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The Chocolate Lovers' Ontological Argument
Or You, Too, Can Prove Anything
(And We Mean Anything).

Valid, Si!
Convincing, No!

THE Proof

For thousands of years - well, maybe a thousand - people as diverse as St. Thomas Anselm and C. S. Lewis have been proposing various ontological arguments. What's odd is that the arguments that have been most convincing are not valid, and the ones that are valid - like the modal proof of Kurt Gödel - are not convincing.

The truth is that while these exercises are at time amusing and entertaining, they are just that - exercises in amusement and entertainment.

Instead, if you want to be a philosopher it's pretty easy to come up with an ontological proof. Particularly if you're a chocolate lover.

Ha? (To quote Shakespeare.) What does that mean?

Well, let us elaborate.

Dr. Edgington's Proof

First consider the following argument. It is a slight recasting of an argument that was first published by the British philosopher Dorothy Edgington in 1995.

Let's hasten to say Dorothy did not say the argument is sound - that is, she did not claim it was both valid and true. But she gave it as an example of problems you can have with logical arguments.

Anyway the argument is as follows:

If God does not exist, then I can pray, but my prayers will not be answered. So I do not pray. Therefore God exists.

Now most people would say this is balderdash, horse hockey, and bullshine. How can this argument possibly be valid?

Well, first we have to determine how to tell if an argument is valid. That's not too hard.

First, write the article in a table with the premises and conclusions clearly stated.

Premise 1       If God does not exist, then I can pray, but God will not answer my prayers.
         
Premise 2       I do not pray.
         
Conclusion       Therefore, God exists.

 

At this point we have to digress a bit. We have to rewrite these English sentences symbolically. That is, we use symbols to represent the individual parts of the arguments. This will be clearer as we go along.

Now, in textbooks or on the Fount of All Knowledge, you learn that to make complex sentences, logic uses four sentential connectives. These represent the English words, "not", "and", and "or" as well as "If-Then" sentences.

Logical Connectives
Symbol Offical
Name
Plain
English
¬ Negation "Not"
"It is false that"
Conjunction "And"
Disjunction "Or"
Material
Implication
"If-Then"

The use of the connectives are best shown in Truth Tables. The "NOT", "AND", and "OR" Tables are pretty much common sense:

Truth Table: "NOT" Statements
A ¬A
TRUE FALSE
FALSE TRUE

 

Truth Table: "AND" Statements
A B A ∧ B
TRUE TRUE TRUE
TRUE FALSE FALSE
FALSE TRUE FALSE
FALSE FALSE FALSE

 

Truth Table: "OR" Statements
A B A ∨ B
TRUE TRUE TRUE
TRUE FALSE TRUE
FALSE TRUE TRUE
FALSE FALSE FALSE

Note that the "OR" in logic means "One of the two or both".

But it's the "If-Then" Table that gives students the most trouble.

Truth Table: "If-Then" Statements
A B A → B
TRUE TRUE TRUE
TRUE FALSE FALSE
FALSE TRUE TRUE
FALSE FALSE TRUE

Perhaps the best way to understand the "If-Then" statements is to realize that if you say:

Do NOT study philosophy OR you will go nuts

... means the same thing as ...

IF you study philosophy THEN you will go nuts

In other words "If-Then" is the same as "Not-Or"

A → B = ¬A ∨ B

... and we see that "Not-Or" has the same Truth Table as "If-Then".

Truth Table: "Not-Or" = "If-Then" Statements
A ¬A B ¬A ∨ B A → B
TRUE FALSE TRUE TRUE TRUE
TRUE FALSE FALSE FALSE FALSE
FALSE TRUE TRUE TRUE TRUE
FALSE TRUE FALSE TRUE TRUE

Armed with the Truth Tables we can now determine if arguments are valid or not.

First we'll abbreviate the individual parts of the argument - called the atomic sentences.

G     God exists.
P     I pray.
A     My prayers will be answered.

Then making allowances for nuances of grammar and verb tenses, we end up with the argument in symbolic form:

Premise 1     ¬G → P ∧ ¬A     If God does not exist, then I pray but my prayers will not be answered.

(Note: "And" and "but" mean the same thing in logic.)
   
Premise 2     ¬P     I do not pray.
   
Conclusion      G   God exists.

 

The Battles of Sir Validahad

OK. Just what do we mean when we say an argument is valid?

A valid argument is one that if all of your premises are TRUE, then your conclusion will also always be TRUE.

Another way to look at it is that a valid argument can have no counterexamples. That is, there are no ways that the premises can be TRUE and the conclusion be FALSE.

This last point is useful. Although when proving validity, every possible way the premises can be made TRUE has to produce a TRUE conclusion. But if you can find even a single counterexample, then the argument is invalid.

So let's look at our argument, but purely in symbolic form:

Premise 1     ¬G → P ∧ ¬A
   
Premise 2     ¬P
   
Conclusion      G

Can we prove this argument valid?

Well, let's start off by pointing out that if Premise 2 is TRUE, then that means ¬P is TRUE. And so P is FALSE.

But if the argument is valid, then Premise 1 must also be TRUE.

Now note that the "Then" part of Premise 1 is the "AND" statement, P ∧ ¬A. But an "AND" statement is FALSE even if only one of the atomic statements is FALSE. Since we just determined P is FALSE, then P ∧ ¬A must be FALSE as well.

Now go back and look at the "If-Then" Truth Table. We see that if the whole "If-Then" statement is TRUE and the "Then" part is FALSE, then the "If" part must also be FALSE. So if Premise 1 is to be TRUE, the "If" part - which is ¬G - must be FALSE.

But hold on! If ¬G is FALSE, then G has to be TRUE!

And G is our conclusion!

So what have we proven?

Well, we've just found out that if Premise 1 and Premise 2 are both TRUE, the conclusion must also be TRUE.

And so our argument is VALID!

A Most Convincing Proof

Honesty compels us to admit this argument doesn't convince many non-believers. But we also know that non-believers are merely (ptui) skeptics who will not accept the Light of the Reavealed Word even if it shines brightly upon their heads. So let's try to convince people who are more receptive.

Like chocolate lovers.

Here, of course, we're in good company. Like many colonial fathers, George Washington loved chocolate. In fact, we have it on record that in 1757 - he ordered 20 pounds of chocolate from England. So if we can craft an argument that convinces chocolate lovers, it would convince George Washington. And so it should convince all loyal Americans.

 

For our new argument, then, we assign - that is we interpret - the symbols to be:

G     God exists.
P     I will not eat chocolate.
A     I will not gain weight.

Notice how we have reversed some ¬ statements. We use P for "I will not eat cholcolate." So ¬P means "I will eat cholcolate." We also are setting A to "I will not gain weight." So ¬A means I will gain weight.

Our new argument, then, is:

Premise 1     ¬G → P ∧ ¬A     If God does not exist, then I will not eat chocolate but will (still) gain weight.
   
Premise 2     ¬P     I eat chocolate.
   
Conclusion      G   God exists.

So is this argument also valid?

At this point, you should remember that in proving our first argument was valid, we simply used the letters, G, P, and A. We didn't need to make any reference to the English sentence or their meaning.

And here, too, we have only G, P, and A as the symbols. And they are used in exactly the same way as our first - and valid argument.

So the proof of this second argument - the Chocolate Lovers' Ontological Argument - will be exactly the same as for our first argument. And so we will prove our new argument is valid as well.

This is the whole point about logic. Validity is due to the structure of the argument, not the meaning. Two arguments that can be represented with the same symbols having the same structure are called isomorphic. If you prove one of the arguments is valid, then you've automatically proven that all its isomorphic arguments are valid as well.

What surprisese some students is that an isomorphic argument can sometimes be used to prove the exact opposite of the conclusion that was just proved! Once more that's because it's not the meaning of the indivdiual parts of the argument that matter, but how they're put together.

We can illustrate such flip-flopping proofs with a new argument - perhaps suited for the more serious minded. We start of defining our symbols as:

G     God does not exist.
 
P     I pray.
 
A     My prayers will not be answered.

Notice that again we've had to flip some of the sentences with the ¬ symbol. So our argument become:

Premise 1     ¬G → (P ∧ ¬A)     If God exists, then I pray and my prayers will be answered.
 
Premise 2     ¬P     I do not pray.
 
Conclusion      G   God does not exist.

Again we have an argument that is isomorphic with a previosly proven argument. So the new argument is also valid.

Just what the heck is going on?

A Minor Difference

Now we must be honest and say that our rendering of Dorothy's proof isn't exactly as she wrote it. Instead in her original proof Dorothy had as Premise 1:

If God does not exist then it is false that if I pray then my prayers will be answered.

... which is symbolized as:

¬G → ¬(P → A)

OK. Now remember that our version of Premise 1 is:

¬G →  P ∧ ¬A

So we now remember that:

A → B

... is the same as:

¬A ∨ B

Although we haven't gone into great detail about how to make logical transformations this is an important part of formal logic. So although we will not go into great detail of making the transformations we will show that you can rewrite Dorothy's original premise:

1     ¬(P → A)     Dorothy's Original Premise: Negation of "If-Then" Statement
             
2     ¬(¬P ∨ A)     "If-Then" = "Not-Or"
             
3     (¬¬P ∧ ¬A)     Moving "Not" Inside Parentheses Negates All Terms and Changes "Or" to "And"
             
4     P ∧ ¬A     Double "Nots" cancel.

We see, then, that Dorothy's premise says:

If God does not exist then I pray and my prayers are answered.

Which is the same as ours!

But ... (you knew there would be a "but").

But if logic is to be useful we can't have it derive oddball arguments. So if you do get an oddball conclusion you have to determine 1) if it really is an oddball statement and 2) if so, find a way to arrive at a non-oddball conclusion.

So let's look at Premise #1 again:

¬G → P ∧ ¬A

Now look at the consequent (the THEN part):

P ∧ ¬A

This can only be TRUE if you pray and God does not answers your prayers. There is no option that if you do not pray, God will not answer your prayers. Omitting this option is putting a rather strange - not to say artificial - restriction on our argument.

So we ask. Is there a better way to phrase Premise 1? One more in accord with natural language?

Well, let's look at this possibility for the THEN part of the premise.

If I pray, then God will not answer my prayers.

... which is symbolized as ...

P → ¬A

Not only is this more like what we really say in ordinary conversation, but it removes the artificial restriction. In other words, this can be TRUE if you do not pray and God does not answer your prayers

With that in mind, let's proceed using this alternative Premise 1.

If God does not exist, then if I pray my prayers will be not be answered.

... is symbolically rendered as:

¬G → P → ¬A

So now our entire argument is:

Premise 1     ¬G → P → ¬A
   
Premise 2     ¬P
   
Conclusion      G

OK. Is this argument valid?

If it's valid then we can go through similar steps as we did before. That is we prove that whenever the premises are TRUE then the conclusion must always be TRUE.

On the other hand remember we mentioned that if an argument is not valid, then there exists at least one counterexample. That is we can assign truth values to the indvidual variables where all of the premises are TRUE, but the conclusion is FALSE. Sometimes this is the easiest approach and we'll try it here.

Well, if there is a counterexample, then the conclusion, G, is FALSE and all premises are TRUE.

Therefore Premise 2, ¬P, must be TRUE. So P must be FALSE.

But P is also the antecedent of P → ¬A, which is the consequent of Premise 1. And again look at the "If-Then" Truth Table. False antecedents always produce TRUE "If-Then" statements.

So if Premise 2 is TRUE, the consequent of Premise 1 is always TRUE.

But the Truth Table also shows that an "If-Then" statement with a TRUE conclusion is always TRUE. So as long as Premise 2 is TRUE, Premise 1 is always TRUE - regardless of whether G is TRUE or FALSE.

In other words we can have both Premise 1 and Premise 2 be TRUE, and yet G can be TRUE or FALSE as we please.

So if we set G to be FALSE what do we have? We have Premise 1 is TRUE, Premise 2 is TRUE, and the conclusion is FALSE. And ...

We have a COUNTEREXAMPLE!

So we have just proven:

The argument is INVALID!!!!!!!

So let's review what we've done.

We've revised the first premise of the original argument to remove an unnecessary and even arbitrary restriction. And in doing so we've rendered the premise more in accord with what we really mean in ordinary conversation. Then - as we expect - what looks like a strange, weird - and invalid - argument is indeed strange, weird, and invalid.

The Shoals of (Red) Herrings

Actually there's one more thing a bit amiss with the first way we formed our argument. And what we did is downright sneaky.

You may have noticed that the sentence that we are calling A only appears once and then in a premise. You never see it again; not in a conclusion, not in another premise.

So if you suspect that the statement we call A is irrelevant to the argument you would be right. This is something we can easily show.

First, let's write the argument and leave A out.

¬G → P
¬P
-------
∴G

Since ¬P is Premise 2, this is taken as TRUE in our tests for validity. If ¬P is TRUE, then P is FALSE.

But P is also the consequent of Premise 1. Since the consequent is FALSE, for Premise 1 to be TRUE, then the antecedent , ¬G, must also be FALSE. So if ¬G is FALSE, G must be TRUE.

Which is our conclusion.

Our argument is valid.

In fact, this "new" argument is a slight variation on a classic type of reasoning called Modus Tollens. Modus Tollens has forerunners that go back to the days of the Ancient Greeks. So the validity of our stripped down argument is nothing new.

So what happens if we stick ¬A back in Premise 1?.

¬G → (P ∧ ¬A)
¬P
-------
∴G

Actually nothing. It doesn't matter if we substitute P ∧ X for P in Premise 1. No matter what X is, the consequent will still be FALSE if P is FALSE. Therefore, ¬G must also be FALSE. Hence G is TRUE - which is our conclusion.

In other words, we can replace ¬A with anything at all! In logic-speak, we've slipped in a red herring.

The New Problem

But in solving one problem, you may suspect we have created a new one.

Let's look again at the stripped down version:

¬G → P
¬P
-------
∴G

We've shown this argument is valid. And yet this argument seems to imply that the existence of God is somehow dependent on the lack of piety of the congregation. This is, of course, absurd (we think).

Happily there is a simple explanation. It turns out that our argument is actually quite limited regarding - not its validity - but it's soundness. That is, even though it's valid the first premise is nothing that we can accept as TRUE.

We can show this by using an alternative for the "If-Then" statement. This is an "Only-If" statement. That is If A then B is the same as A only if B

Now we must admit that claiming that If A then B is identical to A only if B is confusing. A lot of students think A only if B is the same as If B then A.

Nope. Look at it this way. A only if B means that A can only happen if B has happened. That is, if B has not happened then A cannot have happened.

We write this latter statement as:

¬B → ¬A

Or in symbolic language:

A only if B ≡ ¬B → ¬A

But also remember that;

P → Q ≡ ¬P ∨ Q

And with a bit of logical algebra:

A only if B  

  ¬B → ¬A
   

  ¬(¬B) ∨ ¬A
   

  ¬¬B ∨ ¬A
   

  B ∨ ¬A
   

  ¬A ∨ B
   

  A → B

So although it is a bit hard to beleive, A only if B is simply the same as If A then B.

So going back to Premise 1:

If God does not exists then I pray.

...we now see this is the same as saying ...

God does not exists only if I pray.

... or to rephrase it ...

Only if I pray does God exist.

So we see that our premise is not something that anyone can really buy into.

The Indisputable TRUTH

Truth functional or not, sneaky or not, red herring or not, maybe we should stick to our original argument after all. Then we can craft this new and incredibly profound - and valid - argument:

Premise 1     ¬G → ¬C ∧ ¬T     If God does not exist, then the Cubs can never win the World Series, and so Chicago will never have a championship team.
   
Premise 2     C     The Cubs won the World Series.
   
Conclusion      G   God exists.

By now the readers realize the validity of this argument cannot be questioned.

References

"Chocolate Making Demonstrations", George Washington's Mount Vernon, http://www.mountvernon.org/plan-your-visit/activities-tours/chocolate-making-demonstrations/

"Some Remarks on Indicative Conditionals", Barbara Abbott, Proceedings from Semantics and Linguistic Theory, Vol. 14, pp. 1 - 19, (2004). This gives the example of

"On Conditionals", Dorothy Edgington, Mind, Vol. 104, pp. 235 - 329, (1995). This gives the example of

"Ancient Logic", Stanford Encyclopedia of Logic, http://plato.stanford.edu/entries/logic-ancient/