X and Y both offer what appear (to their lights) the strongest considerations that push them to a higher/lower credence on B.
I think this is a good example of something where the text, interpreted literally, leads to bad technique, and doing it right is relying on a skill that’s perhaps invisible (but is part of the long experience of the philosophical tradition).
A core distinction between “non-Bayesian reasoning” and “Bayesian reasoning,” as I see it, is whether hypotheses are judged against themselves or against each other. The first involves desirable properties like consistency and exceeding probability thresholds; the second involves comparison of likelihoods.
Expressed mathematically, this is the difference between the probability of observations given an explanation, P(o|E), and the probability of an explanation given the observations, P(E|o).
So when a good philosopher considers a belief, like ‘the group should order pizza,’ they are reflexively considering lethal consequences of that belief and comparing it against other hypotheses, and attempt to establish the case that there are no lethal consequences and the hypothesis isn’t just consistent with the available evidence, but is more easily consistent with it than other hypotheses.
But when a good lawyer considers a belief, that isn’t what’s happening. They’re doing something like “how do I make the desired explanation seem very high (or low) credence?”. There would be lots of statements like “if my client is innocent, then 1 is equal to 1″ without observations that the mirroring statement “if my client is guilty, then 1 is equal to 1” is equally true. (Against savvy opponents, you might not try to slip that past them.)
The point of looking for cruxes is that it gets people out of lawyer-mode, where they’re looking for observations that their theory predicts, and into look into the dark mode, where they’re looking for observations that their theory anti-predicts. If my belief that the group should order pizza strongly anti-predicts that the pizza place is closed, and you believe that the pizza place is closed, then that’s an obviously productive disagreement for us to settle. And trying to find one where the opponent disagrees keeps it from being empty—”well, my belief that we should order pizza anti-predicts than 1=0.”
For many recondite topics I think about, my credence it in arises from the balance of a variety of considerations pointing in either direction. Thus whether or not I believe ‘MIRI is doing good work’, ‘God exists’, or ‘The top marginal tax rate in the UK should be higher than its current value’ does not rely on a single consideration or argument, but rather its support is distributed over a plethora of issues. Although in some cases undercutting what I take as the most important consideration would push my degree of belief over or under 0.5, in other cases it would not.
One of the things that I think becomes clear in practice is that the ideal form (where a crux would completely change my mind) degrades gracefully. If I think that the group should get pizza because of twelve different factors, none of which could be individually decisive, leading to overwhelming odds in favor of pizza, then I can likely still identify the factor which would most change the odds if it flipped. (And, since those factors themselves likely have quantitative strengths as opposed to ‘true’ or ‘false’ values, this becomes a slope rather than a discrete value.)
This seems to be driving in the right direction—when sorting my beliefs by relevance, the criterion I’m using is ‘ability to change my mind,’ and then checking to see if you actually believe differently. This is somewhat more cooperative and useful than sorting my beliefs by ‘ability to convince my opponent,’ since I don’t have as good access to the counterfactuals of my opponent’s models.
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 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 think this is a good example of something where the text, interpreted literally, leads to bad technique, and doing it right is relying on a skill that’s perhaps invisible (but is part of the long experience of the philosophical tradition).
A core distinction between “non-Bayesian reasoning” and “Bayesian reasoning,” as I see it, is whether hypotheses are judged against themselves or against each other. The first involves desirable properties like consistency and exceeding probability thresholds; the second involves comparison of likelihoods.
Expressed mathematically, this is the difference between the probability of observations given an explanation, P(o|E), and the probability of an explanation given the observations, P(E|o).
So when a good philosopher considers a belief, like ‘the group should order pizza,’ they are reflexively considering lethal consequences of that belief and comparing it against other hypotheses, and attempt to establish the case that there are no lethal consequences and the hypothesis isn’t just consistent with the available evidence, but is more easily consistent with it than other hypotheses.
But when a good lawyer considers a belief, that isn’t what’s happening. They’re doing something like “how do I make the desired explanation seem very high (or low) credence?”. There would be lots of statements like “if my client is innocent, then 1 is equal to 1″ without observations that the mirroring statement “if my client is guilty, then 1 is equal to 1” is equally true. (Against savvy opponents, you might not try to slip that past them.)
The point of looking for cruxes is that it gets people out of lawyer-mode, where they’re looking for observations that their theory predicts, and into look into the dark mode, where they’re looking for observations that their theory anti-predicts. If my belief that the group should order pizza strongly anti-predicts that the pizza place is closed, and you believe that the pizza place is closed, then that’s an obviously productive disagreement for us to settle. And trying to find one where the opponent disagrees keeps it from being empty—”well, my belief that we should order pizza anti-predicts than 1=0.”
One of the things that I think becomes clear in practice is that the ideal form (where a crux would completely change my mind) degrades gracefully. If I think that the group should get pizza because of twelve different factors, none of which could be individually decisive, leading to overwhelming odds in favor of pizza, then I can likely still identify the factor which would most change the odds if it flipped. (And, since those factors themselves likely have quantitative strengths as opposed to ‘true’ or ‘false’ values, this becomes a slope rather than a discrete value.)
This seems to be driving in the right direction—when sorting my beliefs by relevance, the criterion I’m using is ‘ability to change my mind,’ and then checking to see if you actually believe differently. This is somewhat more cooperative and useful than sorting my beliefs by ‘ability to convince my opponent,’ since I don’t have as good access to the counterfactuals of my opponent’s models.
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.