Comment by mitthrowaway2

Addendum: The closest thing I can think of that matches the GP's description is conditional probabilities.

For example, if it rains tonight then I am 70% sure it will rain tomorrow too. I am also 80% sure it will rain tonight. And I am 100% sure that if it doesn't rain tonight, it also will not rain tomorrow. Then I can chain these together: The chance of rain tomorrow is P(Rain tomorrow) = P(Rain tomorrow | rain tonight) * P(Rain tonight) = 0.8*0.7 = 0.56.

You can assign conditional probabilites to the correctness of your model itself. Then the phrasing is more like "I am 70% confident that it will rain tomorrow, conditional on my understanding of meteorology being correct" and "I am 80% sure that I understand meteorology correctly". Then you also need to add in a term for P(Rain | not understanding correctly).