Do you think the Liar's Paradox is just a play on words or more meaningful then that?

Question

My husband and I spent a good 4 hours discussing the Liars Paradox. He believes that it is truly astounding that logicians have discovered statements that condradict each other leaving the statement false but true and true but false. He deducts that you can prove any untrue statement to be true because this paradox exists...

His example:

(A) This statement is not true

(B) The world is flat

(Not A) This statement is true

Either A or B is true. Since we KNOW the world is not flat then Statement A must be true. But since statement Not A exists, statement A cannot be true...making statement B True

I do not see what is so astounding about it and believe it to be a play on words due to the semantic limits of natural language. I also do not believe that just because you can "prove" any statement to be true even if it is false means that it is true. I just do not see what is so great about this paradox. Frankly I need a better arguement for either side...preferably mine:)

Answer 1

First off, lets clarify.

This statement is not true.  

True of false?

If the statement is said to be true than it really is "not true" and therefore it is a false statement.  However if the statement is said to be false than the "not true" part of the statement is in fact telling us the truth, making the statement true.  Hence the contradiction.

Now one cannot dismiss this sentence as irrelevant by calling  it an incomplete thought as JJ posted.  Anything can be a complete thought.  JJ argues that the first word in the sentence (this) cannot stand on its own.  JJ aruges that it needs something, like "this country", or "this painting"  And it has that, "this statement".   "This statement is not true" is complete and the subject is the sentence itself,

Also one cannot dismiss this sentence as "just some meaningless garbage" as King David proposes.  Consider mankind's concept of truth.  We have that concept because we believe things to either be true or false and by definition of truth they cannot be both nor neither at the same time while in reference to the exact same thing, else our concept of truth is not as it seems.  

As Aristotle put it, "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect." (Aristotle, Metaphysics)

Sauwelios statement that the lair's paradox "breaks the law of non-contradiction " is true and that is why this paradox is disturbing.  Just because it doesn't make sense is no reason to dismiss its significance.  

Now what I believe your husband was trying to illustrate with his example is that once a contradiction is allowed in one's logical reasoning it is possible to prove anything through use of that one contradiction.

Lets consider the sentence at question again.

This statement is not true.  

True or false?

Again we find that if the statement is true it is rendered false and vice versa.  We have our contradiction.  If your confused read the top of this post again for clarification.

If we were to illustrate this using logical reasoning we may do so using three different variables.  

Two of the variables will be used to describe our contradiction.

A=This statement is true

B=This statement is not true

The third variable will be something we all assume to be false

C=The Earth is flat

Now consider the following statement.

1)  Either A is true or C is true

We must accept statement 1 as true because we have already accepted A to be true.  However we have also accepted B which leads us to statement 2

2)  A is not true

Now that we have accpeted these two statements we are left with one logical answer for statement 1.

1)  Either A is true or C is true.  Since we have determined A to not be true it follows that C is true, thus the world is flat.

This is the disturbing nature of the lair's paradox.  We all know C to be false, yet through logical reasoning we have shown it to be true.  

I hope this sheds some clarity on the situation.

Answer 2

The more simple version of this to understand goes:

"This statement is false"

Is the statement true or false?

The paradox is in the recursion of the sentence, the fact that it refers to itself. Whichever answer you give invalidates itself. Youre right, this isnt a mind blowing rip in the fabric of reality. I would tell your husband that it is neither true nor false. It is a malformed sentence in that the noun is nonsensical recursion. Its like asking if "bugar pie boo bop" is true or false. Neither, its just some meaningless garbage which will no doubt amaze the minds of our greatest philosophers for decades.

Answer 3

truth is what people beleive to be true, and a paradox has no solution.  so, yes anything can be true or false.  depending on your point of view.

Answer 4

Yes, I believe the Liar's Paradox is more meaningful then a play on words. Indeed, it is the opening to a mind shaft leading to perhaps --ultimate meaning.

“All statements are false” cannot be true because it implies, not a restriction against self-reference as Russell said, but because it implies the necessary truth of the contradictory opposite, “There exists at least one true statement.” Starting with a contradiction-free affirmation, the structures of knowledge can then be made to follow in a necessary and systematic fashion. In this way, the closed system problems that arise in mathematics are avoided.

Answer 5

The liar's paradox is indeed a very helpful philosophical insight.  Now, I must admit that I was not taken very much by your husband's example (and his conclusions), either, but that was less the problem of the liar's paradox.

Let me try to give you an example, where the logic of the liar's paradox holds.  There are those who say that science is the only way to the truth.  Now, no worries, I am a trained scientist, so I don't have anythings against science!  But the logic of this statement is of course paradoxical.  If indeed the statement were true, it would render itself false, since the statement cannot be tested scientifically.  it is actually a metaphysical statement that can only be tested metaphysically.  In other words, if indeed science were the only way to the truth, the statement that this is so would be not a reliable truth statement, since it is not scientific.  

This is a real problem in today's discussion about the role of the scientific method.  As you can see, it's the liar's paradox all over again.  And it is not just a play of words.  

Hope this helps.

Answer 6

That that is is that that isn't isn't is that it it is - Charlie

Tell your husband that paradoxes arise out of dualistic programming.  What is true is and what is not.  What is not true has no existence in the universe or else it would be true.

May I refer the book "Truth vs Falsehood" by Dr. David R. Hawkins

Answer 7

You are right, it is just a play on words and is in no way a paradox.

The phrase "This statement is not true" is not a complete thought.  In this case "This statement" is a verbal placeholder for the preceeding statement of "The world is flat".  Therefore the real full statement of (A) is "The statement 'The world is flat' is not true"

We can see when drawn out to its full logical meaning, statement A is true....

The paradox only comes when only the (A) statement is used, which is a fallacy because the term "this" needs a full thought "this country", "this painting" that is independent of the sentence it is being used in.  "This statement is not true" would be considered paradoxical only because it is alone, and pointing toward itself.....  The problem is that "this statement" is not a full thought, but is rather a verbal simpllification for a necessary preceeding statement.  In the case provided that preceeding statement is supplied showing no paradox, and an easy choice.  Statement B is an untrue statement, Statement A is stating that Statement B is untrue.  Therefore statement A is true, and statement B is untrue.  No paradox.

It could also be argued that the sentence "This statement is untrue" is neither true not untrue (or both).  Considering that the only way to logically fulfill the thought of "this statement" in that sentence is to have an unending chain.  Because the full thought requires the description of what "this statement" is, it would have to be read, in its full form, as "This statement (this statement (this statement (this statement (this statement (this statement....".  where each preceeding "this statement" is describing the "this statement" before it.  One could never logically finish that question because of the unending loop of necessary descriptors, so an answer need never be reached.

Answer 8

think about it this way.

(A) the world is round

(B) the world is not round

(C) statement A is not true (the world is flat)

so basically you have two choices pointing to the wrong answer, and 1 pointing to the right one. Logically, you should pick the 2 answers as there appears to be more evidence. but as we already know the world is round, we can add in this statement

(D) statement B is not true (the world is round)

you now have 2 statements pointing to either answer. which do you choose? paradox

Answer 9

The so-called "Liar's Paradox" is actually an implicit self-contradiction.

"(It is true that) this statement is not false."

It breaks the law of non-contradiction and is therefore nonsensical.