I've argued in several places that traditional Bayesian reasoning is unable to properly handle embeddedness, logical uncertainty, and related issues.

However, in the light of hindsight, it's possible to imagine a Bayesian pondering these things, and "escaping the trap". This is because the trap was largely made out of assumptions we didn't usually explicitly recognize we were making.

Therefore, in order to aid moving beyond the traditional view, I think it's instructive to paint a detailed picture of what a traditional Bayesian might believe. This can be seen as a partner to my post Radical Probabilism, which explained in considerable detail how to move beyond the traditional Bayesian view.

Simple Belief vs Reflective Belief

There is an important distinction between the explicit dogmas of a view (ie, what adherents explicitly endorse) vs what one would need to believe in order to agree with them. One can become familiar with this distinction by studying logic, especially Gödel's incompleteness theorems and Tarski's undefinability theorem. In particular:

Axiomatic systems such as set theory cannot proclaim themselves self-consistent (unless they are, indeed, inconsistent). However, adherents believe the system to be consistent. If they seriously doubted this, they would not use the system.
Such axiomatic systems cannot discuss their own semantics (their map-territory correspondence). However, adherents regularly discuss this.

This becomes especially interesting when we're studying rationality (rather than, say, mathematics), because a theory of rationality is supposed to characterize normatively correct thinking. Yet, due to the above facts, philosophers will typically end up in a strange position where they endorse principles very different from the ones they themselves are using. For example, a philosopher arguing that set theory is the ultimate foundation of rational thought would find themselves in the strange position of utilizing "irrational" tools (tools which go beyond set theory) to come to its defense, EG in discussing the semantics of set theory or arguing for its consistency. $^{1}$

I'll use the term simple belief to indicate accepting the explicit dogmas, and reflective belief to indicate accepting the meta-dogmas which justify the dogmas. $^{2, 3}$ (These are not terms intended to stand the test of time; I'm not sure about the best terminology, and probably won...

Filtered Evidence, Filtered Arguments