Can Logic Exist Without Truth?

[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

 

[nnnnnnnn]

it's false

 

[Galileo]

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]

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]

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]

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 don't 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...

 

[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.

 

[quark]

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##

 

[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.

 

[xJuggleboy]

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 statement 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.

 

[Rogerio]

The Galileu's reply finishes any discussion:

The first statement cannot be true, therefore the third one is true.

 

[closure]

My answer is "true"

 

[AKG]

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?

 

[tsberry901]

Hint # 1

On what premise is logic based?

 

[alias25]

its true because your saying if 1 is true then 2 is false, let's look at one it says: the following statement is true, (if this is true) then the second statement says: the previous statement is false (this must be false) as you have defined statement 1 to be true.

 

[alias25]

you know what don't listen to what i wrote its pobabaly garbage,
noo wait, because if 1. (the following statement is true) is true it implies 2.( the previous statement it false) is true and 2. implies 1. (the following statement is true) is false, then it would be true to say, (the following statement is false) is true this implies 2. (the previous statement is false) is infact false, and since this is false then the previous statement is infact true. so if 1. is true the 2. is false so 3. is true.

 

[Psi 5]

This is a so-called paradox (no such thing as a paradox). It's no different than if I said red is blue and asked if red is blue or red? The given facts conflict so any meaningful answer can't be had because the question doesn't mean anything. Everything I said is a lie except for the last sentence.

 

[tsberry901]

Hint # 2

Logic cannot answer everything. What are it's limitations?

 

[geniusprahar_21]

its true...

 

[Ki Man]

the answer is no

thats like saying: this statement is false.

its an infinitely self-contradicting paradox

 

[tsberry901]

Final Hint:

When is a statement true?

 

[Jimmy Snyder]

Sorry is this has already been posted by someone else:


Whatever the first statement is, it isn't true. For if it were true then it affirms that it is false. Therefor, statement 3 is a conditional in which the premise is not true. Therefor it is a true statement.

 

[DrKareem]

It's a statement that can't be proven to be neither true nor false. This is Godel's incompleteness theorem, which states that in a consistant system, you can construct statements that can't be proved or refuted, thus resulting in a 'paradox'. It's just like the 'All what I'm saying is false' paradox.