Logical Uncertainty

Logical Uncertainty is probabilistic uncertainty about the implications of beliefs. (Another way of thinking about it is: uncertainty about computations.) Probability theory typically assumes logical omniscience, IE, perfect knowledge of logic. The easiest way to see the importance of this assumption is coto consider Bayesian reasoning: to evaluate the probability of evidence given a hypothesis, P(e|h), it's necessary to know what the implications of the hypothesis are. However, realistic agents cannot be logically omniscient.

Motivation

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 & Counterfactuals

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.

See Also: Logical Induction

Logical Uncertainty is probabilistic uncertainty appliesabout the rulesimplications of probabilitybeliefs. (Another way of thinking about it is: uncertainty about computations.) Probability theory typically assumes logical omniscience, IE, perfect knowledge of logic. The easiest way to logical facts which are not yet known to be true or false.

Issee 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 valueimportance of this betassumption is at least $500,000,000, which is 0.5 * $1,000,000, and I should be willingco consider Bayesian reasoning: 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 thatevaluate the probability of tails was 0.5, even though clearly,evidence given all air currents and muscular movements involved in throwing that coin,a hypothesis, P(e|h), it's necessary to know what the probabilityimplications 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.hypothesis are. However, realistic agents cannot be logically omniscient.

References

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

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.

Is the googolth digit of pi odd? ThisThe probability that it is a factodd is, intuitively, 0.5. Yet we know that we knowthis is definitely true or fallsfalse by the rules of logic, even though we do notdon't know if it is true or false. The probability that the fact is true intuitively is 0.5.which. Formalizing this sort of probability is the field of logical uncertainty.

One basicThe problem with this ideathe 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, I don't know which)even); and let Q represent the parity which is not the parity of this digit.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 * $1,000,000.000, and I should be willing to pay that much to take the bet. This logic, of course, leads tois an absurdity. Only expenditure of finite computational power stands between the uncertainty and 100% certainty.

Logical uncertainty is closely related to the proof.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.

One basic problem with thethis idea of logical uncertainty is that it gives non-zero probability to false statements. By the Principle of Explosion, a false statement implies anything. 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 thisthe googolth 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 anything follows. (By the Principle of Explosion, a false statement implies anything.) For example, it followsQ implies that I will win $1 billion. Therefore the value of this bet is at least $500,000,050,000, which is 0.5 * $1,000,000 plus the usual $50 value on a $100 bet at 1-to-1 odds.000. This logic, of course, leads to an absurdity. Only expenditure of finite computational stands between the uncertainty and the proof.

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.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, ifIf 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 anything follows. For example, it follows that I am about to make 1 billion dollars (Principle of Explosion).will win $1 billion. 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 calculatevalue on a $100 bet at 1-to-1 odds. This logic, of course, would leadleads to any desired expected value on the bet, an absurdity. Clearly one could in principle run a supercomputer to calculate this digitOnly expenditure of finite computational stands between the uncertainty and resolve the matter,proof.

References

Questions of Reasoning Under Logical Uncertainty by Nate Soares and no unlimited source of money would pop into existence.Benja Fallenstein.

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 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.

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.