On Plantinga's Ontological Argument
One of the most curious arguments for the existence of God has been presented by St. Anselm, René Descartes, and many other theologians throughout the centuries: the Ontological Argument. The classical formulation of the argument is (1):
1. God is that entity than which nothing greater can be conceived.
2. It is greater to be necessary than not.
3. God must be necessary.
4. God necessarily exists.
Perhaps the most challenging formulation of the argument is presented today by Alvin Plantinga. Dr. William Lane Craig presents Plantinga's argument as (2):
1. It is possible that a maximally great being exists.
2. If it is possible that a maximally great being exists, then a maximally great being exists in some possible world.
3. If a maximally great being exists in some possible world, then He exists in every possible world. ($$)
4. If He exists in every possible world, He must exist in the actual world.
I will discuss this particular formulation at length in this article.
The Classical Take
A brief statement about the classical version of this argument is necessary, particularly about the necessity of "necessary" being an inherently positive quality in and of itself, without regards to its referent in reality. This is not entirely clear; a fantastic counterexample would be certain events in the context of human history, which as an A-time theorist I hold to be necessary facts of existence. Suppose, for instance, that Adam and Eve existed and chose to Fall. Then, unless one is a high-Calvinist, the necessity (by asssumption) of the Fall would be a negative quality, as opposed to a positive one, as the action in the Fall brought death and damnation to Adam, Eve, and subsequently, to all of us. Therefore, it cannot be established that necessity qua necessity is an inherently positive quality of existence.
Other refutations exist of this presentation, from skeptics to believers. Hume famously rejected the argument by stating that it was logically possible to conceive the nonexistence of every entity in reality, i.e. it is logically possible to conceive that the truth value of the existence of every particular entity in the actual world is equal to "false." Geisler and Corduan endorse this objection (3).
Plantinga's Take
Plantinga opens the playing field to the set of all possible worlds. In his presentation, we are asked to imagine every logically possible world, where a possible world is defined to be a world (or state of existence) such that the set of all logical facts describing the world
p1^p2^p3^...^pn
exists without internal contradiction. Now, personally, I agree with the rather minority viewpoint that the only possible world is the actual world, given that I accept the necessity of entities in reality as my primary philosophical axiom (4). But for the sake of argument, I will accept a multiplicity of possible worlds, assumed in this case to be infinite (5). Although counterintuitive if we view Craig's presentation (6) of possible worlds as consisting only of a finite set of facts, we are assuming that the set of facts in one possible world can differ from the set of facts in another possible world arbitrarily. Let's begin the critique with these understandings.
Craig asserts all premises save premise #1 is "relatively uncontroversial," a point which I disagree heavily and will touch upon later. He then goes on to establish a priori warrant for premise #1, stating that the intuitive concept of an omnipotent, omnibenevolent, and omniscient being must be logically incoherent to invalidate the premise. Bill first attempts to show how typical objections along the "maximally great island" lines fail, asserting "there could always be more palm trees and native dancing girls." While true, I sort of have an ill feeling that Craig's criticism is misplaced: certainly an island completely filled with dancing women to the point where nobody could move anywhere but into the ocean would be less great than if four or five women were there to greet me with some freshly cut coconuts.
Bill states the stronger criticism that such concepts are relative to the observer; perhaps, as Bill says, another person would prefer a full resort while another an empty desert island. Indeed, the person in question could be a woman who would prefer men on the island, or, in my case, some other tropical fruit apart from coconuts given my distaste for them. But that doesn't disprove the notion that a maximally great tropical island is logically possible for all humanity (who are interested in these things to begin with). Perhaps the island could contain several resorts sorted for particular tastes, and could contain areas of desertion where those who prefer to be away from civilization could relax maximally. It may be so that not every desire is satisfied by all of our prospective visitors, but the fact remains that a resort island built to maximize every interested person's preferences and leave everyone at least happy that they came is not necessarily impossible logically due to relativity in preference.
A discussion follows regarding quasi-maximal beings. Let's suppose that a logically possible world W1 exists where the maximal being exists and where Fred Sanford, say, is born with omnipotence and omniscience, but not omnibenevolence (Bill offers an example of a quasi-maximal being lacking knowledge of future events). This being may be derived in the same method as the maximal being: given the establishment of a maximal being, we may use the same logic to establish a quasi-maximal being, i.e. the necessary existence in a possible world of a being "one step down" from our maximal one.
Craig correctly states that the maximal being could choose not to create the quasi-maximal being with His creation powers, but why do we presuppose that the maximal being created? Given these premises, it is not logically impossible that a quasi-maximal being exists; perhaps in a possible world Fred Sanford and God existed side-by-side, with the only difference between the two (apart from conscious separation) being that Fred had a bit more heartburn. In this presentation, our quasi-maximal being would be uncreated, as our maximal being is. As another side note, it is possible, logically, that a quasi-maximal being created the maximal one; there is no premise that states that to be created is less in maximality than to have existed in all states of our logically possible world, only that the maximal being's existence in this world is necessary. It is my charge, then, that the challenges stand and that the a priori concept of a maximal being still presents the incoherency that Craig assumes to have been refuted. (**)
Craig then goes on to discuss an a posteriori establishment of Premise #1, but Craig treads carefully here: "I remain uncertain of this argument ... which would require us to reject various nominalistic alternatives to conceptualism such as fictionalism, constructabilism, figuralism, and so forth. Still, prominent philosophers such as Plantinga have endorsed it." (7) This concern is brought forth from Plantinga's a posteriori establishment via means of grounding abstractions metaphysically n the mind of some being, since, as Plantinga argues, they cannot be established in our own.
It took me a while to understand Craig's persistent stomachache over this (ultimately leading to the aforementioned confession of Craig's doubt about the Ontological Argument), but then I remembered that this sort of argumentation from the supposed inability to ground concepts in reality is part of Bahnsen's Transcendental Argument for the Existence of God (TAG).(8)Bahnsen, and Plantinga himself, are generally Reformed, and come from a very rationalist and highly skeptical (in the philosophical sense) worldview that Craig, an evidentialist and classical apologist, tends to shun (TAG, for instance, appears nowhere in Craig's book). I side with Craig here, but a post about this will have to wait.
There is one more difficult premise, however, that Craig accepts without discussion, but that Plantinga elsewhere (9) both understands and attempts to correct: the inductive premise #3, i.e. the premise that if this maximally great being exists in some logically possible world, He exists in all possible worlds, including our actual one.
Loosely, Plantinga describes "maximally excellent" as necessarily including the three omni-'s: benevolence, potence, knowledge. If this Being enjoys maximal excellence in all possible worlds, then this Being is "maximally great." Plantinga wishes to establish premise #3 by establishing that a maximally great Being exists in at least one possible world.
As a side quip, why are the three omni's properties of a maximal being and considered to be tops in evaluating excellence? If I were omniscient - and I'm sorry to go to toilet humor, but you see what I mean - I'd see everyone poop. Not excellent! All kidding aside, who would want to know every detail of the Holocaust, particularly if one were additionally omnibenevolent and omnipotent (but doing the Arminian thing of letting people act on their own power of choice)? But I'm being mean and too speculative, so I will grant Plantinga that the three qualities give a being a great degree of excellence exceeding beings which do not possess these qualities. I'll also be nice and grant that only one being possessing these three qualities can exist in logical possibility in a given possible world.
How are we to establish values for these omni-qualities and for the rest of the qualities of our theoretical being, or any being, for that matter? Plantinga proposes a number (assumed to be bigger than or equal to zero) describing the "excellence" of an extant entity in some possible world, and asks us to sum this excellence number over all possible worlds where the entity in question exists. Such a sum is taken only over the possible worlds containing the assumption of x's existence, for, the concept presupposes existence and therefore an "excellence number" has no value in a world where the truth value of x's existence is false. It's not even zero - it would be, if you remember Algebra, like taking the square root of a negative number while working only with reals. It's simply outside the domain!
But still, how can one metaphysically quantify "excellence" for any particular entity? Nowhere do either Plantinga or Mears propose such a means, but I'll propose a concrete definition: an excellence number represents the number of entities which the entity "x" is, in a sense, "better in more individual respects" than. This would allow God, as omniscient, omnipotent, and omnibenevolent, to have an excellence number equal to the (finite) sum of entities in reality, playing on the undiscussed intuitive notion here that a being with the three omni's is better than all the rest of the entities in reality.
So, taking the sum of excellence over all possible worlds well-defines a function F(x) as such:
F(x) = W1 + W2 + W3 + ... + Wn
This is the "greatness function," as greatness, remember, is to be taken as a representation of the excellence over ALL WORLDS where x exists. Here n represents the number of worlds where x possibly exists. Taking the limit as n goes to infinity gets our "greatness number" for an entity possibly existing in an infinite number of worlds. We assume our case for a maximally great being must exist in an infinite number of possible worlds, then, since if He were to exist in only a finite number of worlds the chance of His existence in the actual world, although logically possible, would be (letting k be the total number of possible worlds)
lim k->infinity (n/k) = 0,
a point which Plantinga and his critic Mears miss in their respective papers.
The conclusion of Plantinga's case for premise #3 is that
lim F(x), x:= "God"
is a number greater than all other greatness valuations F(x') taken over any arbitrary non-Godly entity x' that exists in any possible world.
Here's one killer, built on the idea formulated by Mr. Mears. Note here that F(x) (God's greatness) must be finite to make any sense. For, if
lim F(x) -> infinity,
then the coherency of the maximally great being vanishes logically (10). Therefore, this greatness must be a concrete number. Plantinga's reasoning for this has been refuted above, but Plantinga is still correct for the reason I give in the footnote: natural numbers themselves do not have a greatest upper bound, and although they cannot be used to describe actual metaphysical quantity as e.g. increasing girls indefinitely on our island, the possibility of the comparison with our "greatness function" exists inherently in the definition. Therefore, if it can be demonstrated that the function evaluates to infinity, then God's greatness, e.g. his summation of his "excellency rating" over every world in which He exists, becomes logically incoherent, collapsing the argument. (&&)
Since the existence of God over a finite subset of the (infinite) set of logically possible worlds leads to a zero-probability of His actual existence, as mentioned earlier, we must take the infinite sum. Following my coherent definition of what it means to have excellence, i.e. an integer representing the number of entities that "x" is "more excellent than" in the intuitive sense, then only one logically possible world has an excellence rating of zero for God - the one where only He exists. Therefore there are an infinite number of positive integer representations of excellence-ratings in possible worlds. Letting W1 = 0 (our excellency rating in the one world where only God exists), we have, then:
F(x) = 0 + W2 + W3 + W4 + ... > 0 + 1 + 1 + 1 + .... --> infinity
proving that F(x) = infinity, rendering maximal greatness incoherent by the above argumentation. Therefore, Plantinga's argument fails under my coherent definition of "excellency rating."
Even if Plantinga would object to my definition, he must at the very least represent a function whose infinity-limit tends toward zero if we wish for the summation function F(x) to yield a finite number (to wit, the sum of 1/n^2, n from 1 to infinity, is a finite positive number, and 1/n^2 itself tends to 0 as n rises without bound, fitting our bill).
But this implies that God's excellence by any definition Plantinga wants to use, when ordered, must tend to zero and therefore gets arbitrarily small - and furthermore, must do so in a matter that still yields a finite sum (for, sum 1/n, n from 1 to infinity, is still infinity despite the fact that limit 1/n ->0 as n tends infinitely). As this is intuitively counter to our notion that these values of excellency ought to be large, it seems a tough challenge indeed to define an excellency valuation which both tends to zero when ordered, is such that the sum is still a finite number, and is so that, despite being damned near zero for all but a finite number of possible worlds, it is still greater than all other entities' excellency evaluations in each of those particular possible worlds. And that's if a coherent definition of how to evaluate an "excellency rating," apart from the one I offered, is first given!
One final point, unobserved by neither Plantinga nor Meirs - we have, in the latter part of this paper, only discussed a sum over an infinite number of possible worlds. This is the direct case allowed by Plantinga's presentation. All it would show even if every premise is true and justified is that God exists in an infinite number of possible worlds, and not the actual one, and at most with his definitions this presentation can only establish a probability of God's existence, rather than a logical certainty. (##)
I conclude that Plantinga's presentation of the Ontological Argument has been refuted, pending critiques, comments and discussions from the readers of this blog. I am looking forward to an engaging discussion.
And am hoping nobody fell asleep because of the math. :)
-Darrin
Sources and Notes
(1) St. Anselm. Proslogion, Ch. 2. Retrieved from Wikipedia. See also criticisms of this presentation in Dr. Corduan's response to this post.
(2) Craig, William Lane. Reasonable Faith. Crossway Publishing, p. 184.
(3) Geisler and Corduan, Philosophy of Religion, pp. 147-148. Retrieved from http://www.biblicaldefense.org/Writings/ontological_argument.htm
Corduan notes in the commentary to this post: "Of course, I'm not sure we mean it in the same sense as Hume did, but it's always nice to see it when people realize that there is an unspoken assumption underlying the ontological argument, namely that something exists." (Retrieved 1/19/09, with thanks to Dr. Corduan).
(4) See e.g. Rand, Ayn, "The Metaphysical Versus the Man Made."
(5) Mears, Tyrel. "Sympathy for the Fool," p. 87. If one buys the premise of "possible worlds" in the first place, it logically follows that the number of "possible worlds" is infinite. For, it could be logically possible for a world outside of a proposed finite set of worlds exists where I swivel my chair completely around here at my desk; one where I swivel halfway around; one where I swivel one-forth around; etc. leading immediately to an infinite set of possible worlds if one does not, as I do, hold the necessity of entities and action in the actual world as a foundational premise.
(6) Craig 183.
(7) This quote, as well as the preceding analysis and a bit more proceeding this footnote, are discussed in full on (Craig) pp. 184-189.
(8) Bahnsen mentions the inability of the non-Christian to ground concepts in "The Great Debate" versus Gordon Stein, a debate which I am planning to review in the near future, but Bahnsen's presentation of TAG leans more heavily on the inability to ground the process of reasoning. Personally, I believe the two are interrelated, if not equal processes, and at the very least the former precedes the latter inclusively (see e.g. Ayn Rand, "Introduction to Objectivist Epistemology," or earlier Wittgenstein). See also Gordon Clark's Youtube lectures regarding his critique of Empiricism; both Clark and Bahnsen are free for anyone to watch on that website.
(9) The rest of Plantinga's quotes have been retrieved from Mears' paper, documented earlier (once again here:http://aporia.byu.edu/pdfs/mears-sympathy_for_the_fool.pdf). All quotes from Plantinga are properly annotated in the work, and I have found no indication that Mr. Mears misuses or presents in a nonobjective manner any material written by Dr. Plantinga. The references to Mr. Mears himself which follow from this annotation until the end of my paper are taken, for the most part, from pp. 83-91.
(10) Plantinga, Alvin. God, Freedom, and Evil. P. 91. Plantinga uses the objection mentioned by Craig, about the paradise island that can indefinitely gain more girls and coconuts and get better. But this has been refuted earlier; Plantinga's necessary inherent maximum is present for this concept, as well. However, for the (entirely epistemological) natural numbers, e.g. 1, 2, 3, 4, 5, ..., there is in fact no "greatest number," since infinity is not recognized as a member of the natural number set (11). Thus, with respect to the natural number set, the idea of a "maximal natural number" is incoherent, which properly captures the idea intuited by Craig and Plantinga in the "add more girls!" objection, while not escaping the context of practicality.
(11) See e.g. Royden's Real Analysis, Ch. 1.
(12) ibid.
($$) There is an issue with Plantinga's modal logic presentation of this premise that goes undiscussed here, due to my unfamiliarity with the subject (it's been over a decade since I've seen it!). Refer to http://barefootbum.blogspot.com/2008/05/modal-logic.html for a criticism on this subject. (Thanks to The Barefoot Bum and Dr. Corduan for bringing this issue up in the commentary of this post.)
(**) As a side note, it may be stated that if it can be shown that two omnipotent beings exist in a logically possible world, where one of the two beings is not omniscient and one is, and if it can be logically possible that the nonbenevolent Being would wish to engage the benevolent Being, the dual omnipotence assumed in this case would render a logical impossibility. Perhaps via traveling down the quasi-omniscient chain we may always find such a "Fred Sanford" through inductive establishment on the basis of the establishment of the existence of the maximal being. This would demonstrate the necessary logical incoherency of omnipotence, and would be a logical disproof of God in any possible world.
This is merely postulation, however, since none of my premises have been established in this paragraph; but should they turn out logically sound, then assuming the existence of a maximal being in all logically possible worlds may logically lead to the existence of a quasi-maximal omnipotent, omniscient, but anti-benevolent being in every possible world, who, through the definition of benevolence, would both desire to defeat one another as their highest priority. But since both are omnipotent and omniscient, they both can and can not defeat one another; we would then establish the non-existence of God through the impossibility of the contrary.
(&&) - If we accept this quantity to be possibly infinite despite the intuited objection from the natural number comparison, we run into problems: if any other being had a "greatness valuation" F(x') = infinity, it would be impossible to quantify x' and x in greatness relation (the infinite is countable in any case by definition). As we shall see later in the paper, all it will take is for some other being x' to exist with quality in an infinite number of possible worlds to get an infinite greatness evaluation - let's say that the Devil, for instance, exists in an infinite subset of possible world, and that his valuation of excellency, given the Devil's immense powers described Biblically, is always greater than zero in each world. By the argument which follows this annotation, the Devil would, in this case, have a greatness valuation of infinity, equal to God's; even if the Devil existed in "less" worlds than God (but still existed in an infinite subset of possible worlds), we'd have to equate the Devil and God with greatness, and I don't think we want to say that.
(##) - Again, Plantinga's summation is defined over an infinite number of possible worlds, not all possible worlds. Plantinga has made the mistake in assuming that since the number of possible worlds is infinite, then the sum taken over infinity covers it. But this may not be so. For, by mathematical postulate, we may well-order all the possible worlds in this set; it might be the case that God exists in all odd-numbered worlds W1, W3, W5, W7, W9, .... so that even if Plantinga destroys my case but fails to establish how God's existence in an infinite number of worlds entails His existence in all worlds, we might have that
W1+W3+W5+W7+W9+ ... = F(x)
is a finite number indeed, establishing coherency and validating the argument, but still leaving the evens out of the consideration (the ones where it is possible God does not exist). This leaves only, in this assumed case, the probability of God to be N/2N = 1/2 for our actual world (assuming we don't know which possible world-number it is) even if we assume all of Plantinga's case as otherwise true and valid.
1. God is that entity than which nothing greater can be conceived.
2. It is greater to be necessary than not.
3. God must be necessary.
4. God necessarily exists.
Perhaps the most challenging formulation of the argument is presented today by Alvin Plantinga. Dr. William Lane Craig presents Plantinga's argument as (2):
1. It is possible that a maximally great being exists.
2. If it is possible that a maximally great being exists, then a maximally great being exists in some possible world.
3. If a maximally great being exists in some possible world, then He exists in every possible world. ($$)
4. If He exists in every possible world, He must exist in the actual world.
I will discuss this particular formulation at length in this article.
The Classical Take
A brief statement about the classical version of this argument is necessary, particularly about the necessity of "necessary" being an inherently positive quality in and of itself, without regards to its referent in reality. This is not entirely clear; a fantastic counterexample would be certain events in the context of human history, which as an A-time theorist I hold to be necessary facts of existence. Suppose, for instance, that Adam and Eve existed and chose to Fall. Then, unless one is a high-Calvinist, the necessity (by asssumption) of the Fall would be a negative quality, as opposed to a positive one, as the action in the Fall brought death and damnation to Adam, Eve, and subsequently, to all of us. Therefore, it cannot be established that necessity qua necessity is an inherently positive quality of existence.
Other refutations exist of this presentation, from skeptics to believers. Hume famously rejected the argument by stating that it was logically possible to conceive the nonexistence of every entity in reality, i.e. it is logically possible to conceive that the truth value of the existence of every particular entity in the actual world is equal to "false." Geisler and Corduan endorse this objection (3).
Plantinga's Take
Plantinga opens the playing field to the set of all possible worlds. In his presentation, we are asked to imagine every logically possible world, where a possible world is defined to be a world (or state of existence) such that the set of all logical facts describing the world
p1^p2^p3^...^pn
exists without internal contradiction. Now, personally, I agree with the rather minority viewpoint that the only possible world is the actual world, given that I accept the necessity of entities in reality as my primary philosophical axiom (4). But for the sake of argument, I will accept a multiplicity of possible worlds, assumed in this case to be infinite (5). Although counterintuitive if we view Craig's presentation (6) of possible worlds as consisting only of a finite set of facts, we are assuming that the set of facts in one possible world can differ from the set of facts in another possible world arbitrarily. Let's begin the critique with these understandings.
Craig asserts all premises save premise #1 is "relatively uncontroversial," a point which I disagree heavily and will touch upon later. He then goes on to establish a priori warrant for premise #1, stating that the intuitive concept of an omnipotent, omnibenevolent, and omniscient being must be logically incoherent to invalidate the premise. Bill first attempts to show how typical objections along the "maximally great island" lines fail, asserting "there could always be more palm trees and native dancing girls." While true, I sort of have an ill feeling that Craig's criticism is misplaced: certainly an island completely filled with dancing women to the point where nobody could move anywhere but into the ocean would be less great than if four or five women were there to greet me with some freshly cut coconuts.
Bill states the stronger criticism that such concepts are relative to the observer; perhaps, as Bill says, another person would prefer a full resort while another an empty desert island. Indeed, the person in question could be a woman who would prefer men on the island, or, in my case, some other tropical fruit apart from coconuts given my distaste for them. But that doesn't disprove the notion that a maximally great tropical island is logically possible for all humanity (who are interested in these things to begin with). Perhaps the island could contain several resorts sorted for particular tastes, and could contain areas of desertion where those who prefer to be away from civilization could relax maximally. It may be so that not every desire is satisfied by all of our prospective visitors, but the fact remains that a resort island built to maximize every interested person's preferences and leave everyone at least happy that they came is not necessarily impossible logically due to relativity in preference.
A discussion follows regarding quasi-maximal beings. Let's suppose that a logically possible world W1 exists where the maximal being exists and where Fred Sanford, say, is born with omnipotence and omniscience, but not omnibenevolence (Bill offers an example of a quasi-maximal being lacking knowledge of future events). This being may be derived in the same method as the maximal being: given the establishment of a maximal being, we may use the same logic to establish a quasi-maximal being, i.e. the necessary existence in a possible world of a being "one step down" from our maximal one.
Craig correctly states that the maximal being could choose not to create the quasi-maximal being with His creation powers, but why do we presuppose that the maximal being created? Given these premises, it is not logically impossible that a quasi-maximal being exists; perhaps in a possible world Fred Sanford and God existed side-by-side, with the only difference between the two (apart from conscious separation) being that Fred had a bit more heartburn. In this presentation, our quasi-maximal being would be uncreated, as our maximal being is. As another side note, it is possible, logically, that a quasi-maximal being created the maximal one; there is no premise that states that to be created is less in maximality than to have existed in all states of our logically possible world, only that the maximal being's existence in this world is necessary. It is my charge, then, that the challenges stand and that the a priori concept of a maximal being still presents the incoherency that Craig assumes to have been refuted. (**)
Craig then goes on to discuss an a posteriori establishment of Premise #1, but Craig treads carefully here: "I remain uncertain of this argument ... which would require us to reject various nominalistic alternatives to conceptualism such as fictionalism, constructabilism, figuralism, and so forth. Still, prominent philosophers such as Plantinga have endorsed it." (7) This concern is brought forth from Plantinga's a posteriori establishment via means of grounding abstractions metaphysically n the mind of some being, since, as Plantinga argues, they cannot be established in our own.
It took me a while to understand Craig's persistent stomachache over this (ultimately leading to the aforementioned confession of Craig's doubt about the Ontological Argument), but then I remembered that this sort of argumentation from the supposed inability to ground concepts in reality is part of Bahnsen's Transcendental Argument for the Existence of God (TAG).(8)Bahnsen, and Plantinga himself, are generally Reformed, and come from a very rationalist and highly skeptical (in the philosophical sense) worldview that Craig, an evidentialist and classical apologist, tends to shun (TAG, for instance, appears nowhere in Craig's book). I side with Craig here, but a post about this will have to wait.
There is one more difficult premise, however, that Craig accepts without discussion, but that Plantinga elsewhere (9) both understands and attempts to correct: the inductive premise #3, i.e. the premise that if this maximally great being exists in some logically possible world, He exists in all possible worlds, including our actual one.
Loosely, Plantinga describes "maximally excellent" as necessarily including the three omni-'s: benevolence, potence, knowledge. If this Being enjoys maximal excellence in all possible worlds, then this Being is "maximally great." Plantinga wishes to establish premise #3 by establishing that a maximally great Being exists in at least one possible world.
As a side quip, why are the three omni's properties of a maximal being and considered to be tops in evaluating excellence? If I were omniscient - and I'm sorry to go to toilet humor, but you see what I mean - I'd see everyone poop. Not excellent! All kidding aside, who would want to know every detail of the Holocaust, particularly if one were additionally omnibenevolent and omnipotent (but doing the Arminian thing of letting people act on their own power of choice)? But I'm being mean and too speculative, so I will grant Plantinga that the three qualities give a being a great degree of excellence exceeding beings which do not possess these qualities. I'll also be nice and grant that only one being possessing these three qualities can exist in logical possibility in a given possible world.
How are we to establish values for these omni-qualities and for the rest of the qualities of our theoretical being, or any being, for that matter? Plantinga proposes a number (assumed to be bigger than or equal to zero) describing the "excellence" of an extant entity in some possible world, and asks us to sum this excellence number over all possible worlds where the entity in question exists. Such a sum is taken only over the possible worlds containing the assumption of x's existence, for, the concept presupposes existence and therefore an "excellence number" has no value in a world where the truth value of x's existence is false. It's not even zero - it would be, if you remember Algebra, like taking the square root of a negative number while working only with reals. It's simply outside the domain!
But still, how can one metaphysically quantify "excellence" for any particular entity? Nowhere do either Plantinga or Mears propose such a means, but I'll propose a concrete definition: an excellence number represents the number of entities which the entity "x" is, in a sense, "better in more individual respects" than. This would allow God, as omniscient, omnipotent, and omnibenevolent, to have an excellence number equal to the (finite) sum of entities in reality, playing on the undiscussed intuitive notion here that a being with the three omni's is better than all the rest of the entities in reality.
So, taking the sum of excellence over all possible worlds well-defines a function F(x) as such:
F(x) = W1 + W2 + W3 + ... + Wn
This is the "greatness function," as greatness, remember, is to be taken as a representation of the excellence over ALL WORLDS where x exists. Here n represents the number of worlds where x possibly exists. Taking the limit as n goes to infinity gets our "greatness number" for an entity possibly existing in an infinite number of worlds. We assume our case for a maximally great being must exist in an infinite number of possible worlds, then, since if He were to exist in only a finite number of worlds the chance of His existence in the actual world, although logically possible, would be (letting k be the total number of possible worlds)
lim k->infinity (n/k) = 0,
a point which Plantinga and his critic Mears miss in their respective papers.
The conclusion of Plantinga's case for premise #3 is that
lim F(x), x:= "God"
is a number greater than all other greatness valuations F(x') taken over any arbitrary non-Godly entity x' that exists in any possible world.
Here's one killer, built on the idea formulated by Mr. Mears. Note here that F(x) (God's greatness) must be finite to make any sense. For, if
lim F(x) -> infinity,
then the coherency of the maximally great being vanishes logically (10). Therefore, this greatness must be a concrete number. Plantinga's reasoning for this has been refuted above, but Plantinga is still correct for the reason I give in the footnote: natural numbers themselves do not have a greatest upper bound, and although they cannot be used to describe actual metaphysical quantity as e.g. increasing girls indefinitely on our island, the possibility of the comparison with our "greatness function" exists inherently in the definition. Therefore, if it can be demonstrated that the function evaluates to infinity, then God's greatness, e.g. his summation of his "excellency rating" over every world in which He exists, becomes logically incoherent, collapsing the argument. (&&)
Since the existence of God over a finite subset of the (infinite) set of logically possible worlds leads to a zero-probability of His actual existence, as mentioned earlier, we must take the infinite sum. Following my coherent definition of what it means to have excellence, i.e. an integer representing the number of entities that "x" is "more excellent than" in the intuitive sense, then only one logically possible world has an excellence rating of zero for God - the one where only He exists. Therefore there are an infinite number of positive integer representations of excellence-ratings in possible worlds. Letting W1 = 0 (our excellency rating in the one world where only God exists), we have, then:
F(x) = 0 + W2 + W3 + W4 + ... > 0 + 1 + 1 + 1 + .... --> infinity
proving that F(x) = infinity, rendering maximal greatness incoherent by the above argumentation. Therefore, Plantinga's argument fails under my coherent definition of "excellency rating."
Even if Plantinga would object to my definition, he must at the very least represent a function whose infinity-limit tends toward zero if we wish for the summation function F(x) to yield a finite number (to wit, the sum of 1/n^2, n from 1 to infinity, is a finite positive number, and 1/n^2 itself tends to 0 as n rises without bound, fitting our bill).
But this implies that God's excellence by any definition Plantinga wants to use, when ordered, must tend to zero and therefore gets arbitrarily small - and furthermore, must do so in a matter that still yields a finite sum (for, sum 1/n, n from 1 to infinity, is still infinity despite the fact that limit 1/n ->0 as n tends infinitely). As this is intuitively counter to our notion that these values of excellency ought to be large, it seems a tough challenge indeed to define an excellency valuation which both tends to zero when ordered, is such that the sum is still a finite number, and is so that, despite being damned near zero for all but a finite number of possible worlds, it is still greater than all other entities' excellency evaluations in each of those particular possible worlds. And that's if a coherent definition of how to evaluate an "excellency rating," apart from the one I offered, is first given!
One final point, unobserved by neither Plantinga nor Meirs - we have, in the latter part of this paper, only discussed a sum over an infinite number of possible worlds. This is the direct case allowed by Plantinga's presentation. All it would show even if every premise is true and justified is that God exists in an infinite number of possible worlds, and not the actual one, and at most with his definitions this presentation can only establish a probability of God's existence, rather than a logical certainty. (##)
I conclude that Plantinga's presentation of the Ontological Argument has been refuted, pending critiques, comments and discussions from the readers of this blog. I am looking forward to an engaging discussion.
And am hoping nobody fell asleep because of the math. :)
-Darrin
Sources and Notes
(1) St. Anselm. Proslogion, Ch. 2. Retrieved from Wikipedia. See also criticisms of this presentation in Dr. Corduan's response to this post.
(2) Craig, William Lane. Reasonable Faith. Crossway Publishing, p. 184.
(3) Geisler and Corduan, Philosophy of Religion, pp. 147-148. Retrieved from http://www.biblicaldefense.org/Writings/ontological_argument.htm
Corduan notes in the commentary to this post: "Of course, I'm not sure we mean it in the same sense as Hume did, but it's always nice to see it when people realize that there is an unspoken assumption underlying the ontological argument, namely that something exists." (Retrieved 1/19/09, with thanks to Dr. Corduan).
(4) See e.g. Rand, Ayn, "The Metaphysical Versus the Man Made."
(5) Mears, Tyrel. "Sympathy for the Fool," p. 87. If one buys the premise of "possible worlds" in the first place, it logically follows that the number of "possible worlds" is infinite. For, it could be logically possible for a world outside of a proposed finite set of worlds exists where I swivel my chair completely around here at my desk; one where I swivel halfway around; one where I swivel one-forth around; etc. leading immediately to an infinite set of possible worlds if one does not, as I do, hold the necessity of entities and action in the actual world as a foundational premise.
(6) Craig 183.
(7) This quote, as well as the preceding analysis and a bit more proceeding this footnote, are discussed in full on (Craig) pp. 184-189.
(8) Bahnsen mentions the inability of the non-Christian to ground concepts in "The Great Debate" versus Gordon Stein, a debate which I am planning to review in the near future, but Bahnsen's presentation of TAG leans more heavily on the inability to ground the process of reasoning. Personally, I believe the two are interrelated, if not equal processes, and at the very least the former precedes the latter inclusively (see e.g. Ayn Rand, "Introduction to Objectivist Epistemology," or earlier Wittgenstein). See also Gordon Clark's Youtube lectures regarding his critique of Empiricism; both Clark and Bahnsen are free for anyone to watch on that website.
(9) The rest of Plantinga's quotes have been retrieved from Mears' paper, documented earlier (once again here:http://aporia.byu.edu/pdfs/mears-sympathy_for_the_fool.pdf). All quotes from Plantinga are properly annotated in the work, and I have found no indication that Mr. Mears misuses or presents in a nonobjective manner any material written by Dr. Plantinga. The references to Mr. Mears himself which follow from this annotation until the end of my paper are taken, for the most part, from pp. 83-91.
(10) Plantinga, Alvin. God, Freedom, and Evil. P. 91. Plantinga uses the objection mentioned by Craig, about the paradise island that can indefinitely gain more girls and coconuts and get better. But this has been refuted earlier; Plantinga's necessary inherent maximum is present for this concept, as well. However, for the (entirely epistemological) natural numbers, e.g. 1, 2, 3, 4, 5, ..., there is in fact no "greatest number," since infinity is not recognized as a member of the natural number set (11). Thus, with respect to the natural number set, the idea of a "maximal natural number" is incoherent, which properly captures the idea intuited by Craig and Plantinga in the "add more girls!" objection, while not escaping the context of practicality.
(11) See e.g. Royden's Real Analysis, Ch. 1.
(12) ibid.
($$) There is an issue with Plantinga's modal logic presentation of this premise that goes undiscussed here, due to my unfamiliarity with the subject (it's been over a decade since I've seen it!). Refer to http://barefootbum.blogspot.com/2008/05/modal-logic.html for a criticism on this subject. (Thanks to The Barefoot Bum and Dr. Corduan for bringing this issue up in the commentary of this post.)
(**) As a side note, it may be stated that if it can be shown that two omnipotent beings exist in a logically possible world, where one of the two beings is not omniscient and one is, and if it can be logically possible that the nonbenevolent Being would wish to engage the benevolent Being, the dual omnipotence assumed in this case would render a logical impossibility. Perhaps via traveling down the quasi-omniscient chain we may always find such a "Fred Sanford" through inductive establishment on the basis of the establishment of the existence of the maximal being. This would demonstrate the necessary logical incoherency of omnipotence, and would be a logical disproof of God in any possible world.
This is merely postulation, however, since none of my premises have been established in this paragraph; but should they turn out logically sound, then assuming the existence of a maximal being in all logically possible worlds may logically lead to the existence of a quasi-maximal omnipotent, omniscient, but anti-benevolent being in every possible world, who, through the definition of benevolence, would both desire to defeat one another as their highest priority. But since both are omnipotent and omniscient, they both can and can not defeat one another; we would then establish the non-existence of God through the impossibility of the contrary.
(&&) - If we accept this quantity to be possibly infinite despite the intuited objection from the natural number comparison, we run into problems: if any other being had a "greatness valuation" F(x') = infinity, it would be impossible to quantify x' and x in greatness relation (the infinite is countable in any case by definition). As we shall see later in the paper, all it will take is for some other being x' to exist with quality in an infinite number of possible worlds to get an infinite greatness evaluation - let's say that the Devil, for instance, exists in an infinite subset of possible world, and that his valuation of excellency, given the Devil's immense powers described Biblically, is always greater than zero in each world. By the argument which follows this annotation, the Devil would, in this case, have a greatness valuation of infinity, equal to God's; even if the Devil existed in "less" worlds than God (but still existed in an infinite subset of possible worlds), we'd have to equate the Devil and God with greatness, and I don't think we want to say that.
(##) - Again, Plantinga's summation is defined over an infinite number of possible worlds, not all possible worlds. Plantinga has made the mistake in assuming that since the number of possible worlds is infinite, then the sum taken over infinity covers it. But this may not be so. For, by mathematical postulate, we may well-order all the possible worlds in this set; it might be the case that God exists in all odd-numbered worlds W1, W3, W5, W7, W9, .... so that even if Plantinga destroys my case but fails to establish how God's existence in an infinite number of worlds entails His existence in all worlds, we might have that
W1+W3+W5+W7+W9+ ... = F(x)
is a finite number indeed, establishing coherency and validating the argument, but still leaving the evens out of the consideration (the ones where it is possible God does not exist). This leaves only, in this assumed case, the probability of God to be N/2N = 1/2 for our actual world (assuming we don't know which possible world-number it is) even if we assume all of Plantinga's case as otherwise true and valid.