The General Theory of Notational Relativity

Citation preview

Original fro

HARVARD UNIVE]

“Fhil

s150./ K

e

\|wmmmmm||| e R

HARVARD COLLEGE LIBRARY

HARVARD

UNIVERSITY

o

THE

GENERAL

THEORY

OF

NOTATIONAL

RELATIVITY

—_—eee e

by Henry

M.

Sheffer,

Ph.

D.

Department of Philosophy, Harvard

University

CAMBRIDGE, 1921

Google

MASS.

S50

.|

NG/ 81921

Google

]

%N

‘

———— S——————————————

PREFACE The following pages present,

in mathematical logio. In a volume entitled

writer hopes to publish which Every

then

may be Puture

aprlied

problems

in

Analytic

the

to

the

logic,

presented,

solution

of

mathemsties,

Knowledge,

time

to

Society.

& number

and

Society,

uperpostulates:

Deductive Systeus", (3)

™Deductive

(47

"The

in

»

Systems

p. 269;

™dutually

(6)

Prime

"Principia

1919,

Anericasn

graph

pupers

We

refer

of

Deductive

read

in

uarch,

1910,

Introduction

ib., November,

Elimination

1ib., Merch, 1916,

of

pp.

and

the

abstract:

of

1913,

Postulste

Kodular

Thie

Systems

Association.

systems,

to

We

the

Science

of

Theory;

I.

Finite

Postulates”,

Kerch,

read

quote

it is well known,

the

with

p. £65;

p. 76;

ib.,

was

to

Bulletin of the

Philosophical

paper

before

particular

Existence

Postulates",

Ans=lytica",

187-8.

Philcsophical

"Deductive

is

fundamental

Menzenlehre.

Cese", ib., February, 1915, p. 220;

Merch,

of

reference to the Algebra of Logic",

(5) 287.

the

through &period ot more

time,

following abstracts: (1) "Total Determinations

American Matheus tical

.

which

to the gradusi formulation of this method,

from

Mathematical

special

& new method

churacterized as & sort of Prolegomenon to Postulate Set--is develored in detail, and

than twelve years, Americal

in outline,

in the neur TE?'mr"e“_K ‘this method--

Some of the ideas which led,

were

2

1916,

Review,

before the

the

first

para-

may be determined

by mesns of postulate sets in various ways. Euclidean geometry, for example, is determined by the widely different postulate sets These distinct

of Zilbert, of determinations

Veblen, are all

and of Huntington. 'equivalent?--any two

of the postulste sets are uniquely intertrunslatable. Mey there not be, then, & set of 'superpostulates', of which

Hilbert's, Veblen's, Huntington's, and other vostulate sets are special cases? There is. And,:as a matter of fact, the 'invarient' of these postuluate sets turne out to be of an extraordinsrily simple character".

Cambridge,

April,

Moss.

1921.

Google

INTRODUCTION

3

SYSTEMS

Ordinery euclidean three dimensional geometry 'given' in various waye. In particular, it may be

may be 'given'--

by

the use of "logical principles"--in terms of: 1. The two undefined euclideun-geometric notions, class of solid srheres, and sphere-inclusion; an 2. A smll number of unproved euclidean-geometric ropositions,

and

the

two

each

statable

undefined

notions

By means of "logic",

entirely

in

in 1.

terms

We

shall

spheres,

call

the

of the

two

undefined

sphere-inclusion],

the set of unproved assunptions. By the

the

notions,

only

those

euclidesnbe proved.

[clsass

Huntingtonlan

of

solid

language,

and

propositions in 2., the Huntingtonmian system of euclidean geometry in terms

Huntingtonian language, we propositions

assumptions.

notions--

all the 2.--can

8hall

mean the

Huntingtonian sseumptl_fle'&oneogether with the set &nd

"logio"

all eucli dean-geometric

other than those in 1 can be defined, and ometric propositions-~other than those in

fen\mtington]

of

‘EogicaITy"

Set oF

deducible

of all those from

these

SYSTEM FUNCTIONS Russell has introduced the important generalization of the concept of proposition to that of propositionsl function. We find it useful to coin anslogous terms for certain analogous generalizations. When, in the current literature, the relation "sphere-

inclusion™

is repluced by R, we have no right to oall k &

relation.

R

X,

call

Similarly, we

"golid

must

spheres"

inclusion”

becomes

the

"language

(X,

that the

R]

we

is,

is

and

shaild

be

celled,

a

relational

function.

when the class of "solid spheres" Is replaced by is

K a dlass is

replaced

replaced

R,

Hduntingtonian

function"

therefore

have

by

is

called

propositionsl

the

function. by

K,

the

lsgfga%e

and

the

class

we

RJ.

by

base.

The

assumgtions

may

ca.

e

the name

of

"sphere-

language

(K,

replaced

Huntingtonian

sssml'ff—_m—p ona ctTons. However,

the

relution

function

Huntingtonian that

the

Huntingtonlan

conveniently

functions

When

the

ifle

name

set

of

the

set

nropositions

of

that

we

have

called

Become

now

Huntingtonian

"sssumptional

"the

base.

propositions

tions" is converiently replaced by the name postulates.

ean geometry

name

system

of

funo-

fhe

euclid-

in terms of the Huntingtonian language" tecomes

propositional

Google

functions

that

we

shull

call

"the

Huntingtonisn system funotion in terms of the Euntingtohian

base

(K,

DOUBLE

R]".

Tootnote.

TERMINOLOGY

If we

are

to

avoid

Cf.

Keyser,

fallacious

distinguish between the two sets of system and in gystem function. The

class

Striking

examples

of

the

regerd of this distinction are Einstein's "generalized theory are valid only for of the Zinsteinian steinian

s-}Emema

Einsteinisn In

or,

system

the

results

terms

following

pages

supnosed terms of

with eystem functions. GRAFS are

Although,

either

function

"

assumptional

"

system function

[=base]

function

fdpostulate]

resulting

from

thaul

or

a

dis-

found in many expositions of of relativity". Results that

interpretations are

va

of

a particular

we

shall

or

of the a

be

of snalytic

inconclusive

Ein-

Einsteinian

concerned

or

terms the

"particular

to hold necessurily for "any any Einsteinian language™.

for purposes

irrelevant

always

relationel function language function

confusion

from

in

language” are ian syetem in

class

"the Einsteinian sy_;um function in base" are suppose 0 hold also for

derTveble

base;

we must

1 "

N

system

functions"]

concepts involved in sete of concepts sre:

"

assumption

various

reasoning,

"

relation language

"doctrinal

proof,

both,

Einstein-

exclusively

grafic methods

yet,

for

pur-

poses of exposition, they may well In the following discussion we grafic devices, and shall develop, of "deductive analytic geometry".

be indispenezble. sh&ll employ various on tne grefic side, & sort For the eimrlest case of

volving

the

system functions,

E dyedie,

cases

just

we

we skall

one

X

shall

viz.,for those given in terms of & tase inand

use

introduce

just

one

R,

Schroeder's

new types

K

grufs.

being

For

of diagram.

BASES

finite,

all

the

and

the

other

We shull alwsye symbolize a base involving & K of cardinality n emd an R of degree m, 1. e., an n-element K an foeaty¢ R, by & Rm

or, for typogrefic converience, by

Google

Kn Rm.

ISOTROPY DYADIC

ISOTROPY

Dyedi c Graf

e following be 2 postulate sct for u certain s_,'stem function on base K3 R2: K has just three distinct elements.

1.

2.

There is & unique X clement,

R11

holds.

Rad

fails.

For any

distinct

K elemmts

"1", such thet

«, b,

This pestulste set may be represented diagrammatically

JEER 21

/

7 5]

as in Fig. A, where the X elements are labelled 1, v 2 2, 3. In this figure the element & is found on the left, and the element b at the top. When Rab holds, the square labelled (a,b) is mrked "+"; when Rab fails, the square is marked "-" Factorial

T

Famil

ure

concerned,

are

so

tiee

Postuls tional it

is

represented,

properties

immeterial

are ever represented it

Google

is

whether

by any grafic equally

of & system function

or

not

scheme.

immaterial,

such

4nd,

from

proper-

the

if they stand-

point of the postulationsl emte

are

taken

with

one

6

properties,

permutation

whetler the K eleu-

or

another.

[Footnote.

We use the term "permutation" rather than "order", in the case of & non-finite K, the elements may be with different permutations within one order tyvpe.

becuuse, taken Thus,

ferent

within

1,2,

3, 4,5,

the one three

6, «eo, and 2, 1, 4, 3, 6, 5,

permutations

order

Hence,

of

typeas .)

a

caintable

postuletionally,

elements

of

aur

with permutution 1,

it

system

lesde

does

function

2, 3, as in Fig.

the possible 3! permututions, This

set

to the

auxiliary concepts.

of

not

A,

on

Figs. A-F.

introdwetion

+a., are dif-

elanents

mutter X3

whether

R2

ure

or with

of the

uny other

first

ot

our

We shall cull the set of 3! grufs,

u factorial family alT thoee and only

the

taken

of new

A-F,

of grufs. Lore generall, the set of those n! grafs obtained from a gziven gref

by urrsnging the K elements with all the n! peruutstions constitutes

a

factorial

family

of

grafs.

Thus, when a postulate set is represemted graficelly by & single graf, rather than by the whole factorial fumily of

gr‘t‘LEha s, the purticuler permutation of K elements chosen for the

graf

ie

extra-postulational,

ally irrelevant. Ordinsrily,

neitrer

£11

all

n!

the

grafs

therefore

the n! grafs of = factoridl

identical

with factorial

and

nor

family A-#. of

all

dietinct.

It msy huppen,

& fectorisl

family

postulation-

family are

This

is

the

however,

are

distinct.

case

that

Thus,

1f one of the grafs for » certain postulate set on K3 R2 is Fig. G, the factoriul femily, Figs. G-L, coneists of n!

grafs

which

are

Isotro;

finzally.

&ll

distinct.

it way happen that

sll the n! grafe of a

fuctorial femily are identical. Let us consider, for example, the following set for a system function on buse K3 Z2 :

0.

X hae

2.

For

1.

The

we pleace,

with b".

For

class

any

just

any

X

three

element

distinct

function

K may

& stinct

X

elements.

8,.¢s00000000.. elements

be

a,

b,

Zaa Zad

interpreted

and Zab may be interpreted as

postulate

as

holds. fails.

any

class

"a is identical

This postulute cet ray be revresented

by the graf

of Fgz. 1, where the K elanents are labelled 1, 2, 3.

factariul

are

all

In

femily

identical.

the

consists

Present

system

of

the

six

function,

grafs,

Figs.

whatever

1-6,

The

which

postulational

property holds for amy K element holds alsc for any other;

Google

and whatever pos tuletional faile also for any other.

elenents

7 K element of the

property fails for any That is to say, no two

ére Kz-distinguisheble,

and therefare no ome

elementr is Kz-identifiable. — The grafic equivalent of this state of affaire is identity of all the grafs of the factarial family. will say

of the the

4 system function which hus the property just memtioned be

that

system

said

to

the

have

system

function

postulate

set

is

thut

the

property

function

is

tlerefare mekes

one

of

isotropy,

isotropl

that

fs

identically

the

about eny two of the elements.

eand

*n

we

determined

same

shall

ieo tropic

by

s tstement

&

For the base of an isotropic system finction we shall use Kz, the letter Z, as the initial of "zero", being

always

intended

to

suggest

thet

the

system

function

thus:

3o

dyadic

2?

furnishes

"zero"

information ubout postulational distinctions of elememts. We ehall symbolize an isotropic system function by O (zero), and shall indicete an isotropic system function of, -say, three elewente by writing the zero as a subseript to the humber

with

of

How &

elements

many

three

involved,

isotropic

element

system

K

end

a

method for answering this new corcept.

funct

ions To

are

possibdle

obtain

a general

type of question we introduce &

Vslidands

T eymbol "+", or "-", put into & graf, reans that, for certain combinations of X elements, the given relational function respectively holds, or fails, and thus "+" and "-"

represent, functioual

respectively, positive and yalidation. e shsll call

vali datum,

Herce,

negative validatum. class

Just as we function,

language

it useful velidetum

"+"

is

a

generalized relation to

function,

positive

velidetum,

the rotion relational

the concept a vali detum

neme

TT we

Fig.

7.

&

&

so we find

validatu. letters.

We shall A4 validatum

therefare, ambiguously, either the or the negutive validutuim, "-".

definecd the ngw concept, we shall hemcefarth nsme validatum function by the nore convenient

yslidend:

respectively

"-"

of validetum to that of function we shall mean

& two-valued function whose values are symbolize validatum functions by Greek

furction, o, represents, positive validatwa, "+",

and

of cless to that of function, language to

snd sy stem to systenfunction,

to generalize function. By

Having replace the

negative relstionaleach of these symbols

change

This

in

Fig.

place

figure,

1 by putting validands Oy ; «ese,® g3,

of

the

which

Google

nine

validata,

the

is & graf function,

result

is

we shall call

a hypergraf, (pasff’ive or We

When all negative)

may

now

8

the validands validata, the

write

aut

all

the

are repleced by suitable hypergraf becomes a graf.

3!

hypergrufs

obtainable

from Fig. 7 by arranging the K elements with all the 3!

permutations,

call

Figs.

7-12.

This

sary nine

and sufficient condition validande reduce to two,

of

grafs,

When

this

Figs.

condition

7-12,

is

reduces

hypergrafs of u_ factorisl the hypergrafs isotropic.

tropic

hypergrafs

shows

satisfied,

to

Fig.

euch

13.

of

When

The

conditions: K has just three

1.

7For any

2.

For

forms

and

distinct

K element

any distinct

aa

ab

is

shall

a necesthat

the

= Va2= AL, the

all

Isotropy Superpostulates for K3 e hypergraf of £1g. 12 18 determined

following 0.

that

for their identity as followe:

we

hyper-

the

n!

family sre identical, we shall call Hemce, Fig. 13 remwesents an iso-

hypergraf.

Dysdic

3!

A= oy,= Qy=p =¥y

=Haa §

==y

the

set

a factorial family of hypergrafs. An"examination of these E?pergmx’s

we

&y

«

«

call

by

elements.

X elements

shell

analytically

« .

.

+»

&,

880gq

b, abOlgf

atomic

dyads.

4n

atomic dyad together with o validand--e. g.,.,_aaul:—ugetmx' with Olae, or_é_t_s ogether with Olef--We Bhall call an atomic superpostulate. ‘hen the validand is replaced by & [posiIIve or negative) validatum, the atomic superpostulate becomes an atomic postulate. Thus aa®eqmay become either &a+ or ae-; abOlelmay become, independently, ab+ or ab-.

Hence,

postulate end

2.

To

Fig.

13 is determined

completely by the K3

togetler with the two atomic superpostuletes distirguish

this

type

of atomic

1

superpostulate

from other types to be considered later, we shall call 1 end 2 & set of isotropy The

validands

velues independently, determines

faur

grafs,

atomic superpostulstes.

and Olegmey

assame

ic.

If the

function,

2'2.

Z'

or

"-"

and

represemts

four

system

13

functions.

each of the grafs is iso-

we call the relational functimme 2', 2'', 2''', tem functions represented by Fig. 13 are the

following: (1) faa=+_;

Buce=X3

values

and therefore the hypergraf of Fig.

Since the hypergraf is isotropic, 2'7'7

"+"

Olab=+

In

holds

this

.

three

element

universally.

Google

dyadic

Hence

Z'sb

isotropic mey

be

.

system

interpret-

.

A e€d u8

"a

is

either

(2) Oaa

Base=K3

==

2''2.

sally.,

identical

=-

this

o

with

system

or

different

function

Z''

nor

=t

Base=K3 2'''2. rretable as "s

different

;

Olal

=-

o

;

ool

=t .

from

of

univer-

these

four

system

is

inter-

2s "a is difer-

b".

Each

b".

b".

BasewK3 2''''Q, Z''''ab is interpretable from

fails

In this sytem fuhction Z'''asb is identical with bd". Fig. 1.

(4) Laa==

ent

from

Z''ab is interpretable as "a is neither

with

Qlaa

ofal

In

Hemce,

(3)

identical

;

9

functions

is

determined

by

a postulate set on a distinct buse; i. e., these system functions are non-cobaszl. Therefore, the graf of anyg one of these system Functions cannot be obtained from thut of any otker by = mere permutation of elements; i. e., the four

thut

tions

grafs

ere

non-cofactorial.

cofectoriaIlty imply

We

esch

have

of

grafs

other.

now

answered

t is

and

the

not

diFficult

cobasality

question

&s

of

to

to

system

how

see

funo-

many

iso-

tropic system functions ure possible Wi th a three element K and a dyadic Z. There are just fair such system funcyions. These are interpretable in terms of four basic logical

relatimc

difference.

adic

that we may call

Isotropy

f we

four,

we

truism,

Superpostuls tes

Increuse

obtain

ihe number

the

same

falsism,

for

K4

of K elements

results.

See

16,

when

adic

isotYopy

reduced

to

a finite

from

Figs.

Dyasdic Isotropy Superpostulates for Kn Lore genersally: y finite dyaudic #g.

identity,

14

three

and

n-element

dyadic

n-element

and

15.

to

hypergref,

isotroric

bypergraf on Kn z2, Fig. 17, involves just two validends, e and olef , and is determined analyticslly by the dy0.

X has

2.

PFor any

l.

For

superpostulate

any

For

any

finite

just

n,

K

n

set:

(finite)

element

distinct this

&,

+

distinct +

«

K elemerts

superpostulute

«

a,

+

elements. .

b,

set

«

280lag

abdgf

determines

just four dietinct dyadic system functions, which are the

only dyudic n-element

isotropic

system functions,

viz.,

those which ure interpretoble with the dyadic relstiore

truism,

fulsism,

identity,

Google

snd

difference.

of

Isotro; 30

f

Extensional Definition far, we have defined 1sotropy

of u certain system function;

intensionslly. In Isotropy ¥s

We the

may set

as

a

10

.

in

1i. e., we m!géf':

property

otropy

also defIne it extensionally, thus: of all those and only those syetem

functiors determined by & hypergrsf whose factorial family consis te of hypergrafs which are all identical.

TRIADIC

Irisdic and

ISOTROPY

Graf

Tom Firite isotropic dysdic system functione,

hypergrafs, Let

the

furction

we

proceed

following

on

base

0.

K3

23:

K has

l.

For

2.

PFor

be

just

any

any

To

a

the triadic

postulste

three

K

distinct

element

X

a,

element

Zabe

fails.

b,

for

a

system

elements.

Zaaa

a,

case.

set

¢,

holds.

not

all

identical,

X is-intermretable as any clase we choose,

and Zabe

may be interpreted as

"a, b, o, are all identicsl™.

in

form

This postulute set may be represented by a triadic graf

various

useful

ways.

for

air

The

present

of

inquiry

figure represents a "culic" "slices", (2.v2v?)

of

(£.v2v3).

value

trizdic

iz

1In each

of

vl.

This

of

thet

of

This

ut the bottom, layer (2.v2v3)

layers,

v2

is

most

layer is on

faind

graf

fecilitetes

and also

on

the

elements

2 and

just one

extension

this triadic For ocxample,

v2,

and

graf of Fig. 18, and the system function it represents,

are

vZ, leads to Fig. 19. (Here second and third layers must therefore

its 3%

3!

of

Mg.

16

trisdic

we

tires element triudic

independent va.idunds.

necessury

thrauughout

vl,

should be noted that the interchanged) The triadic

isotropic.

Trisdic h"i%ergrat From the gruf

ing gguerel

it be

3,

the

top

foreshadows &

generulization to be discusesed later. The 2! grafs of the factotial family of graf are easily found to be 21l identical.

of

find

18.

und each layer involves

to tetradic and higher degrees, permusution

we

Pig.

gruf divided into "layers" or

of these

form

graf

thét

80 tuct layer (1.v2vZ) is is on top of (1.v2vZ), und

left, end V2 ut the tcp;

the

grafe,

and

sufficient

hypergrafs

to

hypergref,

the

Fig.

48 in the dyudic

conditi on

of the

Google

proceed for

fuctorisdl

the

correespond-

20, with

cuse,

identity

family

is

of

the

the

obtained

easily, and is found restricted validande

to be the following, are reduced to five:

where

the

27

11 un-

Uy = Yaan= Ay, = Ngq

far = Ny = Xy Vo=

Y123 = Y3y = When

T v,

=Yy, =%

Ya 20u=

qll3=

Sy

= X323

3=TN Ss ¥y,

Dz

3= Yz =

this

comdition

is

=X

2

n

Yp= Yy= ¥aa, =,

28

X,

=y

< Aony

= XY

suatisfied,

each

,

of the

hyper-

grafs.of -the factoriel fumily of Fig. 20 reduces to Fig. 21. Hence, Fig. 21 represents a three eclement triadic isotropic

hypergraf.,

Trisdic

Isotropy

Superpostulates

"_“me“ruypergm by

the

o;

following

triadic

atomic

0+

K has

2,

Ffor

uny

distinct

"

"

"

1.

three

for

K3

s detennined anal ytically superpostulate

distinct

eleuments.

For any X €lement &,.ececesesss

3,

v

4.

6.

This

just

e

.

oW

"

"

"

set

aba, abb, abe,

triads involves Since each

Jjust one of these

"

n

"

"

"

"

superpoe tulate

viz., uaa, aab,

X elements

e,

"

Iriadic

32

non-cobasal

Isotropy

validand. validends

graf,

system

Superpostulstes

IZ we increase four, the result is

¢ number the sume.

may

hypergraf

on Kn 23,

a.bbok

¢,.

a.bodkf,

atomic

assume

functions.

trinds,

for

the

value

"+"

this hypergruf grafs, and

Kn

of X elenents from three See Figs. 22 and 23.

More genersally: Any finite Pig. 24, when reduced to &

isotropic

a.bal

and each of E¥E 86 etomic

or the value "-" independently of any other, determines 2.2.2.2.2, or 32, non-cofactorisl

therefore

a.abol ol

b,

8, b, five

2.880 aq

b,

. b,

involves

set:

to

triadic n-element hyperfinite triadic n-element

Fig.

26,

involves

the

same

FIve valldends, and is determined analytically by the same superpostulate set a8 for K3, except that the number of X

elements is changed from three to any finite number,

TETRADIC Tetradic

et

ISOTROPY Graf

The following be u postulate

Google

n.

set for & system func-

tion

on

base

K4

0.

in

Z4:

K has

1. 2.

4As

12

just

four

distinot

elements.

©For any K element &, Zaaaa holde. PFor any K elements &, b, o, d, not identical, 2Zabod fails.

previous

cases,

K

may

be

taken

as

all

any

class

we plesse, and Zebed may be interpreted as "a, b, ¢, and d are all identical". This postulute set mey be represented by a tetradic graf in various ways. For our present purpose

we shall use the form of tetradic

which

sdic

is

an

extension

case, Fig. 18.

af is

divided

? v2v3v4), into In

and

v4

one be

of

at

In Fig.

16

the

top;

or

we

"slices”, layers

each

employed

for

to

be

all

(1.v2vZv4),

of

in turn,

vZ

(1l.vZv4),

these

is

tri-

(2.vev3v4),

ere divided

found on

leyers

identicel,

and

therefore

and the system function it represents,

Tetradic

26,

the

26, the "faxr dimensional"

layers,

anmd these,

dyadic and

graf given by Fig.

frm

,

the

(44.v2v4).

involves

value of vl, and just one value of v2. The 4! grafs of the factorial family of this

found

graf,

layers,

the

the

cubic

(4.v2v3v4),

dysdic

each

into

of

left,

just

graf

this

will

tetradic

are

isotropic.

Hypergraf

A5 in preceding cases, we gereralize from the graf to the corresponding hypergraf. If we were to replace, in Fig. 26, each "+" and eech "-" by an independent vslidend, we sha1ld obhtuin the generul fair element tetradic hypergraf, with its 4.4.4.4, or 256, unrestricted validanis., The necessary and sufficient condition for the identity of all the

4! hypergrafs of the fuctorial family of this gemral hypergraf

Fiz.

will

27.

be

(In

found

order

to reduce

to

gain

the

256

certuin

validunds

notatiomsl

to

fifteen,

advsntuges,

we reprecent some of the 15 validends subscript, and the others by aF with

by ano with a double a double saubscript)

Tetradic

K4

Isotropy

Amxlyflcdfl',

27 is determined

postulate

set:

SnErFostulates he

tetradic

for

isotroplc hype reraf

by the following atomic

0.

K

hae

Just

1.

unesstly,

2.

na.sbRal

5.

7.

eabadp,

6.

For

four

any

Google

4.

distinct

distinot

se.Bboy

of

180tropy super-

elements.

K elements

a,

b,

7.

ub.ablp

as.booly,

8.

ab.acflac

sb.n@y,

9.

ab.va i,

iig.

c,

d:

10.

ab.vbSpp

13.

ab.ob By

1. 12,

sb.bofy, sb.caf,,

14. 15.

ab.ce fec ab.cafed

13

This superpos tulate set involves 15 atomio tetreds, anae, nasb, r..: abco, abed, und sach of Ensae tetrede in-

volves

Just one Since each

validand. of the 15

validands

may

be

assigned

a

"4"

value or a "-" value indeperermtly of any otker, the number of non-ocofactorisl grafs, and therefore of non-cobasal system funotions, determined by this tetrudic hyprere? s plb. Tetrudic

~—

wken

Isotropy

iny it reduced

to

&

Superpostulates

for

Kn

fintte

n-element

tetrs Tc n-elemnt hypergraf, tetrudic

Fig.

28,

isotropic

hypergraf on Kn 24, Fig. 29, involves the cume 5 vatia-

ands, and is determined unalytically by the same superpostulate set us for K4, except that the number of K elements is changed from fair to any finite number, n. PENTADIC

Pentadic

TR

function

ISOTROPY

Hypergraf

B ‘f&l‘oung be a postulate set for a system on K6

1.

z5:

0.

X

has

PFor

any

just

¥or any K element a,

2,

K

elements

identical,

As in former cases,

soever, are

all

five di stinct

and

Zabode

identical”.

may

be

a,

elements.

zaneaa holds.

b,

¢,

4,

e,

Zubode fails.

not

all

K may be taken as any class what-

This

intermreted

pentadic

as

"a,

postulzte

b,

set

¢,

4,

may

e,

be

represented by & pemtadic graf in various ways, --in particular, by an extension of the drm of graf we have already employed fcr the triadic and the tetradic cases. This mode of

grafing

the

set of 5.6.5, layers,

3,.4, A1l

6,

the

given

pentadic

system

function

or 125, dyedic slices or layers.

five

which

belong

to

type

results

in

a

OFf these 126

(1ii.v4vb),

have exactly one "+" each, viz., (111.11)+, (222.22)4, . . ., (5556.66)+

i=1,

.

2,

the othe otler dyadic layders have "-" throughout. The 5! grafs of the factorial family of this pentadic

Google

grof

are

all

identicel,

and

therefore

this

graf,

&nd

the

14

system functiom it represents, are isotropic. If, in this pentadic graf, we were to replace each "4" snd eech "-" by un independenmt validand, we should obtain the

gereral

five

and

sufficient

5.5.5.6.6,

or

element

23125,

permtadic

unrestricted

condition

hypergraf,

validands.

for

the

idemtity

reduce

the

3126

with

of

The

all

its

necessary

the

5!

hypergrafs of the factorial femily of this gereral hypergraf

will

be

30. (For validands

Perntadic graf,

found

to

validands

to

notasional simplicity, we represent these by the letterg,with double subscripts)

BY.8.€

Isotro

Inalytically,

whose

Sufiegiostuletes e

typical

layers

ve

are

enent

for K6

pentadic

given

in

Fig.

62,

52

isotropic

30,

is

Pig.

hypere

deter-

mined by the following atomic isotropy superpostulate set: 0.

K has

Just

five

distinct

elements.

l.

For any dietinct K 888.88Yaa 6.

elements a, aab.anfi‘g

2,

a88.8b0(of

3.

asa.badfe

4.

aaa.bboly

5.

seaa.bodf,

15.

aab.cd

26.

uhb.unsu

36.

abc.as

26.

abb.cd 5&

52.

abc.de £ g

b,

¢, 4, 16.

Sieeseg

. 25.

Fed

aba.cd

£,

This suparpostulate set involves 62 atomic pemtads, aaaas, saasb, ... , abcdo, abcdd, abcde, amd each P hese penteds involves the

Since

number

cobasal

graf,

Jjust

one

of

system

Any

52

validands

non-cofactorial

functions,

is ,52.

Pentadic

validand.

tkese

Isotropy TInlte

ere

determined

Superpostulates

permtadio

Google

independemt

grafs,

n-element

and

by

for Kn

of

therefore

this

hBypergraf,

each

of

pentadic

when

other,

non-

hyper-

reduced

to

a fintte

Fig.

31,

pentadic

involves

the

n-element

same

isotropic

16

hypergraf

52 vul!flnns

and

is

on Kn

determined

25,

analyticslly by the czme superpostulate as for Kb, except that the uumber of X eleuents is changed from five to any f£ini te number, n. M-ADIC

ISOTROPY If we

proceed

from

finite

pertadic

isotropic

hyper-

grafs to the hexadic und higher degrees, we obtain more complicated Any

but znalogous finite m-adic

represented by a set of determined analytically

superpostulate

results. n-element

We find that: isotropic hypergraf

may

p(n-2) dyadic layers, and may be oy a definlte atomic isotropy

be

set, according to an obvicus generslization

of the method

we kave employed

for m=2,

7, 4, b.

STRATIGRAPFY DYADIC

FIRST

STRATIGRAFY

First se

i.

Strevigrufic Femily of K3 22 From the consideration of the ‘theory we shall now call them, all-isotropic,

e.,

of

eysten

functions

which

have

all

of isotropic, or, system functions,--

their

elame:ts

180tTopic--ve procecd to the study cf the theary of system functions

which

isotropic.

mentel

have

not

necessarily

In order to do

this,

all

dimensions, sented, und

parallel

to

This hypergraf,

if represented in three

wuld be a cube. Let us let us alsc imegine thie

the

vZvZ-axes,

and

it

perpendiculer

set of three dyadic (3.v2v3). Hence,

to

21 muy be regurded from two distinct points of view.

the

sible

we

imagine it so reprecube cut by planes

therefare

the vl-exie. The result will be the luyers, labelled (l.v2®Z), (2.v2vZ), In

elements

a new funda-

corcept. Consider the three element triadic isotropic hyper-

graf of Fig. 2l.

Fig.

their

we introduce

did

may

first

ways in

be

hypergrafs, Fig.

place

our

may

discussion

viewed

viz.,

trisdic

be regarded

of

a

single

isotropy;

as represerting

We shall call 21, the first

single

it

of representing

(1.v2vd),

a

set

(2.v2vZ],

as

one

triudic 1in

of

of

aa;

hyve rgraf,

the

second

three

(3.vival.

pos-

dyadic

place

this set of three dyedic hypergrafs, Btratigtafio fumily of the original

isotrgp

¢

Google

hypergraf.

as

Let us study some of the consequences 21 from the second stendpoint.

Fig.

of

reyurding

16

First Str ufy of K3 22 The a5 e T LySerrirs which ve have just defined us the first

stragigrafic

family

of the

given

triasdic

hypergraf

the importent property of constituting & factorisl

hypergrgrufs.

For example,

if we teke the

Arst

have

femily

of

these

of

dyadic hypergrafs, (1.v2v3), amd write out the 3! hypergrafs obtainabtle from all the permutstions of the X elements, we fird that the intercaunge of elements 2 and 2 lesves the

hypergref invariant, waeress any change elemert

1

2l.

obtair

of

(1.v2v?) We

(2.vevz)

chzrges

the

consists

eractly

or with

hypergref,

of

the

the

(3.vivd).

three

sume

so

in the vosition of

that

the

dyudic

result

if

fretoriul

hypergrafs we

family

of

sturt

with

Pig.

If we tuke the generul dysdic three element hypergraf,

®g. 7, with its nine independent validends, and reguire thut ut least tw of the elanents, say 2 and 2, be isotropic, --i.

e.,

thut

The permutution

hypergréf.unchunged--we cient

condition

for

this

of

2 and

3 shull

leave

find thut the necessury md is

tmt

the

nine

validunds

the

suffi-

be re-

duced to five, exuctly &s in (1.vZv3), Fig. 21. Since we huove as yet imposed no otler condition upon

the five vulidands, the isotropy-value of element 1, i. 6., ite isotropy or non-isotropy, is still entirely undetermined.

Hence, any one of the dyndic hypersrafs of Fig. £l repre=sents w11 those and only those three element dyudic system functions

fuct

gréfs

we

which

indicute

of Mg.

have

by

21 hoe

at leust

siyi:g

two

thut any

elements

one

sbratigrafy 1

) 2,

isotropic.

of tke

dvedic

. Since

Thie

hyper-

the number

of elements whose isotropy value is yet und~termined is one, we cell this tke first strotigrafy of the original trisdic isotropic

hypergraf.

first Stratigrafic Pamily of Kn 23 In 1ixe mencer, Flg. 25 may be viewed us representing elther u single n-element trimdic isotropic hypergruf, as we

qIT ecrlier, lubelled

nypergrefs of

or a set of n dyedic hypergrufd,

(1.VEZv3),

.....,

(D.vev3).

is, ty definition,

the origiral Tals firct

This

set

viz.,

thoso

of n dysdiec

the firet stratigrefic

single n-element tricdic 1sotropie hyperzraf. stretigrafic femily comstitutes u factoriul

fumily of hypergrufs. If we tuie, for exauple, the hypergraf (1.v2v3), and write out the n! hypergrafs

uble

from

11

fomily

the

vermutations

thet the pernutation o £ »ny

Google

cf

the

K

elements,

two of the elements

we

dyudic obtuinfind

£, %,...,

1,

lecves the hypergraf inveriunt,

17

wheress any chauge

in the

position of element 1 chenges the hypergre?,

so that the

of

start

fectoriel family #g.

(2.v2vZ),

rirst

25.

or

1T we were

should

for this

of

find

hypergrafs, which

have

...,

%o—fi.ke

consists

csume

result

of

22

the

of the

(n.vivad).

general

if

we

n-element

n hypergrafs

dyadic

tThat

and

trg

ig.

(i.viv?),

at

necessury

the n®

only

least

&nd

vulidands

the

hyper-

26.

i=1,

those n-1

2,

Hewe,

.., n,

n-element

elements

sufficlent

condition

be reduced to five,

first stratigrafy

of iig.

dyzdic

Zsotropic.

of the

25 represents

system In

functions

otker

given n-element

triudic Tsotropic hypergraf.

exsct-

any one of the dyadic

sny one of these dyadic hype rgrafe has stratigrafy or,

with

16, with its n.n independent validunds, &nd were thatst least n-1 of the n elements be isotropic,

is thst

those

the

of xn

1y es in (1.v2v3), 2ll

(1.v2vZ)

odtain

(2.vevZ),

Stratigrufy

gruf, #g. to require

we

Ve

words,

1¢ ) (n-1)

Dyedic #irst-Stratigrafy Superpostulates for Kn From thece dlugrammatic results we proceed to the ana1ytic vereion. When Fig. 26 was considered =e a single n-element trisdic hypergraf, it wes determined by the isotropy superpostule te set, 1-5, given on page ff . Hence, layer (1.v2v3), #ig. 25, can be determined by the same superpostulste set, provided only we replace a by 1 in each of the five atomic superpostulsies. But layer (2,vive), Fig. 25, belonging to the sume fuctorial fumily as (1.vivd), is determined ‘3{—@ identically the szme superpostulate set, if we merely*I by 2 Likewice, any layer (1.v2v?), i=l, 2, .., n, Mg. 25, is determined by this superpostulite set, if we replace 1 by i. This is only a speciul cuse of the relation which holds vetween any superpostulate set and 1t8 hypergrafic representation; for, any two hypergrufs of a factorial family differ only by o permutation of elements, and not by any formsl superpostulaticnal property. Lence, if we choose to represent a superpostulute set by u single hypergraf rather then by the whole factorial family, the specific permutation of K

elements

factor.

isotropic,

employed

Furthermore,

introduces

un

anslytically

since the given

irrelevant

tricdic hypergraf

is

there is no formally statable difference between

uny two of the elements. Hence, analytically, there is no aifference whatsocver between writing 111, etc., and writing 222,

point

etc.,

or iii,

of view,

any ome of the

the

etc.

In other words,

superpostulate

n dyadic

Google

hypergrafs

set

in

chosen

from the analytic question

determines

arbitrarily,

but--

once

chosen--regarded

as

fixed.

That

is

ic change from abc, where & is a varisble,

not

to

lbe,

where

"oonstent",

but

to

1

is

a

p,bs,

"constant",

where

p,

is

or

to

say,

unalyt-

like b ajd ¢, is

2bc,

&

18

the

where

2

parameter.

is

&

Hence, we muy say, indifferently, FRet The eomplete set

of n dysdio hypergrafs,

one

of

qitions:

these

Fig. 25, or any arbitrarily

hypergrafs,

O.

K hes

1.

is

determined

just n (finite)

by

the

distinct

chosen

following

elements.

For the stratigrafic K element B,, wow K element

2.

p,ep,d A p

2.

con-

Py4,

£aur

there

stratigrafic just

are

Permutivity

"

Google

nine

elements such

are

But the peretill

permutivitiee,

4°+{n-4)°

(1,43 )4(n-4),

, Fig.

undeas

41;

, Pig. 4%;

.

Permutivity

'21‘20)”"'4)0

, Fig.

4%;

" "

(2548,)14(n-4),

, Fig.

44;

(20‘20)0““‘4’0

, Pig.

45; ;

"

4'({n-4)°

, PMg.

46;

"

11&5'{(11—4):,

, FMg.

47;

"

41&ln-4)°

. Fg.

40.

These

eight

permutivities

require

no

Multiplicstionsl Permutivities “Tyadic. e tourth siratlgraty, 4,

30

explenution.

,(n-4), , dtermines,

in rddition to the above eight permutfvities,’also s ninth.

We may impose upon the 26 validands of Fig. 40 enough conditions to mutation and the changes

reduce of the

Fig. 40 to Fig. 48. In this case, elements 1 and 2 &lone changes the

peruutution of the hypergraf.

the perhypergraf,

the elemerts 2 and 4 slone elso But the simultaneous permutation

of

1 und 2, and also of # and 4, leaves the hypergraf unchenged. This

esimultaneous

permutation

ample of vhet we shall

In of

of

sets

¢f

elements

call u multiplicational

is an

ex-

permutivity.

this cs3e, the "multiplication" conslsts of two "factors”, two elemerts each. We shall symbolize this permutivity dy

2.2, or, briefly, by 2% . inum of two

elements,

Just sc isotropy requires u min-

and cyclopermutivity

elements, so a multiplicational permutivity mum of four elements. The ninth permutivi ty determined by 4(

therefore

leads

Permutivity

Por the special to

(4g445)y

2

the two

2 *("'“a

case n-4=4,

permutations,

a minimum of three requires )ln-4)°

, Fig.

the uubigious

(40*40)0

and

&

,

permutivities

Google

of

class

four.

is

48.

(4y+4,)

In & similar way we obtain the trimdic, tetradic,

(m-4)-adic

mini-

)

....

CLASS

S By

PERMUTIVITIES rrecisely

mutivities

unalogous

determired

ctratigrefies,

fies,

3t

the

by

necessery

methods

by the

determining,

dyndic,

we

fifth,the for

trisdic,

each

find

sixth,

...,

of

the

sets

...,

these

per-

s-th

strutigra-

(m-s)-udic

ditions to te inposed upon the respective sete

of

the

con-

of validands.

Cluse 5 Permutivities

= For exauple, among the permutivities determined by tae f£th stratigruty, 5 )(n-5)_ , sre the following [We omit, in

each

cuse,

5y

4 (1y44),

the

the

(25485),

(afitzu),(zvzv),

sixth

stratigrafy,

B

sTnlTarly, rmong

ccse,

the

(1345,),

(1142043,),

(2142,),

&

6 Permutivities

esch

isotropy,

5 ,111o44).

Class

=

muin

6(

nain

la,‘oso).

134(20420)0

{21#5

K

)(n-slo

5 (303,

12102

),

o (330,

51,

),

5

v

deteriined by the [We

onit,

in

"b{n-G)O"].

(3043 )

1, (2443,)) o

21402042005

(492,), (llizgfilo)' (2%e25), (a42)), G lelfiz“),

» 13+(242);

, are, following

(2084 ), (31¢5°).

2

),(1,02

the permutivities

isotropy,

(258450,

"&(n-s)o"].

{20020&20)0

»

4 204(2,42);

(13450, (2y04),

(20020‘20)1

.

{2,,‘20‘29)‘,,

[o2.2. utionsl" per per2 3 [=2.2.2, & — "mltiplicutionsl"

mutivityl. PERMUTIVITY

SUPERPOSTULATE

SETS

“hen we 8dd, to © given strutigrufy superpos tulate set, enough conditions upon the velidunds to deteruine the permutivity of all the stratigrafic elements, we obtain complete (“"cetegorical™) superpos tulute sets. Trese we ciwll call ermutivity supe-postulste Thus we proceed, by s

sets. definite

method,

froz

complele

isotropy superpostulute sets, throsgh incomplete stratigrefy superpostulute sets, to couplele pertativity sugerpostul.te sets.

Google

APPLICATIONS In

the

detailed

tke

preceding

exposition

use

of the

rermutivity.

proof,

new

We

of

sections

presemted

our

method,

now

mention

concepts

shall

we

of

32 a

somewnat

this

method

very

briefly,

isotropy,

involving

stretigrafy,

some of the more inportant applications

Atomic

Poetulute

end

and

without

of this method.

Sets

#,ary In permutiv ty superpostalete set, we replace

all the validands

missible

"4's™

znd

postulates.

Destruti%r\:flzed T

1his

set

by an sppropriete "-'s",

Atomic

we

obtain

Postulate

of atomlc

set of validata,

u

complete

Sets

postulates,

we

set

mey

of

by peratomic

suppress

the

mention of stretigrafic eleuents by "carouflsging" these elements as "constants". [This accounts for the existence of

the

"principle

of

duality"

in

algebrus and projective geometry. elemengs are suppressed, we have

city".

Destratigrofized

T

Deatomized

. e we may combine

such

subjects

as

Where several the "principle

DPostulute

Boolean

stretigrafic of rultipli-

Seis

atomlo postulutes in various

wnys into single non-atonic postulates. e then obtain ordinary postulate sets of current mauthemsmtical logic.

Anslytic Katabaeis Ve may

tnus

proceed,

by definite

stepe,

from

the

isotropy

to stratigrefy, then to permutivity, then to atomic postulates and finelly to "ordinary postulates”.

Anslytic Anebasis S 27 aT% reverse the process, begln with ordinary pstulates, and ond with isotropy super postulates. Waximal

Inde¥endsnoe

A set

of

any

two

p

postulates

such

that

no

p-1

imply the p-th is called an independent set.

B

are

u

very

postulates

imply the whole of B. large

psrt--of

of all the nostulates A.

For

perdence.

shall

this

mean

e

By

of

an

independent

A may imply, however,

B.

Likewise,

a sst

set,

the

Thus, 4

postulates

if A and

does

not

u part--perheps

subset

consisting

except A mey imply & very lerge part of

reason we call such muximal

the

of

independence,

independence,

of postulates

Google

on

such no

the

minimal

other

postulate

hand,

implies

inde-

we

any

part of any other postulute. It is not dif?icult to sesthat any atomic postulite set, ne we have defined it, is maximally independent [See the wriferd abstract entitled "Lutually Prime Postulates”, Bulletin of the Americal Lethem tical Society, warch, 1916, p. 287). Coneistenc;

-z

'e-i_én e atomic postulate

either "+" elonme, self-consistent.

Tnfluence”

or "-" alone. A4ny two atomic

which

consequently

are

always

just,validatum,

It is there?are postulates have

mutually

no poseibility

fere with any other.

involves

one

exclusive.

for one atomic

1. e.,

bound to "spheres

There

postulate

i

be of

to inter

They are therefore bound to be inter-

consistent.

Completeness o superpostulate sets, and thus atomic postulate sets,

ure

constructed

according

to a definite

formula,

and

80 we can ulways esccrtein whether or not we have included all the atomic postulates gereruted by the formula. Hence, W6 know whether or not our postulate set is complete ("cate-

gorical").

Postulational

Ey

the

Te::hnl%ne

use

of

atomic

postulates

we are

emabled

to

solve

the three importent protlems of postulaticnal techaique, viz., coneis tency, independence, und completeness, by a non-interretutional method. And we ehow tit this holds for nomHinfte oo rell ee for.firite cases.

Order A”theory Iypes of ordinal isotropy, strutigrufy, und permutivity can be developed, tu u greut extent parellel to the reletionsl isotropy, Strutigrafy, &nd permutivity expounded above. By mewns of tuis ordinal theory, all of our results nay be extended to the cuce of trunsfinite order types. Cardinoid

The

Numbers

notion of we do not

concept

of

permutivity

gcurdinal number to Jefinme here). The

used especinlly

for those

licetive

seems

tion

of

esome

equivalent

oxiom,

of

ensbles

us

to

generalize

the

that of cardinoid number (which theory of cerdinoid number is

system

functions

Zermelo's

otherwise

axiom,

unsvoidable.

where the LSsunor

of

the

multip-

Equivalence of System Functions = Any two ey tem functionsare shown to be equivalent, i.e uniquely intertrenslatable, if they ere cocardinsl. This

=00 GOogle

holds:

(1)

(2)

for

finite

for

system

(co-ordinel

systew

functions;

34

tunctions;

or non-co-ordinal)

(3) for lindsble system functions a

class

"which

1In8sble Infinite

class. cluse,

hus

the

[We propose

power

of the

to

e 8

cardinel.

Cocardinal ity,

valence.

rother

tionsl

is

inel,

0 far

thun

we

have

with

isotropy,

The genersl

relationsl

been

systems.

not

a

necessary

concerned There

stratigrafy,

and

is

stratigrafy,

havirg

with =~

the same

condition

functions

system

theory

permutivity,

theory of interpretstionsl

isotropy,

a

cach of which is

the Ewo

4iny two cocsrdinold system

valent.

Systems

however,

to cell

continuun"

And we shall speak of & linesbly xnd cf lineubilityl.

(4) for ury two sjstem furctions, kaown

courtable

for

equi-

are also equi-

of

functions

interpreta-

which

relsativity,

leads

just as

to

and permutivity leads to the

general theory of notaticnal relativity,

Speciul Example: Einstein's Relativit e STeTaTs "gererelized Heoty of relativity" may be called: (1)

from the

notational

point

(2)

from

inferpretational

of view,

theory of temsor rels tivity; the

the general theory (1) 1s a special case of whut we theory of notational relativity; what we heve called the general relativity.

point

the

of view,

general the

of motional relativity. have called the general (2) is a special case of theory of interpretational

Foundetions of Deductive Logic Tnstesd of basing deductlive

logic

latee

Muthemtica),

on

such

relations

and

operations &s propositionsl negation, implication, disjunction, or incompatability, and on corresponding "ordinary" postu(e,

g.,

those

of

to base deductive logic propositional isotropy,

superyos tulotional soohical bearings.

Principis

it

is

possible

on the more fundamental notions of strutigrafy, and pertutivity. This

foundation for logic has important philo-

Google

62 by

CAAA'A)

CAAA-0) Catneu) -

(Atncze)

(A1)

£

& z

(ASATATD) (rtncui)

('A%A-3)

w 3 -

.

.

% o)"9 |70

(A1)

"

"0

o

o

o0

2

I

!O

9

9 [™0

=9

0

=0 T

:

0 0

T

=0

Google

UNIVERSITY

I TR0 10 CICCH. QeI {Ql@RIcD lelelelcD

HARVARD

B > Q@@

)

T

2z

[eog

g

|y

g

nvw

naw

|y

g

1€ By

2] 7|7

N v»@

X

ity oy

%

g B €

g

¢

"

ES

@7

o

o T

u AR |

A

g g o

12

|2

UNIVERSITY HARVARD

oo |

v

(nva21)

i, Google

(A1)

//

el

o

"o

argy

g

-0

°0

0 |

2.

n-2)

, —S—— T '

20

B | Bas

K

l

Gac

Bes -8)o¢

BCG

Gee

B

N\ |

(PPavsva) Fig. 32

3.

0-3),

,—/H._—/%

-

.

.

.

-

.

a

£

4

C\ (2

3 |LE4N

E)ad

U

3.%2

t

Ejde

B

(n-3)q

- Ed\

n

©PePsvivs) izeaty

(GO

8lC

Fig. 33

\\ HARVARD UNIVERSITY

(2.+2),

@2, 3

Gau

Bab

“

Goo |

Baa

HARVARD

UNIVERSITY

Google

(A

Fig 35

0-3),

©-3),

(PrPePs-VaVs)

Fig. 36

sutzeary GOOGle

HARVARD

UNIVERSITY

\’7(\@ 3

2

4 ool

)

6_%

3@

ue

-

7

L“@

E/ad

QQ

da

|

!

Cdd

(P-Peps-vavs)

Fig. 3T

oigitzed by (GOC -8lC

HARVARD UNIVERSITY

@at 3, ,

I

“'3)< °

N

E

Ea

’

(PiP2PV4sV) Fig. 38

Go

8lC

HARVARD UNIVERSITY

(et 3,

Wb

©-3),

(PrPeps” VaVs)

Fig 35

i,

Google

HARVARD

UNIVERSIT

4

2

3

4

S

6

n

1

Nao

Moo

MNae

Mo

Mas

2|

Mea

TNbb.

MNoe

Mba

MNbe.

3

Na |

Moo |

M|

Nt

Nee

4|

Mda |

Mas|

Mae |

Maa

Mee

s

Nes

6

@A

|

Mea | Meo|

Mee | Mew

& Nes

L

n (1234.vs\) Fi5.40

izeary

GO

81C

HARVARD UNIVERSIT

y 2 + | - |

3 =

-+

-

3l

+

2l

|

-

-

. o+

> -

2 =

2l

-

|+ |

=

=+

= |

o+

sl

|

Fig 1

s

\

+ |

-]

=

-

|

3

3|+ | = |

=

-

+

2|

2 =

s+ 2|

+

-

—

-

+

3

1

3

'

| - |

-

- |

+

_

o=

-

+

2

1

o | O | o

Oy |

Oe |

Chyy

3|

Oy |

Oy |

e |

Oy

2|

Oy | O |

Fig. 7

Olea |

2

Fig &

o | o, | o

2

s

Fig. 5

2

O |

2

2|

O

Chee

Ou|

i

3

'

!

3|

Ohbe |

3

O | On

Ol

Fig &

3

+

Fig.3

-

Fig 4

!

'

3 =

| -

— | = |

Fig 2

2

-+

2 + | - |

O

Os |

Os

Fig ®

2

3

2

'

Oty

3|

Oy | Os |

Ol

3|

Oy | Ose |

Oy | Oy | O

U]

s | O | Oy

2|

O | Ou | O

O, |

2|

O |

O |

O

Fig 1o

"

Oa | O

Fig 1

i

GO

gle

>

O | O |

T

Oy

—

O

Fgne

o . e~ UNIVERSHF ARVARD

—

4

e

| C

o

s

;

Mo

I

4o

-

JF

s

T q

o9

Nee

ea

¥ n

EPPsPesvy Fig. 41

i,

Google

HARVARD

UNIVERSITY

el

-9,

3.

I

Nel

Nee

T Ter

(PiPsPsPa-Vs™) Fig-42

i,

Google

2

eV

|

Noe| Men]

2]

Moo|

Mee

Mes|

3

2 o e a | ee

aY

s

Nee

M Nee|

|

Mse Mea

Nee

s

A

n

N | Ta| Teo|

Mee

n"‘

e (PPePsPeVs ) Fig 43

sutzeary GOOGle

D UNIVERSITY

2. 1

2,

sy

2.

AL

2. 2

| Moo Mt

3

4

n

lac

2| Mas| Mo 3 4

|

Nea

Mea |

o2

Mee| M [~

Tea | Mee.

S

n

Mae

i

Mee

bigiized by (GO

vglC

H;RV;‘RD i‘]m‘vE‘Rx\"“

—_— ]

24

1

2

Mee|

Nev

2 [ Mab| 3

2.

4

-

M

Mae

3

O-4), 4

Nee

5

3

a

4

N

Mag | Neb|

Teb| Mool

s ©

(n-4),

m‘“

N

.

fl,;

(PP2PsPs- VsV F'B‘ 45

vigitized by (GO

glC

H:PVL‘PDWVEPW'

4

-

2| N 3|

©:3),

Jas|

Mec | Noa

/" Mec

N

Nab|

e e-al

fl ca

Nee

N (PPLPsPsVs V) Fig.

vigitized by (GO

glC

46

H:PVL‘PDWVEPW'

Naa

Mab

Mo

Moe [ Mg

Toa | Noa NI, b

flba

Moe| Mhe

Nes (PP=PsPa- V5 v0) Fig.47

izeary

GO

81C

HARVARD UNIVERSIT

2

.

@9

1 | Tea| et Mec| Medf

asd 2178 Tl Mt M 3 [ Mea|

Meb| Mee|

Nedf

4| Meb| Nea | Nea Mee|

e

|

flu

Mee | Nee

N

Ner (PP2PsPe- Vs V) Fig. 48

izeary

GO

81C

HARVARD UNIVERSIT

2

PO

| O

| Ky

2 s

4l o, |

|

.

o,

Fig 14

(Nd= B Fig 1

3

12

1 [o] o 2|,

n[on] e

3

rig

D

n

Fig

HARVARD

a

17

UNIVERSITY

i elA——————-

,

b

[

(%2)

-

-

-

-

—

-

-

-

+

B

—_—