Part 1: Apologetics for Everyone
Part 2: The Cosmological Arguments

P rofessional philosophers commonly regard the Ontological Argument as the best single logical argument in favor of God’s existence. In fact, many secular philosophers have conceded that the Modal Ontological Argument (the version of the argument under consideration in this article) holds up under even the most rigorous scrutiny. This is particularly impressive when we consider just how many years critics have had to find fault with the argument. It was Anselm of Canterbury who first discovered and defended a version of this argument in his work Proslogion in 1078 AD.

Yet in spite of the tremendous effectiveness and age of the argument, it remains one of the more obscure arguments, especially to the average theist. But why would perhaps the oldest and best argument in favor of God’s existence be the least known? The answer appears to be two fold:

1. The scope of the argument can be challenging to defend.
2. Many people find the argument difficult to understand.

Let’s start with the scope of the argument.

What is the scope of the Ontological Argument?

One of the misunderstandings about the Ontological Argument is that the Ontological Argument claims to prove that God exists. This is false. All that the argument attempts to accomplish is to show that if the idea of God is not logically or metaphysically incoherent, that is, if the idea of God is not irrational, then God does exist. Understanding this is critical to utilizing the argument correctly. For the time being we will not worry too much about this fact, but we will return to this idea in our discussion time. Now, let’s look at the second reason the argument is unfamiliar to many theists.

Many people find the argument difficult to understand. But Why?

1. Part of the reason is that the argument requires some training in the construction of philosophical reasoning.
2. The argument acts almost like a mathematics problem, which some find daunting.
3. Many casual observers think the argument seems “fishy” and then they see the volumes of pop-philosophical criticism of the argument and choose to ignore the argument altogether.

So, my objective in this article is to demystify the Ontological Argument, to break it down in simple terms that anyone can understand. My hope is that everyone will gain a mastery of this argument and will make it a utility in the defense of their faith.

Although the Ontological Argument comes in many forms, in this article we will be examining Alvin Plantinga’s Modal Ontological Argument. I have chosen Plantinga’s version for two reasons:

1. It is relatively simple.
2. It uses modal logic, which if you have read Article 02: The Cosmological Argument you will already be familiar with. If you haven’t, no problem. We’ll cover it again here.

So let’s cover modal logic now and go through some of the definitions that are necessary to understand the argument, including the word Ontology itself.

Modal logic is a method of forming arguments based on three types of objects. We ask, are the objects in question:

1. Impossible Objects: Impossible objects are objects that cannot exist rationally, like a square circle.
2. Contingent Objects: Contingent objects are objects that depend on something else for their existence, like apples depend on apple trees or eggs depend on chickens. In reality all space-time objects are contingent, they depend on physics to exist if nothing else.
3. Necessary Objects: Necessary objects are objects that depend on nothing for their existence. The number 3 might be a necessary object, for example.

In addition to these three types of objects we also need to add another modal logic concept to our repertoire, the idea of possible worlds.

A possible world is not another planet or a parallel dimension. In logical arguments possible worlds are simply descriptions of the way reality could be.

For example, in a possible world Adolph Hitler and the Nazis won World War II. In another possible world, we all pooled our money, bought a lottery ticket, won, and are all millionaires. Both of these examples are worlds which are plausibly true, they could exist. However, according to this definition, a possible world cannot be an impossible world. So there cannot be any world in which logically impossible objects exist. For example, there cannot be a universe in which 2 = 3 or there are 4 sided triangles or, as William Lane Craig puts it, there cannot be a world in which there are married bachelors. These things are logically absurd and cannot exist in any possible world.

Believe it or not, if you understand these four simple logic concepts: impossible objects, contingent objects, necessary objects, and possible worlds, you have all of the required information to reason through the Ontological Argument.

Next let’s define God.

That task might seem daunting, but fortunately the Ontological Argument does this for us. Anselm saw God, in simple terms, as what he called a Maximally Great Being (MGB). He noted that God would always be the greatest being that anyone could imagine. If someone could imagine a greater being, then that being would be God. He went on to say that God would have only great-making properties and no less-making properties. Here are a few examples:

Great-making Properties	Less-making Properties
Timeless	Bounded by time
Space-less	Bounded by space
Immaterial	Bounded by matter
All powerful	Limitedly powerful
All knowing	Limited in knowledge
Eternally living	Mortal
All good	Somewhat good

A maximally great being will have all great-making properties and have those properties to their fullest extent.

Finally, what does the word Ontological itself mean?

Of course, the root word of ‘ontological’ is ‘ontology.’ Although ‘ontology’ sounds like a fancy word, its meaning is actually quite simple: ontology is the study of the nature of being or existing.

So the Ontological Argument simply means a logical argument that, if true, argues in favor of the existence of God. What is so interesting about this argument is that it relies on nothing other than logic to show that if the premises are true, then God must exist.

So, now that we have an understanding of all of the pieces that go into the Ontological Argument, let’s look at the Ontological Argument itself:

Premise 1: It is possible that a maximally great being (MGB) exists.

Premise 2: If it is possible that a MGB exists, then a MGB exists in some possible world.

Premise 3: If a MGB exists in some possible world, then a MGB exists in every possible world.

Premise 4: If a MGB exists in every possible world, then a MGB exists in the actual world.

Premise 5: If a MGB exists in the actual world, then a MGB exists.

Conclusion 1: Therefore, a MGB exists.

Convinced?

Believe it or not, among professional philosophers premises 2 through 5 are incontrovertible; they simply follow from premise 1. In fact, the only premise in contention in the whole argument is premise 1.

Let’s examine the argument more closely to see how it works.

Premise 1:
The first premise makes a truth statement that can either be accepted or rejected. The claim is that it is possible that a maximally great being exists. This premise is quite interesting when we consider the fact that most atheists readily grant that it is possible that God exists, but they just don’t believe that God does exist.

Premise 2:
Premise two basically restates premise 1 using “possible worlds” vernacular. In fact, anything that is possible is possible in some possible world by definition. Nevertheless, utilizing “possible worlds” vernacular makes the rest of the argument simpler to understand.

Premise 3:
This is where things get interesting. Premise three states that if a maximally great being exists in some possible world, he exists in every possible world. But why? The answer is that he is maximally great. If creature A exists in only one possible world, that’s good. If creature B exists in two possible worlds, well, that’s even better. The more worlds that a being exists in, the better the being is. By this reasoning we come to see that a maximally great being must be great maximally and, therefore, must exist in ALL possible worlds.

Premise 4:
Now, in premise four everything begins coming together. For if a maximally great being exists in all possible worlds, and our world is not only a possible world but is the actual world, then a maximally great being exists in the actual world.

Premise 5:
And it only naturally follows that if a maximally great being exists in the actual world, then a maximally great being exists.

Conclusion 1:
Therefore, God exists.

Once this line of reasoning is understood, it is greatly empowering. We come to recognize that we have a logical confirmation of our own faith and a reason for our belief in God. Further, we have a tool for evangelizing.

Now, notice what happens here: if a person admits that it is possible that God exists, they have entered an inescapable syllogism, an air tight logical case that necessarily leads to the fact that God does exist.

But what if the person believes it is impossible that God exists?

Because of the Ontological Argument, most secular or atheistic philosophers find themselves forced into adopting the notion that it is impossible that God exists. However, in order to hold this position with intellectual honesty the atheist must show in what way the very notion of God is logically incoherent.

What does it mean to be logically incoherent?

Simply put, being logically incoherent is synonymous with being impossible. In other words, the atheist must argue in modal logical terms that God is not only not necessary but is also an impossible object. Under the strength of the Ontological Argument a failure to prove that God is an impossible object is the same as logical evidence that God exists.

The last millennium or so has shown just how difficult it is to demonstrate that the idea of God is logically incoherent.

It’s certainly not for a lack of trying, however. In fact, I have catalogued no fewer than fourteen major dissensions to the Ontological Argument. Some of these objections are rather sophisticated while others are really quite silly. Let’s take a look at some of them now.

 

Objection 1: Omnipotence Paradox

The Omnipotence Paradox is an attempt to show that the idea of God is logically incoherent. The objection goes something like this:

“Can God create a rock so big that even he cannot lift it? If he cannot create the rock, then He is not omnipotent. If He can create the rock and cannot lift it, He is not omnipotent.”

On face value, this seems like an interesting argument. But it turns out to be as empty as it is adorable.

The problem with the argument is that it is what is called a “straw man” argument. A straw man argument is an argument which attributes a statement or action to a person who never said that statement or never did that action, and then attacks them for it.

An example of a straw man argument might look like this:

Child: “Can we get a dog?”
Parent: “No.”
Child: “It would protect our home.”
Parent: “Sorry, no.”
Child: “Why do you want to leave us and our house unprotected?”

Not wanting a dog is not evidence that the parent wishes to leave the family and house unprotected. The accusation against the parent is a straw man.

But what does this have to do with Omnipotence Paradox?

Simply this, nowhere does the idea of being maximally great make the promise that a maximally great being can do logically absurd things.

Being maximally great does not give the maximally great being the power to, say, make square circles, or make 2 equal 3. All of these things are impossible objects, which are distinct from contingent or necessary objects.

From a Christian point of view, to give an example, the Bible tells us that there are many things that God cannot do.

1. He cannot lie.
2. He cannot be tempted with sin.
3. He cannot break a promise.

Simply put, maximally great beings must be self-consistent as a part of being maximally great. Doing things that are logically absurd is not consistent with a maximally logical being. Therefore, God is omnipotently capable to living consistent with His own character.

 

Objection 2: The Problem of Evil

The Problem of Evil is one of the most common attempts to show that the idea of God is logically inconsistent. The objection goes like this:

“Look at all the pain and suffering in the world today. If God is maximally powerful, and God is maximally good, then evil should not exist. But evil does exist. Therefore, the only conclusion is that God is either maximally good but not powerful enough to stop the evil or God is maximally powerful but not good enough to want to stop the evil. In either case a maximally great God is incompatible with the actual world, and, therefore, all possible worlds, and so God does not exist.”

Again, this might seem reasonable on the surface. However, because of the work of Alvin Plantinga and others, the Problem of Evil as an argument against God’s existence no longer works among serious philosophers.

Plantinga argues that God could not simultaneously give his creation free will and eliminate evil. Because God is good and humans are free, they are free to choose God and good or to reject God and what is good.

C.S. Lewis in Mere Christianity imagined a world in which evil and suffering were impossible. His thought experiment imagined a world in which a sword turned to a flaccid noodle if you attempted to hit someone. You could not fall out of a tree if the branch broke. Just before thinking and saying something evil the thoughts or words would suddenly change to positive and uplifting words.

Just imagine what kind of physics the world would have to possess in order to make this a reality. It would be so utterly unpredictable that no science or technology could possible exist.

In the end the world would be logically incoherent. The obvious question would be, in this kind of world do we have free will? The answer is certainly, no.

However, because humans have freedom to choose or reject God and what is good, the world cannot be void of evil or suffering.

 

Objection 3: The Problem of Imperfection

Another common objection to the Ontological Argument is the Problem of Imperfection, which is in many ways similar to the Problem of Evil. The Problem of Imperfection objection goes something like this:

“If God is perfect, he could not create something imperfect. But this creation is imperfect. Therefore, God is not perfect and is not maximally great.”

This objection too has been largely abandoned in academic circles.

Three main problems plague The Problem of Imperfection.

1. The definition of “perfection” is not clear. What is perfect to one is not perfect to another.
2. Plantinga points out that perhaps a perfect universe would contradict God’s objective for the creation. For example, what if a perfect creation would bring fewer people to the knowledge of God.
3. Even atheists have pointed out that even if both items 1 and 2 above were solved, logically speaking nothing can be derived from an imperfect universe other than there is an Maximally Great Being that did not intend to make a perfect universe.

All of this, of course, says nothing about Christian theism which largely teaches that God created a perfect world and one or several beings ruined it.

 

Objection 4: Demand for Empirical Evidence

The Empirical Evidence objection goes like this:

“Only scientifically verifiable evidence can show something to be true.”

This is perhaps one of the most commonly held objections to the Ontological Argument and theism in general. As it happens it is also one of the weakest of all the arguments.

Why? Because the statement is self-defeating.

Does the statement “only scientifically verifiable evidence can show something to be true” itself have scientifically verifiable evidence? The answer is, no! In that case the statement itself is not true.

Further, we must remember that the Ontological Argument only requires that it is POSSIBLE that God exists, not that God does exist. In other words, any number of evidences could be forwarded to establish that it is POSSIBLE that God exists—any of the cosmological arguments, for example.

 

Objection 5: Reverse Ontological Argument

One of the more sophisticated objections to the Ontological Argument is the attempt to reverse the argument, which would look like this:

P1: It is possible that a maximally great being (MGB) does not exist.

P2: If it is possible that a MGB does not exist, then a MGB does not exist in some possible world.

P3: If a MGB does not exist in some possible world, then a MGB does not exist in every possible world.

P4: If a MGB does not exist in every possible world, then a MGB does not exist in the actual world.

P5: If a MGB does not exist in the actual world, then a MGB does not exist.

C1: Therefore, a MGB is impossible.

At first blush, this seems to work. But the problem actually occurs in P2:

The problem is that P2 does not logically follow from P1. If P2 is to logically follow from P1, it should be written thusly:

P2: If it is possible that a MGB does not exist, then it is possible that a MGB does not exist in any possible world.

However, to say that a MGB does not exist in any possible world, as we have seen, is the same as saying that a MGB is impossible. But the Reverse Ontological Argument has not in any way shown that the idea of a MGB is logically or metaphysically impossible. Consequently, the Reverse Ontological Argument fails in Premise 2.

 

Objection 6: The Maximally Great Unicorn

Perhaps one of the most common but, frankly, silliest objections against the Ontological Argument, is the Maximally Great Unicorn objection. It formulates the argument this way:

P1: It is possible that a maximally great unicorn (MGU) exists.

P2: If it is possible that a MGU exists, then a MGU exists in some possible world.

P3: If a MGU exists in some possible world, then a MGU exists in every possible world.

P4: If a MGU exists in every possible world, then a MGU exists in the actual world.

P5: If a MGU exists in the actual world, then a MGU exists.

C1: Therefore, a MGU exists.

Why does this argument fail?

The answer is that a unicorn is a physical object. And physical objects are contingent objects, not necessary objects. This is a problem because in some possible worlds space and time either never started to exist or rapidly collapsed into a singularity. A unicorn simply cannot live or even exist in such a possible world.

So in what premise of the MGU argument does this reasoning fail?

The answer is P1 through P3. All three have problems.

P1: Being a maximally great physical being is illogical because to be maximally great is to live without the limitation of physical existence.

P2: It is not possible that a MGU exists because to do so violates P1.

P3: If a MGU cannot exist, it does not follow that it must exist in every possible world. Even if it were possible that a MGU existed in some possible world, it would not follow that a physical being must exist in all possible worlds.

Redirect 01: Sometimes, a person will object claiming, “Your objections may be true for lions and bears, but a unicorn, like God, is mythical. I have a maximally great mythical being and so do you.”

Rebuttal 01: The mythology of a unicorn is that it is a physical horse with a horn. To confirm its mythology to be true would be to confirm that there exists a horse with a horn. Horses with horns are physical objects that exist in time and space. Therefore, a unicorn cannot be a necessary object.

Redirect 02: Oh, no. This is a very special unicorn, a unicorn that is timeless, space-less, immaterial, etc.

Rebuttal 02: Robbing a unicorn of all of the attributes that make it a unicorn and then giving it the attributes of God simply demonstrates that you admit that God exists and prefer to call Him a unicorn.

 

Objection 7: Multiple Maximally Great Beings

Finally, some atheists have argued that it is vastly more likely that we should find multiple maximally great beings existing than one maximally great being. However, this idea is full of problems.

Suppose that maximally great being A thought that unicorns were a most wonderful idea and wished for creation to be full of unicorns. Now, suppose, however, that maximally great being B hated the very idea of unicorns and wished to create a set of physical laws which would make unicorns impossible.

In reality unicorns either exist or do not exist. If they exist, then MGB A is greater than B. If, however, unicorns do not exist, then MGB B is greater. Both cannot be maximally great.

However, those who argue for multiple maximally great beings are not entirely off in left field. It is impossible for multiple maximally great beings to exist, unless they are in perfect agreement. Ironically, for those who object to a maximally great being in favor of multiple maximally great beings, they fall very much in line with Trinitarian theology.

According to Trinitarian theology there is one God in three persons, each fully God, each distinct one from another, each maximally great, each in perfect agreement with one another.

 

Digging Deeper

To get a deeper understanding into the logic behind the ontological argument it’s important to understand logical corollaries. A corollary is a fact that must be true because another fact is already proven. Let’s look at an example:

Fact

1 + 3 = 4

Corollary

4 – 3 = 1

Is there any logical difference here? No.

Is there a perceptual difference here? Yes. We might well perceive them differently, but logically the corollary follows necessarily from the initial fact.

Logic like mathematics uses facts and corollaries.

Logical Fact:

A → □ ◊ A

Symbols:

◊          =          possible or possibly
□          =          necessary or necessarily
→        =          implies
A         =          axiom exists

English Translation:

If something exists, it must be necessarily possible for it to exist.

Example:

A dog exists; therefore, it is possible that a dog can exist.

This is a plain fact of logic that is so basic that it is taken as axiomatic, that is, it is a statement that is assumed to be true in every logical argument and does not have to be proven. In fact, it is formally called  Axiom B:

Axiom B:

A → □ ◊ A

English Translation:

If something exists, it is

necessarily possible for it to exist.

Just like 1+3=4, so Axiom B has a corollary:

Axiom B Corollary:

◊ □ A → A

English Translation:

If it is possible that a necessary object exists, it exists.

This corollary was known to philosophers, but no one had put the implications together. In 1974 Alvin Plantinga published his Modal Ontological Argument. As soon as it was published there was an uproar in nearly every philosophy department the world over. Atheist philosophers were horrified. Why? Because Plantinga had just shown that Axiom B, a completely undisputed and critical axiom, had a corollary that demonstrated that if it is even possible that God exists, He exists.

 

A Deep, Deep Dive

So if you’re still wondering how and why this works from a logical calculus point of view, this section is for you. The purpose of this section is not in any way to explain propositional calculus, the underlying logic of formulating and proving the theorems of modal logic. What follows, therefore, is my best attempt to consolidate and state in relatively simpler terms what I understand to be true about the flow of the logic that necessarily leads to the conclusion that what is possibly necessary must actually exist—the foundational axiomatic corollary of Plantinga’s Modal Ontological Argument. This is not at all to say that what follows is my opinion; rather, it is based upon (and where possible directly quoted from) primary sources. Nevertheless, because it is at times my understanding of the primary sources, errors might exist. I expect that many who read this document will have a greater understanding of the topic than I do; I expect and welcome feedback, corrections, and citations which will improve the accuracy of this section.

In order to explain how the Modal Ontological Argument functions in any meaningful way, we must have a basic understanding of the foundational principles of modal logic—what makes modal logic work and how ideas can be derived from logical axioms. Let us first take a very cursory glance at the syntax of logical language as well as the systems which under-gird modal logic. Once we have taken a quick look at modal logic we can turn our attention toward matters that bear more directly on Alvin Plantinga’s Modal Ontological Argument, which itself is based on a corollary of System S5: Axiom B.

To speak purposefully, about modal logic we first need to understand its vocabulary, the symbols and meanings of the specialized logical calculi of modal logic. What follows is an abridged glossary of terms:

¬          =          negation
◊           =          possible or possibly
□          =          necessary or necessarily
Ʌ          =          and
V          =          or
→         =          implies
↔         =          is equivalent or vise versa
~          =          not
P          =          proposition exists
A          =          axiom exists
R          =          accessible

For the most part, though not entirely, an understanding of these symbols will allow us to read the language of modal logic as plain English. You will, however, in short order find symbols that are not included here. You can go to the link below for a longer list of symbols, although this list too is abridged:

https://en.wikipedia.org/wiki/List_of_logic_symbols

Let us look at a few examples, so that we might understand how the symbols relate to colloquial speech.

1.1 Logic Syntax:

◊P ↔ ~□~P

 

1.1 Literal Syntax:

Possibly (◊) Existing (P) is Equivalent (↔) to Not (~) Necessarily (□) Not (~) Existing (P).

 

1.1 Colloquial Syntax:

Saying something possibly exists is the same as saying that the thing does not necessarily not exist.

 

1.1 Corollary:

If something possibly exists, then it is possible that the thing exists.

Let us try another example:

 

1.2 Logic:

□P ↔ ~◊~P

 

1.2 Literal:

Necessarily Existing is Equivalent to Not Possibly Not Existing.

 

1.2 Colloquial:

Saying something necessarily exists is the same as saying it is not possible that it does not exist.

 

1.2 Corollary:

If something exists necessarily, then it has to exist.

The 1.1 Logic and 1.2 Logic above are not merely examples but also show what is called unary or first place modal operations and their negations. On the left of the ↔ is the operation, and on the right of the ↔ is the negation. Thus, according to the definitions in the glossary above, 1.1 and 1.2 can also and should be written respectively as follows, where ¬ means negation:

◊P ↔ ¬ □¬ P
□P ↔ ¬ ◊ ¬ P

 

Change of Quantifier Rule

Negations are very important in modal logic because they aid in understanding equivalence and, consequently, corollaries. For example:

¬ □¬ P  ↔ ◊P    OR in colloquial English: not necessarily not existing is the same as possibly existing.
¬ ◊ ¬ P ↔ □P    OR in colloquial English: not possibly not existing is the same as necessarily existing.

Notice in the above examples that the terms ¬ □¬ is the same as ◊ and also that the terms ¬ ◊ ¬ is the same exact thing as □. We can begin to see how corollaries can be derived. Here are some further definitions in logical syntax to help in this understanding:

~□P ↔ ◊~P       OR in colloquial English: not necessarily existing is the same as possibly not existing.
□~P ↔ ~◊P       OR in colloquial English: necessarily not existing is the same as not possibly existing.

Notice in these examples we see that ~□ inverts to ◊~ and also that □~ inverts to ~◊. These types of changes are called the Dual Rule, and these types of operations are critical to understanding in what way the Modal Ontological Argument is sound.

You can see the article below (and the website in general) for more information on the topics discussed above:

http://www.manyworldsoflogic.com/modallogic.html

Before we move on, however, we need to look at one other logic rule, which bears on this study.

 

Rule of Contraposition:

Contraposition is a method of restating an assumption such that it inverts the operators and order of the terms. Although this sounds complex, in practice we do this all the time effortlessly. By way of example:

2.1 Assumption: All students are participants.

2.1 Contraposition: No participants are not students.

Notice how the order of the terms ‘students’ and ‘participants’ are reversed, how the quantity ‘all’ became ‘none,’ and the status ‘are’ became ‘are not.’ This is the operation of the Rule of Contraposition. Importantly, contrapositions can also be stated schematically. Consider the following example.

2.2 Assumption: No participants are not students.

2.2 Schematic Contraposition: No non-participants are not non-students.

2.2 Double Negative Reduction: All students are participants.

Notice in 2.2 that the Assumption is already in the negative form as seen in 2.1 Contraposition. In order to simplify 2.2 Assumption to a positive form we must use schematic contraposition. The schematic contraposition introduces double negatives into the system such that a negative might be converted to a positive. Consequently, the schematic contraposition is reduced through the elimination of the double negatives ‘no-non’ and ‘not-non’ to the original positive 2.1 Assumption. Notice that throughout the entire operation the meaning of the original sentence as stated in the 2.1 Assumption never changes. ¹

Everything that we have seen so far is first order modal logic, but modal logic has an entire landscape of systems and axioms. Earlier in this article we mentioned Axiom B—the statement that what what exists implies that it is necessary that it possibly exists. But where is Axiom B in the modal logic landscape? Here is a very brief overview of the modal logic landscape:

First, one point of clarification is that modal logic is distinct from modal logics, the latter of which is a family of logical systems which include Modal (logic of possibility), Deontic (logic of ought), Temporal (logic of tenses and time) and Doxastic (logic of belief). In this appendix we will only be looking at the Modal system, the study of logically follows from something being necessary or possible.

 

System K

System K is named for Saul Kripke, who developed modal logic’s calculus. System K is the most basic form of modal logic, is not particularly powerful in itself, is not at all controversial, and contains the following rules:

 

Necessitation Rule:

A → □A

Literally, A exists implies necessarily A exists.

If A is an axiom of K, then so is □A. This is to say that the laws of logic are necessary and are, therefore, true in all possible worlds. It is also to say that whatever is true axiomatically necessarily exists. This is a critical point in understanding the Modal Ontological Argument.

 

Distribution Axiom:

□ (A → B) → (□A → □B)

Literally, A implies B necessarily implies necessarily A implies necessarily B.

This means simply that necessity distributes to all participants of an implication statement. This will be a rather simple concept to anyone with any experience in algebra.

 

Definition of Possibility:

◊A ↔ ¬ □¬ A

We have already explored this with propositions above, but it bears repeating here.

 

System D

System D is everything that System K has plus one additional axiom.

□A → ◊A

Literally, necessarily A implies possibly A.

So this adds to modal logic the idea that whatever is necessary is possible.

 

System T (aka System M)

System T is System K with Axiom M, and is simply a stronger version of System D, and, in fact, D is contained within M as a corollary. System T/Axiom M acts as follows:

□A → A

Literally, necessarily A implies it is the case that A.

Or whatever is necessarily true is true.

 

System S4

System S4 is Systems K and T plus Axiom 4. System S4 and System S5, which we will cover in a moment have come to be seen as controversial because of some of the “unintuitive” corollary arguments which follow naturally from them. Intuitiveness of axioms or corollaries of axioms does have a role to play in logic; however, the limiting factor of intuitiveness is not the degree to which an axiom or corollary is immediately obviously true, as we might expect, but whether or not it can be explained or exemplified. Let us take a look at System S4:

□A → □□A

Literally, necessarily A implies that it is necessary that necessarily A.

In other words, if A is necessary true it is necessarily necessary that A is true, and A cannot be otherwise only possibly true. System S4 also has the corollary:

◊◊A → ◊A

Literally, possible that possibly A implies possibly A.

Notice how the right and left sides of the equations are related between Axiom 4 and its corollary. It is possible that in the formulation of modal arguments long strings of ◊◊◊◊◊A or □□□□□A might occur. S4 culls these additional possible or necessary terms as long as the terms are identical, that is all ◊ or all □ within the implication statement.

 

System S5

System S5 is simply System T plus Axiom 5, which is itself a stronger form of Axiom 4. Just as was the case in S4, S5 allows for us to limit the number of redundant mixed terms ◊□. So instead of ◊□◊□◊□◊□P, we simply use ◊□P. Instead of going into all of the work necessary to show Axiom 5 in this appendix, we will simply look at the difference between Axiom 4 and Axiom 5, which is known as Axiom B. We come to see, therefore, that Axiom B as we discussed earlier in this article is the axiom which is formed from the difference between Axiom 4 and Axiom 5. It is this difference which has caused so much controversy in the last 40 years or so. Let’s examine it here again:

A → □◊A

Literally, A implies that it is necessary that it is possible that A.

This axiom, of course, carries the controversial corollary:

◊□A → A

Or, literally, possible that necessarily A implies that A exists.

In plain English, if it is possible that a necessary objects exists, it exists. We see, therefore, from S5 Axiom B corollary the basis of Plantinga’s Modal Ontological Argument.

Now we are all up-to-speed on the context and requisite logical syntax and rules which play into a proper understanding of the Modal Ontological Argument for the existence of God. Let us now turn our attention toward understanding in what way the corollary of Axiom B in S5 follows the rules of modal logic.

As we have already seen modal logic uses as its most basic axiom rules for dealing with necessary objects. Consider System K above, which denotes as its first order axiom:

A → □A

Whatever exists axiomatically, necessarily exists. This fact is important as we noted before because necessary propositional objects must exist necessarily. Therefore, where proposition P is a necessary object p, p must necessarily exist. This is important: p below is a necessary object P, as shown here:

p → □p

What is interesting, however, is what happens when we start playing with this assumption. Let’s first use our Rule of Contraposition on this assumption to get:

~□p → ~p

Here what we have done is shown that through contraposition for any necessary propositional object p, if p is not necessary, then p does not exist. This is very interesting. Any necessary object, therefore, exist necessarily or not at all. There is no possibility of contingency. However, when we examine not necessarily p implies not p more closely, we have an interesting opportunity to apply the dual rule to the left hand side of the equation, as follows:

◊~p → ~p

This is a very interesting statement. Not only does not necessarily p imply not p but also possibly not p implies not p for all necessary objects. The ramifications of this implication statement are that if it is even possible that a necessary object p does not exist, it does not exist. This is a strong statement to which we will return later. If possibly not p, then not p, where p is a necessary object by means of definition p → □p. But it also opens up a very intriguing logical operation. We now have negative necessary propositional object p on both sides of the implication statement. That means that we can now apply a schematic contraposition to the implication statement:

◊~~p → ~~p

Schematic contraposition as we have seen is a first step in converting a negative propositional implication statement to a positive statement. How might this be done? By canceling double negatives. To do so gives us the following implication statement for all necessary objects:

◊p → p

In colloquial English if it is possible that necessary object p exists, necessary object p does exist. Therefore, necessary object p exists. And finally we can connect this as a corollary of Axiom B:

(◊p → p) ↔ (◊□P → P)

Therefore:

(◊p → p) ↔ (◊□A → A) is derived from the first order modal logic System K axiom, A → □A, which is the base of S5 Axiom B, A → □◊A, and thus is a corollary of Axiom B.

Trent Dougherty in his article A Defense of the Modal Ontological Argument describes not only all of the above on page 3 but goes on to describe why the Modal Ontological Argument trumps any version of atheism, the affirmative statement that God does not exist.

In logical argumentation the rules which govern which of the two disputing parties has the burden of proof is as follows:

Actuality bears the burden of proof.

Possibility gets the benefit of the doubt.

To state, therefore, that God does not exist is to state an assumption of actuality. To state that God possibly exists is to state an assumption of possibility. Therefore, the burden of proof is on the atheist not the theist, assuming, of course, that the theist understands the underlying modal logic. The argument for the knowledgeable theist is: it is possible that God exists, so God exists, which is the Modal Ontological Argument.

Interestingly the argument for the knowledgeable atheist is: it is not possible that God exists, so God does not exist. This is as close as the atheist can get to a rational position, and unfortunately it is not close enough. But why? Let’s examine the Reverse Modal Ontological Argument as formulated from System K, which incidentally varies from the Reverse Ontological Argument that we covered earlier, which fails at Premise 2. Let propositional object G be a necessary object, God.

G → □G
¬□G → ¬G
◊¬G → ¬G
◊¬G
¬G

This logic is sound and seemingly produces a stalemate between the possibly existing p Modal Ontological Argument and the possibly not existing G in the Reverse Modal Ontological Argument. However, there is one imbalance with this stalemate. There exists, in logical terms, a Symmetry problem between the arguments.

Let us refer to the Modal Ontological Argument as MOA and the Reverse Modal Ontological Argument as ROA. In the case of MOA a person conceives of the possibility of God. In the case of ROA a person conceives of the possibility of no God. Now consider the following statement:

For any sentence S and agent A, if A can conceive ¬S, then A can conceive S.

What this statement tells us is that anyone who can conceive of God not existing—a statement of absence—can also conceive of God existing—an affirmative statement of presence. A couple of ideas bear on this realization. First, the opposite is not at all clearly true. In other words, just because someone can conceive of God does not necessarily imply that someone can conceive of no God. And second, and more importantly, Dougherty says,

“…the opponent of the ontological argument clearly wants ¬ G to be conceivable in support of the main premise ‘◊¬G’ of the atheological ontological argument. However, [this] entails that if that is the case then ◊G has prima facie support. Once that is recognized, then we have reason to believe G and thus ¬◊¬G which defeats the prima facie justification of ◊¬G. I think this asymmetry gives more than a merely procedural advantage to the ontological argument.”

As is the case with every line in this section, there is a great deal more that could be said about this argument. In the end, however, we as Christians should not expect that we should find out God apart from God Himself. What we do have, however, is a perfectly rational basis for our belief in God; indeed, a more rational basis than the atheist has for believing that there is no God, which as Alvin Plantinga quipped, “is all anyone could hope to have.”