In both scenarios, the Walrus chose for the Carpenter's probability to be 60% when he could have chosen for it to be 80%. So what's the difference?

When you provide a Bayesian agent information in order to shift its assessment of the probability of some statement S, you do not only affect its assessment of P(S). You also affect its assessment of the probability of various other statements. The two scenarios are not equivalent because they describe different choices between assessments of the probabilities of various statements beyond the statement that pigs have wings (e.g. statements about evolution).

In addition, a Bayesian agent's knowledge about a statement S is not completely captured by its current assessment of P(S); it is captured by all evidence it has accumulated that is relevant to S, and it is not a priori clear whether or not it is possible to do better than this (although it is; see Chapter 18 of Jaynes).

For example, suppose a Bayesian agent is trying to reason about a coin. Consider the following two pieces of prior knowledge:

• "The coin is fair."
• "There is a 50% chance that both sides of the coin are heads and a 50% chance that both sides of the coin are tails."

Both pieces of prior knowledge lead to an assessment of 50% for the probability that the coin will show heads if the agent flips it. However, after observing such a coin flip, the agent's assessment of the probability that the coin will show heads if the agent flips it is still 50% with the first piece of prior knowledge but will either be 100% or 0% with the second piece of prior knowledge, depending on whether or not the first flip was heads or tails. Hence, before the outcome of the first coin flip, the agent's assessment of P(heads) does not capture everything the agent knows about the coin.