Difference between revisions of "Logical uncertainty"

From Lesswrongwiki
Jump to: navigation, search
(Created page with "Logical uncertainty applies the rules of probability to logical facts which are not yet known to be true or false, because computation effort has not yet been expended. Is th...")
 
(fix wording)
 
(3 intermediate revisions by one other user not shown)
Line 1: Line 1:
Logical uncertainty applies the rules of probability to logical facts which are not yet known to be true or false, because computation effort has not yet been expended.
+
{{stub}}
  
Is the googolth digit of pi odd? This is a fact that we know is definitely true or falls by the rules of logic, even though we do not know if it is  true or false. The probability that the fact is true intuitively is 0.5. Formalizing this sort of probability is the field of logical uncertainty.
+
Logical uncertainty applies the rules of probability to logical facts which are not yet known to be true or false.
  
One basic problem with the idea of logical uncertainty is that it gives non-zero probability to false statements. By the Principle of Explosion, a false statement implies anything. So, if I am asked to bet $100 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 parity of this digit (odd or even, I don't know which) and Q represent the parity which is not the parity of this digit. If Q, then I am about to make 1 billion dollars (Principle of Explosion). Therefore the value of this bet is at very least $500,000,050, which is 0.5 * $1,000,000 plus the usual $50 that one would calculate on a $100 bet at 1-to-1 odds. This logic, of course, would lead to any desired expected value on the bet, an absurdity. Clearly one could in principle run a supercomputer to calculate this digit and resolve the matter, and no unlimited source of money would pop into existence.
+
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. Only expenditure of finite computational power stands between the uncertainty and 100% certainty.
 +
 
 +
Logical uncertainty is closely related to the problem of counterfactuals. Ordinary probability theory relies on counterfactuals. For example, I see a coin that came up heads, and I say that the probability of tails was 0.5, even though clearly, given all air currents and muscular movements involved in throwing that coin, the probability of tails was 0.0. Yet we can imagine this possible impossible world where the coin came up tails.  In the case of logical uncertainly, it is hard to imagine a world in which mathematical facts are different.
 +
 
 +
==References==
 +
[https://intelligence.org/files/QuestionsLogicalUncertainty.pdf Questions of Reasoning Under Logical Uncertainty] by Nate Soares and Benja Fallenstein.

Latest revision as of 13:27, 4 March 2016

Logical uncertainty applies the rules of probability to logical facts which are not yet known to be true or false.

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. Only expenditure of finite computational power stands between the uncertainty and 100% certainty.

Logical uncertainty is closely related to the problem of counterfactuals. Ordinary probability theory relies on counterfactuals. For example, I see a coin that came up heads, and I say that the probability of tails was 0.5, even though clearly, given all air currents and muscular movements involved in throwing that coin, the probability of tails was 0.0. Yet we can imagine this possible impossible world where the coin came up tails. In the case of logical uncertainly, it is hard to imagine a world in which mathematical facts are different.

References

Questions of Reasoning Under Logical Uncertainty by Nate Soares and Benja Fallenstein.