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I meant "personal probability" as the confidence at which people intuit a belief as actually anticipatory (vs. a belief they merely assent to as an association.) This level of confidence is on a sliding scale (vs. all or nothing).
Moat-and-bailey. I don't think there was a suggestion in the above post that you meant with probability something that doesn't follow Kolmogorov's axioms and where you can't directly apply Bayes rule. Especially on LW I think it's valuable to call things that don't follow those axioms and therefore aren't what's usually meant with 'probability', 'probability'.