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

Valid, Si!
Convincing, No!

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.

First consider the following argument. It is a slight recasting of an argument that was first published by philosopher and linguist 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"

We won't go into detail of how to derive the Truth Tables for the connective. But 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

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 God does not answers my prayers.

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

 

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

We start off 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. But note that the "Then" part of Premise 1 is the "AND" statement, P ∧ ¬A. This is FALSE if even one of the statements is FALSE. Since we just determined P is FALSE, then P ∧ ¬A is 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!

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 in their faces. 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 nevertheless still gain weight.
   
Premise 2     ¬P     I eat chocolate.
   
Conclusion      G   God exists.

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.

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 with the same structure - or shape - are called isomorphic. If you prove an argument is valid, then you've proven all its isomorphic arguments are valid as well.

What surprisese some students is that a logical argument can also be used to prove the exact opposite of what it 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     God will not answer my prayers.

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 God answers my prayers.
 
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?

Actually these arguments - although completely valid - are 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.

But that's not all. We can also prove that P can also be anything.

There are various ways to show this. But we'll use a way that at first seems unnecessarily laborious.

Consider if you were an old time rhetorician from Athens. You would stand in the stoa and say something like, "If we have Premise 1 and Premise 2 and Premise 3, then we arrive at the Conclusion." So you can see we can present a logical argument - no matter how long - as a single sentence.

In symbolic logic, we can also have such one-sentence arguments. We create these by connecting the premises with the AND connector, ∧, and then have the connected premises imply the conclusion.

That is, for a set of premises, G1, G2, G3, G4, ..., Gn and conclusions C we write:

G1 ∧ G1 ∧ ... ∧ Gn → C

So how do we use this form of argument to determine if the argument is valid or invalid?

We mentioned that if the argument is invalid, we will have a counterexample. So we can assign all the premises to be TRUE and the conclusion will be FALSE.

But if we have a counterexample, that means in our one-line argument our antecedent - the conjunction of all our premises - will be TRUE, and the consequent - the conclusion - will be FALSE.

We also know that if you have a TRUE consequent and a FALSE conclusion, we know the whole "If-Then" statement is FALSE.

So if our argument is invalid, this "If-Then" form will always be FALSE.

Conversely, if our argument is valid, the whole "If-Then" statement must always be TRUE!

In other words, if you express a valid argument in the one-line "If-Then" form, the "If-Then" statement will always be TRUE. A statement which is always TRUE is called a tautology.

So if the one-line "If-Then" form of an argument is a tautology, the argument is valid.

We can show that our original argument is a tautology by manipulating the symbols according to the rules of logic. Tautaologies always end up being reduced into a statement that can never be false no matter whether the individual parts are TRUE or FALSE. Admittedly it will probably be helpful to consult a logic textbook or website for some details, but everything we're doing below is legitimate:

So we start off with our argument as an "If-Then" statement:

([¬G → (P ∧ ¬A)] ∧ ¬P) G
Conjunction   Conclusion

... and off we go:

1

([¬G → (P ∧ ¬A)] ∧ ¬P) → G     Conjunction of the premises imply the conclusion.
   
2

([¬¬G  (P ∧ ¬A)] ∧ ¬P) → G     "If-Then" = "Not-Or"
   
3

([G ∨ (P ∧ ¬A)] ∧ ¬P) → G     Even number of "Nots" cancel.
   
4

([G ¬P] [(P ¬A) ∧ ¬P]) → G     (A ∨ B) ∧ C = (A ∧ C) ∨ (B ∧ C)
De Morgan's Laws
   
5

([G ∧ ¬P] ∨ [P ∧ ¬P ∧ ¬A]) → G     Remove parentheses and rearrange - (A ∧ B) ∧ C = A ∧ C ∧ B
   
6

([G ∧ ¬P] ∨ [FALSE ∧ ¬A]) → G     Contradiction
A ∧ ¬A = FALSE.
   
7

([(G ∧ ¬P) ∨ FALSE] → G     Conjunction with FALSE
A ∧ FALSE = FALSE.
   
8

(G ∧ ¬P) → G     Disjunction with FALSE - A ∨ FALSE = A.
   
9

¬(G ∧ ¬P) G     "If-Then" = "Not-Or"
   
10

(¬G ∨ ¬¬P) ∨ G     ¬(A ∨ B) = ¬A ∧ ¬B
   
11

(¬G ∨ P) ∨ G     Double "Nots" Cancel
¬¬A = A
   
12

¬G ∨ G ∨ P     Remove Parentheses with Identical Connectives and Rearrange
   
13

TRUE ∨ P     ¬A ∨ A = TRUE
   
14

TRUE     TRUE ∨ A = TRUE

So our one line argument is a tautology, and we once again prove our argument is valid.

This method, though, does take a lot of effort - more than we used before. But now we'll take a step back and show that this type of transformation can help us make sense of the original argument and why our original strange argument is really no big deal.

Note as we proceeded with our transformations, we went through what we called Step 8. That is:

G ∧ ¬P → G

Notice that this is an "If-Then" statement where the "If" part is a conjunction - an AND statement () - and the "Then" part is just another sentence.

In other words, it is an argument with premises and a conclusion:

  G
¬P
-------
∴G

So compare this to the orignal:

Original
Argument
    Transformed
Argument
       
  ¬G → P ∧ ¬A       G
¬P     ¬P
-------     -------
∴G     ∴G

... and what we see it is veeeerrrrrryyyyyy interesting - to quote Artie Johnson.

We started off with a complex argument. Then we show it can be transformed to what is the same argument except the first premise is replaced by its own conclusion!

Which when we put it back into English, we get:

God exists.
 
I do not pray.
-------
Therefore, God exists.

Not a very impressive augment. And yes, you will get the same argument if we start out with the classic arguments like Modus Tollens. The ability to derive such simplistic arguments is what prompted one well-known college professor of the History of Science to point out that conclusions from classic syllogisms are valid, yes, but trivially valid.

You can get an idea of just how trivial, even bordering on pointless, when we consider the following isomorphic arguments:

George Washington lived at Mount Vernon.
 
Napoleon lived in France.
-------
Therefore, George Washington lived at Mount Vernon.

... which is, of course, perfectly valid.

Or we can write:

Preparation H shrinks hemorrhoids without surgery.
 
Drinking and driving don't mix.
-------
Therefore, Preparation H shrinks hemorrhoids without surgery.

... both of which are as valid as our original argument.

At this point you may be thinking we're trying to pull a fast one. Because all valid arguments which are put into the "If-Then" form are tautologies, doesn't that mean we can ultimately derive any other valid argument from another?

A → B ⇨ ⇨ TRUE
TRUE ⇨ ⇨ X → Y

Well, strictly speaking, yes. But because all valid arguments have equal Truth Values, that doesn't mean all arguments are the same. But what we did see that in Premise #1 is the Conclusion, if Premise #1 and Premise #2 are both TRUE, Premise #1 is automatically TRUE.

But more importantly, you could replace either Premise #2 or the Conclusion with their own negations - and the argument would still be valid! In essence even Premise 2 is irrelevant. That is, the argument collapses to:

  G
-------
∴G

So our arguments are equivalent to assuming G to be TRUE and then concluding G is TRUE! So no matter how witty, clever, and amusing our arguments are, ultimately they are circular.

This is all well and good. But none of this brouhaha answered the question: How can all of the original logical argument of Dorothy's which should be invalid nevertheless be valid?

Unless logic is - as we said - balderdash, horse hockey, and bullshine.

Weeeeellllll, we have to say that honesty compels us to point out that we modified Dorothy's original argument a wee bit. Notice we said that "If God does not exist then I pray and my prayers are not answered."

What this does is to make the fact whether my prayers or answered or not independent of my act of praying. That allowed us to do the trick where the second premise being TRUE forces the conclusion to be TRUE as well.

What we need, though, is to make the prayers being answered or not a consequence of whether I prayed or not. So in Premise 1, we need the consequent itself to be an "If-Then" statement.

And indeed that is what Dorothy did - but she did so in a way that the "If-Then" form of the consequent is really a disguised "AND" statement. On the other hand if you restate the first premise in a way that fits better with natural language, the argument can then shown to be invalid. But with the nature of this already overlong essay, we relegate this proof to an appendix which you can read if you click here

So to explain our apparently paradoxical argument, what we have done is simply taken a simple well known argument and slipped in an irrelevancy that has no effect on the original validity. We then construct the symbolic argument to force the conclusion to be TRUE as long as one of the premises is TRUE. The prevents construction of counterarguments, and we have a valid ontological proof.

But 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

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/

 

Appendix

Our rendering of Dorothy's proof isn't exactly as she wrote it. We rewrote it to make it a bit more difficult to refute.

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

So we take Dorothy's original Premise and see what we get:

1     ¬(P → A)     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!

Now if you get a "Not" before a parenthesis, be warned! This might be a deliberate attempt to confusticate the meaning. So what you should do is rephrase the argument in a more simple manner.

Certainly a Premise 1 which fits natural language better is:

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

... 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 valid? Or rather, is there a counterexample?

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

So Premise 2, ¬P is TRUE, and hence P is FALSE.

But P is also the antecedent of the consequent of Premise 1. And 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 conclusions is always TRUE.

So as long as Premise 2 is TRUE, Premise 1 is always TRUE - regardless of the value of ¬G or G.

But G is also the conclusion of the argument. So we can simply set G as FALSE and Premise 2 as TRUE, and we have all premises TRUE and the conclusion FALSE.

That means by using symbolism that makes more sense in natural language our argument:

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

I do not pray.

Therefore God exists.

...then we have just proven:

The argument is INVALID!!!!!!!

How about that?

(Click here to return the essay.)