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26. God Exists and the Moon is Made of Cheese:

Curry’s Paradox

The Question:

If this statement is correct, the moon is made of cheese.

If this statement is correct, Germany borders on China.

If this statement is correct, 12 is a prime number.

If this statement is correct, 12 is not a prime number.

If this statement is correct, God exists.

Are these assertions true?

 

The Paradox:

Surprisingly, yes!

In formal logic, a conditional assertion (A) is a statement of the form “if the antecedent (B) is true, it follows that the consequence (C) is true”. Let us take, for example, the conditional assertion “if it rains, then the street is wet”. If this assertion is true, and if the antecedent (“it rains”) is true, then the consequence (“the street is wet”) is true. Note that the assertion does not claim “only if it rains, is the street wet”. The street may also be wet if it does not rain. After all, someone could have spilled water onto the street or a dog may have peed onto the pavement.

Therefore, by the rules of formal logic the entire conditional assertion is true even if the antecedent is false. This means that the assertion “if it does not rain, then the street is wet” is also true. So, in addition to the assertion “if it rains, the street is wet”, the two other assertions -- “if it does not rain, the street is wet” and “if it does not rain, the street is not wet” – are also true.

In summary, even if the antecedent (B) is false (i.e., it does not rain), and regardless of whether the consequence (C) is true or not (i.e., the street may or may not be wet), the entire assertion (A) itself is still true. The only assertion that is definitely false in this context is “if it rains, the street is not wet.” This accords with intuition since the street must be wet if it rains.

* * *

The problem with the conditional assertions posed at the beginning of this chapter (e.g., “if this statement is correct, 12 is a prime number”) is that the antecedent (B: “if this statement is correct”) refers to the entire assertion (A: “if this statement is correct, 12 is a prime number”). We are once again confronted with a vexing case of self-reference since the word ‘statement’ in the antecedent refers to the entire assertion itself.

Let’s analyze the conditional assertions “if this statement is correct, 12 is a prime number” in detail. We know, of course, that 12 is not a prime number since it can be divided by 1, 2, 3, 4, 6 and 12, but let’s see what happens

First, let us assume that the assertion (A) is true. Since the assertion and the antecedent are one and the same, this is tantamount to saying that the assertion’s antecedent (B) is true. Now, since both the assertion and the antecedent are true, the consequence – “12 is a prime number” – is also true. Hmmm!

Second, let us assume that the assertion (A) is false. Again, since the assertion and the antecedent are one and the same, this is tantamount to saying that the assertion’s antecedent (B) is false. Now comes the crucial point: as explained above, even if the antecedent is false, the conditional assertion remains true. (Oh, the joys of those who trust self-reference: this second assumption, that the assertion (A) is false, is false.)

From here on, it’s easy going. The fact that the assertion (A) is true implies that the antecedent (B) is also true since they are one and the same thing. And since both the assertion and the antecedent are true, the consequence (C) is also true. Hence “12 is a prime number”. Wow!

Whether we start out by assuming that the assertion is true, or that it is false, the implication in both cases is that 12 is a prime number. Similarly, the moon is made of cheese. And Germany borders China. And 12 is not a prime number. And God exists.

 

Background:

The paradox is named after Haskell Curry (1900-1982), an American logician who obtained his PhD in 1930 in Göttingen, Germany, from the then undisputed highpriest of mathematics, David Hilbert. Haskell Curry must not to be confused with the New York magician Paul Curry’s (1917-1986) who invented the ‘missing square puzzle’, a geometrical riddle often also referred to as Curry’s Paradox, which turns out to be simply an optical illusion.  

Our Curry’s Paradox is one more example of self-reference paradoxes.

 

Dénouement:

The crux of the matter is that the assertion ‘if this statement is true…’ does not make clear what is meant by the word statement. If this statement refers to the sentence ‘if this statement is true, then 12 is a prime number’, then…

‘if, “if this statement is true, then 12 is a prime number’ is true”, then 12 is a prime number.’

And this means that…

‘if, “if, ‘“if this statement is true, then 12 is a prime number’” is true, then 12 is a prime number”, is true, then 12 is a prime number.’

And so on, and so on. Hence, ‘this statement’ does not refer to an actual statement, but to an infinite recursion of statements and is, therefore, not defined. So how can a statement that is not defined be true? It cannot. It makes no sense to claim the veracity of statements that are not defined!

 

Technical supplement:

This paradox is especially vexing since it seems to show that any statement one can think of can be proved to be true. ‘God exists’ can be shown to be true, as can ‘God does not exist’. Ludicrous statements like ‘Germany is made of cheese’ and ‘the moon borders China’ can be proved. The number 12 can be revealed – to paraphrase King Hamlet – to be and not to be a prime number.

The disconcerting conclusion would be that in contrast to a certain American president’s conviction that ‘all is fake’, Curry’s Paradox is able to show that all is true. True…but truly ridiculous! So, to quote this president again, ‘what the hell is going on?’ Well, what’s going on is, as pointed out, that ‘the statement’ is not defined.

Comments, corrections, observations: