## 1. Introduction



Bertrand Russell (1872-1970) introduced the phrase
"logical atomism" to describe his philosophy in 1911
(*RA*, 94), and used the phrase consistently throughout the
1910s and 1920s (*OKEW*, 12;
*SMP*, 84; *PLA*, 178; *LA*, 323; *OOP*,
259). Russell's logical atomism is perhaps best described as partly a
methodological viewpoint, and partly a metaphysical theory. 



 
Methodologically, logical atomism can be seen as endorsement of
*analysis*, understood as a two-step process in which one
attempts to identify, for a given domain of inquiry, set of beliefs or
scientific theory, the minimum and most basic concepts and vocabulary
in which the other concepts and vocabulary of that domain can be
defined or recast, and the most general and basic principles from
which the remainder of the truths of the domain can be derived or
reconstructed. 



 
Metaphysically, logical atomism is the view that the world consists in
a plurality of independent and discrete entities, which by coming
together form facts. According to Russell, a fact is a kind of
complex, and depends for its existence on the simpler entities making
it up. The simplest sort of complex, an *atomic fact*, was
thought to consist either of a single individual exhibiting a simple
quality, or of multiple individuals standing in a simple
relation. 



 The methodological and metaphysical elements of logical atomism come
together in postulating the theoretical, if not the practical,
realizability of a fully analyzed language, in which all truths could
in principle be expressed in a perspicuous manner. Such a "logically
ideal language", as Russell at times called it, would, besides logical
constants, consist only of words representing the constituents of
atomic facts. In such a language, the simplest sort of complete
sentence would be what Russell called an "atomic proposition",
containing a single predicate or verb representing a quality or
relation along with the appropriate number of proper names, each
representing an individual. The truth or falsity of an atomic
proposition would depend entirely on a corresponding atomic fact. The
other sentences of such a language would be derived either by
combining atomic propositions using truth-functional connectives,
yielding *molecular propositions*, or by replacing constituents
of a simpler proposition by variables, and prefixing a universal or
existential quantifier, resulting in *general* and
*existential propositions*. According to the stronger form of
logical atomism Russell at times adopted, he held that in such a
language, "[g]iven all true atomic propositions, together with the
fact that they are all, every other true proposition can theoretically
be deduced by logical methods" (*PM2*, xv; cf. *OKEW*,
50). This puts the truth or falsity of atomic propositions at the core
of Russell's theory of truth, and hence, puts atomic facts at the
center of Russell's metaphysics.



Russell also at times suggests that analysis demonstrates that what we take to be essential or necessary properties of things and relations between things are the result of the logical forms the these things, properties and relations are logically constructed to have. This suggests that there are no such connections between simple entities, that all atomic propositions are independent of each other, and that all forms of necessity reduce to (formal) logical necessity. Some commentators interpret these theses to be central to Russell's logical atomism, though explicit commitment to them is scant in his writings.




 
In what follows, various aspects of Russell's logical atomism are
discussed in greater detail. The next section discusses the origins of
logical atomism in the break made by Russell and G.E. Moore from the
tradition of British Idealism, and its development during the years in
which Russell worked on *Principia Mathematica*. In section 3,
we examine Russell's notion of analysis as a philosophical method, and
give various examples of analysis as Russell understood it. In section
4, we turn to a more detailed look at certain metaphysical aspects of
Russell's atomism, and in particular, the nature and classification of
facts, as well certain points of controversy regarding his views. In
particular, we'll examine whether or not Russell's logical
atomism necessarily presupposes a fundamental realm of ultimate
simples, and whether or not the atomic propositions of Russell's
atomism were understood as logically independent. The final section is
dedicated to a discussion of the influence and reception of Russell's
logical atomism within the subsequent philosophical tradition.


## 2. Origins and Development of Russell's Logical Atomism


### 2.1 The Break with Idealism and the Nature of Relations



 In 1959, Russell himself dated his first acceptance of logical
atomism to the years 1899-1900, when he and G.E. Moore rejected the
main tenets of the dominant school of philosophy in Britain at the
time (to which both had previously been adherents), the tradition of
neo-Hegelian Idealism exemplified in works of F.H. Bradley and J.M.E.
McTaggart, and adopted instead a fairly strong form of realism
(*MPD*, 9). Of their break with idealism, Russell wrote that
"Moore led the way, but I followed closely in his footsteps"
(*MPD*, 42). 



 
In 1899, Moore published a paper entitled "The Nature of Judgment", in
which he outlined his main reasons for accepting the new realism. It
begins with a discussion of a distinction made by Bradley between
different notions of idea. According to Bradley, the notion of idea
understood as a mental state or mental occurrence is not the notion of
"idea" relevant to logic or to truth understood as a relationship
between our ideas and reality. Instead, the relevant notion of idea is
that of a sign or symbol representing something other than itself, or
an idea understood as possessing meaning. Bradley understood meaning
in terms of "a part of the content ... [of an idea] cut off,
fixed by the mind, and considered apart from the existence of the
sign" (Bradley 1883, 8). Moore agreed with Bradley that it is not
the mental occurrence that is important to logic. However, with
regard to Bradley's second notion of "idea", Moore accused Bradley of
conflating the symbol with the symbolized, and rejected Bradley's view
that what is symbolized is itself a part of the idea and dependent
upon it. Moore introduces the term "concept" for the meaning of a
symbol; for Moore, what it is for different ideas to have a common
content is for them to represent the same concept. However, the
concept itself is independent of the ideas. When we make a judgment,
typically, it is not our ideas, or parts of our ideas, which our
judgment is about. According to Moore, if I make an assertion, what I
assert is nothing about my ideas or my mental states, but a certain
"connexion of concepts". 



 Moore went on to introduce the term "proposition" for complexes of
concepts such as that which would be involved in a belief or judgment.
While propositions represent the content of judgments, according to
Moore, they and their constituents are entirely independent from the
judging mind. Some propositions are true, some are not. For Moore,
however, truth is not a correspondence relationship between
propositions and reality, as there is no difference between a
proposition--understood as a mind-independent complex--and that which
would make it true (Moore 1899, 5; Moore 1901). The facts of the
world then consist of true propositions, themselves understood as
complexes of concepts. According to Moore, something "becomes
intelligible first when it is analyzed into its constituent concepts"
(Moore 1899, 8). "The Nature of Judgment" had a profound influence
on Russell, who later heralded it as the first account of the "new
philosophy" to which he and Moore subscribed (*MPD*, 42).



 
For his own part, Russell often described his dissatisfaction with the
dominant Idealist (and largely Monist) tradition as primarily having
to do with the nature and existence of relations. In particular,
Russell took issue with the claim found in Bradley and others, that
the notion of a fundamental relation between two distinct entities is
incoherent. Russell diagnosed this belief as stemming from a
widespread logical doctrine to the effect that every proposition is
logically of subject-predicate form. Russell was an ardent opponent of
a position known as the "doctrine of internal relations", which
Russell stated as the view that "every relation is grounded in the
natures of the related terms" (*MTT*, 139). Perhaps most
charitably interpreted (for other interpretations considered by
Russell, see *BReal*, 87), this amounts to the claim that
*a*'s bearing relation *R* to *b* is always
reducible to properties held by *a* and *b*
individually, or to a property held by the complex formed of
*a* and *b*.



 
In the period leading up to his own abandonment of idealism, Russell
was already pursuing a research program involving the foundations of
arithmetic (see, e.g., *AMR*). This work, along with his
earlier work on the foundations of geometry (see *EFG*), had
convinced him of the importance of relations for mathematics. However,
he found that one category of relations, viz., asymmetrical transitive
relations, resisted any such reduction to the properties of the relata
or the whole formed of them. These relations are especially important
in mathematics, as they are the sort that generates series. Consider
the relation of *being taller than*, and consider the fact that
Shaquille O'Neal is taller than Michael Jordan. It might be thought
that this relation between O'Neal and Jordan can be reduced to
properties of each: O'Neal has the property of being 7'2'' tall, and
Jordan has the property of being 6'6'' tall, and the *taller
than* relation in this case is reducible to their possession of
these properties. The problem, according to Russell, is that for this
reduction to hold, there must be a certain relation between the
properties themselves. This relation would account for the ordering of
the various height properties, putting the property of being-6'8''-tall
in between that of being-7'2''-tall and that of being-6'6''-tall. This
relation among the properties would itself be an asymmetrical and
transitive relation, and so the analysis has not rid us of the need
for taking relations as ultimate. Another hypothesis would be that
there is such an entity as the whole composed of O'Neal and Jordan,
and that the relation between the two men is reducible to some
property of this whole. Russell's complaint was that since the whole
composed of O'Neal and Jordan is the same as the whole composed of
Jordan and O'Neal, this approach has no way to explain what the
difference would be between O'Neal's being taller than Jordan and Jordan's
being taller than O'Neal, as both would seem to be reduced to the same
composite entity bearing the same quality (see *POM*,
221-26).



 
Russell's rejection of the doctrine of internal relations is very
important for understanding the development of his atomistic doctrines
in more than one respect. Certain advocates of the claim that a
relation must always be grounded in the "nature" of its relata hold
that in virtue of *a* relating to *b*, *a* must
have a complex nature that includes its relatedness to
*b*. Since every entity presumably bears some relation to any
other, the "nature" of any entity could arguably be described as
having the same complexity as the universe as a whole (if indeed, it
even makes sense on such a picture to divide the world into distinct
entities at all, as many denied). Moreover, according to some within
this tradition, when we consider *a*, obviously we do not
consider all its relations to every entity, and hence grasp *a*
in a way that falsifies the whole of what *a* is. This led
some to the claim that "analysis is falsification", and even to hold
that when we judge that *a* is the father of *b*, and
judge that *a* is the son of *c*, the *a* in the
first judgment is not strictly speaking the same *a* as
involved in the second judgment; instead, in the first we deal only
with *a*-*qua-father-of-b* in the first, and
*a*-*qua-son-of-c* in the second
(cf. *BReal*, 89; *MTT*, 140). 



In contradistinction to these views, Russell adopted what he called
"the doctrine of external relations", which he claimed "may be
expressed by saying that (1) relatedness does not imply any
corresponding complexity in the *relata*; (2) any given entity
is a constituent of many different complexes" (*BReal*,
87). This position on relations allowed Russell to adopt a pluralist
philosophy in which the world is conceived as composed of many
distinct, independent entities, each of which can be considered in
isolation from its relations to other things, or its relation to the
mind. In 1911 Russell claimed that this doctrine was the "fundamental
doctrine" of his realistic position (*BReal*, 87;
cf. *RA*, 92; *POM*, 226), and it represents perhaps the
most important turning point in the development of his logical
atomism.


### 2.2 Propositions in *The Principles of Mathematics*



Russell's first published account of his newfound realism came in the
1903 classic *The Principles of Mathematics* (*POM*).
Part I of *POM* is dedicated largely to a philosophical inquiry
into the nature of propositions. Russell took over from Moore the
conception of propositions as mind-independent complexes; a true
proposition was then simply identified by Russell with a fact
(cf. *MTCA*, 75-76). However, Moore's characterization of a
proposition as a complex of *concepts* was largely in keeping
with traditional Aristotelian logic in which all judgments were
thought to involve a subject concept, copula and predicate
concept. Russell, owing in part to his own views on
relations, and in part from his adopting certain doctrines stemming
from Peano's symbolic logic, sought to refine and improve upon this
characterization. 



 
In the terminology introduced in *POM*, constituents of a
proposition occur either "as term" or "as
concept". An entity occurs "as term" when it can be
replaced by any other entity and the result would still be a
proposition, and when it is one of the subjects of the proposition,
i.e., something the proposition is "about". An entity
occurs *as concept* when it occurs predicatively, i.e., only as
part of the assertion made about the things occurring as term. In the
proposition *Socrates is human*, the person Socrates (the man
himself) occurs as term, but humanity occurs as concept. In the
proposition *Callisto orbits Jupiter*, Callisto (the moon
itself) and Jupiter (the planet) occur as term, and the relation of
orbiting occurs as concept. Russell used the word
"concept" for all those entities capable of occurring as
concept--chiefly relations and other universals--and the
word "thing" for those entities such as Socrates, Callisto
and Jupiter, that can only occur as term. While Russell thought that
only certain entities were capable of occurring as concept, at the
time, he believed that every entity was capable of occurring as term
in a proposition. In the proposition *Wisdom is a virtue*, the
concept wisdom occurs as term. His argument that this held generally
was that if there were some entity, *E*, that could not occur
as term, there would have to be a fact, i.e., a true proposition, to
this effect. However, in the proposition *E cannot occur as term
in a proposition*, *E* occurs as term (*POM*,
44-45).



 
Russell's 1903 account of propositions as complexes of entities was in
many ways in keeping with his views as the nature of complexes and
facts during the core logical atomist period of 1911-1925. In
particular, at both stages he would regard the simple truth that an
individual *a* stands in the simple relation *R* to an
individual *b* as a complex consisting of the individuals
*a* and *b* and the relation *R*. However, there
are a number of positions Russell held in 1903 that were abandoned in
this later period; some of the more important were these: (1) in 1903,
Russell was committed to a special kind of propositional constituent
called a "denoting concept", involved in descriptive and quantified
propositions; (2) in 1903, Russell believed that there was such a
complex, i.e., a proposition, consisting of *a*, *b* and
*R* even when it is not *true*that *a* bears relation *R* to *b*, and
(3) in 1903, Russell believed in the reality of classes, understood as
aggregate objects, which could be constituents of propositions. In
each case, it is worth, at least briefly, discussing Russell's change
of heart.


### 2.3 The Theory of Descriptions



 In *POM*, Russell expressed the view that grammar is a useful
guide in understanding the make-up of a proposition, and even that in
many cases, the make-up of a proposition corresponding to a sentence
can be understood by determining, for each word of the sentence, what
entity in the proposition is meant by the word (*POM*,
46). Perhaps in part because such phrases as "all dogs", "some
numbers" and "the queen" appear as a grammatical unit, Russell came to
the conclusion that they made a unified contribution to the
corresponding proposition. Because Russell believed it impossible for
a finite mind to grasp a proposition of infinite complexity, however,
Russell rejected a view according to which the (false) proposition
designated by


1. All numbers are odd.



 
actually contains all numbers (*POM*, 145). Similarly,
although Russell admitted that such a proposition as (1) is equivalent
to a formal implication, i.e., a quantified conditional of the
form:


2. (*x*)(*x* is a number [?] *x* is
odd)



 
Russell held that they are nevertheless distinct propositions
(*POM*, 74). This was perhaps in part due to the difference
in grammatical structure, and perhaps also because the former appears
only to be about numbers, whereas the latter is about all things,
whether numbers or not. Instead, Russell thought that the proposition
corresponding to (1) contains as a constituent the denoting concept
*all numbers*. As Russell explained them, when denoting concepts 
occur in a proposition, the proposition is not about them but about 
other entities to which the denoting concepts bear a special
relation. So when the denoting concept *all numbers* occurs in
a proposition, the proposition is not about the denoting concept, but
instead about 1 and 2 and 3, etc.



 
In 1905, Russell abandoned this theory in favor of his celebrated
theory of definite and indefinite descriptions outlined in the paper
"On Denoting". What precisely lead Russell to become dissatisfied with
his earlier theory, and the precise nature of the argument he gave
against denoting concepts (and similar entities such as Frege's
senses), are a matter of great controversy, and have given rise to
a large body of secondary literature. For present purposes, it can
merely be noted that Russell professed an inability to understand the
logical form of propositions *about* denoting concepts
themselves, as in the claim that "*The present King of France*
is a denoting concept" (cf. *OD*, 48-50). According to the
new theory adopted, the proposition expressed by (1) was now
identified with that expressed by a quantified conditional such as
(2). Similarly, the proposition expressed by


3. Some number is odd.



 
was identified with the existentially quantified conjunction
represented by


4. ([?]*x*)(*x* is a number & *x*
is odd)



 
Perhaps most notoriously, Russell argued that a proposition involving
a definite description, e.g.,



5. The King of France is bald.



 
was to be understood as having the structure of a certain kind of
existential statement, in this case:


6. ([?]*x*)(*x* is King of France
 & (*y*)(*y* is King of France [?] *x* =
 *y*) & *x* is
 bald)



 
Russell cited in favor of these theories that they provided an elegant
solution to certain philosophical puzzles. One involves how it is that
a proposition can be meaningful even if it involves a description or
other denoting phrase that does not denote anything. Given the above
account of the structure of the proposition expressed by "the King of
France is bald", while France and the relation of *being King
of* are constituents, there is no constituent directly
corresponding to the whole phrase "the King of France". The proposition in
question is false, since there is no value of *x* which would
make it true. One is not committed to a nonexistent entity such as the
King of France simply in order to understand the make-up of the
proposition. Secondly, this theory provides an answer to how it is
that certain identity statements can be both true and informative. On
the above theory, the proposition corresponding to:


7. The author of Waverly = Scott



 
would be understood as having the following structure:


8. ([?]*x*)(*x*
 authored Waverly & (*y*)(*y*
 authored Waverly [?] *x* =
 *y*) & *x* = Scott)



 
If instead, the proposition corresponding to (7) was simply a complex
consisting of the relation of identity, Scott, and the author of
Waverly himself, since the author of Waverly simply *is* Scott,
the proposition would be the same as the uninformative proposition
*Scott = Scott*. By showing that the actual structure of the
proposition is quite a bit different from what it appears from the
grammar of the sentence "The author of Waverly = Scott", Russell
believed he had shown how it might be more informative than a trivial
instance of the law of identity (*OD*, 51-54).



 
The theory of "On Denoting" did away with Russell's temptation to
regard grammar as a very reliable guide towards understanding the
structure or make-up of a proposition. Especially important in this
regard is the notion of an "incomplete symbol", by which Russell
understood an expression that can be meaningful in the context of its
use within a sentence, but does not by itself correspond to a
constituent or unified part of the corresponding
proposition. According to the theory of "On Denoting", phrases such as
"the King of France", or "the author of Waverly" were to be understood
as "incomplete symbols" in this sense. The general notion of an
"incomplete symbol" was applied by Russell in ways beyond the theory
of descriptions, and perhaps most importantly, to his understanding of
classes.


### 2.4 Classes, Propositions and Truth in *Principia Mathematica*



 
In *POM*, Russell had postulated two types of composite
entities: *unities* and *aggregates* (*POM*,
140f). By a "unity" he meant a complex entity in which the constituent
parts are arranged with a definite structure. A proposition was
understood to be a unity in this sense. By an "aggregate", he meant
an entity such as a class whose identity conditions are governed
entirely by what members or "parts" it has, and not by any
relationships between the parts. By the time of the publication of the
first edition of *Principia Mathematica* in 1910, Russell's
views about both types of composite entities had changed drastically.




Russell fundamentally conceived of a class as the extension of a
concept, or as the extension of a propositional function; indeed, in
*POM* he claims that "a class may be *defined* as all
the terms satisfying some propositional function" (*POM*,
20). However, Russell was aware already at the time of *POM*
that the supposition there is always a class, understood as an
individual entity, as the extension of every propositional function,
leads to certain logical paradoxes. Perhaps the most famous, now
called "Russell's paradox", derived from consideration of the class,
*w*, of all classes not members of themselves. The class
*w* would be a member of itself if it satisfied its defining
condition, i.e., if it were not a member of itself. Similarly,
*w* would not be a member of itself if it did not satisfy its
defining condition, i.e., if it were a member of itself. Hence, both
the assumption that it is a member of itself, and the assumption that
it is not, are impossible. Another related paradox Russell often
discussed in this regard has since come to be called "Cantor's
paradox". Cantor had proven that if a class had *n* members,
that the number of sub-classes that can be taken from that class is
2*n*, and also that 2*n* >
*n*, even when *n* is infinite. It follows from this
that the number of subclasses of the class of all individuals, (i.e.,
the number of different classes of individuals) is greater than the
number of individuals. Russell took this as strong evidence that a
class of individuals could not itself be considered an
individual. Likewise, the number of subclasses of the class of all
classes is greater than the number of members in the class of all
classes. This Russell took to be evidence that there is some ambiguity
in the notion of a "class" so that the subclasses of the class of "all
classes" would not themselves be among its members, as it would
seem.



 
Russell spent the years between 1902 and 1910 searching for a
philosophically motivated solution to such paradoxes. He tried
solutions of various sorts. However, in late 1905, after the
discovery of the theory of descriptions, he became convinced that an
expression for a class is an "incomplete symbol", i.e., that while
such an expression can occur as part of a meaningful sentence, it
should not be regarded as representing a single entity in the
corresponding proposition. Russell dubbed this approach the "no
classes" theory of classes (see e.g., *TNOT*, 145), because,
while it allows discourse about classes to be meaningful, it does not
posit classes as among the fundamental ontological furniture of the
world. The precise nature of Russell's "no classes" theory underwent
significant changes between 1905 and 1910. However, in the version
adopted in the first edition of *Principia Mathematica*,
Russell believed that a statement apparently about a class could
always be reconstructed, using higher-order quantification, in terms
of a statement involving its defining propositional function. Russell
believed that whenever a class term of the form
"{*z*|ps*z*}" appeared in some sentence, the sentence
as a whole could be regarded as defined as follows (cf. *PM*,
188):



> 
> *f*({*z*|ps*z*})  
> =*df*  
> ([?]ph)((*x*)(ph!*x* [?] ps*x*)
> & *f*(ph))
> 
> 



 
The above view can be paraphrased, somewhat crudely, as the claim that
any truth seemingly about a class can be reduced to a claim about some
or all of its members. For example, it follows from this contextual
definition of class terms that the statement to the effect that one
class *A* is a subset of another class *B* is equivalent
to the claim that whatever satisfies the defining propositional
function of *A* also satisfies the defining propositional
function of *B*. Russell also sometimes described this as the
view that classes are "logical constructions", not part of the "real
world", but only the world of logic. Another way Russell expressed
himself is by saying that a class is a "logical fiction". While it
may seem that a class term is representative of an entity, according
to Russell, class terms are meaningful in a different way. Classes
are not among the basic stuff of the world; yet it is possible to make
use of class terms in significant speech, *as if* there were
such things as classes. A class is thus portrayed by Russell as a mere
*facon de parler*, or convenient way of speaking about
all or some of the entities satisfying some propositional
function.



 
During the period in which Russell was working on *Principia
Mathematica*, most likely in 1907, Russell also radically revised
his former realism about propositions understood as mind independent
complexes. The motivations for the change are a matter of some
controversy, but there are at least two possible sources. The first is
that in addition to the logical paradoxes concerning the existence of
classes, Russell was aware of certain paradoxes stemming from the
assumption that propositions could be understood as individual
entities. One such paradox was discussed already in Appendix B of
*POM* (527-28). By Cantor's theorem, there must be more classes
of propositions than propositions. However, for every class of
propositions, *m*, it is possible to generate a distinct
proposition, such as the proposition that *every proposition in m
is true*, in violation of Cantor's theorem. Unlike the other
paradoxes mentioned above, a version of this paradox can be
reformulated even if talk of classes is replaced by talk of their
defining propositional functions. Russell was also aware of certain
contingent paradoxes involving propositions, such as the Liar paradox
formulated involving a person *S*, whose only assertion at time
*t* is the proposition *All propositions asserted by S at
time t are false*. Given the success of the rejection of classes
as ultimate entities in resolving the paradoxes of classes, Russell
was motivated to see if a similar solution to these paradoxes could be
had by rejecting propositions as singular entities.



 
Another set of considerations pushing Russell towards the rejection of
his former view of propositions is more straightforwardly
metaphysical. According to his earlier view, and that of Moore, a
proposition was understood as a mind independent complex. The
constituents of the complex are the actual entities involved, and
hence, as we have seen, when a proposition is true, it is the same
entity as a fact or state of affairs. However, because some
propositions are false, this view of propositions posits objective
falsehoods. The false proposition that *Venus orbits Neptune*
is thought to be a complex containing Venus and Neptune the planets,
as well as the relation of orbiting, with the relation occurring as a
relation, i.e., as *relating* Venus to Neptune. However, it
seems natural to suppose that the relation of orbiting could only
unite Venus and Neptune into a complex, if in fact, Venus orbits
Neptune. Hence, the presence of such objective falsehoods is itself
out of sorts with common sense. Worse, as Russell explained, positing
the existence of objective falsehoods in addition to objective truths
makes the difference between "truth" and
"falsehood" inexplicable, as both become irreducible
properties of propositions, and we are left without an explanation for
the privileged metaphysical status of truth over falsehood (see, e.g.,
*NTF*, 152).



 
Whatever his primary motivation, Russell abandoned any commitment to
objective falsehoods, and restructured his ontology of facts, and
adopted a new correspondence theory of truth. In the terminology of
the new theory, the word "proposition" was used not for an objective
metaphysical complex, but simply for an interpreted declarative
sentence, an item of language. Propositions are thought to be true or
false depending on their correspondence, or lack thereof, with
*facts*. 



 
In the Introduction to *Principia Mathematica*, as part of his
explanation of ramified type-theory, Russell described various notions
of truth applicable to different types of propositions of different
complexity. The simplest propositions in the language of *Principia
Mathematica* are what Russell there called "elementary
propositions", which take forms such as "*a* has quality
*q*", "*a* has relation [in intension] *R* to
*b*", or "*a* and *b* and *c* stand in
relation *S*" (*PM*, 43-44). Such propositions
consist of a simple predicate, representing either a quality or a
relation, and a number of proper names. According to Russell, such a
proposition is true when there is a corresponding fact or complex,
composed of the entities named by the predicate and proper names
related to each other in the appropriate way. E.g., the proposition
"*a* has relation *R* to *b*" is true if there
exists a corresponding complex in which the entity *a* is
related by the relation *R* to the entity *b*. If there
is no corresponding complex, then the proposition is false.



 
Russell dubbed the notion of truth applicable to elementary
propositions "first truth". This notion of truth serves as
the ground for a hierarchy of different notions of truth applicable to
different types of propositions depending on their complexity. A
proposition such as "(*x*)(*x* has
quality *q*)" which involves a first-order quantifier,
has (or lacks) "second truth" depending on whether its
instances have "first truth". In this case,
"(*x*)(*x* has quality *q*)" would be
true if every proposition got by replacing the
"*x*" in "*x* has
quality *q*" with the proper name of an individual has
"first truth" (*PM*, 42). A proposition involving
the simplest kind of second-order quantifier, i.e., a quantifier using
a variable for "predicative" propositional functions of
the lowest type, would have or lack "third truth"
depending on whether its allowable substitution instances have second
(or lower) truth. Because any statement apparently about a class of
individuals involves this sort of higher-order quantification, the
truth or falsity of such a proposition will ultimately depend on the
truth or falsity of various elementary propositions about its
members. 



 
Although Russell did not use the phrase "logical atomism" in the
Introduction to *Principia Mathematica*, in many ways it
represents the first work of Russell's atomist period. Russell there
explicitly endorsed the view that the "universe consists of objects
having various qualities and standing in various relations"
(*PM*, 43). Propositions that assert that an object has a
quality, or that multiple objects stand in a certain relation, were
given a privileged place in the theory, and explanation was given as
to how more complicated truths, including truths about classes, depend
on the truth of such simple propositions. Russell's work over the next
two decades consisted largely in refining and expanding upon this
picture of the world.


## 3. Russell's Philosophical Method and the Notion of Analysis



 Although Russell changed his mind on a great number of philosophical
issues throughout his career, one of the most stable elements in his
views is the endorsement of a certain methodology for approaching
philosophy. Indeed, it could be argued to be the most continuous and
unifying feature of Russell's philosophical work (e.g., see Hager
1994). Russell employed the methodology self-consciously, and gave
only slightly differing descriptions of this methodology in works
throughout his career (see, esp., *EFG*,
14-15; *POM*, 1-2, 129-30; *RMDP*,
272-74; *PM*, 59; *IPL*,
284-85; *TK*, 33, 158-59; *OKEW*,
144-45; *PLA*, 178-82, 270-71; *IMP*,
1-2; *LA*, 324-36, 341;
*RTC*, 687; *HWP*, 788-89; *HK*, 257-59;
*MPD*, 98-99, 162-163). Understanding this methodology is
particularly important for understanding his logical atomism, as well
as what he meant by "analysis".



 The methodology consists of a two phase process. The first phase is
dubbed the "analytic" phase (although it should be noted that
sometimes Russell used the word "analysis" for the whole
procedure). One begins with a certain theory, doctrine or collection
of beliefs which is taken to be more or less correct, but is taken to
be in certain regards vague, imprecise, disunified, overly complex or
in some other way confused or puzzling. The aim in the first phase is
to work backwards from these beliefs, taken as a kind of "data", to a
certain minimal stock of undefined concepts and general principles
which might be thought to underlie the original body of knowledge. The
second phase, which Russell described as the "constructive" or
"synthetic" phase, consists in rebuilding or reconstructing the
original body of knowledge in terms of the results of the first phase.
More specifically, in the synthetic phase, one defines those elements
of the original conceptual framework and vocabulary of the discipline
in terms of the "minimum vocabulary" identified in the first phrase,
and derives or deduces the main tenets of the original theory from the
basic principles or general truths one arrives at after analysis. 



 
As a result of such a process, the system of beliefs with which one
began takes on a new form in which connections between various
concepts it uses are made clear, the logical interrelations between
various theses of the theory are clarified, and vague or unclear
aspects of the original terminology are eliminated. Moreover, the
procedure also provides opportunities for the application of Occam's
razor, as it calls for the elimination of unnecessary or redundant
aspects of a theory. Concepts or assumptions giving rise to paradoxes
or conundrums or other problems within a theory are often found to be
wholly unnecessary or capable of being supplanted by something less
problematic. Another advantage is that the procedure arranges its
results as a deductive system, and hence invites and facilitates the
discovery of new results.




 
Examples of this general procedure can be found throughout Russell's
writings, and Russell also credits others with having achieved similar
successes. Russell's work in mathematical logic provides perhaps the
most obvious example of his utilization of such a procedure. It is
also an excellent example of Russell's contention that analysis
proceeds in stages. Russell saw his own work as the next step in a
series of successes beginning with the work of Cantor, Dedekind and
Weierstrass. Prior to the work of these figures, mathematics employed
a number of concepts, *number*, *magnitude*, *series,
limit*, *infinity*, *function, continuity*, etc.,
without a full understanding of the precise definition of each
concept, nor how they related to one another. By introducing precise
definitions of such notions, these thinkers exposed ambiguities (e.g.,
such as with the word "infinite"), revealed interrelations between
certain of them, and eliminated dubious notions that had previously
caused confusion and paradoxes (such as those involved with the notion
of an "infinitesimal"). Russell saw the next step forward in the
analysis of mathematics in the work of Peano and his associates, who
not only attempted to explain how many mathematical notions could be
"arithmetized", i.e., defined and proven in terms of arithmetic, but
had also identified, in the case of arithmetic, three basic concepts
(*zero*, *successor*, and *natural number*) and
five basic principles (the so-called "Peano axioms"), from which the
rest of arithmetic was thought to be derivable.



 
Russell described the next advance as taking place in the work of
Frege. According to the conception of number found in Frege's
*Grundgesetze der Arithmetik*, a number can be regarded as an
equivalence class consisting of those classes whose members can be put
in 1-1 correspondence with any other member of the class. According to
Russell, this conception allowed the primitives of Peano's analysis to
be defined fully in terms of the notion of a class, along with other
logical notions such as identity, quantification, negation and the
conditional. Similarly, Frege's work showed how the basic principles
of Peano's analysis could be derived from logical axioms
alone. However, Frege's analysis was not in all ways successful, as
the notion of a class or the extension of a concept which Frege
included as a logically primitive notion lead to certain
contradictions. In this regard, Russell saw his own analysis of
mathematics (largely developed independently from Frege) as an
improvement, with its more austere analysis that eliminates even the
notion of a class as a primitive idea (see the discussion of classes
in Section 2.4 above), and thereby eliminates the contradictions (see,
e.g., *RMDP*, 276-81; *LA*, 325-27).



 
It was clearly a part of Russell's view that in conducting an
analysis of a domain such as mathematics, and reducing its primitive
conceptual apparatus and unproven premises to a minimum, one is not
merely reducing the vocabulary of a certain theory, but also showing a
way of reducing the metaphysical commitments of the theory. In first
showing that numbers such as 1, 2, etc., could be defined in terms of
classes of like cardinality, and then showing how apparent discourse
about "classes" could be replaced by higher-order
quantification, Russell made it possible to see how it is that there
could be truths of arithmetic without presupposing that the numbers
constitute a special category of abstract entity. Numbers are placed
in the category of "logical fictions" or "logical
constructions" along with all other classes.



 
Russell's work from the period after the publication of *Principia
Mathematica* of 1910 shows applications of this general
philosophical approach to non-mathematical domains. In particular, his
work over the next two decades shows concern with the attempt to provide
analyses of the notions of knowledge, space, time, experience, matter
and causation. When Russell applied his analytic methodology to
sciences such as physics, again the goal was to arrive at a
"minimum vocabulary" required for the science in question,
as well as a set of basic premises and general truths from which the
rest of the science can be derived. We cannot delve into all the
details of Russell's evolving analyses here. However, according to the
views developed by Russell in the mid-1910s, many of the fundamental
notions in physics were thought to be analyzable in terms of
particular sensations: i.e., bits of color, auditory notes, or other
simple parts of sensation, and their qualities and relations. Russell
called such sensations, when actually experienced,
"sense". In particular, Russell believed that the notion
of a "physical thing" could be replaced, or analyzed in
terms of, the notion of a series of classes of sensible particulars
each bearing to one another certain relations of continuity,
resemblance, and perhaps certain other relations relevant to the
formulation of the laws of physics (*OKEW*, 86ff;
*RSDP*, 114-15; *UCM*, 105). Other physical
notions such as that of a point of space, or an instance of time,
could be conceived in terms of classes of sensible particulars and
their spatial and temporal relations (see *TK*,
77; *OKEW*, 91-99). Later, after abandoning the view that
perception is fundamentally relational, and accepting a form of
William James's neutral monism, Russell similarly came to believe that
the notion of a conscious mind could be analyzed in terms of various
percepts, experiences and sensations related to each other by
psychological laws (*AMi* chaps. 1, 5; *OOP* chap. 26;
cf. *PLA*, 277ff). Hence, Russell came to the view that words
as "point", "matter","instant",
"mind", and the like could be discarded from the minimum
vocabulary needed for physics or psychology. Instead, such words could
be systematically translated into a language only containing words
representing certain qualities and relations between sensible
particulars.



 
Throughout these analyses, Russell put into practice a slogan he
stated as follows: "Wherever possible, logical constructions are
to be substituted for inferred entities" (*RSDP*, 115;
cf. *LA*, 326). Rival philosophies that postulate an
*ego* or *mind* as an entity distinct from its mental
states involve inferring the existence of an entity that cannot
directly be found in experience. Something similar can be said about
philosophies that take matter to be an entity distinct from sensible
appearances, lying behind them and inferred from them. Combining
Russell's suggestions that talk of "minds" or "physical objects" is to
be analyzed in terms of classes of sensible particulars with his
general view that classes are "logical fictions", results in the view
that minds and physical objects too are "logical fictions", or not
parts of the basic building blocks of reality. Instead, all truths
about such purported entities turn out instead to be analyzable as
truths about sensible particulars and their relations to one
another. This is in keeping with the general metaphysical outlook of
logical atomism. We also have here a fairly severe application of
Occam's razor. The slogan was applied within his analyses in
mathematics as well. Noting that sometimes a series of rational
numbers converges towards a limit which is not itself specifiable as a
rational, some philosophers of mathematics thought that one should
*postulate* an irrational number as a limit. Russell claimed
that rather than *postulating* entities in such a case, an
irrational number should simply be *defined* as a class of
rational numbers without a rational upper bound. Russell preferred to
reconstruct talk of irrationals this way rather than *infer* or
*postulate* the existence of a new species of mathematical
entity not already known to exist, complaining that the method of
"postulating" what we want has "the advantages of theft over honest
toil" (*IMP*, 71).



 
In conducting an analysis of mathematics, or indeed, of any other
domain of thought, Russell was clear that although the results of
analysis can be regarded as *logical* premises from which the
original body of knowledge can in principle be derived,
epistemologically speaking, the pre-analyzed beliefs are more
fundamental. For example, in mathematics, a belief such as "2 + 2 = 4"
is epistemologically more certain, and psychologically easier to
understand and accept, than many of the logical premises from which it
is derived. Indeed, Russell believed that the results obtained through
the process of analysis obtain their epistemic warrant inductively
from the evident truth of their logical consequences (see, e.g.,
*TK*, 158-59). As Russell put it, "[t]he reason for
accepting an axiom, as for accepting any other proposition, is always
largely inductive, namely that many propositions which are nearly
indubitable can be deduced from it, and that no equally plausible way
is known by which these propositions could be true if the axiom were
false, and nothing which is plausibly false can be deduced from it"
(*PM*, 59; cf. *RMDP*, 282). It is perhaps for these
reasons that Russell believed that the process of philosophical
analysis should always begin with beliefs the truth of which are not
in question, i.e., which are "nearly indubitable". 



 
When Russell spoke about the general philosophical methodology
described here, he usually had in mind applying the process of
analysis to an entire body of knowledge or set of data. In fact,
Russell advocated usually to begin with the uncontroversial doctrine
of a certain science, such as mathematics or physics, largely because
he held that these theories are the most likely to be true, or at
least nearly true, and hence make the most appropriate place to begin
the process of analysis. 



 
Russell did on occasion also speak of analyzing a particular
proposition of ordinary life. One example he gave is "There are a
number of people in this room at this moment" (*PLA*, 179). In
this case, the truth or falsity of this statement may seem obvious,
but exactly what its truth would involve is rather obscure. The
process of analysis in this case would consist in attempting to make
the proposition clear by defining what it is for something to be a
room, for something to be a person, for a person to be in a room, what
a moment is, etc. In this case, it might seem that the ordinary
language statement is sufficiently vague that there is likely no
*one* precise or unambiguous proposition that represents the
"correct analysis" of the proposition. In a sense this is right;
however, this does not mean that analysis would be worthless. Russell
was explicit that the goal of analysis is not to unpack what is
psychologically intended by an ordinary statement such as the previous
example, nor what a person would be thinking when he or she utters
it. The point rather is simply to begin with a certain obvious, but
rough and vague statement, and find a replacement for it in a more
precise, unified, and minimal idiom (see, e.g., *PLA*, 180,
189). 



 On Russell's view, vagueness is a feature of language, not of the
world. In vague language, there is no one-one relation between
propositions and facts, so that a vague statement could be considered
verified by any one of a range of different facts (*Vag*,
217). However, in a properly analyzed proposition, there is a clear
isomorphism between the structure of the proposition and the structure
of the fact that would make it true (*PLA*, 197); hence a
precise and analyzed proposition is capable of being true in one and
only one way (*Vag*, 219). In analyzing a proposition such as
"there are a number of people in this room at this moment", one might
obtain a precise statement which would require for its truth that
there is a certain class of sensible particulars related to each other
in a very definite way constituting the presence of a room, and
certain other classes of sensible particulars related to each other in
ways constituting people, and that the sensible particulars in the
latter classes bear certain definite relations to those in the first
class of particulars. Obviously, nothing like this is clearly in the
mind of a person who would ordinarily use the original English
expression. It is clear to see in this case that a very specific state
of things is required for the truth of the analyzed proposition, and
hence the truth of it will be far more doubtful than the truth of the
vague assertion with which one began the process (*PLA*,
179-80). As Russell put the point, "the point of philosophy is to
start with something so simple as not to seem worth stating, and to
end with something so paradoxical that no one will believe it"
(*PLA*, 193). 


## 4. Ontological Aspects of Russell's Logical Atomism


### 4.1 Russellian Facts: Atomic, Negative and General



 As we have seen, the primary metaphysical thesis of Russell's atomism
is the view that the world consists of many independent entities that
exhibit qualities and stand in relations to one another. On this
picture, the simplest sort of fact or complex consists either of a
single individual or particular bearing a quality, or a number of
individuals bearing a relation to one another. Relations can be
divided into various categories depending on how many relata they
involve: a *binary* or *dyadic* relation involves two
relata (e.g., *a* is to the left of *b*); a
*triadic* relation (e.g., *a* is between *b* and
*c*) involves three relata and so on. Russell at times used the
word "relation" in a broad sense so as to include qualities, which
could be considered as "monadic" relations, i.e., relations that only
involve one relatum. The quality of *being white*, involved,
e.g., in the fact that *a* is white, could then, in this
broader sense, also be considered a relation.



 
At the time of *Principia Mathematica*, complexes in Russell's
ontology were all described as taking the form of *n*
individuals entering into an *n*-adic relation. There he
writes:



> We will give the name of "a complex" to any such object as
> "*a* in the relation *R* to *b*" or "*a*
> having the quality *q*," or "*a* and *b* and
> *c* standing in relation *S*." Broadly speaking, a
> complex is anything which occurs in the universe and is not
> simple. (*PM*, 44)



 As we have seen, at the time of writing *Principia
Mathematica*, Russell believed that an elementary proposition
consisting of a single predicate representing an *n*-place
relation along with *n* names of individuals is true if it
corresponds to a complex. An elementary proposition is false if there
is no corresponding complex. Russell there gave no indication that he
believed in any other sorts of complexes or truth-makers for any other
sorts of propositions. Indeed, he held that a quantified proposition
is made true not by a single complex, but by many, writing, "[i]f
ph*x* is an elementary judgment it is true when it
*points to* a corresponding complex. But
(*x*).ph*x* does not point to a single corresponding
complex: the corresponding complexes are as numerous as the possible
values of *x*" (*PM*, 46).



 
Soon after *Principia Mathematica*, Russell became convinced
that this picture was too simplistic. In the "Philosophy of Logical
Atomism" lectures he described a more complicated framework. In the
new terminology, the phrase "atomic fact" was introduced for the
simplest kind of fact, i.e., one in which *n* particulars enter
into an *n*-adic relation. He used the phrase "atomic
proposition" for a proposition consisting only of a predicate for an
*n*-place relation, along with *n* proper names for
particulars. Hence, such propositions could take such forms as
"*F*(*a*)",
"*R*(*a*, *b*)",
"*S*(*a*, *b*, *c*)"
(cf. *PM2*, xv). An atomic proposition is true when it
corresponds to a positive atomic fact. However, Russell no longer
conceived of falsity as simply *lacking* a corresponding
fact. Russell now believed that some facts are negative, i.e., that if
"*R*(*a*, *b*)" is false, there is
such a fact as
*a*'s *not* bearing relation *R* to *b*.
Since the proposition "*R*(*a*, *b*)" is
affirmative, and the corresponding fact is negative,
"*R*(*a*, *b*)" is false, and, equivalently, its
negation "not-*R*(*a*, *b*)" is true. Russell's
rationale for endorsing negative facts was somewhat complicated (see,
e.g., *PLA*, 211-15); however, one might object that his
earlier view, according to which "*R*(*a*, *b*)"
is false because it lacks a corresponding complex, is only plausible
if you suppose that it must be a *fact* that there is not such
a complex, and such a fact would itself seem to be a negative
fact.



 
By 1918, Russell had also abandoned the view, held at least as late as
1911 (see *RA*, 94) that qualities and relations can occur in a
complex as themselves the *relata* to another relation, as in
"priority implies diversity". Partly influenced by Wittgenstein,
Russell now held the view that whenever a proposition apparently
involves a relation or quality occurring as logical subject, it is
capable of being analyzed into a form in which the relation or quality
occurs predicatively. For example, "priority implies diversity" might
be analyzed as "(*x*)(*y*)(*x* is prior to
*y* [?] *x* is not *y*)" (*PLA*, 205-06;
for further discussion see Klement 2004).



 
Russell used the phrase "molecular proposition" for those propositions
that are compounded using truth-function operators. Examples would
include, "*F*(*a*) &
*R*(*a*, *b*)" and
"*R*(*a*, *b*)
[?] *R*(*b*, *a*)". According to Russell,
it is unnecessary to suppose that there exists any special sort of
fact corresponding to molecular propositions; the truth-value of a
molecular proposition could be entirely derivative on the truth-values
of its constituents (*PLA*, 209). Hence, if
"*F*(*a*)
& *R*(*a*,*b*)" is true, ultimately it
is made true by two atomic facts, the fact that
*a* has property *F* and the fact that *a* bears
*R* to *b*, and not by a single conjunctive fact.



 
However, by 1918, Russell's attitude with regard to quantified
propositions had changed. He no longer believed that the truth of a
general proposition could be reduced simply to the facts or complexes
making its instances true. Russell argued that the truth of the
general proposition "(*x*).*R*(*x*, *b*)"
could not consist entirely of the various atomic facts that *a*
bears *R* to *b*, *b* bears *R* to
*b*, *c* bears *R* to *b*, .... It
also requires the truth that there are no other individuals besides
*a*, *b*, *c*, etc., i.e., no other atomic facts
of the relevant form. Hence, Russell concluded that there is a special
category of facts he calls *general facts* that account for the
truth of quantified propositions, although he admitted a certain
amount of uncertainty as to their precise nature (*PLA*,
234-37). Likewise, Russell also posited *existence facts*,
those facts corresponding to the truth of existentially quantified
propositions, such as
"([?]*x*)*R*(*x*, *b*)". In the case of
general and existence facts, Russell did not think it coherent to make
distinctions between positive and negative facts. Indeed, a negative
general fact could simply be described as an existence fact, and a
negative existence fact could be described as a general fact. For
example, the falsity of the general proposition "all birds fly"
amounts to the fact that there exist birds that do not fly, and the
falsity of the existential proposition "there are unicorns" amounts to
the general fact that everything is not a unicorn. Obviously, however,
the truth or falsity of a general or existence proposition is not
wholly independent of its instances.



 
In addition to the sorts of facts discussed above, Russell raised the
question as to whether a special sort of fact is required
corresponding to propositions that report a belief, desire or other
"propositional attitude". Russell's views on this matter changed over
different periods, as his own views regarding the nature of judgment,
belief and representation matured. Moreover, in some works he left it
as an open question as to whether one need presuppose a distinct kind
of logical form in these cases (e.g., *PLA*, 224-28;
*IMT*, 256-57). At times, however, Russell believed that the
fact that *S* believes that *a* bears *R* to
*b* amounts to the holding of a multiple relation in which
*S*, *a*, *R* and *b* are all
relata (e.g. *NTF*, 155-56; *TK*, 144ff). At other
points, he considered more complicated analyses in which beliefs
amount to the possession of certain psychological states bearing
causal or other relationships to the objects they are about, or the
tendencies of believers to behave in certain ways (see, e.g.,
*IMT*, 182-83; *HK*, 144-48). Depending on how such
phenomena are analyzed, it is certainly not clear that they require
any new species of fact. 


### 4.2 Logical Atoms and Simplicity



 
Russell's use of the phrase "atomic fact", and indeed the very title
of "logical atomism" suggest that the constituents of atomic facts,
the "logical atoms", Russell spoke of, must be regarded as utterly
simple and devoid of complexity. In that case, the particulars,
qualities and relations making up atomic facts constitute the
fundamental level of reality to which all other aspects of reality are
ultimately reducible. This attitude is confirmed especially in
Russell's early logical atomist writings. For example, in "Analytic
Realism", Russell wrote:




> 
> 
> ... the philosophy I espouse is *analytic*,
> because it claims that one must discover the simple elements of which
> complexes are composed, and that complexes presuppose simples, whereas
> simples do not presuppose complexes ... 
> 
> 
>  I believe there are simple beings in the universe, and that these
> beings have relations in virtue of which complex beings are
> composed. Any time *a* bears the relation *R* to
> *b* there is a complex "*a* in relation *R* to
> *b*" ... 
> 
> 
> 
>  You will note that this philosophy is the philosophy of logical
> atomism. Every simple entity is an atom. (*RA*, 94)
> 
> 
> 



 
Elsewhere he spoke of "logical atomism" as involving the
view that "you can get down in theory, if not in practice, to
ultimate simples, out of which the world is built, and that those
simples have a kind of reality not belonging to anything else"
(*PLA*, 270). However, it has been questioned whether Russell
had sufficient argumentation for thinking that there are such simple
beings.



 
In the abstract, there are two sorts of arguments Russell could have
given for the existence of simples, *a priori* arguments, or
empirical arguments (cf. Pears 1985, 4ff). An *a priori*
argument might proceed from the very understanding of complexity: what
is complex presupposes parts. In 1924, Russell wrote, "I confess it
seems obvious to me (as it did Leibniz) that what is complex must be
composed of simples, though the number of constituents may be
infinite" (*LA*, 337). However, if construed as an argument,
this does not seem very convincing. It seems at least logically
possible that while a complex may have parts, its parts might
themselves be complex, and their parts might also be complex, and
so on, *ad infinitum*. Indeed, Russell himself later came to
admit that one could not know simply on the basis of something being
complex that it must be composed of simples (*MPD*, 123).



 
Another sort of *a priori* argument might stem from conceptions
regarding the nature of *analysis*. As analysis proceeds, one
reaches more primitive notions, and it might be thought that the
process must terminate at a stage in which the remaining vocabulary is
indefinable because the entities involved are absolutely simple, and
hence, cannot be construed as logical constructions built out of
anything more primitive. Russell did at some points describe his
logical atoms as reached at "the limit of analysis" (*LA*, 337)
or "the final residue in analysis" (*MPD*, 164). However, even
during the height of his logical atomist period, Russell admitted that
it is possible that "analysis could go on forever", and that complex
things might be capable of analysis "*ad infinitum*"
(*PLA*, 202).



 
Lastly, one might argue for simples as the basis of an empirical
argument; i.e., one might claim to have completed the process of
analysis and to have reduced all sorts of truths down to certain
entities that can be known in some way or another to be
simple. Russell is sometimes interpreted as having reasoned in this
way. According to Russell's well known "principle of acquaintance" in
epistemology, in order to understand a proposition, one must be
acquainted with the meaning of every simple symbol making it up (see,
e.g., *KAKD*, 159). Russell at times suggested that we are
only directly acquainted with sense data, and their properties and
relations, and perhaps with our own selves (*KAKD*, 154ff). It
might be thought that these entities are simple, and must constitute
the terminus of analysis. However, Russell was explicit that
sense data can themselves be complex, and that he knew of no reason to
suppose that we cannot be acquainted with a complex without being
aware that it is complex and without being acquainted with its
constituents (*KAKD*, 153; cf. *TK*, 120). Moreover, Russell continued to use the label "logical atomism" to describe his philosophy long after his epistemology had ceased to center around the acquaintance relation. (For further argumentation on these points, see Elkind 2018.) Indeed,
Russell eventually came to the conclusion that nothing can ever be
*known* to be simple (*MPD*, 123).



 
While there is significant evidence that Russell did believe in the
existence of simple entities in the early phases of his logical
atomist period, it is possible that, uncharacteristically, he held
this belief without argumentation. In admitting that it is possible
that analysis could go on *ad infinitum*, Russell claimed that
"I do not think it is true, but it is a thing that one might argue,
certainly" (*PLA*, 202). In his 1924 piece "Logical Atomism",
Russell admitted that "by greater logical skill, the need for assuming
them [i.e. simples] could be avoided". This attitude may explain in
part why it is that at the outset of his 1918 lectures on logical atomism,
he claimed that "[t]he things I am going to say in these lectures are
mainly my own personal opinions and I do not claim that they are more
than that" (*PLA*, 178). It may have been that Russell was
interested not so much in establishing definitively that there are any
absolutely simple entities, but rather in combating the widespread
arguments of others that the notion of a simple, independent entity is
incoherent, and only the whole of the universe is fundamentally
real. According to Russell, such attitudes are customarily traced to a
wrong view about relations; in arguing for the doctrine of "external
relations", Russell was attempting simply to render a world of simple
entities coherent again. Others had argued that possessing an inner complexity in the form of "nature" was needed to explain the essential properties of things or the necessary connections between them; as we shall see in Section 4.3 below, Russell believed that such apparent essential properties or necessary relations could typically be explained away.



 As his career progressed, Russell became more and more prone to
emphasize that what is important for his philosophical outlook
is not absolute simplicity, but only *relative* simplicity. As
early as 1922, in response to criticism about his notion of
simplicity, Russell wrote:



> 
>  As for "abstract analysis in search of the 'simple' and
> elemental", that is a more important matter. To begin with, "simple"
> must not be taken in an absolute sense; "simpler" would be a better
> word. Of course, I should be glad to reach the absolutely simple, but
> I do not believe that that is within human capacity. What I do
> maintain is that, whenever anything is complex, out knowledge is
> advanced by discovering constituents of it, even if these constituents
> themselves are still complex. (*SA*, 40)



 According to Russell, analysis proceeds in stages. When analysis
shows the terminology and presuppositions of one stage of analysis to
be definable, or logically constructible, in terms of simpler and more
basic notions, this is a philosophical advance, even if these notions
are themselves further analyzable. As Russell says, the only drawback
to a language which is not yet *fully* analyzed is that in it,
one cannot speak of anything more fundamental than those objects,
properties or relations that are named at that level (e.g.,
*LA*, 337).



 
In a later work, Russell summarized his position as follows:



> 
>  If the world is composed of simples--i.e., of things, qualities and
> relations that are devoid of structure--then not only all our
> knowledge but all that of Omniscience could be expressed by means of
> words denoting these simples. We could distinguish in the world a
> stuff (to use William James's word) and a structure. The stuff would
> consist of all the simples denoted by names, while the structure would
> depend on relations and qualities for which our minimum vocabulary
> would have words.
> 
>  
>  This conception can be applied without assuming that there is anything
> absolutely simple. We can define as "relatively simple" whatever we do
> not know to be complex. Results obtained using the concept of
> "relative simplicity" will still be true if complexity is afterward
> found, provided we have abstained from asserting absolute simplicity
> (*HK*, 259)
> 
> 
> 



 Russell concluded that even if there are no ultimate simples, no
fundamental layer of reality that analysis can in principle reach,
this does not invalidate analysis as a philosophical
procedure. Moreover, at a given stage of analysis, a certain class of
sentences may still be labeled as "atomic", even if the facts
corresponding to them cannot be regarded as built of fundamental
ontological atoms (*MPD*, 165). Russell concluded that "the
whole question whether there are simples to be reached by analysis is
unnecessary" (*MPD*, 123). From this vantage point, it might
be argued that Russell's "logical atomism" can be understood as first
and foremost a commitment to analysis as a method coupled with a
rejection of idealistic monism, rather than a pretense to have
discovered the genuine metaphysical "atoms" making up the world of
facts, or even the belief that such a discovery is possible
(cf. Linsky 2003; Maclean 2018). Indeed, Russell continued to use the phrase
"logical atomism" to describe his philosophy in later years of his
career, during the period in which he stressed relative, not absolute
simplicity (*RTC*, 717; *MPD*, 9).


### 4.3 Atomic Propositions, Logical Independence and Necessity



 
Another important issue often discussed in connection with logical
atomism worth discussing in greater detail is the supposition that
atomic propositions are logically independent of each other, or that
the truth or falsity of any one atomic proposition does not logically
imply or necessitate the truth or falsity of any other atomic
proposition. This supposition is often taken to be a central aspect of
the very notion of "logical atomism", perhaps largely because it is
found explicitly in Wittgenstein's *Tractatus
Logico-Philosophicus*, almost certainly the most important account
of a logical atomist philosophy found outside of Russell's
work. Wittgenstein claimed:




> 
>  4.211   It is a sign of a proposition's being elementary
> that there can be no elementary proposition contradicting
> it. (Wittgenstein 1922, 89)
> 
> 
>  
> 5.134   From an elementary proposition no other can be
> inferred. (Wittgenstein 1922, 109)
> 
> 
> 



 The lack of any logical relations between atomic propositions goes
hand in hand with a similar view about atomic *facts*; each
atomic fact is metaphysically independent of every other, and any one
could obtain or fail to obtain regardless of the obtaining (or not) of
any other.


This position also pairs well with a broader view on the nature of necessity: that all necessity reduces to features of logical form. There is nothing in the *form* of one atomic statement that would allow us to infer (or reject) another. Hence, if all necessary connections are a result of logical form, atomic facts must be independent. If what appears to be a simple subject-predicate statement is necessary, it must in fact have a more complicated form when analyzed, so that the necessity of the statement is explained by its form. Similar remarks might be suggested what appear to be simple necessary relational statements. Wittgenstein, at least, explicitly endorsed the position that all necessity is logical necessity (6.37). Some commentators also take the thesis that all necessity is logical necessity (Landini 2010, chap. 4; 2018), or the thesis that *de re* modality is to be rejected in favor of *de dicto* necessity (Cocchiarella 2007, chap. 3), to be key to Russell's logical atomism as well.



Russell's precise positions on these matters is not as clear as one
might hope and nowhere does he treat them at any
length. The few pertinent remarks he does make are either somewhat
ambivalent or seem to work against interpreting him as holding strong
versions of these theses. For example, in 1914, in arguing that
atomic facts are typically known by direct empirical means rather than
by inference, he wrote that "[p]erhaps one atomic fact may
sometimes be capable of being inferred from another, though I do not
believe this to be the case; but in any case it cannot be inferred
from premises no one of which is an atomic fact" (*OKEW*,
48). Here, Russell expressed doubt about the existence of any
relations of logical dependence between atomic propositions, but the
fact that he left it as an open possibility makes it seem that he would
not consider it a defining feature of an atomic proposition that it
must be independent from all others, or a central tenet of logical
atomism generally that atomic facts are independent from one
another. In 1936 he even went so far as to mock Wittgenstein's claims
that atomic facts are independent of one another, and that all
deductions must be formal, by claiming that "no one in fact holds
these views, and a philosophy which professes them cannot be wholly
sincere" (*LE*, 319).



 
Nevertheless, there a number of aspects of Russell's philosophical
positions that lead to the conclusion that they cohere best with
*some* doctrine about the independence of atomic facts or
propositions. Russell did often speak about the *constituents*
of atomic facts as independently existing entities. He writes for
example that "each particular has its being independently of any other
and does not depend upon anything else for the logical possibility of
its existence" (*PLA*, 203). It is not altogether clear what
Russell meant by speaking of particulars or entities as being
logically independent. In contemporary parlance, typically, "logical
independence" is used solely to speak of a relation between sentences,
propositions, or perhaps facts or states of affairs. One possible
interpretation would be to take Russell as holding that any atomic
fact involving a certain group of particulars is logically independent
of an atomic fact involving a distinct group of particulars, even if
the two facts involve the same quality or relation (see, e.g., Bell
and Demopoulos 1996, 118-19). This weakened version of the
independence thesis even has certain attractions over the stronger
principle endorsed by Wittgenstein. Most of the usual counterexamples
given against the thesis that atomic facts or propositions are always
independent involve simple properties that are thought to be
exclusive. Consider, for example, what has come to be called the
"color exclusion problem". The propositions "*a* is red" and
"*a* is blue" do not seem to be independent from one another:
from the truth of one the falsity of the other can seemingly be
inferred. However, the weakened version of the independence principle,
on which only atomic facts involving *different* particulars
are independent, does not entail that it is possible that "*a*
is red" and "*a* is blue" may both be true.



 
It is likely that Russell's contention that particulars are
independent from one another was connected in his mind with his views
on relations. In holding the view that relations among simple
particulars are external, Russell saw himself as denying the view that
when *a* bears *R* to *b*, there is some part of
*a*'s "nature" as an entity that involves its relatedness to
*b*. It might be thought that Russell's doctrine of external
relations committed him at least to certain principles regarding the
modal status of atomic facts (if not the independence principle).
According to certain ways of defining the phrase, what it
*means* for a relation to be *internal* is that it is a
relation that its relata *could not* fail to have; an
*external relation* is one its relata could possibly not
have. Russell then might be seen as committed to the view that atomic
facts (all of which involve particulars standing in relations, in the
broad sense above) are always contingent. While this does not directly
bear on the question of their independence, it would nevertheless
commit Russell to certain tenets regarding the modal features of
atomic facts.



 
However, Russell himself warned against interpreting his position on
relations this way, writing, "the doctrine that relations are
'external' ... is not correctly expressed by saying
that two terms which have a certain relation might not have had that
relation. Such a statement introduces the notion of possibility and
thus raises irrelevant difficulties" (*BReal*, 87).
Complicating matters here are Russell's own rather idiosyncratic and
skeptical views about modal notions. Russell was dissatisfied with the
prevailing conceptions of necessity and possibility among philosophers
of his day, and argued instead against necessity (or possibility) as a
fundamental or irreducible concept (see *NP passim*).



 
Despite Russell's misgivings about modal notions, it is clear enough
from Russell's conception of *logic* that logical relations
between propositions would always obtain in virtue of their form
(*IMP*, 197-98; *PLA*, 237-39). Again, atomic propositions are
of the simplest possible forms, and there is certainly nothing in
their forms that would suggest any logical connection to, or
incompatibility with, other atomic propositions.



 
Perhaps the most illuminating remarks to be found in Russell's work
that would lead one to expect complete logical independence among
atomic propositions involve the claims he made about how it is that
one recognizes a certain class of purported entities as "logical
constructions", and the recommendations he gives about analyzing
propositions involving them. Russell writes:



> 
>  When some set of supposed entities has neat logical properties, it
> turns out, in a great many instances, that the supposed entities can
> be replaced by purely logical constructions composed of entities which
> have not such neat properties. In that case, in interpreting a body of
> propositions hitherto believed to be about the supposed entities, we
> can substitute the logical structures without altering any detail of
> the body of propositions in question (*LA*, 326).



 Russell did not define here what he means by "neat logical
properties", but it is possible to understand what he had in mind by
way of the examples he gave. He cited as "neat properties" of material
objects that it is impossible for two material objects to occupy the
same place at the same time, and that it is impossible for one
material object to occupy distinct places in space at the same time
(*LA*, 329; cf. *AMi*, 264-65; *AMa*,
385). Consider then the propositions "*O*1 is
located at *p*1 at *t*1", and
"*O*1 is located at *p*2 at
*t*1" where "*O*1" is the name of
a physical object, "*p*1" and
"*p*2" represent distinct locations in space, and
"*t*1" the name of a certain instant in time. Prior
to analysis, such propositions appear to be logically incompatible
atomic propositions. However, Russell explains that the logical
necessities involved in cases such as these are due to the nature of
material objects, points and instants as logical constructions. At a
certain point in time, a physical object might be regarded as a class
of sensible particulars bearing certain resemblance relations to one
another occupying a *continuous* region of space. It is
therefore impossible *by definition* for the same physical
object to occupy wholly distinct locations at the same time. When
analyzed, such propositions as "*O*1 is located at
*p*1 at *t*1" are revealed as
having a much more complicated logical form, and hence may have
logical consequences not evident before analysis. We do not have here
any reason to think that truly atomic propositions, those containing
names of genuine particulars and their relations, are not always
independent.



 
Russell strongly intimated that it is a part of the very nature of
logical analysis that if our pre-analyzed understanding of a certain
phenomenon involves the postulation of entities with certain
structural or modal properties, one should seek to replace talk of
such entities with logical constructions specifically constituted so
as to have these features by definition (*PLA*, 272-79;
*LA*, 326-29). A logical construction would typically be
understood as a sort of class; since discourse about classes was
regarded by Russell as a convenience, which would be eliminated in a
fully analyzed language in favor of speaking of their defining
properties and relations, by such a process Russell believed it is
possible to replace commitment to entities having "neat logical
properties" with commitment to those that do not possess such
features. Russell's work on mathematics provides us with what he would have taken as many examples of this phenomenon. If we take "3 > 2" and "3 < 2" at face-value as expressing simple dyadic relations between simple entities *named* "2" and "3", then these appear to be two atomic propositions, one necessary, one impossible, which are not independent of each other. But after analysis, "2" and "3" are revealed not to be names at all, and the necessity, impossibility, and mutual incompatibility of these statements is rendered purely logical.
Assuming that there is a final terminus of analysis in
absolutely simple entities and fully atomic facts, one might suppose
that here the logical necessities and relationships between them
would have completely disappeared.


Russell later summarized the
attitude of his logical atomist period by writing that "it seemed to result that none of the raw material of the world has smooth logical
properties, but that whatever appears to have such properties is
constructed artificially in order to have them" (*IPOM*,
xi). While this is not exactly an endorsement of the claim that atomic
facts are logically independent of one another, it is perhaps the
closest sentiment one can find in his philosophy. It is again perhaps better understood as an endorsement of a methodological maxim. If a certain stage of analysis seems to portray entities as having "neat" logical properties, i.e., necessary features or relations not explained by logical form, this is a sign that more analysis is needed. Perhaps by analyzing the statements further, "busting open" the apparent terms for these things to reveal more complicated logical forms, what appear to be necessary connections between atomic facts will be shown actually to be logically necessary relationships between non-atomic facts owed to their logical forms (cf. Elkind forthcoming).


## 5. Influence and Reception



 Russell's logical atomism had significant influence on the
development of philosophy, especially in the first half of the 20th
century. Nowhere is Russell's influence more clearly seen than in the
work of his pupil Ludwig Wittgenstein. Wittgenstein's *Tractatus
Logico-Philosophicus* appeared in 1921; in it, Wittgenstein
presented in some detail a logical atomist metaphysics. (It should be
noted, however, that there is significant controversy over whether, in
the end, Wittgenstein himself meant to endorse this metaphysics.) In
the *Tractatus*, the world is described as consisting of
facts. The simplest facts, which Wittgenstein called
"*Sachverhalte*", translated either as "states of affairs" or
"atomic facts", are thought of as conglomerations of objects combined
with a definite structure. The objects making up these atomic basics
were described as absolutely simple. Elementary propositions are
propositions whose truth depends entirely on the presence of an atomic
fact, and other propositions have a determinate and unique analysis in
which they can be construed as built up from elementary propositions in
truth-functional ways. 



 
Partly owing to Wittgenstein's influence, partly directly, Russell's
logical atomism had significant influence on the works of the logical
positivist tradition, as exemplified in the works of Carnap, Waismann,
Hempel and Ayer. This tradition usually disavowed metaphysical
principles, but methodologically their philosophies owed much to
Russell's approach. Carnap, for example, described philosophy as
taking the form of providing "the logical analysis of the language of
science" (Carnap 1934, 61). This originally took the form of
attempting to show that all meaningful scientific discourse could be
analyzed in terms of logical combinations beginning with "protocol
sentences", or sentences directly confirmable or disconfirmable by
experience. This notion of a "protocol sentence" in this tradition
was originally modeled after Russellian and Wittgensteinian atomic
propositions. The notion of a "logical construction" was also
important for how such thinkers conceived of the nature of ordinary
objects (see, e.g., Ayer 1952, chap. 3). The view that scientific
language could readily and easily be analyzed directly in terms of
observables gradually gave way to more holistic views, such as Quine's
(see, e.g. Quine 1951), in which it is claimed that it is only a body
of scientific theories that can be compared to experience, and not
isolated sentences. However, even in later works growing out of this
tradition, the influence of Russell can be felt.



 
Besides positive influence, many trends in 20th century philosophy can
be best understood largely as a reaction to Russell's atomistic
philosophy. Ironically, nowhere is this more true than in the later
writings of Wittgenstein, especially his *Philosophical
Investigations* (1953). Among other things, Wittgenstein there
called into question whether a single, unequivocal notion of
simplicity or a final state of analysis can be found (e.g.,
secs. 46-49, 91), and questioned the utility of an ideal language
(sec. 81). Wittgenstein also called into question whether, in those
cases in which analysis is possible, the results really give us what
was meant at the start: "does someone who says that the broom is in
the corner really mean: the broomstick is there, and so is the brush,
and the broomstick is fixed in the brush?" (sec. 60). Much of the work
of the so-called "ordinary language" school of philosophy centered in
Oxford in the 1940s and 1950s can also been seen largely as a critical
response to views of Russell (see, e.g., Austin 1962, Warnock 1951,
Urmson 1956). 



 
Nevertheless, despite the criticisms, many so-called "analytic"
philosophers still believe that the notion of analysis has some role
to play in philosophical methodology, though there seems to be no
consensus regarding precisely what analysis consists in, and to what
extent it leads reliably to metaphysically significant
results. Debates regarding the nature of simple entities, their
interrelations or dependencies between one another, and whether there
are any such entities, are still alive and well. Russell's rejection
of idealistic monism, and his arguments in favor of a pluralistic
universe, have gained almost universal acceptance, with a few
exceptions. Abstracting away from Russell's particular examples of
proposed analyses in terms of sensible particulars, the general
framework of Russell's atomistic picture of the world, which consists
of a plurality of entities that have qualities and enter into
relations, remains one to which many contemporary philosophers are
attracted.