[lit-ideas] Re: Logical Corpuscularism

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  • Date: Mon, 21 Sep 2015 05:39:29 -0400

Was Grice a logical corpuscularian? It seems so. After all, he wrote an
essay, "Definite descriptions in Russell and in the vernacular", by
vernacular meaning Strawson; and while Grice agrees that both Russell and
Strawson make a 'common mistake', Grice feels he has a foot on each camp
(Russell's and Strawson's): Grice's manoeuvre is to re-interpret what Russell

saw as an entailment (and Strawson ignored) as an implicature.
When Myro published (rather posthumously published) his "Logic" he makes
much of the atomic-molecular distinction as applied to propositions, and
it's noteworthy that Grice's claim to fame (among logicians) was his
implicatural analysis of two-place operators, that make up for MOLECULAR
propositions. So a purist could say that Grice was the greatest
molecularist of all. But surely a molecule is composed of corpuscules, too.
Why did Russell describe his philosophy as a kind of "logical atomism"
and not the more correct 'corpuscularism'? After all, Russell just meant
to endorse both a metaphysical view and a certain methodology for doing
philosophy. The metaphysical view amounts to the claim that the world
consists of a plurality of independently existing things exhibiting
qualities
and standing in relations. But as Grice says, every school boy (at least
at Clifton, which he attended) knows that. "There must be more to
Russell's logical atomism." According to logical atomism, all truths are
ultimately dependent upon a layer of atomic facts, which consist either of a
simple particular exhibiting a quality, or multiple simple particulars
standing in a relation.
The methodological view recommends a process of conceptual analysis,
whereby one attempts to define or reconstruct more complex notions or
vocabularies in terms of simpler ones.
According to Russell, at least early on during his logical atomist
phase, such an analysis could eventually result in a language containing
only
words representing simple particulars, the simple properties and
relations thereof, and logical constants, which, despite this limited
vocabulary,
could adequately capture all truths. Russell's logical atomism had a
profound influence on analytic philosophy, including Grice, if only to
criticise it. Ideed, it is arguable that the very name “analytic philosophy”

derives from Russell's defense of the method of analysis. And while
Austin used to say

i. Some like Witters, but Moore's MY man.

Grice could have said:

ii. Some like Witters, but Russell's MY man.

Trust Ayer, who was an 'enfant terrible' to attempt to trump both:

iii. Some like Witters, but Moore and Russell are MY men.

(vide Ayer: Russell and Moore: the anaytical heritage"). Russell
introduced the phrase “logical atomism” to describe his philosophy in 1911.
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 conceptual 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.
(And Grice says Russell is "okay" only he misses to even THINK such a
logically ideal language should invite this or that implicature. Russell
impertinently responded,
"Implicature happens, so?").

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".

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 himself dated his first acceptance of logical atomism to the
vintage year of 1899 (what Grice called "that year of Grice") when he and
Moore
rejected the main tenets of the dominant school of philosophy in Oxford
at the time -- oddly, since both were at Cambridge -- (and to which both
had previously been adherents), the tradition of neo-Hegelian Idealism
exemplified in works of F.H. Bradley, and adopted instead a fairly
strong form
of realism. Of their break with idealism, Russell wrote that "Moore led
the way, but I followed closely in his footsteps".

**************** WHY NEITHER RUSSELL NOR GRICE LIKED BRADLEY -- "MUCH"
------

Moore published an essay 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". 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. 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's "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. 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". Perhaps most charitably interpreted, 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.
This work, along with his earlier work on the foundations of geometry,
had
convinced him of the importance of relations for mathematics. However,
Russell 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

iv. 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. 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-quâ-father-of-b in the first,
and
a-quâ-son-of-c in the second. In contradistinction to these views,
Russell
adopted what he called “the doctrine of external relations”, which he
claimed “may be expressed by
saying that

A) relatedness does not imply any corresponding complexity in the relata.

B) any given entity is a constituent of many different complexes.

This position on relations allowed Russell to adopt a pluralism 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. Russell claimed that this doctrine
was the fundamental doctrine of his realistic position, and it represents
perhaps the most important turning point in the development of his
logical
atomism. Russell's first published account of his newfound realism came
in
the classic The Principles of Mathematics. Part I 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.
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 by Russell, 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

v. Russell is human (versus "Russell is humane"?)

the person Russell (the man himself) occurs as term, but humanity occurs
as concept. In the proposition

vi. Strawson orbits Grice.

Strawson and Grice occur as term, and the relation of orbiting occurs as
concept. Russell used "concept" for all those entities capable of
occurring as concept — chiefly relations and other universals — and
"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

vii. Gluttony is a big vice.

the concept gluttony 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. Russell's 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. 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.
He
is committed to a special kind of propositional constituent called a “
denoting concept”, involved in descriptive and quantified propositions.
He
believes 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. He
also
believes 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. 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.
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. B
ecause 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

viii. All numbers are odd.

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

ix. (x)(x is a number ⊃ x is odd)

Russell held that they are nevertheless distinct propositions. 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 the above contains as a
constituent the denoting concept all numbers. Russell explained denoting
concepts
as entities which, whenever they occur
in a proposition, the proposition is not about them but about other
entities to which they 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. Russell
abandons this
theory in favour of his celebrated theory of definite and indefinite
descriptions. 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. 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". According to the new theory adopted, the proposition expressed
by the
above was now identified with that expressed by a quantified conditional

such as the formalised version. Similarly, the proposition designated by

x. Some number is odd.

was identified with the existentially quantified conjunction represented

by

xi. (∃x)(x is a number & x is odd)

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

xii. The King of France is not bald.

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

xiii. (∃x)(x is King of France & (y)(y is King of France ⊃ x = y) & x
is
not 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. Russell's 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:

xiv. The author of Waverly = Scott

would be understood as having the following structure:

xv. (∃x)(x authored Waverly & (y)(y authored Waverly ⊃ x = y) & x =
Scott)

If instead, the proposition corresponding to the above 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

xvi. Scott = Scott.

(but, as Grice said, "What conversational POINT could THAT have?") By
showing that the actual structure of the proposition is quite a bit
different
from what it appears from the grammar of the sentence

xiv. The author of Waverly = Scott.

Russell believed he had shown how it might be more informative (or
'stronger' as Grice prefers -- The Causal Theory of Perception, 1961,
repr. in
WoW, Way of Words) than a trivial instance of the law of identity, which

intelligent people like Grice or Russell are supposed to KNOW already.
The
theory 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 Russell's theory, phrases such as "the King of France," or
"the
author of the Waverly novels" were to be understood as incomplete symbols

in this way. 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. Russell postulates two
types of
composite entities: unities and aggregates. By a unity Russell 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, Russell means 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, with
Whitehead, of Principia Mathematica 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 The Principles of Mathematics he
claims that a
class may be defined as all the terms satisfying some propositional
function.
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. (Grice was to joke
on this
calling Austin's Play Group, to which he belonged, as "the class of
tutors that have no other class"). 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 2n, and also that
2n >
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 some time searching for a philosophically motivated
solution
to such paradoxes. He tried solutions of various sorts. However, after
the discovery of the theory of descriptions, Russell becomes 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
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.

However, in the version adopted in the first edition of Principia
Mathematica, Whitehead and 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

xv. {z|ψz}

appeared in some sentence, the sentence as a whole could be regarded as
defined as follows:

xvi. f({z|ψz}) =df (∃φ)((x)(φ!x ≡ ψx) & f(φ))

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. 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 (This irritated Grice -- and Hart, "A logician's fairy
tale"). Another way Russell expressed himself is by saying that a class
is a
logical FICTION, an expression he borrowed from Bentham, but never
returned. 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 façon 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,
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. 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

xvii. 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 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. Whatever his primary motivation, Russell
abandons any commitment to objective falsehoods, and restructured his
ontology of facts, and adopted a new Tarski-type correspondence theory
of
truth (also endorsed by Grice just to oppose Strawson's naive
performative
'ditto' theory of 'true').

In the terminology of the new theory, "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, Whitehead Russell described various notions of truth
applicable to different types of propositions of different complexity.

Grice made fun of this when he used the example

xviii. My neighbour's three-year-old child understands Russell's Theory
of

Types.

("Unbelievable, but hardly logically contradictory" -- the implicature
was that only Russell understood his own theory of types). 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”. 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.,

xix. 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

xx. (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

xx. (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.

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 "logical atomism" in the Introduction to
Principia Mathematica, in many ways it represents the first work of
Whitehead's
and Russell's atomist period.

Whitehead and Russell there explicitly endorsed the view that the
universe

consists of objects having various qualities and standing in various
relations.

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.

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.

Russell employed the methodology self-consciously, and gave only slightly

differing descriptions of this methodology in works throughout his
career.

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 "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 is 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 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, 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,
and
thereby eliminates the contradictions.

Cheers,

Speranza
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