Jaynes definitely believed in 0 and 1 probabilities. In Probability Theory: The Logic of Science, equation (2.71), he gives
P(B | A, (A implies B)) = 1
P(A | not B, (A implies B)) = 0
Remember that probabilities are relative to a state of information. If X is a state of information from which we can infer A via deductive logic, then P(A | X) = 1 necessarily. Some common cases of this are
A is a tautology,
we are doing some sort of case analysis and X represents one of the cases being considered, or
we are investigating the consequences of some hypothesis and X represents the hypothesis.
However, Eliezer’s fundamental point is correct when we turn to the states of information of rational beings and propositions that are not tautologies or theorems. If a person’s state of information is X, and P(A | X) = 1, then no amount of contrary evidence can dissuade that person of A. This does not sound like rational behavior, unless A is necessarily true (in the mathematical sense of being a tautology or theorem).
Jaynes definitely believed in 0 and 1 probabilities.
I did not say that he didn’t. I said that he didn’t like Kolmogorov’s axioms. You can also derive Bayes’ rule from Kolmogorov’s axioms; that doesn’t mean Jayes didn’t believe in Bayes’ rule.
I meant that he didn’t think they were the best way to describe probability. IIRC, he thought that they didn’t make it clear why the structure they described is the right way to handle uncertainty. He also may have said that they allow you to talk about certain objects that don’t really correspond to any epistemological concepts. You can find his criticism in one of the appendices to Probability Theory: the Logic of Science.
Jaynes definitely believed in 0 and 1 probabilities. In Probability Theory: The Logic of Science, equation (2.71), he gives
P(B | A, (A implies B)) = 1
P(A | not B, (A implies B)) = 0
Remember that probabilities are relative to a state of information. If X is a state of information from which we can infer A via deductive logic, then P(A | X) = 1 necessarily. Some common cases of this are
A is a tautology,
we are doing some sort of case analysis and X represents one of the cases being considered, or
we are investigating the consequences of some hypothesis and X represents the hypothesis.
However, Eliezer’s fundamental point is correct when we turn to the states of information of rational beings and propositions that are not tautologies or theorems. If a person’s state of information is X, and P(A | X) = 1, then no amount of contrary evidence can dissuade that person of A. This does not sound like rational behavior, unless A is necessarily true (in the mathematical sense of being a tautology or theorem).
I did not say that he didn’t. I said that he didn’t like Kolmogorov’s axioms. You can also derive Bayes’ rule from Kolmogorov’s axioms; that doesn’t mean Jayes didn’t believe in Bayes’ rule.
I don’t know what one thing it means to not like axioms. So I’m not sure what you mean.
I meant that he didn’t think they were the best way to describe probability. IIRC, he thought that they didn’t make it clear why the structure they described is the right way to handle uncertainty. He also may have said that they allow you to talk about certain objects that don’t really correspond to any epistemological concepts. You can find his criticism in one of the appendices to Probability Theory: the Logic of Science.