# Logic Lovers

#### tsberry901

1. The following statement is true:
2. The previous statement is false.
3. If the first statement is true, then the second statement is false.
4. Is the third statement true or false?

Answer to be published in the future

Related General Discussion News on Phys.org

it's false

Science Advisor
Homework Helper
It's true...

#### nnnnnnnn

What about: If the first statement is false, then the second statement is true.

#### nnnnnnnn

The answer to the original question is 'no'.

#### Galileo

Science Advisor
Homework Helper
The first statement cannot be true, therefore the third one is true.

#### gnpatterson

my paradox detector went off, so I assume this is one.

#### KC9FVV

I think it's true...

#### cefarix

tsberry901 said:
1. The following statement is true:
2. The previous statement is false.
3. If the first statement is true, then the second statement is false.
4. Is the third statement true or false?

Answer to be published in the future
The third statement is "If the first statement...". It has no conditions on it so its true in that sense.

#### LarrrSDonald

1. The following statement is true:
2. The previous statement is false.
3. If the first statement is true, then the second statement is false.
4. Is the third statement true or false?
I'd say it's false. If the first statement is true, then so is the second - that's the whole first statement. If the seconds statement wasn't reveiled then this would be a no brainer. With the second statement as it stands, the first and second are paradoxal and neither true nor false (0.5 if one is allowed to be fuzzy) but this doesn't really affect the third statement, if the first statement is true then the second is too and the third states the opposite.

What about: If the first statement is false, then the second statement is true.
Also false, if the first statement is false, then the second is false. Again, neither of them actually is due to the paradox, but if it were false (which it isn't) then the second would be as well by it's definition.

#### AKG

Science Advisor
Homework Helper
The first two sentences essentially create the Liar Paradox. One way to deal with the Liar Paradox is to say that it is meaningless, and since the third sentence makes reference to sentences one and two, you could argue that the third sentence is meaningless, meaning that the answer to the fourth sentence is neither: the sentence is neither true nor false because it is meaningless.

On the other hand, if you ignore the paradox, then you can basically say that either both of the first two sentences are true, both are false, only the first is true, or only the second is true. In all four of these cases, you will be able to deduce the third sentence. If we call sentence one A and sentence two B, then we have:

A --> B
B --> ~A

and we want to say whether the following is true or false:

A --> ~B

If A is false, then A implies anything, so A --> ~B. If A is true, then if B is also true, we have A --> B and B --> ~A, so we get both A and ~A, a contradiction. Seeing as how we have a contradiction, we can derive anything, in particular, we can derive A --> ~B. Finally, if A is true and B is false, then we naturally have A --> ~B.

#### vikasj007

its neither true nor false.........
these statements dont correlate with each other......
if u try working them out u'll find that if the first is true, then second is true, thus first is false, but we assumed that first is true.
if first is assumed to be false then second is false, thus first is true, but is assumed to be false.

so, any ways we take it, the third statement does not satisfy the conditions of either being true or false.........

#### LarrrSDonald

If one is allowed to go fuzzy and have partial truths or falsehoods, the answer is clearly true:

B=A
A=1-B

Is B=1-A?

In the classic liar paradox, both statements are 50% true (as can clearly be seen, nothing else would satisfy it and it can be trivially solved, B=1-B => 2B=1 => B=1/2) should partially true statements be allowed. Thus, A=1-B is 100% true.

I realize that this was probably intended to be a classic rather then fuzzy problem, but I'll mention it anyway as an interesting sidebar.

#### Mr. dude

I think it's false........i think.........i think...... i think................

J

#### jammieg

logic lovers turned logic lunatics

Interesting puzzle, but it would be very kind to define "true" and "false" and "following" without a clear defintion anything is possible within ambiguity.

It is true

#### lueffy

The third statement is a meta statement, you can find something like this in "What the tortoise said to achilles" by Lewis Caroll.......
It's a fun time to think about this paradox, though.......

#### Xargoth

1- The following statement is true

2 - The previous statement is false

3- If the first statement is true, then the second statement is false

If clause failure; The first statement is not True, thus the second part of the statement is irrelevant. -Requirements not met-

4- Is the third statement true or false?

Corrupted Line; 3 :

The statement doesn't support the first statement as False, IF clause fails to answer anything And Line 4 asks something unknown..

ERROR##

Last edited:

#### El Hombre Invisible

Statement 3 is false since it contradicts the 1st statement which states the 2nd statement as being true.

#### croxbearer

I think this problem is similar to two people being interrogated for a crime. And the second one, by mistake, say something which could make him jailed, and saved the first one.

So here goes the dialogue: the first person says," The second one is telling the truth." Then, the second person says, " The first one is lying!" With these statements, we are sure that one of them is lying, and one is telling the truth because they are contradicting with each other.

And since we have a conditional statement that if the first person is telling the truth, then we can conclude that the second one is lying.

Going back to the problem, so I would say that statement 3 is true! #### El Hombre Invisible

Statement 3 is that if statement 1 is true then statement 2 is false. Since statement 2 is that the previous statement (1) is false, then statement 2 really should be that statement 1 is true. Since statement 1 states that statement 2 is true, and (if statement 3 is true) we've just found that statement 2 is actually false, then you have a contradiction. This contradiction comes from the premise that if statement 1 is true than statement 2 is false, therefore the premise is wrong, therefore statement 3 is not true.

X

#### xJuggleboy

tsberry901 said:
1. The following statement is true:
2. The previous statement is false.
3. If the first statement is true, then the second statement is false.
4. Is the third statement true or false?

Answer to be published in the future

3 is a simple If Then statement.

If the first statment is true (wich it may or not be) and the seccond statement contridicts the first (wich it does), then the third statement must be true.

Last edited by a moderator:

#### Rogerio

The Galileu's reply finishes any discussion:

Galileo said:
The first statement cannot be true, therefore the third one is true. C

#### closure

My answer is "true"

#### AKG

Science Advisor
Homework Helper
We can derive that statement 3 is false, and we can derive that it is true. The third sentence says 1 --> ~2, this is equivalent to ~1 v ~2. By the law of excluded middle, we have 1 v ~1. From ~1 we can derive ~1 v ~2. From 1 we can derive 2, from which we derive ~1, from which we derive ~2. ~2 gives us ~1 v ~2, so can indeed conclude 1 --> ~2. On the contrary, we can prove that 1 --> ~2 is false. Again 1 --> ~2 is ~1 v ~2. From 1 we can derive 2, from which we derive ~1, a contradiction. From ~2 we derive 1, from which we derive 2, a contradiction. So ~1 v ~2 results in contradiction, and thus 1 --> ~2 is false. Applying these deduction rules naively, we would have to answer "both" to the given question. This problem is just a more complicated way of asking whether P is true or false given P <--> ~P. In other words, we're just dealing with the Liar Paradox. Is it both true and false? Or neither? Is it even meaningful? Does it even have a truth-value, and if so, is it one of the "classical" truth values, or some other value?

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving