If you know from the outset that these propositions are unrelated, you already know something quite important about the logical structure of the world that these propositions describe.

Jaynes comes back to this point over and over again, and it's also a major theme of the early chapters in Pearl's Causality:

Probabilistic relationships, such as marginal and conditional independencies, maybe helpful in hypothesizing initial causal structures from uncontrolled observations. However, once knowledge is cast in causal structure, those probabilistic relationships tend to be forgotten; whatever judgements people express about conditional independencies in a given domain are derived from the causal structure acquired. This explains why people feel confident asserting certain conditional independencies (e.g., that the price of beans in China is independent of the traffic in Los Angeles) having no idea whatsoever about the numerical probabilities involved (e.g., whether the price of beans will exceed $10 per bushel).

-- Pearl, Causality p. 25