I also notice that I can’t predict whether you’ll look at the “prioritize discussion based on the slope of your possible update combined with the other party’s belief” version that I give here and say “okay, but that’s not double crux” or “okay, but the motion of double crux doesn’t point there as efficiently as something else” or “that doesn’t seem like the right step in the dance, tho.”
I regret it is unclear what I would say given what I have written, but it is the former (“okay, but that’s not double crux”). I say this for the following reasons.
The consideration with the greatest slope need not be a crux. (Your colleague Dan seems to agree with my interpretation that a crux should be some C necessary for ones attitude over B, so that if you changed your mind about C you’d change your mind about B).
There doesn’t seem to be a ‘double’ either: identifying the slopiest consideration regarding ones own credence doesn’t seem to demand comparing this to the beliefs of any particular interlocutor to look for shared elements.
I guess (forgive me if I’m wrong) what you might say is that although what you describe may not satisfy what was exactly specified in the original introduction to double crux, this was a simplification and these are essentially the same thing. Yet I take what distinguishes double crux over related and anodyne epistemic virtues (e.g. ‘focus on important less-resilient considerations’, ‘don’t act like a lawyer’) is the ‘some C for which if ¬C then ¬B’ characteristic. As I fear may be abundantly obvious, I find eliding this distinction confusing rather than enlightening: if (as I suggest) the distinguishing characteristic of double crux neither works as good epistemic tool nor good epistemic training, that there may be some nearby epistemic norm that does one or both of these is little consolation.
I regret it is unclear what I would say given what I have written, but it is the former (“okay, but that’s not double crux”). I say this for the following reasons.
The consideration with the greatest slope need not be a crux. (Your colleague Dan seems to agree with my interpretation that a crux should be some C necessary for ones attitude over B, so that if you changed your mind about C you’d change your mind about B).
There doesn’t seem to be a ‘double’ either: identifying the slopiest consideration regarding ones own credence doesn’t seem to demand comparing this to the beliefs of any particular interlocutor to look for shared elements.
I guess (forgive me if I’m wrong) what you might say is that although what you describe may not satisfy what was exactly specified in the original introduction to double crux, this was a simplification and these are essentially the same thing. Yet I take what distinguishes double crux over related and anodyne epistemic virtues (e.g. ‘focus on important less-resilient considerations’, ‘don’t act like a lawyer’) is the ‘some C for which if ¬C then ¬B’ characteristic. As I fear may be abundantly obvious, I find eliding this distinction confusing rather than enlightening: if (as I suggest) the distinguishing characteristic of double crux neither works as good epistemic tool nor good epistemic training, that there may be some nearby epistemic norm that does one or both of these is little consolation.