The omnipotence paradox is actually a family of related paradoxes, each having to do with the question of what an omnipotent being can do, and particularly whether a being that is able to perform all actions can perform an action that would limit its own ability to perform actions. The argument states that if the being can perform such actions, then it can limit its own ability to perform actions and hence it cannot perform all actions, and if it cannot limit its own actions, then it could never have performed all actions.
This paradox is often formulated in terms of the God of the Abrahamic religions, though this is not a requirement. One version of the omnipotence paradox is the so-called paradox of the stone: "Could an omnipotent being create a stone so heavy that even that being could not lift it?" If so, then it seems that the being could cease to be omnipotent; if not, it seems that the being was not omnipotent to begin with. Other versions of the paradox have been advanced, as the "heavy stone" presents problems with respect to modern physics.
A similar problem occurs in non-theological contexts when accessing legislative or parliamentary sovereignty. Where a specific legal institution is said to be omnipotent in legal power, the question arises over such an institution's ability to regulate itself. 
Some philosophers, such as J. L Cowan, see this paradox as a reason to reject the possibility of any absolutely omnipotent entity. Others, such as Thomas Aquinas, assert that the paradox arises from a misunderstanding of the concept of omnipotence. The paradox can indeed be viewed as a straightforward logical impossibility, in that it frames an inability ("cannot lift it") as an attribute of total ability (omnipotence), rather than its absence or negation.
Still others, such as René Descartes, argue that God is absolutely omnipotent, despite the apparent problem. 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 involve semantic debates over whether language — and therefore philosophy — can meaningfully address the concept of omnipotence itself.
To analyze the omnipotence paradox rigorously, a precise definition of omnipotence must be established. The common definition, "all powerful", is not specific enough to deal with the issues raised by the paradox.
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 which it cannot lift, or it cannot create a stone which it cannot lift. If the being can create a stone that it cannot lift, then it seems that it can cease to be omnipotent. If the being cannot create a stone which it cannot lift, then it seems it is already not omnipotent.
The problem is similar to another classic paradox, the irresistible force paradox: What happens when an irresistible force meets an immovable object? One response to this paradox is that if a force is irresistible, then by definition there is no truly immovable object; conversely, if an immovable object were to exist, then no force could be defined as being truly irresistible. But this way out is not possible in the omnipotence case, because the purpose is to ask if the being's omnipotence makes its own omnipotence impossible.
J.L Cowan attempts to resolve the paradox in "The Paradox of Omnipotence Revisited." He proposes the following:
(1) Either God can create a stone which He cannot lift, or He cannot create a stone which He cannot lift.
(2) If God can create a stone which He cannot lift, then He is not omnipotent (since He cannot lift the stone in question).
(3) If God cannot create a stone which He cannot lift, then He is not omnipotent (since He cannot create the stone in question).
(4) Therefore God is not omnipotent.
Omnipotence implies that God can lift anything, therefore it is illogical to say God can make a stone which He cannot lift. It is however logical to say if God can lift anything, then he is not capable of making a stone He cannot lift. Because He cannot make a stone He cannot lift, omnipotence is negated.
In order to analyze the omnipotence paradox in a rigorous way, one of several definitions of omnipotence must be established as in use. For example, Peter Geach describes four different kinds of omnipotence and distinguishes all of them from the notion of being "almighty".
Types of omnipotence
Peter Geach describes and rejects four levels of omnipotence. He also defines and defends a lesser notion of the "almightiness" of God.
- Y is absolutely omnipotent means that Y "can do everything absolutely. Everything that can be expressed in a string of words even if it can be shown to be self-contradictory," Y "is not bound in action, as we are in thought by the laws of logic." This position is advanced by Descartes. It has the theological advantage of making God prior to the laws of logic, but the theological disadvantage of making God's promises suspect. On this account, the omnipotence paradox is a genuine paradox, but genuine paradoxes might nonetheless be so.
- Y is omnipotent means "Y can do X" is true if and only if X is a logically consistent description of a state of affairs. This position was once advocated by Thomas Aquinas. This definition of omnipotence solves some of the paradoxes associated with omnipotence, but some modern formulations of the paradox still work against this definition. Let X = "to make something that its maker cannot lift". As Mavrodes points out there is nothing logically contradictory about this; a man could, for example, make a boat which he could not lift. It would be strange if humans could accomplish this feat, but an omnipotent being could not. Additionally, this definition has problems when X is morally or physically untenable for a being like God.
- Y is omnipotent means "Y can do X" is true if and only if "Y does X" is logically consistent. Here the idea is to exclude actions which would be inconsistent for Y to do but might be consistent for others. Again sometimes it looks as if Aquinas takes this position. Here Mavrodes' worry about X= "to make something its maker cannot lift" will no longer be a problem because "God does X" is not logically consistent. However, this account may still have problems with moral issues like X = "tells a lie" or temporal issues like X = "brings it about that Rome was never founded."
- Y is omnipotent means whenever "Y will bring about X" is logically possible, then "Y can bring about X" is true. This sense, also does not allow the paradox of omnipotence to arise, and unlike definition #3 avoids any temporal worries about whether or not an omnipotent being could change the past. However, Geach criticizes even this sense of omnipotence as misunderstanding the nature of God's promises.
- Y is almighty means that Y is not just more powerful than any creature; no creature can compete with Y in power, even unsuccessfully. In this account nothing like the omnipotence paradox arises, but perhaps that is because God is not taken to be in any sense omnipotent. On the other hand, Anselm of Canterbury seems to think that almightiness is one of the things that makes God count as omnipotent.
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 differently to each type of being.
One can attempt to resolve the paradox by asserting a kind of omnipotence that does not demand that a being must be able to do all things at all times. According to this line of reasoning, the being can create a stone which it cannot lift at the moment of creation. Being omnipotent, however, the being can always alter the stone later so that it can lift it. Therefore the being is still in some sense omnipotent.
This is roughly the view espoused by Matthew Harrison Brady, a character in Inherit the Wind loosely based upon William Jennings Bryan. In the climactic scene of the 1960s movie version, Brady argues, "Natural law was born in the mind of the Creator. He can change it — cancel it — use it as He pleases!" But this solution merely pushes the problem back a step; one may ask whether an omnipotent being can create a stone so immutable that the being itself cannot later alter it. But a similar response can be offered to respond to this and any further steps.
In a 1955 article published in the philosophy journal Mind, J. L. Mackie attempted 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.
Another common response to the omnipotence paradox is to try to define omnipotence to mean something weaker than absolute omnipotence, such as definition 3 or 4 above. The paradox can be resolved by simply stipulating that omnipotence does not require the being to have abilities which are logically impossible, but only to be able to do anything which conforms to the laws of logic. A good example of a modern defender of this line of reasoning is George Mavrodes.
If a being is accidentally omnipotent, then it can resolve the paradox by creating a stone which it cannot lift and 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 or not 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 (as long as we take omnipotence not to require absolute omnipotence). The omnipotent being is essentially omnipotent, and therefore it is impossible for it to be non-omnipotent. Further, the omnipotent being cannot do what is logically impossible. The creation of a stone which the omnipotent being cannot lift would be an impossibility, and therefore the omnipotent being is not required to do such a thing. The omnipotent being cannot create such a stone, but 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.
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.
Some philosophers maintain that the paradox can be resolved if the definition of omnipotence includes Descartes' view that an omnipotent being can do the logically impossible. In this scenario, the omnipotent being could create a stone which it cannot lift, but could also then lift the stone anyway. Presumably, such a being could also make the sum 2 + 2 = 5 become mathematically possible or create a square triangle. This attempt to resolve the paradox is problematic in that the definition itself forgoes logical consistency. The paradox may be solved, but at the expense of making the logic a paraconsistent logic. This might not seem like a problem if one is already committed to dialetheism or some other form of logical transcendence.
Language and omnipotence
The philosopher Ludwig Wittgenstein is often 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 the succeeding propositions argue that ethics and several other issues are "transcendental" subjects which 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 makes the omnipotence paradox a problem in semantics, the study of how symbols are given 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. But in his later years, Wittgenstein wrote works which are often interpreted as conflicting with his positions in the Tractatus.
C.S Lewis, in his book the "Problem of Pain", holds that the nature of the paradox is internal to the statement. To quote: "This is no limit to His power. If you choose to say God can give a creature free will and at the same time withhold free will from it', you have not succeeded in saying anything about God: meaningless combination of words do not suddenly acquire meaning simply because we prefix to them to other words `God can'" (p. 18). In the end, "not because His power meets an obstacle, but because nonsense remains nonsense even when we talk it about God". (p.18)
Other versions of the paradox
In the 6th century, Pseudo-Dionysius claims that a version of the omnipotence paradox constituted the dispute between St. Paul and Elmyas the Magician mentioned in Acts 13:8, but it is phrased in terms of a debate as to whether or not God can "deny himself" ala 2 Tim 2:13. In the 11th century, St. Anselm 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:
Since the principles of certain sciences, such as logic, geometry and arithmetic are taken only from the formal principles of things, on which the essence of the thing depends, it follows that God could not make things contrary to these principles. For example, that a genus was not predicable of the species, or that lines drawn from the centre to the circumference were not equal, or that a triangle did not have three angles equal to two right angles.
This can be done on a sphere, and not on a flat surface. Note that the later discovery 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 or not an omnipotent being would have the ability to evade the consequences which follow logically from a system of axioms that the being created.
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 you consider the stone's position relative to the sun around which the planet orbits, one could hold that the stone is constantly being lifted. Modern physics indicates that the choice of phrasing about lifting stones should relate to acceleration; however, this does not in itself 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 which 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.
- Suber, P. (1990) The Paradox of Self-Amendment: A Study of Law, Logic, Omnipotence, and Change. Peter Lang Publishing. ((Online))
- Savage, C. Wade. "The Paradox of the Stone" Philosophical Review, Vol. 76, No. 1 (Jan., 1967), pp. 74–79 doi:10.2307/2182966
- Cowan, J. L. "The Paradox of Omnipotence" first published 1962, in The Power of God: readings on Omnipotence and Evil. Linwood Urban and Douglass Walton eds. Oxford University Press 1978 pp. 144–52
- Aquinas Summa Theologica Book 1 Question 25
- Descartes, Rene, 1641. Meditations on First Philosophy. Cottingham, J., trans., 1996. Cambridge University Press. Latin original. Alternative English title: Metaphysical Meditations. Includes six Objections and Replies. A second edition published the following year, includes an additional Objection and Reply and a Letter to Dinet
- Haeckel, Ernst. The Riddle of the Universe. Harper and Brothers, 1900.
- Wittgenstein, Ludwig. Tractatus Logico-Philosophicus (6.41 and following)
- Geach, P. T. "Omnipotence" 1973 in Philosophy of Religion: Selected Readings, Oxford University Press, 1998, pp. 63–75
- Aquinas, Thomas Summa Theologica Book 1 Question 25 article 3
- Mavrodes, George. "Some Puzzles Concerning Omnipotence" first published 1963 now in The Power of God: readings on Omnipotence and Evil. Linwood Urban and Douglass Walton eds. Oxford University Press 1978 pp. 131–4
- Aquinas Summa Theologica Book 1 Question 25 article 4 response #3
- Anselm of Canterbury Proslogion Chap VII in The Power of God: readings on Omnipotence and Evil. Linwood Urban and Douglass Walton eds. Oxford University Press 1978 pp. 35–36
- Hoffman, Joshua, Rosenkrantz, Gary. "Omnipotence" The Stanford Encyclopedia of Philosophy (Summer 2002 Edition). Edward N. Zalta (ed.) Available online. Accessed 19 April 2006.
- Mackie, J. L., "Evil and Omnipotence." Mind LXIV, No, 254 (April 1955).
- The Power of God: Readings on Omnipotence and Evil. Linwood Urban and Douglass Walton eds. Oxford University Press 1978. Keene and Mayo disagree p. 145, Savage provides 3 formalizations p. 138–41, Cowan has a different strategy p. 147, and Walton uses a whole separate strategy p. 153–63
- Gore, Charles, "A Kenotic Theory of Incarnation" first published 1891, in The Power of God: readings on Omnipotence and Evil. Linwood Urban and Douglass Walton eds. Oxford University Press 1978 pp. 165–8
- City of God, Book 5, Chapter 10
- Wittgenstein, Ludwig. proposition 6.5
- Wittgenstein, Ludwig. proposition 7
- D. Z. Phillips "Philosophy, Theology and the Reality of God" in Philosophy of Religion: Selected Readings. William Rowe and William Wainwright eds. 3rd ed. 1998 Oxford University Press
- Hacker, P.M.S. Wittgenstein's Place in Twentieth-Century Analytic Philosophy. 1996 Blackwell
- The Problem of Pain, Clive Staples Lewis, 1944 MacMillan
- Pseudo-Dionysius, "Divine Names" 893B in Pseudo-Dionysius: The Complete Works. trans Colm Luibheid Paulist Press. 1987. ISBN 0-8091-2838-1
- Anselm of Canterbury Proslogion Chap. VII, in The Power of God: readings on Omnipotence and Evil. Linwood Urban and Douglass Walton eds. Oxford University Press 1978 pp. 35–6
- "Cum principia quarundam scientiarum, ut logicae, geometriae et arithmeticae, sumantur ex solis principiis formalibus rerum, ex quibus essentia rei dependet, sequitur quod contraria horum principiorum Deus facere non possit: sicut quod genus non sit praedicabile de specie; vel quod lineae ductae a centro ad circumferentiam non sint aequales; aut quod triangulus rectilineus non habeat tres angulos aequales duobus rectis". Aquinas, T. Summa Contra Gentiles, Book 2, Section 25. trans. Edward Buckner
- Allen, Ethan. Reason: The Only Oracle of Man. J.P. Mendum, Cornill; 1854. Originally published 1784, Available online.Accessed 19 April 2006.