Peirce on Abstraction*

J. Jay Zeman

Events in the history of thought have often moved as elements of drama—now tense, now tragic, now triumphant. And, it would appear, sometimes ludicrous. This latter is the thrust of a parody which Molière visited upon the savants of his day; he pictures a candidate for a medical degree being solemnly asked why opium puts people to sleep. Just as solemnly and sagaciously, the candidate replies

Quia est in eo,
Virtus Dormitiva,
Cujus est natura,
Sensus assopire.

This incisive and revealing answer—that opium puts people to sleep "because there is in it a dormitive power whose nature it is to lull the senses to sleep"— is greeted by a happy congratulatory chorus: "You have responded well indeed, you are most worthy to join our learned brotherhood!"

Peirce refers explicitly to this scene in one location (5.534),1 and mentions opium and its "dormitive virtue" passim, especially in his later writings. The scene from Le Malade Imaginaire, of course, ridicules an affected scholarship which conceals its ignorance even from itself with explanations that don't explain; the implicit argument of the parody is that (1) Opium puts people to sleep; and (2) Opium has dormitive power; are essentially equivalent statements, and that if (2) adds anything to (1), it is the hypostatized fiction of a power which deadens not only the senses of those who actually take the dope, but also the critical abilities of those who speak seriously of such powers. The criticism is not unlike that directed more recently by behaviorists against "mentalism" in psychology, with so-called uncritical hypostatizations such as "mind," "freedom," and even "grammar" taking the place of "dormitive power."

Peirce placed his pragmatist's emphasis on the role of possible observable effects in the explication of concepts (5.402), and he was clear in his opposition to what might be called "Cartesian mentalism" (see, e.g., 5.264 ff.); we might expect him, then, to jeer with Molière at the perversion of explanation expressed in this burlesque of an oral exam. Interestingly and emphatically, however, he does not jeer; he writes that this scene

is a poignant satire, 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. l 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 almost 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.2

This is the mature Peirce speaking, and most of the material we will draw upon is from his later work. Before we get into a detailed study of that, however, we shall look briefly at some of his early mentions of abstraction, and will make clear which of the different historical concepts of abstraction is of concern to him. The detailed study of his treatment of the topic will begin by examining the way abstraction interacts in his thought with his concept of "theorematic reasoning"; we then consider abstraction and "contextualism" or "perspectivalism" in Peirce's thought. After looking briefly at the relationship of abstraction to entia rationis, we move through the contrast between the abstract and the concrete to an examination of abstraction and Peircean thirdness; this includes an indication of how the concept fits into Peirce's semiotic.

Possibility is also associated with abstraction in Peirce's writing; I sketch the outlines of a fit between these concepts. Finally, as a link to more recent thought and a further elucidation of Peircean abstraction, I examine abstraction in connection with some topics from recent linguistic theory and from Gestalt theory.

Peirce mentions abstraction at least as early as the.1867 "On a New List of Categories" (see 1.549); another early reference (1868) is given in 2.422. In these locations, he apparently fails to make an important distinction which he in Par aft in his later writings. as for example. about 1901:

A decrease in supposed information may have the effect of diminishing the depth of a term without increasing its information. This is often called abstraction; but it is far better to call it prescission; for the word abstraction is wanted as the designation of an even far more important procedure, whereby a transitive element of thought is made substantive, as in the grammatical change of an adjective into an abstract noun. This may be called the chief engine of mathematical thought. (2.364)

Prescission is the process in which blue, green, and red balls are all recognized as spherical, and German Shepherds and Chihuahuas are both recognized as dogs. We prescind from the color, say, and, considering only the shapes, recognize the sameness of all balls in this regard. Peirce suggests in this passage that we not think of this process as abstraction, but reserve that term for the process which enables us to speak of the "dormitive virtue" of opium. The 1867 and 1868 locations (1.549, 2.422) appear to ignore the abstraction/prescission distinction. It might be tempting to see this as evidence of an essential difference in treatment of this topic from the early to the late Peirce. The early Peirce recognized the distinction, however, at least so far as the products of the two processes are concerned; in an 1868 letter to the editor of the Journal Speculative Philosophy he writes:

You have apparently understood me as applying the term "abstract" to any concept the result of abstraction. But as I intimated [see 6.620], I adopt that acceptation in which "whiteness" is said to be abstract and "white" concrete. (6.627)

It would seem that Peirce here is following the historical and ambiguous tradition of calling both prescission and the transformation from "white" to "whiteness" abstraction, while being clear about the difference between the results of the processes, and so between the equivocally named processes themselves.

Peirce was later to become very clear about naming the processes; I note one other passage, from a 1904 letter to E. H. Moore:

There are two entirely different things that are often confused from no cause that I can see except that the words abstract and abstraction are applied to both. One is afairesiV leaving something out of account in order to attend to something else. That is precisive abstraction. The other consists in making a subject out of a predicate. Instead of saying, Opium puts people to sleep, you say it has dormitive virtue. This is an important proceeding in mathematics. For example, take all "symbolic" methods, in which operations are operated upon. This may be called subjectal abstractions.3

Peirce also (and more frequently) calls that process "hypostatic abstraction." It is hypostatic or subjectal abstraction that Peirce is interested in; a hint as to why he is interested in it is given in his allusions in these passages to mathematical reasoning; we shall develop this aspect of abstraction at length. Jaakko Hintikka has done us the great service of bringing to our attention4 and tying to contemporary experience one of Peirce's central observations about necessary—which is to say mathematical—reasoning: this is that nontrivial deductive reasoning, even in areas where explicit postulates are employed, always

considers something not implied in the conceptions so far gained [in the particular course of reasoning in question], which neither the definition of the object of research nor anything yet known about could of themselves suggest, although they give room for it.5

Such "theorematic reasoning," to use Peirce's term, introduces its novel elements into the reasoning process in the form of icons, which are then "experimented upon in imagination." That "all necessary reasoning is diagrammatic"6 is a refrain of which the later Peirce seems not to tire; "theorematic reasoning," with its introduction of novelty in diagram, or icon, he contrasts with "corollarial reasoning," which requires the introduction of no new icon or construction (the last word suggests, and is intended to suggest, the constructions employed in the proof of the propositions of Euclidian geometry)7 Hintikka suggests that "Peirce himself seems to have considered a vindication of the concept of abstraction as the most important application of his discovery [of the theorematic/corollarial distinction]."8 Peirce would indeed have agreed that the light shed on necessary reasoning by this distinction helps greatly to illuminate the role of abstraction;9 in fact, so far as the interaction of abstraction and theorematic reasoning is concerned, he remarks that

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 practically important results of mathematics could not in any way be attained without this operation of abstraction.10

For Peirce, as we have noted, all necessary reasoning is diagrammatic, where "a diagram is a representamen which is predominantly 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 lookout 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)

To recast some of our remarks in language that Dewey, say, might have used: where the diagrams involved are already apart of the unproblematic in inquiry, the reasoning involving them is corollarial; where the synthetic reconstruction or generation11 of the needed diagrams is itself part of the problematic, the reasoning is theorematic. The diagrams or icons so supplied and employed are, of course, signs, and so have objects.12 Now, these objects might be abstract, but let us examine a case in which the objects have a nonabstract character. I have a little 3-D puzzle; the assembly steps for it must be executed in a given order. To get it together, I must assemble in my head a model—an icon or related set of icons—of the spatial and temporal relations involved in the assembly. My production of these icons and my experimentation on them are aided by the ongoing assembly of the puzzle itself, but the icons involved are unquestionably distinct from the puzzle, as signs are distinct from their objects. The object of the icons here is the puzzle-in-assembly. Insofar as the icons are novel, the necessary reasoning based on them is theorematic, on my analysis. But it seems reasonable to think of that reasoning as what Peirce calls "Non-abstractional" theorematic reasoning, because, in this case, of the actual existent nature of the object.13 Comparable "abstractional reasoning" might involve an investigation of the relations between properties of puzzles, using statements like "The difficulty of a puzzle is proportional to its complexity," or some such; note, for future reference, the generality of this last statement compared to those about a concrete puzzle. I note, by the way, that the puzzle-in-assembly itself is not "non-abstract" in some absolute sense; more on this anon.

Let us look now at some more of Peirce's remarks:

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.14

However, we might expect that abstractions will differ somewhat from "ordinary things"; after all, "You couldn't load a pistol with dormitive virtue and shoot it into a breakfast roll."15 So, further,

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.16

Peirce, I believe, presents the concept of absolutely primary substance as a logical "limit point," as a regulative ideal; we need not commit ourselves to a search for such substances17—it is enough, I would suggest, that we recognize primacy here as relative; from the perspective of ordinary common sense experience, for example, something I can shoot into a breakfast roll has a primacy that "dormitive virtue" does not; speaking relatively,

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 on a part thereof.18

As a mathematical example, Peirce suggests the relationship between a point and a line;19 if we think of lines, or filaments, as loci of moving points, or particles, then

if the particles be conceived as primary substances, 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.20

It is obvious that what Peirce is calling abstraction figures in a most fundamental way in the definition of mathematical concepts. Note that in a mathematical discipline there is often considerable freedom as to which concepts are considered "primitive"—thus what is considered "primary substance" in the above sense is really a matter of perspective or context. We may contrast this with the absolutistic reductionism of, for example, Principia Mathematica.

What we are saying applies as well in the physical sciences. Classical mechanics nowadays is commonly taught as a meter/kilogram/second system; length, mass, and time are primitives from this viewpoint, and force (as well as other relevant concepts) is defined in terms of them. But I learned theoretical engineering mechanics in a foot/pound/second format; English vs. metric is not at issue in this case. What is is that "pound" here is a unit of force rather than mass. There is no problem: the unit of mass (a "slug," which weighs about 32.2 pounds on the surface of the earth) is defined in terms of length, force, and time, and we are in business. These examples fit well with Peirce's desire to leave absolutely primary substance "for the metaphysicians to wrangle about." This suggests that implicit in Peirce's theory of abstraction is a perspectival approach to reality akin to that made explicit by, say, Charles Morris in his "Objective Relativism."21 ** Indeed, Peirce carries his theory beyond mathematics or theoretical physics:

Atoms are supposed to have existences independent of one another. But in that case according to our definition of an abstraction, a collection of atoms, such as are all the things we see and handle are [sic] abstractions.22

So it would be wrong to think of abstractions as purely and always fictions:

to deny every mode of being to anything whose being consists in some other fact would be to deny every mode of being to tables and chairs, since the being of a table depends on the being of the atoms of which it is composed, and not vice-versa. (4.463)

I wish to emphasize that Peirce is not advocating the naive reductionism which claims that what a table really is is mostly empty space with appropriate molecules at strategic locations; he refuses to "deny every mode of being" to macro-physical objects, and the passage just quoted may be taken as a defense of the reality of at least certain "abstractions." Also, I note that Peirce is not here "arguing to" the existence of tables and chairs. This is a matter of brute secondness, which sets the boundaries within which a theory of abstraction must operate, rather than vice-versa. I would suggest even further that since the evidence for and about atoms is secured and processed mediately compared to the evidence for and about tables and chairs, there are perspectives from which the atoms may be considered the abstractions and the tables and chairs the "primary substances." That that which is considered primary substance in this theory of abstraction is a matter of perspective or context rather than an absolute stance for Peirce finds support in another passage—I note that this passage and the associated preceding ones were written very close together (about 1903). In discussing collections, which he sees as an important class of abstraction, Peirce remarks that

According to the definition [of collection], there must be a collection of luminaries of the day. But there happens to be only one luminary of the day namely, the Sun. Here, then, is a collection having but one member. Is not that collection the Sun itself ? I reply, certainly not. For a collection is an ens rationis. Its being consists in the truth of something. But the Sun is not an ens rationis and its being does not consist in the truth of any proposition. It consists in the act of brute force in which it reacts with everything in its neighborhood.23

Here the sun—which is as much a matter of atoms as is my table—is placed in the position of "primary substance"—primary with respect to the abstraction which is its singleton set. In the context of the previous passages, I take this as strong evidence that at least implicit in Peirce's theory of abstraction is a "radical perspectivalism." This is hardly the most thoroughly examined aspect of Peirce's philosophy, but it fits well with the broader context of his work as well as with the empirical methodology of pragmatism in general.

Note that the above quoted passage speaks of entia rationis; I will mention that Peirce in this period uses the terms 'ens rationis' and 'abstraction' (the product of the process) virtually synonymously; 4.463 (earlier quoted) also uses 'ens rationis'; there he speaks of an ens rationis as having "its being [consist] in some other fact." In 4.463 he clearly speaks of entia rationis as the end products of abstraction. In the lecture just before quoted24 he relates 'collection' and 'abstraction'; in 5.534 he reminds us that "a collection is an hypostatic abstraction, or ens rationis"25; similar juxtapositions and interchangeable usages occur passim. It is clear that the product of hypostatic abstraction is, for Peirce, an ens rationis (see 3.642 for another example) whether he recognized other entia rationis than abstractions might possibly be considered an open question; my present belief, however, is that it is safe to identify the Peircean ens rationis with the product of his hypostatic abstraction, which emerges as a process of exceptionally broad applicability; the nature of this broad applicability will, I think begin to emerge in what follows.

It is very common to contrast the abstract with the concrete; you have perhaps noted that so far, very little has been explicitly said about this although Peirce often speaks of abstraction, he less often places it explicitly in contrast with the concrete. When he does, he is most likely to refer to the predicate which enters into the definition of the abstraction as concrete—he notes that "hard is concrete and hardness is abstract"26; in the first quoted passage of this paper, he speaks of hypostatic abstraction as "the transformation of a concrete predicate into an abstract noun."27 The relationship of the abstract to the concrete is closely connected to that between the abstraction and the "more primary substances" or namable things upon which the abstraction's "being" depends, in that the concrete predication contrasted with a given abstraction is applied to the more primary substances upon which the abstraction in question depends. Although Peirce felt that abstraction was of vital importance in reasoning, he insisted, consistent with the pragmatic maxim (5.402), that the being of abstractions—their semantic meaning—swings on possible effects in the domains where are made the concrete predications associated with those abstractions. Peirce tells us that according to the definition of a proposition,

the Interpretant of it . . . represents the proposition to be a genuine index of a Real Object, independent of the representation .... the definition adds that this Object is a Secondness or real fact. (2.315)

"Real fact" here should not be construed too narrowly; Peirce was always most emphatic about not limiting the real to the actual existent—remember, for example, that we are interested here in the truths of mathematics as well as those of the world of actual existents. Later in the paragraph above cited, Peirce connects experience in a very broad sense "—whether outward experience, or experience of fancies—" with real facts; possible experience is the key to the meaningfulness of propositions; that paragraph goes on:

every kind of proposition is either meaningless or has a Real Secondness as its object. This is a fact that every reader of philosophy should carefully bear in mind, translating every abstractly expressed proposition into its precise meaning in reference to an individual experience. (2.315)

I would suggest that the "Real Seconds" associated with the "meaning in reference to an individual experience" of an abstract proposition are found among the more primary substances upon which the being of the abstraction depends; "real seconds," then, are such relative to appropriate perspectives. Consistent with what I am here suggesting, we find Peirce praising certain pragmatists of his day for

their insistence upon interpreting all hypostatic abstractions in terms of what they would or might (not actually will) come to in the concrete. (6.485)

Abstractly stated propositions must be connected with individual experiences. But as the citations above indicate, each abstraction is interpreted in terms of the whole range of possible events or occurrences associated with the concrete predications on which the abstraction is based, from which it is "necessarily inferred" (see 4.463). An abstraction from this viewpoint is a unification of or a continuum between possible occurrences, and so is a thirdness, a generality, a habit; the being of the abstraction is dependent upon concrete occurrences, but they are dependent on it for their intelligible ordering. Let us look further; discussing symbols, Peirce remarks that

Every symbol is an ens rationis, because it consists in a habit, a regularity; now, every regularity consists in the future conditional occurrence of facts not themselves that regularity. (4.464)

In other words, the being of a habit or regularity consists in the truth of propositions concerning facts, whose being is, in turn, a matter of secondness. Each habit or regularity is, then, an abstraction in the sense we have been developing. But we have already noted that each abstraction may be reasonably considered to be a habit or regularity. This all has the rather remarkable effect of identifying Peirce's theory of abstraction as a theory of thirdness in general, which goes a long way toward explaining the extreme broadness of applicability of abstraction in Peirce's view.

Notice that I have not said "the theory of thirdness." The intelligible wholes which are thirds present us with a variety of aspects from which we may study and theorize about them; the aspects are related, but emphasize different features of the thirdness in question. As an example, consider mathematical functions. A function may be studied as itself an object with its own structure, composition, relatedness to other functions, etc.; from this point of view, it is an abstraction in Peirce's sense. Secondly, it may be studied in terms of its domain and codomain, as a mapping which involves the possible arguments and values of the function. Thirdly, the function may be considered a set of instructions for effecting certain outcomes. This third point of view mediates between that of function as abstract structure and function as a set of domain/codomain ordered pairs. This triadic view, then, considers a thirdness X (in this example, a mathematical function)first, from the point of view in which it is an intelligible body of interconnected relations, capable of being represented by icons and fruitfully studied by experimentation on those icons. As such, it is an abstraction, whose "being" resides in the concrete events whose conditional possibility is predicted by it. X may, then, also be viewed concretely, as the set of those possible events. From this point of view a mathematical function is a set of domain/codomain ordered pairs, and a habit in general is a "bundle" of possible conditioned outcomes or "behaviors." Finally, X may be considered a regularity; in the case of the mathematical function, this was the function considered as a set of instructions for effecting certain outcomes; in case the abstraction connected with X is the dormitive power of opium, the regularity connected with X would emphasize that, in general, when opium is administered under given conditions, the subjects predictably tend to go to sleep.

I feel that Peirce's thought considers thirdness from all of these aspects, and that an adequate overall analysis of thirdness demands that all these aspects be appropriately examined; the theory of abstraction focuses primarily on the first of these perspectives.

It seems to me that the analysis in terms of Peirce's categories I have just suggested is very much like that implicit in one of Peirce's own analyses: this is that of his "third trichotomy of signs"—the third, that is, of the ten trichotomies he discusses in his correspondence with Lady Welby.28 The third trichotomy is a classification of signs based on "The Nature of Their Dynamical Objects" (8.366); I let Peirce comment:

It is usual and proper to distinguish two Objects of a sign, the Mediate without and the Immediate within the sign. The interpretant is all that the sign conveys: acquaintance with its Object must be gained by collateral experience. The Mediate object is the Object outside the sign; I call it the Dynamoid29 Object. The Sign must indicate it by a hint; and this hint, or its substance, is the Immediate Object. Each of these two objects may be said to be capable of either of the three Modalities [possibility, actuality, or necessity], although in the case of the Immediate Object, this is not quite literally true. Accordingly, the Dynamoid Object may be a Possible; when I term the sign an Abstractive; such as the word Beauty; and it will be none the less an abstractive if I speak of "the Beautiful," since it is the ultimate reference, and not the grammatical form, that makes the sign an Abstractive. When the Dynamoid object is an Occurrence (Existent thing or actual fact of past or future), I term the sign a Concretive.... For a sign whose Dynamoid Object is a Necessitant, I have no better designation than a "Collective," which is not quite so bad a name as it sounds to be until one studies the matter . . . .30

It is not surprising that Peirce would have difficulties with the word 'collective' here; he regularly emphasizes that "a collection is an hypostatic abstraction, or ens rationis"(5.534), and here he is contrasting "collectives" with "abstractives." Although in his treatment of existential graphs he experiments with notations which distinguish collections from abstractions in general,31 it is not at all clear that these notational differences correspond to the third trichotomy in any way. I would suggest that Peirce's placement of "collective" in the mediating position in that trichotomy has much the same function as viewing thirdness as regulative has in the analysis I have proposed; taking an object as a collective might—from the viewpoint of the third trichotomy—help me to see it as constituted both by ordering principle and of individual seconds, and so this view would have the collective mediating between the abstractive and the concretive. Perhaps "Regulative" would be preferable to "Collective" as a name for the third in this trichotomy; this would also tend to reflect the connection with necessity which Peirce wishes to make here.

I would like simply to point out that there is another important Peircean triad which is very closely related to those we are examining: this is his classification of logic as abductive, inductive, and deductive; discussion of this must wait till another time.

The theory of abstraction is a theory of thirdness, but as the above analyses suggest, the thirdness of abstraction is a first with respect to certain other viewpoints on thirdness. The "being" of abstractions consists, Peirce tells us, in "what they would or might (not actually will) come to in the concrete."(6.485) This and the analysis of the third trichotomy point to an element of possibility involved in abstractions; in fact, we find Peirce speaking of "an essential part of the doctrine of Existential Graphs [which] treats of the general properties of qualities and relations."32 This doctrine, Peirce tells us, is

the doctrine of substantive possibility, because qualities and relations are possibilities of a peculiar kind. In a secondary sense a quality may be said to exist when it has, as it were, a replica in an existing thing. But strictly speaking, a quality does not exist. For to exist is to be a subject of blind compulsion. A quality not only neither exerts nor suffers such force, but it cannot even be called an idea of the mind. For things possess their qualities just the same, whether anybody thinks so or not. The being of a quality consists in the fact that a thing might be such or such like.33

We are here reexamining some of the ground we have covered before, but from a slightly different viewpoint. The "doctrine of substantive possibility" which Peirce is discussing is pretty clearly the theory of hypostatic abstraction, with emphasis on the entia rationis involved as possibilities. It is not uncommon for Peirce to associate quality with possibility (see, for example, 1.304); we find this emphasis explicit in the notion of substantive possibility. I would suggest that this notion of possibility is—like 'abstraction' itself—a touching point between Peirce and the Scholastic tradition with which he sometimes associates his thought; "possibility" in this sense is potentia esse, potency to existence. I think that this is clear from the way that Peirce relates the abstract with the concrete; another example, employing language even more scholastic in tone than heretofore, supports this:

A quality is an ens rationis, of course. That is, it consists in a certain proposition having a meaning. The term essence means being such as the subject of the essence necessarily is. Quality then has essence. But it has no existence, because it neither exercises nor suffers brute compulsion.34

So quality and hypostatic abstractions are involved with possibility. But we must not assume that all of Peirce's uses of 'possibility' are reducible to, or even similar to this one. We get a monition of this in 4.549, where Peirce mentions sequences of abstractive operations, and seems to imply that based on such sequences may be divisions of the subject matter of logic; he warns us, however,

that the divisions so obtained must not be confounded with the different Modes of Being: Actuality, Possibility, and Destiny (or Freedom from Destiny). On the contrary, the succession of Predicates of Predicates [and so of hypostatic abstractions] is different in the different Modes of Being. (4.549)

The comment of the editors of vol. 4 that the "Modes of Being" are "usually called categories by Peirce" is only partly to the point; the Modes of Being are specifically the "Universes" represented by the "tinctures" in the 1906 "Prolegomenon to an apology for Pragmaticism"—of which 4.549 is a part—and several other contemporary manuscripts by Peirce. In one such manuscript, Peirce has characterized these "Universes of modes of reality" as the Universe of Real Capacities, the Universe of Actual Fact, and the Universe of tendencies (which he explicitly associates with destiny), and comments that

the imperative need of the further manifold differentiation of each of them [the three universes] is ... apparent. It is clear that the differences are not differences of the predicates or significations of the graphs, but of the predetermined objects to which the graphs are intended to refer. Consequently, the Iconic idea of the System requires that they should be represented not by differentiations of the Graphs themselves, but by appropriate visible characters of the surfaces upon which the Graphs are recorded.35

He then goes on to discuss how to go about varying the characters of the surfaces; the tentative result is his "tinctured existential graphs." The point of this is that tinctured existential graphs is, essentially, one of Peirce's attempts—half a century before Kripke and Prior succeeded—to set up what we recognize as a "possible-world semantics" for the existential graphs.36 The "predetermined objects" in the above passage, then, will differ not as dormitive virtue differs from opium and breakfast rolls—since dormitive virtue pertains to the same universes as does opium—but as the namable individuals in one possible world in quantified modal logic differ from those in another. In terms of possibility, this means that we cannot expect the concept of possibility associated with qualities as "substantive possibility" (i.e., hypostatic abstractions) to automatically coincide with that developed via the possible-worlds approach of the "tinctures."

From my last remarks, it should be clear that Peirce saw the existential graphs as having an important role in the study of hypostatic abstraction; unfortunately, space will not permit a detailed development of this intersection of two of the important areas of the late work of Peirce. Don Roberts has discussed many of the aspects of Peirce's "gamma system of graphs" as they affect the theory of abstraction;37 I feel that there is room for considerable study in this area, and Roberts's work gives us a good start on this. One aspect of Peirce's gamma graphs that deserves mention here is his use of (implicitly quantified) "lines of identity" to represent qualities and relations (4.470 ff.); this makes the entia rationis involved the values of quantified variables, and represents a logical step which reasonably follows the naming of those abstractions. This constitutes another link to the earlier-mentioned work of Hintikka, who employs "higher level" quantification in his explication of Peirce's concept of theorematic reasoning.38 There are here strong suggestions of the intimacy of the link between theorematic reasoning and abstraction; there remains, I believe, valuable work to be done in this connection.

As I approach the conclusion of this effort, I would like to comment on some connections between the material I have been investigating and some other matters of contemporary interest, and in the process, cast more light upon hypostatic abstraction itself. First of all, we have seen Peirce both explicitly and implicitly treat the process of abstraction as a grammatical transformation. In contemporary language popular in some circles and controversial in many (but of indubitable import), the movement to an abstraction is like the mapping from a "deep" to a "surface" structure in transformational grammar; indeed, we might consider it an example of such a transformation. Whatever the flux and controversy surrounding transformational grammars, it is clear that in the transformations therein considered, there must be a continuity of meaning in some sense of the term between the "deep" structures and the structures on which they are mapped. (I might insert that for such a theory to make sense, it must also account for differences in use—and so, some of us would say, meaning—between such structures.) The simplest kind of situation here is that in which the deep and the surface structures are both assertions; in many important cases we would expect such assertions to be true and false under the same conditions, that is, to be semantically equivalent. This is so of the active to passive transformation of

(1) Opium puts people to sleep.


(3) People are put to sleep by opium.

It is also true of the kind of transformation central to this paper, from (1) to

(2) Opium has dormitive power.

If anything is clear about Peirce's view of the relationship of (1) to (2), it is that there is a difference between them; that the difference is not semantic we have indicated. To give the question a Jamesian twist: "What, then, is the difference that makes the difference?" It seems to me that the answer to this would be relevant not only to Peirce's theory of abstraction, but to contemporary discussions of language in general.

Actually, for all the head-scratching that this kind of question has caused, the answer, from an authentic Peircean point of view, is close at hand: "Consider what effects . . . ," says the pragmatic maxim (5.402 again). Specifically, consider the effects of a sign insofar as it participates in semiosis—consider its interpretants. Peirce's repeated insistence that the "being" of an abstraction consists in the truth of statements about "more primary substances" is a recognition that semantically—so far as their objects39 are concerned—(1) and (2) are equivalent. But he also insists that we gain something by use of the abstraction—our reasoning about the relations involved in the object of our study is facilitated, sometimes dramatically. This is an effect of the use of the abstractive signs; it is an effect arising in the use of these signs as signs, and so is among their interpretants. The effect is a disposition, a habit regulative of reasoning behavior, and so is a final logical interpretant. (5.491) Here is a point at which it is useful to draw on some Morrisian terminology, specifically, Morris's distinction between the syntactic, semantic, and pragmatic dimensions of semiotic.40 While the abstractive and nonabstractive assertions have the same meaning on the semantic dimension, they differ, often considerably, on the pragmatic dimension, which depends on the relation of the sign to its interpretants. One of the advantages of introducing Morris's terminology is that it is in fairly general use, and so gives us continuity with terminology likely to be employed in the continuing dialogue on language. In fact, we now find Chomsky commenting that

It makes sense, I think, to distinguish what is sometimes called ``grammatical competence" from "pragmatic competence.", . . . Pragmatic competence underlies our ability to use [knowledge of form and meaning in language] along with the conceptual system to achieve certain ends or purposes. It might be that pragmatic competence is characterized by a certain system of constitutive rules represented in the mind, as has been suggested in a number of studies.41

Chomsky uses 'meaning' here in the sense of semantic meaning. The uses to which signs are put are directly correlative with the effects these signs have when used as signs—their interpretants; Chomsky and I are then talking about the same things. That consideration of pragmatic competence is essential for Chomsky in 1980 is indicated by his remark that

I assume that it is possible in principle for a person to have full grammatical competence and no pragmatic competence, hence no ability to use a language appropriately, though its syntax and semantics are intact.42

Whether or not one can have grammatical competence without any pragmatic competence might be argued; nevertheless, it is clear that in any case, without pragmatic competence the language could not be used, and so would be effectively "absent." So my suggestion that the difference made by hypostatic abstraction is a component of meaning on the pragmatic dimension isn't only reasonable in relation to Peirce scholarship, but in the context as well of the most recent thought in the contemporary theory of linguistics. We have seen Peirce state that "An abstraction is something denoted by a noun substantive, something having a name."43 All of Peirce's talk of substance in this context, in fact, is talk of namable things, insofar as they are namable. Naming, as part of language, is a human process whose peduncles merge in prehistory with the mythically expressed efforts of humans to understand and control their environment. Cassirer, commenting on these mythic roots of culture, observes that "He who knows the true name of a god or demon has unlimited power over the bearer of the name."44 So we name entia rationis, and gain power over the intelligibilities implicated in them. Of course, we do not see the bare naming as constitutive of that power; Cassirer again remarks that

It is the name which introduces the first factor of constancy and permanence into this manifold; the identity of the name is the preliminary step, an anticipation of the identity of the logical concept.45

Naming is one aspect of a larger process, that of attending to significant organizations of data in our interactions with our environment; for Peirce,

Attention is a certain modification of the contents of consciousness with respect to a centre. This centre is where there is a strong sense-will reaction, which imparts to the idea the nature of an index (weathercock, sign-post, or other blind, forcible connection between thought and thing). Now, the subject of a proposition is just such an index. Hence the real phenomenon of attending to a quality, say white, or making it the centre of thought, consists in thinking of it as the subject to which the other elements of thought are attributes.(2.428)

He goes on to point out that attention is associated with hypostatic abstraction (rather than with prescission). Peirce's terminology here suggests a way of looking at abstraction which has great import not only for the matter of this paper, but also, I beleive, for philosophical work in general.

Organisms deal adaptively with their environments in cycles involving perception, manipulation of the environment, and consummation;46 it is typical of Dewey and Morris, for example, to emphasize in their thought our ties with these natural movements. This adaptive integration with the environment demands the selection of relevant materials and the creative structuring of these materials into new integral wholes ordering the organism/environment field. That Gestalt psychologists such as Kohler, Wertheimer, and Koffka emphasized the role of such organismic integrations of material in perception is well-known; that the concept of gestalt formation extends well beyond organization of the perceptual field is emphasized by contemporary Gestalt therapists47 and has received careful and articulate—if not well-known—philosophical study.48

The pragmatists have not dealt with the gestalt by name as a philosophical concept, but it is implicit and close to the surface in much of their work; the ending of the irritation of doubt by belief in Pierce's theory (5.358 ff.), for example, is the completion of a gestalt; another example close to the matter of this paper is the creative construction of the icons of theorematic reasoning. In Dewey, whose work is, in many ways, complementary to that of Peirce, we see the gestalt figuring centrally in esthetics and ethics as well as in logic; we have the type of esthetic experience when "An experience has a unity that gives it its name, that meal, that storm, that rupture of friendship."49 In ethics,

Good consists in the meaning that is experienced to belong to an activity when conflict and entanglement of various incompatible impulses and habits terminate in a unified orderly release in action.50

And in logic, Dewey defines inquiry as

the controlled or directed transformation of an indeterminate situation into one that is so determinate in its constituent distinctions and relations as to convert the elements of the original situation into a unified whole.51

This is strongly consonant with Peirce's theory of inquiry, and the examples from esthetic and ethical theory likewise fit with Peircean pragmatism.52

So the concept of the gestalt—under other names—is no stranger to pragmatic thought; nor are other concepts associated with gestalt theory: the ubiquitous pragmatic insistence that the problematic emerges within a largely unproblematic situation, for example, is an important case of the gestaltists' figure/ground polarity. I would suggest that the theory of abstraction is quite susceptible of analysis in gestalt terms; Peirce's association of "attention with respect to a centre"(2.428) with abstraction is directly suggestive of the process which chooses elements from the manifold and organizes them into figure—the process of gestalt formation. The integrative activity of most organisms, including a lot of human activity, involves the structuring of concrete materials in the environment. But in the course of human evolution, we learned that "things" other than actual existents can be named, can be made centers of attention. Peirce's theory of abstraction is an explicit taking-in-account of this fact, and an attempt to locate it in our overall dealings with reality.

Abstraction occurs in specific contexts, relative to perspectives, and so does gestalt formation; both provide unified manipulable wholes as part of the resolution of problematic situations. The wholes are structurings of existent facts, be they about the observed positions of Mars or the anxieties and excitements of human contact. Perls, Hefferline, and Goodman, commenting on the contextual nature of Gestalt Therapy, remark that

as treatment progresses it is frequently necessary to change the emphasis of approach, from the character to the muscle-tension to the habit of rapport to the dream and back again. We believe that it is possible to avoid circling aimlessly if, precisely by accepting all these to give a variety of contexts, one concentrates on the structure of the figure/background, and provides free occasions for the self progressively to integrate the self.53

The employment of abstractions here speaks for itself; the recognition and use in concrete situations of such abstractions is integral to this method, which I see as having vital connections to the theory of abstractions we have been examining.

The material on abstraction and gestalt formation as well as that relating to transformational grammar is, because of space limitations, merely suggestion. Similarly, in what I have done here, I see many suggestions for further work in Peirce scholarship as well as mathematical logic; one Peircean topic that suggests itself to me is that scantily-explored Peircean "methodeutic."54 There is much fruitful work to be done in connection with what Peirce calls abstraction. I can see a number of paths in my own work associated with these matters, and hope that others in the community of "all who inquire" (5.407) will find fruitful fields here as well.

University of Florida


* The Monist 65 (1982), 211-229, and The Relevance of Charles Peirce, ed. Eugene Freeman, La Salle, Illinois: Monist Library of Philosophy, 1983, 293-311.

** Editor's Note: the term "Objective Relativism" was coined by Arthur E. Murphy in 1927, in an article Entitled "Objective Relativism in Dewey and Whitehead," Philosophical Review, 36 (1927), pp. 121-44.

1. The Collected Papers of C. S. Peirce, vols. 1-6, ed. Charles Hartshorne and Paul Weiss, 1931-35; vols. 7 - 8, ed. A. W. Burks, 1958 (Cambridge MA: Harvard University Press). Citations from the Collected Papers are as usual in Peirce scholarship; the first numeral is the volume, and the rest is the paragraph; thus 5.534 is paragraph 534 of volume 5.

2. Charles S. Peirce, The New Elements of Mathematics, vols. 1-4, ed. Carolyn Eisele (The Hague: Mouton, 1976). Citations from this series will be marked by 'Eisele', followed by volume number and page number. The present citation is Eisele 4, p. 160.

3. Eisele 3/2, p. 917 (volume 3 is bound in two parts). Other relevant statements by Peirce are at 2.428, 4.235, 4.463, 4.332.

4. Jaakko Hintikka, "C. S. Peirce's 'First Real Discovery' and its Contemporary Relevance," Monist 63:3 (July, 1980): 304-13.

5. Eisele 4, p. 49.

6. Ibid., but see also 1.162, 4.430 ff., and many other locations in the later work of Peirce, the concept is expressed as early as 1885, in 3.363.

7. See Hintikka for details. (cited in n4, above).

8. Ibid., p. 13.

9. It seems to me that the passages cited by Hintikka on this point (Eisele 4, pp. 49-50, 159-60) are really just indirect evidence for the statement that "Peirce himself seems to have ...."; this does not, however, weaken the argument that abstraction was of key importance to Peirce, and that abstraction and theorematic reasoning work together in the most important kinds of mathematical reasoning.

10. Eisele 4, p. 49.

11. This "generation" is gestalt formation, to use more recent terminology; more on this later.

12. My personal preference in terminology for the second semiosical relatum is for Morris's 'signification'; see Charles Morris, Signification and Significance (Cambridge: MIT Press, 1964). Since we are dealing with Peirce here, I will, however, use 'object', understanding the word in the very broad sense of Peirce's semiotic.

13. There may also, of course, be nonabstractional corollarial reasoning.

14. Eisele 4, p. 161.

15. Ibid., p. 162.

16. Ibid.

17. That this is also Peirce's view is suggested by Eisele 4, p. 164, where, having argued for the reality of abstractions, Peirce states that they "may be real—indeed, a good deal less open to suspicion than are the primary substances." On absolutely primary substance as a "regulative ideal," compare Peirce's concept of truth and the real (5.407).

18. Eisele 4, p. 162.

19. Ibid., pp. 162 - 63.

20. Ibid., p. 163; see also 4.235.

21. Charles Morris, The Open Self (New York: Prentice-Hall, 1948), p. 129 ff.

22. Eisele 4, p. 163.

23. Eisele 3/1, p. 354.

24. Ibid., p. 353.

25. Eisele 4, p. 164.

26. Ibid., p. 160.

27. Ibid.

28. See 8.344 ff. and 8.366; also, see Charles S. Peirce and Victoria Lady Welby, Semiotic and Significs, ed. Charles S. Hardwick (Bloomington: Indiana University Press, 1977), pp. 83-84.

29. Elsewhere, Peirce uses 'dynamic' or 'dynamical'.

30. Peirce and Welby, pp. 83-84.

31. See, for example, 4.411-13; for some discussion of abstraction and the existential graphs, see Don Roberts, The Existential Graphs of C. S. Peirce (= Approaches to Semiotics 27 (The Hague: Mouton, 1974), p. 64ff.

32. Eisele 3/1 p. 350.

33. Ibid., pp. 350-51.

34. Ibid., p. 353.

35. Ms. 300 in the Peirce Microfilms and in Richard Robin, Annotated Catalogue of the Papers of C. S. Peirce (Amherst MA: University of Massachusetts Press, 1967); the quoted passage is on pp. "Bed 38" and "Bed 39" of Ms. 300.

36. See J. Jay Zeman, "Peirce's Logical Graphs," Semiotica 12:3 (1974): 251ff; For considerable detail on tinctured existential graphs, see Roberts, p. 87ff.

37. Roberts, p. 64ff.

38. Hintikka, pp. 6-7, 11-13.

39. Once again, I confess to being more comfortable, personally, with Morris's 'signification' than with Peirce's 'object'.

40. Charles Morris, Foundations of the Theory of Signs (Chicago: University of Chicago Press, 1938), p. 695.

41. Noam Chomsky, Rules and Representations (New York: Columbia University Press, 1980), p. 59.

42. Ibid.

43. Eisele 4, p. 161.

44. Ernst Cassirer, The Philosophy of Symbolic Forms, vol. 2, tr. Ralph Manheim, (New Haven: Yale University Press, 1955), p. 41.

45. Ibid., vol. 3, p. 14.

46. See Morris, Signification and Significance, p. 395 for connections between this and semiotic.

47. A basic document is Frederick Perls, Ralph Hefferline, and Paul Goodman, Gestalt Therapy, (New York: Dell, 1951).

48. See, for example, Aron Gurwitsch, The Field of Consciousness (Pittsburgh, PA: Duquesne University Press, 1964).

49. John Dewey, Art as Experience (New York: Capricorn, 1958 [orig. 1934]), p. 37.

50. John Dewey, Human Nature and Conduct (New York: Modern Library, 1957), p. 196.

51. John Dewey, Logic: The Theory of Inquiry (New York: Holt, 1938), pp. 104-05.

52. For expansion here, see J. Jay Zeman, "The esthetic sign in Peirce's semiotic," Semiotica 19:3/4 (1977): 241-58.

53. Perls, Hefferline, and Goodman, p. 245.

54. Methodeutic is a projected but undeveloped part of Peirce's semiotic which fits approximately where pragmatics does in the Morrisian terminology; he sometimes calls it "speculative rhetoric"; see 2.93, 2.229. For mention of a "method to discover methods" in deductive reasoning, see 3.364, 3.454.