## 1. Introductory Preliminaries



Philosophical reflection upon the notion of omnipotence raises many
puzzling questions about whether or not a consistent notion of
omnipotence places limitations on the power of an omnipotent agent.
Could an omnipotent agent create a stone so massive that that agent
could not move it? Paradoxically, it appears that however this
question is answered, an omnipotent agent turns out not to be
all-powerful. Could such an agent have the power to create or overturn
necessary truths of logic and mathematics? Could an agent of this kind
bring about or alter the past? Is the notion of an omnipotent agent
other than God an intelligible one? Could *two* omnipotent
agents coexist? If there are states of affairs that an omnipotent
agent is powerless to bring about, then how is the notion of
omnipotence intelligibly to be defined? Moreover, an obstacle to
traditional Western theism arises if it is impossible for God to be
morally perfect and omnipotent. If an omnipotent God is powerless to
do evil, then how can he be omnipotent? Rational theology seeks an
analysis of the concept of omnipotence that provides sufficient
granularity about the powers of an omnipotent agent to resolve the
puzzles and apparent paradoxes that surround this concept. If the
notion of omnipotence were found to be unintelligible, or incompatible
with moral perfection, then traditional Western theism would be
false.



According to some philosophers, omnipotence should be understood in
terms of the power to *perform certain tasks*, for instance, to
kill oneself, to make \(2+2=4\), or to make oneself non-omniscient.
However, in recent philosophical discussion, omnipotence has been
analyzed in terms of the power to *bring about certain possible
states of affairs*, understood as propositional entities which
either obtain or fail to obtain (Rosenkrantz & Hoffman 1980; Flint
& Freddoso 1983; and Wierenga 1989). Because we believe that it
can yield an adequate analysis of omnipotence, we take this latter
approach in what follows.



Some philosophers have sought to understand omnipotence in terms of
*infinite power*. One example of such a philosopher is John
Locke. In 1689, he writes that:



> 
> It is Infinity, which joined to our ideas of Existence, Power
> Knowledge, & c. makes that complex Idea, whereby we represent to
> ourselves the best we can, the Supreme Being. (*An Essay Concerning
> Human Understanding*, Book II, Ch. 23, P 35.)
> 



This quotation suggests that omnipotence can be defined as infinite
power. Leaving aside for the moment the historical question of how
Locke understood infinite power, let us explore the suggested
definition of omnipotence in the light of the prevailing mathematical
understanding of infinity.



This mathematical understanding of infinity views it through the lens
of transfinite arithmetic. Transfinite arithmetic presupposes that
there are infinitely many natural numbers. According to transfinite
arithmetic, the cardinal number of the set of natural numbers is
\(\aleph\_0\) (aleph-null). (A *cardinal number* is a number
that can be used to answer the question " How many?") In
other words, if it is asked " How many natural numbers are
there?", the correct answer is \(\aleph\_0\).



The founder of transfinite arithmetic, Georg Cantor (1845-1918),
is also a founding father of set theory. He famously proved that
*the set of real numbers* has a *larger* cardinal number
than *the set of natural numbers*; the set of reals has the
same cardinality as the *power set* (*the set of all
subsets*) of the set of naturals. Cantor further argued that
\(\aleph\_0\) is the *first* (and smallest) transfinite cardinal
number in an infinite series of increasingly larger transfinite
cardinals, \(\aleph\_0,\) \(\aleph\_1,\) \(\aleph\_2\), and so on. But
note that the numerical subscripts of these alephs do not refer to
cardinal numbers, but rather refer to *ordinal* (or ordering)
*numbers*, e.g., first, second, third, and so forth. According
to Cantor, there is no largest transfinite cardinal number, and
although there are infinitely many transfinite cardinal numbers, there
is no cardinal number of them. Rather, the totality of transfinite
cardinals is *absolutely infinite* in a *sui generis*
non-arithmetical sense. Similarly, in Zermelo-Fraenkl set theory (ZF),
and its extension ZFC (ZF + the axiom of choice), the totality of
transfinite cardinal numbers does not qualify as a *set* having
a definite cardinal number of members. Such a set would be too large;
were such a set to exist, paradoxical consequences would ensue akin to
Russell's paradox. ZF avoids these paradoxical consequences by
characterizing the totality of transfinite cardinals as an infinitely
large *proper class*. Within the forms of axiomatic set theory
that have attracted significant support it cannot be proved that one
proper class is larger than another.



Transfinite cardinals may be used to quantify an amount of energy or
force, e.g., \(\aleph\_0\) joules, or \(\aleph\_0\) newtons. Or they may
be be used to state the cardinal number of a totality of objects,
e.g., \(\aleph\_2\) states of affairs. Thus, one [first] way in which
'power' might be interpreted is as a power (or range of
powers) to produce an energy or force quantified by a transfinite
cardinal. One such option is that 'infinite power' means
the power to produce a *specific* transfinite quantity of
joules or newtons, e.g., \(\aleph\_0\). The other option is that
'infinite power' means the power to produce *any*
transfinite quantity of joules or newtons, i.e., \(\aleph\_0,\)
\(\aleph\_1,\) \(\aleph\_2,\) and so on *ad infinitum*. A third,
more radical option, is that 'infinite power' means the
power to produce *more* joules or newtons than can be
quantified by *any* transfinite cardinal number. Such energy or
force would appear to qualify as energy or force than which none
greater is possible.



The alternative way of interpreting 'infinite power' is as
the power to bring about infinitely many states of affairs. The three
options relevant to this interpretation of 'infinite
power' parallel the three options just described. First, that
'infinite power' is the power to bring about a
*specific* transfinite cardinal number of states of affairs.
Second, that 'infinite power' is the power to bring about
*any* transfinite cardinal number of states of affairs. Third,
that 'infinite power' is the power to bring about
*more* states of affairs than *any* transfinite cardinal
number can quantify.



The standard view is that there is no greatest transfinite cardinal,
and that there are proper classes containing more objects than can be
quantified by any transfinite cardinal. Moreover, there could
*not* be an agent who has *greater* power than an
omnipotent agent. Thus, with regards to the two interpretations
offered, there is some reason to conclude that in each case the
*third option* is best.



However, a philosopher who has *positivistic or naturalistic
leanings* might argue that one should posit transfinite cardinals
of *only* those sizes required by prevailing *scientific
theories*. Arguably, the prevailing theory of the s*pace-time
continuum* in physics is committed to the existence of infinite
sets which have the cardinality of the set of space-time points
contained within such a continuum (which is at least \(\aleph\_1\)). If
an argument of that sort is correct, this scientific theory is, in
addition, committed to the existence of the power set of the foregoing
set of space-time points - which is needed to accommodate
continuous and discontinuous linear paths. (This power set has the
cardinality of the set of mathematical functions from reals to reals.)
But scientific theories do *not* appear to require the
existence of any set *larger* *than* that power set. So,
a philosopher who has positivistic inclinations might tentatively
conclude that the cardinal number of this power set (which is at least
\(\aleph\_2\)) is in fact *the largest cardinal*. In a related
vein, W. V. O. Quine dismisses "excessive" magnitudes, for
instance, "\(\beth\_\omega\) and inaccessible numbers," as
mere fancies of "recreational mathematics" bereft of of
ontological import (Quine 1986, p. 400; cf. Quine 1960, Ch. 7). If a
reader is unfamiliar with Quine's illustrations of
scientifically gratuitous magnitudes, please note that \(\beth\_0 =
\aleph\_0\), \(\omega\) (omega) is \(\aleph\_0\)'s [infinite]
*ordinal* counterpart, i.e., \(\aleph\_0\) is the \(\omega\)-th
cardinal number, and \(\beth\_\omega\) is the \(\omega\)-th iteration
of the power set of \(\beth\)0; *inaccessible
numbers* (roughly speaking) are (putative) uncountable large
cardinals which cannot be derived *via* summations of, or power
set operations on, smaller cardinals. Quine judges that such
magnitudes exceed the theoretical demands of the empirical sciences
because admitting them would not contribute to the simplication of our
scientific computations and generalizations. Even so, there is a view
of the cumulative set-theoretical hierarchy to which Quine seems
(reluctantly) willing to accede, at least in part because it provides
a "convenient cut-off" limiting the extent of that
hierarchy. According to this view, all sets are *constructible*
(in a sense defined by Kurt Godel) from more elementary sets, a
position significantly more restrictive than is needed to avoid the
known set-theoretical paradoxes. But, arguably, when examining
omnipotence, given its robust theological and metaphysical dimensions,
a less positivistic or naturalistic, more expansive, criterion of
ontological commitment is appropriate. After all, it would be natural
to assume that the Supreme Being has the power to create any number of
possible worlds governed by different laws of nature. Given that
assumption, it appears that an omnipotent agent's power would
not be circumscribed by theoretical scientific laws governing the
actual world.



Even so, the notions of *more* energy or force than can be
quantified by any transfinite cardinal, and of *more* states of
affairs than can be quantified by any transfinite cardinal, are
mind-boggling. Moreover, arguably, *that much* energy or force,
or the power to bring about *that many* states of affairs, is
impossible. For this reason, with respect to each of the
interpretations of 'infinite power' on offer, there are
doubts about the viability of the corresponding third option. Thus,
with respect to each of these alternate interpretations of
'infinite power', it is an open question which of the
three corresponding options is best. Until greater clarity is attained
about this matter, a definition of omnipotence as infinite power is
problematic.



Moreover, it can be plausibly argued that having the power to bring
about infinitely many states of affairs does not entail omnipotence.
E.g., intuitively, an agent who has the power to move a feather into
any of infinitely many positions, but lacks the power to move a heavy
boulder, would not be omnipotent, despite having the power to bring
about infinitely many states of affairs.



It may be thought that, necessarily, any agent who has the power to
produce an infinitely large amount of energy or force is omnipotent.
Yet, this seems not to be true. After all, it seems possible for there
to be an agent who is infinitely powerful in this sense, but who
nonetheless lacks complete *command and control* of the
energies or forces it has the power to produce. If so, an
agent's possessing the power to produce an infinite amount of
energy or force does *not* entail that the agent in question is
omnipotent. As illustrated by the example described above of an agent
who has the power to move a feather into infinitely many positions,
but who lacks the power to move a heavy boulder, an agent's
(*S*'s) having the power to bring about infinitely many
things of *one* sort, does not guarantee that *S* has
the power to bring about things of *another* sort.



But even if there is some sort of infinite power such that,
necessarily, an agent, *A*, is omnipotent if and only if
*A* has that sort of infinite power, the definitional claim
that omnipotence *is* such power is far *too general* to
provide a *granular* analysis of omnipotence - the kind
of analysis of omnipotence which we are seeking. Proposed granular
analyses of omnipotence offer considerably greater
*specificity* about the powers of an omnipotent agent.



Let us return to the historical question of Locke's
understanding of infinite power. Locke explicitly rejects the notion
that there is an infinite cardinal number. He takes the contention
that there is such a number to be absurd. Rather, Locke appears to
understand infinite power as unlimited power, consistent with the
Aristotelean notion that infinities are "potential" and
never completed, as well as with the empiricist idea that infinity is
a wholly negative notion. The infinite sets discussed earlier are
"actual" or completed infinities. According to one fairly
common understanding of unlimited power, such power includes the power
to bring about or undo necessary states of affairs, e.g., necessities
of logic, mathematics, metaphysics, or ethics. Yet, it does not appear
that Locke means to attribute this power to an omnipotent agent; he
never contends that the Supreme Being has the power to bring about or
undo necessities. In any event, we will critically examine the thesis
that an omnipotent agent has the power to bring about or undo
necessary states of affairs in what follows.



One sense of 'omnipotence' is, literally, that of having
the power to bring about *any* state of affairs whatsoever,
including necessary and impossible states of affairs. Descartes seems
to have had such a notion (*Meditations*, Section 1). Yet,
Aquinas and Maimonides held the view that this sense of
'omnipotence' is incoherent. Their view can be defended as
follows. It is *not* possible for an agent to bring about an
*impossible* state of affairs (e.g., *that there is a
shapeless cube*), since if it were, it would be possible for an
impossible state of affairs to obtain, which is a contradiction (see
Aquinas, *Summa Theologiae*, Ia, 25, 3; and Maimonides,
*Guide for the Perplexed*, Part I, Ch. 15). Nor is it possible
for an agent to bring about a *necessary* state of affairs
(e.g., *that all cubes are shaped*). It is possible for an
agent, \(a\), to bring about a necessary state of
affairs, \(s\), only if possibly, (1) \(a\)
brings about \(s\), and (2) if \(a\)
had not acted, then \(s\) would have
failed to obtain. Because a necessary state of affairs obtains whether
or not anyone acts, (2) is false. As a consequence, it is impossible
for an agent to bring about either a necessary or an impossible state
of affairs. Many philosophers accept the principle that if an agent
has the *power* to bring about a state of affairs, then this
entails that, *possibly*, the agent brings about that state of
affairs. If this principle is correct, then the foregoing
*absolute* sense of 'omnipotence' is incoherent.
Among contemporary philosophers, Earl Conee (1991) rejects this
principle in order to defend the view that an omnipotent being would
have the power to bring about any state of affairs whatsoever.



A second sense of 'omnipotence' is that of *maximal
power*, meaning just that no being could exceed the overall power
of an omnipotent being. It does not follow that a maximally powerful
being can bring about *any* state of affairs, since, as
observed above, bringing about some such states of affairs is
impossible. Nor does it follow that a being with maximal power can
bring about whatever any *other* agent can bring about. If \(a\)
can bring about \(s\), and \(b\)
cannot, it does not follow that \(b\)
is *not overall* more powerful than \(a\), since
it could be that \(b\) can bring about more states of
affairs than \(a\) can, rather than the other way
around. For the remainder of this entry the discussion concentrates on
this *comparative* sense of 'omnipotence' as
maximal power. Within the context of that discussion, it will be
assumed that it is *not* possible for an agent to have the
power to bring about *any* state of affairs whatsoever.



*That a being is omnipotent just provided that its overall power is
not possibly exceeded by any being* may be adopted as the most
general definition of omnipotence in this sense (Hoffman &
Rosenkrantz 2010). Still, the availability of a more fine-grained and
informative analysis, directly applicable to the full range of problem
cases, is highly desirable. A number of prominent proposals for such a
more fine-grained and informative analysis of omnipotence will be
discussed later.



*Power* should be distinguished from *ability*. Power is
ability plus opportunity: a being which has maximal ability but which
is prevented by circumstances from exercising those abilities would
not be omnipotent. Nothing could prevent an omnipotent agent from
exercising its powers, if it were to endeavor to do so.



In the light of the foregoing, is it possible that there be a
*plurality* of coexistent omnipotent agents? Among contemporary
philosophers of religion, Richard Swinburne (2008) holds that a
plurality of coexistent omnipotent agents is possible.



If a plurality of coexistent omnipotent agents were even possible,
then *possibly*, at a time, \(t\), some
omnipotent agent, \(x\), while retaining its
omnipotence, endeavors to move a feather, and at \(t\),
another omnipotent agent, \(y\), while retaining its
omnipotence, endeavors to keep that feather motionless. Intuitively,
in this case, neither \(x\) nor \(y\)
would affect the feather as to its motion or rest. Thus, in this case,
at \(t\), \(x\) would be powerless to
move the feather, and at \(t\), \(y\)
would be powerless to keep the feather motionless! But it is absurd to
suppose that an omnipotent agent could lack the power to move a
feather or the power to keep it motionless. Therefore, neither \(x\)
nor \(y\) is omnipotent. This line of
reasoning appears to reduce the notion of a plurality of coexistent
omnipotent agents to absurdity. If such a *reductio ad
absurdum* is sound, then a plurality of coexistent omnipotent
agents is impossible.



It might be replied that while neither of the omnipotent agents in
question brings about what it endeavors to bring about, each of them
*can* do so, since each of them has the *ability* to do
so; they fail to bring about what they endeavor to bring about only
because they lack the *opportunity* to do so. But earlier
observations about the difference between power and ability and how
each of them is related to omnipotence entail that omnipotence should
be understood in terms of the ability *plus* opportunity sense
of 'can'. If those earlier observations are correct, then,
since neither of the omnipotent agents under discussion *can*
do [in the ability plus opportunity sense] what it endeavors to do,
the possible reply under discussion does not succeed.



Or it might be replied that the possible pair(s) of coexistent
omnipotent agents would *necessarily* avoid stalemates of the
foregoing sort in virtue of the members of each pair resembling one
another in some respect. This reply seems suspiciously *ad
hoc*. It appears that the members of any possible pair of
genetically identical human twins could be stalemated, e.g., in an
arm-wrestling match. Why would not the same be true of a pair of
similar coexistent omnipotent agents? It might be answered that each
of the members of any possible pair of coexistent omnipotent agents
would be necessarily omniscient and necessarily morally perfect.
Moreover, if two omnipotent agents are *necessarily
omniscient*, they *won't disagree* about *any
fact*, and if they are *necessarily morally perfect*, they
*won't disagree* about *what is morally required*
or about *whether they want so to act*. It might then be
inferred that any states of affairs that a pair of coexistent
omnipotent agents would endeavor to bring about at a given time are
*compatible*. In the literature, the controversial *social
trinitarianism* of Richard Swinburne (2008), implies that the
Father, the Son, and the Holy Spirit are a trio of coexistent
omnipotent agents each of whom is necessarily omniscient and
necessarily morally perfect.



However, one might object to the preceding reply on the grounds that
if there is a pair of coexistent necessarily omniscient and
necessarily morally perfect omnipotent agents, then there is a pair of
*incompatible* contingent states of affairs each of which is
*morally optional* for these agents, that is, neither morally
prohibited nor morally required for them. The objection then proceeds
as follows. It may be assumed that *that the feather moves at
t* and *that the feather remains motionless at t* are a
pair of states of affairs of the sort in question. Apparently, also,
possibly, under such circumstances, the state of affairs *that the
omnipotent agents in question are stalemated in their endeavors to
affect the feather as to its motion or rest at t* is morally
optional for those agents. Analogously, in an arm-wrestling match
between \(A\_1 \amp A\_2\), *possibly*, the states of affairs
*that \(A\_1\) wins the match*, *that \(A\_2\) wins the
match*, and *that \(A\_1 \amp A\_2\) are stalemated in the
match* are morally optional for \(A\_1\) and \(A\_2\). In the light
of the foregoing observations, it appears that if it is possible that
there are omnipotent agents of the sort in question, then possibly,
one of them endeavors to make the feather move at \(t\),
while the other endeavors to keep it motionless at
\(t\), *even given their hypothesized necessary
areas of epistemic and moral agreement*. The foregoing defense of
the possibility of a plurality of coexistent omnipotent agents is
persuasive only if there is a cogent reply to this objection.



Would it strengthen such a defense to further require that the
coexistent omnipotent agents are *necessarily aesthetically
perfect*, and hence, *won't disagree* about *what
is aesthetically required* or about *whether they want so to
act*? It appears not. After all, there appear to be incompatible,
contingent, *aesthetically optional* states of affairs, i.e.,
states of affairs which are neither aesthetically required nor
aesthetically prohibited for some agent, and considerations parallel
to those adduced above apply. Moreover, it is not clear that
*futile striving* necessarily has *negative* aesthetic
value, witness, e.g., Camus's *The Myth of Sisyphus*, and
in any case, it can be argued plausibly that *the equipoise of
opposing forces* possibly has *positive* aesthetic value,
implying that opposed volitional activities of coexistent omnipotent
beings would not necessarily be futile.



Further doubts about the possibility of a plurality of coexistent
omnipotent beings are raised by considerations outlined below which
seem to show that if *some* possible world is maximally good,
in other words, is *a best* possible world, then *no*
possible world is [uniquely] *the* best possible world, and
likewise with respect to a possible world that is second best, third
best, and so on.



Here it is assumed that if there is a best possible world, then there
is at least one such possible world containing contingently existing
individual substantial individuals. A parallel assumption is made
about any possible world good enough to be actualized by an
omnipotent, omniscient, morally perfect, aesthetically perfect being,
e.g., by a maximally great divinity such as God. But, with respect to
any possible world containing contingently existing substantial
individuals, it appears that there is *another* possible world
*exactly resembling it* but populated by *different*
contingently existing substantial individuals. It further appears that
the value of one of these worlds is equal to the value of another of
these worlds. So, it appears that if one of these possible worlds is
best, second best, third best, etc., then there is another possible
world of the ordinal rank in question.



Moreover, different possible goods combined in different possible ways
may constitute different, logically independent, possible total goods
of the same value. The following simple example illustrates this
point. Let it be assumed that the pleasure which would be produced by
John's eating a mushroom pizza for dinner tonight and the
pleasure that would be produced by John's eating a garlic pizza
for dinner tonight are logically independent possible goods of the
same value. All other things being equal, a possible world containing
John's eating a mushroom pizza for dinner tonight and
John's not eating a garlic pizza tonight, and a possible world
containing John's eating a garlic pizza for dinner tonight and
John's not eating a mushroom pizza for dinner tonight,
constitute different possible goods of the same value. Generalizing
from examples of this kind, it appears that if some possible world is
best, second best, third best, and so on, then there are
*other* possible worlds, *not exactly resembling them*,
which are best, second best, third best, and so on.



So, for any possible pair of coexistent God-like omnipotent agents, it
appears that one member of that pair could *endeavor to
actualize* a different, equally good, world than the other member
of that pair, even given their hypothesized necessary epistemic,
moral, and aesthetic perfection. In the light of the *reductio ad
absurdum* presented earlier, it appears to follow that such pairs
are impossible.



Leibniz argued that there is a uniquely optimal possible world by
appealing to the *Principle of Sufficient Reason* and the
*Identity of Indiscernibles*, notoriously concluding that the
actual world is *the best of all possible worlds*. Among
contemporary philosophers, both the *Principle of Sufficient
Reason* and the *Identity of Indiscernibles* are
controversial.



Another possible defense of the possibility of a plurality of
coexistent omnipotent agents appeals to quantum mechanics. Quantum
*entanglement* seems to be a unique physical phenomenon whereby
concurrent activities of diverse contingently existing substantial
individuals are directly coordinated in virtue of a necessary linkage
of some sort between those substantial individuals. By drawing an
analogy with this phenomenon, one might argue that there could be
coexistent omnipotent agents who necessarily avoid stalemates. Quantum
mechanics implies that there exist pairs of entangled micro-particles
such that it is causally necessary that one member of the pair is
*spin up* if and only if the other member of the pair is
*spin up*, independently of the locations of those
micro-particles. Albert Einstein skeptically described the theoretical
phenomenon of entanglement as "spukhafte Fernwirkung,"
that is, as "spooky action at a distance." Nowadays,
however, entanglement is an experimentally confirmed part of physics.
Arguably, by analogy with entanglement, if a plurality of coexistent
omnipotent agents is possible, then there could be
"entangled" omnipotent agents, \(A\_1\) and \(A\_2\), such
that it is metaphysically necessary that \(A\_1\) endeavors to bring
about a certain state of affairs if and only if \(A\_2\) endeavors to
bring about the same state of affairs. But quantum mechanics further
implies that there are entangled pairs of micro-particles such that it
is causally necessary that one member of the pair is *spin up*
if and only if the other member of the pair is *spin down*. The
properties of *being spin up* and *being spin down* are
contraries. Thus, if the analogy with quantum entanglement is taken
seriously, then by parity of reasoning, one should conclude that if a
plurality of necessarily cooperating coexistent omnipotent agents is
possible, then *possibly*, there are "entangled"
omnipotent agents, \(A\_1\) and \(A\_2\), such that it is metaphysically
necessary that \(A\_1\) endeavors to act in some way if and only if
\(A\_2\) endeavors to act in a *contrary* way. Given this
conclusion, and in the light of the *reductio ad absurdum* of
the possibility of a plurality of coexistent omnipotent agents
presented earlier, an argument in favor of such a possibility by
analogy with quantum entanglement undermines itself, thereby
reinforcing that *reductio*. So, one cannot credibly defend the
metaphysical possibility of a plurality of coexistent omnipotent
agents by drawing an analogy with the phenomenon of quantum
entanglement.



Finally, could one credibly defend such a metaphysical possibility via
the hypothesis that it is metaphysically possible for there to be a
plurality of necessarily indiscernible omnipotent God-like beings (in
the Leibnizian sense of *indiscernible*)? Because a plurality
of *necessarily* indiscernible objects is of dubious
intelligibility, such a defense would not be credible. Doubts about
the intelligibility of such a plurality arise because of perplexities
concerning the individuation and separation of any pair of necessarily
indiscernible objects, and because positing the existence of a
plurality of such objects is metaphysically extravagant and
gratuitous. Indeed it seems that there is just as much reason to posit
indefinitely many objects of the sort in question as there is to posit
a pair of them, whereas the intelligibility of a *necessarily
self-indiscernible* object is not in doubt. A representative
example of a hypothetical plurality of necessarily indiscernible
objects is a plurality of *necessarily coincident* geometrical
points. In addition to the perplexities concerning the individuation
and separation of any pair of such points, the assumption that it is
metaphysically possible for there to be a plurality of necessarily
coincident points is metaphysically frivolous. Post-Scholastic
ridicule of the sort that was directed at the alleged ontological
excesses of the Schoolmen, e.g., the query "How many angels may
fit upon the point of a needle?", is quite appropriately
directed at metaphysical hypotheses of this sort. For these reasons,
the intelligibility of a plurality of necessarily coincident
geometrical points is suspect; the same is true of the intelligibility
of a plurality of necessarily indiscernible omnipotent God-like
beings.



Could an agent be accidentally omnipotent? At first glance, this
appears possible, but there is the following argument for the opposite
view. On the assumption that God exists, he has necessary existence,
is essentially not temporally limited, and is essentially omnipotent.
But there could not be two coexistent omnipotent agents. Thus, on the
assumption that God exists, an accidentally omnipotent being is
impossible.



This argument against the possibility of accidental omnipotence
presupposes traditional Western theism. However, traditional Western
theism is highly controversial, and *neutrality* about whether
God exists has some advantages. If one is neutral about whether God
exists, then omnipotence should *not* be assumed to be
attributable *only* to the God of traditional Western theism or
*only* to an essentially omnipotent being.


## 2. The Scope of Omnipotence



The intelligibility of the notion of omnipotence has been challenged
by the so-called paradox or riddle of the stone. Can an omnipotent
agent, Jane, bring it about that there is a stone of some mass, \(m\),
which Jane cannot move? If the answer is
'yes', then there is a state of affairs that Jane cannot
bring about, namely, (S1) *that a stone of mass \(m\)
moves*. On the other hand, if the answer is
'no', then there is another state of affairs that Jane
cannot bring about, namely, (S2) *that there is a stone of mass \(m\)
which Jane cannot move*. Thus, it seems that
whether or not Jane can make the stone in question, there is some
possible state of affairs that an omnipotent agent cannot bring about.
And this appears to be paradoxical.



A first resolution of the paradox comes into play when Jane is an
*essentially* omnipotent agent. In that case, the state of
affairs of Jane's being non-omnipotent is impossible. Therefore,
Jane cannot bring it about that she is not omnipotent. Since,
necessarily, an omnipotent agent can move any stone, no matter how
massive, (S2) is impossible. But, as we have seen, an omnipotent agent
is not required to be able to bring about an impossible state of
affairs.



If, on the other hand, Jane is an *accidentally* omnipotent
agent, both (S1) and (S2) *are* possible, and it *is*
possible for some omnipotent agent to bring it about that (S1) obtains
at one time, *and* that (S2) obtains at a different time. Thus,
there is a second solution to the paradox. In this case, Jane's
being non-omnipotent is a possible state of affairs; thus, we may
assume that it *is* possible for Jane to bring it about that
she is non-omnipotent. So, Jane can create and move a stone, \(s\),
of mass, \(m\), while omnipotent, and
*subsequently* bring it about that she is not omnipotent and
powerless to move \(s\). As a consequence, Jane can
bring about both (S1) and (S2), but only if they obtain at different
times.



It might now be conjectured that omnipotence can be analyzed simply as
the power to bring it about that any *contingent* state of
affairs obtains. However, the following list of contingent states of
affairs shows that there can be contingent states of affairs that an
omnipotent agent is powerless to bring about, and hence that this
simple analysis is inadequate:


1. that a raindrop fell;
2. that a raindrop falls at \(t\) (where \(t\)
is a past time);
3. that Parmenides lectures for the first time;
4. that the Amazon River floods an odd number of times less than
four;
5. that a snowflake falls and no omnipotent agent ever exists;
and
6. that Plato freely decides to write a dialogue.



Note that (a) is a past state of affairs. Presumably, it is not
possible for an efficient cause to occur *later* than its
effect. However, an agent's bringing about a state of affairs is
a kind of efficient causation. Therefore, it is not possible for an
agent to bring about anything that is in the *past*. In other
words, it is impossible for *any* agent to have power over what
is past. Hence, no agent, not even an omnipotent one, can bring it
about that (a) obtains. Likewise, despite the fact that (b) can be
brought about *prior* to \(t\), the
impossibility of an agent's having power over what is past
implies that *after t* even an omnipotent agent cannot bring it
about that (b) obtains. In the case of (c), prior to
Parmenides's first lecture, an omnipotent agent can bring about
(c). But once Parmenides has lectured, even an omnipotent agent cannot
bring it about that (c) obtains. As for (d), prior to the
Amazon's third flooding, an omnipotent agent can bring it about
that (d) obtains, while after the Amazon's third flooding, even
an omnipotent agent cannot bring it about that (d) obtains. (e)
introduces a special difficulty. Although it is obvious that (e) could
*not* be brought about by an omnipotent agent, it can be argued
plausibly that it *is* possible for a non-omnipotent agent to
bring about (e) by causing a snowflake to fall, *provided that*
no omnipotent agent ever
 exists.[1]
 But, as we argued earlier, a maximally powerful being need not have
the power to bring about every state of affairs that any other being
could. Lastly, while if the libertarian theory of free will is
correct, an omnipotent agent (who is, of course, *other than
Plato*) cannot bring about (f), apparently a non-omnipotent agent,
namely, Plato, can bring it about that (f) obtains.



Consequently, a satisfactory analysis of omnipotence ought not to
require that an omnipotent agent have the power to bring about (a),
(b), (c), (d), (e), or (f), if it is assumed, *arguendo*, in
the case of (f), that libertarianism is true.



Because of the wide disparity among contingent states of affairs,
(a)-(f), one might despair of finding an analysis of omnipotence
that both deals satisfactorily with all of these states of affairs and
implies that an omnipotent being has, intuitively speaking, sufficient
power. Is such pessimism warranted, or is omnipotence analyzable?



There are at least two approaches to analyzing omnipotence that hold
out some hope of success. The first utilizes the notion of an
*unrestrictedly repeatable state of affairs*, and the second
utilizes the notion of *two worlds sharing their histories up to a
time*. Although these approaches to analyzing omnipotence differ
in important ways, they are in broad agreement on the leading idea
that maximal power has logical and temporal limitiations, including
the limitation that an omnipotent agent cannot bring about, i.e.,
cause, another agent's free decision in the libertarian sense.
In the following two sections, some recent instances of these
approaches are set forth and compared.


## 3. Omnipotence and Unrestricted Repeatability



One attempt to analyze omnipotence in terms of unrestricted
repeatability is the account of Hoffman and Rosenkrantz. According to
their approach, by identifying certain features of (a)-(f), we
can find a feature that none of them possesses, and in terms of which
an analysis of omnipotence can be stated. To begin, unless it is
possible for *some* agent to bring about a given state of
affairs, an omnipotent agent ought not to be required to be able to
bring about that state of affairs. But (a) is not possibly brought
about by any agent.



Next, while (b) and (c) are possibly brought about by some agent, they
are not *repeatable*: it is not possible for either one of them
to obtain, subsequently fail to obtain, and then obtain again. Note
that if, because (a) is not possibly brought about by someone, an
omnipotent agent is not required to be able to bring about (a), then
for the same reason, that agent is also not required to be able to
bring about impossible or necessary states of affairs. Moreover, if,
because (b) and (c) are not repeatable, an omnipotent agent is not
required to bring about (b) or (c), then for the same reason, that
agent is also not required to be able to bring about impossible or
necessary states of affairs. These reasons for not requiring an
omnipotent agent to have the power to bring about impossible or
necessary states of affairs cohere with our earlier independent
arguments for these restrictions.



Third, while (d) *is* repeatable, it is *not*
*unrestrictedly* repeatable, that is, it cannot obtain, then
fail to obtain, then obtain again, and so on, eternally.



Fourth, while (e) *is* unrestrictedly repeatable, it is a
*complex* state of affairs, namely, a *conjunctive*
state of affairs whose second conjunct is *not* repeatable.
These examples suggest a hypothesis about repeatability and its
relation to power, namely, that an omnipotent agent should
*not* be required to have the power to bring about
*either* a state of affairs that is not unrestrictedly
repeatable, *or* a conjunctive state of affairs one of whose
conjuncts is not unrestrictedly repeatable.



Lastly, although (f) *is* unrestrictedly repeatable, (f) is
another type of complex state of affairs. In particular, it is
identifiable with or analyzable as a conjunctive state of affairs.
This state of affairs has three conjuncts, the second of which is not
possibly brought about by anyone. The conjunctive state of affairs in
question can be informally expressed as follows: Plato decides to
write a dialogue; and there is no *antecedent* sufficient
causal condition of Plato's deciding to write a dialogue; and
there is no concurrent sufficient causal condition of Plato's
deciding to write a dialogue. Because it is impossible for an agent to
have power over what is *past*, the second conjunct of this
state of affairs is not possibly brought about by anyone. Thus, an
omnipotent agent ought not to be required to have the power to bring
about a state of affairs that is identifiable with or analyzable as a
conjunctive state of affairs one of whose conjuncts is not possibly
brought about by anyone.



According to the account of Hoffman and Rosenkrantz, incoporating
these ideas, omnipotence can be analyzed in terms of the following
three definitions.



(D1)

The period of
time \(t\) is a sufficient interval for \(s =\_{df} s\)
is a state of affairs such that: it is possible that \(s\)
obtains at a time-period which has the duration of
\(t\).


For example, any period of time with a duration of 7 seconds is a
sufficient interval for the state of affairs *that a ball rolls for
7 seconds*.



(D2)
A state of
affairs, \(s\), is unrestrictedly repeatable \(=\_{df}\)
\(s\) is possibly such that: \(\forall n\exists
t\_1\exists t\_2\exists t\_3 \ldots \exists t\_n [(t\_1 \lt t\_2 \lt t\_3 \lt
\ldots \lt t\_n\) are periods of time which are sufficient intervals
for \(s \amp s\) obtains at \(t\_1 \amp s\) doesn't obtain at
\(t\_2 \amp s\) obtains at \(t\_3 \amp \ldots \amp s\) obtains at
\(t\_n)\) if and only if \(n\) is
 odd].[2]



For instance, the state of affairs *that a ball rolls for 7
seconds* is unrestrictedly repeatable.



(D3)

\(x\) is omnipotent at \(t =\_{df} \forall s\)(if it is possible for
some agent to bring about \(s\) then at \(t\) \(x\) has it within his
power to bring about \(s)\). 


In (D3), \(x\) ranges over agents, and \(s\)
over states of affairs that satisfy the following
condition:



(C)
(i) \(s\)
is unrestrictedly repeatable, and of the form
'in \(n\) minutes, \(p\)',
& \((p\) is a complex state of affairs \(\rightarrow\) (each of
the parts of \(p\) is unrestrictedly repeatable &
possibly brought about by someone)), or (ii) \(s\) is
of the form '\(q\) forever after', where
\(q\) is a state of affairs which satisfies
 (i).[3]



In applying (D3) to states of affairs like (e) and (f) it should be
observed that a *conjunct* of a conjunctive state of affairs is
a *part* of such a complex state of
 affairs.[4]


 (C) (ii) refers to a state of affairs of the form '\(q\)
forever after', where \(q\) is a
state of affairs satisfying (i). An example of a state of affairs of
this kind is *in two minutes, a ball rolls forever after*. Two
situations in which this state of affairs may obtain are, first, that
the ball will start rolling in two minutes, and then continue to roll
forever after, and second, that the ball will start rolling earlier
than that, for instance, two minutes earlier, will be rolling in two
minutes, and will continue to roll forever after.


As intended, (D3) does not require an omnipotent agent to have the
power to bring about either impossible or necessary states of affairs,
or states of affairs such as (a)-(f). Furthermore, (D3) does not
unduly limit the power of an omnipotent agent, since an agent's
bringing about a state of affairs can always be "cashed
out" in terms of that agent's bringing about an
unrestrictedly repeatable state of affairs that it is possible for
some agent to bring about. That is, necessarily, for any state of
affairs, \(s\), if an agent, \(a\),
brings about \(s\), then either \(s\) is
an unrestrictedly repeatable state of affairs which it is possible for
some agent to bring about, or else \(a\) brings about
*s by* bringing about \(q\), where \(q\)
is an unrestrictedly repeatable state of affairs
which it is possible for some agent to bring about. For instance, an
omnipotent agent can bring about the state of affairs, *that in one
hour, Parmenides lectures for the first time*, by bringing about
the state of affairs, *that in one hour, Parmenides lectures*,
when this lecture is Parmenides's first. And although the former
state of affairs is a nonrepeatable one that (D3) does *not*
require an omnipotent agent to be able to bring about, the latter
state of affairs *is* an unrestrictedly repeatable state of
affairs that (D3) *does* require an omnipotent agent to be able
to bring about.


## 4. Omnipotence and the Shared Histories Approach



The alternative approach to analyzing omnipotence in terms of two
worlds sharing their histories up to a time is exemplified by the
accounts of Flint and Freddoso, and Wierenga. As we shall see,
although these two accounts are similar, they differ in certain
significant respects.



Flint and Freddoso's account of what it is for an agent \(S\)
at a time \(t\) to be omnipotent in a
possible world \(W\) is formulated as follows.




\(S\) is omnipotent at \(t\) in \(W\)
if and only if for any state of affairs \(p\)
and world-type-for-\(S\) \(Ls\) such
that \(p\) is not a member of \(Ls\), if there is a
world \(W^\*\) such that


1. \(Ls\) is true in both \(W\) and \(W^\*\), and
2. \(W^\*\) shares the same history with \(W\) at \(t\),
and
3. at \(t\) in \(W^\*\) someone actualizes \(p\),
then \(S\) has the power at \(t\)
in \(W\) to actualize \(p\)
(Flint & Freddoso 1983, p. 99).




The notion of a world-type-for-\(S\) \(Ls\) is to be
understood in the following way. A *world-type* is "a set
which is such that for any *counterfactual of freedom*, i.e.,
any proposition which can be expressed by a sentence of the form
'If individual essence \(P\) were instantiated in
circumstances \(C\) at time \(t\) and
its instantiation were left free with respect to action \(A\),
the instantiation of \(P\) would
freely do \(A\)'--either that counterfactual
or its negation is a member of the set" (Flint & Freddoso
1983, p. 96). It may also be stipulated "that for any two
members of the set, the conjunction of those two members is a member
of the set as well" (Flint & Freddoso 1983, pp.
96-97). Moreover, "a world-type is *true* just in
case every proposition which is a member of it is true" (Flint
& Freddoso 1983, p. 97). In addition, it is presupposed that
"for any free agent \(x\), there will be a set of
all and only those true counterfactuals of freedom (or true negations
of such counterfactuals) over whose truth-value \(x\)
has no control" (Flint & Freddoso 1983, p. 97). A set of
this kind is referred to as *the world-type-for-x*. Finally,
'\(Lx\)' designates the true-world-type-for-\(x\).



The notion of actualization employed in this account of omnipotence
calls for some explanation. If an agent, \(S\), brings
about a state of affairs, \(p\), then \(S\)
actualizes \(p\). However, this
account presupposes that an agent may [*weakly*] actualize
another agent's making a *free* decision without
*bringing about* or *causing* that decision. In
particular, it is assumed that an agent may weakly actualize a
decision that is free in the libertarian sense by bringing about the
antecedent of a true "counterfactual of freedom."



The basic idea of this account of omnipotence is that an agent is
omnipotent just when he can actualize any state of affairs that it is
possible for someone to actualize, except for certain
"counterfactuals of freedom", their consequents, and
certain states of affairs that are "accidentally
impossible" because of the past.



With respect to so-called counterfactuals of freedom, this account
presupposes that some of them, for example,



If Jessica were offered the grant, then she would freely decide to
accept it,



are *true*. Some philosophers hold the contrary view that a
subjunctive conditional of this kind is *necessarily*
*false*. Why do these philosophers reject the claim that some
"counterfactuals of freedom" are true? Presumably, what
distinguishes a subjunctive conditional from a corresponding material
conditional is that only the former expresses a strong or necessary
connection of some sort between the conditions specified by the
antecedent and the consequent. Seemingly, the only kinds of strong or
necessary connections available in this case are relations [broadly
speaking] of either *causation* or *entailment*.
Consequently, it appears that the subjunctive conditional under
discussion is necessarily false, since if Jessica *freely*
decides to accept the grant [in the relevant libertarian sense], then
her making that decision is neither *caused* nor
*entailed* by her being offered the grant. If the foregoing
line of reasoning is correct, then the notion of a true
"counterfactual of freedom" is incoherent. Since Flint and
Freddoso's account of omnipotence presupposes that there are
such "counterfactuals of freedom," it can be argued that
this account is incoherent.



Moreover, it can be argued that a state of affairs discussed earlier
provides a counter-example to Flint and Freddoso's account of
omnipotence, namely:



(e)
A snowflake falls
and no omnipotent agent ever exists. 


A non-omnipotent agent can bring about or actualize (e) by bringing it
about that a snowflake falls when in fact no omnipotent agent ever
exists. But, it is clear that an omnipotent agent cannot bring about
or actualize (e). For although an omnipotent agent can bring it about
that a snowflake falls, surely, an omnipotent agent cannot bring it
about that *no omnipotent ever exists*, nor would this conjunct
of (e) obtain if there were an omnipotent agent. Moreover, we may
assume that there are possible worlds, \(W\) and
\(W^\*\), such that \(W\) and \(W^\*\) share the same
history up to a time \(t\), no omnipotent agent ever
exists in \(W^\*\), and a contingently omnipotent agent, Oscar, is
omnipotent for the first time at \(t\) in \(W\).
We may also assume that \(W^\*\) is a world in which
at \(t\) some non-omnipotent agent actualizes (e). On
the other hand, evidently, if in \(W\), Oscar is
omnipotent at \(t\), then at \(t\) Oscar
cannot actualize (e). Note that since the second conjunct of (e) is
not unrestrictedly repeatable, this is consistent with Hoffman and
Rosenkrantz's account of omnipotence; their account does not
require an omnipotent agent to be able to bring about a conjunctive
state of affairs one of whose conjuncts is not unrestrictedly
repeatable. On the other hand, Flint and Freddoso's account of
omnipotence implies that in \(W\), at \(t\)
Oscar has the power to actualize (e). This
implication holds for the following reasons. First, (e) is not a
member of a world-type-for-Oscar, inasmuch as (e) is neither a
"counterfactual of freedom," the negation of one, nor a
conjunction of such "counterfactuals of freedom." Second,
we may assume that a world-type-for-Oscar is true in both \(W^\*\) and
\(W\), since the assumption that an agent is not
omnipotent in one possible world, and is omnipotent for a time in
another possible world, does not necessitate any difference in the
world-type for that agent which is true in those worlds. Third, it is
possible for someone at \(t\) to actualize (e) in a
world, \(W^\*\), that has the same history up to \(t\)
as *W*.[5]
 Thus, arguably, Flint and Freddoso's account of omnipotence
requires that in \(W\) an omnipotent agent, Oscar, at
\(t\) has the power to actualize (e), when Oscar lacks
this power. If this is right, then their account does not provide a
logically necessary condition on
 omnipotence.[6]



A counter-example of this kind assumes that an analysis of omnipotence
should allow for the possibility of an omnipotent agent other than
God. Given this assumption, (e) seems to provide a counter-example to
Flint and Freddoso's account of omnipotence, but not to the
account of Hoffman and Rosenkrantz.



Let us now turn to Wierenga's account of omnipotence. The basic
idea of Wierenga's account of omnipotence is that an agent is
omnipotent if and only if he can do anything that it is possible for
him to do, given the past. According to this account, we can analyze
what it is for an agent, \(A\), to be omnipotent at \(t\)
in a world \(W\) in terms of what it
is *possible for A* to strongly actualize at \(t\)
in worlds having the same history as \(W\)
up to \(t\).



Wierenga's account of omnipotence, like Flint and
Freddoso's, relies on the intuitive idea that two possible
worlds can share the same past or history up to a certain point in
time, and then diverge. According to Wierenga's account, two
worlds of this kind share an *initial segment*, where \(S (W,
t)\) is an initial segment of a possible world \(W\) up
to a time \(t\) (Wierenga 1989, pp. 18-20).
Unlike Flint and Freddoso's account, Wierenga's account is
not stated in terms of what an agent can actualize, but rather in
terms of the narrower notion of what an agent can *strongly
actualize*. An agent, \(A\), strongly actualizes
just those states of affairs that \(A\) brings about
*directly* or those actions that \(A\) does not
do *by* doing something else (Wierenga 1989, pp. 20-23).
Of course, an agent may actualize a state of affairs
*indirectly* by strongly actualizing *another* state of
affairs. Wierenga's account of omnipotence is formulated as
follows.



> 
> A being \(x\) is omnipotent in a world \(W\)
> at a time \(t =\_{df}\) In \(W\) it is
> true both that (i) for every state of affairs \(A\), if
> it is possible that both \(S (W, t)\) obtains and that \(x\)
> strongly actualizes \(A\) at \(t\),
> then at \(t, x\) can strongly actualize \(A\),
> and (ii) there is some state of affairs which \(x\)
> can strongly actualize at \(t\).
> (Wierenga 1989, p. 25)
> 



This account of omnipotence may be vulnerable to a counter-example of
the following kind. Arguably, there could be an agent, \(x\),
such that: \(x\) has a wide range of
powers, \(x\) is essentially limited to these powers,
and \(x\) essentially lacks a power, \(P\),
which an omnipotent agent ought to possess. Of
course, \(x\) would not be omnipotent. Yet,
Wierenga's account of omnipotence paradoxically implies that \(x\)
would be omnipotent. Hence, it can be argued that
Wierenga's account does not provide a logically sufficient
condition for omnipotence. The assumption that there could be a
non-omnipotent agent that is essentially limited in its powers can be
defended as follows. An omnipotent agent has the power to overrule (or
supersede) any law of nature (a mere *physical necessity*). For
example, God has the power to overrule the law of gravity by bringing
it about that a mountain floats in midair without any physical cause.
Yet, arguably, it is possible for there to be a non-omnipotent agent
who essentially lacks the power to overrule any law of nature. For
example, it can be argued that there could be a physical or material
agent who is *essentially* subject to certain laws of nature.
Surely, such an agent would lack the power to overrule any law of
nature, and so would not be omnipotent.



A similar, though weaker, sort of objection concerns McEar, a
hypothetical agent who essentially has the power to do only
*one* thing, namely, scratch his ear. It may be objected that
Wierenga's analysis of omnipotence falsely implies that McEar
would be omnipotent. But it might be replied that an agent such as
McEar is impossible. It can be cogently argued that, necessarily, if
McEar has the power to scratch his ear, then he *also* has the
power to move a part of his body to scratch his ear, for instance, his
arm (Wierenga 1989, pp. 28-29). So, it appears that there could
not be an agent that has the power to do only one thing. In reply to
the stronger sort of objection discussed earlier, it may be suggested
that, necessarily, for *any* power, if an agent lacks that
power, then an omnipotent being could give that agent that power
(Wierenga 1989, p. 29). The difficulty with such a reply is that there
could be a non-omnipotent agent who *essentially* lacks the
power to overrule any law of nature, and hence that not even an
omnipotent agent could give this non-omnipotent agent that power.


## 5. Omnipotence and Divine Moral Perfection



It has been argued that the traditional God has incompatible
attributes, namely, necessary existence, essential omnipotence,
essential omniscience, and essential moral perfection (Pike
 1969).[7]
 The contention has been that it is impossible for God to have the
power to bring about evil, while non-omnipotent (and morally
imperfect) beings may have this power. The precise form of such an
argument varies depending on what precisely the relation between God
and evil is assumed to be. However, generally speaking, it is argued
that divine moral perfection and omnipotence are incompatible because
divine omnipotence entails that God has the power to bring about evil,
whereas divine moral perfection entails that God is powerless to bring
about evil.



One can respond to arguments of this kind as follows. Assume that if
God exists, then this is a best possible
 world.[8]
 In that case, if God exists, there could not be an evil unless it
were necessary for some greater good, in which case any state of
affairs containing evil incompatible with there being a maximally good
world is *impossible*. But it be may be assumed that it is not
possible for *any* agent to bring about an impossible state of
affairs. Thus, if God exists, any moral evil, that is, any evil
brought about by anyone, and any natural evil, or any evil which has
an impersonal, natural cause, must be necessary for some greater
good.



Suppose that God exists and that some other person, for example, Cain,
brings it about that an evil, \(E\), exists. There are
two possibilities that need to be considered here. The first is that
Cain's decisions and actions are causally determined, as are all
occurrences in the created universe. Then, given our assumptions,
since Cain's bringing it about that \(E\) exists
is necessary for some good which more than compensates for \(E\)'s
existence, it is consistent with God's
moral perfection that God [remotely] brings it about that Cain brings
it about that \(E\) exists.



The second possibility is that Cain's decision to do evil is
uncaused by anything other than Cain and free in the libertarian
sense. In that case, God did *not* [remotely cause Cain freely
to] bring it about that \(E\) exists, while [let us
assume] Cain *did* freely bring it about that \(E\)
exists. If so, then it must be the case that
God's creating Cain and permitting Cain freely to do what he
chooses to do [in the context of the entire creation] brings about
more good than his *not* creating Cain and thus *not*
permitting him freely to do what he chooses to do. It might be
objected that if Cain can bring about a state of affairs that God
cannot, namely, *that E exists*, then God is not omnipotent.
But, as we have seen, an agent's being omnipotent does not
require of that agent that it be able to bring about *every*
state of affairs which *any* other agent can bring about. It
*does*, of course, require that an omnipotent agent have more
power than any other agent. And God, of course, *would* have
more power than Cain, even though Cain could bring about something
that God could not. For there are many more states of affairs that God
could bring about and that Cain could not, than *vice versa*.
At this point, it might further be objected that an omnipotent agent,
one that was morally imperfect, who *could* bring it about that
\(E\) exists, as well as all the other states of
affairs that God could bring about, would be more powerful than God.
But recall that if God exists, then he exists eternally in every
possible world. Recall, too, that apparently there cannot be more than
one omnipotent agent. Thus, it appears that if God exists, then an
omnipotent agent who is morally imperfect is *impossible*.
Thus, this second objection is based on an assumption that seems to be
impossible, namely, that if God exists there could exist another
omnipotent agent who is morally imperfect and who is therefore more
powerful than God.



Of course, if God exists, then any evil state of affairs, \(s\),
which *is* incompatible with a maximally good
world is *impossible*. And if \(s\) is
impossible, then neither God nor any other agent has the power to
bring it about that \(s\) obtains. God would lack the
power to bring it about that \(s\) obtains because of
his moral perfection, and any created agent would lack the power to
bring it about that \(s\) obtains either because (i)
God would not create an agent who had the power to bring it about that
\(s\) obtains, or (ii) God would not permit any created
agent to bring it about that \(s\) obtains. Thus, to
the extent indicated, if God's attributes impose moral
restrictions on the nature of the universe and on what he can bring
about, then they impose parallel restrictions on what any other agents
can bring about.



The foregoing line of reasoning implies that God's moral
perfection and omnipotence are not
 incompatible.[9]



This argument about God and the possibility of evil has been disputed
by theists such as Alvin Plantinga, who do not hold that God's
existence implies the existence of a maximally good world, but do hold
that God seeks to create as good a world as he
 can.[10]
 Theists such as Plantinga allow for there to be evil that is
 *unnecessary* for any greater good that outweighs it. An evil
of this kind involves free decisions of non-divine agents, which God
does not prevent, but which these other agents can prevent. Plantinga
contends that God is not wrong to permit an evil of this kind, since
God cannot bring about a vital good, the existence of free human
agents, without there being such an evil. Alternatively, it might be
argued that God does no wrong in this sort of case, because he does
not know how to do better (knowledge of the future free actions of
created agents being impossible). However, as an omnipotent God is
*not* required to have power over the free decisions of
non-divine agents, it follows that on these views, his omnipotence and
moral perfection are compatible, roughly to the extent indicated
earlier in our discussion of the view that God's existence
implies a maximally good world. Of course, nothing that has been said
here answers the question of how much, if any, evil is compatible with
the existence of the traditional God. This question is central to the
problem of evil for theism.