Best question

2008-08-17 19:19

The joke said to be Kit Fine’s favorite goes as follows in one version.

An angel came down for a meeting of the American Philosophical Association. Greeting the assembled philosophers, the angel offered to answer a single question for them. Immediately the philosophers set to arguing about what they should ask. So the angel said, “Alright, you figure out what you want to ask. I’ll come back tomorrow.” And he left the philosophers to deliberate.

Some of the philosophers favored asking conjunctive questions, but others argued persuasively that the angel probably wouldn’t count this as a single question. One philosopher wanted to ask “What is the best question to ask?”, in the hope that some day another angel might make a similar offer, at which point they could then ask the best question. But this suggestion was rejected by those who feared that no such opportunity would arise and did not want to waste their only question.

Finally, the philosophers agreed on the following question: “What is the ordered pair whose first member is the best question to ask, and whose second member is the answer to that question?” Satisfied with their decision, the philosophers awaited the angel’s return the next day, whereupon they posed their question. And the angel replied: “It is the ordered pair whose first member is the question you just asked, and whose second member is the answer I am now giving.” And then he disappeared.

We can formalize the question and answer in the joke by thinking of a question as a function from short answers to propositions. For example, the meaning of the question “What did Waldo eat?” maps the meaning of the answer “tofu” to the proposition that Waldo ate tofu, the meaning of the answer “peppers” to the proposition that Waldo ate peppers, and so on. This approach has a long history.

If the philosophers’ question is Q, then

Q(q,a) = q is the best question to ask ∧ q(a),

where (q,a) is any question-answer pair. If the angel’s short answer is A, then

A = (Q,A).

Hence the proposition that the angel’s answer asserts is

Q(A) = Q is the best question to ask ∧ Q(A).

This equation does not pin down a unique proposition Q(A) but only tells us that Q(A) entails that Q is the best question to ask. That is, the angel could have been dispensing brilliant wisdom or spewing blatant contradiction, but all we know the angel said for sure is that Q is the best question to ask.

Exercise: what are the types of Q and A, such that Q can apply to A yet be part of A?