Omnipotence paradox

The omnipotence paradox is a family of paradoxes that arise with some understandings of the term "omnipotent". The paradox arises, for example, if one assumes that an omnipotent being has no limits and is capable of realizing any outcome, even a logically contradictory one such as creating a square circle. Atheological arguments based on the omnipotence paradox are sometimes described as evidence for countering theism. Other possible resolutions to the paradox hinge on the definition of omnipotence applied and the nature of God regarding this application and whether omnipotence is directed toward God himself or outward toward his external surroundings.
The omnipotence paradox has medieval origins, dating at least to the 12th century. It was addressed by Averroës and later by Thomas Aquinas. Pseudo-Dionysius the Areopagite (before 532) has a predecessor version of the paradox, asking whether it is possible for God to "deny himself".
The most well-known version of the omnipotence paradox is the so-called "paradox of the stone": "Could God create a stone so heavy that even He could not lift it?" This phrasing of the omnipotence paradox is vulnerable to objections based on the physical nature of gravity, such as how the weight of an object depends on what the local gravitational field is. Alternative statements of the paradox that do not involve such difficulties include "If given the axioms of Euclidean geometry, can an omnipotent being create a triangle whose angles do not add up to 180 degrees?" and "Can God create a prison so secure that he cannot escape from it?".
Overview.
A common modern version of the omnipotence paradox is expressed in the question: "Can [an omnipotent being] create a stone so heavy that it cannot lift it?" This question generates a dilemma. The being can either create a stone it cannot lift, or it cannot create a stone it cannot lift. If the being create a stone that it cannot lift, then it is not omnipotent because there is a weight threshold beyond its own power to lift. If the being create a stone it cannot lift, then there is something it cannot create, and is therefore not omnipotent. In either case, the being is not omnipotent.
A related issue is whether the concept of "logically possible" is different for a world in which omnipotence exists than a world in which omnipotence does not exist.
The dilemma of omnipotence is similar to another classic paradox—the irresistible force paradox: "What would happen if an irresistible force were to meet an immovable object?" One response to this paradox is to disallow its formulation, by saying that if a force is irresistible, then by definition there is no immovable object; or conversely, if an immovable object exists, then by definition no force can be irresistible. Some claim that the only way out of this paradox is if the irresistible force and immovable object never meet. But this is not a way out, because an object cannot in principle be immovable if a force exists that can in principle move it, regardless of whether the force and the object actually meet.
Types of omnipotence.
Augustine of Hippo in his City of God writes "God is called omnipotent on account of His doing what He wills" and thus proposes the definition that "Y is omnipotent" means "If Y wishes to do X then Y can and does do X". 
The notion of omnipotence can also be applied to an entity in different ways. An essentially omnipotent being is an entity that is necessarily omnipotent. In contrast, an accidentally omnipotent being is an entity that can be omnipotent for a temporary period of time, and then becomes non-omnipotent. The omnipotence paradox can be applied to each type of being differently.
In addition, some philosophers have considered the assumption that a being is either omnipotent or non-omnipotent to be a false dilemma, as it neglects the possibility of varying degrees of omnipotence. Some modern approaches to the problem have involved semantic debates over whether language—and therefore philosophy—can meaningfully address the concept of omnipotence itself.
Proposed answers.
Omnipotence does not mean breaking the laws of logic.
A common response from Christian philosophers, such as Norman Geisler or William Lane Craig, is that the paradox assumes a wrong definition of omnipotence. Omnipotence, they say, does not mean that God can do anything "at all" but, rather, that he can do "anything that is possible according to his nature." The distinction is important. God cannot perform logical absurdities; he cannot, for instance, make 1+1=3. Likewise, God cannot make a being greater than himself because he is, by definition, the greatest possible being. God is limited in his actions to his nature. The Bible supports this, they assert, in passages such as Hebrews 6:18, which says it is "impossible for God to lie."
A good example of a modern defender of this line of reasoning is George Mavrodes. Essentially, Mavrodes argues that it is no limitation on a being's omnipotence to say that it cannot make a round square. Such a "task" is termed by him a "pseudo-task" as it is self-contradictory and inherently nonsense. Harry Frankfurt—following from Descartes—has responded to this solution with a proposal of his own: that God can create a stone impossible to lift and also lift said stoneFor why should God not be able to perform the task in question? To be sure, it is a task—the task of lifting a stone which He cannot lift—whose description is self-contradictory. But if God is supposed capable of performing one task whose description is self-contradictory—that of creating the problematic stone in the first place—why should He not be supposed capable of performing another—that of lifting the stone? After all, is there any greater trick in performing two logically impossible tasks than there is in performing one?
If a being is "accidentally omnipotent", it can resolve the paradox by creating a stone it cannot lift, thereby becoming non-omnipotent. Unlike essentially omnipotent entities, it is possible for an accidentally omnipotent being to be non-omnipotent. This raises the question, however, of whether the being was ever truly omnipotent, or just capable of great power. On the other hand, the ability to voluntarily give up great power is often thought of as central to the notion of the Christian Incarnation.
If a being is "essentially omnipotent", then it can also resolve the paradox. The omnipotent being is essentially omnipotent, and therefore it is impossible for it to be non-omnipotent. Further, the omnipotent being can do what is logically impossible—just like the accidentally omnipotent—and have no limitations except the inability to become non-omnipotent. The omnipotent being cannot create a stone it cannot lift.
The omnipotent being cannot create such a stone because its power is equal to itself—thus, removing the omnipotence, for there can only be one omnipotent being, but it nevertheless retains its omnipotence. This solution works even with definition 2—as long as we also know the being is essentially omnipotent rather than accidentally so. However, it is possible for non-omnipotent beings to compromise their own powers, which presents the paradox that non-omnipotent beings can do something (to themselves) which an essentially omnipotent being cannot do (to itself). This was essentially the position Augustine of Hippo took in his "The City of God":For He is called omnipotent on account of His doing what He wills, not on account of His suffering what He wills not; for if that should befall Him, He would by no means be omnipotent. Wherefore, He cannot do some things for the very reason that He is omnipotent.
Thus Augustine argued that God could not do anything or create any situation that would, in effect, make God not God.
In a 1955 article in the philosophy journal "Mind", J. L. Mackie tried to resolve the paradox by distinguishing between first-order omnipotence (unlimited power to act) and second-order omnipotence (unlimited power to determine what powers to act things shall have). An omnipotent being with both first and second-order omnipotence at a particular time might restrict its own power to act and, henceforth, cease to be omnipotent in either sense. There has been considerable philosophical dispute since Mackie, as to the best way to formulate the paradox of omnipotence in formal logic.
Paradox is meaningless: the question is sophistry.
Another common response is that since God is supposedly omnipotent, the phrase "could not lift" does not make sense and the paradox is meaningless. This may mean that the complexity involved in rightly understanding omnipotence—contra all the logical details involved in misunderstanding it—is a function of the fact that omnipotence, like infinity, is perceived at all by contrasting reference to those complex and variable things, "which it is not". An alternative meaning, however, is that a non-corporeal God cannot lift anything, but can raise it (a linguistic pedantry)—or to use the beliefs of Hindus (that there is one God, who can be manifest as several different beings) that whilst it is possible for God to do all things, it is not possible for all his incarnations to do them. As such, God could create a stone so heavy that, in one incarnation, he could not lift it, yet could do something that an incarnation that "could" lift the stone could not.
The lifting a rock paradox ("Can God lift a stone larger than he can carry?") uses human characteristics to cover up the main skeletal structure of the question. With these assumptions made, two arguments can stem from it:
The act of killing oneself is not applicable to an omnipotent being, since, despite that such an act does involve some power, it also involves a "lack" of power: the human person who can kill himself is already not indestructible, and, in fact, every agent constituting his environment is more powerful in some ways than himself. In other words, all non-omnipotent agents are concretely "synthetic": constructed as contingencies of other, smaller, agents, meaning that they, unlike an omnipotent agent, logically can exist not only in multiple instantiation (by being constructed out of the more basic agents they are made of), but are each bound to a different location in space contra transcendent omnipresence.
C. S. Lewis argues that when talking about omnipotence, referencing "a rock so heavy that God cannot lift it" is nonsense just as much as referencing "a square circle"; that it is not logically coherent in terms of power to think that omnipotence includes the power to do the logically impossible. So asking "Can God create a rock so heavy that even he cannot lift it?" is just as much nonsense as asking "Can God draw a square circle?" The logical contradiction here being God's simultaneous ability and disability in lifting the rock: the statement "God can lift this rock" must have a truth value of either true or false, it cannot possess both. This is justified by observing that for the omnipotent agent to create such a stone, it must already be more powerful than itself: such a stone is too heavy for the omnipotent agent to lift, but the omnipotent agent already can create such a stone; If an omnipotent agent already is more powerful than itself, then it already is just that powerful. This means that its power to create a stone that is too heavy for it to lift is identical to its power to lift that very stone. While this does not quite make complete sense, Lewis wished to stress its implicit point: that even within the attempt to prove that the concept of omnipotence is immediately incoherent, one admits that it is immediately coherent, and that the only difference is that this attempt is forced to admit this despite that the attempt is constituted by a perfectly irrational route to its own unwilling end, with a perfectly irrational set of 'things' included in that end.
In other words, the 'limit' on what omnipotence 'can' do is not a "limit" on its actual agency, but an "epistemological boundary" without which omnipotence could not be "identified" (paradoxically or otherwise) in the first place. In fact, this process is merely a fancier form of the classic Liar Paradox: If I say, "I am a liar", then how can it be true if I am telling the truth therewith, and, if I am telling the truth therewith, then how can I be a liar? So, to think that omnipotence is an "epistemological" paradox is like failing to recognize that, when taking the statement, 'I am a liar' self-referentially, the statement is reduced to an actual failure to lie. In other words, if one maintains the supposedly 'initial' position that the necessary conception of omnipotence includes the 'power' to compromise both itself and all other identity, and if one concludes from this position that omnipotence is epistemologically incoherent, then one implicitly is asserting that one's own 'initial' position is incoherent. Therefore, the question (and therefore the perceived paradox) is meaningless. Nonsense does not suddenly acquire sense and meaning with the addition of the two words, "God can" before it. Lewis additionally said that, "Unless something is self-evident, nothing can be proved." This implies for the debate on omnipotence that, "as in matter, so in the human understanding of truth: it takes no true insight to destroy a perfectly integrated structure, and the effort to destroy has greater effect than an equal effort to build; so, a man is thought a fool who assumes its integrity, and thought an abomination who argues for it. It is easier to teach a fish to swim in outer space than to convince a room full of ignorant fools why it cannot be done."
Language and omnipotence.
The philosopher Ludwig Wittgenstein is frequently interpreted as arguing that language is not up to the task of describing the kind of power an omnipotent being would have. In his "Tractatus Logico-Philosophicus", he stays generally within the realm of logical positivism until claim 6.4—but at 6.41 and following, he argues that ethics and several other issues are "transcendental" subjects that we cannot examine with language. Wittgenstein also mentions the will, life after death, and God—arguing that, "When the answer cannot be put into words, neither can the question be put into words."
Wittgenstein's work expresses the omnipotence paradox as a problem in semantics—the study of how we give symbols meaning. (The retort "That's only semantics," is a way of saying that a statement only concerns the definitions of words, instead of anything important in the physical world.) According to the "Tractatus," then, even attempting to formulate the omnipotence paradox is futile, since language cannot refer to the entities the paradox considers. The final proposition of the "Tractatus" gives Wittgenstein's dictum for these circumstances: "What we cannot speak of, we must pass over in silence".
Wittgenstein's approach to these problems is influential among other 20th century religious thinkers such as D. Z. Phillips. In his later years, however, Wittgenstein wrote works often interpreted as conflicting with his positions in the "Tractatus", and indeed the later Wittgenstein is mainly seen as the leading critic of the early Wittgenstein.
Other versions of the paradox.
In the 6th century, Pseudo-Dionysius claims that a version of the omnipotence paradox constituted the dispute between Paul the Apostle and Elymas the Magician mentioned in Acts 13:8, but it is phrased in terms of a debate as to whether God can "deny himself" ala 2 Tim 2:13. In the 11th century, Anselm of Canterbury argues that there are many things that God cannot do, but that nonetheless he counts as omnipotent.
Thomas Aquinas advanced a version of the omnipotence paradox by asking whether God could create a triangle with internal angles that did not add up to 180 degrees. As Aquinas put it in "Summa contra Gentiles":
This can be done on a sphere, and not on a flat surface. The later invention of non-Euclidean geometry does not resolve this question; for one might as well ask, "If given the axioms of Riemannian geometry, can an omnipotent being create a triangle whose angles "do not" add up to "more" than 180 degrees?" In either case, the real question is whether an omnipotent being would have the ability to evade consequences that follow logically from a system of axioms that the being created.
A version of the paradox can also be seen in non-theological contexts. A similar problem occurs when accessing legislative or parliamentary sovereignty, which holds a specific legal institution to be omnipotent in legal power, and in particular such an institution's ability to regulate itself.
In a sense, the classic statement of the omnipotence paradox — a rock so heavy that its omnipotent creator cannot lift it — is grounded in Aristotelian science. After all, if we consider the stone's position relative to the sun the planet orbits around, one could hold that the stone is "constantly" lifted—strained though that interpretation would be in the present context. Modern physics indicates that the choice of phrasing about lifting stones should relate to acceleration; however, this does not in itself of course invalidate the fundamental concept of the generalized omnipotence paradox. However, one could easily modify the classic statement as follows: "An omnipotent being creates a universe that follows the laws of Aristotelian physics. Within this universe, can the omnipotent being create a stone so heavy that the being cannot lift it?"
Ethan Allen's "Reason" addresses the topics of original sin, theodicy and several others in classic Enlightenment fashion. In Chapter 3, section IV, he notes that "omnipotence itself" could not exempt animal life from mortality, since change and death are defining attributes of such life. He argues, "the one cannot be without the other, any more than there could be a compact number of mountains without valleys, or that I could exist and not exist at the same time, or that God should effect any other contradiction in nature." Labeled by his friends a Deist, Allen accepted the notion of a divine being, though throughout "Reason" he argues that even a divine being must be circumscribed by logic.
In "Principles of Philosophy," Descartes tried refuting the existence of atoms with a variation of this argument, claiming God could not create things so indivisible that he could not divide them.