There's this thing called the Godel's theorem that states that any logical system must be incomplete. The proof is simply to propose a self-referential statement like: "this statement is false", and then show that it can neither be true or false. Therefore, they say, logic can't do everything.

Of course, many people don't just stop there, and extrapolate this to "logic is powerless" and to "you can't really know anything", which is what such people want to believe, for they don't live logically and know nothing. I suspect that Godel himself fell close to being in this category. But I digress...

How can the aforementioned statement be logically described? Saying that it is true or that it is false obviously won't work. At least not directly so. Let's first examine what logic means by the concepts of "true" and "false". Why is 2=2 true, for instance? Because it is compliant with the law of identity, which states that a thing is itself, and cannot simultaneously be something else without being equivalent to it. In other words, 2 can not be both 2 and 3, because if 2=2 and 2=3, then 2 and 3 refer to the same entity or concept. It is valid to define several aliases to a concept and I could, if I wanted to, use both symbols to represent "two".

Logic can then be defined as the application of the law of identity (and the concept of equivalence that it implies) to statements about real entities in order to determine whether these statements correspond to reality. Logically, if a statement violates the law of identity, it is false. Otherwise it is true. Then we define this to mean that true statements correspond to reality and that false ones do not.

A statement violating the law of identity is also called a contradiction. Thus "contradiction" is also another way of saying "false", or "unreal", or "an invalid concept". Logic then, is the set of rules, derived from the law of identity, designed for manipulating statements in search of contradictions. 2+2=5 is not true because it implies 2=5-2=3, which is a contradiction.

So what does "this statement is false" evaluate to? "this statement is false"="false" and "this statement is false"="true" both create contradictions. But contradictions of what? When you say 2=3, then either 2 and 3 are both the same concept, or the statement is false. What a self-referential statement does is create a "true"="false" contradiction.

This means that as long as we are dealing with "this statement is false", we can not use the words "true" and "false", because they are invalidated through its evaluation. Likewise, you can't use a paper ruler to measure the width of a flame.

Note, however, that only the words were invalidated. Defining 2 and 3 to be equivalent tokens does not make the concepts they refer to equivalent. Similarly, the concepts of "true" and "false" still exist, even through the words can no longer be used. So invent new ones. It is clear that "this statement is false" is a contradiction, therefore it is false in concept, as in "not corresponding to reality".

The lesson here is: don't measure fire with a paper ruler, and never listen to morons who tell you that there are ultimate limits to your knowledge.