Conjunctions and thresholds

Consider some positive epistemic or doxastic concept E, say knowledge or belief. Suppose that (maybe for a fixed context) E requires a credence threshold t₀: a proposition only falls under E when the credence is t₀ or higher.

Unless the non-credential stuff really, really cooperates, we wouldn’t expect to have closure under conjunction for all cases of E. For if p and q are cases of E that just barely satisfy the credential threshold condition, we wouldn’t expect their conjunction to satisfy it.

Question: Do we have any right to expect closure under conjunction typically, at least with respect to the credential condition? I.e., if p and q are randomly chosen distinct cases of E, is it reasonable to expect that their conjunction falls above the threshold?

Simple Model: The credences of our Es can fall anywhere between t₀ and 1. Let’s suppose that the distribution of the credences is uniform between t₀ and 1. Suppose, two, that distinct Es are statistically independent, so that the probability of the conjunction is the product of the probabilities.

Then there is a simple formula for the probability that the conjunction of randomly chosen distinct Es satisfy the credential threshold condition: (p₀log p₀ + (1 − p₀))/(1 − p₀)². (Fix one credence between p₀ and 1, and calculate the probability that the other credence satisfies the condition; then integrate from p₀ and 1 and divide by 1 − p₀.) We can plug some numbers in.

• At threshold 0.5, probability of conjunction above threshold: 0.61

• At threshold 0.75, probability of conjunction above threshold: 0.55

• At threshold 0.9, probability of conjunction above threshold: 0.52

• At threshold 0.95, probability of conjunction above threshold: 0.51

• At threshold 0.99, probability of conjunction above threshold: 0.502

And the limit as threshold approaches 1 is 1/2.

So, it’s more likely than not that the conjunction satisfies the credential threshold, but on the other hand the probability is not high enough that we can say that it’s typically the conjunction satisfies the threshold.

But the model has two limitations which will affect the above.

Limitation 1: Intuitively, propositions with positive epistemic or doxastic status are more likely to have a credence closer to the low end of the [t₀, 1] interval, rather than being uniformly distributed over it. This is going to make the probability of the conjunction meeting the threshold be lower than the Simple Model predicts.

Limitation 2: Even without being coherentists, we would expect that our doxastic states to “hang together”. Thus, typically, we would expect that if p and q are propositions that have a credence significantly above 1/2, then typically p and q will have a positive statistical correlation (with respect to credences), so that P(p ∧ q)>P(p)P(q), rather than being independent. This means that the Simple Model underestimates the how often the conjunction is above the threshold. In the extreme case that all our doxastic states are logically equivalent, the conjunction will always meet the threshold condition. In more typical cases, the correlation will be weaker, but we would still expect a significant credential correlation.

So it may well be that even if one takes into account Limitation 1, taking into account Limitation 2 will allow one to say that typically conjunctions of Es meet the threshold condition.

Acknowledgment: I am grateful to John Hawthorne for a discussion of closure and thresholds.