Let’s say we’re talking about something complicated. Assume that any proposition about the complicated thing can be reformulated as a series of conjunctions.
Suppose Alice thinks P with 90% confidence (and therefore not-P with 10% confidence). Here’s a fully general counterargument that Alice is wrong:
Decompose P into a series of conjunctions Q1, Q2, … Qn, with n > 10. (You can first decompose not-P into R1 and R2, then decompose R1 further, and decompose R2 further, etc.)
Ask Alice to estimate P(Qk | Q1, Q2, … Q{k-1}) for all k.
At least one of these must be over 99% (if we have n = 11 and they were all 99%, then probability of P would be (0.99 ^ 11) = 89.5% which contradicts the original 90%).
Argue that Alice can’t possibly have enough knowledge to place under 1% on the negation of the statement.
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What’s the upshot? When two people disagree on a complicated claim, decomposing the question is only a good move when both people think that is the right way to carve up the question. Most of the disagreement is likely in how to carve up the claim in the first place.
Let’s say we’re talking about something complicated. Assume that any proposition about the complicated thing can be reformulated as a series of conjunctions.
Suppose Alice thinks P with 90% confidence (and therefore not-P with 10% confidence). Here’s a fully general counterargument that Alice is wrong:
Decompose P into a series of conjunctions Q1, Q2, … Qn, with n > 10. (You can first decompose not-P into R1 and R2, then decompose R1 further, and decompose R2 further, etc.)
Ask Alice to estimate P(Qk | Q1, Q2, … Q{k-1}) for all k.
At least one of these must be over 99% (if we have n = 11 and they were all 99%, then probability of P would be (0.99 ^ 11) = 89.5% which contradicts the original 90%).
Argue that Alice can’t possibly have enough knowledge to place under 1% on the negation of the statement.
----
What’s the upshot? When two people disagree on a complicated claim, decomposing the question is only a good move when both people think that is the right way to carve up the question. Most of the disagreement is likely in how to carve up the claim in the first place.