paradox of the stone

8/10/2019 Paradox of the Stone

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THE PARADOX OF THE STONE

A. (i)

Either

God can create

a

stone which He cannot lift, or He

cannot create

a

stone which

He cannot lift.

(2)

If God can create a stone which He cannot

lift,

then

He is

not

omnipotent (since

He

cannot

lift the

stone

in

question).

(3)

If

God cannot create a stone

which He cannot lift, then He is

not omnipotent (since He cannot create the stone in question).

(4) Therefore, God is not omnipotent.

Mr.

Mavrodes has offered

a

solution to the familiar paradox above;,

but

it

is erroneous. lMlavrodes states that he

assumes the existence of

God,2

and then

reasons (in

pseudo-dilemma fashion) as follows. God is

either

omnipotent or He is not. If we assume that He is not omnip-

otent,

the task

of creating

a

stone

which

He

cannot

lift is

not self^

contradictory.

And

we can conclude

that

God

is

not omnipotent on the

grounds that both His ability and His inability to perform this task

imply that He is not omnipotent.

But to prove His non-omnipotence

in

this way is trivial. To be

significant [the paradoxical argument]

must

derive this same

conclusion from

the

assumption

that God is omnip-

otent;

that

is,

it

must show that the

assumption

of

the

omnipotence

of

God leads to a

reductio. However, on

the

assumption

that

God is

omnipotent,

the task

of creating

a

stone which God cannot

lift is

self-contradictory. Since inability to perform a self-contradictory

task

does not imply a limitation on the

agent,

one

of

the

premises

of the

paradoxical argument-premise

A(3)-is false.

The

argument is,

in

consequence, either insignificant

or unsound.

There are

many objections to this solution.

First,

the

paradoxical

argument need not be represented as a

reductio;

in A

it is

a

dilemma.

Mavrodes'

reasoning implies

that

the

paradoxical argument

must

either

assume

that God is omnipotent or assume

that He is not

omnip-

otent. This

is

simply

false: neither

assumption

need be

made,

and

neither is made

in

A. Second, a

stone which God cannot lift is

self-contradictory-on

the

assumption

that God

is

omnipotent-only

1

George I. Mavrodes, Some Puzzles Concerning Omnipotence, Philo-

sophicalReview,LXXII

(i963),

22I-223. The heart of his solution

is

contained

in pars. 6, 7, and i i.

2

See n.

2,

p.

22I.

74





THE PARADOX

OF THE STONE

if God is omnipotent is necessarily

true. Russell can

lift any stone

is a contingent

statement. Consequently, if we assume

that Russell can

lift any stone we

are thereby committed only to saying

that creating a

stone which

Russell cannot lift

is a

task which in fact

cannot

be per-

formed by Russell

or anyone else.

Third,

if

God

is

omnipotent is

necessarily true-as Mavrodes must

claim for his solution

to work-

then

his

assumption

that

God

exists

begs

the

question

of the

para-

doxical argument. For what

the argument really tries

to establish is

that the existence of

an

omnipotent

being is logically impossible.

Fourth, the claim that inability to

perform

a

self-contradictory

task is

no limitation on

the agent is not entirely uncontroversial.

Descartes

suggested

that

an

omnipotent

God must be able to

perform

such

self-

contradictory

tasks as

making

a

mountain without

a

valley

and

arrang-

ing that the sum of one and two is

not three.3 No doubt

Mavrodes and

Descartes

have

different theories

about the nature of contradictions;

but

that is

part

of the

controversy.

Mavrodes has been

led

astray

by version

A

of the

paradox,

which

apparently

seeks to prove that God

is

not omnipotent.

Concentration

on

this

version, together

with the inclination to say that God

is

by

defini-

tion

omnipotent,

leads

straight

to

the

conclusion

that the

paradox

is

specious.

For

if

God is

by

definition

omnipotent,

then, obviously,

creating

a

stone

which

God

(an

omnipotent being

who can

lift

any

stone) cannot

lift is

a task

whose description

is

self-contradictory.

What the

paradox

of

the

stone

really seeks to prove

is that the notion

of

an

omnipotent

being

is

logically inconsistent-that

is,

that the

3Harry G. Frankfurt,

The Logic of Omnipotence,

Philosophical Review,

LXXIII

(i964),

262-263.

The relevant passage from Descartes is quoted by

Frankfurt in

a

long footnote.

Mavrodes assumes (on

his significant interpretation

of the paradox) that

creating

a

stone which

God

cannot lift

is a

self-contradictory task, and contends

that God therefore cannot perform it. This forces

him onto the second horn

of

dilemma A, which

he tries to

break by arguing

that

inability to perform

a

self-contradictory task

does not imply

a

limitation

on the agent. Frankfurt also

assumes that creating

a stone which God cannot lift is a self-contradictory

task, but contends with

Descartes (for the sake

of argument) that God can

nevertheless

perform

it.

This

forces

him

onto

the first

horn

of the

dilemma,

which he tries to break with the following argument. If God can perform the

self-contradictory

task of creating a stone which

He cannot lift, then

He can

just

as

easily perform

the additional self-contradictory

task of

lifting

the stone

which

He

(creates

and) cannot

lift.

Frankfurt's

fundamental error

is the

same

as

Mavrodes': both suppose

that on any significant

interpretation the paradox

sets for God the self-contradictory task of creating

a

stone which God (an

omnipotent being who

can

lift

any stone)

cannot

lift.

75





C. WADE SAVAGE

existence

f an omnipotent

eing,

God or any other, s

logically

mpossible. t

tries to

do this by focusing on the

perfectly

consistent task

of creating

a stone which

the creator

cannot

lift. The essence

of the

argument is

that an omnipotent

being must

be able to perform

this task

and

yet

cannot

perform

the task.

Stated

in

its

clearest form,

the paradoxical

argument of

the stone

is

as follows.

Where

x is any being:

B.

(i)

Either x can create

a

stone which

x cannot lift,

or x cannot

create

a

stone

which

x

cannot lift.

(2) If x can create a stone which x cannot lift, then, necessarily,

there is at least one task

which x cannot

perform

(namely,

lift

the stone

in

question).

(3)

If

x cannot

create

a stone

which x cannot

lift, then,

necessarily,

there is

at least one task

which

x cannot perform

(namely,

create the

stone

in

question).

(4)

Hence,

there

is at

least

one

task

which x cannot perform.

(5) If x is an omnipotent being, then x can perform any task.

(6)

Therefore,

x

is

not omnipotent.

Since

x is

any being,

this

argument proves

that the existence of

an

omnipotent

being,

God

or

any other,

is

logically

impossible.

It

is

immediately

clear that Mavrodes' solution will not apply

to

this version

of the

paradox.

B

is obviously

a significant,

nontrivial

argument.

But

since

it

does not

contain

the word

God,

no critic

can

maintain that B assumesthat God is omnipotent. For the same reason,

the

point

that a

stone

which God cannot

lift

is self-contradictory

is

simply

irrelevant.

Notice

also

that B is

neutral

on the

question

of

whether

inability

to

perform

a

self-contradictory

task

is a limitation

on

the

agent's power.

We

can,

however,

replace

every

occurrence

of

task with task whose description

is not

self-contradictory

without

damaging

the

argument

in

any

way.

The

paradox

does

have a

correct

solution, though

a different

one

from that offered by Mavrodes. The two solutions are similar in that

both consist

in

arguing

that an

agent's

inability to create

a

stone which

he cannot

lift

does not

entail a

limitation on

his power.

But here

the

similarity

ends.

For,

as

we shall see

presently,

the

basis of the correct

solution

is

not that creating

a

stone

which the creator

cannot

lift

is

a

self-contradictory

task

(which

it is

not).

Consequently,

the

correct

76





THE PARADOX OF THE

STONE

solution side-steps the question

of whether

an agent's inability to

perform a self-contradictory

task is a limitation

on his power.

The fallacy

in the paradox of the stone lies in the falsity of the

second horn-B(3)-of

its dilemma: x

can create a

stone which

x

cannot

lift does indeed entail that there

is a task which

x cannot

perform and, consequently,

does entail

that x is not omnipotent.

However,

x cannot create

a

stone which

x cannot lift

does not entail

that there is a task which

x cannot perform and, consequently,

does

not entail that

x is not omnipotent. That

the entailment

seems to hold is

explained

by the misleading

character

of the statement x cannot

create

a

stone which x

cannot lift.

The phrase cannot create a

stone seems to imply that

there is a task

which x cannot

perform and,

consequently, seems to

imply that x is

limited in power. But this

illusion vanishes

on analysis:

x cannot create a stone

which x cannot

lift can only

mean If x

can create a stone, then x can

lift it. It is

obvious that the latter statement

does

not entail that x is limited

in

power.

A

schematic

representation

of B(i)-B(3) will bring

our point into

sharper focus. Let S

=

stone, C

=

can

create, and

L - can lift;

let x be any being; and let the universe of discourse be conceivable

entities.

Then

we obtain:

C.

(I) (]y) (Sy Cxy -Lxy)

v

-(3y)

(Sy Cxy -Lxy).

(2) (]y)

(Sy*

Cxy -Lxy)

D

(]y)

(Sy -Lxy).

(3)

-(3Y)

(SY CXY -LXY)

O (3Y)

(SY

-

CXY)

That

the second alternative

in

C( i)

is

equivalent to (y) [ (Sy

-

Cxy)

D

Lxy]

schematically explains our interpretation of x cannot create a stone

which x cannot

lift as

meaning

If

x can

create

a

stone,

then

x

can

lift it. It

is now quite

clear where the fallacy in the

paradoxical

argument lies. Although

C(2) is logically

true,

C(3) is not.

(3y)(Sy

-Cxy -Lxy)

logically implies (3y)(Sy

--Lxy).

But

-

(y) (Sx

-

Cxy

-

Lxy)

does not

logically

imply (]y)

(Sy

-

Cxy)

;

nor does

it

logically imply

(]v)

(Sy

-

Lxy).

In

general, x cannot

create

a

stone which x cannot

lift

does

not

logically

imply

There

is

a task which x cannot perform.

For some reason

the

above analysis

does not

completely

remove

the

inclination

to think that an

agent's inability

to create a stone which he

himself

cannot

lift does entail his inability

to perform some task,

does

entail

a

limitation on

his

power.

The

reason becomes clear when

we

consider the task

of creating

a

stone which

someone other

than the

77





C. WADE SAVAGE

creator cannot lift. Suppose

that

y cannot

lift

any stone heavier than

seventy pounds. Now

if

x cannot create

a

stone which

y

cannot lift,

then

x

cannot create

a

stone

heavier

than

seventy pounds, and is indeed

limited in power. But

suppose that y is omnipotent and can lift stones

of

any poundage.

Then

x's inability

to

create

a stone

which y cannot

lift does not necessarily

constitute a limitation on x's power. For x may

be able to create stones of

any poundage, althoughy

can lift

any stone

which x

creates.

If

y can

lift

stones of any poundage,

and

x cannot

create

a

stone heavier than

seventy pounds,

then

x

cannot

create a

stone

whichy cannot lift,

and x is

limited

in

power.

But

if

x can create

stones of

any poundage, andy

can

lift

stones of any poundage, then x

cannot create

a

stone

which

y

cannot

lift, and

yet

x is

not thereby

limited

in

power. Now it

is easy to see that precisely parallel considera-

tions obtain where x is both stone-creator and

stone-lifter.

The logical facts above

may

be summarized

as

follows.

Whether

x

- y

or x

#

y,

x's

inability

to

create

a

stone which

y

cannot lift con-

stitutes

a

limitation on

x's power only

if

(i) x

is unable to create

stones

of

any poundage,

or

(ii)

y

is

unable to

lift

stones of

any poundage. And,

since either (i) or (ii)

may

be

false,

x cannot create a stone which

y

cannot lift does not entail x is limited in power. This logical point

is obscured, however, by the normal context of our discussions of

abilities and inabilities. Since such discussions are normally restricted

to beings who

are

limited

in

their stone-creating, stone-lifting, and

other

abilities,

the

inability

of

a

being

to create

a

stone which he him-

self or some other

being

cannot

lift

normally

constitutes

a

limitation

on

his power. And this

produces the

illusion

that

a

being's inability to

create

a

stone which he

himself

or some other

being

cannot

lift

neces-

sarily

constitutes a limitation on his power, the illusion that x cannot

create

a

stone which

y

cannot lift

(where

either

x

==

y

or

x

7

y)

entails

x

is

limited

in

power.

Since our discussions

normally

concern

beings

of limited

power,

the

erroneous belief that x cannot

create

a

stone which x cannot

lift

entails x is limited

in

power

will

normally

cause

no

difficulty.

But

we

must beware when

the

discussion turns

to

God-a

being

who

is

presumably unlimited

in

power. God's inability to create

a stone

which

He cannot lift is a limitation on His power only if (i) He is unable to

create

stones of

any

poundage,

or

(ii)

He is

unable

to lift

stones

of

any

poundage-that

is,

only

if

He is limited

in

His

power

of

stone-creating

or His

power

of

stone-lifting.

But until

it

has been

proved

otherwise-

and it is

difficult to

see

how

this

could

be done-we are

free

to

suppose

that

God suffers

neither of

these

limitations. On

this

supposition

God's

78





THE PARADOX OF THE

STONE

inability

to

create

a

stone

which

He

cannot lift

is

nothing

more

nor

less than a

necessary

consequence of two facets of His

omnipotence.4

For if

God is

omnipotent,

then

He can create stones of

any poundage

and

lift stones of any

poundage.

And

God can create stones

of any

poundage, and God can lift

stones of any

poundage entails God

cannot create a stone

which

He

cannot lift.

C. WADE SAVAGE

University of

California,

Los

Angeles

4

Mavrodes apparently sees

this point in the

last three paragraphs of

his

article. But

his

insight

is

vitiated

by

his

earlier

mistaken

attempt

to

solve

the

paradox.

79

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