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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 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 it is false that if I pray, then my prayers will 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 it is false that if I pray then my prayers will be answered.
         
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 Official
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 it is false that if I pray my prayers will be answered.
   
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.

This means 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.

So if we can show that it is impossible to create at least one counterexample, then we know the argument is valid.

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, remember. If the argument is valid, then there is no counterexample.

But if there is a counterexample, then there is at least one way we can make all the premises to be TRUE and the conclusions be FALSE.

So if Premise #2:

¬P

... is TRUE.

This in turn means that:

P

... would be FALSE.

But if P is FALSE, then we again know from the "IF-THEN" Truth Table that

P → A

... must be TRUE.

So that means that the THEN part of Premise #1:

¬(P → A)

... must be FALSE.

But in that case, for Premise #1:

¬G → ¬(P → A)

...to be TRUE, then the "IF-THEN" Truth Table tells us that the "IF" part:

¬G

... must also be FALSE.

But if ¬G if FALSE, then that means that:

G

... which is also the conclusion can only be TRUE.

OK. What have we done?

Well, we've just shown that if both Premises are TRUE, then the conclusion can never be FALSE. That is there are no counterexamples.

And if both Premises are TRUE, then the conclusion MUST be TRUE.

And so our argument is VALID!

∴QED

A Most Convincing Proof

Honesty compels us to admit Dorothy's 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 do not eat chocolate.
A     I will not gain weight.

Notice how we have reversed some ¬ statements. We use P for "I will not eat chocolate." So ¬P means "I will eat chocolate." 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 it is false that if I do not eat chocolate then I will not 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 it 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 and have 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 surprises some students is that isomorphic arguments can have the opposite conclusions! Once more we emphasize 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. We start by 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 becomes:

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

Note this argument is isomorphic with a previously proven argument. So the new argument is also valid - even though it reaches a completely different conclusion.

Just what the heck is going on?

A Minor Change

Now you may have noted the phrasing of the argument - particularly Premise #1 - is a bit awkward. That is we say:

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

But we may ask, isn't this better stated and is the same as:

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

Uh - better stated, maybe. The same, no. These two expressions are not equivalent. As we'll show later, the difference is important.

Well, then what does Dorothy's original Premise really say?

To answer that, though, we'll have to make some transformations of the original premise.

Making logical transformations is an important part of formal logic. So although we will not go into general detail of how to make the transformations, we will show the specifics needed to rewrite Dorothy's original Premise #1 to something else. AND we can do that just by transforming the "THEN" part.

1     ¬(P → A)     "THEN" Part of 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:

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

... or symbolically:

¬G → ¬(P → A)

... is the same as:

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

... which is symbolically:

¬G → P ∧ ¬A

And it is this last version of the premise that let's us show that our argument - if not actually invalid - is nevertheless complete hogwash.

A Premise is True ONLY IF It's True.

So let's look at Premise #1 again:

¬G → ¬(P → A)

... which we just showed is equal to:

¬G → P ∧ ¬A

It's not easily apparent but there is indeed a major problem - not with the validity of the argument - but the soundness of the argument.

Remember an argument is sound if it is valid but also that all of the premises are TRUE!

And Premise #1 is JUST NOT TRUE!

And how do we show that?

First we have to realize that:

If A then B

... is the same as:

A only if B

Ha? (Again Shakespeare). Don't you mean "A only if B" is the same as "If B then A"?

No. We mean "A only if B" is the same as "If A then B".

You don't believe it?

Well, when we say that A only if B we mean that A can only happen if B also happens. That is, if B has not happened then A cannot have happened.

That is:

A only if B ≡ ¬B → ¬A

But also remember that;

A → B ≡ ¬A ∨ B

Now let's run through a bit of simple logical algebra:

If A then B  

  A only if B
   

  ¬B → ¬A
   

  ¬(¬B) ∨ ¬A
   

  ¬¬B ∨ ¬A
   

  B ∨ ¬A
   

  ¬A ∨ B
   

  A → B

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

Now let's go back to Premise #1.

Dorothy's original Premise was:

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

... which we showed was the same as:

If God does not exist then I pray and my prayer is not answered.

... which in the "only-if" form is:

God does not exist only if I pray and my prayer is not answered.

We see immediately the premise states that only if you pray and your prayer is not answered does God not exist.

But hold on! Suppose some misguided soul prays from something crazy - like the sun not to rise the next morning. And if it does rise? Well, according to Premise #1, that is conclusive proof God doesn't exist!

This might be a bit clearer if we look at the premise in symbolic form again:

¬G → P ∧ ¬A

Note that if God does not exist, then ¬G is TRUE. So for the premise to be TRUE, then both P and ¬A must be TRUE. That is, P must be TRUE and A must be FALSE. That means if God does not exist you must pray! Not only must you pray, but your prayer must go unanswered!

Not, as Eliza Doolittle said, bloody likely.

 

Out of the Ordinary

So we can see now that the original Premise #1 has been craftily worded. When the last part of the premise is written in the "not-(IF-THEN)" form. It obscures the real meaning.

The warning sign is when you see the phrase "It is false that" before a fairly complex expression. If you do, beware! The arguer may be trying to confuse you.

So we ask. Can we select a new Premise 1? One that is more d'accord with natural language and what most people can accept as TRUE?

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

If I pray, then my prayers will not be answered.

... 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 that only if a prayer is not answered, does God not exist.

With that in mind, let's analyze the argument using this alternate Premise #1.

If God does not exist, then if I pray then 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. We'll see if it is possible to have a counterexample.

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 "If" part - that is, the antecedent - of P → ¬A. False antecedents always produce TRUE "If-Then" statements.

So the "THEN" part - the consequent or conclusion - of Premise 1 is TRUE (and regardless of whether A is TRUE or FALSE).

So if Premise 2 is TRUE, the conclusion 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 what if we do go and set G to be FALSE?

.

Well, that means Premise 1 is TRUE, Premise 2 is TRUE, and the Conclusion is FALSE. And ...

We have a COUNTEREXAMPLE!

Specifically, our counterexample (actually, counterexamples) is:

God exists (G):     FALSE
I pray (P):     FALSE
My prayers are answered (A):     TRUE or FALSE

So we have just proven:

The argument is INVALID!!!!!!!

So let's review what we've done.

As originally stated, Dorothy's Premise #1 places an unsound restriction on the concept of an all powerful deity. You can see this clearly now since in the counter example whether the prayer is answered (A) can be TRUE or FALSE. So we've revised the first premise to remove the restriction.

But most importantly, the new premise is more in accord with what we really say - and mean - in ordinary conversation. With a more realistic argument - as we expect - what looks like a strange, weird, and invalid argument is indeed a strange, weird, and invalid argument.

The Shoals of (Red) Herrings

At this point, it's worth elaborating on the point that it doesn't matter if the prayer is answered or not. And why we can eat chocolate instead of praying.

You may have noticed that the atomic sentence that we are calling A only appears once and then in the first 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 use something other than A. Let it be anything else, like X.

So substitute X for P in Premise 1 in any of our arguments. No matter what the TRUTH of FALSITY of X is, the validity or non-validity of the arguments will not change.

In other words, the A - whether it's praying or eating chocolate - is in logic-speak a red herring.

 

The Indisputable TRUTH

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:

G     God exists.
P     The Cubs will not win the World Series this year.
A     The Cubs will never have a championship team.

And let's put these sentences into an argument:

Premise 1     ¬G → P ∧ ¬A     If God does not exist, then the Cubs will not win the World Series this year and so will never have a championship team.
   
Premise 2     ¬P     The Cubs won the World Series!
   
Conclusion      G   God exists.

By now the readers will 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).

"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/