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The Calculus of Individuals and Its UsesAuthor(s): Henry S. Leonard and Nelson GoodmanReviewed work(s):Source: The Journal of Symbolic Logic, Vol. 5, No. 2 (Jun., 1940), pp. 45-55Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2266169.

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Tax

JoURNAL

r Saouc

LOGIC

Volume

5.

Number 2,

June

190

THE CALCULUS

OF

INDIVIDUALS

AND ITS

USES1

HENRY

S.

LEONARD AND NELSON

GOODMAN

I. An

individualor whole

we

understand

o be whatever

s

represented

n

any

given discourse

by signs belonging to

the lowest

logical type of

which that

discoursemakes

use. What

is

conceived

as

an

individual

nd what as

a

class

is

thus relative

to

the

discourse

within

whichthe conception

ccurs. One task of

applied logic

is to

determinewhich entities re to be construed

s

individuals

and

which

as

classes

when the

purpose s the development

f a comprehensive

systematicdiscourse.

The

concept

of an

individual

nd

that

of a classmay be

regarded s

different

devicesfor istinguishing

ne

segment

f the totaluniverse rom

ll that

remains.

In bothcases, the

differentiatedegment s potentially

ivisible, nd

may even

be physically

discontinuous.

The

difference

n

the concepts ies in

this: that

to conceive a

segment

s a

whole

or

individual

offers o suggestion

s to what

these

subdivisions,

f

any,

must

be, whereas to

conceive

a segment s a class

imposes

a definite

cheme

of

subdivision-into

subclassesand

members.2

The relations

of

segments

f

the

universe

are treated

n traditional

ogistic

at two places, first n its theoremsconcerning he identityand diversityof

individuals, nd

second

n

its

calculus

of membership

nd class-inclusion.

But

further

elations

of

segments

nd

of

classes

frequently emand consideration.

For example,

what s

the

relation f

theclass of

windows o the

class ofbuildings?

No member f

either

lass

is a

member f the other,

nor are

any of the segments

isolated

by the

one concept

denticalwith egments

solated by the other.

Yet

the classes

themselves

have

a

very

definite

elation

n

that

each

window

s

a

part

of some

building.

We cannot

express

his

fact n

the

anguage

of

a

logistic

which

acks

a

part-whole

elation

between

ndividualsunless,bymaking

use of

some special physicaltheory,we raise thelogicaltypeof each window nd each

building

to the

level

of a

class-say

a

class

of

atoms-such

that

any

class of

atoms

that

is a

window

will

be

included

class-inclusion)

n

some class

that

is

a

building.

Such an

unfortunate

ependence

of

logical

formulation

pon

the

discovery

nd

adoption

of a

special physical

theory,

r even

upon

the

presump-

tion that

such a

suitable

theory

ould in

every

ase

be

discovered

n

the

course

of

time,

indicates

serious

deficiencies

n

the

ordinary ogistic.

Furthermore,

raising

of

type

ike

that

illustrated

bove

is often

precluded

n

a constructional

systemby

other

onsiderations

overning

he choice

of

primitive

deas.

Received

July 28,

1939.

1

A

somewhat

elaborated

version

of

a

paper

read

in

Cambridge,

Mass.,

before a

joint

meeting

of the Association

for

Symbolic Logic

and the

American

Philosophical

Association,

Eastern

Division,

on

December

28,

1936.

2

The

relation

is

somewhat

analogous

to

the

more

familiar

one between

classial

and

serial

concepts,

dealing

as

they

do

with the same

material,

but

in a manner that makes

the

latter

more

highly

specialized.

45

8/10/2019 The Calculus of Individuals and Its Uses 1940

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46

HENRY

S.

LEONARD

AND

NELSON

GOODMAN

The

ordinary ogistic

defines

o

relations

between

ndividuals

xcept

dentity

and

diversity.

A

calculus

of

ndividuals hat

ntroduces ther

elations,

uch

as

thepart-whole

elation,

would

obviously

be

very

convenient;'but what

chiefly

concernsus

in

this paper is the

general

applicability

f

such a

calculus

tothe

solutionofcertain ogico-philosophical roblems.

The

calculus

of

ndividualswe shall

employ

s

formally

ndistinguishable

rom

the general

theory

f

manifolds

eveloped

by

Lesniewski.4

Leiniewski's

pur-

pose, quite

differentrom

urs,

was to

establish

a

general

theory

f

manifolds

that would not

be

subject

to

Russell's

paradox;

but

since he

excludes

henotion

of

a null

class,

his

formal

ystem

s

virtually

he

same as that which

we

interpret

as

a calculus

of

individuals.

Inasmuch

as

his

system

s rather

naccessible,

lacksmany useful

definitions,

nd is set forth

n

the

language

of

an

unfamiliar

logical doctrine

nd

in words

rather

han

symbols,

we

shall

attempt in Part

II)

to restate the calculus in more useable form,with additionaldefinitions,

practical

notation

nd

a

transparent

nglish

terminology.

In

Part

III

we

shall

explain

how

this

calculus enables us to

describe

enerally

ertain

mportant,

ut

oftenneglected

properties

f

relations,

nd

thereby ontributes o the

clarifica-

tion

of

many

philosophical

roblems.

II. The

general

features

f

the

abstract

calculus

may perhaps

be

most

readily

apprehended

by

comparison

with

the

Boolean

algebra

of

classes. It

involves

operations

of

addition,

multiplication,

nd

negation,

a

part-whole

relation

analogousto class-inclusionnd an element nalogousto theBoolean universal

class.

It differsrom he

Boolean

analogue

n

waysconsequent

pon the

refusal

to

postulate

a null

element,

lthough

the

primitive

elation

of

"discreteness"

may be

correlatedwith

the Boolean

function

x*

y

0".

In

the

ight

f

this

nalogy,

he

characteristic

ropositions f

the

calculus

may

be

generally

escribed: To

any

analytic

proposition

f

the

Boolean

algebra

will

correspond

postulate

or

theorem f this

calculus

provided

that,

when n the

Boolean

proposition

very expression

f

the

form x

y=0"

is

replaced by

an

expression

f

the

form

x

is

discrete

from

y",

no

reference

o

the null

element

remains nd everyproduct nd negation s eitherdeduciblyunequal to thenull

element

or else

is

conditionally

ffirmed

o

be

unequal

to it.

From

the

three

postulates

presented

elow)

of the formal

alculus,

enough

heorems

ave

been

deduced to indicate

that

this characterization

s

accurate.

Some

illustrative

theorems

ppear

n the

sequel.

For the

formal

stablishment

f

the

calculus,

the

symbolism

nd

logistic

of

Whitehead nd

Russell's

Principia

mathematics

ave

been

employed

n

order o

secure correlationwith other

logical

doctrines.

Only

the one

primitive

dea

already

mentioned s

required:

the

dyadic

propositional

unction,

r

relation,

written xly" and here nterpretedomeanthat theindividualswhich re its

arguments

ave no

part

n

common,

hat

they

re discrete.'

In

our

nterpreta-

3Since

this paper

was

presented, the

convenience of

such a calculus of

individuals has

been well

illustrated

by Dr. J. H.

Woodger's

Axiomaticmethod in

biology 1937).

4

In

0

podstawach

matematyki

in Polish),

Przeglqd

filozoficzny,

vols.

30-34 (1927-31).

6

Legniewski

employs

discreteness as

his primitive

relation in

the final version

of his

system. See

Chapter X of

his

above-mentioned

paper.

8/10/2019 The Calculus of Individuals and Its Uses 1940

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CALCULUS

OF

INDIVIDUALS

AND

ITS USES

47

tion,furthermore,

arts and commonparts

need not necessarily

e spatial

parts.

Thus

in our applications

of

the calculus to philosophic

roblems, wo concrete

entities,

o be taken

as

discrete,have

not onlyto be spatially

discrete,

ut also

temporally

iscrete,

iscrete

n

color,etc., etc.

In termsof the one primitive dea just described,otherconceptsmay be

defined s follows:

I.01

X