Published on September 16, 2024 8:11 AM GMT
I notice that reasoning about logical uncertainty does not appear more confusing to me than reasoning about empirical one. Am I missing something?
Consider the classical example from the description of the tag:
Is the googolth digit of pi odd? The probability that it is odd is, intuitively, 0.5. Yet we know that this is definitely true or false by the rules of logic, even though we don't know which. Formalizing this sort of probability is the primary goal of the field of logical uncertainty.
The problem with the 0.5 probability is that it gives non-zero probability to false statements. If I am asked to bet on whether the googolth digit of pi is odd, I can reason as follows: There is 0.5 chance that it is odd. Let P represent the actual, unknown, parity of the googolth digit (odd or even); and let Q represent the other parity. If Q, then anything follows. (By the Principle of Explosion, a false statement implies anything.) For example, Q implies that I will win $1 billion. Therefore the value of this bet is at least $500,000,000, which is 0.5 $1,000,000, and I should be willing to pay that much to take the bet. This is an absurdity.
I don't see how this case is significantly different from an empirical incertainty one:
A coin is tossed and put into an opaque box, without showing you the result. What is the probability that the result of this particular toss was Heads?
Let's assume that it's 0.5. But, then just as in the previous case, we have the same problem: we are assigning non-zero probability to a false statement. And so, by the same logic, if I am asked to bet on whether the coin is Heads or Tails, I can reason as follows: There is 0.5 chance that it is Heads. Let P represent the actual, unknown, state of the outcome of the toss (Heads or Tails); and let Q represent the other state. If Q, then anything follows. For example, Q implies that I will win $1 billion. Therefore the value of this bet is at least $500,000,000, which is 0.5 $1,000,000, and I should be willing to pay that much to take the bet. This is an absurdity.
It's often claimed that important difference between logical and empirical uncertainty is that in the case with the digit of pi, I can, in principle, calculate whether its odd or even if I had arbitrary amount of computing power and therefore become confident in the correct answer. But in case of opaque box, no amount of computing power will help.
First of all, I don't see how it addresses the previous issue of having to assign non-zero credences to wrong statements, anyway. But, beyond that, if I had a tool which allowed me to see through the opaque box, I'd also be able to become confident in the actual state of the coin toss, while this tool would not be helpful at all to figure out the actual parity of googolth digit of pi.
In both cases the uncertainty is relative to my specific conditions be it cognitive resources or acuity. Yes, obviously, if the conditions were different I would reason differently about the problems at hand, and different problems require different modification of conditions. So what? What is stopping us from generalize this two cases as working by the same principles?
Discuss
