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HamletHenna comments on Causal Reference - Less Wrong
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This is very different from how I think about it. Could you expand a little? What do you mean by "when the truth-value of the conjecture is unknown"? That neither C nor ¬C is in your bound agent's store of known theorems?
Let S1, S2 be purported single-conclusion proofs of a statement C.
If I know C is false, the purported proofs are trivially independent because they're fully determined incorrect?
Why is S1 independent of S2 given C is true? Are you saying that learning S2⊢C puts C in our theorem bank, and knowing C is true can change our estimation that S1⊢C , but proofs aren't otherwise mutually informative? If so, what is the effect of learning ⊨C on P(S1⊢C)? And why don't you consider proofs which, say, only differ after the first n steps to be dependent, even given the truth of their shared conclusion?
I meant that the agent is in some state of uncertainty. I'm trying to contrast the case where we are more certain of either C or ¬C with that where we have a significant degree of uncertainty.
Yeah, this is just the trivial case.
I was talking about the simple case where there are no other causal links between the two proofs, like common lemmas or empirical observations. Those do change the causal structure by adding extra nodes and arrows, but I was making the simplifying assumption that we don't have those things.