Peirce's Continuous Predicates

Peirce’s Continuous Predicates

FRANCESCO BELLUCCI Abstract Around 1906, Peirce discovered that the logical analysis of a proposition comes to an end when a “continuous predicate” is found. Continuous predicates are those predicates that cannot be analyzed, or, which is the same, are only analyzable into parts all homogeneous with the whole. This paper examines Peirce’s concept of continuous predicate and its relevance to his theory of logical analysis.

Keywords: Charles S. Peirce, Combination, Continuity, Continuous Predicate, Existential Graphs, Leading Principles, Logical Analysis, Triadicity.

A rose is a rose is a rose is a rose.

Gertrude Stein

Introduction

As is well known, according to Charles S. Peirce one of the principal tasks of logic is the

analysis of reasoning (CP 4.134, 1893). This was indeed the explicit purpose of his logical

algebras and graphical logic, and Peirce often credits himself with possessing a special gift for

logical analysis (SS 114, 1909). Yet he surprisingly also holds that “absolute completeness of

logical analysis is no less unattainable [than] is omniscience. Carry it as far as you please, and

something will always remain unanalyzed.” (R 1454, 1902)1 The question thus arises as to what

Peirce could have meant by saying that, in logical analysis, something remains unanalyzed. This

question, important though it is for determining both the extent and the limit of Peirce’s notion of

logical analysis, has received little attention in Peirce scholarship. The hypothesis that I shall

develop here is that that which remains unanalyzed in logical analysis is what Peirce comes to

call a “continuous predicate.” Continuous predicates are those predicates that cannot be analyzed,

or, which amounts to the same, are only analyzable into parts all homogeneous with the whole.

These predicates are indeed said to be “very important in logical analysis,” for “when we have

carried analysis so far as to leave only a continuous predicate, we have carried it to its ultimate

elements.” (SS 72, 1908)

Despite the importance Peirce accords to it, the notion of continuous predicate has not

received the attention in secondary literature that it warrants. Murray Murphey, who was the first

to write about it, rightly suggests that continuous predicates have a fundamental role in Peirce’s

late synechistic philosophy. However, Murphey goes on to claim that, from his theory of

continuous predicates, Peirce “drew the conclusion that relations constitute continuous

connections among their correlates.” (1961, 318) It will be shown that this claim does not in any

way follow from Peirce’s argument. Kelly Parker’s account (1998, 67-71), referring as it does to

Peirce’s peculiar conception of combination, seems more satisfactory; but like Murphey he

makes an erroneous inference, purporting that when two concepts are connected by a continuous

predicate, “[t]hey arrive as one idea, but may be distinguished if necessary.” (Ibid., 95). By

contrast, Helmut Pape correctly considers Peirce’s continuous predicates as “not-analyzable

form[s] of pure combination” (Pape 1990, 58; see also Pape 1989, 277-279), though he does not

spell out the details of his suggestion. Taking Pape’s remark as my starting point, I will try to

give some indications that might help facilitate a deeper understanding of what is an

incontestably important notion of Peirce’s mature logical thought and, in so doing, also to point

out the error behind Murphey’s and Parker’s interpretations.

For all the significance he attaches to continuous predicates, Peirce himself writes

surprisingly little about them. In his published and unpublished papers the phrase “continuous

predicate” (with its variants “continuous conception,” “continuous relation,” and “continuant

sign”) is used very little, no more than five times. Four out of five of these occurrences appear in

letters or sketches of letters dating from 1906 to 1908, while another is in a manuscript from late

October 1908. With the exception of these five, to my knowledge there are no other references to

continuous predicates in Peirce’s work, nor are there any more in-depth treatments that would

help elucidate his use of them. It is therefore reasonable to ask why such a “very important”

concept was only given attention for such a brief and limited period of time.

Part of the answer, I believe, has to do with Peirce’s Existential Graphs. In roughly the

same years in which continuous predicates make their appearance, Peirce begins talking about

two “continuous graphs.” In his work on Existential Graphs the Sheet of Assertion and the Line

of Identity are said to be continuous, for any part of a Sheet of Assertion is itself a Sheet of

Assertion, as any part of a Line of Identity is itself a Line of Identity. Several scholars (Zeman

1968, Roberts 1973, Pietarinen 2006, Burch 2011) have stressed the importance of continuity in

Peirce’s graphic logic, but none of them ever ventured to speculate on the existence of a

theoretical connection between continuous graphs and continuous predicates. I shall argue that

such a connection actually does exist, for the former prove to be the iconic representation of the

latter. If this hypothesis is correct, it will mean that Peirce does not abandon continuous

predicates after 1908; quite the contrary, they just merge with the continuous graphs within the

diagrammatic system of Existential Graphs that he is then developing and perfecting.

This approach will allow us to understand what continuous predicates evolved into, but

not where they evolved from. This constitutes the other part of the answer. I shall argue that, in a

sense, continuous predicates are well present in Peirce’s mind as early as his first works on the

theory of inference, for they are nothing more than a late reformulation—updated to the logic of

relatives and to the definition of continuity—of Peirce’s early theory of logical leading principles.

But once we have gained some understanding of continuous predicates and of these related

concepts, there emerges the question of their connection to Peirce’s characteristic ideas of

combination and triadicity, for Peirce is very clear that the idea of combination is a triadic idea.

Thus, in being pure forms of combination, continuous predicates and leading principles must

express genuinely triadic relations.

This paper is organized as follows: section one introduces Peirce’s five occurrences of the

term “continuous predicates”; section two attempts to analyze them, to connect them to the

problem of logical analysis and to show what I believe is wrong in Murphey’s and Parker’s

accounts; section three deals with Peirce’s continuous graphs and places his continuous

predicates within his graphic logic; section four traces the antecedents of continuous predicates

back to Peirce’s early theory of inference; section five deals with the question of Peirce’s peculiar

conceptions of combination and triadicity; and section six offers some brief concluding remarks.

1. The five occurrences

The topic of continuous predicates appears in Peirce’s writings in 1906 and disappears in

1908. To my knowledge, Peirce did not mention continuous predicates more than five times in all

of his life. If additional occurrences are found, they will probably belong to the period 1906-

1908, and, still more probably, to late 1908. The first written reference to this notion occurs in a

letter written, but apparently never sent, to Lady Welby on March 9, 1906 (SS 195-201). Peirce is

trying to defend his view that common nouns are in fact unsaturated predicates or rhemata, i.e.,

predicates from which one or more subjects have been removed. According to this analysis, if

from the proposition “man is an animal” we remove the subject, what we are left is “— is an

animal,” or “something is an animal.” Peirce goes on to say,

If I throw off from this the “Something is” still what remains is

Something is an animal,

because the conception “Something is” is continuous. That is, it is the same as

“Something is something that is,” or “Something is Something that is Something

that is,” and so on ad infinitum. (SS 198, 1906)

After this fleeting apparition in 1906, continuous predicates disappear, only to reappear

two years later in a more developed form. The second written reference to the concept occurs in

R 611 (October 28-31, 1908), titled “Logic. Chapter I. Common Ground”:

Thus one assertion may have any number of Subjects. Thus, in the assertion “Some

roses are red,” i.e. possess the color redness, the color redness is one of the

Subjects; but I do not make “possession” a Subject, as if the assertion were “Some

roses are in the relation of possession to redness,” because this would not remove

relation from the predicate, since the words “are in” are here equivalent to “are

subjects of,” that is, are related to the relation of possession of redness. For to be in

relation to X, and to be in relation to a relation to X, mean the same thing. If

therefore I were to put “relation” into the subject at all, I ought in consistency to put

it in infinitely many times, and indeed, this would not be sufficient. It is like a

continuous line: no matter what one cuts off from it a line remains. So I do not

attempt to regard “A is B” as meaning “A is identical with something that is B.” I

call “is in the relation to” and “is identical with” Continuous Relations, and I leave

such in the Predicate. (R 611:14-15, 1908)

The third reference occurs in a letter sent to Lady Welby and dated December 14, 1908.

As in R 611 just quoted, Peirce is here concerned with the definition of the subject of a

proposition. It is worth quoting in full:

When we have analyzed a proposition so as to throw into the subject everything that

can be removed from the predicate, all that it remains for the predicate to represent

is the form of connection between the different subjects as expressed in the

propositional form. What I mean by “everything that can be removed from the

predicate” is best explained by giving an example of something not so removable.

But first take something removable. “Cain kills Abel.” Here the predicate appears as

“____ kills ____.” But we can remove killing from the predicate and make the latter

“____ stands in the relation ____ to ____.” Suppose we attempt to remove more

from the predicate and put the last into the form “____ exercizes the function of

relate of the relation ____ to ____” and then putting the function of relate to the

relation into another subject leave as predicate “____ exercizes ____ in respect to

____ to____.” But this “exercizes” expresses “exercizes the function.” Nay more, it

expresses “exercizes the function of relate,” so that we find that though we may put

this into a separate subject, it continues in the predicate just the same. Stating this in

another form, to say that ‘A is in the relation R to B’ is to say that A is in a certain

relation to R. Let us separate this out thus: “A is in the relation R1 (where R1 is the

relation of a relate to the relation of which it is the relate,) to R to B.” But A is here

said to be in a certain relation to the relation R1. So that we can express the same

fact by saying “A is in the relation R1 to the relation R1 to the relation R to B,” and

so on ad infinitum.2 A predicate which can thus be analyzed into parts all

homogeneous with the whole I call a continuous predicate. It is very important in

logical analysis, because a continuous predicate obviously cannot be a compound

except of continuous predicates, and thus when we have carried analysis so far as to

leave only a continuous predicate, we have carried it to its ultimate elements. (SS,

71-72, 1908)

The fourth occurrence is found in a letter addressed to P. E. B. Jourdain, dated December

5, 1908 (NEM 3:879-888). Peirce has just mentioned his chef d’oeuvre (his Existential Graphs)

and introduced his notion of logical valency. He then proceeds to define the subject of a

proposition as follows:

[E]verything in a proposition that possibly can should be thrown into the subjects,

leaving the pure predicate a mere form of connection, such as ‘is,’ ‘possesses (as a

character),’ ‘stands in the dyadic relation ____ to ____,’ ‘and’ = ‘is at once ____

and ____,’ etc. A Pure Predicate should be ‘continuous’ or ‘self-containing’ as ‘A

is coexistent with B’ = ‘A coexists with something that coexists with B,’ and ‘A

possesses the character ϱ’ = ‘A possesses the character of possessing the character

ϱ’ and ‘A stands in the relation λ to B’ = ‘A stands in the relation of being in the

relation of standing in it to the relation λ to the relation of a relation to its correlate

to B,’etc. (NEM 3:885-886, 1908)

The fifth and last appearance of continuous predicates is once again contained in a letter,

probably addressed to Lady Welby, dated December 1908 and never sent (CP 8.342-376, EP

2:481-491). Here Peirce is dealing with the classification of signs. After having distinguished

descriptive signs from designative signs, Peirce introduces what he calls “copulant signs” and

defines them as those signs that “neither describe nor denote their Objects, but merely express

universally the logical sequence of these latter upon something otherwise referred to.” (CP 8.350,

EP 2:484, 1908)3 When they are pure, copulant signs are said to be “continuant”:

[T]he sign “A is red” can be decomposed so as to separate “is red” into a Copulative

and a Descriptive, thus: “A possesses the character of redness.” But if we attempt to

analyze “possesses the character” in like manner, we get “A possesses the character

of the possession of the character of Redness”; and so on ad infinitum. So it is, with

“A implies B,” “A implies its implication of B,” etc. So with “It rains and hails,” “It

rains concurrently with hailing,” “It rains concurrently with the concurrence of

hailing,” and so forth. I call all such sign Continuants. They are all Copulants and

are the only pure Copulants. These signs cannot be explicated: they must convey

Familiar universal elementary relations of logic. We do not derive these notions

from observation, nor by any sense of being opposed, but from our own reason. (CP

8.352, EP 2:485, 1908)

It should be evident that, despite the slight terminological shift from “continuous

conception” through “continuous predicate” and “continuous relation” to “continuant sign,”4

Peirce’s idea remains essentially unchanged. Having identified these five references to the

conception, let us take a closer look at Peirce’s argument.

2. Continuous predicates and logical analysis

In several places, Peirce is very clear that the real purpose of logic is the analysis of

reasoning (CP 2.532, 1893; CP 4.134, 1893; R 1147:14, n.d. but c.1900) and thus, arguably, of

the principal objects that constitute reasoning. Analysis is conceived by Peirce quite standardly as

the process of decomposing something into its constituent parts: “if one concept can be

accurately defined as a combination of others, and if these others are not of more complicated

structure than the defined concept, then the defined concept is regarded as analyzed into these

others.” (CP 1.294, c.1905). What is truly original in Peirce’s thinking, however, is the way he

conceives the ultimate constituent parts obtained through logical analysis: these are said to

possess a special kind of “elementarity.” (R 300:49, 1908) What does the term encompass?

In the “Prolegomena” of 1906 Peirce states that “no analysis, whether in logic, in

chemistry, or in any other science, is satisfactory, unless it be thorough, that is, unless it separates

the compound into components each entirely homogeneous in itself, and therefore free from the

smallest admixture of any of the others.” (CP 4.548, 1906) The purpose of logical analysis is thus

to find those elements that are uniform in composition, i.e., not composed of heterogeneous or

dissimilar parts. Yet, as mentioned, Peirce also thought that absolute completeness of logical

analysis is unattainable, for something will always remain unanalyzed (R 1454, 1902). In a 1909

letter to James Peirce is even more precise:

I find myself bound, in a way which I discovered in the sixties, to recognize that there are

concepts which, however we may attempt to analyze them, will always be found to enter

intact into one or the other or both of the components into which we may fancy that we

have analyzed them. (NEM 3:851, 1909)

Absolute completeness of logical analysis is impossible: there are concepts that always remain

unanalyzed, i.e., concepts that resist logical analysis. In other words, if we have analyzed A into

B and C, and if we then find that either B or C, or both B and C, contains A, we must conclude

that A has, in fact, not been analyzed, for it remains intact in those elements into which we had

thought we had analyzed it.

It is evident that what in the “Prolegomena” Peirce calls a “component entirely

homogeneous in itself” must be identified with that something which “always remains

unanalyzed.” For if Peirce were talking about different things, then either a logical component

entirely homogeneous in itself would be analyzable—but this contradicts the assertion that

analysis is satisfactory when such an element is reached; or that which remains unanalyzed would

not be homogeneous in itself, and thus mixed with other components—but then it would be

analyzable, for analysis is exactly what separates the different components that are mixed in a

heterogeneous compound. Thus, if we are to interpret Peirce consistently, we must say that while

an absolutely complete logical analysis is unachievable, a satisfactory logical analysis is

nevertheless possible in so far as can we discover those elements of a logical compound that are

not composed of other elements but are entirely homogeneous in themselves; when such things

are encountered, we can say that analysis must stop, for a component entirely homogeneous in

itself always remains unanalyzed and is thus elementary in the required sense.

Let us see what this means in the case of the analysis of propositions. According to

Peirce, a proposition has a certain number of subjects and one predicate. For example, the

proposition “Cain kills Abel” has two subjects (“Cain” and “Abel”) and a predicate (“____ kills

____”). This is but one possible analysis of the proposition. Other analyses are also possible:

Every proposition has one predicate and one only. But what that predicate is

considered to be depends upon how we choose to analyze it. Thus, the proposition

God gives some good to every man

may be considered as having for its predicate either of the following rhemata:

— gives — to —

— gives some good to —

— gives — to every man

God gives — to —

God gives some good to —

God gives — to every man

— gives some good to every man

God gives some good to every man. (CP 4.438, 1903)

What the predicate is thus depends on what we choose to consider as a subject. In the first

example, “God,” “some good,” and “every man” are the subjects, while “____ gives____ to

____” is the predicate. In the last, the whole proposition is the predicate, while the subject is,

arguably, the Universe of Discourse to which the proposition refers. (EP 2:173, 1903) But Peirce

is convinced that an “ultimate” analysis of the proposition exists. This ultimate analysis is that

which throws into the subject everything that can be removed from the predicate: “the proper way

in logic is to take as the subject whatever there is of which sufficient knowledge cannot be

conveyed in the proposition itself, but collateral experience on the part of its interpreter is

requisite.” (NEM 3:885, 1908) “Collateral” here means exactly “known otherwise than through

this very proposition.” Just as I must already know John in order to fully understand whatever

assertion about him, so one must already have some conception of God, of something good and

of men in order to fully understand the proposition that “God gives some good to every man.”

But, and this is exactly where continuous predicates enter the picture, we must also have some

collateral experience of the act of giving, for otherwise we would not understand the proposition

at all. Here is Peirce’s example:

Thus the statement, “Cain killed Abel” cannot be fully understood by a person who

has no further acquaintance with Cain and Abel than that which the proposition

itself gives. Of course, Abel is as much a subject as Cain. But further, the statement

cannot be understood by a person who has no collateral acquaintance with killing.

Therefore, Cain, Abel, and the relation of killing are the subjects of this proposition.

(SS 70, 1908)

“The proper way in logic” is to take “killed” as a further subject. For this purpose,

hypostatic abstraction is called for. Hypostatic abstraction5 is the logical operation that turns

“predicates from being signs that we think or think through, into being subjects thought of. We

thus think of the thought-sign itself, making it the object of another thought-sign.” (CP 4.549,

1906) In other words, hypostatic abstraction is the operation whereby “a transitive element of

thought is made substantive, as in the grammatical change of an adjective into an abstract noun”

(CP 2.364, 1901), or in the logical change of a predicate into a subject (NEM 3:917, 1904).

Hypostatic abstraction is the means to complete the process of throwing into the subject all that

we possibly can, and is thus absolutely necessary if we want to reach the pure form of the

proposition.6

Indeed, when everything that possibly can be removed from the predicate has been, the

proposition is given its ultimate logical form. Once the proposition “Cain kills Abel” is turned

into “Cain is to Abel in the relation of killing,” the predicate of this second version of the

proposition becomes “____ is to ____ in the relation of ____.” Now, if we try to analyze this

predicate further, i.e., if we try to hypostatically abstract the concept of relation and to turn it into

a further subject of the proposition, we get something like this: “Cain is to Abel in relation to the

relation of killing,” in which the predicate is “____ is to ____ in relation to ____ of ____.” But

this latter predicate simply means “to be in relation to,” and therefore we see that making

“relation” a further subject would result in essentially the same predicate, for both of these

predicates express the same concept, that of “being related to.” Only, the latter is redundant.

Hence, it was in the preceding step that we already came to the end of logical analysis: the end of

analysis is reached when further steps produce only redundancy.

Those predicates that remain after everything that could possibly be given in a collateral

experience has been removed are called by Peirce “continuous predicates.” This expression is

chosen to establish an analogy with Peirce’s notion of continuity. As is well known, continuity is

something for which Peirce had been trying his whole life to provide a definition and a

mathematical expression.7 After rejecting the view that continuity is infinite divisibility

(maintained until 1884) and that it is Cantor’s perfect concatenation (with which he was

somewhat sympathetic in the period 1884-1892), Peirce comes to understand a continuum as a

supermultitudinous multiplicity composed of parts all homogeneous with the whole (PM 191-

200, 1897; PM 215, 1908). The parts of a continuous line are continuous lines themselves, and

the same is true of the parts of each part (R 955, c.1897). Accordingly, continua cannot be

composed of actual, definite point-like parts, for a point is precisely something without parts; a

continuum is instead composed of potential and indefinite parts, and it always contains a

multiplicity of such parts that exceeds any multitude (PM 138, 1903; NEM 4:325, 1906). When a

potential point in a continuum becomes actual, a topical singularity emerges, and continuity

breaks off (PM 187, 1898; PM 215, 1908). A perfect continuum is one without such topical

singularities (CP 4.642, 1908; PM 215, 1908).

Now, it is in this sense that Peirce says that a continuous predicate is analyzable only into

parts that are homogeneous with the whole. We have seen that in the logical analysis of a

proposition there is a point beyond which nothing new is reached. Peirce proves this by showing

that what results from the hypostatic abstraction of a continuous predicate contains that very

element that was supposed to be thereby analyzed. So if we attempt to analyze “to be in relation

to” as “to be in some relation to the relation to,” we find that the very concept we wanted to

analyze (“being related to”) remains intact in the elements into which we have analyzed it (“being

related to the relation to”). To express this peculiar notion of unanalyzability or elementarity,

Peirce uses the analogy with continuity: just as a Peircean continuum, these predicates are

homogenous in themselves, for each of their parts has the same properties as the whole. This is

Peirce’s reason for calling these predicates continuous. As the operation of hypostatic abstraction

only produces redundancy, these predicates are, when considered as composed or analyzable,

“composed” of themselves and “analyzed” through themselves. The analogy is with a continuous

line: no matter what one cuts off from a continuous line, a continuous line remains; no matter

what one cuts off (abstracts) from a continuous predicate, a continuous predicate remains. These

predicates are simple and cannot be further analyzed, so that when the logical analysis of the

proposition has thrown into the subject all that can possibly be removed from the proposition,

what remains is a not-analyzable pure logical form. But if we try to analyze it, then we are forced

to say that what results is composed of itself. “To be____,” “to coexist with____,” “to be in

relation to____,” “to imply____” are equivalent to “to be ____ that is____,” “to coexist with the

coexistence of ____,” “to be in relation to the relation to____,” “to imply the implication

of____,” respectively. A predicate so unanalyzable or, which is the same, analyzable only as

composed of itself, is termed a continuous predicate.

A continuous predicate thus represents the ultimate and elementary logical form of every

proposition, i.e., the pure form of connection of the things the proposition is about. “to be ____,”

“to be identical to ____,” and “____ to possess the character of ____” are the pure logical forms

of the categorical proposition “A is B,” or “A possess the character B”; “to coexist with ____” is

the pure logical form of the copulative proposition “A and B”; “to imply ____” is the pure logical

form of the hypothetical proposition “if A, then B.” All these predicates contain nothing

“material” (things and qualities that are possibly given in a collateral experience), but just the

logical relation that connects the objects spoken about. As such, continuous predicates mark the

end of logical analysis. A continuous predicate, indeed, is “entirely homogenous in itself” and

“free from the smallest admixture of any of the other[]” components of the proposition, for if it

contained some other logical entity, this latter would be hypostatically abstractable from it. When

a continuous predicate is reached, it is useless to pursue the analysis further, for that very

continuous predicate we are trying to analyze will always be found to enter intact into the

components into which we have tried to analyze it. It is in this peculiar sense that continuous

predicates are those elements that always remain unanalyzed in logical analysis. We may thus say

that absolute completeness of logical analysis is impossible exactly because the elementary forms

of logic are continuous, so that the analysis has to be considered as satisfactory when a

continuous predicate is reached.

We are now in a position to assess whether Murphey’s and Parker’s claims about

continuous predicates are justified.8 After quoting the passage on continuous predicates in R 611

(our second occurrence), Murphey states that from the “seemingly trivial fact” that there exist

logical continuous relations Peirce drew two important principles of his synechistic philosophy.

“First,” Murphey says, “he drew the conclusion that relations constitute continuous connections

among their correlates.” (1961, 318) But what does that mean? Does Murphey mean that all

relations constitute continuous connections among their relata? If this were the case, every

predicate would be continuous in Peirce’s sense, and the difference between continuous and non-

continuous predicates would disappear. But Peirce is very clear that not all the predicates are

continuous; “____ kills ____” is not continuous, for it can be turned into a subject through

hypostatic abstraction; “____ is in relation to ____,” by contrast, is continuous, for “relation” is

not abstractable from it.9 If, on the other hand, Murphey recognizes this distinction, then what

does it mean to say that a continuous predicate forms a continuous connection among its relata?

From the fact that logical analysis is satisfactory when any continuation of it would only produce

redundancy, i.e., when a “continuous predicate” is found, it does not follow that these predicates

represent real continuities among things—whatever that might mean.

In support of his interpretation, Murphey cites CP 6.143 (1892; also at W 8:150), where

Peirce claims that “ideas can nowise be connected without continuity” , but Murphey fails to

explain the connection between this claim and the 1906-1908 thesis about continuous relations.

Peirce’s 1892 argument for the continuity of ideas, concerning as it does the continuity of time

and the continuous flow of inferences through a finite interval of time (CP 6.107-111, W 8:137-

138, 1892), is very different from his 1906-1908 argument for an ultimate logical analysis of the

proposition. It thus appears that Murphey’s citing Peirce’s cosmology of the 1890s sheds no light

on the idea of continuous predicates and instead obscures what continuous predicates are

intended to represent, which is not a real continuity between correlates or ideas, but the ultimate

logical form of propositions.

Secondly, Murphey states that, according to Peirce’s idea of continuous predicates,

“relations cannot be created except as specifications of already existing relations. And since

relations constitute continuous connections among things, new relations are only possible as

specifications of antecedently existing continuous connections.” (1961, 318-319) If Murphey’s

view were correct, we should say that, for Peirce, the concept of relation specifies itself into a

series of concepts similar to it. Yet, this is not Peirce’s idea. Peirce’s idea is that logical analysis

should cease before the point of redundancy is reached, that is, the point at which “to be in

relation to” is represented as “to be in relation to the relation to.” The concept of relation is not

specifiable or analyzable, for, if we try to make it more specific or to analyze it, we found that

that very concept remains unanalyzed in one or both of the components into which we thought

we had analyzed it. Peirce’s talk of “analyzing the concept of relation through a combination of

the very concept with itself” is more a reductio ad absurdum of any attempt to carry logical

analysis beyond a certain point than it is a suggestion that these concepts require an infinite

analysis or specification. “To be in relation to the relation to” is not, as Murphey says, the

specification of an antecedently existing continuous relation, but the proof that the concept of

“relation” that it is supposed to analyze is on the contrary not an analyzable logical form. To

think otherwise is to get Peirce’s point exactly backwards and to assign to him the very absurdity

he wanted to warn us against.

Parker appears to make a mistake similar to Murphey’s first one. He seems to infer from

Peirce’s theory of continuous predicates that the relata of a continuous relation “are connected

from the beginning” (1998, 95). According to Parker, the relata connected by a continuous

relation behave like the neighboring infinitesimals in a Peircean continuum: as two infinitesimal

parts of a continuum are, in one sense, identical, but, in another sense, they occur in an ordered

relation, likewise the subject and predicate of “The rose is red” are “immediately connected, or

welded together. They arrive as one idea, but may be distinguished if necessary.” (Ibid.) But to

say this is to confuse the parts of a continuum, which are indeed immediately connected and

welded together in the continuum, with the relata of a propositional relation, which are connected

by the continuous relation that constitutes the proposition’s pure form. This confusion derives

from two different senses that the word “connection” has in this context. The first sense concerns

the parts of a continuum, which are immediately connected and thus at once different and the

same. The second sense concerns the relata that are connected by a continuous relation but are

not immediately connected. These are two distinct claims. Parker’s failure in distinguishing them

leads him to erroneously infer that, since subject and predicate are connected by a continuous

relation, they are continuous with one another, thus falling into the error committed by Murphey.

But Parker also argues convincingly that, according to Peirce, the most elementary

relations imply combination and triadicity, and he rightly remarks that continuous predicates

represent this kind of triadic combination (1998, 65-71). Before entering directly upon this

subject, however, it would help to have a clearer picture of what continuous predicates evolved

into and from what they evolved.

3. Continuous graphs

If continuous predicates are so important in logical analysis, why are they mentioned so

infrequently? The question Peirce stops talking about continuous predicates after 1908 is

particularly puzzling. As anticipated in the introduction, continuous predicates are by no means

abandoned. Instead, they are incorporated within the system of Existential Graphs (EGs), where

they are represented by what Peirce calls “continuous graphs.”10 The earlier occurrence of the

phrase “continuous graphs” dates from 1906. Then, arguably, the doctrine of continuous

predicates was developed alongside the studies of the graphs. But since Peirce ceased to talk

about continuous predicates in 1908, shifting his focus to continuous graphs and continuing to

develop them even after that date (see R 670, 1911), it is probable that he decided on graphs as

the exclusive means by which to express his ideas about continuous predicates.

Among Existential Graphs there are two that are remarkable for being truly

continuous both in their Matter and in their corresponding Signification. There

would be nothing remarkable in their being continuous in either, or in both respects;

but that the continuity of the Matter should correspond to that of the Signification is

sufficiently remarkable to limit these Graphs to two; the Graph of Identity

represented by the Line of Identity, and the Graph of coëxistence, represented by

the Blank. (R 293, NEM 4:324, 1906)

The reason why Peirce states that the Graph of Coexistence (Sheet of Assertion) and the

Graph of Identity (Line of Identity) possess a material continuity is easy to see: every part of the

Sheet of Assertion is itself a Sheet of Assertion, as every part of a Line of Identity is itself a Line

of Identity. Since their parts are all homogeneous with the whole, these two graphs possess a

material continuity. But what does it mean that these graphs’ material continuity corresponds to

their formal continuity?

Let us begin with the Sheet of Assertion. First, the Sheet of Assertion, also called the

Blank (R 293, 1906), the Phemic Sheet (R 300, 1908; R 500, 1911) and the Sheet of Truth (R

514, 1909), and it is said to represent the “Universe of Discourse,” that is, the collection of

individuals to which every proposition as such refers (R 300:48, 1908). Secondly, the Sheet does

not simply represent the domain of discourse; it also says something about this domain. And in

saying something, it is not a mere collection of individuals, but is itself a graph, and then an

assertion (CP 4.397, 1903; R 295:41, 1906). But, thirdly, when it is considered neither as an

assertion, nor as a set of individuals, but as a rhema, it must express a logical predicate: the

predicate of coexistence.

A blank, considered as a medad, expresses what is well-understood between

graphist and interpreter to be true; considered as a monad, it expresses “— exists”

or “— is true”; considered as a dyad, it expresses “— coexists with —” or “and.”

(CP 4.466, 1903)

When it is considered as the Universe of Discourse, the Sheet represents a collection of

individuals. When it is considered as an assertion (a medad), the Sheet represents all that is

virtually taken for true about this universe. But when it is considered as a predicate, the Sheet

expresses the coexistence of what is thereupon scribed and thus says that the truth of a graph

coexists with the truth of any other graph. This explains why Peirce calls the Blank not only the

Sheet of Assertion, the Phemic Sheet or the Sheet of Truth, but also the Graph of Coexistence.

Since it says that whatever is scribed is true, it thereby asserts the coexistence, or conjunction, of

what is so scribed.

The other continuous graph, the Line of Identity, plays different roles too. When it is

attached to a spot, it functions as an existential quantifier. But when it is drawn alone upon the

Sheet of Assertion, it is to be considered as a graph, and thus as an assertion.11 It asserts that

something, the individual denoted by one of its terminal points, is identical with another

something, the individual denoted by the other of its terminal points. Indeed, a dot, in the Beta

part of EGs, signifies an individual, that is, an indefinite “something” that belongs to the

Universe of Discourse. The lines of identity have the task of identifying these individuals, so that

when two dots are conjoined by a line this is the representation that “something is identical with

something else.” But since any portion of the line is itself a line, Peirce says that the line of

identity expresses something like this: “Something (its one end) is something that is something

that is something that is something that is something that is something that is something that is

(its other end). Only instead of a finite series of ‘somethings,’ there is not an infinite number (for

that would not suffice) but a continuum of somethings!” (R 670: 23 variant, 1911).12 It is evident

that the predicate of this proposition13 is precisely the continuous predicate of identity: “____ is

identical to ____.”

Therefore Peirce’s fundamental idea about continuous graphs is that they express logical

predicates: “the continuous Graphs do not express Existential Predicates but only Logical

Predicates” (R 499s:33, 1907). These predicates are of course the continuous predicate of

coexistence, or conjunction (“____ coexists with ____,” “____ and ____”) and the continuous

predicate of identity (“____ is identical to ____”). Although in the secondary literature the

importance of continuity in Peirce’s EGs has long been recognized (Zeman 1968; Roberts 1973,

124-125; Pietarinen 2006, 159-166; Burch 2011), the connection between EGs’ continuous

graphs and continuous predicates has never been noticed. Yet, as shown, this connection is as

close as it could possibly be. It is worth noting, however, that not all continuous predicates

mentioned by Peirce in the 1906-1908 texts have a corresponding continuous sign in EGs. As

explained in R 293 quoted above, Peirce believes that the correspondence between formal and

material continuity is possible only in the case of coexistence and identity.

Peirce repeatedly states that the purpose of EGs is the minute analysis of propositions (EP

2:279, 1903; CP 4.552, 1906; R 30s:5, 1905; R 27s:4, n.d.) and arguments (R 455:2, 1903; R

27s:5, n.d.; R 500:10, 1911) into their ultimate elements.14 The system of EGs provides such an

analysis by representing certain logical forms of connection by means of certain visible forms of

connection (R 300:34 variant). Accordingly, what is found to be elementary in the propositional

structure must be represented in EGs by an elementary visual sign. Since what is most elementary

in a proposition is its “void form” or continuous predicate, this form must be represented by a

corresponding visually continuous sign. Just as the Graph of Coexistence and the Graph of

Identity are materially “indecomposable” (R 30s:8, 1905), so the relations that they express (the

continuous predicates of coexistence and identity) are “logically simple relation[s]” (EP 2:381,

1906). We can see that the Sheet of Assertion and the Line of Identity really capture Peirce’s

fundamental idea that there exists a logical form that is not analyzable, and which then is, in

Peirce’s terms, continuous. The two continuous graphs’ material continuity is nothing more than

the icon of Peirce’s logical continuity.

4. Leading principles

We have seen that continuous predicates came to be incorporated within the system of EGs.

But continuous predicates were also a late reformulation of something with which Peirce had

been concerned since his early studies on the theory of inference. My hypothesis, which I will

defend in this section, is that Peirce’s early notion of logical leading principles is the antecedent

of his mature notion of continuous predicates.

According to Peirce, Kant’s Falsche Spitzfindigkeit der vier syllogistischen Figuren, in

which the indirect figures of the syllogism were proved to be reducible to the first one by

immediate inference only, was mistaken. In Peirce’s “natural classification” of arguments the

second and the third figure are reducible to the first only through the employment of the very

figure that is to be reduced. The principles involved in the different syllogistic figures cannot then

be reduced to a combination of other, more primitive principles, but always enter as parts into the

reduction proof itself. These irreducible principles are called leading principles, and they

constitute (critical) logic’s principal concern.

In the 1867 paper “On the Natural Classification of Arguments,” an argument is defined

as “a body of premises considered as such” (CP 2.461, W 2:23, 1867). The term “premise,” in

turn, refers to “something laid down … and not to anything only virtually contained in what is

said or thought, and also exclusively to that part of what is laid down which is (or is supposed to

be) relevant to the conclusion.” (Ibid.) An inference is a judgment that, if the premisees are true,

then the conclusion is true. Any judgment of this kind is based upon a leading principle (CP

2.462, W 2:23, 1867), so that “[t]he leading principle contains, by definition, whatever is

considered requisite besides the premises to determine the necessary or probable truth of the

conclusion,” and “nothing can be eliminated from the leading principle except by being

expressed in the premises.” (CP 2.465, W 2:23-24, 1867).

Peirce’s point is that there is always something that cannot be eliminated from the leading

principle:

It can be shown that there are arguments no part of whose leading principle

can be transferred to the premises, and that every argument can be reduced to such

an argument by addition to its premises. For, let the premises of any argument be

denoted by P, the conclusion by C, and the leading principle by L. Then, if the

whole of the leading principle be expressed as a premise, the argument will become

L and P

∴ C.

But this new argument must also have its leading principle, which may be denoted

by L'. Now, as L and P (supposing them to be true) contain all that is requisite to

determine the probable or necessary truth of C, they contain L'. Thus L' must be

contained in the leading principle, whether expressed in the premise or not. Hence

every argument has, as portion of its leading principle, a certain principle which

cannot be eliminated from its leading principle. Such a principle may be termed a

logical principle. (CP 2.466, W 2:24, 1867)

Peirce presents this point better through the example he gives in his 1880 “On the Algebra

of Logic.” If we take the enthymematic argument

Enoch was a man,

Therefore, Enoch died,

the leading principle of this argument can be extracted from it and laid down as a further premise:

All men die,

Enoch was a man,

Therefore, Enoch died.

Now, the leading principle of this latter argument is that nota notae est nota rei ipsius. If

we lay it down as a further premiss, we have the following, expanded argument:

Nota notae est nota rei ipsius,

All men die, and Enoch was a man,

Therefore, Enoch died.

But, Peirce says, “this very same principle of the nota notae is again active in the drawing

of this last inference” (CP 3.166, W 4:167, 1880). This new argument has in turn its leading

principle, but this latter is nothing more that the principle of nota notae previously laid down.

Laying it down as a further premise only produces another argument whose own leading

principle is again the nota notae. As Thompson observes, “such procedure leads to an infinite

regress,” for the nota notae “is also the leading principle of any subsequent stage” (1953, 8). The

only way to avoid such an infinite regress is to recognize that an argument is complete when its

leading principle contains nothing “material,” or nothing that can be extracted and laid down as a

premise. Peirce calls a leading principle of this kind logical.15

Exactly as with continuous predicates, one can remove (abstract) from a leading principle

as much as can be removed (abstracted), but once a pure logical principle is found, analysis must

stop: the pure logical form of reasoning always remains unanalyzed. As continuous predicates,

each leading principle is proved valid through (is composed of) itself, and adding it as a further

premise (abstracting and changing it into a subject) does not modify its structure and primitive

logical power. I therefore propose that Peirce’s logical leading principles be considered as the

antecedents of his 1906-1908 continuous predicates. Leading principles and continuous

predicates behave precisely the same with respect to logical analysis: they are “elementary” or

“unananlyzable” logical forms that are found to enter intact into the parts into which we try to

analyze them.

5. Combination and triadicity

So far we have seen what Peirce’s 1906-1908 continuous predicates are, how they merge

with the continuous graphs in his EGs and what were their antecedents in his early theory of

inference. What remains to investigate is Peirce’s peculiar notion of “combination,” for it appears

to have been central to his late concern with continuous predicates. Of course, what follows does

not attempt to be an exhaustive analysis of Peirce’s concept of combination—which would

require an account of his phaneroscopical and metaphysical doctrines. I seek, more modestly, to

explore the connections between Peirce’s notion of combination, his doctrine of continuous

predicates and his Reduction Thesis (Burch 1991, 1997).

Peirce states that continuous predicates represent “form[s] of connection” (SS 71, 1908;

NEM 3:885, 1908) or “copulant” signs (CP 8.352-360, EP 2:485-487, 1908). Since anything

possessing content has been removed from it, a continuous predicate represents the pure form of

combination of the elements of the proposition, its constitutively unifying element. This has been

acknowledged by Parker, who, despite the above-mentioned shortcomings of his interpretation,

aptly recognizes that “[t]he most basic relations (of mere co-being or of identity, for example)

imply genuine combination and triadicity” (1998, 71). This was first observed by Helmut Pape:

Peirce developed a theory of continuous predicates according to which any assertion can

be reduced to an ultimate, not-analyzable form of pure combination of the terms in the

proposition. This ultimate form of combination of the terms in the proposition … is called

by Peirce continuous predicate. As one can easily see, this is the proof that there is a

general relation to which any relationship among predicates can be reduced. The

continuous predicate is the form of forms. (Pape 1990, 58; see also Pape 1989, 277-279)

If Parker and Pape are right, as I think they are, continuous predicates, in being elementary forms

of combination, must involve triadicity. Let us briefly see why.

In 1905 Peirce argues that combination cannot be a composed idea: “the general idea of a

combination must be an indecomposable idea. For otherwise it would be compounded, and the

idea of combination would enter into it as an analytic part of it.” (R 908, EP 2:364) Since any

composed idea whatsoever must presuppose the very idea of combination itself, this latter cannot

in turn be a composed idea, or we are launched into an infinite regress. But Peirce also claims

that combination is a triadic relation: “this idea [of combination] is a triad; for it involves the

ideas of a whole and of two parts (ibid.); “the word ‘combination’ means precisely something

involving a triadic relation” (NEM 3:830, 1905); “Combination is a triadic relation between the

two elements (for every Combination results from successive couplings) and the result, and is in

so far genuine that it cannot be analyzed into any Combination of dyadic relations” (EP 2:391,

1906); “composition is itself a triadic relationship, between the two (or more) components and

the composite whole” (CP 6.321, 1908); “combination is essentially a triadic relation” (NEM

3:763, n.d.). Hence, it follows that triadic relations are indecomposable or unanalyzable relations

(EP 2:176, 1903), and one can easily perceive that this latter claim corresponds to the negative

component of what Burch (1991, 1997) calls “Peirce’s Reduction Thesis,” the thesis that triadic

relations cannot be constructed through combiningonly monadic and dyadic relations. Triadic

relations are unanalyzable, simple forms of combination, for “that which combines two will by

repetition combine any number. Nothing could be simpler; nothing in philosophy is more

important.” (CP 1.298, 1905)

Continuous predicates, in being pure forms of combination, are thus triadic relations, even

when they are expressed in dyadic form (“____ is coexistent with ____,” “____ is identical to

____,” “____ is in relation to ____”; see Parker 1998, 71). They are triadic exactly because they

involve the pure idea of combination, and this, Peirce claims, is a genuinely triadic idea.16 But if

continuous predicates are triadic relations, then the continuous graphs that represent them in EGs,

and the logical leading principles that are their “progenitors,” must share the same relational

form.

That this is indeed the case is easily shown. Take coexistence, which is the continuous

predicate that, in EGs, is represented by the continuous graph of coexistence. Peirce says that a

copulative proposition “P and Q” (whose pure logical form is the continuous predicate of

coexistence or co-being) “predicates the genuinely triadic relation of tri-coexistence, ‘P and Q

and R coexist.’ For to say that both A and B is true is to say that something exists which tri-

coexists with true replicas of A and B.” (EP 2:281, 1903) The assertion of the conjunction of two

propositions A and B is equivalent to the assertion that there is something (arguably, a universe of

discourse) for which A and B are both true. Hence, the graph of coexistence is to be considered as

a triad (R 499s:34, 1907), for it expresses the genuine triadic relation between the conjunction

and the conjuncts. The same is true of identity, which is the continuous predicate that, in EGs, is

represented by the continuous graph of identity. Peirce says that the “graph of teridentity” (R

292, 1906; R 300, 1908), which expresses the identity of three individuals, is even more primitive

than that of identity: “the concept of teridentity is not mere identity. It is identity and identity, but

this ‘and’ is a distinct concept, and is precisely that of teridentity.” (CP 4.561, 1906) Teridentity

is not a special case of identity. Rather, it is the relation of identity which is a degeneration of the

genuine relation of teridentity. Indeed, while it is impossible to produce the triad of teridentity by

means of a combination of dyads of identity (for, according to the Reduction Thesis, a triadic

relation cannot be constructed through combination of dyadic relations only), it is perfectly

possible to produce the dyad of identity by means of a combination of triads of teridentity: “[w]e

must hereafter understand [the line of identity] to be potentially the graph of teridentity” (CP

4.583, 1906); in this way, “no other triad than that of teridentity seems to be needed” (R 490:10,

1906; see Brunning 1997; Burch 2011, 31-40). Co-existence and identity are thus genuinely

triadic relations, and the graphs that express them are to be considered as essentially triadic even

when they are expressed otherwise (R 499s:34-35, 1907).

Leading principles express triads as well, for the drawing of a conclusion from a set of

premises “is itself a triadic relation.” (CP 1.346, 1903) Peirce first states in 1865 that there are

three kinds of logical leading principles. The leading principle of deduction is that “the symbol of

the symbol is itself a symbol of the same object”; that of induction is that “[t]he symbol of an

object has the same predicates as its object”; that of abduction is that “[t]he symbol which

embodies any form is predicable of the same subjects as the form itself.” (W 1:186-187, 1865)

These are nothing more than generalizations of the principle that nota notae est nota rei ipsius,

or, as Peirce prefers to say, the interpretant of a sign is a sign of the same object of which the first

sign is a sign. Among Peirce’s numerous definitions of the sign, the following, contained in a

draft of a letter to Lady Welby dated July 1905, is probably one of the best:

A Sign would be a Priman, Secundan to something termed its Object, and if

anything were to be in a certain relation to the sign, called Interpretant, the Sign

actively determines the Interpretant to be itself in a relation to the same Object,

corresponding to its own. (R 793:11, 1905)

From this perspective, Peirce’s many attempts to define what a sign is turn out to be

nothing more than definitions of the most general logical leading principle of reasoning. The

relation between a sign, its object and its interpretant (leading principle) is thus the most

important genuinely triadic relation of logic. Just as Peirce’s mature notion of continuous

predicate, a logical leading principle is a pure relation of combination. In it the sign plays the role

of “combinant” (R 145:28, n.d.), “nectent” (R 284:48, 1905) and “copulant” (EP 2:485, 1908)

element or medium (R 793, 1905), for it brings its object and its interpretant into relation. What

combines the elements in a proposition, and different propositions into other propositions and

arguments, is a continuous relation. And since continuous predicates, continuous graphs and

leading principles are triadic relations, this immediately gives us a further reason to appreciate his

claim that “[c]ontinuity represents Thirdness almost to perfection.” (CP 1.337, c.1875)

6. Conclusion

If the proposed reconstruction is correct, the importance of Peirce’s continuous predicates

is hard to overestimate, not only because they represent the outcome of his peculiar conception of

logical analysis, but also because they serve as the cornerstone for several aspects of his mature

logical thought as well as some of the most brilliant and original of his earlier views. Peirce wrote

to his former Johns Hopkins student Christine Ladd-Franklin in 1891:

My work in philosophy has consisted in an accurate analysis of concepts, showing

what is and what is not essential to the subject of analysis. Particularly, in logic, my

motive for studying the algebra of the subject, has been the desire to find out with

accuracy what are the essential ingredients of reasoning in general and of its

principal kinds. (CP 8.316)

In 1891, Peirce had not yet invented the system of EGs, and here he is thus referring to his works

on the algebra of logic. But it is important to note the emphasis already laid upon “the essential

elements of reasoning,” which would become a main theme in his subsequent studies on graphic

logic.

In conclusion, what I hope to have shown is that, for Peirce, the essential elements of

thought possess a very distinctive feature: they are simple, and this simplicity resides in their

being unanalyzable or, which according to Peirce amounts to the same, in their being analyzable

only into components identical to the whole. When we try to analyze a simple element of

thought, we find that it enters intact into the parts into which we are trying to analyze it. This

means that no matter how many times we try to divide, decompose or analyze it, it always

remains as before, untouched and unanalyzed. Peirce thought that the peculiar nature of these

elements, which is called “elementarity,” is best understood through analogy with a continuous

line: just as a continuous line is composed of parts that are in turn continuous lines, so an

elementary predicate is only analyzable into parts that are in turn elementary predicates. We may

thus understand why, according to Peirce, an absolutely complete logical analysis is “no less

unattainable [than] is omniscience”: it is unattainable because continuous predicates and the idea

of combination that they involve alway remain unanalyzed. But, at the same time, we understand

what it means for a satisfactory analysis to be possible: logical analysis is satisfactory when a

“Peircean elementary element” or “continuous predicate” is found.17

Università degli Studi di Siena

[email protected]

References

Burch, R. W. 1991. A Peircean Reduction Thesis: The Foundations of Topological Logic, Texas

Tech University Press, Lubbock, TX, 1991.

_____. 1997. “Peirce’s Reduction Thesis,” in N. Houser, D. Roberts, and J. Van Evra, eds.,

Studies in the Logic of Charles Sanders Peirce, Indiana University Press, Bloomington,

IN, pp.234-251.

_____. 2011. “The Fine Structure of Peircean Ligatures and Lines of Identity,” Semiotica 186,

pp.21–68.

Havenel, J. 2008. “Peirceʼs Clarifications of Continuity,” Transactions of the Charles S. Peirce

Society 44, 1, pp.86-133.

Maddalena, G. 2009. Metafisica per assurdo. Peirce e i problemi dell’epistemologia

contemporanea, Rubbettino, Catanzaro.

Moore, M. E. 2007. “The Genesis of the Peircean Continuum,” Transactions of the Charles S.

Peirce Society 43, 3, pp.425-469.

Murphey, M. G. 1961. The Development of Peirce’s Philosophy, Harvard University Press,

Cambridge, MA; Hackett, Indianapolis, IN, 1993.

Pape, H. 1989. Erfahrung und Wirklichkeit als Zeichenprozeβ, Suhrkamp, Frankfurt am Main.

_____. 1990. “Einleitung,” in C. S. Peirce, Semiotische Schriften Band 2, C. Kloesel and H.

Pape, eds., Suhrkamp, Frankfurt am Main, pp.7-79.

Parker, K. A. 1998. The Continuity of Peirce’s Thought, Vanderbilt University Press, Nashville,

TN.

Peirce, C. S. 1931–1958. The Collected Papers of Charles S. Peirce, 8 vols. C. Hartshorne, P.

Weiss and A. W. Burks, eds., Harvard University Press, Cambridge, MA. (CP)

_____. 1976. The New Elements of Mathematics, C. Eisele, ed., 4 vols., Mouton, The Hague.

(NEM)

_____. 1977. Semiotic and Significs. The Correspondence Between Charles S. Peirce and

Victoria Lady Welby, C. S. Hardwick, ed., Indiana University Press, Bloomington, IN.

(SS)

_____. 1982– . Writings of Charles S. Peirce: A Chronological Edition, 7 vols., E. Moore, C.

J. W. Kloesel et al., eds., Indiana University Press, Bloomington and Indianapolis, IN.

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_____. 1992. Reasoning and the Logic of Things, K.L. Ketner, ed., Harvard University Press,

Cambridge, MA. (RLT)

_____. 1998. The Essential Peirce. Volume 2 (1893-1913), the Peirce Edition Project, ed.,

Indiana University Press, Bloomington and Indianapolis, IN. (EP 2)

_____. 2010. Philosophy of Mathematics. Selected Writings, M. E. Moore, ed., Indiana

University Press, Bloomington and Indianapolis, IN. (PM)

Pietarinen, A.-V. 2006. Signs of Logic: Peircean Themes on the Philosophy of Language, Games

and Communication, Springer, Dordrecht.

Potter, V. G., and P. B. Shields. 1977. “Peirce's Definitions of Continuity,” Transactions of the

Charles S. Peirce Society 13, 1, pp.20-34.

Roberts, D. 1973. The Existential Graphs of Charles S. Peirce, Mouton, The Hague.

Robin, R. S., ed. 1967. Annotated Catalogue of the Papers of Charles S. Peirce, University of

Massachusetts Press, Amherst, MA.

_____. 1971. “The Peirce Papers. A Supplementary Catalogue,” Transactions of the Charles

S. Peirce Society 7, 1, pp.37-57.

Shin, S.-J. 2010. “Peirce’s Two Ways of Abstraction,” in M. E. Moore, ed., New Essays on

Peirce’s Mathematical Philosophy, Open Court, Chicago, IL, pp.41-58.

Thompson, M. 1953. The Pragmatic Philosophy of C. S. Peirce, University of Chicago Press,

Chicago, IL.

Zeman, J. J. 1968. “Peirce’s Graphs—the Continuity Interpretation,” Transactions of the Charles

S. Peirce Society 4, 3, pp.144-154.

_____. 1982. “Peirce on Abstraction,” Monist 65, 2, pp.211-229.

Notes

                                                                                                                                                                                                                                                                                                                                                                                                         1 “R” refers to Peirce’s unpublished manuscripts according to catalog numbers given in Robin

1967 and 1971. References are by manuscript number and, when available, page number.

2 Although the text here reads “A is in the relation R1 to the relation R1 to the relation R to B,”

Peirce probably meant to write “A is in the relation R2 to the relation R1 to the relation R to B.”

3 Here I quote the text as revised by Peirce himself and published at EP 2:484. Peirce had

originally written that copulant signs “neither describe nor denote their Objects, but merely

express the logical relations of these latter to something otherwise referred to,” and this is the text

that is provided at CP 8.350.

4 It should be evident that these terms are used practically interchangeably. However, a note of

caution seems in order. In his 1897 “Logic of Relatives” Peirce distinguishes between a relation

and a relative. The former “is a relationship considered as something that may be said to be true

of one of the objects” (CP 3.466, 1897), i.e., the representation of a relational fact from the point

of view of one of the correlates. The latter is the nominal form that corresponds to each relation,

and is Peirce’s 1870 way of expressing relations. Yet, Peirce gradually abandons the nominal

form, settling on a verbal one: what logic needs is only the blank form of a proposition, and this

is exactly what Peirce’s nominal relatives develop into. Peirce calls it a rhema. Although

sometimes Peirce speaks of “relative rhemata” and “non-relative rhemata” (CP 3.420-421, 1892),

when no specification is given (as in some works on the graphs or in connection with the

exposition of the triplet rhema-proposition-argument) “rhema” generally means a relative, be it

monadic, dyadic, triadic or polyadic. That “rhema” is equivalent to “concept” and “conception” is

shown by Peirce’s several discussions of the “composition of concepts” (CP 1.294, 1905; R 284,

1905; CP 4.572, 1906; R 292, 1906; R 293, 1906; R 300, 1908; R 643, 1909; R 646, 1910),

                                                                                                                                                                                                                                                                                                                                                                                                         where it is clear that by “concept” Peirce means a “concept having a strict valency” (CP 5.469,

1907), that is, a rhema. Although rhemata are sometimes straightforwardly identified with

predicates (CP 4.560, 1906), at other places Peirce is more precise; at CP 4.438 (1903), for

example, Peirce says that a “blank form of proposition produced by such erasures as can be filled,

each with a proper name, to make a proposition again, is called a rhema, or, relatively to the

proposition of which it is conceived to be a part, the predicate of that proposition.” It thus appears

that “rhema” and “predicate” are used interchangeably, but that a rhema is better called a

predicate when considered in the context of a proposition. Peirce’s use of “sign” at EP 2:485,

1908, cannot be explained along these lines, and must await the discussion of Peirce’s concept of

sign given in §4.

5 Not to be confused with the other kind of abstraction that Peirce calls prescissive abstraction or

prescission. See CP 2.364, 1901; NEM 3:917, 1904; CP 4.332, c. 1905; see also Zeman 1982 and

Shin 2010.

6 Accordingly, the need for hypostatic abstraction arises only at a certain stage of the process of

subtracting content from a predicate. A rhema is obtained from a proposition removing all that

can possibly be considered as a subject, but this is not at all hypostatic abstraction. Once all the

subjects have been removed but there still remains something in the predicate that is abstractable,

hypostatic abstraction completes the process and thereby furnishes us with the “complete”

analysis of the proposition.

7 Some Peirce scholars have given a periodization of the development of Peirce’s ideas about

continuity. Potter and Shields (1977) have recognized four main periods: pre-Cantorian until

1884, Cantorian from 1884 to 1894, Kantistic from 1895 to 1908, and post-Cantorian from 1908

                                                                                                                                                                                                                                                                                                                                                                                                         until 1911. Havenel (2008) has updated Potter and Shields’ work by dividing the Kantistic period

into an Infinitesimal period (until 1897) and a Supermultitudinous period (until the middle of

1908), and by stressing the importance of topology in the last period (1908-1913). Other useful

periodizations of Peirce’s ideas on continuity, which anyway roughly coincide with Havenel’s,

may be found in Moore 2007 and Maddalena 2009.

8 I owe the main points concerning Murphey’s claims about continuous predicates to

an anonymous referee for this journal.

9 Another way of saying this is that continuous predicates combine “standard” predicates that

have been hypostatically abstracted. It is the form of connection that is continuous in Peirce’s

sense, not the predicates so combined.

10 I'm indebted to an anonymous referee of this journal for suggesting to me that continuous

predicates "just merge[d] with some related developments, such as the logic of EGs.” By

contrast, I take responsibility for the thesis about the connection continuous predicates-

continuous graphs.

11 Which Burch calls the “medad of identity” (2011, 38).

12 Peirce wrote to Kehler in 1911: “The line of identity can be regarded as a graph composed of

any number of dyads ‘— is —.’ or as a single dyad. … [I]dentity is continuous … It is the nature

of identity. That is its entire meaning.” (R L231, NEM 3:164, 1911)

13 Which Burch calls the “dyad of identity” (2011, 39).

14 See also CP 4.424, 1903; NEM 3:349, 1903; CP 4.564, 1906; R 498, 1906; R 499, 1906; R

300, 1908; CP 4.619, 1908; NEM 3:165-168, 1911.

15 See also RLT 132, 1898.

                                                                                                                                                                                                                                                                                                                                                                                                         16 One may reasonably object that, since any predicate whatsoever combines things and objects,

then every predicate, no matter how expressed, has triadic structure. Now, in a sense it is true that

any predicate, in being a species of combination, has triadic structure. But not all predicates are

pure forms of combination (i.e., without content). Let us take the dyadic predicate “____ kills

____.” What is genuinely triadic in it is not its content (“killing”), which is indeed hypostatically

abstractable, but its ultimate logical form or continuous predicate (which combines the

propositional elements: the killer, the killed, and the act of killing). The same is true of triadic

predicates. Let us take the triadic predicate “____ gives ____ to ____”. Although it is expressed

as a triad, what is genuinely triadic in it is not simply its content (“giving”), which is in turn

hypostatically abstractable, but its ultimate logical form (which combines the propositional

elements: the giver, the object given, the person to whom it is given, and the act of giving; see CP

1.363, 3.424, R 292). In order to show that the genuinely triadic and combining element of any

predicate is pure (i.e., without content), Peirce invents the graph of teridentity, for once the graph

of teridentity is employed, “no other triad than that of teridentity seems to be needed” (R 490,

1906). On this more anon.

17 I am very grateful to Prof. Helmut Pape, with whom I discussed several of the topics treated in

this paper. I also express my gratitude to the anonymous referees for their useful comments and

suggestions.