introduction to the Liar's Paradox was an episode of the old Star
. In it the Enterprise
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
. 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
The most common expression of the Liar's Paradox is a single sentence:
This sentence is false.
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:
This sentence is true.
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:
sentence is green.
This sentence is
The green sentence is true, and the red one is false. We can also avoid
self-reference with sentences like these:
The sentence below is true.
The sentence above is false.
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
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:
sentence is false.
This green sentence
This true sentence is false.
This false sentence is false.
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:
green sentence is false.
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:
green sentence has six words.
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.