Logical endpoints

More on Aaronson (see previous post for context):

I was struck by this commment Scott made on Gil Kalai’s blog:

Yes, I admit, I do have the moral philosopher’s (or for that matter, the mathematician’s) habit of trying to take stated principles to their logical conclusions, even if many people would regard those conclusions as “irrelevant” or “absurd.” (To take a different example: “People should have the right to own whatever weapon they want, since merely owning it doesn’t harm anyone.” “OK then, what about nuclear missiles?” “That’s irrelevant and absurd! I was talking about guns.”) Is this habit something I should apologize for?

and this reddit comment he quotes approvingly:

I think the reason Dworkin comes up in discussions like this is because her thinking is the logical endpoint of mainstream feminist theory.
It goes something like this:
1) Women are systematically oppressed by men
2) If 1 is true, how can a woman ever consent to sex or practically anything else with men? Any “consent” a woman gives will be given under duress because she is being systematically oppressed.
3) If any “consent” a woman gives is under duress (because every decision and choice a woman makes is under duress because she’s being systematically oppressed), then women can never ever give consent in any dealing with men.
Dworkin, to her credit, was so logical that she came to this conclusion and accepted it. All logical thinkers will probably come to this conclusion which is why nerds and STEM people will like and understand Dworkin. She’s logical. She makes sense.

For my own part, I find this idea of taking political and moral principles to their logical conclusions to be very weird.  And I don’t think it’s “the mathematician’s habit,” as Scott says.  At least, it’s not this mathematician’s habit.  Being a mathematician doesn’t incline me to apply Boolean operations to ethical principles; on the contrary, I think being a mathematician makes me more alive than the average person to the difference between mathematical assertions (which do behave really well under logical operations) and every other kind.

In particular, I don’t find the argument by the reddit commenter very compelling.  There are lots of feminists (I think almost all feminists!) who sound nothing like Andrea Dworkin, and who pretty obviously think that there exists sex between men and women that isn’t rape.  Is that because they can’t do logic?  I am a STEM person and a feminist and I think systematic sexism exists in the world and I don’t think heterosexual sex is rape.  Is that because I can’t do logic?

No — it’s because I think there are very few assertions about sex, power and feminism which stand in a relation of authentic logical entailment.

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24 thoughts on “Logical endpoints

  1. JSE says:

    P.S: I also think that Scott Aaronson’s imaginary interlocutor has the stronger side in their argument about allowable weapons.

  2. Jon Awbrey says:

    There’s a perennial question about the relationship of logic to the other two classical normative sciences, ethics and aesthetics. C.S. Peirce held that aesthetics came first, ethics second, and logic third in order of practical dependence, or something like that.

  3. Melissa Tacy says:

    I have always thought of the habit of running an argument to a logical endpoint as being mathematical. At least my tendency to do so has increased as I have trained as a mathematician. In my mind I compare it to taking an extreme example to test a conjecture or some intuition. It forces you to very carefully state what it really is that you mean. In the guns example by taking as far as warheads you are highlighting the “owning doesn’t hurt anyone” part of the argument and questioning whether this is indeed your interlocutor’s intended argument. Perhaps they mean that owning a gun (or a particular class of guns) presents a low risk to others compared to more extreme weapons. Once you have cleared that up you may still not agree with them but at least you understand their opinion better. This kind of test can also be used to carefully examine your own opinions and why you hold them.

    However I think this is quite a different thing than being attracted to extreme viewpoints, like the reddit argument. I don’t see that as a particularly mathematical trait at all.

  4. Callan McGill says:

    I agree with Melissa, I don’t think of this style of argument, which I have found common amongst mathematicians, as useful for providing one with moral or political principles but merely as an “intuition pump” for others to tighten theirs. If one has a simple principle by which to argue a particular point it is worthwhile to tweak the parameters of that principle as a method to gauge efficacy and nuance.

    That said, I see the real problem as being that ethics and politics are usually more complicated than such easily stated principles (even those as seemingly benign as the golden rule) and the mathematician is merely tacitly observing this when using end point logic for clarification.

  5. One difference between the gun example and the Dworkin characterization is that in the former the logical extreme is intended to come out as an absurdity in order to challenge the premise. Like Melissa, I’d think that makes the gun version a more mathematical use of running to logical endpoints, and the difference in how the logical endpoint works in the overall argument is really important: here, it’s a way of rejecting simplifications and adding nuance and specificity to claims, more or less as Jordan says he approaches ethics.

  6. As Melissa Tacy says above, I think that *considering* how arguments look when pushed to extremes may well be a mathematical habit, and can often be enlightening. But Scott Aaronson’s approach that if one accepts the moderate version one should also accept the extreme seems baffling to me too. To take a mathematical comparison: it seems like the expectation that a statement like “x >> y” should remain true even in the limiting case where x goes to 0.

    He’s not the only intelligent person I’ve heard more-or-less explicitly espousing this principle, though. I’d be interested to hear them explain their position.

  7. Hi Jordan, thank you for engaging thoughtfully. I think anyone would agree that, in “soft” subjects, what looks at first like a contradiction can always be clarified via further discussion, arguments, and examples. E.g., “sure, systematic sexism exists in the world, but in saying that, we mean A,B,C, rather than D,E,F; in particular, this explains why, despite the existence of systematic sexism, you shouldn’t feel any guilt about expressing romantic interest in the following ways…” By contrast, saying that the resolution is that “there are very few assertions about sex, power and feminism which stand in a relation of authentic logical entailment” just seems to make matters *worse* for me. That’s like saying: “the more you value intellectual coherence, the harder this is going to be for you; if you’re *too* logical then you might as well not even try.”


    PS. “How Not To Be Wrong” was a great book! My favorite part was the proof of the Buffon’s needle theorem using linearity of expectation.

  8. It’s definitely a minority, but I do know mathematicians who not very cognizant to, as you put it, the difference between mathematical assertions and every other kind. This does lead them to some pretty odd conclusions about the real world at times…

  9. I think Modus Ponens has a place in ethics, etc. In this case, someone “proves” A implies B, where A is moderate and B is extreme. If the prover had an aversion to extreme views, he might suspect that something is amiss — perhaps he should be more careful with definitions, and maybe special cases (e.g., nuclear weapons) will reveal the need for additional hypotheses. If this argument were part of a dialog between opposing views, the opponent would gladly point out these defects. After some debate and proof editing, I think both sides would come to an agreement provided both sides believed the hypotheses.

  10. Daniel Biss says:

    I think the entire dispute here is contained in Scott’s jump from “authentic logical entailment” to “intellectual coherence”.

    For what it’s worth, I share Jordan’s appreciation for Scott’s recent thoughtful and brave writing, as well as Jordan’s disagreement that “authentic logical entailment” is the same thing as “intellectual coherence”.

    Maybe I should just be stating my own opinions, though, rather than pretending to know exactly what Jordan thinks :)

  11. Alison Miller says:

    I found the Reddit comment illuminating, despite Jordan’s valid objections, because it describes a personality/character trait which I have seen in others (and to some extent in myself), and explains why Dworkin’s work is particularly appealing to them (I wouldn’t call this trait being “logical”, though — I’d call it something more like “purity”.) I suspect it is disproportionately common among young people interested in Math/CS (but less so among successful academics).

  12. Jon Awbrey says:

    Here’s some thoughts on the links among Aesthetics, Ethics, and Logic from an essay I wrote out on one of my many returns to grad school some years ago —

    Inquiry Driven Systems • Logic, Ethics, Esthetics

  13. Pierre Menard says:

    “Being a mathematician doesn’t incline me to apply Boolean operations to ethical principles; on the contrary, I think being a mathematician makes me more alive than the average person to the difference between mathematical assertions (which do behave really well under logical operations) and every other kind.”

    While non-mathematical assertions are different from mathematical assertions, they are not so different to escape the laws of logic. A denial that the statement “people should have the right to own whatever weapon they want” logically entails “people should have the right to own missiles” is nothing short of a denial of the distinction between sense and nonsense.

    “I am a STEM person and a feminist and I think systematic sexism exists in the world and I don’t think heterosexual sex is rape. Is that because I can’t do logic?”

    That is entirely possible, especially in view of your writings above. To be more precise, it is not that you are fundamentally incapable of doing logic, but rather you may be refusing to do it due to your remarkable belief that ethical statements do not “behave well” under logical operations.

    An alternative explanation is that you (as well as everyone else in this discussion, it seems) is guilty of some imprecision. Feminism is not a precisely enumerated set of propositions; on the contrary, there are are various varieties of it, e.g., first-wave, second-wave, and so forth, whose proponents disagree on key points. The variety of feminism that Dworkin inhibits may involve beliefs in certain propositions (for example, that women are oppressed and that oppressed people cannot meaningfully consent) that you may not share, despite the fact that both of you label yourselves as feminists.

  14. JSE says:

    Scott: sorry to be slow to respond! I think

    “the more you value intellectual coherence, the harder this is going to be for you; if you’re *too* logical then you might as well not even try.”

    is only partially a good paraphrase of what I’m saying. I think of myself as valuing “intellectual coherence,” but as Daniel says we may just have different ideas about what constitutes intellectual coherence; e.g., in my usage, “intellectual coherence” could be a property of something like a novel, which, whatever it is, is definitely not a collection of propositions with asserted truth values.

    The second part, no, that’s not my view. Certain things are harder for me than they are for the average person; I’m not, for instance, a natural athlete. But I would never say that means I shouldn’t try!

    Glad you liked the book. You’re the first person ever to mention the section on Barbier’s proof of Buffon’s theorem, which is one of my favorite parts too… what an amazing proof. It makes you feel there’s some hope for the world.

  15. Sniffnoy says:

    I’m not really clear on what the alternative is. If you don’t take them to ther logical conclusions, you’re saying they’re not really true statements. Which, I think, is maybe exactly the right thing to do — whenever you hear someone use words like “always” in cases where this appears to lead to absurd results, maybe you should draw the conclusion that they don’t really have a good idea of what “always” means and intend something milder.

  16. Jan says:

    Logical reasoning is one of the weapons with which patriarchy defends its numerous privileges. It is a time-honoured strategy for men when discussing with women to attack them on the grounds of being irrational or non-logical. Any appeal to logic or rational thought is therefore inherently fraught with sexism and misogyny. I am glad to see JSE attempting to preëmpt this strategy by simply highlighting the non-logical nature of ethical discussions.

  17. Jan says:

    Having said that, let me analyze the example about ownership of weapons from Scott’s original comment.

    Scott’s argument seems to be that, if we allow the argument “merely owning a weapon doesn’t harm anyone” as valid, then we may draw the following conclusions: (1) we have to allow the argument in other contexts as well, (2) the conclusion of the argument will be identical or analogous, regardless of the change in context. On the other hand, these two conclusions obviously lead to (3) nuclear missile ownership should be permitted by law — a conclusion that, from the way Scott presents it, we are meant to perceive as problematic.

    I think that there is no debate about whether the logical conjunction of (1) and (2) implies (3). The question is rather, seeing as how we disagree with (3) (at least for the sake of argument), which one of the premises (1) and (2) fails.

    It is my opinion that it is precisely premise (2) that fails. In almost any ethical discussion, there are multiple considerations coming into play, and the fact that weapon ownership in and of itself does not cause any casualties is merely one of these considerations. Premise (2) is precisely saying that with which JSE takes issue, and I believe completely rightly so: this premise is saying that the principle quoted above irrevocably, inexorably, and logically entails the desired conclusion, as if there were no further need for context or consideration of other “principles”.

    Now when I take a step back and ask myself: what was the point that Scott was actually trying to make, then I realize that Scott was actually on the right side of the debate all along! It seems to me that Scott is arguing that anybody who tries to settle a debate about ethics by invoking a principle such as “merely owning a weapon doesn’t hurt anybody” is pretending that the outcome of the debate follows *logically* from this principle. But this is precisely what JSE is denying that can be done.

    So I think that actually, you guys are all on the same page, and this debate is a lot like the famous discussion between Wittgenstein and Turing about the nature of contradiction in mathematics, in which Wittgenstein kept insisting that Turing and he were in complete agreement about everything, except Turing didn’t realize it yet. With which, of course, Turing disagreed.

  18. Sniffnoy says:

    I think it’s worth also reposting here the following comment Sarah Constantin made on Facebook and then later reposted on Slate Star Codex:

    Here’s the thing. The idea of “microaggressions” is actually a necessary concept for a phenomenon that exists. The idea that “even enlightened people can be bigoted” is a necessary corrective (it wasn’t so long ago that people seriously believed that a feminist white woman could not be racist, for instance.) The idea that the threat of rape informs a lot of everyday interactions — I think that’s just obviously *true*. The problem isn’t that the feminists who come up with these concepts are dastardly villains.

    The problem is a subtle one. It’s that feminism, like a lot of ideologies, is based on moral assumptions that are horrifying if taken to extremes. Things like “one should harm others as little as possible.” Sounds nice, right? except think about what “as little as possible” means. Taken literally, it means you should kill yourself so you can be certain you will never harm anyone again. “Minimize the chance that you will cross a sexual boundary”, taken literally, means you should castrate yourself.

    People who aren’t literal thinkers, I’ve found, don’t understand that “all” or “minimize” or “100%” are VERY VERY EXTREME WORDS. People who are literal thinkers, like me, have to be very careful around ideologies that tell us to do anything “as much as possible.”

    My own hack is adopting a moral system that treats my own life as intrinsically valuable. A lot of the major religions do that, as Leah points out. (Judaism says you should “live by the law, not die by it.”) If you look carefully, you’ll see that the Wizard’s Oath in the Diane Duane books is a moral imperative which, if you followed it perfectly, would *not* result in suicide or other self-destructive behavior. Kant’s categorical imperative is “safe” in that sense; so is the Golden Rule. I think there are individual feminists (Audre Lorde, for instance) who want you to live by the law and not die by it.

    But altruism in general is *not fucking safe* unless you put guardrails around it.

  19. JSE says:

    I think that’s a useful comment, but I also think that people who are “literal thinkers” in the way Constantin means should recognize that it is *they*, not other people, who are failing to take words for what they really mean. If I tell you

    “Always look both ways before crossing the street”

    and your response is

    “That’s terrible advice, what if I were being chased by a murderer bent on killing me who was only five steps behind? You must mean you should *often* look both ways before crossing the street”

    it means you’re using language in a nonstandard way. Which is fine! It just means the person who doesn’t understand what “always” means is you.

  20. Sniffnoy says:

    Well, it’s a matter of context. Obviously “always” isn’t usually meant literally; but when you’re not using words literally, you’re relying on common sense, and common sense needs calibration data. Most of the time we say “always” in a non-literal sense and it’s not a problem, but when I go to the doctor and the form asks if I’ve ever experienced shortness of breath, I check yes, though I haven’t in years; because while it seems absurd to interpret “ever” literally there, I don’t have enough of a sense of what actually is meant to do otherwise. May as well play it safe.

    When it comes to the sort of scrupulosity-inducing feminist writings we’re discussing here, I think there’s a number of factors pushing someone towards a literal or cautious interpretation. These are the ones I can think of right now.

    1. Who’s hurt? If you’re told to always look both ways before crossing the street, that’s for your own protection, not for the protection of others. You’re free to ignore it if you please.

    2. Opacity. In the crossing the street case, it’s pretty clear why you’re being told to always look both ways, so you can infer what was meant and adapt it properly even if it’s not literally true. (Or maybe it’s not so clear! I feel like it maybe takes a few close calls to realize just how easy it is to screw up if you don’t actually look. But this is beside the point.) In the other case — well, again, it should be pretty clear, and if it’s not, it should become clear through discussion (see point #5). But recently “social justice” sorts have adopted something of a doctrine of opacity — “lived experience”. Since you don’t have their lived experience, you will never understand. In such a case, it doesn’t seem too unreasonable to conclude that you’d best just do as they say (“Shut up and listen!”, as they say) and not try to adapt it, as you’ll surely get it wrong if you do.

    3. Your common sense is sexist. We know people are unconsciously biased. As such, common sense — the opaque output of your brain — should be considered untrustworthy. By contrast, reason and principle are free of such flaws. So if it comes down to feminist principle contradicting common sense, well, the latter must just be the product of your sexist unconscious mind. You should discard it.

    4. Compression of badness. What better way to induce scrupulosity than to talk about minor offenses with the same langauge as grave ones? But we’re constantly told that even small violations can only be because we “don’t think of women as people”, etc.

    But I think the biggest most important one is:

    5. Context of discussion and the echo chamber. Yeah, “always” shouldn’t always be taken literally; but when you’re having what’s supposed to be a serious discussion, things are different. Plain language frequently just doesn’t suffice there; you have to be more technical. So I at least assume people are being more literal when they’re having a serious discussion.

    Now, does that mean that when someone in a serious discussion says “You should always look both ways before crossing the street”, they really think that you should always look both ways before crossign the street? No, probably not. People are not generally going to be so careful with what they say, and you can’t expect people to get things exactly right the first time. And in a case like the street-crossing example where it’s pretty clear anyway, it’s not a problem.

    But it isn’t always so clear to everyone, which is why a normal part of a sensible discussion is bringing up counterexamples to how things were stated, and attempting to reformulate the statement to accommodate these, to better get at what the original speaker meant. But inside the echo chamber comprising most feminist spaces a person is likely to encounter, attempting to do this — trying to get a better understanding, even just asking for clarification — marks you as an enemy. Which is why I think that while scrupulosity may be the object-level problem, the more serious problem supporting it is the echo chamber. Many of these principles seem as if nobody’s ever run a first-pass check for counterexamples. And instead of adapting the principles to be more correct, people opaquely alter the meanings of words to make the original correct as stated! This is really not helpful to those who are already confused.

    Now, in fact, I actually think that trying to be totally precise here is actually the wrong solution. The problem is just too complex, and it’s a lot of little social cues stuff that is hard to explain verbally. The right solution is to do the opposite and embrace imprecision — as well as forgiveness for those who screw up but were acting in good faith (and back off as soon as a problem becomes apparent, apologize if necessary, and learn from their mistakes). (It may be that a lot of people are taking the forgiveness part as implict. But a lot of us aren’t!) But such imprecision should be clearly marked, and admissions of ignorance made explicit! Right now we have imprecise things stated as if they were precise.

    Are people discussing ordinarily deliberately and consciously being imprecise? I doubt it. Seems to me that generally, those who haven’t studied math, programming, analytic philosophy, etc., have no idea just how hard being precise really is, and honestly think they’re being precise when they’re not. Not to mention the terrible abuse of words. Most of these people really need to read 37 ways that words can be wrong. So much in there that can help avoid pathological discussions.

    But again, I’m not saying everyone has to be able to do this sort of thing immediately. It’s not a natural way of thinking for most people. But maybe if feminist spaces had better norms of discussion, people would learn to be able to do this, and the more careful thinkers would hold more sway within them. But right now those are places to stay away from if you want a serious discussion.

  21. Pluviann says:

    IIRC Dworkin was referring to sex within marriage, at the time of writing many states and countries had laws that explicitly allowed men to use their wives sexually at any time, without consent, and held it to be impossible that a man could rape his wife. Dworkin wasn’t talking about an ill-defined ‘oppression’ she was saying that if you are legally unable to say ‘no’ then your ‘yes’ is meaningless.

  22. […] a mathematician who wrote the excellent popular book How Not to Be Wrong.  (Ellenberg had also criticized me last year for my strange, naïve idea that human relations can be thought about using […]

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