Consider Statement A.

Statement A:  "Statement A is not true."

Is Statement A true?

Statement A is not true.  Argument 1 explains why.

Argument 1:
SUPPOSE Statement A is true.
Then the proposition that Statement A states is true.
But Statement A states that Statement A is not true.
So, Statement A is not true, contrary to our initial supposition.
So, IN FACT, Statement A is not true.

Unfortunately, Statement A can't be not true either.

Argument 2:
2.1  SUPPOSE Statement A is not true.
2.2.  Then the proposition that Statement A states is not true.
2.3    But Statement A states that Statement A is not true.
2.4   So Statement A is not not true, contrary to our initial supposition.
2.5  So, IN FACT, Statement A is not not true.

Your mind should be blown.  You should not be saying, "That's puzzling."  You should be saying, "My mind is exploding."

Argument 1 proves that Statement A is not true.  Argument 2 proves that Statement A is not not true.  We have proved a contradiction, which is impossible.

A fundamental principle of logic is the Law of the Excluded Middle, which states that every statement is either true, or not true, and never both.  It is just a straightforward consequence of the definition of the word "not."  For any well-defined characteristic (say, "onchyness"), and for any thing (say, "Ralph") , if Ralph is not onchy, then...Ralph is not onchy!

However important you think this paradox is, it is more important.  If contradictions are possible, then anything's possible.

So how can we resolve the paradox?

One approach is never to utter or think Statement A or statements like it.  Then we never have to worry about the paradox.

It's easy to avoid Statement A, but how do we know what statements are similar enough to Statement A to cause a problem?  Strategy Alpha will keep us safe.

Strategy Alpha:  Never make any statement that refers to a statement.

That rules out Statement Alpha and other statements liable to cause the same problem.

Unfortunately, Strategy Alpha rules out many useful and harmless statements.  For example:
Statement B:  Two plus two is five.
Statement C:  Statement B is not true.
According to Strategy Alpha, Statement C is inadmissible--we can't say Statement B is false.  So Strategy Alpha is too stringent.

Statement A not only refers to a statement, it refers to itself, Statement A.  Perhaps the root problem is this self reference.

Strategy Beta:  Never make a statement that refers to itself.

Strategy Beta does rule out Statement A, but it suffers from the opposite problem to Strategy Alpha:  it's too weak to prevent the paradox.

Statement D:  Statement E is not true.
Statement E:  Statement D is true.

Statement D can't be true.
Argument 3:
SUPPOSE that Statement D is true.
Then Statement E is not true.
Then Statement D is not true--which contradicts our original supposition.
So, IN FACT, Statement D is not true.

But Statement D can't be not true.
Argument 4:  SUPPOSE that Statement D is not true.
Then Statement E is true.
Then Statement D is true--which contradicts our original supposition.
So, IN FACT, Statement D is not not true.

Again we face paradox.  Hofstadter illustrates this version of the paradox with Escher's drawing of two hands drawing each other.

Preventing a statement from referring to any other statement is too strong.  Preventing a statement from referring to itself is too weak--if we only prevent it from directly referring to itself.  We need to rule out indirect self reference as well.

Strategy Gamma:  Don't make any statement that refers to any statement that...that refers to any statement that refers to the original statement.

That'll do it.

But there are still two problems with Strategy Gamma.

1. To implement Strategy Gamma we have to keep careful track of all our statements to make sure that there are no chains of self reference.  That vigilance could be irksome for any lengthy argument; but, if we ignore Strategy Gamma, we can't be sure we won't be led into contradiction.
2. Strategy Gamma only avoids the Epimenides Paradox.  It is an emergency stopgap.  What we ultimately want is to resolve the paradox.  The Epimenides Paradox appears to prove that contradictions are possible, or that the Law of the Excluded Middle is false.  But if that's true, then we can never be sure that avoiding the Paradox is enough to avoid paradox; all our arguments are suspect.
How can we resolve the paradox?

### Meaninglessness,

#### or, "It depends what your definition of is is."

Statement F:  Statement F is true.

Statement F doesn't lead us into any paradoxes.  Supposing it's true only confirms that it's true.  Supposing that it's not true only confirms that it's not true.  But we still have the problem of decidng which it is--is Statement F true or not true?

It seems that there's no way to tell.  How can that be?  What extra information could possibly turn up that would help us to decide?!  Since Statement F is only about Statement F, and we know Statement F, it seems we should be able to decide, right now and for sure, whether or not it's true.  F is for Fishy.

Perhaps Statement F is meaningless:  it doesn't express any proposition at all.  Perhaps Statement A is meaningless.  Perhaps that's the resolution of the paradox--Statement A is neither true nor not true, just meaningless.

To accomodate this possibility, from now on let's consider not "Statements" but "Sentences".

Sentence A:  Sentence A is not true.

Unfortunately, it is not proper to say that Sentence A is neither true nor not true, but meaningless.

Sentence G:  Zebras are white.

You might think Sentence G is both true and not true.
Argument 5:
Zebras are partially white; therefore Sentence G is true.
Zebras are only partially white; therefore Sentence G is not true.
Therefore, Sentence G is both true and not true.

We seem to have violated the Law of the Excluded Middle again.

Similarly, you might think that Sentence G is neither true nor not true.  It's not true, because zebras are only partially white; it's not not true, because zebras are partially white.

Nevertheless, Sentence G does not pose a contradiction.  The Law of the Excluded Middle says that Ralph is either onchy or not onchy; but it applies only if onchyness is well defined.

Sentence H:  That suit is smart.

Argument 6:
The suit looks good; therefore Sentence H is true.
The suit does not have a high IQ; therefore Sentence H is not true.
Therefore, Sentence H is both true and not true.

Argument 6 doesn't overturn the Law of the Excluded Middle, since it uses two different meanings for the word "smart".  If we specify that "smart" means "good-looking", then Sentence H is just true.  If we specify that "smart" means "intelligent", then Sentence H is just not true.

Similarly, but more subtly, the seeming contradiction in Argument 5 stems from a shift in the definition of "white".  If we define "white" to mean "completely white" (ignore the circularity, you know what I mean) then Sentence G is just not true.  If we define "white" to mean "partially white" then Sentence G is just true.  As long as we stick to a single definition, we don't face any contradiction.

Similarly, but more subtly, when we deal with Sentence A we must be careful about our definition of the word "true".

Sentence I:  Green virtue swims under fastness.

Is Sentence I true or not true?

Definition i:  "True" means "expressing a correct proposition".

According to Definition i, Sentence I is not true, since, being meaningless, it doesn't express any proposition at all.

Definition ii:  "True" means "not expressing an incorrect proposition".

According to Definition ii, Sentence I is true, since it doesn't express an incorrect proposition, since it doesn't express any proposition at all.

Even though Sentence I is meaningless, we can still determine whether or not it's true, so long as we define carefully what we mean by "true".

Let us adopt Definition i, which seems more natural.

Recall that, in our attempt to resolve the Epimenides Paradox, we supposed that Sentence A is neither true nor not true, because it is meaningless.  With Definition i in hand, however, we can see that "meaninglessness" is not a distinct third possibility.  Rather, meaninglessness is a form of untruth.

If Sentence A is meaningless, then, using Definition i, Sentence A is not true.  But we have seen that supposing that Sentence A is not true implies, via Argument 2, that Sentence A is not not true.  Thus, our attempt to use meaninglessness to resolve our dilemma fails.  The paradox lives.

Or does it?

### Meaninglessness, Mark II

Let us from now on adopt Definition i of "true".

And suppose that Sentence A is meaningless.

Then Sentence A is not true.

Earlier, we saw in Argument 2 that supposing that Sentence A is not true implies that Sentence A is not not true, enveloping us in contradiction.

Argument 2 seems ironclad, but in fact it is flawed, as we can see now that we recognize the possibility of meaninglessness.

The flaw is step 2:
2.2   Then the proposition that Sentence A states is not true.

If Sentence A is meaningless, then Sentence A doesn't state any proposition at all--it's just a string of words.  Therefore 2.2 doesn't hold, therefore Argument 2 doesn't hold, and therefore no contradiction arises.

To summarize.

1. Sentence A is meaningless.
2. Therefore, Sentence A is not true.
3. And Argument 2, which attempts to prove that Sentence A is also not not true, fails:  Argument 2 refers to the proposition which is expressed by Sentence A, but Sentence A doesn't express any propositions.
Indeed, we can now prove that Sentence A is
1. not true, and
2. meaningless.
Argument 7:
7.1   Sentence A is either not true or not not true, but not both.  (Law of the Excluded Middle)
7.2   Sentence A is not true.  (By Argument 1.  Do you see why Argument 1 still holds even though Argument 2 does not?)
7.3   If Sentence A is meaningful, then Sentence A is not not true (by Argument 2)--contradicting 7.2.
7.4  Therefore, Sentence A is meaningless.

Thus, we have resolved the Epimenides Paradox.

Or have we?  In fact, meaninglessness embroils us in even worse confusion than before!  See why?

### Meaninglessness:  Ugh

There are two problems with the meaninglessness approach.

First:  Sentence A is weird, but it doesn't seem meaningless.  It has a clear subject, a sentence.  And it has a clear predicate adjective, untruth.  And we know that truthfulness is just the kind of thing that we ordinarily talk about when we talk about sentences.  So we have an independent argument that Sentence A is perfectly meaningful.

I merely mention this first problem, because the second problem is decisive anyway.

Namely:  We have decided that Sentence A is meaingless and not true.  I.e., we have concluded
i.  Sentence A is a meaningless sentence.
ii.  Sentence A is not true.
We think that i and ii are both true.
But expand i:
i'.  "Sentence A is not true" is a meaningless sentence.

Return to Argument 7, which we had hoped had settled this whole matter; reexamine 7.2.  Behold!  It's Sentence A!  But how can an argument employ a sentence that that same argument proves is meaningless!?  It can't.

So what do we do now?