p(X) is a measure of my uncertainty about outcome X - "p(X) is correct" is the outcome where I determined my uncertainty about X correctly. There are also outcomes where I incorrectly determined my uncertainty about X. I therefore need to have a measure of my uncertainty about outcome "I determined my uncertainty correctly".

Also I need a sketch of the argument that went from the probability of one proposition being 1-epsilon to the probability of a different proposition being smaller than 1-epsilon.

The argument went from the initial probability of one proposition being 1-epsilon to the updated probability of the same proposition being less than 1-epsilon, because there was higher-order uncertainty which multiplies through.

A toy example: We are 90% certain that this object is a blegg. Then, we receive evidence that our method for determining 90% certainty gives the wrong answer one case in ten. We are 90% certain that we are 90% certain, or in other words - we are 81% certain that the object in question is a blegg.

Now that we're 81% certain, we receive evidence that our method is flawed one case in ten - we are now 90% certain that we are 81% certain. Or, we're 72.9% certain. Etc. Obviously epsilon degrades much slower, but we don't have any reason to stop applying it to itself.