Summary: CFAR proposes double crux as a method to resolve disagreement: instead of arguing over some belief B, one should look for a crux (C) which underlies it, such that if either party changed their mind over C, they would change their mind about B.

I don't think double crux is that helpful, principally because 'double cruxes' are rare in topics where reasonable people differ (and they can be asymmetric, be about a considerations strength rather than direction, and so on). I suggest this may diagnose the difficulty others have noted in getting double crux to 'work'. Good philosophers seem to do much better than double cruxing using different approaches.

I aver the strengths of double crux are primarily other epistemic virtues, pre-requisite for double crux, which are conflated with double cruxing itself (e.g. it is good to have a collaborative rather than combative mindset when disagreeing). Conditional on having this pre-requisite set of epistemic virtues, double cruxing does not add further benefit, and is probably inferior to other means of discussion exemplified by good philosophers. I recommend we look elsewhere.

What is a crux?

From Sabien's exposition, a crux for some belief B is another belief C which if one changed one's mind about C, one would change one's mind about B. The original example was the impact of school uniforms concealing unhelpful class distinctions being a crux for whether one supports or opposes school uniforms.

A double crux is a particular case where two people disagree over B and have the same crux, albeit going in opposite directions. Say if Xenia believes B (because she believes C) and Yevgeny disbelieves B (because he does not believe C), then if Xenia stopped believing C, she would stop believing B (and thus agree with Yevgeny) and vice-versa.

How common are cruxes (and double cruxes)?

I suggest the main problem facing the 'double crux technique' is that disagreements like Xenia's and Yevgeny's, which can be eventually traced to a single underlying consideration, are the exception rather than the rule. Across most reasonable people on most recondite topics, 'cruxes' are rare, and 'double cruxes' (roughly) exponentially rarer.

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.

Thus if I meet someone else who disagrees with me on (say) whether God exists, it would be remarkable if our disagreement hinges on (for example) the evidential argument of evil, such that if I could persuade them of its soundness they would renounce their faith, and vice versa. Were I persuaded the evidential argument from evil 'didn't work', I expect I would remain fairly sceptical of god's existence; were I to persuade them it 'does work', I would not be surprised if they maintained other evidence nonetheless makes god's existence likely on the total balance of evidence. And so on and so forth for other issues where reasonable people disagree. I suspect a common example would be reasonably close agreement on common information, yet beliefs diverging based on 'priors', comprised of a melange of experiences, gestalts, intuitions, and other pieces of more 'private' evidence.

Auxiliary challenges to double crux

I believe there are other difficulties with double crux, somewhat related to the above:

Crux-asymmetry

As implied above, even in cases where there is a crux C for person X believing B, C may not be a crux for B for person Y, but it might be something else (A?). (Or, more generally, X and Y's set of cruxes are disjoint). A worked example:

Carl Shulman and I disagree about whether MIRI is doing good research (we have money riding on it). I expect if I lose the bet, I'd change my mind substantially about the quality of MIRI's work (i.e. my view would be favourable rather than unfavourable). I don't see this should be symmetrical between Shulman and I. If he lost the bet, he may still have a generally favourable view of MIRI, and a 'crux' for him maybe some other evidence or collection of evidence.

X or Y may simply differ in the resilience of their credence in B, such that one or the other's belief shifts more on being persuaded on a particular consideration. One commoner scenario (intra-EA) would be if one is trying to chase 'hits', one is probably more resilient to subsequent adverse information than the initial steers that suggested a given thing could be hit.

A related issue is when one person believes they are in receipt of a decisive consideration for or against B their interlocutor is unaware of.

'Changing one's mind' around p=0.5 isn't (that) important

In most practical cases, a difference between 1 and 0.51 or 0 and 0.49 is much more important than between 0.49 and 0.51. Thus disagreements over confidence of dis/belief can be more important, even if they may not count as 'changing one's mind': I probably differ more with a 'convinced Atheist' than an a 'doubter who leans slightly towards Theism'.

Many arguments and considerations are abductive, and so lend strength to a particular belief. Thus a similar challenge applies to proposed cruxes - they may regard the strength, rather than direction, of a given consideration. One could imagine the 'crux' between me and the hypothetical convinced Atheist is they think that the evidential problem of evil provides overwhelming disconfirmation for Theism, whilst I think its persuasive, perhaps decisive, but not so it drives reasonable credence in Theism down to near-zero.

Sabien's exposition recognises this, and so suggests one can 'double crux' over varying credences. So in this sample disagreement, the belief is 'Atheism is almost certain', and the crux is 'the evidential argument from evil is overwhelming'. Yet our language for credences is fuzzy, and so what would be a crux for the difference between (say) 'somewhat confident' versus 'almost certain' hard to nail down in a satisfactory inter-subjective way. An alternative where a change of raw credence is 'changing ones mind' entails all considerations we take to support our credence in a given a belief are cruxes.

Intermezzo

I suggest these difficulties may make a good diagnosis for why double cruxing has not always worked well. Anecdata seems to vary from those who have found it helpful to those who haven't seen any benefit (but perhaps leaning towards the latter), and remarks along the lines of wanting to see a public example.

Raemon's subsequent exegesis helpfully distinguishes between the actual double crux technique. and "the overall pattern of behaviour surrounding this Official Double Crux technique". They also offer a long list of considerations around the latter which may be pre-requisite for double cruxing working well (e.g. Social Skills, Actually Changing your Mind, and so on).

I wonder what value double crux really adds, if Raemon's argument is on the right track. If double cruxing requires many (or most) of the pre-requisites suggested, all disagreements conditioned on meeting these pre-requisites will go about as well whether one uses double crux or some other intuitive means of subsequent discussion.

A related concern of mine is a 'castle-and-keep' esque defence of double crux which arises from equivocating between double crux per se and a host of admirable epistemic norms it may rely upon. Thus when defended double crux may transmogrify from "look for some C which if you changed your mind about you'd change your mind about B too" to a large set of incontrovertibly good epistemic practices "It is better to be collaborative rather than combative in discussion, and be willing to change ones mind, (etc.)" Yet even if double cruxing is associated with (or requires) these good practices, it is not a necessary condition for them.

Good philosophers already disagree better than double cruxing

To find fault, easy; to do better, difficult

- Plutarch (paraphrased)

Per Plutarch's remark, any shortcomings in double crux may count for little if it is the 'best we've got'. However, I believe I can not only offer a better approach, but this approach already exists 'in the wild'. I have the fortune of knowing many extraordinary able philosophers, and not only observe their discussions but (as they also have extraordinary reserves of generosity and forbearance) participate in them as well. Their approach seems to do much better than reports of what double cruxing accomplishes.

What roughly happens is something like this:

  1. X and Y realize their credences on some belief B vary considerably.

  2. X and Y both offer what appear (to their lights) the strongest considerations that push them to a higher/lower credence on B.

  3. X and Y attempt to prioritize these considerations by the sensitivity of credence in B is to each of these, via some mix of resilience, degree of disagreement over these considerations, and so forth.

  4. They then discuss these in order of priority, moving topics when the likely yield drops below the next candidate with some underlying constraint on time.

This approach seems to avoid the 'in theory' objections I raise against double crux above. It seems to avoid some of the 'in practice' problems people observe:

  • These discussions often occur (in fencing terms) at double time, and thus one tends not to flounder trying to find 'double-cruxy' issues. Atheist may engage Theist on attempting to undermine the free will defence to the argument from evil, whilst Theist may engage Atheist on the deficiencies of moral-antirealism to prepare ground for a moral argument for the existence of god. These may be crux-y but they may be highly assymetrical. Atheist may be a compatibilist but grant libertarian free will for the sake of argument, for example: thus Atheist's credence in God will change little even if persuaded the free will defence broadly 'checks out' if one grants libertarian free will, and vice versa.

  • These discussions seldom get bogged down in fundamental disagreements. Although Deontologist and Utilitarian recognise their view on normative ethics is often a 'double crux' for many applied ethics questions (e.g. Euthanasia), they mutually recognise their overall view on normative ethics will likely be sufficiently resilient such that either of them 'changing their mind' based on a conversation is low. Instead they turn their focus to other matters which are less resilient, and thus they anticipate a greater likelihood of someone or other changing their mind.

  • There appears to be more realistic expectations about the result. If Utilitarian and Deontologist do discuss the merits of utilitarianism versus (say) kantianism, there's little expectation of 'resolving their disagreement' or that they will find or mutually crucial considerations. Rather they pick at a particular leading consideration on either side and see whether it may change their confidence (this is broadly reflected in the philosophical literature: papers tend to concern particular arguments or considerations, rather then offering all things considered determinations of broad recondite philosophical positions).

  • There appear to be better stopping rules. On the numerous occasions where I'm not aware of a particularly important consideration, it often seems a better use of everyone's time for me to read about this in the relevant literature rather than continuing to discuss (I'd guess 'reading a book' beats 'discussing with someone you disagree with' on getting more accurate beliefs about a topic per unit time surprisingly often).

Coda: Wherefore double crux?

It is perhaps possible for double crux to be expanded or altered to capture the form of discussion I point to above, and perhaps one can recast all the beneficial characteristics I suggest in double crux verbiage. Yet such a program appears a fool's errand: the core of the idea of double crux introduced at the top of the post is distinct from generally laudable epistemic norms (c.f. Intermezzo, supra), but also the practices of the elite cognisers I point towards in the section above. A concept of double crux so altered to incorporate these things is epiphenomenal - the engine which is driving the better disagreement is simply those other principles and practices double crux has now appropriated, and its chief result is to add terminological overhead, and, perhaps, inapt approximation.

I generally think the rationalist community already labours under too much bloated jargon: words and phrases which are hard for outsiders to understand, and yet do not encode particularly hard or deep concepts. I'd advise against further additions to the lexicon. 'Look for key considerations' captures the key motivation for double crux better than 'double crux' itself, and its meaning is clear.

The practices of exceptional philosophers set a high bar: these are people selected for, and who practice heavily, argument and disagreement. It is almost conceivable that they are better than this than even the rationalist community, notwithstanding the vast and irrefragable evidence of this group's excellence across so many domains. Double crux could still have pedagogical value: it might be a technique which cultivates better epistemic practices, even if those who enjoy excellent epistemic practices have a better alternative. Yet this does not seem the original intent, nor does there appear much evidence of this benefit.

In the introduction to double crux, Sabien wrote that the core concept was 'fairly settled'. In conclusion he writes:

We think double crux is super sweet. To the extent that you see flaws in it, we want to find them and repair them, and we're currently betting that repairing and refining double crux is going to pay off better than try something totally different. [emphasis in original]

I respectfully disagree. I see considerable flaws in double crux, which I don't think have much prospect of adequate repair. Would that time and effort be spent better looking elsewhere.

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I’m the person affiliated with CFAR who has done the most work on Double Crux in the past year. I both teach the unit (and it’s new accompaniment-class “Finding Cruxes") at workshops, and semi-frequently run full or half day Double Crux development-and-test sessions on weekends. (However, I am technically a contractor, not an employee of CFAR.)

In the process of running test sessions, I’ve developed several CFAR units worth of support material or prerequisite for doing Double Crux well. We haven’t yet solved all of the blockers, but attendees of those full-day workshops are much more skilled at applying the technique successfully (according to my subjective impression, and by the count of “successfully resolved" conversations.)

This new content is currently unpublished, but I expect that I’ll put it up on LessWrong in some form (see the last few bullet point below), sometime in the next year.

I broadly agree with this post. Some of my current thoughts:

  • I’m fully aware that Double Crux is hard to use successfully, which is what motivated me to work on improving the usability of the technique in the first place.

  • Despite those usability issues, I have seen it work effectively to the point of completely resolving a disagreement. (Notably, most of the instances I can recall were Double Cruxes between CFAR staff, who have a very high level of familiarity with Double Crux as a concept.)

  • The specific algorithm that we teach at workshops has undergone iteration. The steps we teach now are quite different than those of a year ago.

  • Most of the value of Double Crux, it seems to me, comes not from formal application of the framework, but rather from using conversational moves from Double Crux in “regular” conversations. TAPs to “operationalize”, or “ask what would change your own mind” are very useful. (Indeed, about half of the Double Crux support content is explicitly for training those TAPs, individually.) This is, I think, what you're pointing to with the difference between "the actual double crux technique. and 'the overall pattern of behaviour surrounding this Official Double Crux technique'".

  • In particular, I think that the greatest value of having the Double Crux class at workshops is the propagation of the jargon "crux". It is useful for the CFAR alumni community to have a distinct concept for "a thing that would cause you to change your mind", because that concept can then be invoked in conversation.

  • I think the full stack of habits, TAPs, concepts, and mindsets that lead to resolution of apparently intractable disagreement, is the interesting thing, and what we should be pursuing, regardless of if that stack "is Double Crux." (This is in fact what I'm working on.)

  • Currently, I am unconvinced that Double Crux is the best or “correct” framework for resolving disagreements. Personally, I am more interested in other (nearby) conversational frameworks,

  • In particular, I expect that non-symmetrical methods for grocking another person’s intuitions, as Thrasymachus suggests, to be fruitful. I, personally, currently use an asymmetrical framework much more frequently than I use a symmetric Double Crux framework. (In part because this doesn't require my interlocutor to do anything in particular or have knowledge of any particular conversational frame.)

  • I broadly agree with the section on asymmetry of cruxes (and it is an open curriculum development consideration). One frequently does not find a Double Crux, and furthermore doesn't need to find a Double Crux to make progress: single cruxs are very useful. (The current CFAR unit currently says as much.)

  • There are some non-obvious advantages to finding a Double Crux though, namely that (if successful), you don't just agree about the top-level proposition, but also share the same underlying model. (Double Crux is not, however, the only framework that produces this result.)

I have a few points of disagreement, however. Most notably, how common cruxes are.

I suggest the main problem facing the 'double crux technique' is that disagreements like Xenia's and Yevgeny's, which can be eventually traced to a single underlying consideration, are the exception rather than the rule.

My empirical experience is that disputes can be traced down to a single underlying consideration more frequently than one might naively think, particularly in on-the-fly disagreements about "what we should do" between two people with similar goals (which, I believe, is Double Crux's ideal use case.)

For many recondite topics I think about, my credence it in arises from the balance of a variety of considerations pointing in either direction.

While this is usually true (at least for sophisticated reasoners), it sometimes doesn't bear on the possibility of finding a (single) crux.

For instance, as a very toy example, I have lots of reasons to believe that acceleration due to gravity is about 9.806 m/s^2: the textbooks I've read, the experiments I did in highschool, my credence in the edifice of science, ect.

But, if I were to find out that I were currently on the moon, this would render all of those factors irrelevant. It isn't that some huge event changed my credence about all of the above factors. It's that all of those factors flow into a single higher-level node and if you break the connection between that node and the top level proposition, your view can change drastically, because those factors are no longer important. In one sense it's a massive update, but in another sense, it's only a single bit flipped.

I think that many real to life seemingly intractable disagreements, particularly when each party has a strong and contrary-to-the-other's intuition, have this characteristic. It's not that you disagree about the evidence in question, you disagree about which evidence matters.

Because I think we're on the moon, and you think we're on earth.

But this is often hard to notice, because that sort of background content is something we both take for granted.

Next I'll try to give some realer-to-life examples. (Full real life examples will be hard to convey because they will be more subtle or require more context. Very simplified anecdotes will have to do for now.)

You can notice something like this happening when...

1) You are surprised or taken aback at some piece of information that they other person thinks is true:

"Wait. You think if we had open borders almost the same number of people would immigrate as under the current US immigration policy?!" [This was a full Double Crux from a real conversation, resolved with recourse to available stats and a fermi estimate.]

Whether or not more people will imigrate could very well change your mind about open borders.

2) You have (according to you) knock-down arguments against their case that they seem to concede quickly, that they don't seem very interested in, or that don't change their overall view much.

You're talking with a person who doesn't think that decreasing carbon emissions is important. You give them a bunch of evidence about the havoc that global warming will wreak, and they agree with it all. It turns out (though they didn't quite realize it themselves) that they're expecting that things are so bad that geo-engineering will be necessary, and it's not worth doing anything short of geoengineering. [Fictionalized real example.]

The possibility and/or necessity of geoengineering could easily be a crux for someone in favor of carbon-minimizing interventions.

3) They keep talking about considerations that, to you, seem minor (this also happens between people who agree):

A colleague tells you that something you did was "rude" and seems very upset. It seems to you that it was a little abrasive, but that it is important that actions of that sort be allowed in the social space. Your colleague declares that it is unacceptable to be "rude." It becomes clear that she is operating from a model whereby being "rude" is so chilling to the discourse that it effectively makes discussion impossible. [Real, heavily simplified, example.]

If this were true it might very well cause you to reassess your sense of what is socially acceptable.

Additionally, here are some more examples of Cruxes, that on the face of it, seem too shallow to be useful, but can actually move the conversation forward:

If there were complete nuclear disarmament, more people would die violently. [Real example from a CFAR workshop, though clouded memory.]

If everyone were bisexual, people would have more sex. [I'm not sure if I've actually seen this one, but it seems like a plausible disagreement from watching people Double Crux on a nearby topic.]

CFAR wants to reach as many people as possible. [Real example]

For each of these, we might tend to take the proposition (or its opposite!) as given, but rather frequently, two people disagree about the truth value.

I claim that there is crux-structure hiding in each of these instances, and that instances like these are surprisingly common (acknowledging that they could seem frequent only because I'm looking for them, and the key feature of some other conversational paradigm is at least as common.)

More specifically, I claim that on hard questions and in situations that call for intuitive judgement, it is frequently the case that the two parties are paying attention to different considerations, and some of the time, the consideration that the other person if tracking, if born out, is sufficient to change your view substantially.

. . .

I was hoping to respond to more points here, but this is already long, and, I fear, a bit rambly. As I said, I'll write up my full thoughts at some point.

I'm curious if I could operationalize a bet with Thrasymachus about how similar the next (or final, or 5 years out, or whatever) iteration of disagreement resolution social-technology will be to Double Crux. I don't think I would take 1-to-1 odds, but I might take something like 3-to-1, depending on the operationalization.

Completely aside from the content, I'm glad to have posts like this one, critiquing CFAR's content.

I've attended a CFAR workshop. I agree with you that Double Crux has all of these theoretical flaws, but it actually seems to work reasonably well in practise, even if these flaws make it kind of confusing. In practise you just kind of stumble through. I strongly agree that if the technique was rewritten so that it didn't have these flaws, it would be much easier to learn as the stumbling bit isn't the most confidence inspiring (this is when the in person assistance becomes important).

One of the key elements that haven't seen mentioned here is this separation between trying to persuade the other person and trying to find out where your point of view differs. When you are trying to convince the other person it is much easier to miss, for example, when there's a difference in a core assumption. Double Crux lets you understand the broad structure of their beliefs so that you can at least figure out the right kinds of things to say later to persuade them that won't be immediately dismissed.

I think the second paragraph is good point, and it is a big part of what I think makes Double Crux better, perhaps, from the standard disagreement resolution model.

A specific sub-point that I don't want to be lost in the sea of my previous comment:

A related concern of mine is a 'castle-and-keep' esque defence of double crux which arises from equivocating between double crux per se and a host of admirable epistemic norms it may rely upon. Thus when defended double crux may transmogrify from "look for some C which if you changed your mind about you'd change your mind about B too" to a large set of incontrovertibly good epistemic practices "It is better to be collaborative rather than combative in discussion, and be willing to change ones mind, (etc.)" Yet even if double cruxing is associated with (or requires) these good practices, it is not a necessary condition for them.

I think there's a third path here, which is something like "double crux may be an instrumentally useful tool in causing these admirable epistemic norms to take root, or to move from nominally-good to actually-practiced."

I attempted in the original LW post, and attempt each time I teach double crux, to underscore that double crux has as its casus belli specific failure modes in normal discourse, and that the point is not, actually, to adhere rigidly to the specific algorithm, but rather that the algorithm highlights a certain productive way of thinking and being, and that while often my conversations don't resemble pure double crux, I've always found that a given marginal step toward pure double crux produces value for me.

Which seems to fit with your understanding of the situation, except that you object to a claim that I and CFAR didn't intend to make. You interpreted us (probably reasonably and fairly) as doing a sort of motte-and-bailey bait-and-switch. But what I, at least, meant to convey was something like "so, there are all these really good epistemic norms that are hard to lodge in your S1, and hard to operationalize in the moment. If you do this other thing, where you talk about cruxes and search for overlap, somehow magically that causes you to cleave closer to those epistemic norms, in practice."

It's like the sort of thing where, if I tell you that it's an experiment about breathing, your breathing starts doing weird and unhelpful things. But if I tell you that it's an experiment about calculation, I can get good data on your breathing while your attention is otherwise occupied.

Hopefully, we're not being that deceptive. But I claim that we're basically saying "Do X" because of a borne-out-in-practice prediction that it will result in people doing Y, where Y are the good norms you've identified as seemingly unrelated to the double crux framework. I've found that directly saying "Do Y" doesn't produce the desired results, and so I say "Do X" and then feel victorious when Y results, but at the cost of being vulnerable to criticism along the lines of "Well, yeah, sure, but your intervention was pointed in the wrong direction."

(This is Dan from CFAR.)

Most of the "data" that we have on double crux is of the informal type, from lots of experience doing double crux, trying to teach people double crux, facilitating double cruxes, watching people try to double crux, etc. But some of the data consists of numbers in a spreadsheet. Here are those numbers.

At workshops, when we teach double crux we have people pair off, find a topic which they and their partner disagree about, and try to double crux. We typically give them about 20 minutes, and have a few staff members floating around available to help. At the end of the class, participants get a handout with three questions:

  • How easy was it for you and your partner to find an interesting disagreement to apply the technique to? (0 = very hard, 10 = very easy)

  • Was your conversation valuable / productive / something that you learned from? (0 = not at all, 10 = very much)

  • Did you and your partner find a double crux? (No, Sort Of, Almost, Yes)

With a sample of 124 people, the averages on these were:

6.96 Easy to find an interesting disagreement

7.82 Conversation was valuable

49% Yes found a double crux

The value of the conversation rating was 8.08 among those who found a double crux ("Yes", n=61), 8.14 among those who easily found a disagreement (rating of 7 or higher, n=86), and 8.35 among those who both easily found a disagreement and found a double crux (n=43). (In contrast with 7.56 among those who didn't find a double crux (n=63) and 7.08 among those who had difficulty finding a disagreement (n=38).)

Thanks for presenting this helpful data. If you'll forgive the (somewhat off topic) question, I understand both that you are responsible for evaluation of CFAR, and that you are working on a new evaluation. I'd be eager to know what this is likely to comprise, especially (see various comments) what evidence (if any) is expected to be released 'for public consumption'?

[These don't seem like cruxes to me, but are places where our models differ.]

[...]

a crux for some belief B is another belief C which if one changed one's mind about C, one would change one's mind about B.

[...]

A double crux is a particular case where two people disagree over B and have the same crux, albeit going in opposite directions. Say if Xenia believes B (because she believes C) and Yevgeny disbelieves B (because he does not believe C), then if Xenia stopped believing C, she would stop believing B (and thus agree with Yevgeny) and vice-versa.

[...]

Across most reasonable people on most recondite topics, 'cruxes' are rare, and 'double cruxes' (roughly) exponentially rarer.

It seems like your model might be missing a class of double cruxes:

It doesn't have to be the case that, if my interlocutor and I drew up belief maps, we would both find a load-bearing belief C about which we disagree. Rather, it's often the case that my interlocutor has some 'crucial' argument or belief which isn't on my radar at all, but would indeed change my mind about B if I were convinced it were true. In another framing, I have an implicit crux for most beliefs that there is no extremely strong argument/evidence to the contrary, which can match up against any load-bearing belief the other person has. In this light, it seems to me that one should not be very surprised to find double cruxes pretty regularly.

Further, even when you have a belief map where the main belief rests on many small pieces of evidence, it is usually possible to move up a level of abstraction and summarize all of that evidence in a higher-level claim, which can serve as a crux. This does not address your point about relatively unimportant shifts around 49%/51%, but in practice it seems like a meaningful point.

I guess my overall impression is that including the cases you specify in a double cruxy style look more like epicycles by my lights rather than helpful augmentations to the concept of double crux.

Non-common knowledge cruxes

I had a sentence in the OP on crux asymmetry along the lines of 'another case may be is X believes they have a crux for B which Y is unaware of'. One may frame this along the lines of an implicit crux of 'there's no decisive consideration that changes my mind about B' for which a proposed 'silver bullet' argument would constitute disagreement.

One of the past-times of my mis-spent youth was arguing about god on the internet. A common occurrence was Theist and Atheist would meet, and both would offer their 'pet argument' which they took to be decisive for A/Theism. I'm not sure they were high-quality discussions, so I'd not count it as a huge merit if they satisfy double crux.

I guess this ties back to my claim that on topics on which reasonable people differ, decisive considerations of this type should be very rare. One motivation for this would veer along social epistemiological lines: a claim about a decisive consideration seems to require some explanation as others nonetheless hold the belief the decisive consideration speaks against. Explaining why your interlocutor is not persuaded is easy - they may simply have not come across it. Yet on many of the sort of recondite topics of disagreement one finds that experts are similarly divided to the laity, and usually the experts who disagree with you are aware of the proposed decisive consideration you have in mind (e.g. the non-trivial proportion of economists who are aware of the Laffer curve yet nonetheless support higher marginal tax rates, etc.). Although it is possible there's some systemic cause/bias/whatever which could account for why this section of experts are getting this wrong, it seems the more common explanation (also favoured by an outside view) is that you overrate the importance of the consideration due to inadequate knowledge of rebutting/undercutting defeaters.

I probably have a non-central sample of discussions I observe, so it may be the case there are a large family of cases of 'putative decisive considerations unknown by one party'. Yet in those cases I don't think double cruxing is the right 'next step'. In cases where the consideration has in mind is widely deemed to settle the matter by the relevant body of experts, it seems to approximate a 'lets agree to check it on Wikipedia case' (from a semi recent conversation, "I'm generally quite sympathetic to this particular formulation of the equal weight view on disagreement" "Oh the field has generally turned away from that due to work by Bloggs showing this view can be dutch-booked?" "If that is so that's a pretty decisive show-stopper, can you point me to it?" /end conversation). In cases where the decisive consideration X has in mind is not held as decisive by the expert body, that should be a red flag to X, and X and Y's time, instead of being spent inexpertly hashing out this consideration, is better spent looking the wider field of knowledge to see likely more sophisticated treatment of the same.

Conjunctive cruxes

I agree one could summarise the case in where many small pieces of evidence provide support for the belief could be summarized into some wider conjunction (e.g. "I'd believe god exists if my credence in the argument from evil goes down by this much, and credence in the argument from design goes up by this much") which could be a crux for discussion.

Yet in such cases there's a large disjunction of conjunctions that would lead to a similar shift in credence, as each consideration likely weights at least someone independently on the scales of reason (e.g. I'd believe god exists if my credence in AfE goes down by X, and credence in the argument from design goes up by Y, or credence in AfE goes down by X-e, and credence in AfD goes up by Y+f, or credence in AfE goes down by X-e, credence in AfD goes up by Y, and credence in argument from religious disagreement goes down by Z, etc. etc.) Although there are not continua in practice due to granularity in how we store credences (I don't back myself to be more precise than the first significant digit in most cases), the size of this disjunctive set grows very rapidly with number of relevant considerations. In consequence, there isn't a single neat crux to focus subsequent discussion upon.

I don't think discussion is hopeless in these cases. If X and Y find (as I think they should in most relevant cases) their disagreement arises from varying weights they place on a number of considerations that bear upon B and ¬B, they can prioritize considerations to discuss which they differ the most on, and for which it appears their credences are the least resilient (I guess in essence to optimise expected d(credence)/dt). This is what I observe elite cognisers doing, but this doesn't seem to be double crux to me.

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.

  1. 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).

  2. 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 the key contribution of Double Crux is "diagram what the argument even is, before making it." If you try to make your argument at the same time as you clarify what the precise topic is, you risk getting confused more easily. "Get on the same page first, then debate" also is a practical way to motivate collaborative rather than competitive discussion, in a way that still retains a driving force towards clarity (whereas many other "be nice to each other" priming techniques point you away from being clear.)

gjm and I recently completed an Actual Example of Double Crux.

I posted it to Meta instead of the front page because it was more of a "talk about the community itself" post (which we're trying not to have on the front page). But linking it here seemed like a reasonable compromise.

I have looked for this mysterious "post to Meta" option, and I am confused—is it a thing that is available only to mods? I have neither been able to post to Meta, nor peruse things in Meta, and am afraid it is because I am uniquely stupid.

<3 In the menu on the top left, there's a 'meta' button, and the page it leads you to looks a lot like the front page but with all different posts. When you're in meta, there's an extra 'new post' button under 'recent meta posts', and this will post things to meta.

I like most of this; it seems like the sort of post that's going to lead to significant improvements in people's overall ability to do collaborative truth-seeking, because it makes concrete and specific recommendations that overall seem useful and sane.

However,

principally because 'double cruxes' are rare in topics where reasonable people differ

disagreements like Xenia's and Yevgeny's, which can be eventually traced to a single underlying consideration, are the exception rather than the rule

and similar make me wish that posts like this would start by getting curious about CFAR's "n" of several hundred, rather than implicitly treating it as irrelevant. We've been teaching double crux at workshops for a couple of years now, and haven't stopped the way we've stopped with other classes and concepts that weren't pulling their weight.

My sense is that the combined number of all of the double-crux-doubters and double-crux-strugglers still does not approach, in magnitude, the number of people who have found double crux moderately-to-very useful and workable and helpful (and, in specific, the number o