Two Puzzles About Truthfulness
1. Suppose you have strong evidence that L are the true laws of nature, where L is a system of deterministic laws. You also have strong evidence that the universe started in the exact microstate P. Your have a choice of either affirming or denying the conjunction of L and P. You want to speak truly. What should you do?
Intuitively, you should affirm. But what would happen if you denied?
Since L is deterministic, L & P either logically entails that you affirm, or it logically entails that you don't affirm. Let's consider both possibilities.
If L & P entails that you don't affirm, then it is logically guaranteed that you speak falsely if you affirm, while you might speak truly if you deny (since you are not 100% sure of L & P).
If, on the other hand, L & P entails that you do affirm, then it is logically guaranteed that you speak truly if you deny, while you might speak falsely if you affirm.
So even though you have strong evidence that L & P is true, and your only goal is to speak the truth, it looks like you should deny L & P.
2. You are giving testimony. You intend to answer all questions truthfully, to the best of your knowledge. One of the questions is whether you intend to answer all questions truthfully, to the best of your knowledge. Should you say yes, given your goal of answering all questions truthfully?
Intuitively, you should. Since you intend to answer all questions truthfully, the correct answer to the current question is yes. And you know that it is.
But what would happen if you said no?
One might think that you would then knowingly give a false answer. But if that's right, then it would no longer be true (if you answered no) that you intend to answer all questions truthfully. And that would make your answer correct! So, on the assumption that by answering no you would knowingly giving a false answer, you would also give a true answer. Contradiction. So you wouldn't knowingly give a false answer by answering no.
Would you knowingly give a true answer? That also seems wrong. You do intend to answer all the other questions truthfully. And if you knowingly give a true answer to the current question, then you intend to answer all questions truthfully. And then the correct answer to the current question is yes, not no. So you can't knowingly give a true answer by saying no.
Could you unknowingly give a true answer by saying no? That is, could you give a true answer (because you really don't intend to answer all questions truthfully) even though you believe the answer is false (because you believe you do intend to answer all questions truthfully)? No. If you believe you are giving a false answer, you can hardly intend to answer all questions truthfully.
The only remaining option is that you would unknowingly give a false answer by saying no. That is, you would give a false answer (because you really do intend to answer all questions truthfully) even though you believe the answer is true (because you believe you don't intend to answer all questions truthfully). That looks consistent. But wait. Now that you realize that you would give a false answer by saying no, you can hardly believe that you would thereby give a true answer.