quantity" (Eisele 4,1934), Peirce goes on to note that

modern mathematicians recognize as the truly essential characteristic of their science that . . . it concerns itself with pure hypotheses without caring at all whether they correspond to anything in nature or not, or at least, disregards such correspondence entirely after its hypotheses are formed.

From this it follows that whenever a student of any science has occasion to argue, that supposing certain definite hypotheses to be true, then, no matter what else may be true or false, a certain conclusion will inevitably be found true, he is taking the place of the mathematician whose essential function it is to determine whether what that student is saying [about the conclusions claimed to follow] be true or not. (Eisele 4, 194)

This gives us the connection between necessary reasoning and mathematics; mathematics is, then, as Peirce’s father remarked, "The science which draws necessary conclusions" (4.229). And, although as we have suggested, Peirce held that "It does not seem to me that mathematics depends in any way upon logic" (4.228), he contended

that logic cannot possibly attain the solution of its problems without great use of mathematics. Indeed, all formal logic is merely mathematics applied to logic. (4.228)

Thus, the relationship between logic and mathematics would appear to be much more like that between physics and mathematics than that proposed by the logicist; logic, like physics, is an observational science, with the difference being in the nature of the observables. Peirce expands somewhat:

In order to prove that the procedure he recommends really [leads invariably to the truth], the logician has to show that as long as certain definite assumptions are supposed true, then, no matter what may be the case besides them, the line of reasoning he recommends will be the speediest road to the truth. Thus logic must appeal to mathematics, or else, what amounts to the same thing, must invade the domain of mathematics, in order to make certain of the truth that it essentially seeks. (Eisele 4,194)

Note the similarity of Peirce’s treatment of logic to the way he speaks of science in general in our last lengthy quotation.

So necessary reasoning is mathematical reasoning, and logic must draw on math for its study of necessary reasoning. I would like to go in more detail into Peirce’s views on the nature of necessary reasoning. Following his own lead in the study of necessary reasoning, we, find him saying that his

first real discovery about mathematical procedure was that there are two kinds of necessary reasoning, which I call the Corollarial and the Theorematic. (Eisele 4, 49)

This discovery of Peirce’s has been discussed by Jaakko Hintikka (1980) among others, and enters into my (1982) examination of abstraction in Peirce’s thought; it is an example of Peirce’s keen observation and clear insight into his own process. He bases the terms "theorematic" and "corollarial" on the terminology connected with Euclid’s Elements (ibid.); corollaries there are generally supported, he points out, by a different kind of argument than are the more important theorems:

The peculiarity of theorematic reasoning is that it considers something not implied at all in the conceptions so far gained, which neither the definition of the object of research nor anything yet known about could of themselves suggest, although they give room for it. (ibid.)

Peirce is here sharing a brilliant insight into the reasoning process. The process of deductive reasoning has, we might presume, been thoroughly studied—after all, logic is one of the oldest and most venerable of academic disciplines. But, as Julian Huxley is supposed to have remarked on seeing the details of Darwin’s theory, "How stupid of us not to have noticed!" Non-trivial deductive reasoning is hardly a matter of robotic, linear, left-hemispheric thinking alone. It involves a creative moment, a moment in which the deducer "constructs," to use Peirce’s term, a term which is based on the constructions so familiar in Euclidean geometry. We may draw an example from that particular mathematical science, remembering that it is neither the special subject matter nor the specific kind of diagram we here construct that is at stake—it is the process, the method.

Let us then ask how we go about proving a basic but non-trivial proposition of Euclidian geometry: That the interior angles of a triangle total to 180 degrees. So long as we just look at the triangle, making no changes in our diagram, we also make no progress in our proof. But when we move to the construction of a line parallel to a base through the opposite vertex, we see that propositions involving parallel lines solve the problem. The construction is by no means implied by the problem or by the postulates of geometry, but it is permitted by them.

Corollarial reasoning, for Peirce, will involve the construction of no new diagram, but will proceed on the basis of diagrams already constructed. Peirce’s insight into the creative element involved in non-trivial necessary reasoning will have consequences, I believe, for our understanding of the reasoning process in general, including cases we don’t think of as primarily mathematical. I feel that the commonest views about deductive reasoning see it as essentially what Peirce calls corollarial; recognition of the need to construct a diagram or image—an icon of some sort—in non-trivial cases opens the door to a multitude of insights in logic.

The diagram formed in this process "is a representamen which is predominately an icon of relations" (4.418), and

one can make exact experiments upon uniform diagrams; and when one does so, one must keep a bright outlook for unintended and unexpected changes thereby brought about in the relations of different significant parts of the diagram to one another. Such operations upon diagrams, whether external or imaginary, take the place of the experiments upon real things that are performed in chemical and physical research . . . experiments upon diagrams are questions put to the nature of the relations concerned. (4.530)

So although Peirce contrasts mathematics with observational sciences, as dealing with hypothetical states of affairs which have no necessary relationship to whatever we may find in "real experience" (and, as we have noted, logic is to be classified with the observational sciences rather than with math), he still sees an experimental element as essential to mathematical reasoning Within its own problematic field, mathematics is experimental and experiential; relative to physics or logic, the field of math is a field of hypotheses and their relationships, but within that field of hypothesis, we see in action the abduction - deduction - induction triad which embodies empirical inquiry. Peirce’s "discovery" of theorematic reasoning and its role in deduction exemplifies this.

I must comment that my contact with Peirce’s work has tuned me in to my own process of inquiry in some very fruitful ways; by and large, my own observations of this process in myself tend to confirm Peirce’s views; and an important element of my teaching of logic and its mathematics is an invitation to my students to check into their own process for evidence about these matters. This is phenomenological experimentation.

Peirce’s theory on these matters is an intricately interwoven fabric. His insights on theorematic and corollarial reasoning have important ties in his thought to another area of his theory which he also saw as vital:

the operation of abstraction, in the proper sense of the term … turns out to be so essential to the greater strides of mathematical demonstration that it is proper to divide all theorematic reasoning into the Non-abstractional and the abstractional. I am able to prove that the most important results of mathematics could not in any way be attained without this operation of abstraction. (Eisele 4, 49)

Abstraction, for Peirce, is an operation on signs which employs a name for an "object" which cannot, in itself, be said to exist. Peirce often uses the example given in the statement

Opium has dormitive virtue. (5.534 and passim)

This is supposed to be trivial, indeed laughable, since it says no more nor less than

Opium puts people to sleep.

However, this triviality is misleading; commenting on the opium example, Peirce, effectively, notes that we are all supposed to see it as ridiculous,

because everybody is supposed to know well enough that the transformation from a concrete predicate to an abstract noun in an oblique case, is a mere transformation of language that leaves the thought absolutely untouched. I knew this as well as everybody else until I had arrived at that point in my analysis of mathematics where I found that this despised juggle of abstraction is an essential part of every really helpful part of mathematics and since then, what I used to know so very clearly does not appear to be at all so. (Eisele 4,160)

The transformation from concrete to abstract may not change the transformed assertion at all semantically, but it can have profound effects on the pragmatics of the sign use involved, which is to say on the effects, or the interpretants of the signs. Speaking of the "dormitive virtue" of opium may not add much to what we are saying, but, Peirce insists, there are many non-trivial uses of abstraction: cases where it does make a significant difference in the pragmatics of sign use. He emphasizes that mathematics is a key locus of such use; I would point out that any user of English employs such abstractions daily—words like ‘duty’, ‘law’, ‘organism’, ‘institution’, and indeed, any name of a collection are such abstractions; it would be very difficult to function linguistically without them.

Peirce tells us that abstractions are defined in terms of other "things," things concrete relative to the abstraction in question; but his theory of abstraction is perspectival—he does not hold, as an absolute reductionist might, that there is some ontologically basic class of individuals in terms of which abstractions are to be defined:

An abstraction is something denoted by a noun substantive, something having a name; and therefore, whether it be a reality or whether it be a figment, it belongs to the category of substance, and is in proper philosophical terminology to be called a substance, or thing....

An abstraction is a substance whose being consists in the truth of some propositions concerning a more primary substance.

By a primary substance I mean one whose being is independent of what may be true of anything else. Whether there is any primary substance in this sense or not we may leave to the metaphysicians to wrangle about.

By a more primary substance I mean one whose being does not depend on all that the being of the less primary substance does, but only a part thereof. (Eisele 4, 161, 2)

A very simple mathematical example given by Peirce involves the definition of lines, or filaments, in terms of "moving" particles:

if the particles be conceived as primary substance, the filaments are abstractions, that is, they are substances the being of any one of which consists in something being true of some more primary substance or substances none of them identical with this filament. (163)

Peirce’s theory of abstraction is a description of sign use in naming and manipulating mathematical objects, but it applies in a much broader area of semiosis as well. An abstraction is one aspect of a universal I have argued in some detail (see Zeman 1982) that Peirce’s theory of abstraction is an important aspect of his theory of generals, or universals—in his own terminology, of his theory of "thirdness." The use of abstractions also illustrates a key semiotical distinction: that between meaning as semantic and meaning as pragmatics—the pragmatic dimension of semiotic considers the interpretants, or the effects of sign use; two semantically equivalent statements may have very different effects. As an example, consider the difficulty a peace officer would have if he were required to be completely concrete in giving a suspect his "Miranda rights"; the formula usually associated with this situation is quite abstract, but is far more intelligible to us than would be a semantically equivalent concrete statement.

The adequate elaboration of this theory would require far more space than is available in a survey of his work such as this; he did see abstraction as a very important part of the reasoning process, however, and I have tried to indicate here some of the connections of this part of his theory to other key areas of his thought.

The thought of Peirce was in many ways ahead of its time. He was at the forefront of work in the mathematics of logic, and anticipated developments only recently rediscovered. His mathematical and philosophical work on logic remains a rich lode to be tapped by the researcher and the teacher in logic and in mathematics, and perhaps even more importantly, by those who would become aware of their own experiential process in order to live more fully.

University of Florida




Frege, Gottlob

1964. "The Concept of Number,"Philosophy of Mathematics, ed. Paul Benacerraf and Hilary Putnam, Englewood Cliffs, NJ: Prentice-Hall.

Hintikka, Jaakko

1980. "C. S. Peirce’s ‘First Real Discovery’ and its Contemporary Relevance," Monist 63:3 (1980), 304-13.

Peirce, Charles Sanders

1931-58. The Collected Papers of C S. Peirce, ed. Charles Hartshorne and Paul Weiss, and Arthur Burks, Cambridge, Mass: Harvard.

1976. The New Elements of Mathematics, ed. Carolyn Eisele, The Hague: Mouton, vol. 1-4.

Prior, A. N.

1958. "Peirce’s Axioms for Propositional Calculus," Journal of Symbolic Logic 23 (1958), 135-6.

Robin, Richard

1967. Annotated Catalogue of the Papers of Charles S. Peirce, U. of Mass Press.

Zeman, J. Jay

1964. The Graphical Logic of C S. Peirce, Doctoral Dissertation, The University of Chicago.

1967. "A System of Implicit Quantification," Journal of Symbolic Logic 32 (1967), 480-504.

1974. "Peirce’s Logical Graphs, Semiotica 12 (1974), 239-56.

1982. "Peirce on Abstraction," Monist 65:2 (1982), 211-29.



1. The original prparation of material for this paper wasfor a conference on "The Birth of Mathematical Logic" at Fredonia College, SUNY in March 1983. This paper appeared in the Transactions of the Charles S. Peirce Society 22 (1986), 1-22.

2. References to the work of Peirce are drawn from Peirce 1931 (The Collected Papers) and Peirce 1976; the former are as usual in Peirce scholarship, with volume and paragraph number to the left and right respectively of a point--thus 4.512 would be paragraph 512 of volume 4. References to Peirce 1976 are from volume 4 of that set, and are indicated by "Eisele 4".

3. For an expansion on the important notion of "Hypothetical" in Peirce and for its relationship to the "if-then" of classical mathematical logic, see Jay Zeman, "Peirce and Philo," Studies in the Logic of Charles Sanders Peirce, Bloomington: 1997, Indiana University Press, 402-17 (Note added September, 1998).

4. The Font employed is simply Symbol TTF (September, 1998).