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hilosophy cience
VOL. 28
April,
I
96
I
NO. 2
AN INTRODUCTION TO SIMPLICITY
RICHARD
RUDNER
Michigan
State University
The four papers
which follow this introductory essay were presented
at
the 1960 meetings of the A.A.A.S. in New York City. They were given at
a session
devoted to
the topic of Simplicity
of Scientific Theories,
and
spons-
ored jointly by Section
L and the Philosophy of Science Association.
For
good reasons,
each of the papers appears in this issue essentially
in
the form in
which it was offered at the session.'
Each of the papers was written
independently of the others-indeed, the
occasion of the meeting was the
first time any of the authors had access to
all of the other papers. None of
the
essays, therefore,
evidences the luxury, afforded the present
introductory
remarks,
for
reflection on the other papers
of the session,
or for response
to the
lively
discussion
which followed
the reading
of the
papers,
or for
second thoughts in general.
In this light, and
considering the breadth
of interpretation which
has
traditionally marked
the notion of Simplicity, the relevance
which
each
of
the
papers
has for the others is a relatively fortuitous
but nonetheless happy
circumstance for which we must be grateful.
I shall comment briefly
below
on the
relationships which I find noteworthy
among
the
essays.
I
think,
however, that it will
be more appropriate to begin by focusing attention
on
the
importance
of contemporary treatments of Simplicity
and
on what
appears
to be a
pivotal distinction
between two of the relevant
senses
in which
the
term is
currently being
used.
Whatever may be the case for the serenity or unselfconsciousness with which
practicing scientists go about the business
of accepting or rejecting
theories,
it will
surely not be
denied that the problem of constructing
an adequate
philosophical rationale
for such practice remains in its perennial
state
of
crisis.
The recent
past has
witnessed monumental
and illuminating attempts (such
as those
by Reichenbach
and
Carnap)
to provide
that
rationale
essentially
in
the form of a logic
of induction. For the purposes of our present
concern
it is
not necessary to
rehearse considerations
of the cogency
of the
objections
1
Owing
to
its
length
it
was necessary for Professor
Bunge to actually
read
a somewhat
com-
pressed version of the paper he contributed. The complete version is published here.
109
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RICHARD S.
RUDNER
that
advocates of objective
or statistical
theories of
induction have hurled
against advocates
of logical
theories
of
confirmation or, conversely, to
consider the objections which advocates of the latter view have hurled against
those
of the
former.
It
is
not even necessary
to
go
into the
arguments which
partizans of neither
of these points of view
have
leveled
against both, nor
into those which
partizans of either
have leveled
against
such third forces
as
recent theories
of Subjective probability.
The
unhappy fact is that in
the matter of cogent objections against
theories of inductive
inference, the
recent
philosophical
literature provides
us
with an
embarrassment of riches.
The reason that none
of
these
considerations
need
detain
us for
the present,
however, is to be found
in
the
fact that
even if
any
of
the types of programs
for inductive logic mentioned above
could
be
brought
to the successful
consummation its proponents apparently envisage, we would still not have
been provided with a complete
or
general
basis
for
choice among theories.
There are weights other than
that
of
evidential
strength
whose assessment
is a necessary condition
for
rational
(i.e.
scientifically reliable) choice among
hypotheses.2 One of these
additional
weights
we
may
refer to as the cost
associated with the acceptability
of
any
hypothesis;
and
philosophers
and
many
scientists
(e.g.,
some
who are concerned
with
Decision
Theory)
have
in
recent
years
come to
give
the
explication
of this
notion
something
like the
attention
it has
always
warranted.
Whatever
the
importance
or
the
poignancy
of
the
problems which
attend the explication
of
cost, however,
our concern here
is not with it but with still a third weight whose explication is also a necessary
condition for
the achievement
of
an
adequate
theory
of
inductive
inference: I
refer,
of course,
to
simplicity.
Now,
allusions
to
simplicity
in
the
literature
of Science and of
Philosophy
are innumerable
and
immensely
varied
in intent and
nuance;
and
before
any
fruitful consideration
of
the topic
or its
importance
can be
undertaken
it
is
necessary
to delimit
to
some
extent
the
range
of
our
attention. This can
be
accomplished by
fitting,
with
a
minimum
of
procrustean ferocity,
all of the
varied
references
to
simplicity
which we are
heir to
under
a
relatively
un-
complicated
classificational
schema.3 Uses
of
'simplicity' tilen, may
be
classified either as
Ontological (i.e., extra-linguistic)
or Descriptional (i.e.,
linguistic). Sub-classifications
under
these main rubrics are
Subjective (i.e.,
psychological)
and
Objective (i.e.,
non-psychological). Moreover,
under the
rubric, Descriptional,
it
is also
fruitful to
distinguish
Notational and
Logical
(or Structural)
as further subclassifications.
A few
examples
will
be sufficient
2
In
making
this claim
I
do
not,
of
course,
intend
to minimize the
importance
of the attainment
of an adequate
measure
of evidential
strength
for
hypotheses
as a
desideratum of
Philosophy
of Science.
I have urged
more fully
than is
appropriate
here
my
conclusions about
the
insuffi-
ciency of any measure
of evidential
strength in [23]
and [24],
and more
recently
in a
paper,
The
Reducibility
of
Types
of Weights
in the
Acceptance
of Scientific Theories: Evidence, Cost,
and Formal Simplicity, read in Stanford at the 1960 Meetings of the International Congress for
Logic,
Methodology
and Philosophy
of
Science.
3
The one
we are about
to
employ
is
suggested
though
not in
precisely
this form in
Chapter
1
of
Dr. Ackermann's searching
thesis,
[1].
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RICHARD
S. RUDNER
since
our interest is not
in how people psychologically respond
to
logical
properties
of theories,
we
may
characterize
our field of
attention
as Objective-
Logical Simplicity.
Hereafter
in these comments
references
to simplicity,
unless otherwise
qualified,
are
intended
as references
to
Objective-Logical
Simplicity.
Realization
of the
importance
of considerations
of simplicity
for
the Philos-
ophy
of
Science
is a phenomenon
of the relatively
recent past.
This is not
altogether
surprising
in
view of the
fact
that advances
in
Logic, upon
application
of
which much
of the significant
work accomplished
has
depended,
are themselves
phenomena
of this century.
Despite
the
importance of
achieving
an adequate explication
of the
concept, sustained
and
significant
work
on its
accomplishment
has thus
far been
undertaken
by only
a relatively
small
circle
of philosophers. In the quite recent past this circle has slowly widened as
interest
in the problem
has
come to be
quickened
or inspired
under
the impetus
of the
positive
and
detailed results
achieved
especially
by Professor
Goodman.
In
any case,
however
slowly
launched,
work by
an increasing
number
of able
men,
is now
under
way
and
we can look
forward
with hopeful
excitement
to the solution
of
problems
about simplicity
which
once appeared
well
nigh
insuperable.
Perhaps the
importance
of
attaining an
adequate
explication
of simplicity
can best
be
indicated by pointing
out some aspects of
its connection
with
systematism.
On
this
score,
Goodman's
opening
remarks
in a recent
article
are as illuminating and pithy as any which have been made on the topic:
All scientific
activity
amounts to the
invention
of and the choice among
systems
of
hypotheses.
One of
the primary considerations
guiding this process
is
that of simplicity.
Nothing
could
be
much
more
mistaken
than the traditional
idea that we
first
seek
a
true system
and then,
for the sake of elegance
alone, seek
a
simple
one. We
are inevitably
concerned
with simplicity as soon
as
we
are concerned
with system
at
all;
for
system
is achieved just
to
the extent
that
the basic vocabulary
and set
of
first principles
used
in
dealing with
the given subject
matter
are
simplified.
When
simplicity
of basis vanishes
to
zero-that is,
when
no term
or
principle
is derived
from
any
of
the
others-system
also
vanishes
to zero. Systematization
is the
same thing
as
simplification
of
basis.
Furthermore,
in the choice
among
alternative
systems,
truth and
simplicity
are
not
always clearly
distinguishable
factors.
(p.
1064, [12]).
System is
no mere
adornment
of
Science,
it is its
very
heart.
To
say
this
is
not merely
to
assert
that
it
is not the business of Science
to
heap up
unrelated,
haphazard,
disconnected
bits
of
information,
but to
point
out
that
it
is an
ideal
of science
to
give
an
organized
account of the
universe-to
connect,
to
fit
together
in
logical
relations
the
concepts
and
statements
embodying
whatever
knowledge
has been
acquired.
Such
organization
is,
in
fact, a necessary condition for the accomplishment of two of Science's chief
functions:
explanation
and
prediction.
The
work
that has
been
done,
and
the
work
currently
being
done
so
far
as
it is
manifest,
on
objective-formal
simplicity
cannot
plausibly
be
viewed
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RICHARD
S. RUDNER
of
systemic
simplicity
would
have an
obvious focus on
the
simplicity
properties
of sets of
postulates. Thus, a
normal first
impulse
might be to
say that of
two
otherwise equally
adequate theories the
one with the
fewer
postulates
was
objectively the
simpler. But
little
reflection is needed
to show
both that
this suggestion
is
unhelpful and
also that its
very lack
of promise
leads naturally
to consideration of
the
simplicity of a
theory's set
of primitive
predicates.
For, the finite number
of
postulates of any
theory
can be
trivially reduced
to
1
by the
simple
operation of
conjunction.
By the
criterion of
number of
postulates
every theory
would
be
equivalent to some
theory which
was
maximally
simple.
Nor would
it
be possible
to ameliorate
this
unwelcome
result by
any evident
stipulation
regarding
the number of
conjuncts in a
set
of
postulates. For
if
the
import
of
such a
stipulation is, for
example, that a
postulate whose form is
(1) f*gX
is
less
simple
than a
postulate
whose
form is
(2)
hx
then the
defectiveness of that
stipulation
becomes clear as
soon as it
is realized
that
it
is
always
trivially possible to
construct, i.e.,
to
define or
explicate a
predicate,
h, such that
(3)
hx=-
(f-
g$)
will be logically true. Accordingly, any postulate of finitely many conjuncts
is
trivially reducible to
a
postulate
of
one
conjunct
and
by
such
a
criterion
all
postulates
must be
regarded
as
equally simple.
Even this
unelaborate
example
indicates that to
get
at a
relevant
sense of
'simplicity'
we must
go
beyond
considerations of
the number
of,
or
gross logical
structure
of,
postulates
and come to
grips
with
the
logical structure
of
the
predicate
bases of
theories.
Since
it is
plausible
to assume that
the theories
we are
interested
in
all
share a common
logical
apparatus
this
means that
attention turns to
the formal
simplicity
of the
extra-logical predicates.
And
this
is,
indeed,
the route
which
Goodman follows.
In
the
course of
several
years
of work and
through
a
process
of increasingly successful modifications he has been able to construct a
calculus
of
predicate
simplicity
which
provides
a measure of
the
simplicity
of
predicate
bases
of
every
relevant
logical
kind.5
In
general,
and
necessarily
vaguely,
Goodman's
assignments
of
simplicity
values
may
be
thought
of as
depending
on
the manner in
which the
extra-logical
predicates
of a
theory
organize, by
virtue of such of
their
logical properties as
reflexivity
or
symmetry,
the
entities
comprising
the total extension of
the
theory.
In
coming to understand
the
import
of
Goodman's
work it is
especially
important
to avoid
a confusion
(not
always
avoided
by
earlier
commentators
on his
work (see [1], [20], [25], and again especially [9]) between the simplicity
of a basis and its
power.
The sets of
predicates
of two
systems,
S and S'
5
For an
explanation of the crucial
notion of relevant kind, see
[12],
and
especially
[9].
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AN INTRODUCTION
TO SIMPLICITY
115
are equally
powerful
if the sets are interdefinable.
Suppose that no predicate
of the set in S is defined
by any other in S. If the
power of a
basis were the
same thing as its simplicity,
no simpler basis for
S .
. .
[could
be arrived
at than] . .
.
by taking
all the predicates of S as primitive
(p. 430
[9]). But,
it is precisely
the greater simplicity of
an S' whose primitive basis
is narrower
(i.e., whose
basis systematizes through
defining the remainder of
the predicates
by a subset of the total
number in the system,)
over an S whose
basis is the
widest possible which
we desire to measure.
In the last analysis what
we
are after is the economy
of a system; and just as
we get
an indication of the
economy of an automobile
not from
its having gone a certain
distance but
from how
much gasoline it requires
to go that
distance, so too with the
economy
of systems. The power of a
system is strictly analogous
to the distance
driven of our car in that knowing it alone will not give us a measure of
economy.
To arrive at the economy
of a system we require also
some measure
of the simplicity of its basis-and
it is this that Goodman's calculus
attempts
to
provide.
What
has been said above
must here
suffice as
an
indication of
the sort
of
concern
that
the
topic
of
simplicity
of
predicate
bases
involves.
In
connec-
tion
with
inductive simplicity
I
shall not linger on
any exposition since three
of
the
four panel papers which
follow scrutinize the major work
done on it.
Perhaps,
however, some further
light may
be
thrown
on
its
relationship
to
predicate
simplicity and
the propriety with
which it has been placed
in the
category of Objective-Logical Simplicity, by the following considerations.
First,
as Dr. Ackermann points
out
in his
paper
(and
in
more
complete
detail
in [1]), the concept of inductive
simplicity as
elaborated by Jeffreys,
Popper,
and
Kemeny
comes to
depend
on
some
such
notion as
the
number
of
freely adjustable parameters
which occur
in
alternative
hypotheses.
Now,
Ackermann's discussions have
revealed
very grave
defects
in
these
specific
treatments of
inductive
simplicity
on,
so to
speak,
their
own
grounds. But,
the
viability
of some
specific
treatment
is not what
is at issue
here.
Even
if
such treatments were
otherwise
wholly successful,
it
is
doubtful
that
they
would furnish
any
tenable
criterion
or test of
simplicity-especially
of formal
simplicity. Thus, for example, there seems to be no good reason for believing
that
a
hypothesis
with
n
+
1
parametric expressions
is less
simple
in the sense
of
Goodman's
calculus
than a
hypothesis
with
n
parametric
expressions.
Goodman's calculus
of
simplicity
is,
of
course,
not
even
applicable
to
hypo-
theses.
Moreover,
the obvious
suggestion
to
classify
one
hypothesis
as
simpler
than another
if
the sum
of
the
complexity
values of its
predicates
is
less than that
of the other
is,
on
a
little
reflection,
seen to
be of
no
avail.
Apart
from
the
fact
that the sets
of
predicates,
which
are constituents
of
each
of
the alternative
hypotheses
to be
assessed,
will
not
in
general
be
identifiable
with sets of primitive predicates of theories,
there
are
perhaps
more decisive
reasons
for the failure
of the
suggestion.
For one
thing,
it
will
not in
general
be the
case that
the set
of
predicates
from the
hypothesis
of n
+
I
parameters,
will have
a
higher
complexity
value
than sets
of
predicates
from alternative
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116 RICHARD
S. RUDNER
hypotheses of less than
n + I parameters.
This will depend among other
things, on whether or
how some of the
predicates in such sets are definable
by others in the same
set. Thus, the criteria urged by
proponents of inductive
simplicity will yield
judgments
in
conflict with what might
be thought of
as
the obvious application of Goodman's
concept in the same
situations.
Of
course,
all
that this indicates is that
the two concepts6 are
not identical;
and
this fact would
probably not
discomfort the proponent of inductive
simplicity. He would
very likely maintain
that he had always been aware his
technique measured
some other type of
simplicity than simplicity of predicate
bases-one whose
assessment is, nevertheless, at least equally
important. But
if
some other kind of
simplicity, then what
kind? Ontological simplicity surely
is not at issue here; and
if it were, could
scarcely be defended. Again, despite
our initial characterization of inductive simplicity as falling within the category
of
Formal Simplicity, it
is puzzlingly difficult to make out
just what formal
or
logical properties
of
hypotheses, i.e.,
statements,
are involved.
If
con-
siderations
of
gross logical structure (such
as those discussed
earlier
in
connection
with
postulational simplicity) are
at issue,
then
we
are
at
once driven
to
the formal structure
of the constituent predicates. And
here
the
logical
characteristics, other
than those indexical of simplicity in Goodman's
sense,
seem
to
have relevance
to such measures as power rather than
simplicity.
On
the other
hand,
if
the
relevant formal properties of hypotheses
are construed
as those
having to do with their logical
strength, then Goodman's
and Barker's
criticisms (and especially the counter-examples adduced by the former,) in
the
papers which follow show decisively
that it is a mistake both to identify
degree of logical
strength of a hypothesis with simplicity and
also to advocate
the
choice of the
simplest hypotheses in
the relevant situations.
Goodman's arguments and especially
his suggestion that
simplicity
of
hypotheses is associated with their
projectibility or the entrenchment of
their
predicates, are illuminating and
stimulating. Yet, the
suggestion,
if
cogent,
seems to me
only to
show that in situations, to which some
proponents
of
inductive simplicity have addressed
themselves, projectibility, or predicate
entrenchment, rather than the criteria
adduced by those proponents will
order
our selection of hypotheses. What is not shown is that either projectibility
or
entrenchment
are
identical with, or even reliable indices of,
simplicity
in
any
tenable
sense. To be sure, one might
take the course
of
just identifying
entrenchment with simplicity in some
special sense, but in
the absence
of
any independent
evidence for supposing
that degree
of entrenchment
is a
function
of
the
simplicity, in any plausible sense, of the predicates
entrenched,
this
would
seem to be
merely a way of undesirably
trivializing
the
entire
problem.
In
point of
fact, Goodman's explication of entrenchment
(see [13],
Chapter 4), gives us no
reason to suppose any
such
connection.
All of
these considerations persuade me
that whatever
the
cogent
criteria
may
be
for
ordering our selection of hypotheses
in
those situations
(e.g.,
6
I
am
assuming
here that
the
proponents
of inductive
simplicity
have been after some one
concept-an assumption
I
shall
shortly question.
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AN INTRODUCTION TO SIMPLICITY
117
curve fitting ) that
have hitherto been
held to involve
inductive simplicity,7
it would be misleading to classify
them
as
considerations of
Objective-Logical
Simplicity.
One
final
category remains whose involvement
might warrant the claim
that objective measures
of simplicity are being sought by proponents of
inductive simplicity. We have not thus
far discussed Objective-Notational
Simplicity-but here too, such an obvio-us
objective characteristic of a formula-
tion as its length, seems whole uninteresting
and unequal to the burden any
portentous theory of inductive simplicity would
make it bear. Patently, with
the relevant qualifications, a hypothesis
with n parametric
expressions is
shorter
than one with
n + 1. Again, patently, it seems sensible to
speak of such
shorter hypotheses
as being objectively notationally more simple
than longer
ones. But (as Goodman points out) any hypothesis is trivially reducible to
one
of minimal length,
i.e., may be formulated
in as brief notational compass
as
any other;
so
that
on this criterion all hypotheses become
in
effect
objectively
equally simple.
And,
in any case, perusal of the literature really
precludes
the belief
that what
the proponents of inductive
simplicity have
been after
is an ordering of
hypotheses with respect to their brevity.
The
fact is that
those who held that there is a weight, properly
called
simplicity, which
must be assessed in curve fitting situations,
seem
initially to have been impelled to their analyses
by recognition of the
influence,
not of objective characteristics
of hypotheses,
but of such subjectivecharacter-
istics of descriptions as familiarity, or manipulability, or elegance, or in-
telligibility, etc.
No doubt, too, the psychological responses
these terms
signalize have, through the processes of
socialization that
entrants to the
community of
scientists undergo, come to
be
fairly
standard among scientists;
and
perhaps
this fact made the notion that
there was some objective
character-
istic
of descriptional
simplicity operative in
curve fitting seem
more
plausible
than
can
actually
be warranted. Whatever
its etiology, however,
such a
conclusion
seems especially misleading
in view
of
the fact
that
Goodman
has
provided us with a
quite distinct explication
of objective simplicity.
What
I
have been
saying, then,
can be
summed
up by pointing
out
that
insofar as the considerations which influence hypothesis acceptance in the
curve fitting situation can properly be
called considerations
of
simplicity
they
are
subjective, while, on the other
hand,
insofar
as
they
are
objective
they
are
only misleadingly called considerations
of simplicity.
In
what has
gone before
I have already indicated
that
the
papers
of Good-
man, Ackermann,
and
Barker,
which
follow are
related
through
the
fact that
7
I
am
assuming
that these will be situations
in which well articulated
and
independently
confirmed theories,
which have
as consequenceshypotheses that fit the data,
are not available.
If such theories are
available we have another
kind of
(broadly speaking,) inductive
situation
and considerations
of say, the simplicity of
the predicate bases of
such theories
may well be
quite relevant. These latter kinds of cases, of course, are not in point here as is clearly revealed
by the fact
that not
the formal properties of
the curve fitting hypothesis but rather those
of
the
theory from
which it is derived
become relevant.
For an incisive
discussion of this
point
see
Ackermann, [1], especially
the
last section of Chapter
II.
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118
RICHARD
S. RUDNER
each
critically
addresses some
proponent of inductive simplicity. They also
appear
to
be
united,
it
should be
mentioned,
in
holding that simplicity is
an important weight
in
scientific acceptances or rejections. In this respect,
Bunge's paper
serves as an
admirable foil for the other three (and,
accordingly,
served
to
spark
a
lively
discussion at the
actual session) as well as to provide
an
extensive
survey (one, moreover, valuably informed by the thoroughness
of
his
knowledge
of
physical science)
of various
possible interpretations of
'simplicity.'
I
do not
find
his
arguments for the conclusion, that Simplicity
is not an important weight
in the
acceptability of
theories,
wholly compelling
ones. Nevertheless,
those
arguments together
with the
erudition arrayed in
their
support
stimulate and
require
the most
serious
consideration. For me
the
conclusions fail to
carry
conviction on two
counts.
Though it would be
inappropriate to argue these here, perhaps I can with propriety indicate what
they
are:
First,
Professor
Bunge
seems to
arrive
at
his conclusion, concerning the
insignificance
of the
simplicity
as a
weight,
on the
basis of its failing to be
a reliable
sign
of
truth. But this seems
to
me to be
an
irrelevance,
even
if
accurate.
As indicated
above, systematization
seems to me
as much a desidera-
tum
of science
as
is
truth and
nothing
that Professor
Bunge
writes seems
ponderably
to assail the conclusion
that
simplicity
is
an
important
measure
of
systematization.
Second,
I
find Professor
Bunge's
discussion
sometimes vitiated
by
the
opacity of some of his categories. In particular, I find myself still quite unclear
about
what
he intends
by
semantic,
epistemological
and
metaphysical
simplicity.
It
is doubtless
inevitable,
that
the
carrying
out
of
so
extensive a
treatment
of various
types
of
simplicity
in
the short
compass
of an
article,
prohibits
full elucidation
of
every
category.
These, however,
seem so crucial
to his
discussion,
and so
interesting
in their
own
right,
that
it is to
be
hoped
that Professor
Bunge
will find
the
time to tell
us
more about them.
REFERENCES
[1]
ACKERMANN, Robert.
Simplicity
and the Acceptability
of Scientific Theories, Doctoral
Dissertation,
Michigan
State
University,
1960.
[2] BARKER, S. F., Induction and Hypothesis, New York, 1957.
[31
GOODMAN, Nelson, Axiomatic Measurement
of Simplicity, The
J7ournal
of
Philosophy,
LII
(1955),
pp.
709-722.
[4]
GOODMAN, Nelson, An Improvement
in the Theory
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of Symbolic
Logic, XIV
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228-229.
[5]
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of
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XIV
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32-41.
[6]
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New Notes on
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[7]
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[1]
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