Professor Quillibrace closed the book.
Mr Blottisham looked thoughtful.
Miss Stray looked amused.
“Well,” said Blottisham, “I think we can all agree that Bellwether is badly organised.”
Quillibrace sighed.
“We have just finished discussing a badly organised book.”
“Yes.”
“And now?”
“A badly organised town.”
Miss Stray smiled.
“I think Bellwether is rather more interesting than that.”
“Why?”
“Because the town doesn't simply contain contradictions.”
Blottisham nodded.
“Correct.”
“It seems to produce them.”
“Also correct.”
“Without anyone necessarily making a mistake.”
Blottisham frowned.
“That can't be right.”
“Why not?”
“Because if something produces a contradiction, someone must have done something wrong.”
Quillibrace looked at him.
“Must they?”
“Yes.”
“Consider the sentence written on the bench.”
“THIS SENTENCE IS FALSE.”
“Yes.”
“If it is true, it is false. If it is false, it is true.”
“Exactly.”
“So who made the mistake?”
Blottisham thought about it.
“The sentence.”
Quillibrace waited.
Blottisham continued.
“It should have been more careful.”
Miss Stray laughed.
Quillibrace rubbed his forehead.
“The sentence is not a person, Blottisham.”
“Then whoever wrote it.”
“Why?”
“Because they knew it was going to cause trouble.”
“Perhaps causing trouble was the point.”
Blottisham looked suspicious.
“That's what philosophers always say when they've made a mess.”
Miss Stray opened the book again.
“I think there is an important difference between a contradiction and a paradox.”
Blottisham brightened.
“Excellent. What is it?”
“A contradiction is a conflict between propositions.”
“Yes.”
“A paradox is something that appears to follow from apparently acceptable reasoning, yet leads to an unacceptable, impossible, or self-undermining conclusion.”
Blottisham nodded.
“So a paradox is a contradiction that has done its homework.”
Quillibrace paused.
“Yes.”
Blottisham looked delighted.
“I've understood it.”
“I would not go that far.”
Miss Stray turned a page.
“The population sign is interesting.”
“The one that says 1,000 including Clara?”
“Yes.”
“At least that one is easy.”
“How?”
“If Clara is included, there are 1,000 people.”
“Yes.”
“But if the population was already 1,000 before she arrived, then there must have been 999 before her.”
“Yes.”
“So the sign isn't wrong.”
“No.”
“It simply depends on when the population was counted.”
Quillibrace nodded.
“Good.”
Blottisham looked pleased.
“Therefore Bellwether has solved the problem.”
“What problem?”
“Counting.”
Miss Stray smiled.
“I think Bellwether has created another one.”
“The sign changing between 999, 1,000 and 1,001?”
“Yes.”
“That's just bad bookkeeping.”
Quillibrace looked at her.
“Is it?”
Miss Stray shook her head.
“No. The interesting thing is that the sign appears to be counting different things at different times.”
Blottisham frowned.
“People?”
“Perhaps.”
“Or possible people?”
“Perhaps.”
“Or people who have arrived?”
“Yes.”
“Or people who will arrive?”
“Yes.”
Blottisham looked troubled.
“That is cheating.”
“Why?”
“Because a population should consist of people who are actually there.”
Quillibrace raised an eyebrow.
“Should it?”
Blottisham looked at him.
“Of course.”
“Why?”
“Because that's what population means.”
Quillibrace smiled.
“Then perhaps the town's problem is not that it cannot count.”
“What is it?”
“It does not agree with you about what counts as a person who is there.”
Blottisham stared at him.
“That is considerably worse.”
Miss Stray nodded.
“And the barber gives us another version of the same problem.”
“The barber's paradox.”
“Yes.”
“He shaves every man who doesn't shave himself.”
“And does he shave himself?”
“If he does, he shouldn't.”
“And if he doesn't?”
“He should.”
Blottisham thought for a moment.
“Perhaps he shaves nobody.”
“That doesn't solve it.”
“No?”
“No. The definition still creates the problem.”
Blottisham looked disappointed.
“Perhaps he has an electric razor.”
Quillibrace closed his eyes.
“Why?”
“Because then the razor shaves him.”
“The barber doesn't.”
“Precisely.”
Miss Stray laughed.
Quillibrace opened his eyes.
“You have changed the subject.”
“That's a legitimate strategy.”
“Not here.”
“Why not?”
“Because the paradox concerns the relation between the barber and the set of men he shaves.”
Blottisham looked thoughtful.
“So the razor is irrelevant.”
“Yes.”
“That seems unfair to the razor.”
Quillibrace said nothing.
Miss Stray turned another page.
“I think the deepest thing in Bellwether is that the characters keep trying to force a question into a form in which it must have an ordinary answer.”
Blottisham looked at her.
“Such as?”
“Is the sentence true or false?”
“Yes.”
“Is the book finished or unfinished?”
“Yes.”
“Is Clara inside the town or outside it?”
“Yes.”
“Has she arrived or not?”
“Yes.”
Blottisham frowned.
“You've noticed something.”
“I think the town keeps exposing the assumptions hidden inside apparently simple questions.”
Quillibrace nodded.
“And that is why the paradoxes work.”
Blottisham leaned back.
“So the town isn't impossible.”
“No.”
“It is our way of describing it that is impossible.”
“Sometimes.”
“Then Bellwether is a language problem.”
Miss Stray looked at him.
“Not necessarily.”
“Why not?”
“Because the contradictions arise from relations between things as well as from what is said about them.”
“The clocks?”
“Yes.”
“The signs?”
“Yes.”
“The road?”
“Yes.”
“The population?”
“Yes.”
Blottisham considered this.
“So it's a world in which the relations have become unreliable.”
Quillibrace smiled.
“Now you're being interesting again.”
Blottisham nodded.
“I occasionally surprise myself.”
There was a pause.
Miss Stray looked at the final page.
“What about the last question?”
“Which one?” asked Quillibrace.
“Did Clara ever actually arrive?”
Blottisham answered immediately.
“Yes.”
Quillibrace looked at him.
“How do you know?”
“She was there.”
“But was she there before she arrived?”
“Yes.”
“And if she was already there, what does arriving mean?”
Blottisham hesitated.
“Moving from not being there to being there.”
“Was she not there?”
“She was.”
“Then she didn't arrive.”
Blottisham frowned.
“But she did.”
Quillibrace waited.
Blottisham looked increasingly annoyed.
“This is absurd.”
“Yes.”
“So which is it?”
Quillibrace smiled.
“That is the question.”
Blottisham looked at Miss Stray.
“Do you know?”
“No.”
“You?”
“No.”
He looked at the closed book.
“I don't like paradoxes.”
Miss Stray smiled.
“Why?”
“Because they don't let you be right.”
Quillibrace shook his head.
“On the contrary. That's precisely what makes them dangerous.”
Blottisham thought about this.
“So the problem with a paradox is that I can reason correctly and still arrive somewhere I don't want to be.”
“Yes.”
“That is extremely unfair.”
“It is also useful.”
“How?”
“It tells us that something in our assumptions needs examining.”
Blottisham nodded slowly.
“So a paradox isn't necessarily a mistake.”
“No.”
“It can be a warning.”
“Yes.”
“A warning that we've asked the wrong question.”
“Sometimes.”
“Or defined something badly.”
“Yes.”
“Or assumed that two things must be mutually exclusive when they aren't.”
“Yes.”
“Or assumed that a thing must have one definite status when it can have different statuses depending on the relation.”
Miss Stray smiled.
“Yes.”
Blottisham looked pleased.
“I think I understand paradoxes now.”
Quillibrace raised an eyebrow.
“What have you understood?”
“That whenever something is impossible, I should blame the definitions.”
Quillibrace sighed.
“Not always.”
Blottisham frowned.
“Then whenever something is impossible, I should blame the town.”
Miss Stray laughed.
Quillibrace stood.
“Perhaps we should leave.”
They walked towards the door.
Blottisham stopped.
“Wait.”
“What is it?” asked Miss Stray.
He pointed at the door.
There were two signs.
One said:
EXIT
The other said:
NOT AN EXIT
Blottisham looked at Quillibrace.
“Which one do we use?”
Quillibrace considered them.
“The door is the same.”
“Yes.”
“So either sign might be wrong.”
“Yes.”
Blottisham nodded.
“I've got it.”
He opened the door.
On the other side was the Senior Common Room.
He looked back at the others.
“I think we're still inside Bellwether.”
Quillibrace looked through the doorway.
Miss Stray smiled.
Blottisham closed the door.
Then opened it again.
The corridor was still there.
He closed it.
Opened it.
Still there.
He looked at Quillibrace.
“That's not a paradox.”
“No?”
“No.”
“What is it?”
Blottisham smiled.
“A badly organised university.”
For a moment there was silence.
Then Miss Stray laughed.
Quillibrace shook his head.
And somewhere, very faintly, a clock began counting backwards.