THE OMNIPOTENCE PARADOX AS A PROBLEM OF INFINITE REGRESS TZACHI ZAMIR

Department o[ Philosophy Tel-Aviv Unwersity, Tel-Aviv, Israel PREAMBLE

In this paper, I argue that attempts to explain away the Omnipotence Paradox by referring to the way in which its standard presentation is temporally confused fail. I offer an alternative formulation, which presents the paradox as a problem of infinite regress. Unlike epistemic concepts, the concept of ability becomes self-contradictory when maximally extended. Finally, I argue that this peculiarity plagues another modal concept: that of possibility. Can an omnipotent Being create a stone that It cannot lift? A now popular reply to this medieval paradox is that, with appropriate limitations, it can. I will argue that it cannot. My main concern will be to show that the traditional presentation of the paradox is wrong, and that it requires reformulation in terms of a problem of infinite regress (I). I shall then address possible objections to my suggestion (II). In the third and final section, I shall argue for some general conceptual conclusions that may be drawn from the omnipotence paradox.

Call a being 'omnipotent' if it can succeed in accomplishing any logically consistent task. Since performing a task in a particular way can be itself formulated as a task, it seems fair to

Sophia Vo138 No 1 1999, March-April.

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require of omnipotent beings to be able to accomplish any logically consistent task in any logically consistent way. By 'logically inconsistent' tasks, I mean tasks like drawing a round square or naming the largest prime number. Most of the philosophers who have discussed the paradox regard these demands as wrong in one way or another and argue that, therefore, no being - omnipotent or not - should be required to meet them. Creating unliftable stones is a logically consistent task (it can, for example, be easily met by nonomnipotent beings like us). A being that fails this test cannot be omnipotent. Under its standard formulation, the Omnipotence paradox supposedly yields two possibilities which lead to the same result: either the Being cannot create the stone (thereby proving itself not omnipotent), or the Being can create the stone (again proving itself not omnipotent since a stone exists which it cannot lift). Therefore, omnipotent beings do not exist. A possible refutation of this is to claim that an omnipotent Being can create such a stone and thereby cease to be omnipotent. This is not paradoxical or strange because there is no contradiction between a Being that is omnipotent on t and not so on t'. 1 An omnipotent Being has the ability to voluntarily lose Its omnipotence, much as a pianist has the ability to voluntarily lose her playing ability by cutting off her hands. Granted, an omnipotent Being cannot create such a stone and remain omnipotent, but this does not affect the Being's omnipotence any more than Its inability to comply with any contradictory demand. Since the demand to create such a stone was ipso facto a demand to give up omnipotence, the Being cannot meaningfully be challenged both to retain and give up omnipotence. The answer to the original question would be positive: the Being can non-paradoxically create such a stone and the paradox is explained away. 2 This answer is, on the face of it, plausible, and has been used to show that according to some notions of omnipotence,

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the alleged paradox is not a genuine paradox at aU.3 However, I will attempt to show that it entails an infinite regress, which proves that the Being cannot create the stone. My proposed reformulation would, thereby, retain the idea that omnipotence involves paradox. My argument is as follows: Granted that it is not contradictory to think that an omnipotent Being can lose one of Its abilities, thereby becoming non-omnipotent (through, for example, creating the stone or losing Its ability to lift existing stones), what would stop It from regaining Its lost ability? In opposition to the pianist mentioned earlier, who unfortunately cannot bring back her lost playing ability, an omnipotent Being has, by definition, all abilities. These include no doubt, the ability to cancel limitations that It had previously set upon Itself. It should be obvious that if the Being retains an ability to restore Its ability to lift the stone, then It can in fact lift the stone. This means that there will exist no point in time in which it could be truly asserted of the Being that it cannot lift the stone. The Being, therefore, fails to comply with the original challenge to create a stone that It cannot lift. + So the refuter of the paradox has to show that the Being can create the stone, while not being able to regain the abilities It necessarily loses by performing that action, s This seems to be simple: If the Being really intends to stop being omnipotent, It also has to eliminate Its ability to restore Its lost abilities. However, this would mean that when the Being decides to stop being omnipotent, It has to lose not one ability, but two. This involves two distinct actions. 6 The Being has to: (1)

Stop being omnipotent (by, for example, creating the stone).

And: (2)

Eliminate Its ability to restore the abilities It loses by performing)

But surely, an omnipotent Being which has lost only two of Its abilities, namely, the ability to lift up a certain stone and 3

the ability to regain that ability, still retains all the rest of Its abilities. Amongst these, no doubt, is the ability to restore the second ability It has lost. However, one might say, if the Being has really eliminated Its ability to restore the ability It has lost in (1), then It has also eliminated Its ability to restore that ability as well. From this it would, of course, follow that we were wrong to assume that the Being only performed two actions when It created the stone. It necessarily had to perform a third action. The Being also had to: Eliminate Its ability to restore the second ability It had lost by performing (2). One can easily see how this process goes on, yielding the unfortunate result that the Being has to perform an endless number of actions so that It can eventually become nonomnipotent. But if they are indeed endless, then the Being cannot finish doing them and get on with being non-omnipotent. This means that It cannot create the stone and that It was therefore not omnipotent from the start. The paradox is thereby reinstated. However, what if the Being could eliminate any ability to restore eliminated abilities? Alternatively, eliminate Its ability to revise any of Its actions? This ability itself could not be later regained since any restoring of lost abilities was eliminated by it. This way, the Being need not perform an endless number of actions, but only two. Namely: (a)

Stop being omnipotent (through losing an ability to lift a certain stone). (b) Eliminate Its ability to restore any ability It loses.

This would bypass the infinite regress argument but would unfortunately fall short of answering the paradox, which would be reformulated thus: 'Can an omnipotent Being create a stone It cannot lift, and not eliminate Its ability to restore any ability It loses?' There is no reason to regard this demand as self-contradictory or empty (non-omnipotent

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beings can easily comply with it). It does not challenge the Being in the same illegitimate ways that the vacuous demands mentioned above do. 7 A negative answer to this question would expose the Being to be non-omnipotent from the start (since a task exists that It cannot perform). A positive answer, on the other hand, would simply succeed in relocating the paradox. The reason for this is that a positive reply means that the Being can somehow eliminate the set of all abilities connected with the lifting of the stone or with restoring such abilities if they are eliminated (call this set S). It also maintains some of Its other abilities to restore eliminated abilities. Therefore, according to this hypothesis, in order to create an unliftable stone, the Being supposedly performs two actions: (I) Creates a stone that It cannot lift. (n) Eliminates S. However, if S is composed solely of the infinite (1), (2), (3) type abilities, then eliminating S is not enough for the elimination of the stone-lifting ability. The reason is that in such a case the Being retains Its ability to restore sets of abilities that It eliminates. Therefore, eliminating the stone-lifting ability via the elimination of S necessitates that S should include as members of it the elimination of an infinite number of abilities involved by restoring the S set itself once it is eliminated. This involves eliminating the ability to restore S, eliminating the ability to restore the eliminated ability to restore S, and so on. S becomes incredibly big. This makes it possible to simply reformulate the paradoxical question. The demand of the omnipotent Being would be to be able (like non-omnipotent beings) to create the stone and not lose the ridiculously large number of eliminated abilities included in S. A negative answer now appears mandatory. The Being is exposed as non-omnipotent from the start thereby reinstating the paradox.

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II I have argued that attempting to explain away the paradox by assuming that an omnipotent Being can enter a state in which it is not omnipotent cannot work since it entails an infinite regress that stems from the omnipotent Being's unique nature. The following general objections could be raised against my argument: (I)

One could argue that God is not only omnipotent but also necessarily successful, therefore He can create such a stone. (II) One could argue that the Being does not in fact have to perform an infinite number of tasks. (RI) One could argue that performing an infinite number of tasks is possible. (IV) One could accept the validity of the paradox but deny that its conclusion is necessarily that the Being does not exist. Variations of these objections will be addressed below. One could object to the above formulation of the paradox through claiming that an omnipotent God is also always successful. Given such a necessary efficacy of God, then if He wishes at t to lose His ability to lift the stone at t', then - contra the infinite regress argument - He necessarily succeeds. Such an objection, however, begs the question since the paradox investigates whether or not the Being can in fact succeed in performing certain tasks. The paradox challenges the sufficiency of omnipotence to necessary success, hence necessary success cannot simply be assumed. A more subtle variation on such an objection is to claim that God is both omnipotent and necessarily successful (without assuming that the former implies the latter). Presented with such a concept of God, the infinite regress argument will fail, since (again) if the divine will wishes to eliminate any of God's abilities, It necessarily succeeds. Such an argument

would succeed in saving God's omnipotence at the price of introducing a notion that is no less vulnerable to paradoxical constructions. For consider the following: 'Can a necessarily successful Being (a Being that necessarily succeeds in performing any logically consistent task) succeed in devising tasks that It can never successfully perform?' A negative answer would entail that the Being is not necessarily successful. A positive reply would entail that such a necessarily successful Being - that is, a being which is successful at all possible worlds - is necessarily unsuccessful at some possible worlds (hence not necessarily successful from the start). A second possible objection can be based on the claim that the infinite regress argument involves an inadequate description of the Being's actions when It becomes nonomnipotent. Eliminating or limiting an ability does not necessarily involve performing an endless number of actions. As the case of the pianist above shows, non-omnipotent beings can in fact eliminate or limit some of their abilities without being involved in an infinite regress. Why should there be a specific problem for omnipotent Beings? This objection rests on falsely carrying over the meaning of 'eliminating (or limiting) an ability' from non-omnipotent contexts to omnipotent ones. Strictly speaking, in order to ascribe to any being - omnipotent or not - a loss of an ability, there must be no existing defeaters to that claim. For a non-omnipotent being, this need not be problematic given knowledge of possible defeaters, and the fact that there are not, usually, an infinite number of them. Losing the ability to walk because of some neck injuries, for example, has - in the present state of medical knowledge - no existing defeaters. 8 However, things are different for a Being that has all abilities. Since the ability to restore an eliminated ability is not only itself an ability which the Being must, by definition, possess, but also an actually existing defeater, the omnipotent Being has to somehow eliminate it. The infinite regress springs from

the fact that this elimination, instead of settling the matter, is exactly what constitutes the coming into being of a new defeater. Another objection that argues for the same conclusion is to accept the adequacy of the description as a possible one, but to deny that it is actually true. When one looks at a coloured surface, for example, this action can be described as involving looking at an infinite number of colored points. However, from the fact that it is possible to describe the action as such, it in no way follows that the action in fact requires actually performing an infinite number of actions. Therefore, although the Being's action can be described as involving an infinite number of separate actions, it does not necessarily follow that It has to actually perform them. In answering this, one may claim that there is a false analogy between the case of looking at a colored surface, and creating the stone. There is no prima facie reason to think that it is impossible to look at an infinite number of points (if they are in fact infiniteg), although it is indeed impossible to look at each one individually. However, there is no reason to think that one needs to look at all of its individual points in order to look at a coloured surface. In the case of the infinite regress above, on the other hand, the Being starts by performing one action - creating the stone - then another - eliminating Its ability to restore the lost ability to lift the stone and so on ad infinitum. The problem is not a temporal one. It is, rather, that each successive action creates a new defeater, making a 'final' action impossible. This is also the reason why allowing the Being to start with other actions than the creation of the stone does not circumvent the difficulty. A third possible critique is to argue that there is no reason to suppose that performing an infinite number of tasks is impossible. J.E Thomson gave strong arguments against such a claim, l~ However, even if one rejected Thomson's reasons, an ability to perform an infinite number of tasks would not

save the Being's omnipotence. The paradoxical question would then be reformulated in terms we have already met earlier: a demand to create the stone without performing an infinite number of tasks. Non-omnipotent beings can easily meet such a demand, omnipotent Beings - as it turned out could not. Finally, one could claim that all that the argument shows is that an omnipotent Being cannot lose Its omnipotence. This would mean that if a Being is omnipotent, It is necessarily omnipotent. If one regards 'omnipotence' as meaning 'necessary omnipotence' from the start, then the infinite regress argument is irrelevant. An omnipotent Being cannot lose Its omnipotence, and therefore It cannot create the stone in question. This, though, is not a limitation on Its abilities but a consequence of Its special nature. All that the so called 'paradox' proves is that if a necessarily omnipotent Being exists, then the existence of such a stone is logically impossible. 11 Such a move seems attractive to some, since it ultimately accepts the validity of the paradox, but instead of drawing conclusions concerning the abilities of the Being; it makes ontological limitations on the world in which the Being exists. This hardly makes it a persuasive move, however, since the inability to break free of these ontological limitations is precisely what revealed the Being to be non-omnipotent from the start, and thereby exposed the concept of omnipotence as self-contradictory. It is indeed true that if such a Being exists, then the existence of such a stone is logically impossible. But it is the exactly the inability to create such a stone (an inability that - to the omnipotent Being's disgrace - non-omnipotent beings do not share) that makes the existence of such a Being impossible. A close variation of this objection is to Claim that from the beginning the paradox is not genuine since there can exist no object X that can satisfy the description 'a stone that cannot be lifted by an omnipotent Being'. This boils down to the

same conclusion as the one reached by the previous 'necessary omnipotence' objection (though this time without the conditional clause): the existence of the sort of stone the paradox postulates is logically impossible. The 'paradox' is no challenge to the Being since it does not meet the initial requirement of demanding the Being to perform a logically consistent task. For someone who wishes to defend the traditional formulation of the stone paradox this last objection might prove insurmountable. But since my purpose here is not to defend the traditional formulation - a formulation that could be easily dismissed by the sort of rejection of the paradox with which we begun - but to suggest a close reformulation of it such an objection can be circumvented. My suggested formulation of the paradox in terms of a problem of infinite regress enables seeing that the omnipotence paradox does not, in fact, depend on the idea that a previously non-existent stone has to be created. The paradox can be formulated as a demand to eliminate an ability to lift an already existent stone (if the Being cannot eliminate such ability, It is not omnipotent. To mistakenly suppose that It can, is to over-look the infinite-regress which, as we have seen, such elimination involves). It would hardly make much sense to say that the existence of such stones is 'logically impossible'. If the Being cannot lose any of Its abilities (that is, it is 'necessarily omnipotent') then - as was traditionally concluded from the paradox - it would turn out that such a being lacks many abilities that non-omnipotent beings possess. In fact, if being able to call aloud the name of a number is admitted as an ability that can be eliminated, non-omnipotent beings are able to eliminate an infinite number of abilities by losing the power of speech. That is, they possess an infinite number of abilities that a being, which is supposed to have all abilities, does not. Such an objection would therefore entail defending a rather strange concept of omnipotence that is hardly worthy of its name.

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Ill What general conceptual conclusions can be drawn from this? If the Omnipotence Puzzle is to be regarded as more than an intellectual exercise, it should surely reveal something about the concept of ability. The infinite regress argument, if it is correct, proves that, unlike non-omnipotent beings, an omnipotent Being cannot lose any of Its abilities. 12 The interesting point that emerges concerns a peculiarity of the concept of ability when maximally extended to include all abilities. This peculiarity can be shown through contrasting abilities with knowledge. If A knows some tunes and B knows all tunes, than B knows all the tunes known to A and of course many more. This asymmetry remains if the notion of knowledge is maximally extended to include everything knowable. Limited cases of ability are also asymmetrical: If A can lift some stones and B can lift all stones, than B can lift any stone A can. However, unlike knowledge, in the case of ability the asymmetry is broken when maximally extended: if A has s o m e abilities and B has all abilities, it paradoxically turns out that B cannot have all of A's abilities. The omnipotence puzzle shows that, unlike the epistemic concepts of knowledge or belief, the modal concept of ability collapses when maximally extended. Because some uses of the notion of possibility are synonymous with ability, it is hardly surprising that similar infinite regress problems can be constructed pertaining also to this modal concept: can a being which can realize any possibility realize a state of affairs (S) in which it can permanently realize fewer possibilities? An argument analogous to the one above would lead to a negative answer here too. If realizing S requires permanently losing the possibility to realize A (call this possibility P), this can only be truly achieved through losing the possibility (P') of regaining P as a realizable possibility after it has been lost, and so on. Here too, the infinite number of actions that the Being would be required to perform 11

would render the move to S impossible. As in the case of abilities, a Being for which anything is possible paradoxically cannot realize possibilities that are easily accessible to a less fortunate being (e.g. our former pianist which had easily enough entered such an S state). Some notions of t3ossibility, however, do not require anyone to do anything. An obligation to perform an infinite number of actions would, therefore, not seem to be problematic for all cases of possibility. Unfortunately, this does not save the concept of possibility from contradicting itself when maximally extended. This can be shown through the following: Suppose that anything is possible, that is, all conceivable (logically consistent) possible worlds exist. Call possible worlds that have access only to some - but not to all - other possible worlds 'third' worlds. For third possible worlds, not everything is possible. Call possible worlds to which all other possible worlds are accessible 'rich' worlds. If all possible worlds exist, then both third and rich possible worlds exist. So far, none of this is problematic. But if all logically consistent possible worlds exist, then there is no way through which to exclude the existence of a possible world - call it a 'benefactor' world - which in some strange, affirmative action, way allows itself to be accessible only to third worlds. Given the previous assumptions, the existence of benefactor worlds and rich worlds is mutually exclusive. Therefore, if all conceivable possible worlds exist, it cannot be the case that all possible worlds exist. Unlike other concepts, something strange happens to modal concepts when linked to the notion of 'all'. 13

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This observation was made by Igal Kvart: 1982, 'The Omnipotence Puzzle', Logic and Analysis, and David E Shrader: 1979, 'A Solution to the Stone Paradox', Syntbese. Some theorists demand that the Being be 'necessarily omnipotent', meaning that It cannot lose Its omnipotence. I will discuss such a possibility below. A different objection which I have sometimes heard theorists use (though never noted in the literature) is that the Being can create the stone and remain omnipotent through Its ability to destroy the unlikable stone. The elimination of the stone is tantamount to restoring omnipotence and so, as long as the Being retains an ability to destroy the stone It never loses Its omnipotence (that is, whenever the Being's omnipotence would be questioned, there would exist no stone that It could not lift). There are several problems with this suggestion, the most important of which is that, at best, it would simply require changing the paradoxical demand from creating an unlikable stone to creating an indestructible one. Kvart (Ibid.) used this to reject the notion of what he calls 'a merely omnipotent' being. Such a being can non-paradoxically lose Its omnipotence. One could obviously claim that the Being still succeeds in entering a state in which It temporarily cannot lift the stone ('temporarily' referring here to the point in time between creating the stone and restoring the ability to lift it). Circumventing this objection simply requires clearing a possible ambivalence in the formulation of the paradoxical task. The Being is required to create a stone that It can never lift. I am assuming here that creating the stone involves losing an ability. This assumption is not crucial to the argument and, if challenged, would merely change the original question from which the paradox begins. For example: Can an Omnipotent being lose Its ability to lift stones? A negative answer would show It to be non-omnipotent; a positive one immediately has to face the infinite regress argument that follows above. I shall assume that such actions can be individuated and counted. The reader might wonder why a similar reformulation, relat-

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ing abilities (1) and (2) into the paradoxical question, was not suggested instead of the infinite regress argument which was introduced at that stage. The answer is that in opposition to the question above, such a formulation could have been rejected as a meaningless demand, since in that case it does seem contradictory to challenge the Being to create the unliftable stone while retaining an ability to lift it. In fact, one should draw a distinction here between actual and possible existing defeaters. A possible medical procedure that would bring back the ability to walk can be envisaged, and therefore is a possible defeater. This distinction is, however, irrelevant to the above argument. Strictly speaking, they are not infinite. The coloured surface actually contains a finite number of points of any specific size. An unlimited ability to divide spatial points further does not entail that there are an infinite number of them. This fallacy constitutes one of Zeno's paradoxes of motion. J.E Thomson: 1954, ' Tasks and Super-Tasks', Analysis, (15) pp. 1-13. Kvart (Ibid.) and others make this point. Though It can, non-problematically lose all of them through suicide. This paper owes much to helpful and insightful comments by Ruth Manor, Anat Matar, Ruth Weintraub, and an anonymous reader.

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