American
Philosophical
Volume
Quarterly
2, April
1980
17, Number
V. A MODAL MODEL FOR PROVING THE
EXISTENCE OF GOD
E. MAYDOLE
ROBERT
the first section of this paper, I employ an
to show that the
IN ontological
type argument
a
in
standard
of
God
system of
implies,
possibility
existence
the
and
modal
logic,
uniqueness
quantified
of God. In the second section, I argue that the very
In the third
idea of a supreme being ismeaningful.
I use
section,
a
type
cosmological
to prove
argument
that it is possible for God to exist. In the fourth and
of
final section, I briefly discuss the plausibility
in
used
the
first
section.
modal
the
logic
adopting
I. The
of God Implies the Existence
God
Possibility
of
In keeping with the idea of God as a supreme being,
Anselm
St.
thought of God as a being no greater than
which can be conceived. This suggests two closely
related, yet distinct things about what itmeans to be
supreme. It suggests first that it is not possible for
there
to be
something
is greater
which
than
this be abbreviated
by "Sx." That there is a
supreme being is then expressed by "(3x)Sx"; and
that it is possible for a supreme being to exist is
expressed by "0(3x)Sx".
The proof that "0(3x)Sx"
implies "(3x)SxT is
carried out in an S5 system of quantified modal logic
called "CM."1 CM is the modal extension of the
natural deduction
order logic with
system of first
' '
of
is the only modal
identity
Copi's Symbolic Logic.2
"~ D ~." The
primitive of CM, with 'O' defined as
are
modal inference rules of CM
Necessity Elimina?
tion (NE: [JA/A) and Necessity
Introduction
(NI:
that
every assumption on which A
A/[JA, provided
depends is completely modalized).3
The proof that "0(3x)Sx" implies "(3x)Sx" can now
be stated as follows :
Let
1. 0(3x)Sx
2. ~(3x)Sx
~
D ~(3x)Sx
3.
D
4.
(3x)Sx
~
5. D
(3x)Sx
a supreme
being; and thus that a supreme being could not be
greater than itself. It suggests secondly that it is not
possible for there to be something distinct from a
supreme being than which that supreme being is not
among distinct
greater; and thus that co-supremity
is
beings
impossible.
<
<
Let
"(V))"
only
"there
"Gxy"
=
<-',
', 'v',
stand
for
"not,"
?=>',
"and,"
if," "it
exists
be
>, <D',
'O',
"(37)",
"or," "if, then,"
"it is possible
is necessary
that,"
a
y," and "for every
for "* is greater
short
and
6.
"if and
that,"
Let
j," respectively.
"* = /'
than /'
and
~0(3y)(~x=y
there
turns
namely:
out,
i,def.ofO'
NI
~
(3x)Sx
3, 4 Conj.
IP.
3-5
is of necessity
interestingly,
(x)(z)((Sx
one
supreme
only
to be a theorem
Sz)3
x = z). Here
being,
of CM,
is the
proof:
1. (Sx Sz)
2. [(~ 0(3y)Gyx
for "x is identical with y" The idea of a supreme
being x can then be formally expressed as follows :
~0(3y)Gyx
~
2,
(3x)Sx
That
God,
premise
(- 0(3y)Gyz
~Gxy).
1The
-
0(3y)(~x=y
0(3y)(~z=y
i,def. W
~Gxy))
Gzy))]
could also be carried out in other S5 systems of modal
natural deduction
and Hughes
system Q-M,
logic, such as Massey's
axiomatic
is described
in Appendix
G of Gerald J. Massey's Understanding Symbolic Logic (New York,
system LPC + S5. Q-M
in Hughes
and Creswell's An Introduction toModal Logic (New York,
1970). LPC + S5 is described
1972), ch. 8.
' '
2
=
Because
does not figure essentially
in the proof of "(3x)Sx" from
Irving M. Copi, Symbolic Logic, 4th edition.
(New York, ' 1973).
'
in the sense that any predicate
could replace = without
in the proof, a logic with
"0(3x)Sx"
any alteration
identity is not actually
' '
=
required. Yet there are two reasons for using a first order logic with identity rather than a plain first order logic. The first is that
occurs in "Sx." The second and main reason is that "(3z)(z = x)" is a theorem schema of a first order
logic with identity, but not of plain
first order logic. The theoremhood
of "(3z)(z = *)" figures in my reply to subsequently
to my proof of "(3x)Sx" from
discussed objection
"0(3x)Sx."
3A wff is
if and only if every predicate
letter and every occurrence
of a variable of the wff occurs within
the
completely modalized
scope of a modal operator.
proof
and Creswell's
135
~ ~
3?
~(3y)Gyz
4. (Vy)~Gyz
~ Gxz
5.
4,
=
8.
7, UI
~Gxz)
=
9. Gxz v x
z
x?z
10.
6, DN, NE, QN
~Gxy)
z
~(~*
8, DeM, DN, Com
9 DS
5,
11. (Sx Sz) 3 x = z
12. D(Vx)(Vz)((Sx
Sz)z>
x=
def.'O'
2, Simp.,
~(3y)(~x=y
Gxy)
7. (Vy)~(~x=y
2-10, CP
n,UG,NI
theorem of CM is Anselm's
interesting
cannot exist
which
that
says
Principle,
perfection
Stated differently,
this principle
says
contingently.4
that it is necessarily the case that a perfect or supreme
being exists only if such a being necessarily exists :
The proof is:
\J((3x)Sxzd a(3x)Sx).
Another
3.
NI
1,
1-2, CP
(3x)Sxz> n(3x)Sx
4. D((3*)S*3
3,NI
\J(3x)Sx)
it also
conceived,
proves
such as a perfect
superlative,
Gaunilo
the
a
raise
similar
existence
of
island. Might
to my
objection
any
not
argument?
Could we not, he might ask, use the modal method of
arguing to show, say, that the possibility of a perfect
island,
an
island
The
implies
answer
conceived,
island?
no
the
greater
existence
than
of
can
which
a
perfect
unique
island
x no greater
than which
one such that it is not possible
can
greater,
be
~
0(3y)(Iy
0(3y)(Iy
Gyx)
~x=y
be
for any
the idea
conceived,
:
is not
completely
not make
does
is the
modalized,
and
we
cannot
use
NI to infer "D - (3x)Pxn from "~(3x)Px"
in order
to prove that "0(3x)Px"
implies "(3x)Px"> although
"is
Does
meaningful?
The
main
greater
it make
in this idea
ingredient
than."
Is this predicate
to
in other
sense,
words,
in terms
of ontological
things
greatness?
as
it were
self evident:
writes
though
compare
Anselm
...
sense.
predicate
if one
observes
he will
the
or no,
nature
that
not
of
all
he perceives,
things
are embraced
in a
of dignity
them
; but that certain
among
single degree
an
are
For
he
who
of
by
degree.
inequality
distinguished
to wood,
in its nature
is superior
doubts
that the horse
excellent
than a horse,
and a man more
does
assuredly
not deserve
~Gxy).
Let this be abbreviated by "Px." "/V says that x is a
the
perfect island. Now "/*" does" not occur within
"~
~
a
in
So
of
modal
scope
operator
(3x)
(3x)Px"
Px"
The proof that "0(3x)Sx" implies "(3x)Sx" will be
an empty formalism if the idea of a supreme being
whether
island to be
as follows
formally
expressed
-
Ix
can
Idea of a Supreme Being
II. The Very
is "no."
Let "Ix" be short for "x is an island." Then
of an
be
0(3y)(By
Gyx)
~x=y
0(3y)(By
~Gxy)"
"
and "Bx" is short for "x is a being." Since ~ (3x)S^x"
is not completely modalized,
he would argue, we
cannot use NI to infer "(3x)S#x" from "0(3x)S*x".
But Gaunilo's disciple overlooks the fact that "Bx"
is logically superfluous in CM. To say that # is a being
is logically the same as saying that there is something
which is identical with x. In other words, "Bx" can be
=
=
x)." But "(3z)(z
x)" is prova?
replaced by "(3z)(z
ble in CM, which makes "S^x" logically equivalent to
"Sx." "0(3x)S?x" implies "(3x)S?x" after all.
The monk Gaunilo objected to St. Anselm's onto
logical argument on the ground that if it proves the
existence of a being no greater than which can be
then
-
"Bx
-
z)
1. (3x)Sx
2. n(3x)Sx
QUARTERLY
we were able to make that kind of move in proving
that "0(3x)Sx" implies "(3x)Sx".
A disciple of Gaunilo might
by
counterobject
that if incomplete modalization
prevents
claiming
the inference of a perfect island from the possibility
of a perfect island, then it also prevents the inference
of a supreme being from the possibility of a supreme
to the expression of "x is a perfect
being. Parallel
island" by "Px," he might say, we should express "x
is a supreme being" by "S?x" rather than by "Sx,"
where "5V' is short for
2, Simp., def. 'O'
3,DN,NE,QN
UI
~
~
6.
~
PHILOSOPHICAL
AMERICAN
136
Hartshorne
the name
of a man.5
joins with Anselm
thus:
the reply
if you ask about
the import of "greater,"
than y insofar as x is, and y is not, something
is, x is greater
to be than not to be." Greater
thus
"which
it is better
And
means
ation
more
superior,
and respect.6
4
See Charles Hartshorne's
The Logic ofPerfection (La Salle, Illinois, 1962), especially
as a premise of his modal argument
for the existence of God.
5 St.
ch. IV.
Anselm, Monologium,
6
Anselm's Discovery (La Salle, Illinois, 1965), pp. 25-26.
Charles Hartshorne,
pp. 49-57.
excellent,
Hartshorne
more
worthy
assumes Anselm's
of admir?
Principle
A MODAL
not seem to help much
But it would
greater
excellent
than"
more
as
than"
of
as
or,
if we
these
then
puzzling,
"is
"is greater
consider
are
terms
other
equally
puzzling. This manner of explication may have been
to Anselm and other medieval
perfectly acceptable
who
viewed
reality
through neo
philosophers
Platonic
an
constituting
It is not perfectly
possessed.
not
do
wear
we
What
require
are
which
terms
those
is an
analysis
clearly
us
tells
Hartshorne
respects.
we
that
may
compare
existence
of
whole,
they
to those of us
acceptable
same
cite obvious
and Malcolm
Hartshorne,
Plantinga
ways in which things may be compared in particular
to the degree of
even
and
excellence,
137
things with respect to that which ismore worthy of
our admiration and respect. We frequently do claim,
and meaningfully
so, that one person ismore worthy
ordered
things being graded according
perfection,
who
as
glasses
with
OF GOD
in
the same work
in the same time and was
accomplish
not
other
but did
fuel,
respects
satisfactory,
require
would
be a superior
engine.8
"is
than." For
perfect
to explicate
to," "is more
superior
of "is
it sometimes
is, in terms
terms
in
than"
EXISTENCE
THE
PROVING
FOR
MODEL
glasses.
in
than"
metaphysical
of "is greater
familiar
meaningful,
and,
our
admiration
and
than
respect
The
another.
loving and charitable person ismore worthy of our
admiration and respect than the greedy millionaire
who continues to swindle his fellows, not because he
the extra
needs
but
goods,
to
he wants
because
simply
from
free
pad his already secure nest egg. Put differently, we
metaphysical
particular
hopefully,
could say that the loving and charitable person is
trappings.
with respect to the
in terms
than"
of "is greater
Hartshorne's
mention
greater than our greedy millionaire
and
of our admiration
of "more worthy of admiration and respect" strikes property of being worthy
me as a step in the right direction. Plantinga takes it respect ;even though our millionaire might be greater
with respect to the having of wealth.
a
step further.
Plantinga cites wisdom, intelligence, power, moral
as wisdom
such qualities
displays
so
one
who
does
is
than
far
forth,
courage
greater,
a cat let's say, is not as great a being
Furthermore,
of intelligence
in that the latter has properties
man,
A
man
who
that
knowledge
have
might
and
the former
one
course
lacks. Of
and
not.
as a
and
being
but little courage
and
intelligence
be hard to say
then itmight
just the reverse;
as
if either was
But
such qualities
the greater.
and the like
moral
excellence,
courage
power,
wisdom
another
which,
wisdom,
are what
more
we might
call "great making"
these a being
the greater,
has,
it is.7
of
equal,
properties;
all else
the
being
specifies an entire range of "great making"
Malcolm
be
properties
If a housewife
then
has a set of extremely
dishes,
fragile
to those of another
set like
they are inferior
in all respects
that they are not fragile.
except
of the first set are dependent
for their continued
as dishes
them
Those
on
existence
not. There
between
set are
; those of the second
gentle
handling
is a definite
in common
connection
language
the notions
independence
was
which
and
dependent
of dependency
and
To
say
superiority.
on nothing
whatever
and
inferiority,
that something
was
superior
in any
to ("greater
that was dependent
than") anything
is quite
in keeping
with
the everyday
way upon anything
use of the terms
and "greater."
Correlative
"superior"
are
with
the notions
of dependence
and
independence
the notions
fuel,
that
excellence,
and
an
of limited
engine's
limited by
and
is a limitation.
this
operation
unlimited.
engine
requires
same thing
to say
on as that it is
is dependent
its fuel supply. An
7Alvin
Plantinga,
8Norman
Malcolm,
An
It is the
engine
that could
a paper
"God and Possible Worlds,*
"Anselm's Ontological
Arguments,"
and
courage,
the
as
like
great
making
properties. These are particular respects inwhich one
thing may be said to be greater than another. Thus,
xmight be greater than y with respect to intelligence,
but ymight be greater than x with respect to courage.
Malcolm
goes further still. He tells us in effect that
one thing x may be greater than another thing y with
respect to being less limited by or dependent on z. If
y is limited by z and x is not, or if y ismore limited by
z than x is limited by z, then x will be greater than y
with respect to being less limited by z. Conceivably,
does not say so, one thing x might
although Malcolm
than
greater
another
thing
y with
are
unquestionably
to compare
things
There
which
of which would
to being
respect
than x with
less limited by z1? yet y may be greater
respect to being less limited by z2
other
many
in particular
ways
respects,
by
each
a locution of the form "x is
generate
we
to property
P." Thus,
y with
respect
that Beethoven
is greater
than Montavanni
than
greater
say
might
with respect to composing. Or, Ty Cobb is greater
than Lefty Grove with respect to hitting a baseball.
The point of these examples is not somuch that they
are
true,
noncontroversial,
make
sense,
being
that
about. Rather,
they
are
to get complete
possible
a
greater
would
than
easy
not
be
about Beethoven
Montavanni.
probably
read at Davidson
12, 1974.
College on November
no. 1 (January
The Philosophical Review, vol. LXIX,
or
it is that they
It may
meaningful.
agreement
composer
even less agreement
decidable,
readily
to secure agreement
And
be reached over
i960), pp. 41-62.
AMERICAN
138
or Bach
Beethoven
whether
we
In fact,
want
might
is the
to say
PHILOSOPHICAL
and
lacks,
greater
composer.
Bach
is not greater
that
QUARTERLY
a
that
than
greater
is
y
condition
necessary
x
that
x
for
at
possesses
least
to
be
one
with respect to composing, and also
is not greater than Bach with respect
In doing
not be
so, we would
than Beethoven
that Beethoven
to composing.
combination
of great making
properties which y
lacks. This suggests that the concept of being greater
than should be expressed in disjunctive normal form.
not
"x
is
Let "F(x, y, z)" be short for "x is greater than y with
because
than
ourselves,
contradicting
greater
y with respect to composing" does not entail "y is respect to z." Then the definition of "x is greater than
greater than x with respect to composing." Yet, if it /' should, as suggested, have the following form :
to say that Beethoven
is meaningful
is not greater
v
than Bach with respect to composing?and
it is?
[F(x,y,Pu)
.....F(x,y,P??]
to say that Beethoven
v...
then it is also meaningful
is
[F(x,y,P2l)
...*F(x,y,P2n2)]
?
than Bach with
to composing.
greater
respect
v[F(x,y,Pu)
...>F{x,y,Pknk)]
these
Although
sial,
they
claims
comparison
nonetheless
are
be
controver?
Their
meaning
may
meaningful.
is guaranteed
by the fact that they are commonly
used and understood
by speakers of ordinary
language.
I have dwelled so far on stressing that things may
in particular
be compared
respects. What must be
is that things may be compared
shown, however,
to show that it is
That
is, we want
absolutely.
x
to
that
is
say
greater than y, without
meaningful
"in
adding
To
or
this
that
x is greater
that
say
respect."
than
y is not
and y regardless or independently
be
in particular
compared
x will be greater
than
y
On
respects.
respects.
particular
the
contrary,
because x is greater
than y precisely
in various
x
to compare
of how x and ymay
as
Plantinga,
quoted above, is suggesting this when he says that
more great making properties a being has, the greater,
all else being equal, it is.Malcolm
is suggesting the
same
another
save
he
when
not
The
set of dishes
if the first is like the second
being
is that
in all
in all respects
in
a
of Plantinga
the suggestions
are
things
relevant
compared
to
is superior
fragile.
trouble with
Malcolm
same
one
that
says
ways
particular
and
the
never,
rarely,
perhaps
are
save the way
they
being
and
respect.
Plantinga
where each "/y represents a great making property
to compare x and
with respect which it ismeaningful
y. We will also want each of these great making
properties to be such as to guarantee that two things
may be equally great, and that "is greater than" is
irreflexive,
edge,
and
anti-symmetric
I believe
that power,
kindness,
transitive.9
courage,
knowl?
intelligence,
truthfulness,
beauty,
lov
gentleness,
ingness, and spiritedness should be counted among
the great making
the great
properties. But what
making properties are exactly, and how the predicates
that represent
them should be conjoined
in the
form of "is
disjuncts of the suggested definitional
greater than" are questions beyond the scope of the
answers
Their
paper.
present
would
us with
provide
a specification
of what "is greater than" actually
means; and that is something that will have to wait
until
further
concerning
variety
evidence
is
"is greater
how
of contexts,
and
acquired
than"
functions
including
the
religious,
analyzed
in a wide
the moral,
and the aesthetic. Even though I am not presently in
the position of being able to provide anything more
than
the
definitional
schema
of
need not for that reason decline
greater
is in fact meaningful.
than"
"is greater
than"
from saying
Many
we
that "is
phrases
of
our
are
even
would have great difficulty, then, in even
and
Malcolm
understood,
language
meaningful
users
are not
to
when
of the language
in a position
being able to say that one thing is absolutely greater
is needed is an analysis of "is explicitly define those phrases. I think that the phrase
than another. What
"is greater
is used
than"
and
understood?hence,
greater than" which takes into account not merely
the
that
;
one, but all great making properties, or a sufficient
and,
thus,
very idea of a supreme
meaningful
sense.
subset thereof. We want to be able to say that x is being makes
than
greater
to say that a sufficient
than
y
is
that
combinations
9This
Indeed,
x
a certain
combina?
properties which y does not,
possess certain great making
x does
which
properties
x possesses
y because
tion of great making
even though y may
possess
we
not. Moreover,
any
one
of
a
number
properties
III. The
want
for x to be greater
condition
of great making
also
which
to preclude
is not meant
the possibility
formal analysis
it is quite conceivable
that an informal explication may
of
y
Possibility
of God
are interested
We
in demonstrating
that the
statement "It is possible for God to exist" is true of the
to possible world semantics,
actual world. According
of providing
prove more
an informal
satisfactory
of what "is greater than" means.
explication
in the long run than a formal analysis.
A MODAL
one
that
of demonstrating
statement
for
"0^4,"
any
the
way
ment
actual world
"A"
of
even though
to be false of
which is accessible to the actual world,
one or more of these premises happens
actual
world.10
valid
That
and
if
is,
each
"Bx,...,
"Bx,"
premise
is
BJA"
...,
"Bn"
is
true of the same possible world W, then 'A9will also
be true of W. But \VAy is true of Wand Wis accessible
to
the
actual
then
world,
is true
"<>A"
of
the
actual
world.
So, ifwe had a deductively valid argument for the
existence of God, all the premises of which are true of
some possible world W which
to the
is accessible
actual world, we could then correctly infer both that
God exists in M^and that it is possible for God to exist
in the actual world. Or ifwe had an argument for the
existence of God which is not deductively
valid, but
to a deductively
which
could be extended
valid
OF GOD
not
and
139
some
of
existence
causing
God.11
the
world
EXISTENCE
necessity,
valid
"Bx"
deductively
the
the
of
such
that
each
"Bx,...,
BJA"
. . .,
some
"Bn" is true in
possible
argument
premises
the
state?
possibility
is true
a deductively
is by constructing
THE
PROVING
FOR
MODEL
in others
its own
of itself
being
having
it from another,
but rather
receiving
their necessity.
This all men
speak of as
The first thing to get straight about this argument
is what Aquinas means by necessity. He does not
mean
he suggests that a
logical necessity. Rather,
one
cannot
that
is
be generated and
necessary being
cannot be corrupted. In the idiom of possible worlds,
a necessary being for Aquinas
is one that exists in a
possible world only if it neither begins to exist nor
ceases to exist in that or any other possible world ; it
is a being which either eternally exists or eternally
fails to exist in every possible world where it exists.
But a necessary being in this sense need not exist in all
possible worlds. I prefer to say that such beings are
temporally-necessary.
will
ally-necessary
in
either
possible
or
generated
world's
language,
be
are
which
Beings
called
not
tempor?
temporally-contingent.
a temporally-contingent
Thus,
can
be
is one which
being
can
a
be
corrupted.
Again,
temporally-contingent
the addition
of one or more
by
premises,
being is one which either begins to exist in some
and if the premises of this new, extended argument
possible world or ceases to exist in some possible
are all true in some possible world W which
is world.
It could happen, however,
that in a given
too we could
to the actual
then here
accessible
world,
possible world a temporally-contingent
being exists
correctly infer both that God exists in W and that it eternally.
is possible for God to exist in the actual world.
can
We
the ideas of temporal-necessity
and
express
I believe that St. Thomas Aquinas' Third Way can
more
as follows.
Let
temporal-contingency
precisely
to prove that it is possible
be appropriately modified
the expression "tx < t2" be short for the expression
for God to exist. Aquinas argued as follows :
"time t2 is later than time ?l5"and let "Txt" be short
"x is realized (exists) at time t." The notion of
for
The
third way
is taken from possibility
and necessity
and
can
thus:
then be defined
are
runs thus. We
to
that
find in nature
be
generated
being
things
possible
=
< h * Txt2
to be generated
and not to be, since they are found
and
"Gen(x)"
~Txttf\
^"(B^H^K'i
to be corrupted,
it is possible
and consequently,
for them
can be defined
The notion of being corrupted
to be and not to be. But it is impossible
for these always
:
similarly
argument
to exist,
for
then
even
because
can
that which
not-be
now
there
would
be
some
time
then
at one
time
if this were
true
at
can not-be,
if everything
Now
in
existence.
nothing
Therefore,
there was
nothing
is not.
in existence,
to exist only
does not exist, begins
Therefore
if at one
already
existing.
in existence,
it would
have
been
to exist ; and thus
to have begun
for anything
that which
through
something
was
time nothing
impossible
now
nothing
Therefore,
must
exist
But
would
be
in existence?which
is absurd.
not
are merely
but there
all beings
possible,
of
which
the
existence
is
necessary.
something
caused
necessary
thing either has its necessity
every
or not. Now
to go on to
it is impossible
another,
in necessary
have
their necessity
infinity
things which
. . .Therefore,
we cannot
caused
but admit
by another
by
10
A possible world WY is accessible
toWx just in
intuitively, W2 is accessible
see Hughes
and Creswell, An Introduction
11
Thomas Aquinas,
Summa Theolagica,
"Cor(x)"
And
=
given
concepts
#"(3*i)
these
of
porally-contingent,
being
two
@h)(h
notions,
< h
we
Txtx
can
temporally-necessary
then
~Txt2)n
define
and
the
tem?
respectively.
=
D ~Gen(*)
D ~Cor(*)"
"TemNecW"
= df"
v OCor(x)"
"TemCon(*)"
^"OGen^)
As expected, "(x) (TemNec (x) = ~TemCon(#))"
is a logical truth. Moreover,
for any possible world
where "(Vi1)(V/2)(/1 < t2 v t2 < tx v tx = t2)" and
v (3t)~ Txt)" are true, such as the
"(Vx)((3t)Txt
=
actual world, "(V*)(TemNec(x)
((it) Txt v
true.
is
also
(Vt)~Txt))"
The Third Way can now be modified and extended
to possible world W2 if and only if every statement
true of Wx is possibly
true of W2. More
case someone living inWx can conceive of what W2 is like. For more on the
relation,
accessibility
toModal Logic, especially pp. 75-80.
p. 3, art. 3.
AMERICAN
I40
PHILOSOPHICAL
to a possible world which is accessible to but
distinct
from the actual world. The argument
possibly
to apply
runs
:
thus
( i ) Some temporally-contingent
being presently
tem?
(2) There have been only finitely many
to
date.
beings
porally-contingent
(3) Every temporally-contingent
being begins to
some
at
time
ceases
and
to exist
at
some
time.
which
(4) Everything
time
ceases
and
to exist at some
begins
to exist
at
some
time
exists
for
a finite period of time.
(5) If everything exists for only a finite period of
time, and there have been only finitely many
then
things,
existed.
a
was
there
time
when
nothing
(6) If there was a time when nothing existed,
then nothing presently exists (since things
begin to exist only through something else
which already exists, and if there was a time
when nothing
existed then nothing
could
begin to exist at any time thereafter).
(7) Everything which exists exists for some time
or other.
has a sufficient reason for its
(8) Everything
existence either in itself or beyond itself.
cannot
be an infinite regress of
(9) There
sufficient reasons (for failure to reach an
an
is not
explanation
for
(11) Every
a
whose
as
essence,
existence?one
is
it were,
being without any limitations.
(12) A being without any limitations
than
greater
any
other
IV.
is a
being which
its own
for
to
a
exist?is
x is greater
(The
than
relation
necessarily
y,
then
y is not
than
greater
greater
is
than
anti-symmetric.)
Therefore,
( 15) There
This
moreover,
is a supreme being.
argument
are
is deductively
consistent,
mutually
some possible world W.
12See Nicholas
Rescher,
valid.
and
It is plausible
A Theory of Possibility,
Its premises,
thus
true
modal
able,
being.
of being
The
proof
out
carried
of
Logical
that God
in
a
very
logic. Such
considerable
is necessarily
to be greater
(13) It is not possible for anything
than itself.
(14) It is necessarily the case that for all x and y, if
x.
we
since
world,
are
here
evident analytic truths, and hence true in all possible
worlds. Only
(12) requires special justification.
Assume x is a being without limitations inW. Then
x possesses every great making
in W. In
property
x
not
the
in
Wof
possesses
property
particular,
being
limited in world Wx by anything. In other words, if
x is a being without
in W, then x
any limitations
possesses every great making property inW. But the
property of not being limited inWl is a great making
property of W. So it is true inW that it is true inWl
that x is unlimited. But for any statement/?, if it is true
in world a that p is true in world ?, then p is true in
world ?.12 Hence, x is unlimited in world Wl. Now if
x is unlimited inWl, then inWl, x is greater than any
other being inWl ; otherwise x would be limited by
not possessing a great making property possessed by
something else. Hence it is true inWl that x is greater
is an arbitrarily
than every other being. Since Wl
selected possible world,
it follows that it is true in
every possible world that x is greater than every other
the case that x
it is necessarily
being. Consequently,
is greater than every other being. So ( 12) is true inW.
being.
temporally-necessary
reason
actual
being is a sufficient
temporally-necessary
sufficient
to the
accessible
and now conceiving of W from the actual world. So
(15) is true of W.
Statements
(1), (2), (3), (6) and (10) can be thought
of as logically-contingent
facts about W\ whereas,
explanation).
( 1o) No temporally-contingent
reason
H^is
(4)> (5)> (7)> (8)> (9). (ii), (13) and (14) are self
exists.
exist
QUARTERLY
therefore,
Assumptions
is possible
strong
logics have
only
system
been
if actual was
of
quantified
the subject of
in recent
It is reason?
controversy
years.
to request
the certification
of any
particular system of quantified modal logic whenever
that system is employed in a demonstration,
especially
if the logic is as strong as CM. The task of fully
the use of CM, however, far exceeds the
justifying
I intend to do instead is
scope of this paper. What
direct my attention to a particularly
important mode
of reasoning upon which the rejection of CM might
well be based.
CM has the Barcan Formula (BF) as a theorem.
(BF):D ((V*)Di4=> nOfx)A)
So in order to be able to accept CM as a viable modal
true, that is,
logic (BF) will have to be necessarily
to assume
that
(Pittsburgh,
1975), especially
pp.
111, 112, and
122.
A MODAL
true in all possible worlds.13 Yet Plantinga
have
argued
to contrary.
persuasively
and others
to
According
Plantinga,
... is
Formula
by the Barcan
proposition
expressed
seen to be false or at any rate not necessarily
true.
easily
no material
world W* where
No doubt
there is a possible
are
in which
world W*
the only objects
exist?a
objects
and God. Now
such things as propositions,
sets, numbers
The
as a set is essentially
have been a material
such a thing
no set could
true
immaterial
object.
; for
surely
It is therefore
that
(38)
Every
immaterial.
of our immaterial
feature
accidental
is not a merely
true
in every
world.
is
true,
world;
necessarily
(38)
we
are
W*
of which
Hence
it is true in this world
And
speaking.
course
of
same
the
regard;
God. Hence
is essentially
Everything
(38')
true
are
sets
in W*.
not
for properties,
goes
in
unique
this
and
propositions,
immaterial.
constitutional
W*.
More
is false
in W*,
[the actual
is an
in view
of
in which
world]
injustice
world
terial
ment
to Plantinga's
contains
W*
but without
that his
argument,
The
numbers.
only
then boils down as follows
( 1) Everything
(2) Numbers
object.
the possibility
of worlds
like a
there exist material
objects.14
Let us assume for simplification,
an
immaterial
doing
imma?
argu?
:
is a number.
are
necessarily
serious
W*.
are
difficulties
true
are
words,
in every possible world where
It is true
not.
suggests
present.
. '.
immaterial.
everything
is immaterial.
(4) Necessarily
nonstandard
that
necessarily
models.
are
there
even
Second,
we
diverse
ways
some
reservation
about
the
existence
of
a
possible
world whose domain is different from the domain of
the actual world. Indeed, if (BF) is adopted, even
tentatively,
as a necessary
truth
of an
S5
structure
of
from
know
set
of construing
'"
V
cluster,
material
of
individuals
i
cluster
'"
V
'
n
4
numbers
composite
the
V
3
The
this
are
world
consisting
the
material,
of the objects within
is a particular
material
of two
other
objects
2
object.
i. 3
than
is a
is a
cluster of three objects other than those comprising
and
Compelling
though it may seem, this argument is
not without
its difficulties. In the first place, I have
any model
the "numbers" which make up the standard models
of number
theory. Third, and most
importantly,
number theory does not at all specify that the objects
of itsmodels must be immaterial. Indeed, a subset of
a possible world with infinitely many material objects
for number
might well be a model
theory, with
certain material objects ofthat world being numbers.
a possible world with
Imagine
infinitely many
material objects represented by the dots below.
IV
true.
that
of number theory drawn from the actual world will
be immaterial. This is because there are only finitely
many material objects in the actual world and that
the domain of any model for number theory must be
infinite. But number
theory does not uniquely
determine
the objects of its models.
the
First,
incompleteness results of G?del and others tell us that
if number theory has a standard model then it also
12
(1) is the constitutional
assumption about W*. (2) is
viewed as a logical truth, hence true inW*. (3) is said
to follow from (1) and (2); (4) from (3) and (BF).
Reductio
conclusion:
(4) is true of W*. But (4) is
really false of W* ;otherwise, the actual world would
have to be exclusively
it is not.
immaterial, which
true
Final conclusion:
is
neither
of
W*, nor
(BF)
essentially
immaterial
they exist? Number
of course
(essentially)
is necessarily
Everything
are
numbers
immaterial.
. '.
(3)
let us
However,
that ( i ) is true of
that numbers
? In other
immaterial
has
everything
141
about
assumption
Is it really
But
Necessarily,
OF GOD
assume for the purpose of argument
theory
(39)
EXISTENCE
possible worlds, then precisely the same objects exist
in every possible world, albeit with differing inessen?
tial attributes. So instead of rejecting (BF) outright,
the rejection of his
Plantinga
ought to consider
theory
set is essentially
This
is also
THE
PROVING
FOR
MODEL
2. And
so on.
could
Plantinga
argue,
of
course,
that
there might
be material models
of number
theory, but that the objects of such a model are not
really numbers. He would then have to specify how
to individuate numbers
of number
independently
a
Until
such
is
theory.
specification
forthcoming,
Plantinga's
case
against
13
true in world Wii and only if A is true in all
Properly
speaking, A is necessarily
possible worlds
to accessibility
will only serve to unnecessarily
the subsequent discussion, however.
complicate
14
Alvin Plantinga,
The Nature ofNecessity (Oxford,
1974), p. 59.
(BF)
remains
accessible
tenuous.
toW. Constant
reference
1
I now
turn
argument
Let
"Nx"
to
short
be
"x ismaterial."
are,
the
of whether
question
is valid. Does
Plantinga's
(3) follow from (1) and (2)?
is a number"
"x
for
formal renditions
The
PHILOSOPHICAL
AMERICAN
142
and
"Mx"
for
of (1) and
(3)
:
respectively
(If) (ix)Nx
(30 (V*)D -Mx
are we
possibilities :
to translate
But how
are three
(2)? There
(2fa) (ix)U(Nx ZD~Mx)
(2fb) D(Vx)(iVxD ~Mx)
(2fc) (Vx)(Nxz>D ~Mx)
argument
Plantinga's
and
mally
does
not
(BF) is stated
so far
as
I can
a number
is necessarily
or
essentially
infor?
ascertain,
a choice by him of one of these three ways
that
of saying
immaterial.
that Plantinga
tells us is that x has a property P
if
and
essentially
only if x has P in every possible
world inwhich it exists.15 This, however, is insufficient
All
for
enabling
us
to choose
among
(3f) is implied by (2fc) and (if) ;and the argument
will be valid if (2) translates as (2fc). I personally
believe that (2fa) is the appropriate translation of (2).
But that issue aside, what reasons might Plantinga
have for suggesting that a statement as strong as (2fc)
is true in W*? He suggests that (2) is not merely an
accidental feature of W*, but necessarily true? So if
(2) translates as (2fc), then (2fc) must be necessarily
true,
the above
three
ways
of possibly construing
(2). Yet the argument against
of these three
(BF) crucially
depends on which
and
hence
true
in W*.
Now
a
case
strong
can
be
for claiming
that both (2fa) and (2fb) are
true
in any possible world. It is not
if
necessarily true,
at all clear to me, however,
that (2fc) is necessarily
made
true.
against
involve,
QUARTERLY
as I
argued
First,
number
earlier,
theory
suggests
the possibility of material numbers. Second, even if
material numbers are not possible, (2fc) will fail to be
true just in case there are possible worlds
necessarily
Wx and W2 such that there is something in both Wx
and W2 which is an immaterial number inWx and a
material
worlds
in W2.
non-number
seems
there
argument
Plantinga's
plausible.
(BF) can be made
That
are
such
against
valid, but only at the cost of being
unsound.
To
against
seriously
your
the
question
position
does
soundness
not,
as we
of an argument
all
know,
prove
(?f) is implied by neither (2fa) and (if), nor by
(2fb) and (if). So if either (2fa) or (2fb) is the modal
is
translation
of (2) then Plantinga's
argument
the weakness
of
your position. Notwithstanding
the
and
CM
argument
Plantinga's
against (BF),
logic
its equivalents are still in need of being certified. Until
they are, the best I can claim to have provided in this
paper is a modal model for proving the existence of
invalid.
God.
alternatives
Davidson
15
Ibid., p. 6o.
we
adopt.
College
Received
April 12, 1979