Paradox of the liar

**bthizle1** · December 22, 2008, 03:41 PM

statement 1: statement 2 is false.

statement 2: statement 1 is true.

case 1 (assuming 1 is true):

If 1 is true, then 2 is false, then 1 is false, then 2 is true.

case 2 (assuming 1 is false):

if 1 is false, then 2 is true, then 1 is true, then 2 is false.

So, 1 and 2 are both true & false.

According to the principle of bivalence each one of these propositions is either true, or false, however not both. So, the liar violates this principle through a contradiction. What do we have here....a rejection of the principle of bivalence or Aristotle's law of non-contradiction?

Edit:
By the way, the law of non-contradiction states that: "One cannot say of something that it is and that it is not in the same respect and at the same time.”

Principal of bivalence states: every proposition takes exactly one of two truth values (e.g. truth or falsehood).

**Da Skinna** · December 22, 2008, 04:50 PM

I would have to ask where the paradox is.

**Viking Prince** · December 23, 2008, 03:37 AM

No paradox is possible when the information is insufficient to be testable. Neither statement is testable, therefore it is impossible to determine whether the law of non-contradiction is true for either statement independant of the loop.

**Odovacar** · December 23, 2008, 03:58 AM

The liar says: I lie.
1, If he lies --> he doesnt say the truth, so he cannot lie
2, if he doesnt lie--> his statement is true, so he lies, but a lie is a false statement, so then he doesnt lie.

However, lieing means: I deny a statement which I think to be true
not: I deny a statement which is true

So, its possible that this paradox is not a pardox as the liar denies a statement that is not true, as he thinks.

What if the statement: "creteans lie" is true (1st statement)
So, if the cretean thinks: creteans speak the truth (2st statement, an error by the cretean)
and he will lie: he will say: creteans lie (3st statement, a denial of his errenous statement which believes to be true)
So, in the end he will say that which is true, even though he wished to lie.

**bthizle1** · December 23, 2008, 12:38 PM

Originally Posted by Odovacar

The liar says: I lie.
1, If he lies --> he doesnt say the truth, so he cannot lie
2, if he doesnt lie--> his statement is true, so he lies, but a lie is a false statement, so then he doesnt lie.

However, lieing means: I deny a statement which I think to be true
not: I deny a statement which is true

So, its possible that this paradox is not a pardox as the liar denies a statement that is not true, as he thinks.

What if the statement: "creteans lie" is true (1st statement)
So, if the cretean thinks: creteans speak the truth (2st statement, an error by the cretean)
and he will lie: he will say: creteans lie (3st statement, a denial of his errenous statement which believes to be true)
So, in the end he will say that which is true, even though he wished to lie.

Well, that would be the same as stating there's opinionated truths and falsities within our existence. (Basically truth is based upon perspective) Very nice observation Odovacar! However does this mean that we cannot hold anything to be universally true?

**Viking Prince** · December 23, 2008, 04:36 AM

The fact that both can be either true or false is not in itself a paradox. As I had stated, nether statement can be confirmed with an outside test and thus it is not posssible to determine the validiity of either case 1 or case 2.

Odovacar has shown another means by which it can be true.

Again the real issue is we cannot know the truthfulness of either statement through another test outside of the loop.

**Odovacar** · December 23, 2008, 03:56 PM

Well, that would be the same as stating there's opinionated truths and falsities within our existence. (Basically truth is based upon perspective) Very nice observation Odovacar! However does this mean that we cannot hold anything to be universally true?

My example -or as I remember rather my logic teachers example- tried to show that this logical paradox is not necessarily a paradox.
Other paradoxes may have more validity.

But I show you another one, with a possible solution.

There is a barber who only shaves those who do not shave themselves.
Can the barber shave himself?
1, If he shaves himself, he will be someone who shaves himself, then he can't
2, If he doesnt shave himself, he will be someone who doesnt shave himself, thus he can.

But..what if he take the dimension of time into this?

What if never shaved himself before? Then he is someone who doesnt shave himself. As such he can shave himself, but only once.
After then he must follow his general rule.

As you can see formal logic doesnt deal with time and doesnt really deal with materials.
Formal logic only deals wth the forms of logical judgment and statements, not with their matter.

We can solve a syllogism about Socrates
(like Socrates is human. Humans are mortal, Socrates is mortal)
But we cant decide from pure logic whether Socrates exists or not, who is Socrates, and thus we cant really decide a statements validity using logical alone.

I don't connect this problem with universal truths.
Some universal truths seem to be exist.
I think geometrical statements, logical principles like the contradiction thesis (you can't predicate two contradictory statements about the same subject at the same time) have universal validity at least in the present universe.