Both of these puzzles fall apart if you understand the concepts in Argument Screens Off Authority, A Priori, and Bayes Theorem. Essentially, the notion of “defeat” is extremely silly. In Puzzle 1, for example, what you should really be doing is updating your level of belief in T based on the mathematician’s argument. The order in which you heard the arguments doesn’t matter—the two Bayesian updates will still give you the same posterior regardless of which one you update on first.
Puzzle 2 is similarly confused about “defeat”; the notion of “misleading evidence” in Puzzle 2 is also wrong. If you look at things in terms of probabilities instead of the “known/not known” dichotomy presented in the puzzle, there is no confusion. Just update on the mathematician’s argument and be done with it.
But that’s the thing: you don’t “know” (T). You have a certain degree of belief, which is represented by a real number between 1 and 0, that (T) is true. You can then update this degree of belief based on (RM) and (TM).
Both of these puzzles fall apart if you understand the concepts in Argument Screens Off Authority, A Priori, and Bayes Theorem. Essentially, the notion of “defeat” is extremely silly. In Puzzle 1, for example, what you should really be doing is updating your level of belief in T based on the mathematician’s argument. The order in which you heard the arguments doesn’t matter—the two Bayesian updates will still give you the same posterior regardless of which one you update on first.
Puzzle 2 is similarly confused about “defeat”; the notion of “misleading evidence” in Puzzle 2 is also wrong. If you look at things in terms of probabilities instead of the “known/not known” dichotomy presented in the puzzle, there is no confusion. Just update on the mathematician’s argument and be done with it.
Well, puzzle 2 is a puzzle with a case of knowledge: I know (T). Changing to probabilities does not solve the problem, only changes it!
But that’s the thing: you don’t “know” (T). You have a certain degree of belief, which is represented by a real number between 1 and 0, that (T) is true. You can then update this degree of belief based on (RM) and (TM).