rudner an introduction to simplicity

8/12/2019 Rudner an Introduction to Simplicity

1/11

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


2/11

110

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].


3/11


4/11

112

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


5/11


6/11

114

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].


7/11

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


8/11

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.


9/11

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.


10/11

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

of Simplicity. The

Yournal

of Symbolic

Logic, XIV

(1949),

pp.

228-229.

[5]

GOODMAN, Nelson, The Logical Simplicity

of

Predicates. The Journal of Symbolic

Logic,

XIV

(1949),

pp.

32-41.

[6]

GOODMAN,

Nelson,

New Notes on

Simplicity,

The

J7ournal

of

Symbolic

Logic,

XVII

(1952),

pp.

189-191.

[7]

GOODMAN,

Nelson, On the Length of Primitive Ideas, The Journal of Symbolic Logic,

VIII (1943),

p.

39.

[8]

GOODMAN, Nelson, On

the Simplicity

of

Ideas, The

Yournal

of Symbolic Logic,

VIII

(1943),

pp.

107-121.


11/11

AN INTRODUCTION

TO

SIMPLICITY

119

[9]

GOODMAN,

Nelson, Recent Developments in

the

Theory

of Simplicity, Philosophy

and

Phenomenological

Research,

XIX (1959),

pp.

429-446.

[10]

GOODMAN,

Nelson, Sequences. The

J3ournal

of Symbolic Logic,

VI

(1941),

pp.

150-153.

[1]

GOODMAN,

Nelson, The Structure of Appearance, Cambridge, 1951.

[12] GOODMAN,

Nelson,

The Test of Simplicity,

Science, CXXVIII

(1958),

pp.

1064-1069.

[13]

GOODMAN, Nelson,

Fact, Fiction,

and Forecast,

Cambridge,

1955.

[14] JEFFREYS,

Harold and Dorothy WRINCH,

On

Certain Fundamental Principles

of

Scientific

Inquiry,

Philosophical Magazine,

XLII

(1921),

pp.

369-390.

[15]

JEFFREYs,

Harold, Scientific

Inference,

Cambridge,

1957.

[16]

JEFFREYS,

Harold,

Theory of Probability,

Oxford,

1948.

[17]

KEMENY, John

G.,

A Logical Measure

Function,

The

J7ournal

of Symbolic

Logic, XVIII

(1953),

pp.

289-308.

[18] KEMENY,

John G.

A

New

Approach

to Semantics-Part

I. The

J7ournal

of Symbolic

Logic,

XXI (1956),

pp.

1-27.

[19]

KEMENY,

John G.

A Philosopher Looks

at

Science, Princeton,

1959.

[20]

KEMENY, John G. Two Measures of Simplicity, The Journal of Philosophy, LII (1955),

pp.

722-733.

[21]

KEMENY, John G.,

The Use

of

Simplicity

in

Induction,

Philosophical

Review,

LVII

(1953),

pp.

391-408.

[22]

POPPER,

Karl R.

The Logic of Scientific

Discovery, New

York, 1959.

[23] RUDNER,

Richard S.,

The Scientist Qua

Scientist Makes

Value

Judgments.

Philosphy of

Science, XX (1953),

pp.

1-6.

[24]

RUDNER,

Richard

S., Value Judgments

in the Acceptance

of Theories. Philipp

Frank,

ed. The Validation of

Scientific Theories,

Boston, 1956.

[25]

SUPPES,

Patrick,

Nelson Goodman

on the

Concept

of

Logical

Simplicity.

Philosophy

of Science,

XXIII (1956),

pp.

153-159.

[26]

SVENONIus,

Lars, Definability and Simplicity. The J7ournal of Symbolic Logic, XX

(1955),

pp.

235-250.

rudner an introduction to simplicity

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