Omnipotence paradox

Detail depicting Averroes, who addressed the omnipotence paradox in the 12th century, from the 14th-century Triunfo de Santo Tomás by Andrea da Firenze (di Bonaiuto).

The omnipotence paradox is a family of semantic paradoxes that explores what is meant by 'omnipotence'. If an omnipotent being is able to perform any action, then it should be able to create a task that it is unable to perform. Hence, this being cannot perform all actions (i.e. it is not omnipotent), a logical contradiction. The most well-known version of the omnipotence paradox is the so-called paradox of the stone: "Could an omnipotent being create a stone so heavy that even they could not lift it?"[1] 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 Riemannian 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?"

The omnipotence paradox is medieval, dating at least to the 12th century. It was addressed by Averroës (1126–1198) and later by Thomas Aquinas.[2] Pseudo-Dionysius the Areopagite (before 532) has a predecessor version of the paradox, asking whether it is possible for God to "deny himself".

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 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 it cannot lift, then it seems it is already not omnipotent.[1]

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

Main article: Omnipotence

Peter Geach describes and rejects four levels of omnipotence. He also defines and defends a lesser notion of the "almightiness" of God.

  1. "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."[3] This position is advanced by Descartes. It has the theological advantage of making God prior to the laws of logic. Some claim that it in addition gives rise to the theological disadvantage of making God's promises suspect. On this account, the omnipotence paradox is a genuine paradox.
  2. "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.[4] 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 that he could not lift.[5] 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. But this brings about a new problem that if God is bound by logic he therefore cannot be the author of logic.
  3. "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 that are inconsistent for Y to do, but might be consistent for others. Again sometimes it looks as if Aquinas takes this position.[6] Here Mavrodes' worry about X= "to make something its maker cannot lift" is no longer 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."[3]
  4. "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.[3]
  5. "Y is almighty" means that Y is not just more powerful than any creature; no creature can compete with Y in power, even unsuccessfully.[3] 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 make God count as omnipotent.[7]

St Augustine 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.[8]

Some Philosophers, such as René Descartes, argue that God is absolutely omnipotent.[9] 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.[10] Some modern approaches to the problem have involved semantic debates over whether language—and therefore philosophy—can meaningfully address the concept of omnipotence itself.[11]

Proposed answers

A common response from Christian philosophers, such as Norman Geisler or Richard Swinburne 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's 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." This raises the question, similar to the Euthyphro Dilemma, of where this law of logic, which God is bound to obey, comes from. According to these theologians, this law is not a law above God that he assents to but, rather, logic is an eternal part of God's nature, like his omniscience or omnibenevolence. God obeys the laws of logic because God is eternally logical in the same way that God does not perform evil actions because God is eternally good. So, God, by nature logical and unable to violate the laws of logic, cannot make a boulder so heavy he cannot lift it because that would violate the law of non contradiction by creating an immovable object and an unstoppable force.

Another common response is that since God is supposedly omnipotent, the phrase "could not lift" does not make sense and the paradox is meaningless.[12][13] 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 couldn't lift it, yet could do something that an incarnation that could lift the stone could not.

Other responses claim that the question is sophistry, meaning it makes grammatical sense, but has no intelligible meaning. 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:

  1. Lifting covers up the definition of translation, which means moving something from one point in space to another. With this in mind, the real question would be, "Can God move a rock from one location in space to another that is larger than possible?" For the rock to be unable to move from one space to another, it would have to be larger than space itself. However, it is impossible for a rock to be larger than space, as space always adjusts itself to cover the space of the rock. If the supposed rock was out of space-time dimension, then the question would not make sense—because it would be impossible to move an object from one location in space to another if there is no space to begin with, meaning the faulting is with the logic of the question and not God's capabilities.
  2. The words, "Lift a Stone" are used instead to substitute capability. With this in mind, essentially the question is asking if God is incapable, so the real question would be, "Is God capable of being incapable?" If God is capable of being incapable, it means that He is incapable, because He has the potential to not be able to do something. Conversely, if God is incapable of being incapable, then the two inabilities cancel each other out, making God have the capability to do something.

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.

Thomas Aquinas asserts that the paradox arises from a misunderstanding of omnipotence. He maintains that inherent contradictions and logical impossibilities do not fall under the omnipotence of God.[14] J. L Cowan sees this paradox as a reason to reject the concept of 'absolute' omnipotence,[15] while others, such as René Descartes, argue that God is absolutely omnipotent, despite the problem.[9]

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’s too heavy for it to lift is identical to its power to lift that very stone. While this doesn’t 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 if 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.[12] 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.

John Christian Uy said that it is just the same as someone with double-bladed sword (accidentally omnipotent), or sword and a shield (essentially omnipotent). Therefore, an accidentally omnipotent deity CAN remove its omnipotence while an essentially omnipotent deity CANNOT do anything that would make it non-omnipotent. Both however, have no limitations so far other than the essential omnipotent being who cannot do anything that makes it non-omnipotent—like making someone equal with it, lowering or improving itself (for omnipotence is the highest) etc. It could, however, make someone with a great power, though it cannot be 99%, because Omnipotence is infinite, because that created being is not equal with it. Overall, God in the Christian Bible, is essentially omnipotent.

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).[16] 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.[17]

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 that the being have abilities that are logically impossible, but only be able to do anything that conforms to the laws of logic. A good example of a modern defender of this line of reasoning is George Mavrodes.[5] 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 stone

For 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?[18]

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 or not the being was ever truly omnipotent, or just capable of great power.[8] 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.[19]

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 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.[20]

Thus Augustine argued that God could not do anything or create any situation that would, in effect, make God not God.

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, 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."[21]

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".[22]

Wittgenstein's approach to these problems is influential among other 20th century religious thinkers such as D. Z. Phillips.[23] In his later years, however, Wittgenstein wrote works often interpreted as conflicting with his positions in the Tractatus,[24] 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 St. Paul and Elymas 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.[25] In the 11th century, St. Anselm argues that there are many things that God cannot do, but that nonetheless he counts as omnipotent.[26]

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.[27]

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 or not 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.[28]

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.[29] 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.

The paradox also appears in popular culture. In an episode of The Simpsons, Homer Simpson asks Ned Flanders, "Could Jesus microwave a burrito so hot that He Himself could not eat it?" In one strip of the webcomic Saturday Morning Breakfast Cereal, a child is seen asking a priest "Could God make an argument so circular that even He couldn't believe it?"

In the Marvel Comics Runaways, Victor Mancha, the technorganic android created by Ultron, is shown as unable to process correctly paradoxes: as such, it's known that a small number of well known paradoxes may force his logic in a permanent loop, shutting his functions down until someone steps in to give Victor the proper solution. As such, his peers stop him once by asking "Could God make a sandwich so big that even he couldn't finish it?", and reboot his mind by explaining him a simplified version of the God as essentially omnipotent solution ("Yes. God could make a sandwich so big that even he couldn't finish it, and eat it all").

In the book Bart Simpson's Guide to Life this question is phrased as, "If God can do anything, could he create a hot dog so big that even he couldn't eat it?"

See also

Notes

  1. 1 2 Savage, C. Wade. "The Paradox of the Stone" Philosophical Review, Vol. 76, No. 1 (Jan., 1967), pp. 74–79 doi:10.2307/2182966
  2. ↑ Averroës, Tahafut al-Tahafut (The Incoherence of the Incoherence) trans. Simon Van Den Bergh, Luzac & Company 1969, sections 529–536
  3. 1 2 3 4 Geach, P. T. "Omnipotence" 1973 in Philosophy of Religion: Selected Readings, Oxford University Press, 1998, pp. 63–75
  4. ↑ Aquinas, Thomas Summa Theologica Book 1 Question 25 article 3
  5. 1 2 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. 131–34
  6. ↑ Aquinas Summa Theologica Book 1 Question 25 article 4 response #3
  7. ↑ 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
  8. 1 2 Hoffman, Joshua, Rosenkrantz, Gary. "Omnipotence" The Stanford Encyclopedia of Philosophy (Summer 2002 Edition). Edward N. Zalta (ed.). (Accessed on 19 April 2006)
  9. 1 2 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
  10. ↑ Haeckel, Ernst. The Riddle of the Universe. Harper and Brothers, 1900.
  11. ↑ Wittgenstein, Ludwig. Tractatus Logico-Philosophicus (6.41 and following)
  12. 1 2 The Problem of Pain, Clive Staples Lewis, 1944 MacMillan
  13. ↑ Loving Wisdom: Christian Philosophy of Religion by Paul Copan, Chalice Press, 2007 page 46
  14. ↑ "Summa Theologica". Ccel.org. Retrieved 2012-05-10.
  15. ↑ 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
  16. ↑ Mackie, J. L., "Evil and Omnipotence." Mind LXIV, No, 254 (April 1955).
  17. ↑ 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
  18. ↑ Frankfurt, Harry. "The Logic of Omnipotence" first published in 1964 in Philosophical Review and now in Necessity, Volition, and Love. Cambridge University Press November 28, 1998 pp.1–2
  19. ↑ 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. 165–68
  20. ↑ "NPNF1-02. St. Augustine's City of God and Christian Doctrine".
  21. ↑ Wittgenstein, Ludwig. proposition 6.5
  22. ↑ Wittgenstein, Ludwig. proposition 7
  23. ↑ 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
  24. ↑ Hacker, P.M.S. Wittgenstein's Place in Twentieth-Century Analytic Philosophy. 1996 Blackwell
  25. ↑ Pseudo-Dionysius, "Divine Names" 893B in Pseudo-Dionysius: The Complete Works. trans Colm Luibheid Paulist Press. 1987. ISBN 0-8091-2838-1
  26. ↑ 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
  27. ↑ "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
  28. ↑ Suber, P. (1990) The Paradox of Self-Amendment: A Study of Law, Logic, Omnipotence, and Change. Peter Lang Publishing
  29. ↑ Allen, Ethan. Reason: The Only Oracle of Man. J.P. Mendum, Cornill; 1854. Originally published 1784. (Accessed on 19 April 2006)

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