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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?

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    $\begingroup$ Um, any introductory proof course or textbook will heavily cover propositions, contradictions, tautologies, truth tables, etc. $\endgroup$
    – Jonathan Hebert
    Commented Sep 6, 2016 at 13:32
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    $\begingroup$ An this falls under none of those rubrics. $\endgroup$
    – Rene Schipperus
    Commented Sep 6, 2016 at 13:36
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    $\begingroup$ I would change the "not this one" to "namely this one" -- that is, your analysis shows that the given statement must be false, so the easiest way to make it false is to make it the only false statement. $\endgroup$
    – Barry Cipra
    Commented Sep 6, 2016 at 13:37
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    $\begingroup$ @ReneSchipperus this is a question about logic. How is this not appropriate for math.SE? $\endgroup$
    – user307169
    Commented Sep 6, 2016 at 13:41
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    $\begingroup$ I would phrase your supposition as "exactly one statement in the book other than this one,is false." $\endgroup$
    – David K
    Commented Sep 6, 2016 at 13:51

2 Answers 2

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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.

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  • $\begingroup$ You are correct! I misread the statement. But how does this solve things? $\endgroup$
    – Jonathan Hebert
    Commented Sep 6, 2016 at 13:40
  • $\begingroup$ I count two false statements in this exercise, so if you find a third one somewhere then $(j)$ is a paradox. $\endgroup$
    – David K
    Commented Sep 6, 2016 at 13:55
  • $\begingroup$ @DavidK, nm, I get it now. Maybe that is how the problem is supposed to be done. I count 3 false statements though (b, c, d). $\endgroup$
    – user307169
    Commented Sep 6, 2016 at 14:20
  • $\begingroup$ @JonathanHebert, I updated my answer. $\endgroup$
    – user307169
    Commented Sep 6, 2016 at 14:28
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    $\begingroup$ @JonathanHebert, I dislike that convention but that's obviously moot. In any case it must be easy to find 3 (and actually more, making the problem even easier) false statements in the book. I think the problem does have to be done that way, i.e., it needs to be shown that my #1 above can't be true. If we do it in an "isolated" setting, then suppose (j) is the only false statement in the book. We already know #2 when true leads to a paradox. So we must have #1 true. But if (j) is the only false statement in the book then #1 is certainly true and there is no paradox. $\endgroup$
    – user307169
    Commented Sep 6, 2016 at 14:40
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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.)

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