Look At You, Now Look At Us

Logic Level 2

If this statement is true, then the other statement is also true.
If this statement is false, then the other statement is also false.

How many of the statements above is/are true?

0 1 2 This is an impossible scenario

3 solutions

Ivan Koswara
Jul 22, 2016

A statement in the form "if this statement is true, then P" always implies that the statement itself is true (and thus P is also true). To see this, note that the only way the statement can be false is if the antecedent "this statement is true" is true, and the consequent "P" is false. But "this statement is true" being true contradicts the fact that the statement should be false. Thus the statement cannot be false, and by law of excluded middle, it must be true.

Thus the first statement alone is enough to determine that both statements are indeed true, so there are two statements that are true.

To see that this is consistent with the second statement, since the second statement is true, the antecedent "this statement is false" is false. Thus the second statement is indeed true (because the premise is false; this is known as vacuous truth). Thus this assignment is indeed consistent, not causing a paradox.

You said: "A statement in the form 'if this statement is true, then P' always implies that the statement itself is true"

And what if I say the following?:

1) "if this statement is true, then 'I am the Super Man' "

or better...

2) "if this statement is true, then 'this statement is false'"

It still implies that the statement itself is true?

It looks like an absurd, no?

Claudio Augusto Rolim - 4 years, 8 months ago

[1] First statement = true
[2] Second statement = impossible scenario

See the explanation here , here and here .

Pi Han Goh - 4 years, 8 months ago

@Pi Han Goh , actually it is exactly what I think, thanks! The answer can be either true or "impossible scenario".

But I disagree that implies that the statement itself is true (because it can be "impossible scenario" as well, like the [2] you said above!).

So, in my opinion, in this exercise, either "2" or "This is an impossible scenario" are valid answers. Don't you think so?

Claudio Augusto Rolim - 4 years, 8 months ago

@Claudio Augusto Rolim – Please explain why the answer to this question can also be "this is an impossible scenario".

Pi Han Goh - 4 years, 8 months ago

@Pi Han Goh – I think I got confuse in the comment above, now I'm understanding better and I seeing that I made a mistake above! Thanks for helping me and explaining :)

Claudio Augusto Rolim - 4 years, 8 months ago

Thank you for your solution! =D =D

Pi Han Goh - 4 years, 10 months ago

Vacuous truth may be right in contemporany logic. To me , I never found an explanation which is right enough and well justified for why a vacuous truth is to be considered sound.

It's in my opinion merely an artifice which lacks sense done in mathematics. As such if the antecedent is false in S1 it would lack sense anyway.I never licked problems which sue that "vacuous truth" argument for a solution. They seem to me to be accepted by people because they are officialy accepted and since I anyway don't understand it makes me feel that people don't know what they are talking about when they are speaking about "vacuous truth" anyway.

A A - 4 years, 10 months ago

Tom Verhoeff
Nov 10, 2016

Because "A => B" is equivalent to "not A or B" and a statement is either True or False (I would assume in this setting), the first statement translates into "Statement 1 is False or Statement 2 is True", and the second into "Statement 2 is True or Statement 1 is False".

Since "or" is commutative, the statements are identical and have the same truth value. If that truth value would be False, then (by De Morgan), we would have Statement 1 is True and Statement 2 is False. This implies that the truth values of the two statements differ, which contradicts their equality. Hence, both statements are True. And this is indeed a consistent scenario.

Romain Milon
Jul 16, 2016

If the first statement is true, then the second is too. If the second is true, then nothing happens. Thus, we have two statements true.

Furthermore, if the second statement is false, then what happens is too and the first isn't false (do not conclure that the first is necessarily true !).

I would argue either 1 or 2 is correct. To see why, let's step through this.

If this statement is true, then the other statement is also true.
If this statement is false, then the other statement is also false.

Working with statement 2 first, let's assume it is false, which then triggers the conditional. But since it is false, that means, 1 would be true. But if 1 is true, then by applying 1, 2 must also be true. This is a contradiction.

From this, we know 2 must be TRUE.

Now, if statement 1 is false, then, the conditional is meaningless, (it becomes: "If this statement is true, which it isn't, but if it were, then the other statement would be true.") So it is a conditional that is never triggered, so we cannot determine its validity, maybe it would have caused the other statement to be true in some alternate universe, we will not know because it is not the case. So, there is no direct contradiction.

If statement 1 is true, then statement 2 is true which has already been concluded. Again, no contradiction.

So, Statement 1 could either be TRUE or FALSE, but Statement 2 must be TRUE.

The answer is Cannot be determined, 1 or 2 are both valid scenarios.

Thoughts?

Kyle Manley - 4 years, 10 months ago

Well , according to the accepted truth table of the implciation your reasoning anyway is wrong explained by Koswara's solution for understanding anyway. That because the statement P implies false just when P true and Q false and right for all other possibilities , even when P's truth value is false!

The statement "if this statement is true then ...." can't be false as it results in a contradiction so it must be true and because of that the 2 statement are true. Nonetheless your reasoning that the implciation would be meaningless for such truth values in which statement 1 is false is right by my reasoning too anyway.

A A - 4 years, 10 months ago

Look At You, Now Look At Us

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