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