## Liar's paradox...an attempt to solve it.

• 4.7k
Everyone knows the liar's paradox. It is simply the impossibility of assigning a truth value (true/false) to the statement: This statement is false. If it is true then it must be false and if it is false it must be true...round and round we go in a circle.

The end result being that the liar statement is neither true nor false. The most common inference drawn from this juncture is that the liar statement is not a statement/proposition at all.

However consider the following reasoning:

This statement is false is neither true nor false. So there is a logical equivalence between the two statements below:

1) This statement is false

AND

2) This statement is neither true nor false

Notice however that statement 2 does have a truth value. It is TRUE - the liar statement IS neither true nor false. But statement 2 is logically equivalent to statement 1 (the liar statement).

Therefore the liar statement This statement is false is TRUE by virtue of the above logical equivalence.

• 2.3k
Can you show how these statements are equivalent?
• 4.7k

A(the liar statement) =This statement is false

B=This statement is neither true nor false

As per how the paradox is known A cannot be true and cannot be false. In other words A can neither be true nor can A be false. But this is statement B. And statement B is true as shown above. Therefore A (the liar statement) must also be true
• 352
What rules of logic did you use to get from 1) to 2)? They appear to be saying difference things.

Statement 2 doesn't seem to make sense. Try rewriting it in a way that makes it easier to discuss it with logic.
• 8.3k
As per how the paradox is known A cannot be true and cannot be false. In other words A can neither be true nor can A be false. But this is statement B. And statement B is true as shown above.

When you say that the statement "this statement is false" is neither true nor false you're not saying that the statement "this statement is false" means "this statement is neither true nor false". You're conflating statement A with a (different) statement about A. A and B are not logically equivalent.

I think a way to approach the problem is to consider what it means for a statement to be false and then rephrase the statement with this in mind. For example, "this statement does not correspond to a fact" (correspondence theory) or "this statement does not cohere with some specified set of sentences" (coherence theory). We then might say "'this statement does not correspond to a fact' does [not] correspond to a fact" or "'this statement does not cohere with some specified set of sentences' does [not] cohere with some specified set of sentences". Are these contradictions?
• 13.8k
We discussed this a bit in another thread . . . although with my crap memory, I can't recall much of what we ended up saying, even though I know a couple posts were interesting.

At any rate, in my opinion, "This statement is false" is equivalent to assigning "F" to "This statement." But "This statement" isn't actually a proposition. So the solution on my view is that "This statement is false" doesn't actually say anything--it functions kind like a context-free pronoun, where we have no idea what the pronoun is referring to, and so it's not true or false.
• 3.2k
A symbolic statement is talking about itself. This must assuredly will lead to real world problems as is often the case in paradoxical symbolism and art.

Now, if someone said, "Everything I say is a lie", I would take him at his word knowing sometimes need v will tell a lie.

There is always going to be a problem when symbols are given real status beyond just being a symbol.
• 6.9k
1) This statement is false

AND

2) This statement is neither true nor false

Dude, these statements contradict each other. ~p . (p.~p)
• 4.7k

A: This statement is false is the original Liar statement.

Upon logical analysis we end up in a true-false never-ending loop. The end result being that we cannot assign a truth value to it i.e.

the Liar statement is neither true nor false.
Therefore the Liar statement can be rephrased as
B: This statement is neither true nor false.

But B is true - the Liar statement is neither true nor false. That is to say B is TRUE. Since B is just a different version of A (they're logically equivalent) it follows that A (the original Liar statement) is also TRUE.
• 8.3k
the Liar statement is neither true nor false.
Therefore the Liar statement can be rephrased as
B: This statement is neither true nor false.

Again, no. You're conflating statement A with a (different) statement about A.

Consider the sentence "this sentence has four words". You can't say that because this sentence has five words that it can be rephrased as "this sentence has five words". That'd be a different sentence.
• 4.7k
You're right. If I think of anything new I'll come back. Thanks
• 4.7k
Again, no. You're conflating statement A with a (different) statement about A.

A: THIS (A) statement is false is
1. Not true
2. Not false
3. Not both true AND false

The only option left is that A is "neither true nor false.

So now we have the new statement:

THIS (A) statement is neither true nor false. This new statement is TRUE for A is neither true nor false. Thus even A (liar statement) is true
• 8.3k

A: THIS (A) statement is false is
1. Not true
2. Not false
3. Not both true AND false

The only option left is that A is "neither true nor false.

So now we have the new statement:

THIS (A) statement is neither true nor false. This new statement is TRUE for A is neither true nor false. Thus even A (liar statement) is true

You're still conflating. You have one sentence, "this sentence is false", and you have another sentence, "this sentence is neither true nor false". They're not the same sentence.

That the first one is neither true nor false (according to you) is not that it means "this sentence is neither true nor false".

To give another analogy, the sentence "this sentence has five words" is true, but it would be wrong to then say that "this sentence has five words" and "this sentence is true" are logically equivalent.
• 2.3k
Aside from Michael's point, (which can be summed up as the conclusion isn't its proposition), your conclusion that the second sentence has a truth value is also incorrect.

If the statement is true or false (has a truth value) then it is not true that it's neither true nor false. It might be true if the 2nd sentence would say that "that sentence is neither true nor false" referring to the 1st sentence.
• 4.7k

The Liar statement: A: This statement is false

Available truth value options for the Liar statement are:

1. True
2. False
3. True and false
4. Neither true nor false

Option 1 and 2 are not possible because of the well-known true-false loop.
Option 3 is a contradiction so again, not possible
The only option available is 4, neither true nor false.

Therefore, This statement is false is saying exaclty what B: This statement is neither true nor false.

But B is a TRUE statement. Therefore the liar statement, which is equivalent to B, is also true.
• 1.2k
Therefore the liar statement, which is equivalent to B, is also true.

As others have been pointing out, the liar statement (A) is not equivalent to B. At the very least, you have to provide an argument for why the two statements are (supposedly) equivalent.
• 2.3k
But B is a TRUE statement. Therefore the liar statement, which is equivalent to B, is also true.

If B is TRUE, then it is not "neither true nor false".
• 4.7k
As others have been pointing out, the liar statement (A) is not equivalent to B. At the very least, you have to provide an argument for why the two statements are (supposedly) equivalent

Let me try to explain it as clearly as possible.

The liar statement: A: This sentence is false

Please note that A is only concerned about the truth value of A, nothing less and nothig more.

Let us now assess what possible options of truth value are there for A:

1. True
2. False
3. Both true and false
4. Neither true nor false
The above 4 choices are jointly exhaustive and mutually exclusive.

Option 1 and 2 are impossible for the reason that a true-false loop results. Option 3 is impossible because its a contradiction. The only option available is 4 which is ''neither true nor false''.

Note again A is only about the truth value of A, nothing less nothing more.

Therefore it is acceptable to replace ''false'' in A with ''neither true nor false'' since the former equates with the latter as I've shown above.

Therefore A can be replaced with B: This statement is neither true nor false. This however is a TRUE statment about A. Therefore, the original Liar statement, A, must also be true by virtue of it being equivalent to B.
• 1.2k
Let me try to explain it as clearly as possible.

Repeating the exact same argument is not an explanation, and it is certainly not any more convincing.

This statement is neither true nor false. This however is a TRUE statment about A.

This is not a statement about A at all; it is a statement about B. As soon as you change "false" to "neither true nor false," you have a different statement, and the two are not equivalent.
• 4.7k
If B is TRUE, then it is not "neither true nor false".

I think you're right. B cannot be true in a direct way. However...

It can be
1. True
2. False
3. Both true and false
4. Neither true nor false
B can be 2(false). So again, can I make a claim that I've opened a new option [2. False] for the Liar statement which was written off in the original paradox?
• 1.2k
B can be 2(false). So again, can I make a claim that I've opened a new option [2. False] for the Liar statement which was written off in the original paradox?

Again, nothing about B has any bearing whatsoever on what you can claim about A. They are two different statements.
• 4.7k
X-) LOL. Thanks for your time guys. I'm wrong.
• 13.8k
Since B is just a different version of A (they're logically equivalent

No they're not, but what's your argument or demonstration of that?
• 2.3k
Let us now assess what possible options of truth value are there for A:

1. True
2. False
3. Both true and false
4. Neither true nor false
The above 4 choices are jointly exhaustive and mutually exclusive.

Normally, only 1 and 2 are considered truth values. 3 is impossible under classical logic as it violates the law of non-contradiction. 4 is the absence of a truth value.
• 4.7k

Below is a new look at the Liar's paradox.

The Liar statement A: This statement is false.

Options available for the truth value of A
1. True
2. False
3. True AND false
4. Neither true nor false

1 and 2 result in the true-false loop and 3 is a contradiction. The last option available is 4 which is neither true nor false.

Note A only concerns itself with the truth value of A. Nothing more nothig less. Therefore we can rephrase A as
B: This statement is neither true nor false.
I'm not making any illegitmate claims about A. It's only about the truth value of A. So, A is neither true nor false.

Let us now check the possible truth values of B.
It is:
1. True
2. False
3. True AND false
4. Neither true nor false

It cannot be 1 in a direct manner as it leads to a true-false loop. It can't be 2 because that leads to the original Liar statement. It cannot be 3 as its a contradiction. The last option is 4 (neither true nor false). Notice that this is what B states. So it must be that A (the Liar statement) is true since B is nothing more than a rephrased version of A.
• 8.3k
B is nothing more than a rephrased version of A.

No, it isn't. B is a different sentence.

You can't go from "'this sentence is false' is neither true nor false" to "'this sentence is false' means 'this sentence is neither true nor false'" (which is what you're doing).

You can't just rephrase A like this, just as I can't rephrase "this sentence has five words" to "this sentence is true".
• 2.3k
Options available for the truth value of A
1. True
2. False
3. True AND false
4. Neither true nor false

Let's write this a bit less perfunctory.

Options are:

1. "The sentence "this statement is false" is true"
2. "The sentence "this statement is false" is false"
3. "The sentence "this statement is false" is true and false"
4. "The sentence "this statement is false" is neither true nor false"

Going with option 4, we see that statement B is a statement about A, as such it cannot refer to itself by using the pronoun "this". It should be "that statement is neither true nor false". If it refers to itself again, we get the same problem I pointed out earlier:

If the statement (in this case B) is true or false (has a truth value) then it is not true that it's neither true nor false.
• 13.8k
Note A only concerns itself with the truth value of A.

I don't agree with that part.

"Is false" "Is true" etc, in my opinion, are equivalent to "spelling out" that we're assigning T or F to some proposition. For example, if we assign T to "The cat is on the mat," it's the same as saying "The cat is on the mat is true" (or vice versa). So "This statement is true" should be the same as assigning T to "This statement."

The subject matter should be whatever the proposition is about, whatever we're assigning true or false to That would putatively be "This statement," but "this statement" isn't a proposition. and unfortunately it functions like a context-barren, free-floating pronoun. We can't include "is true" as part of the proposition, because that's the truth-value assignment (that we're just spelling out and supposedly appending to a proposition), not part of the proposition itself.

With "This statement is neither true nor false," you're not spelling out the truth-value assignment and appending it to a (supposed) proposition.
• 4.7k
So you're saying that B: This statement is neither true nor false is

1. Not true
2. Not false
3. Not true and false
4. Not neither true nor false

It appears to me that the above 4 options are jointly exhaustive and mutually exclusive. I'm at a loss to find out what sort of truth value B has.
• 8.3k

No, what we're saying is that A. "this sentence is false" and B. "this sentence is neither true nor false" are not logically equivalent. They're different sentences.
• 4.7k
Also...

''This statement is false'' is a claim about the truth value of itself. Analysis results in a truth value, if one may call it that, of neither true nor false.

That is to say: ''This statement is neither true nor false'' is a logically acceptable rephrased version of ''This statement is false''. Note, like A, B makes a claim only about truth value of given statement.
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal