Who Stole the Tarts? Part VI

N.b. another part intervened between what was solved in the last post and the current one but it did not concern logic as such, but an incongruity with one of the characters used in story. As such I have skipped over it:

"That certainly cost me a lot of work finding the pepper," said the King angrily, "and I doubt that the tarts will be all that much the better for it! Pepper indeed!" continued the King. "Why don't you use blotting paper while you're at it?" he added sarcastically.

"I do," replied the Queen, "but not much."

"Very funny!" said the King. "Anyway, now you have your pepper back, so will you please make me the tarts?"

"Without sugar?" said the Queen.

"What's the matter, isn't the jam sweet enough?" asked the King impatiently.

"I need sugar for the dough, and my sugar has been stolen!"

"Oh, not again!" said the King wearily. "These tarts will never get made!"

Well, recovering the sugar turned out to be a relatively simple affair. The sugar was found in the house of the Duchess, and as events proved, it was stolen by either the Duchess or the Cook, but not both. They made the following statements at the trial:

DUCHESS: The cook did not steal the sugar.
COOK: The Duchess stole the sugar.

The one who stole the sugar was lying. (It is not given whether the other one lied or told the truth.)

Which one stole the sugar? Also, was the other one lying or telling the truth?

(Source: Alice in Puzzle Land: A Carrollian Tale for Children Under Eighty by Raymond Smullyan)

The simplest way to tackle this is to consider the implications of guilt for both the Duchess and the Cook. If the Duchess stole the sugar, then it really is the case that the cook did not steal the sugar, but who stole the sugar has to be lying. The only alternative is that the Cook stole the sugar. And, of course, for the second part, anyone in Wonderland might be talking nonsense, even not to their own advantage: the Duchess was also lying.