Ontology is the branch of metaphysics that deals with the nature of being.
The classic ontological argument for the existence of God runs as follows:
- God is the greatest imaginable being.
- All else being equal, a being or entity that exists is greater than one that doesn't.
- Therefore, God exists.
In this argument, existence is given as one of God's attributes as part of the definition: if X is God, then X has the property of existence. This is logically equivalent to "if X does not exist, then X is not God." It does not prove that there are any entities that actually match the definition.
Existence can hardly ever be considered an attribute, as something nonexistent cannot have attributes. Therefore, making conclusions about existence of an entity based on its properties is not logically sound. In short, this argument boils down to "show me a god, and I'll show you an existing god." It is a form of circular reasoning because the existence is built into the assumptions.
Here are some examples of this proof that highlight the fallacy.
- Let us define a unicorn as a magical equine being that has one horn, and that exists.
- By the above definition, such a being must necessarily exist.
- Therefore unicorns exist.
- Shangri-La is the greatest place on earth.
- A place that exists is greater than one that doesn't.
- Therefore, Shangri-La exists.
- Hercules is the greatest warrior in history.
- A warrior that existed is greater than one that did not.
- Therefore, Hercules existed.
A comprehensive and shorter refutation can be found in positiveatheism.org 
Immanuel Kant: Existence is not a predicate
Immanuel Kant put forward a key refutation of the ontological argument in the Critique of Pure Reason. It is explicitly directed primarily against Descartes but also against Leibniz. His criticism was anticipated in Pierre Gassendi's Objections to Descartes' Meditations. Kant's refutation consists of several separate but interrelated arguments. They are shaped by his central distinction between analytic and synthetic judgments. In an analytic judgment, the predicate expresses something that is already contained within a concept and is therefore a tautology; in a synthetic judgment, the predicate, or claim, links the concept to something outside it that is not already logically implied by it. New knowledge consists of synthetic judgments.
Kant first questions the intelligibility of the very concept of an absolutely necessary being, considering "whether I am still thinking anything in the concept of the unconditionally necessary, or perhaps rather nothing at all". He examines one way of understanding the concept, which looks to examples of necessary propositions, e.g. "a triangle has three angles". But he rejects this account for two related reasons. First, no absolutely necessary judgments will ever yield an absolute necessity for things and their existence: e.g., "a triangle has three angles" yields only the conditioned necessity that, if a triangle exists, then necessarily three angles exist. Thus even if we defined a concept of a thing X so that "X exists" were a necessary judgment, all that would follow is the conditioned necessity that, if X exists, then necessarily X exists. Second, since contradictions arise only when we keep the subject and cancel the predicate (e.g., keeping God and canceling omnipotence), and since judgments of nonexistence cancel both the subject and the predicate, therefore no judgment of nonexistence can involve a contradiction. Kant concludes that there is a strong general case against the intelligibility of the concept of an absolutely necessary being.
Second, Kant argues that if we include existence in the definition of something, then asserting that it exists is a tautology. If we say that existence is part of the definition of God, in other words an analytic judgment, then we are simply repeating ourselves in asserting that God exists. We are not making a synthetic judgment that would add new information about the real existence of God to the purely conceptual definition of God.
Third, Kant argues that "'being' is obviously not a real predicate" and cannot be part of the concept of something. That is, to say that something is or exists is not to say something about a concept, but rather indicates that there is an object that corresponds to the concept, and "the object, as it actually exists, is not analytically contained in my concept, but is added to my concept". For objects of the senses, to say that something exists means not that it has an additional property that is part of its concept but rather that it is to be found outside of thought and that we have an empirical perception of it in space and time. A really existing thing does not have any properties that could be predicated of it that differentiate it from the concept of that thing. What differentiates it is that we actually experience it: for example, it has shape, a specifiable location, and duration. To give an example of Kant's point: the reason we say that horses exist and unicorns do not is not that the concept of horse has the property of existence and the concept of unicorn does not, or that the concept of horse has more of that property than the concept of unicorn. There is no difference between the two concepts in this regard. And there is no difference between the concept of a horse and the concept of a really existing horse: the concepts are identical. The reason we say that horses exist is simply that we have spatio-temporal experience of them: there are objects corresponding to the concept. So any demonstration of the existence of anything, including God, that relies on predicating a property (in this case existence) of that thing is fallacious.
Thus, in accordance with the second and third arguments, the statement "God is omnipotent" is an analytic judgment that articulates what is already contained in and implied by the concept of God, i.e. a particular property of God. The statement "God exists" is a synthetic judgment of existence that does not assert something contained in or implied by the concept of God and would require knowledge of God as an object of that concept. What the ontological argument does is attempt to import into the concept of God, as though it were a property, the synthetic assertion of the existence of God, thereby illegitimately and tautologously defining God as existing. In other words, it begs the question by assuming what it purports to prove.
But, fourth, Kant argues that the concept of God is in any case not the concept of one particular object of sense among others but rather an "object of pure thought", of something that by definition exists outside the field of experience and of nature. With regard to unicorns, we can specify how we could determine that unicorns exist, i.e., what spatio-temporal experience of them would look like. With regard to the concept of God, there is no way for us to know it as existing in the only legitimate and meaningful way we know other objects as existing. We cannot even determine "the possibility of any existence beyond that known in and through experience".
A piece of parody, Gasking's Proof for the Non-existence of god is as follows:
- The creation of the universe is the greatest achievement imaginable.
- The merit of an achievement consists of its intrinsic greatness and the ability of its creator.
- The greater the handicap to the creator, the greater the achievement (would you be more impressed by Turner painting a beautiful landscape or a blind one-armed dwarf?)
- The biggest handicap to a creator would be non-existence
- Therefore if we suppose that the universe is the creation of an existing creator, we can conceive a greater being — namely, one who created everything while not existing.
- Therefore, God does not exist.
Rejection of the Second Premise
Assuming that existence and non-existence can actually be properties of something, there is no logical justification for existence being greater than non-existence
Modal Ontological Argument
This argument was advanced by Alvin Plantinga, one of the most prominent Christian philosophers in the world. It is based on axiom S5 of modal logic. In this argument, God is either necessary or impossible.
1) It is proposed that a being has maximal excellence in a given possible world W if and only if it is omnipotent, omniscient and wholly good in W; and
2) It is proposed that a being has maximal greatness if it has maximal excellence in every possible world.
3) Maximal greatness is possibly exemplified. That is, it is possible that there be a being that has maximal greatness. (Premise)
4) Therefore, possibly it is necessarily true that an omniscient, omnipotent and perfectly good being exists.
5) Therefore, it is necessarily true that an omniscient, omnipotent and perfectly good being exists. (By S5)
6) Therefore, an omniscient, omnipotent and perfectly good being exists.
Of course, premise 3 can be altered, offering the opposite conclusion:
3) It is not possibe that there is a being that has maximal greatness.
4) Therefore, possibly it is necessarily false that an omniscient, omnipotent and perfectly good being exists.
5) Therefore, it is necessarily false that an omniscient, omnipotent and perfectly good being exists.
6) Therefore, an omniscient, omnipotent and perfectly good being does not exist.
Some have objected that Plantinga's argument merely re-assumes that existence is a property and continues the argumentation by tautology.
The obvious logical fallacy committed by Pastor Plantinga here, is the use of the "possible" premise. He baldly asserts that all his "possible" premises are true. He doesn't even know that truth MUST be proven before it is asserted as true.
Plantinga commits the logical fallacy known as "Begging the Question", because one only has the epistemic right to accept it if one understands the nested modal logic operators. And if one understands them, then one understands that "possibly necessarily" is exactly the same as "necessarily". People wrongly assume that he uses Classical logic in his argument. Instead, he uses the more elusive Modal logic in order to MAGNIFY the confusion factor and trick you into accepting his nonsense.
Plantinga's Ontological argument is based on axiom S5 of modal logic.
1. A is larger than B
2. B is larger than C
3. A is necessarily larger than C
1. A is larger than B
2. B is smaller than C
3. A is possibly larger than C
Possibly Necessarily P implies Necessarily P
- A statement is necessarily true if it can not possibly be false.
- This means that a statement which is possibly false can not possibly be necessarily true.
- From that it follows logically that if it is possible for a statement to be necessarily true then it can not possibly be false.
- This means: If it is possible for a statement to be necessarily true then it is necessarily true.
- A being is a necessary being if it is logically impossible for that being not to exist.
- This means that if it is logically possible for a being not to exist this being can not possibly be a necessary being.
- Therefore it is possible for a being to be necessary if and only if that being can not possibly not exist.
- Therefore it is possible for a being to be necessary if and only if that being is necessary.
The recognition that it is logically possible for something to be necessarily true, is equal to the recognition that it is logically necessary for it to be true.
His trick is revealed: Pastor Plantinga's premise is EQUAL to his conclusion!
To deny premise (3) amounts to asserting that it is logically impossible that there is a being that exemplifies maximal greatness — thus the argument appears to demonstrate that either the existence of God is logically impossible or it is logically necessary. However, this is only the case if one conceives God as this maximally great being.
Therefore we can not tell that Pastor Plantinga's first premise is true unless we KNOW God exists. Hence the circularity of his argument. And this begs the question because Pastor Plantinga is using a deceptinve sleight of hand known as: ASSUMING YOUR PREMISE IS TRUE AND FORCING IT IN THE CONCLUSION. So we see that saying that it is logically possible, is definitely NOT possible unless we know that God exists.
So Pastor Plantinga is trying to deceive us by FIRST assuming that God exists.
Interestingly, Plantinga himself does not think the modal ontological argument is always a good proof of the existence of God. It depends on what his interlocutor thinks of the possibility premise. Nonetheless, Plantinga has suggested that because we do not have any evidence against the possibility premise, it might be reasonable to suppose it has a probability of 50/50.
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 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
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. (##)
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.
A final parody of the argument:
1. It is possible that an all-encompassing red sky exists.
2. If it is possible that an all-encompassing red sky exists, then an all-encompassing red sky exists in some possible world.
3. If an all-encompassing red sky exists in some possible world, then it exists in every possible world (being all-encompassing, you see).
4. If it exists in every possible world, it must exist in the actual world.
and, 5. If you look up and see a blue sky then you're not seeing things correctly.
A Modal Ontological Argument For The Non-Existence Of God
Like the modal arguments presented by theists, Pollock starts with the premise that God is defined as perfect, and that perfection implies necessary existence. He does not, however, assume that one can jump immediately to Pg, because, as we have seen, definition cannot directly imply instantiation. Rather, he uses the only logical conclusion that we can draw from the definition, that N(Eg->Pg).
Modal arguments tend to be more arduous for the casual reader.
1.g=Df (the x such that Px) God is defined as a perfect being. (premise)
2.N(Eg->Pg) We reformulate (1) by saying that God’s existence necessarily entails its perfection. All we did here was explain in terms of existence what (1) means. (from 1)
3.N(x)(Px->NEx) Let us assume, as the Ontological arguments do, that the perfection of x necessarily implies the existence of x, for all x. (premise)
4.N(Pg->NEg) Instantiating the principle in (3) for God. (from 3)
We now see that (2) and (4) can be combined into one proposition. If Eg implies Pg, and Pg implies NEg, then Eg implies NEg – the existence of God implies the necessary existence of God. (from 2 and 4)
We have to take a break here. As Pollock explains in his development, this is the furthest that we can take (1) by logical means. Even assuming the truth of the premise of the ontological arguments in (3), it is impossible to arrive at Eg, the proposition that God exists.
Rather, the best we can do is the proposition that IF God exists, then NEg necessarily obtains. This is important for two reasons: one because it shows that we cannot arrive at Eg, and two because we will use this conclusion again at the end of our argument.
6.(g=Df the x such that Px) -> N(Eg->NEg)
Here we simplify the first half of our argument in one proposition. (from 1 to 5)
7.~ [(g=Df the x such that Px) -> Eg]
We can explain this proposition in two ways. The first is to remember, as I discussed before, that a definition cannot entail actual existence. The other is to point out that we already showed that we cannot logically obtain Eg from (1). Either way, it is a fact that Eg is unattainable from the definition alone. (premise)
8.NEg iif [(g=Df the x such that Px) -> Eg]
This is obtained from the definition of logical necessity. Something is logically necessary iif it follows logically from its definition. (premise)
If something is only logically necessary iif it follows logically from its definition, and God’s existence does not follow logically from its definition, then God’s existence is not logically necessary. (from 7 and 8)
But we saw in (5) that it is necessary that if God exists, he exists necessarily: N(Eg->NEg). Since it is not the case that NEg, it is logically necessary that God does not exist. (from 5 and 9)