**Are Josiah
Royce’s Communities of Interpretation Homomorphic with Boolean Algebras?**

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**A Discussion
Paper Submission**

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**Abstract**

The short answer to my
title question is a surprising yes, but with a qualification. A given community
of interpretation, as defined and discussed in Royce’s *The Problem of
Christianity*, does exhibit the same lattice structure as Boolean algebra,
but only a particular form of Boolean algebra, notably the algebra Royce defines
in his essay “The Relation of the Principles of Logic to the Foundations of
Geometry.”[1]
In this essay, I want to expand upon the this claim. This requires a sojourn
into Royce’s logic, and then a reading of *The Problem of Christianity *in
light of this logic. The results, among other consequences, lend considerable
support to the thesis that logic is a foundational discipline for Royce, and
thus occupies a central place in Royce’s overall philosophical doctrines.

**Are Josiah
Royce’s Communities of Interpretation Homomorphic with Boolean Algebras?**

The short answer to my
title question is a surprising yes, but with a qualification. A given community
of interpretation, as defined and discussed in Royce’s *The Problem of
Christianity*, does exhibit the same lattice structure as Boolean algebra,
but only a particular form of Boolean algebra, notably the algebra Royce defines
in his essay “The Relation of the Principles of Logic to the Foundations of
Geometry.”[i]
In that essay, Royce derives a Boolean algebra from a more general and inclusive
system known as System S, and the
characteristic relation is not the typical binary relation stipulated by most
Boolean systems, but instead is a triadic relation which is formally analogous
with Royce’s characterization of the logical properties of interpretation.
Furthermore, the unique Boolean elements, 0 and 1, i.e. the least and the
greatest elements of any Boolean algebra, are formally analogous to the origins
and ends, respectively, of communities, and hence produces in their constitution
the familiar lattice structure found in Boolean algebras.

In this essay, I want
to expand upon the foregoing claims. This requires a sojourn into Royce’s
logic, and then a reading of *The Problem of Christianity *in light of this
logic. What follows will, among other consequences, lend considerable support
to the thesis that logic is a foundational discipline for Royce, and thus
occupies a central place in Royce’s overall philosophical doctrines.

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**Royce’s Logic**

The context of Royce’s philosophical and logical project is the rapid development of modern symbolic logic and inquiries into the foundations of mathematics. Just a half century before Royce’s most important logical writings, George Boole had put logic on a new ground by introducing into it the procedures of abstract algebra. Peirce and others continued this algebraic tradition, while simultaneously inventing a new graphical logic and developing quantification, an advance which Frege also invented independently of Peirce. Set theory was just underway following the work of Cantor, and consistent axioms were not given for it till after Royce had died. Finally, with the development of the order theory of Dedekind, the logical foundations of mathematics were beginning to be explored. In short, when Royce was writing on the subject, logic and the foundations of mathematics were undergoing a revolution in content and methods, and the dust had not yet settled.

Royce observed in this revolution a new opportunity to address perennial philosophical problems. Specifically, Royce was after a new theory of categories which specified the conditions of the possibility of rational activity, i.e. inquiry, and, under a considerable influence of Dedekind and other mathematicians, the notion of order became for him the central and unifying concept in his new theories.[ii] In a remarkable passage, Royce connects his new Science of Order with problematics in both Plato and Kant:

They are problems regarding, *nor* the
methods by which the thinker succeeds, nor

yet the norms of correct thinking viewed as
norms, but rather the *Forms*, the *Categories*, the *Types of
Order*, which characterize any realm of objects which a thinker
has actually succeeded in mastering, or can possibly succeed in mastering, *Theory
of the Forms of any Orderly Realm of Objects*, real or ideal.[iii]

In other instances, Royce likens his science of order to Peirce’s logic of relatives and Russell’s calculus of relations.[iv]

Royce’s emphasis on the study of relations in his logic certainly justifies the parallels he draws. Order, he argues, is nothing but particular kinds of relations which allow the disciplines of science, logic, and mathematics to function.[v] For instance, the asymmetrical, binary (what Royce calls dyadic) and transitive relation of implication grounds our ability to make inferences between propositions, just as the similar relation which obtains between the natural numbers grounds the operations of simply arithmetic.

A study of order, and particularly order systems, that is, collections or sets of elements under an ordering relation, is therefore germane to the study of logic, mathematics, and in fact any discipline which makes use of inferences of any kind. Today, the modern set theoretic definitions of numbers and number systems have to a large degree made good on Royce’s general intuition with respect to mathematics. Royce, however, is not content with grounding mathematics in ordered sets of elements. His concern is much wider and more general. A theory of order, and in particular, the definition of an extremely general and inclusive ordered system, may be the basis of a rich metaphysics and other core areas of philosophy. He avers:

It is, therefore, not a matter of mere accident or of mere play of words that, if a man publishes a book called simply ‘A Treatise on Order,’ or ‘The Doctrine of Order,’ we cannot tell from the title whether it is a treatise on social problems or an preserving an orderly social order against anarchy or with studying those unsymmetrical and transitive relations, those operations and correlations upon which the theories of arithmetical, geometrical, and logical order depend. The bridge that should connect our logic and mathematics with our social theories is still unfinished. The future must and will find such a bridge.[vi]

Royce’s System S is his attempt at defining a system of order general and inclusive enough to function as the foundation of this bridge.

**System
S and Boolean Algebra**

Boole intended his original calculus to capture the most fundamental logical operations. After the development of first order logic, however, Boolean algebras are generally considered as a collection of abstract algebras with definite relations to logic and set theory, and which have concrete applications in computer science. For example, a Boolean algebra is isomorphic to an algebra of sets, i.e., we may interpret the elements of a Boolean algebra as sets; and, the operations of Boolean algebra have analogues in the connectives of first-order logic.

A typical way to
define a Boolean Algebra is to stipulate a set *K* of at least two
elements, 0 and 1, partially ordered under a binary relation *R*, and such
that the following operations are permitted upon the elements:

*u *+ *v *= *v *+*u*,
*u* · *v* = *v
· u, *
(commutativity)

*u* + (*v* + *w*) = (*u* +
*v*) + *w*, *u* ·
(*v* · *w*) = (*u*
· *v*)
· *w*, (associativity)

*u* ·
(*v *+ *w*) = *u* · *v*
+ *v* · *w*, *u* + (*v
*· *w*) = *u* + *v*
· *v* + *w*,
(distributivity)

*u* ·
(*u* + *v*) = *u*, *u* +
(*u* · *v*) = *u,*
(absorption)

*u* + (-*u*) =
1 *u* ·
(-*u*) = 0, (complementation)

Numerous consequences may be
derived from these axioms, not the least of which is that the elements 0 and 1
are the least and the greatest elements in *K*. Moreover, a subset of *K*,
{*u*,*v*} has a least upper bound defined by *u *+
*v* (the sum, or supremum) and a greatest lower bound defined by *
u *· *v *(the product or
infinum). Roughly, a Boolean algebra is *complete* if every subset has a
supremum and an infinum.[vii]

Royce derives a
complete Boolean algebra upon a different basis: through the definition of a
more general and inclusive order system, System
S. S’s
laws and principles may be not be defined in first order logic since Royce
implicitly quantifies over relations and sets. A second-order definition is too
cumbersome for our purposes, so I will remain within the informal language of
S. This language uses
a, b,
c, d
… to symbolize collections; *a*, *b*, *c*, … to symbolize
elements in collections. Collections may stand under O, E, and F relations,
relations which are n-ary, or what Royce calls polyadic.

1. For any collection a and any collection b, if a is an “O-collection,”

symbolized by O(a), then O(ab). This defines an operation of adjunction.

2. For any collection
b, for any element *b _{n}*
of b, and for any collection
d, if

O(d*b _{n}*)
and O(b), then O(d).
This defines an operation of adjunction.

3. There exists an element *x*.

4. For any element *x*, there exists an
element *y*, such that *x *¹*
*y.

5. For any element *x*, for any element
*y*, there exists a *z*, such that if *x *
¹ *y*, then O(*xyz*)
and ~O(*xz*) and ~O(*yz*).[viii]

6. For any element *x*, for any
collection *a*, there exists a *
y*, such that if O(a*x*), then
O(*xy*) and O(*xya _{n}*).[ix]

The rough structure of the system so defined is that of an infinitely large O-collection, which by virtue of axioms 5 and 6 is continuous, i.e., it is dense and includes its limits. Indeed, Royce nicknames his System the “logical continuum.”[x]

Certain
“transformations” of this “logical continuum” lead to promising results. The
relational properties of O-collections are identical to special cases of Boolean
multiplication: O(*abc..*) = *a ·
b · c
· …* = 0 . Because of this analogy with Boolean operations, other
relations akin to those in Boolean algebra may be defined. If, for example, O(*ab*),
then *a *totally excludes* b* and vice-versa, elements which Royce
consequently labels *obverses*. It then follows directly that if ~*a*,
then *b*, and if *~b, *then *a*. This is to say that a binary
O-relation implies two conditional statements of the foregoing form. Further,
if O(*abc*) then, if ~*b*, then *a *and *c*. In other
words, if *a,b,* and *c* exclude one another in their totality, then
it follows that if we replace one of the elements with its negative or obverse,
the other two may “overlap” or “coexist.” Royce calls ~*b* in this
situation the “mediator” of *a* and *c*, and symbolizes it as
follows: F(~*b/ac*).[xi]
Finally, if one designates an element of this triad the *origin*, say *a*,
then the F-relation assumes a binary form, *with respect to the origin*,
and also becomes asymmetrical and transitive. Such a relation can therefore be
the basis of partially or totally ordered sets. Royce symbolizes such relations
as follows: ~*b *-<_{a} c .[xii]
This appears to be a modification of Peirce’s symbolization of illation.

Royce derives the
whole of the Boolean operations by converting the O-collection of
S into a partially ordered set, that is
an F-collection, as follows. He arbitrarily selects an element and designates
it the 0-element. By the axioms S, the
obverse of 0 exists, which we may designate as 1, and the totality of the
remaining elements become so ordered by F-relations that for all elements *x*,
*x *-<_{o} 1. In more informal language, every element of set is
implied by 0 and implies 1 so that every *x* is “between” 0 and 1. The
usual operations of Boolean algebra are then verifiable by virtue of the
construction of this partially ordered set.[xiii]
Axioms 5 and 6 ensure that this Boolean algebra is complete.

Royce’s derivation of
a complete Boolean algebra is unusual in two respects, both of which he
attributes to the work of British mathematician A.B. Kempe.[xiv]
First, the 0-element is arbitrarily selected from the elements of
S, and second, the set is ordered not
by a binary relation, but by a triadic relation. However, the inclusion of 0 in
the relation reduces the triadic relation to binary. The 0-element
consequently functions as merely an *origin*, in terms of which the
direction of the asymmetry of the binary relation is defined. It follows from
these unusual features of Royce’s System, that one may define an infinite number
of complete Boolean algebras from System S,
each ordered with respect to the particular element selected as the origin. In
other words, System S “contains” the
partially ordered set in terms of which the Boolean operations are defined in an
infinite number of ways.

It is notable that
Royce thinks that his System also “contains” *any* possible ordered set,
and so avers that his System is a statement of a complete system of categories.[xv]
Since all rational activity, as Royce puts it, is dependent upon ordering
relations, and all ordering relations may be, so he argues, defined in term of
his System, his claim is *prima facie* plausible. It turns on his ability
to derive various ordered sets. Before he died, he successfully derived the
order system of common metric geometry,[xvi]
and in unpublished notes attempted to derive projective geometry. The
definitive mathematical investigation of his System he has left to us to
pursue.

Let us leave that
difficult task for another time, and grant Royce his claim: all possible order
systems may be derived from System S.
What does this imply for his late religious metaphysics as it is stated in *
The Problem of Christianity*, specifically his chapters on the logical
structure of interpretation and communities of interpretation? Is it even
reasonable to pursue such a line of inquiry?

**On Logic and Metaphysics**

In July of 1913 Royce
wrote an extraordinary letter to Warner Fite in which he argued that his theory
of community, as presented in *The Problem of Christianity,* is a
development and application of a special case of the epsilon relation, the
relation of “belonging” or “being a member of” that stands at the basis of set
theory. He writes to Fite:

Yet I am not sure that you see precisely what
the *most* elementary logical problem involved in my theory of the
“community” really is. Of this elementary problem, in my book, I said
nothing explicit, because my audience was popular… But *you* take an
interest in such formalities; for you know that they throw, or may throw,
light on life, So, I say to you what I could not say in my book.[xvii]

Royce proceeds to discuss in some
detail the formal properties of epsilon-relation, noting that it is
nonreflexive, intransitive, and also asymmetrical. These properties express
themselves at the concrete level of the community, respectively, in the
following ways. Since one cannot be a member of oneself, then if “salvation” is
equivalent to belonging to a community, it is impossible for one to save
oneself. Further, if a community A is a member of community B, it is doesn’t
follow necessarily that a member *a *of A is a member of B. *a* may
be a member of the community of Americans, but *a* is not thereby a member
of the United Nations. Finally, consideration of the asymmetry of the
e-relation illustrates “that if I am
e of a given community, and if I love
that community, and if that community loves me, still our loves are *different*
loves and lie on different levels, the relation of lover and beloved being, in
the two cases quite distinct relations.”[xviii]
In other words, “[T]he community does not depend upon me in the way that I
depend upon it.”[xix]

If it is the case that
Royce utilizes the formal properties of the epsilon relation in developing his
*The Problem of Christianity*, might it not be the case that other elements
from his science of order are also present? It is reasonable to suppose so.[xx]
The fact that the science of order is intended to bridge the gap between the
exact sciences and social theory only lends further support to this thesis.

What follows is a
brief reading of *The Problem of Christianity*,[xxi]
with special attention paid to Royce’s metaphysics of community in light of
Royce’s science of order, and specifically System
S and Royce’s triadic Boolean
algebras.

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**The Problem of Christianity**

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As with most
pragmatic thinkers, Royce begins with a concrete problem: is it possible for a
“modern man”[xxii]
to “consistently be, in creed, a Christian?”
[xxiii]
By this query Royce means to explore the possibility of holding a particular
religious-metaphysical-ethical doctrine and yet accept the contemporary results
and methods of the sciences.[xxiv]
Royce’s path towards a solution to his problem involves not only a phenomenology
of religious experience, but an examination of the leading ideas of
Christianity, the metaphysical basis and import of these ideas, and the
possibility of their union in a modern, skeptical, scientific mind.
Accordingly, *PC *is divided into two major parts: a empirical-historical
interpretation of the leading ideas of Christianity, in particular, their
interpretation as a doctrine of life, and a metaphysical investigation of the
import of these ideas. Together, the social-psychological and metaphysical
leadings of the ideas Royce asserts are central to Christianity open up a
coherent philosophy, and consequently, a coherent field of activity, that covers
all facets of the modern life, from religious activity to scientific inquiry.
This ordered field of activity is visible in broad outlines by the end of *PC*.
But its full clarity only emerges through its logical formulation. I will
indicate how and where Royce’s logic of order informs Royce’s discussion as I
proceed through the exposition.

The natural, human individual is born into a dire predicament, argues Royce. The development of moral and ethical consciousness, he writes, is necessarily “tainted” by a process that involves social strife, or “the original sin of social contentiousness.”[xxv] In childhood and infancy, the individual is a mass of inherited and instinctual, unorganized, and in many cases conflicting, impulses. These impulses are neither good nor evil, since there is no conscious direction of them into organized conduct oriented towards selected ends. They lack moral and ethical quality. It is the task of parents and others to train these tendencies towards behavior and activity that follows certain mores, customs, or laws which organize the family or community, as well as benefit the individual. This training is more or less coercive, however, occurring through various forms of criticism. It therefore produces in the developing self an awareness of itself as having individual tendencies toward action, of having its own conduct, and in many cases, tendencies and conduct that run contrary to the will of the other, whether this other is parental or societal. As the self develops and matures, the meaning and place of its conduct in social life becomes more apparent to itself; it is able to take the position of the other with reference to its own actions and judge their moral/ethical quality. Further, the individual may began to recognize its dependence on the community, the need for its cooperation if it is to sustain itself and achieve its own life plans. In any case, the power of the will of the community is exponentially greater than the will of the individual, and cannot be ignored.

As a result of this process, an inner division begins to form in the developing individual: the individual will of the self, or its “self-will,”[xxvi] and the will of the other, or social life in general, become embodied in a form of self-consciousness that is fractured. This is the genesis of the “moral burden,” a burden which cannot be avoided, as human being is, by nature, or “by the flesh,”[xxvii] a social creature. The intensity of the moral burden, of course, may vary, depending upon the individual’s consciousness and estimate of the nature of the division. In some cases, the social will may not only be contrary to the ideals of the self-will, but be morally repugnant. In others, one may identify more closely with the social will, and see one’s own impulses and tendencies as “evil.” In still other cases, both wills may seem equally morally praiseworthy, yet still opposed. But never, according to Royce, can one side simply negate the other. One cannot dismiss one’s own will, any more than one can dismiss the superior power of the social will. Conversely, complete identification with either side is impossible.[xxviii]

In terms of
Royce’s theory of order, the formal character of the divided self is that of an
O-collection. Two, or more, possibilities of action, *a*, *b*, *c*…*n*,
stand in relations to one another such that their Boolean product is 0; in
Royce’s symbolism O(*abc…n*), or O(a).
In other words, they exclude one another in their totality.[xxix]
The self as O-collection is consequently no real self at all, but a fragmented
“collection” of mutually contradictory impulses or possible modes of action.

This divided self can only be reconciled through loyalty to a cause that unites many individuals into a determinate community. In earlier work on loyalty,[xxx] Royce explains that a cause, to which others are also loyal, provides a both a ground and a goal of activity, thereby providing, on the one hand, the social unit with an ordering principle capable of harmonizing disparate wills, and on the other, a coherent “plan of action” which unifies the fractured life of the individual.[xxxi] For example, the goal of winning a football championship harmonizes the wills of the players who remain loyal to this cause. Individual players recognize that they must cooperate with the others to achieve their goal, but this does not mean abandoning their own wills, since, presumably, they want to achieve the same goal. The cause may require modification of their immediate impulses and tendencies, but this modification is welcomed, as it is necessary to achieve their goal. The task of the divided self is to find a cause and be loyal to it. Only loyalty can unify the self, give it meaning and a life-plan. Only the loyal individual is an actual, individual person.[xxxii]

In the context of *PC,* Royce reinterprets
and develops his earlier work on loyalty to mean membership in a divinely
instituted community – the Kingdom of Heaven. Membership is accomplished through
the miracle of love which comes via the grace of the divine, that is, through
the spirit.[xxxiii]
Embodied as the origination and flourishing of a practical and thoroughgoing
devotion to a cause, the miracle of love transforms the self from an interior
battlefield into a unified person with a new type of consciousness. Royce
writes,

[F]or Christianity, the salvation of man means the destruction of his natural self, - the sacrifice of what his flesh holds dearest, - the utter transformation of the primal core of the social self.[xxxiv]

This new type of self loves both itself and the members of the community, but this love is on a different “level” than is the love the self has for the ideal cause that unites the self and community. Loving devotion to the cause, the new “center of gravity” of the self, provides a conscious standpoint whereby the warring wills of the self and the community are encountered not as enemies, but as potential partners in cooperation. The self, instead of identifying with the ideals of one side or another, or worse, both, stands in a position which is “between” the two, and can thereby mediate its own conflicting impulses such that they are harmonized and put into conformance with the needs of the cause. In this context of Christianity, the cause is the realization of the Kingdom of Heaven, or the Beloved Community. The will of the self and of other are now encountered as potential sources of action that may be coordinated to realize this Kingdom.

Again, from the context of Royce’s theory of order, the movement from the fragmented self to a unification and membership in a determinate community is formally represented in the transformation of O-collections to asymmetrical or directed F-collections. The mutually exclusive obverses – contrasting wills in the fragmented self and in fragmented communities – are reconciled and mediated in relation to one another through the introduction of a new kind of action, an obverse, which is consonant with a particular purpose and cause. This mode of action, as Royce’s logic of order demonstrates, is not a particular action within the O-collection. Instead, it is a new mode of action, a mode implied by the structure and content of the original O-collection.

Returning now to the notion of the of the Kingdom of Heaven, Royce notes that this Kingdom is itself invisible; that is, has no being in actuality. Yet, it does have being as an “actual potentiality.”[xxxv] The religious mission of the historical Christian Church, i.e., the Christian community, is to realize, in the present, this Kingdom. Such a mission, however, is fraught with peril. The church is invisible, in part, because of the “deeds of treason” against the spirit of the Kingdom.[xxxvi] These deeds have played their part in preventing the actual realization of the Kingdom of Heaven. One need not list the atrocities committed in the name of Christianity. They are evident. Much the same is true for all large and complex communities. Yet, it is precisely these treasonous deeds that unlock the conditions which are required for the eventual realization of the Kingdom of Heaven. Royce explains what he means by this in his analysis of the process of atonement.

The idea of
atonement is not unrelated to the issue of grace. As noted above, the divided
self, the “lost and morally burdened” individual, cannot be saved by his own
devices, just as no O-collection can be transformed into an F-collection without
the emergence of a new mode of action, a new interpretation, which is not
contained in the original O-collection. The miracle of love, given by grace, is
required to lift him into membership within the Beloved Community and reconcile
his self-will with the will of the other. Similarly, the treasonous individual,
who by his actions has sinned against the spirit of the community, thereby
fracturing it, cannot effect his own atonement. Repentance and forgiveness do
not go far enough, for the treasonous deed, the evil action, remains. It can
never be undone, and the sinner, by his own lights, “consigns herself to the *
hell of the irrevocable*.”[xxxvii]
He cannot be reconciled to himself or his action. What is needed is a new
creative act by a member of the community and guided by its spirit that not only
provides forgiveness, but actually creates a new, and better, situation for both
the sinner and the community. In other words, the community must be improved in
some respect by a new, creative deed such that a good is realized which, until
now, was impossible. Only in this way can the “sinner” be reconciled to his own
act, and the crack in the community repaired. For now, he and the community
understand that all is not lost because of his action, but that some new good
has emerged.[xxxviii]

In Royce’s analysis lies a complicated theory regarding the meaning and solution to the so-called “problem of evil.” This is not the place for a critical examination of Royce’s doctrine which is most clearly expressed in his earlier essay, “The Problem of Job.”[xxxix] Suffice it to say, that Royce thinks evil, most clearly expressed for him in terms of division, estrangement and opposition, is a necessary condition for a higher good through a deeper and more complex reconciliation. This fact, again, follows from the logic: treason against the community is akin to splintering an F-collection into an O-collection by adjoining to it contrasting elements. Reconciling these elements to one another and the entire collection requires novel modes of action – mediators – to transform the O-collection to a new, more complex F-collection. In other words, the ongoing realization of the Kingdom of Heaven requires new contrasts – and new “treasonous acts” for its perfection.

This Kingdom, when realized, would be a social organization, where the wills of many would harmonized into the will of one, each mediated through the other through loyalty to the cause and the spirit of the community. This idea of community, the last of the four “leading ideas,” is by far the most important for Royce’s work. His interpretation of the Kingdom of Heaven after Paul’s social interpretation, rather than Jesus’s (“the Kingdom of Heaven is within you,”) has momentous consequences.[xl]

Royce defines a community primarily in relation to what he calls the time-process.[xli] Our families, our countries, even our sports teams, are united, in part, by common, historical events. These events, not now present, but preserved in the memory of the community, serve as originary points of reference for communities. 9/11 is contained both in the collective memory of America, and in the minds of individual citizens. Births and deaths of family members contribute to the sense and life of the family, as do all the significant events in the history of the family. The life and death of Jesus stands as the central historical event of the Christian community. The expectation of the coming of the Kingdom of Heaven further unifies the community. Royce writes:

Then, *with reference
to the ideal common past and future in question, I say that these selves
constitute a community*….[This definition] furnishes both the most exact,
the most widely variable, and the most important of the motives which warrant
us in called a community a real unit. The Pauline metaphor of the body and the
members finds [in this definition] its most significant basis, - a basis
capable of exact description.[xlii]

The unifying power of historical events and future goals is contingent upon the memory and expectation of each member, and, specifically, the interpretation each gives of the history and direction of their lives. In fact, the constant interpretation of oneself in light of past actions and events, and future plans and goals is identical to the structure of the self, according to Royce. Each of us, he avers, interprets her past self to her future self from the present moment.[xliii] In those cases where a past event(s) and/or future event(s) are included in the interpretations of several distinct individuals, communities may exist. It is the conscious inclusion of identical historical events and future goals through the interpretative activities of loyal members that distinguish Roycean communities from the common social group.

Royce’s elaboration of the nature of community is directly based on the logic of Boolean algebra as it is developed by means of F-relations in Royce’s System S. There are two salient properties of in the structure of communities, just as there are two salient properties in Royce’s Boolean-ordered set. First, a community is an ordered union of individuals which are directed from a common post toward a fulfilling a specific purpose. Communities are, then, formally F-collections, and the order involved is identical to the triadic ordering relation of Royce’s Boolean algebra. Specifically, the formal nature of interpretation, those conscious acts which unite the past activities of the community with its future activities, and which Royce goes as great pains to highlight, are triadic, asymmetrical, and transitive relations. In fact, in numerous instances Royce directly calls interpretations “mediators,” the same term he uses to describe triadic F-relations.[xliv]

Second, there are two exceptional elements in any community, and also in a Boolean algebra: the origin and goal of the community, and the 0 and 1 elements in the algebra. These elements are strictly analogous. In the community, a historical event marks the origin of the community, and, in conjunction with the goal of the community, gives it sense and direction. The entire community is therefore ordered by its relation to these elements. They are its limits. The same holds true in the Boolean algebra. The 0-element and its obverse the 1-element bound the system of F-relations and determine the asymmetry or direction of the F-relations. Further, just as the 0-element is contained in every possible F-relation in the system, the historical origin is contained within every relation between members of a community; in other words, the origin of each is equivalent to the logical product of the totality of elements and relations in the system. Similarly, each element and relation in both the community and F-collection are contained within the 1-element, or goal of the community. For the 1-element or the goal or cause of the community must include the totality of elements and relations which are directed toward its realization. The goal or cause, or the 1-element, is the completion of the system. In Boolean terms, it is the logical sum of the entire system or community and as such functions as the superior limit. Communities, like the system of Boolean elements, thus stand “between” the limits of 0 and 1, origin and purpose, and they are asymmetrical or directed.

The four Christian ideas which Royce argues constitute the essence of Christianity – morally burdened individual, atonement, salvation, and community – are therefore woven together in such a manner as to be simultaneously a “doctrine of life,” but also a logical system of O- and F-relations which captures the structure of this life, and transformation and growth of this life through O-relations to F-relations. By virtue of the logical basis of these Christian ideas, Royce implies that it is possible to recognize them universally, thus demonstrating that they contain in addition to their specific religious meaning a purely “practical and human” meaning. The necessary development of the self involves it in a process which produces a division between self and other, a division which it embodies within itself and its fractured consciousness. Originating from some hidden abyss, only loyalty to the cause of a living community reconciles this inner and outer division, thus producing a higher form of consciousness and an integrated life. This is, according to Royce, the universal meaning of the symbolic stories and legends found in the Christian scriptures, and embodied in the Christian Church.

Royce’s project
does not end here, for his problematic requires him to give an argument that the
structure of communities, which may now be understood as partially ordered sets
submitting to Boolean operations, has a “more than human meaning and
foundation.”[xlv]
In view of his audience, in *PC*, Royce focuses mostly on the fact that the
triadic relation of interpretation may be capable of a metaphysical
generalization. I will not follow these arguments here. Suffice it to say,
however, that in light of a thorough knowledge of his science of order, Royce
has some stronger arguments at his disposal than those that he puts forth in *
PC*. For every extension of System S,
i.e., every derivation of an order system that is then actually discovered as
obtaining in some region of the universe, is further support that the order
which underlies communities, also underlies all other possible relations of
entities.

**Conclusion: **

** **I will
end by noting two features of Royce’s logic which are implicit in *PC*, but
which gain a new clarity and force in light of the foregoing considerations.
Royce clearly envisions universe composed of a plurality of communities with
myriad of relations between them. Indeed, he is often seen, despite his early
metaphysical monism, to be a champion of social pluralism. Furthermore, it is
not only possible, but also is frequently the case, that from the perspective of
one community other communities may appear strange or irrational. Both of these
facts are modeled and partially explained by Royce’s science of order. A
plurality of communities is satisfied by the fact that
S contains the partial order of Boolean
algebra in an infinity of ways by virtue of the arbitrary selection of origins
and ends, or 0 and 1 elements. Moreover, an event which functions as the origin
of one community may not function in the same way for another. Consequently,
the direction of the asymmetry, and consequently the forms of interpretation,
between origin and end vary as events play different roles for different
communities. The ordering relations of one are not, *from the internal
perspective of one community*, the same as others. Nevertheless, the order
systems are homomorphic; in other words, from the perspective of an observer
which stands outside each community, it is clear that both are structured by
Boolean laws; only the origins and ends, the 0 and 1 elements are
different.

** **

** **

** **

** **

** **

** **

** **

**Notes**

[1]
Royce, “The Relation of the Principles of Logic to the Foundations of
Geometry,” in *Royce’s Logical Essays*, ed. Daniel Robinson,
(Iowa: The Wm. C. Brown Company, 1951), 379 – 441.

[i]
Royce, “The Relation of the Principles of Logic to the Foundations of
Geometry,” in *Royce’s Logical Essays*, ed. Daniel Robinson,
(Iowa: Wm. C. Brown Company, 1951, 379-441.)

[ii]
For a substantial discussion of Royce’s philosophy of logic and a
description of his project see his “The Principles of Logic,’ in *
Royce’s Logical Essay*, especially Chapters I and III.

[iii] Royce, “The Principles of Logic,” 312.

[iv]
Royce, “The Problem of Truth in Light of Recent Discussion,” in *
Royce’s Logical Essays*, 73.

[v]
Royce, “Order,” in *Royce’s Logical Essays*,” 205.

[vi]
*Ibid*., 223.

[vii]
The method of characterizing a Boolean algebra given here is borrowed
from Thomas Jech, *Set Theory*, *The Third Millenium Edition*
(New York: Springer, 2003), 78-82.

[viii]
“~” symbolizes “not.” ~O(*ab*) means “the collection consisting of
*a *and *b* is not an O-collection.”

[ix] Royce, “The Relation of the Principles of Logic to the Foundations of Mathematics,” 392.

[x]
Royce, *The Letters of Josiah Royce*, ed. John Clendenning,
(Chicago: The University of Chicago Press, 1970), 610.

[xi]
Royce actually draws a bar over the top of the letter instead of placing
the negation symbol in front of the letter as I do here. Royce’s
derivation of F-relations from O-relations is a modification and
generalization of Christine-Ladd Franklin’s work on Boolean algebra some
22 years prior. Cf., “On the Algebra of Logic*,*” in *The
Studies in Logic by Members of the Johns Hopkins University*, ed.
C.S. Peirce, (Boston: Little, Brown and Company, 1883), 17-71.

[xii] Royce, “The Relations of the Principles of Logic to the Foundations of Geometry,” 405-407.

[xiii]
*Ibid*., 428.

[xiv]
*Ibid*., 382-383.

[xv]
*Ibid*., 384.

[xvi]
Cf. *Ibid*, 429-437.

[xvii]
Josiah Royce, *The Letters of Josiah Royce, *604.

[xviii]
*Ibid*., 605.

[xix]
*Ibid*., fn 607.

[xx]
Others have agreed with this thesis, notably Frank Oppenheim in *
Royce’s Mature Philosophy of Religion*, (Indiana: The University of
Indiana Press, 1987).

[xxi]
*The Problem of Christianity* will hereafter be referred to as *
PC*.

[xxii] Royce’s use of the term “man” undoubtedly denotes the both woman and man. It should be read as such in subsequent quotations of this term.

[xxiii]
*PC*, 62.

[xxiv]
This is not an unfamiliar task. In a related and generalized form,
William James explored it over a decade before Royce. Summarized in *
The Varieties of Religious Experience,* his solution amounts to the
championing of a religious life which attends to the “fringes” and the
“influx” of immediate and personal experience, the only locus James
thought the divine would and could disclose itself. This resolution of
the problem involves a rejection of traditional religious doctrine,
dogma, and institution, and in particular, metaphysical, theological,
and ethical doctrines that are anything but radically empirical. These
hollow relics of primary experience often prevent the immediate
experiential influx that he thought to be central to religion, whatever
its particular form. James’s private, experientially-based religious
life offers no contradiction to scientific inquiry and its practical
empiricism, but neither does it offer a creed that attempts to set the
whole of life and the universe into a coherent narrative structure,
complete with a metaphysic and what Royce calls a “doctrine of life.”
James’s solution, as presented in the *Varieties*, hence fails to
answer Royce’s particular form of the question; Royce seeks not only a
reconciliation of the religious experience and modernity, but a
mediation of a particular *creed* and modernity. In this context,
creed denotes a religious, ethical and metaphysical doctrine. This is
what the alienated, nomadic, modern mind seeks, according to Royce – a
spiritual homeland which makes room for both the influx and leadings of
primary experience and a communal life which is able, through constant
interpretation of its experience, to give intellectual form and meaning
to the whole of being. James’s solution, from the perspective of Royce,
stops short of this goal by arbitrarily restricting the religious life
to the private “percolations” of the solitary soul.

[xxv]
*Ibid*., 113.

[xxvi] Royce never defines with precision what “self-will” may be. Often, however, it is described as capricious, chaotic and unpredictable. Given other of Royce’s writings, self-will most likely refers to the immediate “pushings and pullings” in the flux of immediate experience of the Jamesian variety. Immediate feeling with its direction might be a better way of expressing “self-will.”

[xxvii]
*Ibid*., 115.

[xxviii] Royce draws this analysis of the moral burden from Paul’s epistle to the Romans, particularly, chapter 7.

[xxix] It is noteworthy that Royce believes that the ultimate interpretation of the elements of S are alternatively “modes of action,” or later “interpretations.” I will use these two terms interchangeably in what follows. Cf. “The Principles of Logic,” 373-375.

[xxx]
Royce, *The Philosophy of Loyalty*, (Nashville and London:
Vanderbilt University Press, 1995.)

[xxxi] The term cause is pregnant with meaning. It is plausible that Royce intends the term, in part, to call up all of Aristotle’s four causes, final, efficient, material, formal, though he does not discuss the term in this manner.

[xxxii]
One is tempted here to describe Royce’s “true person” in terms drawn
from the literature of existentialism. Only the committed individual
authentically exists on Royce’s view. Martin Buber’s view, for example,
that the I is a “person” only when it stands in relation to a “living
center,” and that only through creative service to this “Thou” is the
person free, is substantially identical to Royce’s position. Compare *
PC*,* *202 and Buber’s * I and Thou,* (New York:
Touchstone, 1996), * *esp. 95,113.

[xxxiii]
Loyalty (and love) is not blind obedience and is not inconsistent with
critique. Recent debates in the U.S. have highlighted this distinction
with reference to the issue of patriotism. In many cases, both
patriotism and loyalty to a cause require critique. See *The
Philosophy of Loyalty, *especially the first two chapters.

[xxxiv]
Royce*, PC*, 194.

[xxxv] I borrow this phrase from Pierce to highlight that Royce believes in “real possibilities,” including a universal “set.”

[xxxvi] The other reason follows from the logical impossibility of an actual infinite set; such sets are real possibilities, but do not exist in fact. Cf. Royce, “The Principles of Logic,” 375.

[xxxvii]
Royce, *PC*, 162.

[xxxviii]
Agency is thus a product of the contrasting structure of O-collections.
It, as Royce shows in his logic, directly implies agency in the form of
possible mediators and F-collections. But why a certain F-collection is
formed, or when it is formed, is, as Royce rightly notes, a mystery. It
is in the terms of *PC* the “miracle of love,” and the “origin of
life.”

[xxxix]
Royce, “The Problem of Job,” *The Basic Writings of Josiah Royce*,
833-844.

[xl]
Royce argues that Jesus’s parables and remarks concerning the Kingdom of
Heaven are *prima facia *ambiguous and vague. They were intended,
he argues, to be developed and interpreted by the community.

[xli]
Royce*, PC*, 249.

[xlii]
*Ibid*., 252, 256.

[xliii] The structure of self is consequently best described as a process of interior dialogue.

[xliv]
Cf. *Ibid*., 301.

[xlv]
*Ibid*., 231.