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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 times 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 which was first published by the British philosopher Dorothy Edgington in 1995.

Let's hasten to say that Dorothy did not claim that 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 particularly when they are based on the infamous "If-Then" statements.

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 argument is balderdash, horse hockey, and bullshine. How can it 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 the English sentences symbolically. That is, we use symbols to represent the individual parts of the arguments. And the symbols are combined into formulas which represent the English in an unambiguous and precise manner. If this doesn't make sense, it 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" constructions.

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". This is called the Inclusive OR as opposed to the Exclusive OR which means one of the the two but not both. After a while the Inclusive OR becomes easy to use and understand.

On the other hand, the "If-Then" Table 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

We won't go into details of all the wild and whacky sentence that you can construct based on this table. Instead we'll just point out that the best way to understand the underlying logic 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

We can prove this by showing that a Truth Table for"Not-Or" is the same as a Truth Table for "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 our ontological 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 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.

 

And now, as the psychiatrist said, we can begin, yes?

 

The Battles of Sir Validahad

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

A valid argument is simply 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 is FALSE.

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, first we'll start off with the simplest premise which is Premise #2.

¬P

But if ¬P is TRUE, then we know that:

P

... must 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

... has to be FALSE.

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

G

... can only be TRUE.

And G is the conclusion.

OK. What have we done?

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

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. 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. But most importantly, 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. Two arguments that can be represented with the same symbols and have the same structure are called isomorphic. If you prove an 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! We can show this 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?

Theologically Correct?

One thing you might be thinking is what a minister once said before his nightly sermonette. "God answers all prayers," he quipped. "But some people can't just take "No" for an answer."

If we accept the Good Reverend's words, then we see our original argument is flawed. That is, it's wrong to say God does not answer all prayers.

Hearing this objection, though, will make philosophy professors smile behind their hands. They'll say, well, that may be true regarding beliefs of one particular creed. But we are testing the validity of the argument as stated. In logic we are permitted to assign the values TRUE or FALSE to each atomic sentence, including A. And by doing so we have shown the argument is VALID.

The minister - who took a logic course at the seminary - does not disagree. But he insists that his revised argument is more in keeping with ordinary thinking. And formulating the new argument is easy. We simply add another premise.

And the new premise is "All prayers are answered." That is, we stick A after Premise 2. That is the argument we should analyze.

Premise 1     ¬G → ¬(P → A)     If God does not exist, then it is false that if I pray then my prayers will be answered.
 
Premise 2     ¬P     I pray.
 
Premise 3     A     All prayers are answered.
 
Conclusion      G   God exists.

So is this argument valid?

Well, let's see.

If God always answers all prayers, then Premise 3, A, like all premises, is always TRUE.

Since A is always TRUE, the "If-Then" Truth Table tells us the sentence P → A must also be TRUE.

That in turn means that ¬(P → A) is always FALSE.

But note that ¬(P → A) is the consequent of Premise 1. So the only way Premise 1 can be TRUE is to have ¬G be FALSE.

And if ¬G is FALSE, then G must be TRUE.

And G is the CONCLUSION OF OUR NEW ARGUMENT!

And this means....?

It means we've proven that if all the premises are TRUE, then the conclusion must also be TRUE! And so here are no counterexamples.

Therefore even if we accept that God answers all prayers, our oddball argument is still VALID.

Once more we have to ask, what the hey is going on?

A Minor Change

Now you may have noticed the phrasing of the original 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.

In fact, 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 simpler. 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 original Premise #1:

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

... which is 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

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 for 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!

But things get even stranger if you again look at the Premise in symbolic form.

¬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. In other words, if God does not exist not only must all your prayers go unanswered, but you still have to pray!

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 in a way that obscures the real meaning.

The warning sign is the phrase "It is false that" placed before a fairly complex expression. If you see that, beware! The arguer may be trying to pull a fast one.

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 #1.

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

... which is symbolized as ...

P → ¬A

Not only is this more like what we say in ordinary conversation, but it removes the artificial restriction that only if a prayer is not answered, then God does 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.

Remember if a argument is valid then if the premises are TRUE the conclusion must be TRUE.

Then that means 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.

But a TRUE consequent means the entire "If-Then" statement is also TRUE. So Premise 1 is TRUE.

So if Premise 2 is TRUE, Premise 1 is TRUE.

Now note one little thing.

For Premise #1 to be TRUE, all that was required is for Premise 2 also be TRUE. That is, ¬P must be TRUE or equivalently, P must be FALSE.

But this means - and pardon us if we shout:

G or A CAN BE TRUE OR FALSE!

So what if we do get if we 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

And 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. But this new argument does not have this problem since in the counterexample the prayer be answered or not as the pleasure of the deity wishes.

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

The Shoals of (Red) Herrings

You'll remember that we also proved our original argument is valid even if the prayers are always answered. At this point, it's worth exploring this point further. And in doing so we'll see why 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.

Hm. This does seem strange. And what does this tell us about our premise?

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 of the arguments does 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 it is false that if Cubs will not win the World Series this year then they 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/