Is this statement actually a paradox?

Question

I'm grading assignments for an introductory proof class, and the following exercise is in the book:

"Are the following propositions? If so, give their truth value."

"There are more than three false statements in this book, and this statement is one of them."

The answer key says that this is a paradox, so it is not a proposition.

Now, obviously we can't be expected to actually know the truth value, but why is this a paradox? Suppose that exactly one statement in the book other than this one is false. Then the problem statement is false and there seems to be no paradox. Thanks to the conjuction, this statement can be false without contradicting itself.

The statement cannot possibly be true, however, because then it must contradict itself and it is a paradox in that case. A "conditional" paradox?

Answer 1

Are you sure it doesn't say more than three statements? Your analysis of the problem as you've written it is correct as far as I can tell. Could be a typo in the book or the key then.

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Re: your question about how this solves things.

Suppose the statement is true. Then the book has more than 3 false statements and this statement is one of them. Therefore the statement is false (because of the and this statement is one of them). Paradox.

Suppose statement (j) is false. Then at least one of the following must be true:

1. The book has 3 or fewer false statements.
2. Statement (j) is not one of them.

Suppose #2 is true. Then statement (j) is true. Paradox.

Suppose #1 is true. (Thanks @DavidK for the hint.) We already know that statements (b), (c), and (d) in exercise 1 are false.

• (b) simplifies to "It is the case that $\pi$ is a rational number," which is false.
• (c) counterexample: $x = 2\pi$.
• (d) counterexample: $x = 0$ and $y = i$.

So (b), (c), and (d) are already known to be false, and since we're assuming that (j) is false, then we have 4 false statements in the book. This makes #1 false. Paradox.

Answer 2

In order for a statement to be a proposition is must be either true or false(, but not both).

If this statement were true then there must be three incorrect statements in the text. However, that would mean that there are exactly two other incorrect statements since the statement claims to be false. However, since it is true and there are only two wrong statements it must be false. This is a contradiction.

It this statement were false then there are not three incorrect statements or this statement is true. So, by the the excluded middle, this is equivalent to there not being three incorrect statements.

A conceivable situation, as you tried to get at, might be that this is the only false statement. (Note that your assumption that there is exactly one wrong statement except this one is incorrect by the analysis given above.)