Ordering Chaos

The Liar's Paradox

My introduction to the Liar's Paradox was an episode of the old Star Trek. In it the Enterprise is hijacked and taken to a planet populated entirely by androids. The androids plot to infiltrate human society and control it. Kirk and the crew disarm the androids by taxing their understanding with illogical behavior. At the climax, Kirk and the other kidnapped crew members convince Norman, the android leader, that Harry Mudd is an inveterate liar. In fact, they tell him, he never speaks the truth. Mudd then delivers the coup de grâce. Addressing Norman, he says, "I am lying." Norman goes into overload. If Mudd is lying, then he must be telling the truth, but if he is telling the truth, then he must be lying. Norman's circuits fry, and the crew of the Enterprise escapes.

The most common expression of the Liar's Paradox is a single sentence:
It is a paradox because assuming it is true leads to the conclusion that it is false, and assuming it is false leads to the conclusion that it is true. There is no such contradiction with the similar statement:
Nevertheless, the truth-value of the above sentence cannot be determined. We can only determine that assigning a truth-value does not lead to a contradiction. Some have suggested that the fault lies in the self-referential nature of the sentence. Yet there is no problem determining the truth-value of sentences like these:
The green sentence is true, and the red one is false. We can also avoid self-reference with sentences like these:
Clearly, to determine the truth-value of the first sentence, we must first determine the truth-value of the second. But the truth-value of the second relies on the first. Again, assigning a truth-value leads to logical contradiction.

In the case of the colored sentences, the sentence refers to its own color-value, but not to its own truth-value. Since we can evaluate the color-value independently of the truth-value, we can determine whether the sentences are in fact true. The truth-value of sentences cannot be determined if they depend on evaluating their own truth-value (directly or indirectly) in order to determine their truth-value. There is no logical difference among the following sentences:
The truth-value of the sentences is both an attribute and a meaningful claim. The attribute depends on evaluating the claim, and evaluating the claim depends on determining the attribute. Strictly speaking, therefore, the truth-value cannot be determined. Put another way, being self-contradictory does not make these sentences false any more than being non-self-contradictory makes corresponding statements true.

One more example:
This sentence is peculiar in that it refers to itself with an inaccurate attribute. Does the inaccurate attribute cause the self-reference to fail? In other words, can we say that the sentence does not refer to itself because it is not green? What is the truth value of a sentence like this one:
Does the inaccurate attribute make it false? The sentence plainly has six words, but it is not green. But it does not claim to be green, it merely points to the green sentence that turns out to be red.