For AO1 you need to know:

Deductive proofs & ‘a priori’
Anselm’s 1st argument in Proslogion 2
Anselm’s 2nd argument in Proslogion 3
Descartes’ ontological argument
Malcolm’s ontological argument
Gaunilo’s objection of the perfect island
Kant’s objection that existence is not a predicate

For AO2 you need to be able to debate:

The extent to which ‘a priori’ arguments for God are persuasive
The extent to which different religious views on the nature of God impact on arguments for the existence of God.
The effectiveness of the ontological argument for God
Whether the ontological argument is more persuasive than the cosmological/teleological arguments
The effectiveness of the challenges to the ontological argument
The extent to which objections to the ontological argument are persuasive

AO1 Ontological Argument

AO1: The Ontological argument’s status as an (a priori deductive) proof

Ontological arguments are a priori, meaning they are based purely on reason, not the application of reason to experience.
They begin with a definition of God:
The greatest conceivable being (Anselm).
Supremely perfect being (Descartes).
An unlimited being (Malcolm).
The argument then involves a purely conceptual analysis of the logical implications contained within this definition.
The argument is deductive, meaning that if the premises are true, the conclusion must be true.
The truth of the premises logically entails the truth of the conclusion, rather than merely supporting it.
A priori deductive arguments are often associated with rationalism (e.g., Descartes), the view that knowledge of reality can be gained through reason alone.
Though, they can also be accepted by empiricists, who may regard God as a unique case where logical truth and matters of fact converge.
Deductive arguments aim at certainty.
If successful, they demonstrate what must be true, not merely what is probably true.
They are not defeasible in the same way as inductive arguments, since no further evidence could overturn a valid deduction.
Therefore, they can only be challenged by either rejecting the premises (denying soundness) or by denying that the conclusion follows from them with necessity (denying validity).
Such arguments rely on analytic entailment, where what is contained within a concept determines what must be true of it.
This is how the ontological argument claims that once the concept of God is properly understood, existence (or necessary existence) is entailed by it.
It intends to show that God’s non-existence is a contradiction.
That would establish the proposition “God exists” as an analytic truth (true by definition), the key criterion of which is that it cannot be denied without contradiction.
Standard deduction operates through analytic entailment of categories:
E.g.: “All men are mortal; Socrates is a man; therefore Socrates is mortal”
However in the ontological argument, the deduction proceeds not merely through categories but ‘modal’ concepts, such as possibility and necessity.
A necessary being is one that cannot fail to exist.
If such a being is even possible, then it must exist in all possible worlds, including the actual world.
The claim is that a priori reason discovers that analytic entailment functions just as decisively in this unique case.

AO1: Anselm’s ontological argument

Anselm’s ontological argument involves a purely a priori analysis of the concept of God
It’s intended to be deductive, so that the conclusion follows from the premises with necessity.
Anselm uses the illustration of a painter who has an idea of what they will paint in their mind before painting it in reality.
This illustrates the distinction between our idea of something existing in the mind alone, verses existing both in the mind and in reality.
Anselm then points to Psalm 14:1 “the fool says in his heart, ‘there is no God’.”
An atheist says they do not believe in God, but they at least have an idea of God in their mind.
In Proslogion 2, Anselm argues:
P1. God is the greatest conceivable being (by definition)
P2. It is greater to exist in reality than the mind alone
P3. God exists in the mind
C1. Therefore, God exists in reality
The force of Anselm’s argument is that God cannot be an idea that exists in the mind alone (as an Atheist thinks).
That would be incoherent, since then we could conceive of something greater, i.e., God also existing in reality.
Yet, God is the greatest being, so conceiving of anything greater is incoherent.
So, our idea of God must therefore be of a being that exists in reality.
To say that God does not exist in reality is effectively to say that the greatest being is not the greatest being. It is self-contradictory.
Malcolm and Hartshorne were responsible for renewing interest in the Ontological argument in the 20th century. They point to Proslogion chapter 3 as containing the strongest presentation of the argument.
There, Anselm argues:
P1. A necessary being whose nonexistence is impossible is greater than a contingent being whose nonexistence is possible.
C1. Therefore, God (as the greatest conceivable being), necessarily exists.
Malcolm interprets Anselm’s term ‘greater’ as referring to degrees of limitation, such as dependence on other things for existence.
God must be unlimited, without any of the contingencies of ordinary beings which make their non-existence possible.
So, a being greater than which none may be conceived is one whose nonexistence is impossible.
Hartshorne calls this insight “Anselm’s discovery”.
In his replies to critics, Anselm captures the key move: if such a being is logically possible, then it must exist.

AO1: Descartes’ Ontological argument

Descartes aimed to strengthen the ontological argument through founding it on his rationalist epistemology.
Rationalism claims that we can gain absolutely certain knowledge of some truths a priori.
Anselm is often called the father of Scholasticism, a theological movement influenced by Aristotle’s approach to argumentation.
At its core is subject-predicate analysis. Propositions are combinations of subjects and predicates which assert something as true or false.
Descartes rejected scholasticism.
He instead argued that the foundation of knowledge was intuition.
Intuition operates through direct intellectual awareness (clear and distinct perception), rather than the step-by-step analysis of logical propositions or sylogisms.
Intuition provides absolute certainty. We can bring ideas before our mind and apprehend truths about them due to the clarity and distinctness with which they are perceived
He illustrates that we intuitively know it is impossible to bring a triangle before our mind separated from it having three sides.
Similarly, we intuitively know it impossible to think of a supremely perfect being separated from existence.
We thus rationally appreciate that God contains the perfection of existence.
Descartes is claiming the intellectual means by which we know God’s existence has the same level of epistemological warrant as when we know mathematical truths.
The parallel occurs in two steps:
First, the idea of a supremely perfect being is found “within me just as surely as the idea of any shape or number”.
So the concept itself is just as certain as that of a triangle etc.
Secondly, our understanding that it belongs to God’s nature that God “always exists” is “no less clear and distinct” than the case when we prove of any shape or number that some property belongs to its nature.
So knowing God’s existence is just as warranted as when we know that a triangle has three sides.
Descartes did put it into the form of a deductive argument:
P1: The idea of a supremely perfect being is one which contains all perfections.
P2: Existence is a perfection
C3: God exists
The argument is deliberately short, suggestive of Descartes’ view that God’s existence can be known intuitively.
For Descartes, knowing God’s existence isn’t a matter of reasoning from premises to a conclusion.
We simply intuitively know that God exists, just as we know a triangle has three sides.

AO1: Malcolm’s ontological argument

Norman Malcolm’s version of the ontological argument is a priori and deductive.
It proceeds through conceptual analysis of the definition of God rather than appealing to experience.
It also employs modal logic, which analyses concepts such as necessity and possibility.
The argument aims to show that, if the concept of God is coherent, then God’s existence is not merely possible but necessary, so that the conclusion follows with logical necessity rather than probability.
The definition Malcolm starts with is that God is an unlimited being.
He thought ‘unlimited’ was a better term than ‘greatest’, because it is less subjective and captures what Anselm meant by the ‘greatest’ being.
A limited being depends for its existence on something else, which means it is capable of non-existence.
An unlimited being has no dependencies and thus cannot fail to exist (it contains the impossibility of non-existence; necessary existence).
So the definition of ‘unlimited’ implies non-dependence, and thus necessary existence.
Defining God as an unlimited (non-dependent) being is crucial for Malcolm’s argument, as it implies:
God cannot have contingent existence: because God would then depend on external conditions.
God cannot have contingent non-existence: because contingency involves dependence, and God is defined as a non-dependent being.
This leaves only two possibilities: either God exists necessarily or God is impossible.
Malcolm then formulates this insight into a deductive modal argument:
P1. God’s existence is either necessary or impossible.
P2. If God exists, then God exists necessarily (God cannot come into or go out of existence).
P3. If God does not exist, then God’s existence is impossible.
P4. God is not impossible (the concept of God is not self-contradictory).
C1. Therefore, God exists necessarily.
Since Leibniz, proponents of the ontological argument have accepted that it depends on God being a logically coherent concept.
Malcolm makes that a clear premise of the argument in P4.
There are lots of debates about whether God is coherent, but Malcolm thinks there haven’t been any clear proofs of incoherence.
Malcolm concludes the only way one could reject God’s existence is to deny that God is coherent.
This aims to vindicate Anselm’s original insight that, if the concept of God is coherent at all, then God must exist

AO2 ontological content

AO2: Gaunilo’s ‘lost island’ objection

Gaunilo’s ‘reductio ad absurdem’ critique argues absurdity results if we apply Anselm’s logic to another case: the greatest possible island.
Following P2, if it’s greater to exist, then this island must exist.
This would work for the greatest possible version of anything, absurdly implying reality would be ‘overloaded’ with greatest versions of every possible thing.
Gaunilo is denying that the conclusion of God’s existence follows from the premises, so he is denying deductive validity.

Counter

However, Anselm replies that his argument is only intended to work in the case of God.
Descartes develops this line of thought by arguing that God’s essence includes necessary existence, whereas contingent beings like islands do not.
An island is contingent by definition (land enclosed by water), so its existence depends on water.
Therefore, like all things we observe, no matter how great or perfect an island is, it will still be contingent.

Evaluation

This reply defeats Gaunilo because a priori reasoning can only demonstrate existence of a necessary being.
A priori reasoning solely involves the analysis of the definition of concepts.
The definition of contingent beings is that their existence depends on something else.
The existence-conditions of contingent beings are therefore separate to their definition.
So, a priori reasoning cannot determine the existence of contingent things.
E.g.:
I can know a priori that a perfect island depends on water to exist, but not whether that water does exist.
So I can’t know a priori whether a perfect island, or any other contingent thing, exists.
For necessary beings, by contrast, there is no supra-definitional existence requirement.
The ontological argument therefore works exclusively in the case of the greatest conceivable being.
So, there’s a relevant difference between God and contingent things like Islands which explains why the logic works exclusively for God.
Its logic cannot be tested where it doesn’t belong, through reference to contingent beings.

AO2: Gaunilo’s critique that God is beyond our understanding

Gaunilo objects to P3, the claim that God is in our mind/understanding.
He invokes the classical notion that God is beyond our understanding.
In that case, Anselm can’t go on to conclude that God being the greatest being requires that he is not just in our understanding, but also in reality.

Response:

Anselm replies with an analogy: just because we cannot look directly at the sun, doesn’t mean we can’t see by sunlight.
Similarly, just because we can’t understand what God is, doesn’t mean we can’t understand that whatever God is, God is the greatest conceivable being.
We can understand that in a hierarchy, like greatness, there must be a highest point or ‘intrinsic maximum’, to use Plantinga’s modern term.
That’s all we need to understand for Anselm’s argument to work.
When we combine that with the premises that it is greater to exist, we can then deduce the conclusion that the greatest conceivable being exists.
We don’t need to understand what a maximally great being actually is, only that whatever it is, it is the pinnacle of the greatness hierarchy.

Evaluation:

Firstly, Gaunilo’s critique doesn’t work against Anselm’s 2nd form of his argument, which doesn’t rely on a premise about God existing in the mind.
Secondly, Gaunilo misunderstands P3 and commits a straw man fallacy, attacking a claim Anselm didn’t make.
In P3, Anselm uses ‘in intellectu’, meaning ‘can be conceived or thought about’, not ‘fully understood’.
Anselm didn’t mean God is in the mind in the sense of us having full knowledge of God’s substantive nature.
He just meant that we understand something proper to the form of God; that God is maximally great.
So, Anselm’s defence works because knowing God to be the greatest being, and knowing what it is to be the greatest being, are two distinct things.

AO2: Kant’s 2nd critique: existence is not a predicate

The ontological argument claims that denying God’s existence is an incoherent denial of what God is (a maximally great/perfect being).
Kant responds that this misunderstands what existence is by treating it like a predicate (a description of what a thing is).
He illustrates: there’s no conceptual difference between 100 coins in reality versus only in the mind.
If existence were a predicate, it would describe a quality of the real coins.
They would then have more qualities and thus be conceptually different to the mental coins.
But, they are not: 100 coins is just 100 coins, defined by the predicates of 100, round, shiny, etc.
So, existence is not a predicate.
This attacks the premise that existing is greater or more perfect.
If existence is not part of what God is, we can deny God’s existence without incoherently contradicting what God is.

Response:

However, Descartes’ epistemology doesn’t rely on Anselmian scholastic attribution of predicates to subjects, but on intuition.
We rationally appreciate that God is inseparable from existence, just as a triangle is inseparable from three sides.
So Kant’s criticism fails to target Descartes’ formulation.
Malcolm also defends Anselm:
Kant is correct, but only about contingent existence.
The reason for the existence of a contingent thing is dependence on something else, so is external and not a defining part of it.
However a necessary being contains the reason for its existence within itself.
So, necessary existence is a defining quality of a thing, in a way contingent existence is not.
So necessary existence is a predicate.

Evaluation:

So, both Anselm and Descartes’ approaches succeed against Kant’s criticism.
Kant makes the same mistake as Gaunilo, thinking an argument for a necessary being could be undermined by showing it fails when applied to contingent things like coins.

AO2: Kant’s 1st critique: necessity doesn’t imply existence

Gaunilo criticised the conflation of necessity in mental judgement with necessity in reality.
The Island (and Kant’s 100 coins) misapplied this to contingent things.
Kant’s first critique refines Gaunilo’s underlying point by applying it to necessity itself:
A triangle necessarily has three sides.
This proves that if a triangle exists, then it necessarily has three sides.
Similarly, God is defined as necessarily existing.
But again, this only shows that if God exists, then God exists necessarily.
So here Kant grants that necessary existence can be predicated of God, but denies this establishes God’s actual existence.
If God exists, it’s contradictory to deny God’s necessity.
But if God does not exist, then neither does God’s necessity.

Counter:

Malcolm responds that Kant’s criticism is incoherent because a necessary being must exist.
If God is a necessary being, then God must exist.

Evaluation

However, Hick uses modern distinctions between types of necessity to counter Malcolm.
Anselm & Malcolm claim a maximally great/unlimited being cannot have contingent dependencies.
They then treat non-contingency as entailing logical necessity.
Hick objects that non-contingency only means that God is defined as a non-dependent, eternal, self-explaining being (aseity), which Hick calls ‘ontologically necessary’.
This does not entail that such a being must exist, but only how it would exist if it did.
Its existence is not logically required but would be a metaphysical ‘sheer fact’ (Hick).
So, the ontological argument cannot establish the incoherence of God’s non-existence.
It at most proves that if God exists, then God exists in a metaphysically special way (with aseity).
Gaunilo and Kant’s point that conceptual necessity doesn’t entail actual existence was right, but needed these newer distinctions between types of necessity to make clear why.