# Logically speaking, is this exam question paradoxical?

By itself the statement

Gasses show ideal behavior at high pressure and low temperature.

is false. This being the case

All statements are true.

is also false.

Since we're asked to evaluate which one of these statements is false, is this a fair question?

• @lulu I believe A & C are also true – John Dorian Feb 25 '16 at 22:40
• @K.Power C is true, we assume that molecules of ideal gases do not attract each other. – Nikunj Feb 25 '16 at 22:46
• @Nikunj Quite right, I confused myself by somehow thinking the question was asking which of these statements are not false. I've removed my garble. – K.Power Feb 25 '16 at 22:56

The intended meaning of E) is that all statements are true, of if you prefer, that no statement is false. This is what the reasonable reader understands. English is not formal logic.

It's very common on multiple-choice questions for the last choice to be "none of the above". This is such a question.

The structure of this particular question is not perfectly logical, either in mathematical terms or in plain English. But it is very clear what is meant: either one of the first four choices is a false statement, and must be chosen, or the first four statements are all true; E is the correct choice in that last case and only in that case.

You can argue this all day long with whoever designed the test, but the best you can hope for is that they'll word the question more carefully next time. I would not expect to get any credit for any answer other than the intended one.

Well, if we're being thoroughly logical...

the question states which one of the questions is false. That doesn't mean "exactly one question is false". It suggests and it but doesn't state it. It doesn't state that any question is false.

So e) All statements are true is false if there is any other statement that is false.

So as c) is false, e) is false and is a valid answer.

If all other statements were true, statement e) is unresolvable. I think logically such unresolvable statements (statement that are true if they are true and false if they are false) can be declared to be false (or true) by fiat.

So e) is a safe bet. The instructor can not argue it is true. And in fact it is false.

• The teacher can not argue it is true. s/he can (and almost certainly will) however argue that by choosing it you are arguing it is true rather than false. Pointing out that the questions stated to choose the false question and you chose it because it was false will probably not do you any good. – fleablood Feb 26 '16 at 1:42

(E) could be construed as begin not a "statement about real and ideal gases", but rather a statement about statements in general, hence not covered by the original statement.

As for fairness, that's subjective. My opinions are:

• If it's a logic exam, the question is fair. In logic, you're trained to work only with what's written explicitly, to suppress colloquial interpretation of English, to deal carefully with statements about statements.
• If it's a chemistry exam, the question is severely flawed: I disagree Yuval Filmus's response

The intended meaning of E) is that all statements are true, of if you prefer, that no statement is false. This is what the reasonable reader understands. English is not formal logic.

In English it's quite natural to introduce a qualification on a subject (statements qualified to be following and about real and ideal gases") and then omit the qualification in nearby later references, so (E) could be interpreted as "All statements (about real and ideal gases) (following the question sentence) are true". At best (E) admits multiple interpretations.

My answer would be E on the grounds that we are told that exactly one of the statements is false, and therefore E cannot be true,so it must be the false one.The matter that E is a sentence about sentences, including itself, does not in this case render it paradoxical.

• The question doesn't say exactly one, but rather asks to point out which ones are false. – Pedro Tamaroff Feb 26 '16 at 1:23
• It says "Which ONE of the..." not "which ones." In mathematical English "which one" always means that there is exactly one. – DanielWainfleet Feb 26 '16 at 1:47