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The Liars Paradox - Resolved

Cory Duchesne

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If you're not familiar with the Liars Paradox (which I will refer to as "LP" in this thread), a quick read on google should make it clear.

The paradox is as follows:

"this statement is false".

This is linguistically interesting, since it leads to an eternal regression where the statement ends up being true when it is false and vice versa. So it appears to violate the law of non contradiction.

But I have resolved it.

First, let's be clear on what it is we mean by true and false before we tackle the statement, and also be conscious of how we interpret the statement, as it can be seen in at least two ways, and one must not unconsciously confuse the two perspectives together, as that is what creates the contradiction, the sense of paradox.

So let's look at it this way:

1) The statement is false insofar as the statement itself exists when it tries to say it doesn't. So it is false in that sense.

2) The statement is true insofar as it correctly denies the validity of it's attempt at denying itself.

So we can say the statement is true from one perspective, and false from another perspective. The context in which the statement is true is not the same context in which it is false. So there is no contradiction in the Liars Paradox statement, despite superficial appearances.

Paradox resolved!

Another approach, which is just as reasonable, is to say that it is neither true nor false because of a failure to refer. The words 'this statement' refers to something outside of the sentence which is not specified.
 
The Liar's Paradox is most often presented in beginning logic studies to help students understand the need for clear external referents. In sentential logic, propositions are limited to assertions, e.g., A is B. Implicit is that A is also A (the simplest case of identity), but in either kind of statement, the assertion is not about the assertion, it's about A. An assertion about itself is undefined rather than paradoxical.
 
Cory Duchesne said:
If you're not familiar with the Liars Paradox (which I will refer to as "LP" in this thread), a quick read on google should make it clear.

The paradox is as follows:

"this statement is false".

This is linguistically interesting, since it leads to an eternal regression where the statement ends up being true when it is false and vice versa. So it appears to violate the law of non contradiction.

But I have resolved it.

First, let's be clear on what it is we mean by true and false before we tackle the statement, and also be conscious of how we interpret the statement, as it can be seen in at least two ways, and one must not unconsciously confuse the two perspectives together, as that is what creates the contradiction, the sense of paradox.

So let's look at it this way:

1) The statement is false insofar as the statement itself exists when it tries to say it doesn't. So it is false in that sense.

2) The statement is true insofar as it correctly denies the validity of it's attempt at denying itself.

So we can say the statement is true from one perspective, and false from another perspective. The context in which the statement is true is not the same context in which it is false. So there is no contradiction in the Liars Paradox statement, despite superficial appearances.

Paradox resolved!

Another approach, which is just as reasonable, is to say that it is neither true nor false because of a failure to refer. The words 'this statement' refers to something outside of the sentence which is not specified.
Click to expand...

I'm not following you. And I think it's because you used the wrong example. I believe the correct example of the liar's paradox is: "I always lie", thus creating a paradox, because if it is true that I always lie, then the moment I said that, I was telling the truth (contradiction).
 
Ron_Tomkins said:
I'm not following you. And I think it's because you used the wrong example. I believe the correct example of the liar's paradox is: "I always lie", thus creating a paradox, because if it is true that I always lie, then the moment I said that, I was telling the truth (contradiction).
Click to expand...

"I always lie" is not a contradiction. It's simply false, as long as I sometimes tell the truth.
 
phildonnia said:
"I always lie" is not a contradiction. It's simply false, as long as I sometimes tell the truth.
Click to expand...

Then I guess that's not the paradox. I can't remember right now, but it was something like "Everything I say is a lie"

... yeah, I think that's it.

So if in fact, everything I say is a lie, then that statement would be a lie and it would turn into "Not everything I say is a lie"

If the statement is true, then I have just broken the "always telling lies" rule by making a true statement.
 
Ron_Tomkins said:
Then I guess that's not the paradox. I can't remember right now, but it was something like "Everything I say is a lie"

... yeah, I think that's it.

So if in fact, everything I say is a lie, then that statement would be a lie and it would turn into "Not everything I say is a lie"

If the statement is true, then I have just broken the "always telling lies" rule by making a true statement.
Click to expand...

So, do you like cake or not?

(I like my post number.)
 
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Ron_Tomkins said:
I'm not following you. And I think it's because you used the wrong example. I believe the correct example of the liar's paradox is: "I always lie", thus creating a paradox, because if it is true that I always lie, then the moment I said that, I was telling the truth (contradiction).
Click to expand...

The example I used is used in uni POL courses, but with your example, it's fairly easy to resolve as well.

If all you ever did was tell the truth, then saying, "I always lie" is simply a single instance of a lie, and hence you statement is merely false, without any truth value.

If all you ever did was lie, then saying, "I always lie" is in fact a single instance of telling the truth, and hence your statement is again merely false, without any truth value.
 
"Everything I say is a lie" can be resolved when you realize that some utterances do not have truth values; for instance, a burp.

Hence, the statement is a lie, but the lack of a binary form means it isn't a paradox.

(This post has been removed.)
 
schrodingasdawg said:
the cake is a lie
the cake is a lie
the cake is a lie
the cake is a lie

ETA: Aw man, it turned my all caps into no caps. I can understand why this feature is in place though...
Click to expand...
Ah, I just posted A=0 B=100 in another thread, and the "B" was changed into "b". I edited it and saved it three times before I gave up; now I understand what was happening. Thanks.

Didn't change my "A" though.
 
marplots said:
"Everything I say is a lie" can be resolved when you realize that some utterances do not have truth values; for instance, a burp.
Click to expand...

But a "burp" never has a true value. The same cannot be said of statements such as "Everything I say is a lie". Such statement could very well be true or it could be a lie.
 
1. "This and the following statement are false."

2. "2 plus 2 equals 5"

The first assertion can not be true. If it were, it would contradict its own message: " This and....are false". So it has to be false. But in order to keep it in the false category, the second statement needs to be true. If the second assertion were also false, then the first one would be true, and we just saw that it has to be false. Thus it is true that "2 plus 2 equals 5".

An essay on consciousness by Nicholas Humphrey included in the book Intelligent Thought mentions this paradox, or fallacy or whatever it is. How can it be proven to be mistaken? Aside from taking 2 oranges, adding 2 more, and counting that we now have 4 oranges. Just on an intellectual level.

edited to add:

I suspect that the falsehood of those assertions lies somewhere near where the falsehood of St. Anselm's ontological argument also resides. Human imagination is capable of dreaming up the most improbable ideas. But that does not make them true, here on planet Earth.
 
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Some statements are neither true nor false. Examples include

- The one in the OP (this statement is false).
- The following statement is true. The previous statement is false.
- Everything I say is a lie (which is nothing but a modification of the first example).

One way to explain it is to say they are internally inconsistent.
 
Cory Duchesne said:
1) The statement is false insofar as the statement itself exists when it tries to say it doesn't. So it is false in that sense.
Click to expand...


You're mistaken. The statement "This sentence is false." does not try to deny it's own existence, and so is not false for that reason.

Cory Duchesne said:
2) The statement is true insofar as it correctly denies the validity of it's attempt at denying itself.
Click to expand...

That makes the statement true. But if it's true, then it must be correct in saying that it is false, in which makes it false. But if it's false, then it is not correct in saying that it is false, which makes it true. But if it's true...

So logically, you're second statement should be followed by a third statement outlining the consequences of the second statement, which in turn should be followed by a fourth statement, and so on in an infinite series.

3) The statement is false insofar as it incorrectly denies the validity of it correctly denying the validity of it's attempt at denying itself.

4) The statement is true insofar as it correctly denies the validity of it incorrectly denying the validity of it correctly denying the validity of it's attempt at denying itself.

5) The statement is false insofar as [...]

I don't think you've quite succeeded in resolving the paradox. Let's just say it's truth value is indeterminate and leave it at that. :p

DallasDad said:
The Liar's Paradox is most often presented in beginning logic studies to help students understand the need for clear external referents. In sentential logic, propositions are limited to assertions, e.g., A is B. Implicit is that A is also A (the simplest case of identity), but in either kind of statement, the assertion is not about the assertion, it's about A. An assertion about itself is undefined rather than paradoxical.
Click to expand...

But using external referents doesn't preclude this kind of paradox. For example...

The next sentence is true. The previous sentence is false.

Both sentences are making a truth claim about something external to themselves (the other sentence) yet the liar's paradox remains.
 
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