Two Truths and a Lie

Statement 1 is contradicting Statement 2.

And since 2 of the Statements are true, only 1 of the 2 statements is true.

Which means Statement 3 must be true.

And since Statement 3 said that Statement 2 is true, Statement 2 must be true as well.

So the false Statement is Statement 1.

This is regarding the bonus question you gave. I think the first one will be true and the other two will be false if two of them had to be false.

The third statement says the second statement is true, which binds the two statenments together. If one is true, the other must be true, and vice versa.

Let us try to figure out the incorrect statement, in the question.

• If statement (1) is false and the other two are true, then statement (1) is false, statement (2) is true, and statement (3) is also true. The logical system is consistent.

• If statement (2) is false and the other two are true. then statement (1) is true, statement (2) is false, and statement (3) is also true. But statement (3) states that statement (2) is true: thus, we have a logical contradiction.

• If statement (3) is false and the other two are true, then statement (1) is true, statement (2) is true, and statement (3) is false. But statement (2) is true, which implies that statement (3) is true, which is not the case in this particular system. Thus, this logical system is inconsistent.

The only consistent logical system being the one where (1) is false, the result follows.

For the bonus :

• If statement (1) is true and the other two are false, then statement (1) is true, statement (2) is false, and statement (3) is also false. The logical system is consistent.

• If statement (2) is true and the other two are false, then statement (1) is false, statement (2) is true, and statement (3) is false. But statement (2) is true, which implies that statement (3) is true, which is not the case in this particular system. Thus, this logical system is inconsistent.

• If statement (3) is true and the other two are false, then statement (1) is false, statement (2) is false and statement (3) is true. But statement (3) is true, which implies that statement (2) is true, which is not the case. Thus, this logical system is inconsistent.

Thus, the only consistent logical system is the one where (1) is true.

If the first statement is true, second and third statements are false. Therefore, first statement must be false. Then the second and third statements are true, which is the case.

1 and 2 are mutually exclusive, and 2 and 3 are equivalent.

So we have either :

• 1 is false, 2 and 3 are true (question),

• or 1 is true, 2 and 3 are false (bonus)

I predicted that 1 is false and it made 2 and 3 look true which means that 1 is false

Since the third statement states that the second statement is true, if one is false than the other one is false, and if one is true than the other is true. Since the second one states that the first one is false, if it is false, that means the first one is true, and the other way around. Since the problem states that two are true and one is false, the first one must be false.

Since Statement 1 ("This statement is true") does not refer or involve any other statement, it can be either true or false without starting something. The other two refer to each other, so that's where you have to think. It is like a chain. The third statement says the second one is true, the second one says the first one is false, and the first doesn't refer. Since two have to be true, the third and second statements won't contradict if they were true, so it is reasonable to assume the third and second statements are true.

We want to assign 'true' and 'false' to the statements in such away that there is exactly one 'false' statement.

First of all suppose #1 is false. Then simply by reading #2 we can see that #2 is true. Then simply by reading #3 we can see that #3 is also true.

At first glance this would seem to solve the problem and let us say $\boxed{\text{\#1 is the false statement}}$

However to complete the problem we need to show that there are no other solutions. So we must show that if we assume that either #2 or #3 is false, at least one of the other statements is also false.

First, suppose #2 is false, then simply by reading #1 we see that #1 is also false.

Now suppose #3 is false, then by simply reading #3 we can see that #2 is also false.

So no contradictions, paradoxes or multiple solutions arise and our solution is complete.

Statement 1 can be either true or false -- it's arbitrary, so it doesn't contribute to the truth of the overall argument. Symbolically: $(S1 \implies S1) \implies (S1 \lor \neg S1)$

Statement 3 implies statement 2 is true. Symbolically: $S3 \implies S2$

Statement 2 implies statement 1 is false and implies statement 3 is true as only two statements can be true. Symbolically: $S2 \implies \neg S1, S2 \implies S3$

Using the rules of replacement, the overall truth of the argument can be represented as: $(S2 \iff S3) \implies \neg S1$

Which is true, when statement 1 is false, statement 2 is true and statement 3 is true. Hence, statement 1 is false.

And for the bonus, only way $(S2 \iff S3) \implies \neg S1$ can be true but with 2 statements false is when only statement 1 is true and statement 2 and 3 are false.

Assume 1 is false. Then 2 is true. Then 3 is true. This checks out. Looking at the answer choices, we figure out there can only be one solution (or a paradox). Therefore 1 is the solution.

Statement 2 states that statement 1 making it being a possible answer Statement 3 states that statement 2 is true, which makes statement 1 false still, making statement 1 the answer.

For the bonus question, statement 1 would be true as statement 2 and 3 both contradict it, indicating they are both false

when one of the statements is true, Statement 1 is to be true.

I predicted that 1 is false and it made 2 and 3 look true which means that 1 is false.

same answer , wether 2 R ,1F or 2F,1R.

Take Statement 3 as true. Then 2 will be also true. If 2 is true, 1 must be false.

This statement is FALSE!

It's a paradox...