Double crux is one of CFAR's newer concepts, and one that's forced a re-examination and refactoring of a lot of our curriculum (in the same way that the introduction of TAPs and Inner Simulator did previously). It rapidly became a part of our organizational social fabric, and is one of our highest-EV threads for outreach and dissemination, so it's long overdue for a public, formal explanation.

Note that while the core concept is fairly settled, the execution remains somewhat in flux, with notable experimentation coming from Julia Galef, Kenzi Amodei, Andrew Critch, Eli Tyre, Anna Salamon, myself, and others. Because of that, this post will be less of a cake and more of a folk recipe—this is long and meandering on purpose, because the priority is to transmit the generators of the thing over the thing itself. Accordingly, if you think you see stuff that's wrong or missing, you're probably onto something, and we'd appreciate having them added here as commentary.

Casus belli

To a first approximation, a human can be thought of as a black box that takes in data from its environment, and outputs beliefs and behaviors (that black box isn't really "opaque" given that we do have access to a lot of what's going on inside of it, but our understanding of our own cognition seems uncontroversially incomplete).

When two humans disagree—when their black boxes output different answers, as below—there are often a handful of unproductive things that can occur.

The most obvious (and tiresome) is that they'll simply repeatedly bash those outputs together without making any progress (think most disagreements over sports or politics; the people above just shouting "triangle!" and "circle!" louder and louder). On the second level, people can (and often do) take the difference in output as evidence that the other person's black box is broken (i.e. they're bad, dumb, crazy) or that the other person doesn't see the universe clearly (i.e. they're biased, oblivious, unobservant). On the third level, people will often agree to disagree, a move which preserves the social fabric at the cost of truth-seeking and actual progress.

Double crux in the ideal solves all of these problems, and in practice even fumbling and inexpert steps toward that ideal seem to produce a lot of marginal value, both in increasing understanding and in decreasing conflict-due-to-disagreement.


This post will occasionally delineate two versions of double crux: a strong version, in which both parties have a shared understanding of double crux and have explicitly agreed to work within that framework, and a weak version, in which only one party has access to the concept, and is attempting to improve the conversational dynamic unilaterally.

In either case, the following things seem to be required:

  • Epistemic humility. The number one foundational backbone of rationality seems, to me, to be how readily one is able to think "It's possible that I might be the one who's wrong, here." Viewed another way, this is the ability to take one's beliefs as object, rather than being subject to them and unable to set them aside (and then try on some other belief and productively imagine "what would the world be like if this were true, instead of that?").
  • Good faith. An assumption that people believe things for causal reasons; a recognition that having been exposed to the same set of stimuli would have caused one to hold approximately the same beliefs; a default stance of holding-with-skepticism what seems to be evidence that the other party is bad or wants the world to be bad (because as monkeys it's not hard for us to convince ourselves that we have such evidence when we really don't).1
  • Confidence in the existence of objective truth. I was tempted to call this "objectivity," "empiricism," or "the Mulder principle," but in the end none of those quite fit. In essence: a conviction that for almost any well-defined question, there really truly is a clear-cut answer. That answer may be impractically or even impossibly difficult to find, such that we can't actually go looking for it and have to fall back on heuristics (e.g. how many grasshoppers are alive on Earth at this exact moment, is the color orange superior to the color green, why isn't there an audio book of Fight Club narrated by Edward Norton), but it nevertheless exists.
  • Curiosity and/or a desire to uncover truth. Originally, I had this listed as truth-seeking alone, but my colleagues pointed out that one can move in the right direction simply by being curious about the other person and the contents of their map, without focusing directly on the territory.

At CFAR workshops, we hit on the first and second through specific lectures, the third through osmosis, and the fourth through osmosis and a lot of relational dynamics work that gets people curious and comfortable with one another. Other qualities (such as the ability to regulate and transcend one's emotions in the heat of the moment, or the ability to commit to a thought experiment and really wrestle with it) are also helpful, but not as critical as the above.

How to play

Let's say you have a belief, which we can label A (for instance, "middle school students should wear uniforms"), and that you're in disagreement with someone who believes some form of ¬A. Double cruxing with that person means that you're both in search of a second statement B, with the following properties:

  • You and your partner both disagree about B as well (you think B, your partner thinks ¬B).
  • The belief B is crucial for your belief in A; it is one of the cruxes of the argument. If it turned out that B was not true, that would be sufficient to make you think A was false, too.
  • The belief ¬B is crucial for your partner's belief in ¬A, in a similar fashion.

In the example about school uniforms, B might be a statement like "uniforms help smooth out unhelpful class distinctions by making it harder for rich and poor students to judge one another through clothing," which your partner might sum up as "optimistic bullshit." Ideally, B is a statement that is somewhat closer to reality than A—it's more concrete, grounded, well-defined, discoverable, etc. It's less about principles and summed-up, induced conclusions, and more of a glimpse into the structure that led to those conclusions.

(It doesn't have to be concrete and discoverable, though—often after finding B it's productive to start over in search of a C, and then a D, and then an E, and so forth, until you end up with something you can research or run an experiment on).

At first glance, it might not be clear why simply finding B counts as victory—shouldn't you settle B, so that you can conclusively choose between A and ¬A? But it's important to recognize that arriving at B means you've already dissolved a significant chunk of your disagreement, in that you and your partner now share a belief about the causal nature of the universe.

If B, then A. Furthermore, if ¬B, then ¬A. You've both agreed that the states of B are crucial for the states of A, and in this way your continuing "agreement to disagree" isn't just "well, you take your truth and I'll take mine," but rather "okay, well, let's see what the evidence shows." Progress! And (more importantly) collaboration!


This is where CFAR's versions of the double crux unit are currently weakest—there's some form of magic in the search for cruxes that we haven't quite locked down. In general, the method is "search through your cruxes for ones that your partner is likely to disagree with, and then compare lists." For some people and some topics, clearly identifying your own cruxes is easy; for others, it very quickly starts to feel like one's position is fundamental/objective/un-break-downable.


  • Increase noticing of subtle tastes, judgments, and "karma scores." Often, people suppress a lot of their opinions and judgments due to social mores and so forth. Generally loosening up one's inner censors can make it easier to notice why we think X, Y, or Z.
  • Look forward rather than backward. In places where the question "why?" fails to produce meaningful answers, it's often more productive to try making predictions about the future. For example, I might not know why I think school uniforms are a good idea, but if I turn on my narrative engine and start describing the better world I think will result, I can often sort of feel my way toward the underlying causal models.
  • Narrow the scope. A specific test case of "Steve should've said hello to us when he got off the elevator yesterday" is easier to wrestle with than "Steve should be more sociable." Similarly, it's often easier to answer questions like "How much of our next $10,000 should we spend on research, as opposed to advertising?" than to answer "Which is more important right now, research or advertising?"
  • Do "Focusing" and other resonance checks. It's often useful to try on a perspective, hypothetically, and then pay attention to your intuition and bodily responses to refine your actual stance. For instance: (wildly asserts) "I bet if everyone wore uniforms there would be a fifty percent reduction in bullying." (pauses, listens to inner doubts) "Actually, scratch that—that doesn't seem true, now that I say it out loud, but there is something in the vein of reducing overt bullying, maybe?"
  • Seek cruxes independently before anchoring on your partner's thoughts. This one is fairly straightforward. It's also worth noting that if you're attempting to find disagreements in the first place (e.g. in order to practice double cruxing with friends) this is an excellent way to start—give everyone the same ten or fifteen open-ended questions, and have everyone write down their own answers based on their own thinking, crystallizing opinions before opening the discussion.

Overall, it helps to keep the ideal of a perfect double crux in the front of your mind, while holding the realities of your actual conversation somewhat separate. We've found that, at any given moment, increasing the "double cruxiness" of a conversation tends to be useful, but worrying about how far from the ideal you are in absolute terms doesn't. It's all about doing what's useful and productive in the moment, and that often means making sane compromises—if one of you has clear cruxes and the other is floundering, it's fine to focus on one side. If neither of you can find a single crux, but instead each of you has something like eight co-cruxes of which any five are sufficient, just say so and then move forward in whatever way seems best.

(Variant: a "trio" double crux conversation in which, at any given moment, if you're the least-active participant, your job is to squint at your two partners and try to model what each of them is saying, and where/why/how they're talking past one another and failing to see each other's points. Once you have a rough "translation" to offer, do so—at that point, you'll likely become more central to the conversation and someone else will rotate out into the squinter/translator role.)

Ultimately, each move should be in service of reversing the usual antagonistic, warlike, "win at all costs" dynamic of most disagreements. Usually, we spend a significant chunk of our mental resources guessing at the shape of our opponent's belief structure, forming hypotheses about what things are crucial and lobbing arguments at them in the hopes of knocking the whole edifice over. Meanwhile, we're incentivized to obfuscate our own belief structure, so that our opponent's attacks will be ineffective.

(This is also terrible because it means that we often fail to even find the crux of the argument, and waste time in the weeds. If you've ever had the experience of awkwardly fidgeting while someone spends ten minutes assembling a conclusive proof of some tangential sub-point that never even had the potential of changing your mind, then you know the value of someone being willing to say "Nope, this isn't going to be relevant for me; try speaking to that instead.")

If we can move the debate to a place where, instead of fighting over the truth, we're collaborating on a search for understanding, then we can recoup a lot of wasted resources. You have a tremendous comparative advantage at knowing the shape of your own belief structure—if we can switch to a mode where we're each looking inward and candidly sharing insights, we'll move forward much more efficiently than if we're each engaged in guesswork about the other person. This requires that we want to know the actual truth (such that we're incentivized to seek out flaws and falsify wrong beliefs in ourselves just as much as in others) and that we feel emotionally and socially safe with our partner, but there's a doubly-causal dynamic where a tiny bit of double crux spirit up front can produce safety and truth-seeking, which allows for more double crux, which produces more safety and truth-seeking, etc.


First and foremost, it matters whether you're in the strong version of double crux (cooperative, consent-based) or the weak version (you, as an agent, trying to improve the conversational dynamic, possibly in the face of direct opposition). In particular, if someone is currently riled up and conceives of you as rude/hostile/the enemy, then saying something like "I just think we'd make better progress if we talked about the underlying reasons for our beliefs" doesn't sound like a plea for cooperation—it sounds like a trap.

So, if you're in the weak version, the primary strategy is to embody the question "What do you see that I don't?" In other words, approach from a place of explicit humility and good faith, drawing out their belief structure for its own sake, to see and appreciate it rather than to undermine or attack it. In my experience, people can "smell it" if you're just playing at good faith to get them to expose themselves; if you're having trouble really getting into the spirit, I recommend meditating on times in your past when you were embarrassingly wrong, and how you felt prior to realizing it compared to after realizing it.

(If you're unable or unwilling to swallow your pride or set aside your sense of justice or fairness hard enough to really do this, that's actually fine; not every disagreement benefits from the double-crux-nature. But if your actual goal is improving the conversational dynamic, then this is a cost you want to be prepared to pay—going the extra mile, because a) going what feels like an appropriate distance is more often an undershoot, and b) going an actually appropriate distance may not be enough to overturn their entrenched model in which you are The Enemy. Patience- and sanity-inducing rituals recommended.)

As a further tip that's good for either version but particularly important for the weak one, model the behavior you'd like your partner to exhibit. Expose your own belief structure, show how your own beliefs might be falsified, highlight points where you're uncertain and visibly integrate their perspective and information, etc. In particular, if you don't want people running amok with wrong models of what's going on in your head, make sure you're not acting like you're the authority on what's going on in their head.

Speaking of non-sequiturs, beware of getting lost in the fog. The very first step in double crux should always be to operationalize and clarify terms. Try attaching numbers to things rather than using misinterpretable qualifiers; try to talk about what would be observable in the world rather than how things feel or what's good or bad. In the school uniforms example, saying "uniforms make students feel better about themselves" is a start, but it's not enough, and going further into quantifiability (if you think you could actually get numbers someday) would be even better. Often, disagreements will "dissolve" as soon as you remove ambiguity—this is success, not failure!

Finally, use paper and pencil, or whiteboards, or get people to treat specific predictions and conclusions as immutable objects (if you or they want to change or update the wording, that's encouraged, but make sure that at any given moment, you're working with a clear, unambiguous statement). Part of the value of double crux is that it's the opposite of the weaselly, score-points, hide-in-ambiguity-and-look-clever dynamic of, say, a public political debate. The goal is to have everyone understand, at all times and as much as possible, what the other person is actually trying to say—not to try to get a straw version of their argument to stick to them and make them look silly. Recognize that you yourself may be tempted or incentivized to fall back to that familiar, fun dynamic, and take steps to keep yourself in "scout mindset" rather than "soldier mindset."


This is the double crux algorithm as it currently exists in our handbook. It's not strictly connected to all of the discussion above; it was designed to be read in context with an hour-long lecture and several practice activities (so it has some holes and weirdnesses) and is presented here more for completeness and as food for thought than as an actual conclusion to the above.

1. Find a disagreement with another person

  • A case where you believe one thing and they believe the other
  • A case where you and the other person have different confidences (e.g. you think X is 60% likely to be true, and they think it’s 90%)

2. Operationalize the disagreement

  • Define terms to avoid getting lost in semantic confusions that miss the real point
  • Find specific test cases—instead of (e.g.) discussing whether you should be more outgoing, instead evaluate whether you should have said hello to Steve in the office yesterday morning
  • Wherever possible, try to think in terms of actions rather than beliefs—it’s easier to evaluate arguments like “we should do X before Y” than it is to converge on “X is better than Y.”

3. Seek double cruxes

  • Seek your own cruxes independently, and compare with those of the other person to find overlap
  • Seek cruxes collaboratively, by making claims (“I believe that X will happen because Y”) and focusing on falsifiability (“It would take A, B, or C to make me stop believing X”)

4. Resonate

  • Spend time “inhabiting” both sides of the double crux, to confirm that you’ve found the core of the disagreement (as opposed to something that will ultimately fail to produce an update)
  • Imagine the resolution as an if-then statement, and use your inner sim and other checks to see if there are any unspoken hesitations about the truth of that statement

5. Repeat!


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. In particular, we believe that embracing the spirit of this mental move has huge potential for unlocking people's abilities to wrestle with all sorts of complex and heavy hard-to-parse topics (like existential risk, for instance), because it provides a format for holding a bunch of partly-wrong models at the same time while you distill the value out of each.

Comments appreciated; critiques highly appreciated; anecdotal data from experimental attempts to teach yourself double crux, or teach it to others, or use it on the down-low without telling other people what you're doing extremely appreciated.

- Duncan Sabien

[1]One reason good faith is important is that even when people are "wrong," they are usually partially right—there are flecks of gold mixed in with their false belief that can be productively mined by an agent who's interested in getting the whole picture. Normal disagreement-navigation methods have some tendency to throw out that gold, either by allowing everyone to protect their original belief set or by replacing everyone's view with whichever view is shown to be "best," thereby throwing out data, causing information cascades, disincentivizing "noticing your confusion," etc.

The central assumption is that the universe is like a large and complex maze that each of us can only see parts of. To the extent that language and communication allow us to gather info about parts of the maze without having to investigate them ourselves, that's great. But when we disagree on what to do because we each see a different slice of reality, it's nice to adopt methods that allow us to integrate and synthesize, rather than methods that force us to pick and pare down. It's like the parable of the three blind men and the elephant—whenever possible, avoid generating a bottom-line conclusion until you've accounted for all of the available data.

The agent at the top mistakenly believes that the correct move is to head to the left, since that seems to be the most direct path toward the goal. The agent on the right can see that this is a mistake, but it would never have been able to navigate to that particular node of the maze on its own.

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A suggestion: don't avoid feelings. Instead, think of feelings as deterministic and carrying valuable information; feelings can be right even when our justifications are wrong (or vice versa).

Ultimately, this whole technique is about understanding the causal paths which led to both your own beliefs, and your conversational partner's beliefs. In difficult areas, e.g. politics or religion, people inevitably talk about logical arguments, but the actual physical cause of their belief is very often intuitive and emotional - a feeling. Very often, those feelings will be the main "crux".

For instance, I've argued that the feelings of religious people offer a much better idea of religion's real value than any standard logical argument - the crux in a religious argument might be entirely a crux of feelings. In another vein, Scott's thrive/survive theory offers psychological insight on political feelings more than political arguments, and it seems like it would be a useful crux generator - i.e., would this position seem reasonable in a post-scarcity world, or would it seem reasonable during a zombie outbreak?

7Brendan Long
This feels like a point but I'm having trouble thinking of specific examples. What would a crux like this look like and how could you resolve the disagreement involved with it?

Let's use the school uniforms example.

The post mentions "uniforms make students feel better about themselves" as something to avoid. But that claim strongly suggests a second statement for the claimant: "I would have felt better about myself in middle school, if we'd had uniforms." A statement like that is a huge gateway into productive discussion.

First and foremost, that second statement is very likely the true cause for the claimant's position. Second, that feeling is something which will itself have causes! The claimant can then think back about their own experiences, and talk about why they feel that way.

Of course, that creates another pitfall to watch out for: argument by narrative, rather than statistics. It's easy to tell convincing stories. But if one or both participants know what's up, then each participant can produce a narrative to underlie their own feelings, and then the real discussion is over questions like (1) which of those narratives is more common in practice, (2) should we assign more weight to one type of experience, (3) what other types of experiences should we maybe consider, and (4) does the claim make sense even given the experience?

Th... (read more)

I always thought of school uniforms as being a logical extension of the pseudo-fascist/nationalist model of running them. (I mean this in the pre-world war descriptive sense rather than the rhetorical sense that arose after the wars) Lots of schools, at least in America, try to encourage a policy of school unity with things like well-funded sports teams and school pep rallies. I don't know how well these policies work in practice, but if they're willing to go as far as they have now, school uniforms might contribute to whatever effects they hope to achieve. My personal opinion is in favor of school uniforms, but I'm mostly certain that's because I'm not too concerned with fashion or displays of wealth. I'd have to quiz some other people to find out for sure.
Are the uniforms at US schools reasonably practical, comfortable and do they have reasonable colour, e.g. not green ? As a girl of socialism, I experienced pioneer uniforms, which were not well designed. They forced short skirts on girls, which are impractical in some weather. The upper part, the shirt, needed to be ironed. There was no sweather or coat to unify kids in winter.. My mother once had to stand coatless in winter in a wellcome row for some event. I can also imagine some girls having aesthetic issues with the exposed legs or unflattering color. But what are the uniforms in the US usually like ?
What's wrong with green?
Well, it's not easy.
Schools in the US are run locally and are very diverse. Some schools (mostly private ones) require uniforms, but as far as I know the majority of public schools do not. There are dress codes -- all schools have policies about what's acceptable to wear to school -- but within those guidelines you can generally wear what you want.
1[DEACTIVATED] Duncan Sabien
I should note that my own personal opinions on school uniforms are NOT able-to-be-determined from this article.

Anecdotal data time! We tried this at last week’s Chicago rationality meetup, with moderate success. Here’s a rundown of how we approached the activity, and some difficulties and confusion we encountered.


Before the meeting, some of us came up with lists of possibly contentious topics and/or strongly held opinions, and we used those as starting points by just listing them off to the group and seeing if anyone held the opposite view. Some of the assertions on which we disagreed were:

  • Cryonic preservation should be standard medical procedure upon death, on an opt-out basis
  • For the average person, reading the news has no practical value beyond social signalling
  • Public schools should focus on providing some minimum quality of education to all students before allocating resources to programs for gifted students
  • The rationality movement focuses too much of its energy on AI safety
  • We should expend more effort to make rationality more accessible to ‘normal people’

We paired off, with each pair in front of a blackboard, and spent about 15 minutes on our first double crux, after the resolution of which the conversations mostly devolved. We then came together, gave feedback, switche... (read more)

5[DEACTIVATED] Duncan Sabien
Very useful. I don't have the time to give you the detailed response you deserve, but I deeply appreciate the data (and Eli says hi).

Personally, I am still eagerly waiting for CFAR to release more of their methods and techniques. A lot of them seem to be already part of the rationalist diaspora's vocabulary -- however, I've been unable to find descriptions of them.

For example, you mention "TAP"s and the "Inner Simulator" at the beginning of this article, yet I haven't had any success googling those terms, and you offer no explanation of them. I would be very interested in what they are!

I suppose the crux of my criticism isn't that there are techniques you haven't released yet, nor that rationalists are talking about them, but that you mention them as though they were common knowledge. This, sadly, gives the impression that LWers are expected to know about them, and reinforces the idea that LW has become a kind of elitist clique. I'm worried that you are using this in order to make aspiring rationalists, who very much want to belong, come to CFAR events, to be in the know.

Decided to contribute a bit: here's a new article on TAPs! :)

TAPs = Trigger Action Planning; referred to in the scholarly literature as "Implementation intentions". The Inner Simulator unit is CFAR's way of referring to what you actually expect to see happen (as contrasted with, say, your verbally stated "beliefs".) Good point re: being careful about implied common knowledge.
Here's a writeup on the Asana blog about Inner Simulator, based on a talk CFAR gave there a few years ago.

I'm excited to try this out in both strong and weak forms.

There are parallels of getting to the crux of something in design and product discovery research. It is called Why Laddering. I have used it when trying to understand the reasons behind a particular person's problem or need. If someone starts too specific it is a great way to step back from solutions they have preconceived before knowing the real problem (or if there is even one).

It also attempts to get to a higher level in a system of needs.

Are there ever times that the double crux have resulted in narrowing with each crux? Or do they generally become more general?

There is the reverse as well called How Laddering which tries to find solutions for more general problems.

It sounds like the 'reverse double crux' would be to find a new, common solution after a common crux has been found.

In programming, we call that The XY Problem.
I hadn't heard of that before. Thanks!
Nice association. I see this model as building on Laddering or the XY problem, because it also looks for a method of falsifiability. It's closer to a two-sided use of Eric Ries' Lean Startup (the more scientific version), where a crux = leap of faith assumption. I've called the LoFA a "leap of faith hypothesis", and your goal is to find the data that would tell you the assumption is wrong. The other product design thinker with a similar approach is Tom Chi who uses a conjecture -> experiment -> actuals -> decision framework. In all of these methods, the hard work/thinking is actually finding a crux and how to falsify it. Having an "opponent" to collaborate with may make us better at this.

Thanks for writing this up! One thing I particularly like about this technique is that it seems to really help with getting into the mindset of seeing disagreements as good (not an unpleasant thing to be avoided), and seeing them as good for the right reasons - for learning more about the world/your own beliefs/changing your mind (not a way to assert status/dominance/offload anger etc.)

I feel genuinely excited about paying more attention to where I disagree with others and trying to find the crux of the disagreement now, in a way I didn't before reading this post.

It's interesting how you assume that disagreements are not likely to lead to bad real-world consequences.
I don't think I was assuming that, but good point - there are of course lots of nuances to whether disagreements are good/bad/useful/problematic in various ways. I definitely wasn't meaning to say "disagreements are always a good thing", but rather something much weaker, like "disagreements are not always a bad thing to be avoided, and can often be a good opportunity to learn more about the world and/or your own reasons for your beliefs, and internalising this mindset more fully seems very useful." I don't think this means we should try and create disagreements where none exist already, or that the world wouldn't be a better place if people agreed more. But assuming a lot of disagreements already exist, identifying those disagreements can be a very good thing if you have good tools for resolving/making better sense of them. So when I say I'm excited about finding more disagreements, I mean that given the assumption that those disagreements already exist, and would have any potential bad real-world consequences regardless of whether I'm aware of them or not.
Yep, these... clarifications make the statement a lot more reasonable :-) It's just that the original unconditional claim had a lot of immediate counterexamples spring to mind, e.g.: * When there is a power disparity between the two disagreeing (e.g. arguing with cops is generally a bad idea) * When you make yourself a target (see e.g. this) * When it's just unnecessary (imagine two neighbours who get along quite well until they discover that one is a Trumper and one is a Clintonista)
Ooh, I hope you're not upset because I disagree with you! Your examples, numbered for easier reference: I'm not sure what disagreement you had in mind with respect to [2]; maybe whether 'forking someone's repo' was a sexual reference? For [1], I can think of counter-counter-examples, e.g. where a copy suspects that you've committed a crime and you know you haven't committed that crime. If you could identify a shared crux of that disagreement you might be able to provide evidence to resolve that crux and exonerate yourself (before you're arrested, or worse). For [3], I can also think of counter-counter-examples, e.g. where because both neighbors got along well with each other before discovering their favored presidential candidates, they're inclined to be charitable towards each other and learn about why they disagree. I'm living thru something similar to this right now with someone close to me. I agree with Jess and think that discovering disagreement can be a generally positive event, regardless of the overall negative outcomes pertaining to specific disagreements. Saying 'water is good to drink' doesn't imply 'you can't hurt or kill yourself by drinking too much water'.
It's hard to upset me :-) Re [2] the disagreement was about whether that joke was (socially) acceptable. One can always come up with {counter-}examples -- iterate to depth desired -- but that's not really the point. The point is to which degree a general statement applies unconditionally. I happen to think that "a disagreement is a positive event" is a highly conditional observation. Oh, and I don't recommend double-cruxing cops. I suspect this will go really badly.
Disagreements can lead to bad real-world consequences for (sort of) two reasons: 1) At least one person is wrong and will make bad decisions which lead to bad consequences. 2) The argument itself will be costly (in terms of emotional cost, friendship, perhaps financial cost, etc). In terms of #1, an unnoticed disagreement is even worse than an unsettled disagreement; so thinking about #1 motivates seeking out disagreements and viewing them as positive opportunities for intellectual progress. In terms of #2, the attitude of treating disagreements as opportunities can also help, but only if both people are on board with that. I'm guessing that is what you're pointing at? My strategy in life is something like: seek out disagreements and treat them as delicious opportunities when in "intellectual mode", but avoid disagreements and treat them as toxic when in "polite mode". This heuristic isn't always correct. I had to be explicitly told that many people often don't like arguing even over intellectual things. Plus, because of #1, it's sometimes especially important to bring up disagreements in practical matters (that don't invoke "intellectual mode") even at risk of a costly argument. It seems like something like "double crux attitude" helps with #2 somewhat, though.
See my reply to Jess.
Disagreements are not always bad, however what happens in the real world most of the time is that the disagreements are not based on rational thought and logic, but instead on some fluffy slogans and "feelings". People don't actually go deep into examining whether their argument makes sense and is supported by sound facts and not things like narrative fallacy. In fact, when you point out to other people that what they are saying is not supported by any logical arguments, they get even more defensive and irrational.
What you're missing is a rather important concept called "values".

I'm going to go out and state that the chosen example of "middle school students should wear uniforms" fails the prerequisite of "Confidence in the existence of objective truth", as do many (most?) "should" statements.

I strongly believe that there is no objectively true answer to the question "middle school students should wear uniforms", as the truth of that statement depends mostly not on the understanding of the world or the opinion about student uniforms, but on the interpretation of what the "should" m... (read more)

I think you're basically making correct points, but that your conclusion doesn't really follow from them.

Remember that double crux isn't meant to be a "laboratory technique" that only works under perfect conditions—it's meant to work in the wild, and has to accommodate the way real humans actually talk, think, and behave.

You're completely correct to point out that "middle school students should wear uniforms" isn't a well-defined question yet, and that someone wanting to look closely at it and double crux about it would need to boil down a lot of specifics. But it's absolutely the sort of phrase that someone who has a well-defined concept in mind might say, at the outset, as a rough paraphrase of their own beliefs.

You're also correct (in my opinion/experience) to point out that "should" statements are often a trap that obfuscates the point and hinders progress, but I think the correct response there isn't to rail against shoulds, but to do exactly the sort of conversion that you're recommending as a matter of habit and course.

People are going to say things like "we should do X," and I think letting that get under one's skin from the outset is unproductive, whereas going, ah, cool, I can think of like four things you might mean by that—is it one of these? ... is a simple step that any aspiring double cruxer or rationalist is going to want to get used to.

I've waited to make this comment because I wanted to read this carefully a few times first, but it seems to me that the "crux" might not be doing a lot here, compared to simply getting people to actually think about and discuss an issue, instead of thinking about argument in terms of winning and losing. I'm not saying the double crux strategy doesn't work, but that it may work mainly because it gives the people something to work at other than winning and losing, and something that involves trying to understand the issues.

One thing that I have not... (read more)

I think that crux is doing a lot of work in that it forces the conversation to be about something more specific than the main topic, and because it makes it harder to move the goal posts partway through the conversation. If you're not talking about a crux then you can write off a consideration as "not really the main thing" after talking about it.
2[DEACTIVATED] Duncan Sabien
I agree with both you and atucker—I think the crux is doing the important things they cite, but I also think that a significant amount of the value comes from the overall reframing of the dynamic.

Hi, some friends and I tried to practice this a couple days back. So I guess main takeaway points (two years later! Haha):

  1. It's hard to practice with "imagined" disagreements or devil advocates; our group often blanked out when we dug deeper. Eg one thing we tried to discuss was organ donation after death. We all agreed that it was a good idea, and had a hard time imagining why someone wouldn't think it was a good idea.

  2. Choice of topic is important - some lighter topics might not work that well. We tackled something lighter afterwards - "Apple products

... (read more)

This set of strategies looks familiar. I've never called it double crux or anything like that, but I've used a similar line in internet arguments before.

Taking a statement that disagrees with me; assuming my opponent is sane and has reasons to insist that that statement is true; interrogating (politely) to try to find those reasons (and answering any similar interrogations if offered); trying to find common ground where possible, and work from there to the point of disagreement; eventually either come to agreement or find reasons why we do not agree that d... (read more)

I hope very minor nitpicks make acceptable comments. I apologize if not.

In the section about Prerequisites, specifically about The belief in an objective truth, the example "is the color orange superior to the color green" does not function as a good example in my opinion. This is because it is not a well-posed problem for which a clear-cut answer exists. At the very least, the concept of "superior" is too vague for such an answer to exist. I would suggest adding "for the specific purpose we have in mind", or removing that example.

Happy Easter!

I'm seeing similarities between this and Goldratt's "Evaporating Cloud". You might find it worthwhile to read up on applications of EC in the literature on Theory of Constraints, if you haven't already.

(This is Dan from CFAR)

Here are a few examples of disagreements where I'd expect double crux to be an especially useful approach (assuming that both people hit the prereqs that Duncan listed):

2 LWers disagree about whether attempts to create "Less Wrong 2.0" should try to revitalize Less Wrong or create a new site for discussion.

2 LWers disagree on whether it would be good to have a norm of including epistemic effort metadata at the start of a post.

2 EAs disagree on whether the public image of EA should make it seem commonsense and relatable ... (read more)

Does anyone else, other than me, have a problem with noticing when the discussion they're having is getting more abstract? I'm often reminded of this fact when debating some topic. This is relating to the point on "Narrowing the scope", and how to notice the need to do this.

0[DEACTIVATED] Duncan Sabien
A general strategy of "can I completely reverse my current claim and have it still make sense?" is a good one for this. When you're talking about big, vague concepts, you can usually just flip them over and they still sound like reasonable opinions/positions to take. When you flip it and it seems like nonsense, or seems provably, specifically wrong, that means you're into concrete territory. Try just ... adopting a strategy of doing this 3-5 times per long conversation?
This seems useful and simple enough to try. I'll set up an implementation intention to do this next time I find myself in a long conversation. It also reminds me of the reversal test, a heuristic for eliminating status-quo bias. Bostrom, Ord (2006)

Interesting. Some time ago I was planning on writing some things on how to have an argument well, but I found a lot of it was already covered by Eliezer in "37 ways words can be wrong". I think this covers a lot of the rest of it! Things like "Spot your interlocutor points so you can get to the heart of the matter; you can always unspot them later if they turn out to be more crucial than you realized."

One thing I've tried sometimes is actively proposing reasons for my interlocutor's beliefs when they don't volunteer any, and seeing i... (read more)

Yeah, when you ask leading questions ("You are saying X is true, do you think so because you believe that Y happens?"), people tend to get unreasonably suspicious :-/

If double crux felt like the Inevitable Correct Thing, what other things would we most likely believe about rationality in order for that to be the case?

I think this is a potentially useful question to ask for three reasons. One, it can be a way to install double crux as a mental habit -- figure out ways of thinking which make it seem inevitable. Two, to the extent that we think double crux really is quite useful, but don't know exactly why, that's Bayesian evidence for whatever we come up with as potential justification for it. But, three, pinning down su... (read more)

This is well written and makes me want to play.

I think the cartoon could benefit from concrete, short, A and B. It also may benefit from rewording the final statements to "If I change my mind about B, that would change my mind about A too"?

Take some of your actual double crux sessions, boil down A and B into something short, try the comic out with the concrete example?

Has anyone here tried building a UI for double crux conversations? The format has the potential to transform contentious conversations and debates, but right now it's unheard of outside of this community, mostly due to difficulties in execution.

Graph databases (the structure used for RoamResearch) would be the perfect format, and the conversations would benefit greatly from a standardized visual approach (much easier than whiteboarding or trying to write every point down). The hard part would be figuring out how to standardize it, which would involve having several conversations and debating the best way to break them down.

If anyone's interested in this, let me know. 

3[DEACTIVATED] Duncan Sabien
I believe Owen Shen has tried this; I cannot at the moment recall his LW username but it might just be some permutation of "Owen Shen."
I think you mean lifelonglearner.
Thanks! Got in touch and he recommended Kialo, which appears to be a good basic implementation. It's missing some flexibility - specifically the ability to connect every topic - but that might be a feature.  I envisioned creating a giant knowledge web, where any topic can link to any other topic. For example, a discussion on The Trolley Dilemma might have arguments with edges connecting to discussions on Utilitarianism and Consequentialism. The site's format doesn't allow linking discussions, each discussion has one topic and the arguments flow down from there.  This creates redundancy - the site has many topics with similar arguments inside - but it also makes each individual discussion easier to keep track of - it may be difficult to jump from topic to topic in a discussion. I'll try it out on some conversations and see how it works. Based on other posts in this thread, double crux conversations rarely go exactly as envisioned.

How much will you bet that there aren't better strategies for resolving disagreement?

Given the complexity of this strategy it seems to me like in most cases it is more effective to do some combination of the following:

1) Agree to disagree 2) Change the subject of disagreement 3) Find new friends who agree with you 4) Change your beliefs, not because you believe they are wrong but because other people believe they are wrong. 5) Violence (I don't advocate this in general, but in practice it's what humans do when they have disagreed through history)

6[DEACTIVATED] Duncan Sabien
The word "better" is doing a lot of work (more successful? Lower cost?), but in my personal experience and the experience of CFAR as a rationality org, double crux looks like the best all-around bet. (1) is a social move that sacrifices progress for happiness, and double crux is at least promising insofar as it lets us make that tradeoff go away. (2) sort of ... is? ... what double crux is doing—moving the disagreement from something unresolvable to something where progress can be made. (3) is absolutely a good move if you're prioritizing social smoothness or happiness or whatever, but a death knell for anyone with reasons to care about the actual truth (such as those working on thorny, high-impact problems). (4) is anathema for the same reason as (3). And (presumably like you), we're holding (5) as a valuable-but-costly tool in the toolkit and resorting to it as rarely as possible. I would bet $100 of my own money that nothing "rated as better than double crux for navigating disagreement by 30 randomly selected active LWers" comes along in the next five years, and CFAR as an org is betting on it with both our street cred and with our actual allotment of time and resources (so, value in the high five figures in US dollars?).
I'd take your bet if it were for the general population, not LWers... My issue with CFAR is it seems to be more focused on teaching a subset of people (LWers or people nearby in mindspace) how to communicate with each other than in teaching them how to communicate with people they are different from.
0[DEACTIVATED] Duncan Sabien
That's an entirely defensible impression, but it's also actually false in practice (demonstrably so when you see us at workshops or larger events). Correcting the impression (which again you're justified in having) is a separate issue, but I consider the core complaint to be long-since solved.
I'm not sure what you mean and I'm not sure that I'd let a LWer falsify my hypothesis. There are clear systemic biases LWers have which are relatively apparent to outsiders. Ultimately I am not willing to pay CFAR to validate my claims and there are biases which emerge from people who are involved in CFAR whether as employees or people who take the courses (sunk cost as well as others).
2[DEACTIVATED] Duncan Sabien
I can imagine that you might have hesitated to list specifics to avoid controversy or mud-slinging, but I personally would appreciate concrete examples, as it's basically my job to find the holes you're talking about and try to start patching them.

Nice to see the Focusing/Gendlin link (but it's broken!)

Are you familiar with "Getting To Yes" by Ury and the Harvard mediators? 

They were initially trained by Marshall Rosenberg, who drew on Gendlin and Rogers. I don't know if Double Crux works, but NVC mediation really does.  Comparison studies for effectiveness would be interesting!

For more complex/core issues, Convergent Facilitation by Miki Kashtan is brilliant, as is by Dominic Barter, who is very insightful on our unconscious informal "justice systems".

Fast Consensing, Sociocracy and Holacracy are also interesting.

Yes, but the idea is that a proof within one axiomatic system does not constitute a proof within another.


2 points:

  1. I've used something similar when evaluating articles: I ask "what statement of fact would have to be true to make the main vague conclusion of this article correct"? Then I try to figure out if that fact is correct.


For instance: (wildly asserts) "I bet if everyone wore uniforms there would be a fifty percent reduction in bullying." (pauses, listens to inner doubts) "Actually, scratch that—that doesn't seem true, now that I say it out loud, but there is something in the vein of reducing overt bullying, maybe?"

... (read more)

If someone uses different rules than you to decide what to believe, then things that you can prove using your rules won't necessarily be provable using their rules.

No, really, what? What "Different rules" could someone use to decide what to believe, besides "Because logic and science say so"? "Because my God said so"? "Because these tea leaves said so"?
Yes, but as it happens that kind of difference is unnecessary in the abstract. Besides the point I mentioned earlier, you could have a logical set of assumptions for "self-hating arithmetic" that proves arithmetic contradicts itself. Completely unnecessary details here.
Unfortunately, yes.

What if the disagreeing parties have radical epistemological differences? Double crux seems like a good strategy for resolving disagreements between parties that have an epistemological system in common (and access to the same relevant data), because getting to the core of the matter should expose that one or both of them is making a mistake. However, between two or more parties that use entirely different epistemological systems - e.g. rationalism and empiricism, or skepticism and "faith" - double crux should, if used correctly, eventually lead ... (read more)

Is there good reason to believe that any method exists that will reliably resolve epistemological disputes between parties with very different underlying assumptions?
Alcohol helps :-P
Not if they're sufficiently different. Even within Bayesian probability (technically) we have an example in the hypothetical lemming race with a strong Gambler's Fallacy prior. ("Lemming" because you'd never meet a species like that unless someone had played games with them.) On the other hand, if an epistemological dispute actually stems from factual disagreements, we might approach the problem by looking for the actual reasons people adopted their different beliefs before having an explicit epistemology. Discussing a religious believer's faith in their parents may not be productive, but at least progress seems mathematically possible.
Not particularly, no. In fact, there probably is no such method - either the parties must agree to disagree (which they could honestly do if they're not all Bayesians), or they must persuade each other using rhetoric as opposed to honest, rational inquiry. I find this unfortunate.
Could you elaborate on that? Sorry, I just don't get it.
It's a hint at Aumann's theorem.
Oh, I wasn't aware that they had to be Bayesian for that rule to apply, thanks for the help.

This looks like a good method to derive lower-level beliefs from higher-level beliefs. The main thing to consider when taking a complex statement of belief from another person, is that it is likely that there is more than one lower-level belief that goes into this higher-level belief.

In doxastic logic, a belief is really an operator on some information. At the most base level, we are believing, or operating on, sensory experience. More complex beliefs rest on the belief operation on knowledge or understanding; where I define knowledge as belief of some in... (read more)

This is my response in which I propose a different approach.

For the author and the audience: what are your favourite patience- and sanity-inducing rituals?

0[DEACTIVATED] Duncan Sabien
For me, sanity always starts with remembering times that I was wrong—confidently wrong, arrogantly wrong, embarrassingly wrong. I have a handful of dissimilar situations cached in my head as memories (there's one story about popsicles, one story about thinking a fellow classmate was unintelligent, one story about judging a student's performance during a tryout, one story about default trusting someone with some sensitive information), and I can lean on all of those to remind myself not to be overconfident, not to be dismissive, not to trust too hard in my feeling of rightness. As for patience, I think the key thing is a focus on the value of the actual truth. If I really care about finding the right answer, it's easy to be patient, and if I don't, it's a good sign that I should disengage once I start getting bored or frustrated.

Correct me if I'm wrong. You are searching for a sentence B such that:

1) if B then A
2) if not B, then not A. Which implies if A then B.

Which implies that you are searching for an equivalent argument. How can an equivalent argument have an explanatory power?

"Aluminium is better than steel!" cries Alice. "Steel is better than aluminium!" counters Bob. Both of them continue to stubbornly hold these opinions, even in the face of vehement denials from the other. It is not at once clear how to resolve this issue. However, both Alice and Bob have recently read the above article, and attempt to apply it to their disagreement. "Aluminium is better than steel because aluminium does not rust," says Alice. "The statement 'aluminium does not rust, but steel does' is an equivalent argument to 'aluminium is better than steel'". "Steel is better than aluminium because steel is stronger than aluminium," counters Bob. "Steel can hold more weight than aluminium without bending, which makes it a superior metal." "So the crux of our argument," concludes Alice, "is really that we are disagreeing on what it is that makes a metal better; I am placing more importance on rustproofing, while you are showing a preference for strength?"
For this example you don't need any double cruxes. Alice and Bob should have just defined their terms, specifically the word "better" to which they attach different meanings.
5[DEACTIVATED] Duncan Sabien
True, but they then could easily have gone on to do a meaningful double crux about why their chosen quality is the most important one to attend to.
That is true. In a disagreement where the root of the disagreement is applying different meanings to the word 'better', properly defining that term would identify the true disagreement straight away. The double crux method, by seeking equivalent statements for each position, brings that disagreement in terminology to light almost immediately (where a word-by-word process of definitions might well get mired down in the definition of 'steel' and whether or not it includes small amounts of chromium - which might be interesting and informative on its own, but does nothing to resolve the disagreement). This appears to suggest that double crux, applied properly, will work in every case where the true disagreement is a matter is inconsistent definition of terms (as above). I'd go further, and say that the double crux method will also work in cases where the disagreement is due to one of the debaters having made an error in a mathematical equation that he believes supports his argument. So, when you don't know the root cause of the argument, double crux is probably at least as fast a route to finding that cause as a careful definition of all terms, and probably faster.
Let me rephrase: does the double crux method contains any improvement that is not already covered by tabooing terms? Or simply saying "why do you think this is the case?" "Steel is better than aluminum because aluminum is worse than steel" is also equivalent, adheres to the letter of the prescription but does not move the discussion forward. What I'm trying to prove wrong is that the logical prescriptions, given as explanation of what a crux is, do not really capture anything substantial.
In this particular argument, no. (In fact, if both participants are willing to examine their own chain of reasoning and consider that they might be wrong, then asking "why do you think this is the case?" sounds like a perfect first step in the double crux method to me) In cases where the disagreement is due to (say) Bob making a mathematical error, tabooing terms is unlikely to reveal the error, while double crux seems likely to do so. So, as a general disagreement-solving technique, it seems powerful as it can be applied to a wide variety of causes of disagreement, even without knowing what the cause of the disagreement actually is.
0[DEACTIVATED] Duncan Sabien
If I'm understanding correctly, I think you've made a mistake in your formal logic above—you equated "If B, then A" with "If A, then B" which is not at all the same. The search for a double crux encourages each side to adopt the causal model of the other (or, in other words, to search through the other's causal models until they find one they can agree is true). I believe "If B, then A," which is meaningfully different from your belief "If ¬B then ¬A." If each of us comes around to saying, "Yeah, I buy your similar-but-different causal model, too," then we've converged in an often-significant way, and have almost always CLARIFIED the underlying belief structure.
No, he only inferred "If A, then B" from "If not B, then not A" which is a valid inference.
0[DEACTIVATED] Duncan Sabien
... but then he went on to say "How can an equivalent argument have explanatory power?" which seemed, to me, to assume that "if B then A" and "if A then B" are equivalent (which they are not).
I read that statement as implying that argument A is equivalent to argument B. (Not (1) and (2), which are statements about arguments A and B) And, if A implies B and B implies A, then it seems to me that A and B have to be equivalent to each other.
(This is Dan from CFAR) This is easier to think about in the context of specific examples, rather than as abstract logical propositions. You can generally tell when statement B is progress towards making the disagreement about A more concrete / more tractable / closer to the underlying source of disagreement. I typically think of the arrows as causal implication between beliefs. For example, my belief that school uniforms reduce bullying causes me to believe that students should wear uniforms. With logical implication the contrapositive is equivalent to the original statement (as you say). With causal implication, trying to do the contrapositive would give us something like "If I believed that students should not wear uniforms, that would cause me to believe that uniforms don't reduce bullying" which is not the sort of move that I want to make in my reasoning. Another way to look at this, while sticking to logical implication, is that we don't actually have B-->A. Instead we have (B & Q & R & S & T ... & Z) --> A. For example, I believe that students should wear uniforms because uniforms reduce bullying, and uniforms are not too expensive, and uniforms do not reduce learning, and uniforms do not cause Ebola, etc. If you take the contrapositive, you get ~A --> (~B or ~Q or ~R or ~S or ~T ... or ~Z). Or, in English, I have many cruxes for my belief that students should wear uniforms, and changing my mind about that belief could involve changing my mind about any one of those cruxes.
The requirement that B is crucial for A is equivalent to "If A then B", not "if B then A". For example: A = self-driving cars would be safer B = the chances that a bug would cause all self-driving cars to crash spectacularly on the same day are small. If A is true, then B must be true, because if B is false, A is clearly false. But B does not imply A: even if B is true (there could be zero chance of such a bug), A could still be false (self-driving cars might be bad at avoiding crashes due to, say, object recognition being too slow to react in time). So B being crucial for A means that A implies B.
I hear what you're saying, which is what I hinted at with point n° 2, but "if B then A" is explicitely written in the post: last paragraph in the "How to play" section. It seems to me you're arguing against the original poster about what "being crucial" means logically, and although I do not agree on the conclusion you reach, I do agree that the formulation is wrong.
0[DEACTIVATED] Duncan Sabien
I'm quite confident that my formulation isn't wrong, and that we're talking past each other (specifically, that you're missing something important that I'm apparently not saying well). What was explicitly written in the post was "If B then A. Furthermore, if not B then not A." Those are two different statements, and you need both of them. The former is an expression of the belief structure of the person on the left. The latter is an expression of the belief structure of the person on the right. They are NOT logically equivalent. They are BOTH required for a "double crux," because the whole point is for the two people to converge—to zero in on the places where they are not in disagreement, or where one can persuade the other of a causal model. It's a crux that cuts both ways—B's true state implies A's trueness, but B's false state is not irrelevant in the usual way. Speaking strictly logically, if all we know is that B implies A, not-B doesn't have any impact at all on A. But when we're searching for a double crux, we're searching for something where not-B does have causal impact on A—something where not-B implies not-A. That's a meaningfully different and valuable situation, and finding it (particularly, assuming that it exists and can be found, and then going looking for it) is the key ingredient in this particular method of transforming argument into collaborative truth-seeking.
Ah, I think I've understood where the problem lies. See, we both agree that B --> A and -B --> -A. This second statement, as we know from logic, is equivalent to A --> B. So we both agree that B --> A and A --> B. Which yields that A is equivalent to B, or in symbols A <--> B. This is what I was referring to: the crux being equivalent to the original statement, not that B --> A is logically equivalent to -B --> -A
0[DEACTIVATED] Duncan Sabien
I'm probably rustier on my formal logic than you. But I think what's going on here is fuzz around the boundaries where reality gets boiled down and translated to symbolic logic. "Uniforms are good because they'll reduce bullying." (A because B, B --> A) "Uniforms are bad, because along with all their costs they fail to reduce bullying." (~A because ~B, ~B --> ~A) Whether this is a minor abuse of language or a minor abuse of logic, I think it's a mistake to go from that to "Uniforms are equivalent to bullying reduction" or "Bullying reductions result in uniforms." I thought that was what you were claiming, and it seems nonsensical to me. I note that I'm confused, and therefore that this is probably not what you were implying, and I've made some silly mistake, but that leaves me little closer to understanding what you were.
A: "Uniforms are good" B: "Uniforms reduce bullying" B->A: "If uniforms reduce bullying, then uniforms are good." ~B->~A : "If uniforms do not reduce bullying, then uniforms are not good." "A is equivalent to B": "The statement 'uniforms are good' is exactly as true as the statement 'uniforms reduce bullying'." A->B: "If uniforms are good, then it is possible to deduce that uniforms reduce bullying." ...does that help?
2[DEACTIVATED] Duncan Sabien
Yep. Thanks. =) I was misunderstanding "equivalency" as "identical in all respects to," rather than seeing equivalency as "exactly as true as."
There's confusion here between logical implication and reason for belief. Duncan, I believe, was expressing belief causality -- not logical implication -- when he wrote "If B, then A." This was confusing because "if, then" is the traditional language for logical implication. With logical implication, it might make sense to translate "A because B" as "B implies A". However, with belief causality, "I believe A because I believe B" is very different from "B implies A". For example: A: Uniforms are good. B: Uniforms reduce bullying. C: Uniforms cause death. Let's assume that you believe A because you believe B, and also that you would absolutely not believe A if it turned out that C were true. (That is, ~C is another crux of your belief in A.) Now look what happens if B and C are both true. (Uniforms reduce bullying to zero because anyone who wears a uniform dies and therefore cannot bully or be bullied.) C is true, therefore A is false even though B is true. So B can't imply A. B is only one reason for your belief in A, but other factors could override B and make A false for you in spite of B being true. That's why you can have multiple independent cruxes. If any one of your cruxes for A turns out to be false, then you would have to conclude that A is false. But any one crux being true doesn't by itself imply that A is true, because some other crux could be false, which would make A false. So with belief causality, "A because B" does not mean that B implies A. What it actually means is that ~B implies ~A, or equivalently, that A implies B -- which in that form sounds counter-intuitive even though it's right. So for B to be a crux of A means only (in formal logical implication) that A -> B, and definitely not that B -> A. In fact, for a crux to be interesting/useful, you don't want a logical implication of B -> A, because then you've effectively made no progress toward identifying the source of disagreement. To make progress, you want each crux to be "saying
In a pure-logic kind of way, finding B where B is exactly equivalent to A means nothing, yes. However, in a human-communication kind of way, it's often useful to stop and rephrase your argument in different words. (You'll recognise when this is helpful if your debate partner says something along the lines of "Wait, is that what you meant? I had it all wrong!") This has nothing to do with formal logic; it's merely a means of reducing the probability that your axioms have been misunderstood (which is a distressingly common problem).
I guess we now can get rid of .


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"In essence: a conviction that for almost any well-defined question, there really truly is a clear-cut answer."

Every question is formulated using words, and words are either defined by other words, or defined by pointing to a number of examples, which means that all words are ultimately defined by pointing to examples. Pointing to examples does not and cannot make anything "well-defined", so no question is well defined, nor does any question have a really truly clear-cut answer.

This comment might look like a joke, but it is not; I think it is pretty much as true as anything can be (since the point is that nothing is ever absolutely precise, the argument can't claim absolute precision itself.)

4[DEACTIVATED] Duncan Sabien
I disagree that no question is well-defined, but I think it's because we have different bars for what constitutes "well." I'm with you on the fact that we can't reach absolute precision, but I think a question like "how many grasshoppers are alive on Earth right now" is well-defined, and to the extent that it's not, it becomes well-defined once you (e.g.) clarify the boundaries between species and what counts as a grasshopper (and maybe define what you mean by alive in the context of this question). Similarly, as Peteris points out above, the question form of "middle school students should wear uniforms" isn't well-defined yet, because the should is doing all the work. But something like "Will a policy of requiring middle school students to wear uniforms result in a net increase in happiness?" seems sufficiently well-defined to me that I feel confident saying a definitive answer exists (even if we may not be able to find it). I second Gram_Stone in that I was surprised that you were predicting people would label your point as disingenuous. You seemed pretty serious and straightforward to me; I just think you're looking for more rigor than discussion and debate actually require in practice.
As I said responding to Gram_Stone, the original formulation sounded somewhat less serious than the final one, so perhaps I shouldn't have predicted that given that I modified the comment. I wouldn't say that I'm looking for "more rigor" than needed. In fact, I am totally in favor of taking discussion seriously. In fact, to some extent my point is that sometimes people dismiss discussion and argument on the grounds that the terms aren't well defined, and I think it is undesirable to do this -- I am willing to take arguments seriously even with badly defined terms, and basically because "well-defined" is a matter of degree in the first place. The problem with saying that we can make things "well-defined" by clarifying boundaries follows from my original argument. You cannot get perfect precision in what counts as a grasshopper, for example -- if you had before you the entire evolutionary series that resulted in grasshoppers, you could draw no boundary except by artificial stipulation. Likewise with defining what you mean by alive -- there will always be able to be some grasshoppers on the borderline where it is not clear whether they are alive or not. I am not saying that you can never answer any question - there are two chairs in my room right now, not three or four. But despite that, none of those terms are absolutely well defined. I am also objecting to saying "this is well defined and that isn't" based on your personal impression of what has a clear meaning, when other people might have a different impression. So for example, there is no objective reason why your question about the "net increase in happiness" is more well defined than the question about whether middle school students should wear uniforms, because it is not necessarily more clear what happiness is, than what it to means to say someone should do something. It might seem more clear to you; but "should" might seem more clear to someone else.
3[DEACTIVATED] Duncan Sabien
Gotcha. That all makes sense/seems correct. I think I'm still optimistic about being able to move a given open question toward the "chair" situation, in a context where I'm discussing it with another human (in other words, I think that the actual criteria is "both people in the disagreement agree that the question is now sufficiently well-defined for them to enjoy wrestling with it"). I think you're right that lines between grasshoppers and non-grasshoppers (or alive and dead) will always be to some extent arbitrary, and sometimes extremely arbitrary, but that picking them anyway for the sake of the discussion is both possible and productive.
Sorry if this is an obvious 101 question (I'm new here), but isn't there a difference between some of those examples? I would also say that "how many grasshoppers are there?" has an answer that we simply don't know. But I can't think of "orange is superior to green" being true or false. I can think of ways that it could be better for a purpose (like if we're deciding what color the hunting vests should be), but not what the truth conditions of "orange is superior to green" would actually be. If you're having an argument over school uniforms, presumably you're saying "Having a uniform will help this school fulfill its mission in X way" because you're trying to inform the other person's opinion on a particular policy. Is there a context where an argument over whether orange is superior to green is broader than "better for this specific purpose" but is about something other than subjective aesthetics, and is an attempt to convince one's interlocutor? (Nobody's going to say "You're right, I do find orange things nicer to look at than green things, although I thought it was the other way around until you pointed that out to me", are they?) (This isn't meant to be an argument from personal incredulity -- I just can't think of a better way to word it. In fact, I'm not really trying to argue for anything, more like seeking clarification.)
Your comment is a response to Duncan's comment (I guess you haven't worked out how that works yet). "Nobody's going to say "You're right, I do find orange things nicer to look at than green things, although I thought it was the other way around until you pointed that out to me", are they?" Once I said, "food tastes better when you just eat without paying attention to the taste, and basically that happens because your body is telling you it is great, while if you pay attention, you figure out it wasn't as great as your body was saying." I was eating with someone and they responded by saying there was no way this could be true. A few minutes later, after noticing his experience while eating, he said, "You know, I think you were right."
0[DEACTIVATED] Duncan Sabien
What I was gesturing at with the "orange superior to green?" example is something very large, very subjective, very fuzzy, with a lot of uncertainty in interpretation, but that nevertheless could be boiled down to something concrete. Things that might make a color superior to another color: more people rate it more highly, it's more prevalent on buildings and clothes and in pictures, it produces greater levels of happiness when registered in the brain, it's cheaper to produce at the same price, it has more effective uses within a given society, people will pay more for an otherwise-identical version of a thing that's in that color, etc. etc. etc. To get at what someone might "really mean" when they say something like orange is superior to green, I'd propose a whole bunch of these as hypotheses, and then try some form of pseudomath or other concrete weighting of the three or four most relevant categories. Once I had something like "Okay, so it's the subjective ratings of 7 billion people plus the results of brain scans when exposed, and if those come up 50/50 then we'll pull in, as a representative data point, how many different deliberate uses of each color there are in the continent of Europe," I'd feel like yes, there was in fact a true answer to the question (but not one we'd ever be able to find in practice).
Genuine question: why do you anticipate that we'll assume that you're being disingenuous?
Because it sounded like a joke when I reread it. (But as it stands above, it is a bit different from what it was then because after reading it I rewrote it to take a bit of that away.)