If an agent assigns 50% credence to the proposition that not all ravens ever observed are black, countable additivity entails that the agent must be nearly certain the first non-black raven appears within some finite initial segment of observations
A subjective degree of belief, used in Bayesian epistemology to represent how strongly an agent believes a proposition.
entails(describes a logical relationship between statements)
Logically forces or guarantees; if A entails B, then whenever A is true, B must also be true.
finite initial segment(mathematics and logic)
A limited, countable starting portion of something longer (like the first 1,000 observations out of potentially infinite observations).
proposition(Used in the context of a semantic theory sensitive to differences in subject matter.)
The content expressed by a sentence, individuated at least in part by the subject matter of the sentence and the contents of its subsentential expressions.
the raven problem(epistemology and inductive reasoning)
A famous puzzle in philosophy about whether observing non-black things (like white shoes) gives evidence that all ravens are black—it seems illogical but probability theory suggests it does.
Countable additivity is not as innocent as it looks: it rules out the possibility that any agent is indifferent over a countably infinite set of mutully exclusive possibilities. De Finetti (1970, 1972) famously argued that we ought to reject countable additivity since it is conceivable that God could pick out a natural number “at random” and with equal (zero) probability. For another example, suppose you assign 50% credence to the proposition \(\neg B\) that not all ravens that will ever be obse