Which Alice?

And That's the Beautiful Part of It!
"Actually," said Alice, "I heard something like that paradox
before: It is the story of Epimenides the Cretan, who said, 'All
Cretans are liars.' If Epimenides is telling the truth, then he is lying,
and if he is lying, then he is telling the truth. So we have a paradox."
"Not true!" said Humpty Dumpty, decisively. "That is a popular
fallacy! This is one of these things which seems like a paradox, but
really isn't one."
"Will you please explain?" asked Alice.
"In the first place, what do you mean by a liar?—one who lies all
the time, or one who lies some of the time?"
"I had never thought of that before," admitted Alice. "I guess one
who lies even some of the time should be called a liar."
"Then there is certainly no paradox," replied Humpty Dumpty.
"Epimenides' statement could be true, which would then mean only
that all Cretans sometimes lie. Then Epimenides, being a Cretan, also
sometimes lies, but that does not mean that the particular statement
in question is a lie. Thus there is no paradox."
"I see," said Alice. "Then I guess I had better define a liar as one
who always lies. Then we get a paradox?"
"No, even then we don't," replied Humpty Dumpty. "This time
we do know that Epimenides' statement can't be true, because if it
were, then all Cretans would always lie, and Epimenides, being a
Cretan, would always lie, hence would have lied when he made that
statement. So if the statement were true, it would also have to be
false, which is a contradiction."
"Then it is a paradox!" said Alice.
"No! No!" said Humpty Dumpty. "A contradiction only arises if
we assume the statement true; if the statement is false, there is no
contradiction!"
"Will you please explain that?" asked Alice.
"Well, what does it mean for the statement to be false? It means
that it is not the case that all Cretans are liars—in other words, that
at least one Cretan sometimes tells the truth. So all that emerges
from Epimenides' having made the statement is that he is lying and
that at least one Cretan sometimes tells the truth. This is no
paradox!"
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