It seems like it might be useful to set aside discussion of the practice of “double crux” and just focus on the concept of a “crux” for a while. My impression is that “crux” is a very useful concept to have (even without double crux), that it is a simpler/easier concept to get than double crux, and that lots of LWers who have heard of it still don’t have a very clear idea of what a “crux” is. (Relatedly, at CFAR workshops these days, the concept of a crux gets introduced before the class on double crux.)
Something is a crux for a belief for a person if changing their mind about the crux will change their mind about that belief. “Cruxiness” is actually a matter of degree, e.g. going from “90% sure of X” to “60% sure of X” is changing your mind less than a shift from 90% to 10%, but more than a shift from 90% to 85%.
A typical belief has many cruxes. For example, if Ron is in favor of a proposal to increase the top marginal tax rate in the UK by 5 percentage points, his cruxes might include “There is too much inequality in the UK”, “Increasing the top marginal rate by a few percentage points would not have much negative effect on the economy”, and “Spending by the UK government, at the margin, produces value”. If he thought that more inequality would be good for society then he would no longer favor increasing the top marginal rate. If he thought that increasing the top marginal rate would be disastrous for the UK economy then he would no longer favor increasing it (even if he didn’t change his mind about there being too much equality). If he thought that marginal government spending was worthless or harmful then he would no longer favor increasing taxes.
There are technically a whole bunch more cruxes, such as most radically skeptical scenarios. If Ron became convinced that he was dreaming and that “the UK” was just a figment of his imagination, then he would no longer favor increasing UK tax rates. So other cruxes of Ron’s include things like “I am not dreaming”, “I am not a Boltzmann brain,” “The UK exists”, etc. But cruxes like these are uninteresting and typically don’t come to mind. The “interestingness” of a crux is also a matter of degree, and of context, in a way that is more complicated than just probabilities and hard to make precise.
To take another example: suppose that Melissa thinks she has a brilliant startup idea of making gizmos, and believes that it’s the best career path for her to take right now. One crux is whether gizmos can be manufactured for less than a certain price. Another is whether many people are willing to buy gizmos for a certain price. If Melissa learned that many entrepreneurs had tried making gizmos and all of them had failed, and some of them were highly capable people who had the skills that she thinks of as her biggest strengths and had the ideas that she thinks of as her biggest insights about gizmos, that would also change her mind about this startup idea. If Melissa explained her startup idea to 5 specific friends of hers (who she sees as having good judgment, and expertise relevant to startups or to gizmos) and all 5 advised her against the startup, then that would also change her mind. So those also point to cruxes, of an “outside view” flavor rather than a “business model” flavor. And there are various other potentially interesting/relevant cruxes (e.g., she wouldn’t start the company if she discovered that she had a $10 million per year job offer from Google, or that she had cancer, or that gizmos are terrible for people even if they’re willing to pay a bunch of money for them).
Cruxes are really important. Lots of useful thinking involves trying to figure out what your cruxes are, or trying to gain information about a particular crux, or checking if the topic that you’ve been thinking about is a crux. These aren’t the only useful kinds of thinking, of course (it’s often also useful to try to get the lay of the land, to follow up on something you feel uneasy about, etc., etc.). But they’re useful, and underutilized, and having a crystallized concept with a one-syllable label makes it easier to do more of them.
One issue is that people often spend a lot of time arguing about things that aren’t cruxes for either of them. Two people who disagree about whether to increase the top marignal tax rate might get into a back-and-forth about the extent to the higher rate will lead to rich people hiding their money in offshore banks, when the answer to that question wouldn’t shift either of their views. Maybe they’re talking about that topic because it seems like a it should be an important consideration (even though it isn’t crucial for either of them). Maybe one of them mentioned the topic briefly as part of a longer argument, and said something about it that the other person disagreed with and therefore responded to. Maybe one of them guessed that it was a crux for the other person and therefore chose to bring it up. Whatever the reason, this subtopic is mostly a waste of time and a distraction from a potentially more interesting conversation that they could be having. Focusing on cruxes helps to avoid these sidetracks because each person is frequently checking “is this a crux for me?” and occasionally asking “is that a crux for you?” (or some reduced-jargon alternative, like “So if you imagine one world where raising the top rate would mostly just lead to rich people hiding their money in offshore banks, and another world where that didn’t happen, would your view on raising the rate be the same in both of those worlds?”).
Focusing on your cruxes also flips around the typical dynamic of a disagreement. Normally, if Alice and Bob disagree about something, then for the most part Alice is trying to change Bob’s mind and Bob is trying to change Alice’s mind. If Alice is used to thinking about her cruxes, Alice can instead mostly be trying to change Alice’s mind. “Raising the top marginal tax rate seems to me like a good idea, but Bob thinks otherwise—maybe there’s something I’m missing?” Alice understands Alice’s mind a lot better than Bob does, so she has the advantage in looking for what sorts of information might shift her views. Bob is helping her doing it, collaboratively, noticing the places where her thinking seems funny, or where she seems to be missing information, or where her model of the world is different from his, and so on. But this looks very different from Bob taking the lead on changing Alice’s views by taking his best guesses, often going mainly on his priors about what things tax-rate-increasers tend to be wrong about.
Obviously this is not the best way to approach every disagreement. In cases like negotiation you have other goals besides “improve my model of this aspect of the world”, if you don’t have much respect for Bob’s thinking then it may not be worth the trouble, and in online discussions with many people and lots of time lag this approach may be impractical. But in cases where you really care about getting the right answer (e.g., because your career success depends on it), and where the other person’s head seems like one of the better sources of information available to you, focusing on your cruxes in a conversation about disagreements can be a valuable approach to take.
(I still haven’t gotten into “double crux”, and am not planning to.)
I’m not sure whether these remarks are addressed ‘as a reply’ to me in particular. That you use the ‘marginal tax rate in the UK’ example I do suggests this might be meant as a response. On the other hand, I struggle to locate the particular loci of disagreement—or rather, I see in your remarks an explanation of double crux which includes various elements I believe I both understand and object to, but not reasons that argue against this belief (e.g. “you think double crux involves X, but actually it is X*, and thus your objection vanishes when this misunderstanding is resolved”, “your objection to X is mistaken as Y”, etc.) If this is a reply, I apologise for not getting it; if it is not, I apologise for my mistake.
In any case, I take the opportunity to suggest to concretely identify one aspect of my disagreement:
A typical belief has many cruxes. For example, if Ron is in favor of a proposal to increase the top marginal tax rate in the UK by 5 percentage points, his cruxes might include “There is too much inequality in the UK”, “Increasing the top marginal rate by a few percentage points would not have much negative effect on the economy”, and “Spending by the UK government, at the margin, produces value”. If he thought that more inequality would be good for society then he would no longer favor increasing the top marginal rate. If he thought that increasing the top marginal rate would be disastrous for the UK economy then he would no longer favor increasing it (even if he didn’t change his mind about there being too much equality). If he thought that marginal government spending was worthless or harmful then he would no longer favor increasing taxes.
This seems to imply agreement with my take that cruxes (per how CFAR sees them) have the ‘if you change your mind about this, you should change your mind about that’, and so this example has the sequence think-esque characteristic that these cruxes are jointly necessary for ron’s belief (i.e. if Ron thinks ¬A, ¬B, or ¬C, he should change his mind about the marginal tax rate). Yet by my lights it seems more typical considerations like these exert weight upon the balance of reason, but not of such strength that their negation provides a decisive consideration against increasing taxes (e.g. it doesn’t seem crazy for Ron to think “Well, I don’t think inequality is a big deal, but other reasons nonetheless favour raising taxes”, or “Even though I think marginal spending by the UK government is harmful, this negative externality could be outweighed by other considerations”).
I think some harder data can provide better information than litigating hypothetical cases. If the claim that a typical belief has many cruxes, one should see that if one asks elite cognisers to state their credence for a belief, and then state their credences for the most crucial few considerations regarding it, the credence for the belief should only be very rarely higher than the lowest credence among the considerations. This is because if most beliefs have many (jointly necessary) cruxes which should usually comprise at least the top few considerations, and thus this conjunction is necessary (but not sufficient) for believing B, and P(one crux) >= P(conjunction of cruxes). In essence ones credence in a belief should be no greater than ones weakest crux (I guess usually the credence in the belief of a sequence-thinking argument should generally approximate a lower credence set by P(crux1)*P(crux2) etc, as these are usually fairly independent.)
In contrast, if I am closer to the mark, one should fairly commonly see the credence for the belief be higher than the lowest credence of the set of important considerations. If each consideration offers a bayesian update favouring B, a set of important considerations that support B may act together (along with other less important considerations) to increase its credence such that one is more confident of B than of some (or all) of the important considerations that support it.
I aver relevant elite cognisers (e.g. superforecasters, the philosophers I point to) will exhibit the property I suggest. I would also venture that when reasonable cognisers attempt to double crux, their credences will also behave in the way I predict.
I agree that it would be good to look at some real examples of beliefs rather than continuing with hypothetical examples and abstract arguments.
Your suggestion for what hard data to get isn’t something that we can do right now (and I’m also not sure if I disagree with your prediction). We do have some real examples of beliefs and (first attempt at stating) cruxes near at hand, in this comment from Duncan and in this post from gjm (under the heading “So, what would change my mind? ”) and Raemon (under the heading “So: My Actual Cruxes”). And I’d recommend that anyone who cares about cruxes or double crux enough to be reading this three-layers-deep comment, and who has never had a double crux conversation, pick a belief of yours, set a 5 minute timer, and spend that time looking for cruxes. (I recommend picking a belief that is near the level of actions, not something on the level of a philosophical doctrine.)
In response to your question about whether my comments were aimed at you:
They were partly aimed at you, partly aimed at other LWers (taking you as one data point of how LWers are thinking about cruxes). My impression is that your model of cruxes and double crux is different from the thing that folks around CFAR actually do, and I was trying to close that gap for you and for other folks who don’t have direct experience with double crux at CFAR.
For my first comment: the OP had several phrases like “traced to a single underlying consideration” which I would not use when talking about cruxes. Melissa’s current belief that she should start a gizmo company isn’t based on a single consideration, it’s a result of the fact that several different factors line up in a way that makes that specific plan look like an especially good idea. So of course she has several different cruxes. Similarly with views on marginal tax rates.
For my second comment: ‘Primarily look for things that would change your own views, not for things that would change the other person’s views’ is one of the core advantages of focusing on cruxes, in my opinion, and it didn’t seem to be a focus of the OP. It’s something that’s missing from your suggested substitute (“Look for key considerations”) and from your discussion of the example of how experts philosophers handle disagreements. e.g., If Theist is the one pressing the moral argument for the existence of God, because Theist guesses that it might shift Atheist’s views, then that is not a conversation based on cruxes. Whereas if Atheist is choosing to focus the discussion on that argument because Atheist thinks it might shift their own views, then it sounds like it is very similar to a conversation based on cruxes.
On the question of whether cruxes are all-or-nothing or a matter of degree: I think of “crux” as a term similar to “belief”. It suggests sharp category boundaries when in fact things are a matter of degree, but it’s often a good enough approximation and it’s easier for a person to think about, learn, and use the rest of the framework if they can fall back on the categorical concept. Replacing “look for cruxes” with “look for considerations to which your beliefs have relatively high credence sensitivity” also seems like a decent approximation. Doing a Value of Information calculation also seems like a decent approximation, at least for the subset of considerations that are within the model. I could say more to try to elaborate on all of this, but it feels like it really needs some concrete examples to point at. If a discussion like this was happening at a workshop, I’d elaborate by looking at the person’s attempts to come up with cruxes and giving them feedback.
(I’ll repeat here: this comment is about cruxes, not about double crux in particular.)
(This is Dan from CFAR.)
It seems like it might be useful to set aside discussion of the practice of “double crux” and just focus on the concept of a “crux” for a while. My impression is that “crux” is a very useful concept to have (even without double crux), that it is a simpler/easier concept to get than double crux, and that lots of LWers who have heard of it still don’t have a very clear idea of what a “crux” is. (Relatedly, at CFAR workshops these days, the concept of a crux gets introduced before the class on double crux.)
Something is a crux for a belief for a person if changing their mind about the crux will change their mind about that belief. “Cruxiness” is actually a matter of degree, e.g. going from “90% sure of X” to “60% sure of X” is changing your mind less than a shift from 90% to 10%, but more than a shift from 90% to 85%.
A typical belief has many cruxes. For example, if Ron is in favor of a proposal to increase the top marginal tax rate in the UK by 5 percentage points, his cruxes might include “There is too much inequality in the UK”, “Increasing the top marginal rate by a few percentage points would not have much negative effect on the economy”, and “Spending by the UK government, at the margin, produces value”. If he thought that more inequality would be good for society then he would no longer favor increasing the top marginal rate. If he thought that increasing the top marginal rate would be disastrous for the UK economy then he would no longer favor increasing it (even if he didn’t change his mind about there being too much equality). If he thought that marginal government spending was worthless or harmful then he would no longer favor increasing taxes.
There are technically a whole bunch more cruxes, such as most radically skeptical scenarios. If Ron became convinced that he was dreaming and that “the UK” was just a figment of his imagination, then he would no longer favor increasing UK tax rates. So other cruxes of Ron’s include things like “I am not dreaming”, “I am not a Boltzmann brain,” “The UK exists”, etc. But cruxes like these are uninteresting and typically don’t come to mind. The “interestingness” of a crux is also a matter of degree, and of context, in a way that is more complicated than just probabilities and hard to make precise.
To take another example: suppose that Melissa thinks she has a brilliant startup idea of making gizmos, and believes that it’s the best career path for her to take right now. One crux is whether gizmos can be manufactured for less than a certain price. Another is whether many people are willing to buy gizmos for a certain price. If Melissa learned that many entrepreneurs had tried making gizmos and all of them had failed, and some of them were highly capable people who had the skills that she thinks of as her biggest strengths and had the ideas that she thinks of as her biggest insights about gizmos, that would also change her mind about this startup idea. If Melissa explained her startup idea to 5 specific friends of hers (who she sees as having good judgment, and expertise relevant to startups or to gizmos) and all 5 advised her against the startup, then that would also change her mind. So those also point to cruxes, of an “outside view” flavor rather than a “business model” flavor. And there are various other potentially interesting/relevant cruxes (e.g., she wouldn’t start the company if she discovered that she had a $10 million per year job offer from Google, or that she had cancer, or that gizmos are terrible for people even if they’re willing to pay a bunch of money for them).
Cruxes are really important. Lots of useful thinking involves trying to figure out what your cruxes are, or trying to gain information about a particular crux, or checking if the topic that you’ve been thinking about is a crux. These aren’t the only useful kinds of thinking, of course (it’s often also useful to try to get the lay of the land, to follow up on something you feel uneasy about, etc., etc.). But they’re useful, and underutilized, and having a crystallized concept with a one-syllable label makes it easier to do more of them.
How are cruxes relevant in disagreements?
One issue is that people often spend a lot of time arguing about things that aren’t cruxes for either of them. Two people who disagree about whether to increase the top marignal tax rate might get into a back-and-forth about the extent to the higher rate will lead to rich people hiding their money in offshore banks, when the answer to that question wouldn’t shift either of their views. Maybe they’re talking about that topic because it seems like a it should be an important consideration (even though it isn’t crucial for either of them). Maybe one of them mentioned the topic briefly as part of a longer argument, and said something about it that the other person disagreed with and therefore responded to. Maybe one of them guessed that it was a crux for the other person and therefore chose to bring it up. Whatever the reason, this subtopic is mostly a waste of time and a distraction from a potentially more interesting conversation that they could be having. Focusing on cruxes helps to avoid these sidetracks because each person is frequently checking “is this a crux for me?” and occasionally asking “is that a crux for you?” (or some reduced-jargon alternative, like “So if you imagine one world where raising the top rate would mostly just lead to rich people hiding their money in offshore banks, and another world where that didn’t happen, would your view on raising the rate be the same in both of those worlds?”).
Focusing on your cruxes also flips around the typical dynamic of a disagreement. Normally, if Alice and Bob disagree about something, then for the most part Alice is trying to change Bob’s mind and Bob is trying to change Alice’s mind. If Alice is used to thinking about her cruxes, Alice can instead mostly be trying to change Alice’s mind. “Raising the top marginal tax rate seems to me like a good idea, but Bob thinks otherwise—maybe there’s something I’m missing?” Alice understands Alice’s mind a lot better than Bob does, so she has the advantage in looking for what sorts of information might shift her views. Bob is helping her doing it, collaboratively, noticing the places where her thinking seems funny, or where she seems to be missing information, or where her model of the world is different from his, and so on. But this looks very different from Bob taking the lead on changing Alice’s views by taking his best guesses, often going mainly on his priors about what things tax-rate-increasers tend to be wrong about.
Obviously this is not the best way to approach every disagreement. In cases like negotiation you have other goals besides “improve my model of this aspect of the world”, if you don’t have much respect for Bob’s thinking then it may not be worth the trouble, and in online discussions with many people and lots of time lag this approach may be impractical. But in cases where you really care about getting the right answer (e.g., because your career success depends on it), and where the other person’s head seems like one of the better sources of information available to you, focusing on your cruxes in a conversation about disagreements can be a valuable approach to take.
(I still haven’t gotten into “double crux”, and am not planning to.)
Hello Dan,
I’m not sure whether these remarks are addressed ‘as a reply’ to me in particular. That you use the ‘marginal tax rate in the UK’ example I do suggests this might be meant as a response. On the other hand, I struggle to locate the particular loci of disagreement—or rather, I see in your remarks an explanation of double crux which includes various elements I believe I both understand and object to, but not reasons that argue against this belief (e.g. “you think double crux involves X, but actually it is X*, and thus your objection vanishes when this misunderstanding is resolved”, “your objection to X is mistaken as Y”, etc.) If this is a reply, I apologise for not getting it; if it is not, I apologise for my mistake.
In any case, I take the opportunity to suggest to concretely identify one aspect of my disagreement:
This seems to imply agreement with my take that cruxes (per how CFAR sees them) have the ‘if you change your mind about this, you should change your mind about that’, and so this example has the sequence think-esque characteristic that these cruxes are jointly necessary for ron’s belief (i.e. if Ron thinks ¬A, ¬B, or ¬C, he should change his mind about the marginal tax rate). Yet by my lights it seems more typical considerations like these exert weight upon the balance of reason, but not of such strength that their negation provides a decisive consideration against increasing taxes (e.g. it doesn’t seem crazy for Ron to think “Well, I don’t think inequality is a big deal, but other reasons nonetheless favour raising taxes”, or “Even though I think marginal spending by the UK government is harmful, this negative externality could be outweighed by other considerations”).
I think some harder data can provide better information than litigating hypothetical cases. If the claim that a typical belief has many cruxes, one should see that if one asks elite cognisers to state their credence for a belief, and then state their credences for the most crucial few considerations regarding it, the credence for the belief should only be very rarely higher than the lowest credence among the considerations. This is because if most beliefs have many (jointly necessary) cruxes which should usually comprise at least the top few considerations, and thus this conjunction is necessary (but not sufficient) for believing B, and P(one crux) >= P(conjunction of cruxes). In essence ones credence in a belief should be no greater than ones weakest crux (I guess usually the credence in the belief of a sequence-thinking argument should generally approximate a lower credence set by P(crux1)*P(crux2) etc, as these are usually fairly independent.)
In contrast, if I am closer to the mark, one should fairly commonly see the credence for the belief be higher than the lowest credence of the set of important considerations. If each consideration offers a bayesian update favouring B, a set of important considerations that support B may act together (along with other less important considerations) to increase its credence such that one is more confident of B than of some (or all) of the important considerations that support it.
I aver relevant elite cognisers (e.g. superforecasters, the philosophers I point to) will exhibit the property I suggest. I would also venture that when reasonable cognisers attempt to double crux, their credences will also behave in the way I predict.
I agree that it would be good to look at some real examples of beliefs rather than continuing with hypothetical examples and abstract arguments.
Your suggestion for what hard data to get isn’t something that we can do right now (and I’m also not sure if I disagree with your prediction). We do have some real examples of beliefs and (first attempt at stating) cruxes near at hand, in this comment from Duncan and in this post from gjm (under the heading “So, what would change my mind? ”) and Raemon (under the heading “So: My Actual Cruxes”). And I’d recommend that anyone who cares about cruxes or double crux enough to be reading this three-layers-deep comment, and who has never had a double crux conversation, pick a belief of yours, set a 5 minute timer, and spend that time looking for cruxes. (I recommend picking a belief that is near the level of actions, not something on the level of a philosophical doctrine.)
In response to your question about whether my comments were aimed at you:
They were partly aimed at you, partly aimed at other LWers (taking you as one data point of how LWers are thinking about cruxes). My impression is that your model of cruxes and double crux is different from the thing that folks around CFAR actually do, and I was trying to close that gap for you and for other folks who don’t have direct experience with double crux at CFAR.
For my first comment: the OP had several phrases like “traced to a single underlying consideration” which I would not use when talking about cruxes. Melissa’s current belief that she should start a gizmo company isn’t based on a single consideration, it’s a result of the fact that several different factors line up in a way that makes that specific plan look like an especially good idea. So of course she has several different cruxes. Similarly with views on marginal tax rates.
For my second comment: ‘Primarily look for things that would change your own views, not for things that would change the other person’s views’ is one of the core advantages of focusing on cruxes, in my opinion, and it didn’t seem to be a focus of the OP. It’s something that’s missing from your suggested substitute (“Look for key considerations”) and from your discussion of the example of how experts philosophers handle disagreements. e.g., If Theist is the one pressing the moral argument for the existence of God, because Theist guesses that it might shift Atheist’s views, then that is not a conversation based on cruxes. Whereas if Atheist is choosing to focus the discussion on that argument because Atheist thinks it might shift their own views, then it sounds like it is very similar to a conversation based on cruxes.
On the question of whether cruxes are all-or-nothing or a matter of degree: I think of “crux” as a term similar to “belief”. It suggests sharp category boundaries when in fact things are a matter of degree, but it’s often a good enough approximation and it’s easier for a person to think about, learn, and use the rest of the framework if they can fall back on the categorical concept. Replacing “look for cruxes” with “look for considerations to which your beliefs have relatively high credence sensitivity” also seems like a decent approximation. Doing a Value of Information calculation also seems like a decent approximation, at least for the subset of considerations that are within the model. I could say more to try to elaborate on all of this, but it feels like it really needs some concrete examples to point at. If a discussion like this was happening at a workshop, I’d elaborate by looking at the person’s attempts to come up with cruxes and giving them feedback.
(I’ll repeat here: this comment is about cruxes, not about double crux in particular.)