12

FINE LINES AND GRAY AREAS

HOW LOGIC PUSHES US TO BLACK AND WHITE IF WE’RE NOT CAREFUL

ONE NIGHT DURING MY first term at university another fresher was found in the kitchen just before midnight eating a bowl of cereal. He explained that, according to the best-before date, his milk was about to go off at midnight. I’m afraid we laughed at him rather heartily, and in his belief that a best-before date is so precise, or that milk could suddenly go off at midnight like Cinderella’s carriage turning into a pumpkin.

Unfortunately many more serious problems arise from our attempts to deal with things that are on a sliding scale. If we’re not careful logic pushes us to extremes, so if we want to avoid adopting extreme positions all the time we have to do something more human. A human approach is more nuanced than that. It turns out that our brains are able to process gray areas in subtle ways that don’t seem entirely logical but do seem to make sense. Rather than use logic to push out the sense, we should find the logic inside the human nuances. There are various different ways of dealing with gray areas that stem from different logical interpretations. In this chapter we’ll discuss these different approaches, and the pitfalls of adopting the simplest approach of drawing a line. Allowing for some uncertainty can seem unsettling, but it can avoid both the extremes and the anomalies of drawing a line.

One of my favorite moments in Jane Austen’s Pride and Prejudice is when Elizabeth asks Mr. Darcy how and when he fell in love with her. He replies,

I cannot fix on the hour, or the spot, or the look, or the words, which laid the foundation. It is too long ago. I was in the middle before I knew that I had begun.

He is unable to draw a line and say that is the exact line in between not being in love with her and being in love with her. Where could such a line possibly be? He can only know that at some point he was not in love with her, and at some later point he definitely was. This is understandable because being in love with someone is a somewhat hazy and nebulous concept that grows gradually (except in cases of love at first sight, if that really exists), with a lot of gray area in between the definite “no” and definite “yes”.

Gray areas are an important part of the human experience, but are not very well dealt with by logic. Logic is about getting rid of ambiguity. The law of the excluded middle forces us to put the whole gray area in with black or in with white. That is better than the sort of black and white thinking that pretends the whole gray area doesn’t exist, but it can have the same effect as it can push people to think only the black or the white if they try to pursue logic to its rigid conclusion.

The language we use in everyday life seems to be getting pushed further and further to extremes and certainties. People say that something is “the best thing ever” (or the worst). They try to reassure me by saying “Everything is going to be fine”, they advertise events to me by saying “You will not want to miss this!” Admittedly, “You might not want to miss this” doesn’t sound so good. But I worry about the world turning into certainties that are almost certainly flawed. We should understand different ways of dealing with gray areas and become better at working with their nuance, instead of longing for the false promise of black and white clarity.

CAKE

I am not immune to extreme thinking myself. Here is the kind of reasoning that makes me prone to getting fat:

• It won’t hurt to eat one small piece of cake.

• And however much cake I’ve already eaten, it can’t hurt to eat one more mouthful.

Unfortunately, logically this means that it is fine to eat any amount of cake, as long as you go one mouthful at a time. And unfortunately this is what I am prone to do.

It is another example where being strictly logical is not entirely helpful. The only way to avoid thinking it’s okay to eat infinite cake is to decide it’s not even okay to eat one small piece of cake. I am much better at avoiding cake altogether than eating a small amount and stopping. The trouble is that everyone around me is likely to point out that one small piece by itself is fine.

This is an example where the logic of the situation pushes us to one of two extreme positions, either

• it is not okay to eat any cake at all, or

• it is okay to eat infinite amounts of cake.

The trouble is the gray area. There is no strict line we can draw between a sensible amount of cake and “too much”. Parents are liable to try and draw those lines to stop children gorging on cake, but children aren’t fooled–they easily see that those lines are arbitrary, and they try to push them by having one more mouthful, and one more. Or they try to push bedtime by an extra two minutes, and then another extra two minutes, by saying they need to go to the toilet, or fetch a toy, or drink some water, or any number of other spurious requests. But really, the bedtime itself is spurious–it is an arbitrary line that has been set in a gray area between “sensible bed time” and “much too late”.

One way to get round this logic is just to shrug and say just because something is logically implied, that doesn’t mean I’m going to believe it. But that is unsatisfactory as it would allow for all sorts of other illogical thinking, such as believing two things that cause a contradiction.

The idea of believing all the logical implications of your other beliefs is called “deductive closure”–a set of statements is deductively closed if it also contains everything you can deduce from all the statements in the set. So my set of beliefs is only deductively closed if I believe all the implications of my beliefs. I think this is an important part of being a logical human being, and I’ll come back to it in the last chapter.

So, if I want to be rational, what can I do about gray areas? I might have to let go of the most obvious logical approach and learn to deal with something more complex.

DRAWING A LINE

Gray areas are often dealt with like bedtime–an arbitrary line is drawn somewhere in the gray area and a rule is made. Where you put the line in the gray area depends on how dire the consequences of the extremes are. If one of the extremes is very dire then the line in the gray area will probably need to be further away from that extreme, to give a buffer zone around it. For example, some roller coasters have a minimum height requirement for safety reasons. If you’re too small then the safety harness won’t fit around you enough to keep you safe. In that case the consequences are very dire (injury or even death) and so the limit should probably be at the tall end of the gray area, to be safe.

With the cake, I am trying not to get fat, so I should probably draw the line safely within the “most probably won’t make me fat” part of the gray area, rather than somewhere in the “might not make me fat but it’s not clear” part. This is especially true because I am liable to stray over my line a bit, so I should put a little buffer zone in to be on the safe side.

One contentious area of drawing lines I’ve had to deal with a lot professionally is grade boundaries in exams. In the UK system, students graduate from university with a degree that is classed as first class, “upper second” class, “lower second” class, or third class, otherwise known as first, 2:1, 2:2 or third. But where should the boundaries be drawn? I have spent long and contentious hours in examiners’ meetings battling this out, in what is essentially a futile exercise placing a line in a gray area. No matter where you place it, someone will argue that it’s unfair to the person just below it, and as a result the line tends to shift further and further down. There is no logical place to put that line. I think the only logical thing to do is get rid of the lines and publish either averages on a fully sliding scale or percentiles instead.

A more contentious gray area arises when talking about race. I will not call this “literal” black and white, because we’re actually all shades of brown and pink. As discussed in Chapter 4, Barack Obama is often called “black” although one of his parents was black and one was white. So arguably it’s just as valid to call him white. However, once we understand that “black” here is being used to mean “non-white”, we see why it makes some sense to call Obama the first black president of the United States. Which is to say, we have found the sense in which it makes sense.

Where should we draw a line between black and white? If we draw it nearer the white side, this could acknowledge that only people who look really white enjoy the privileges of white people. But it could constitute exclusion of anyone non-white as “other”. At least talking about white people and non-white people is a genuine dichotomy, where black and white is a false one.

SEXUAL HARASSMENT

It can be especially hard to draw lines when dealing with people who do not respect boundaries. This can happen if you are the kind of person who likes being generous and being able to help people. Unfortunately people are liable to ask more and more of you. More seriously, it arises in situations of micro-aggressions and sexual harassment: how serious does someone’s misconduct need to be before you should take action and report them?

Some forms of physical contact are generally accepted, such as a handshake. Others are clearly not, such as groping. But where do we draw the line in between? Is touching someone’s shoulder appropriate? Their back? Their waist? Their hip? Do we have to draw a literal line on our bodies to indicate where can count as friendly and where counts as harassment? This is a difficult dilemma for those who have been made uncomfortable by someone’s behavior, especially if they’re in a position of vulnerability or on the lower side of a power differential. If you report someone touching your shoulder you will almost certainly be told you are overreacting. So at what point is it worth taking action? If you accept one action then you can feel like the next small escalation isn’t that much worse. But all those small escalations add up.

Manipulative people can actually exploit this to take advantage of people who are prone to being kind, generous or accepting. As soon as you give in once, even a little bit, they know that the boundary can move by a little bit, and so by applying that logic repeatedly, they can think that the boundary is totally movable. If you put your foot down and stop them at some point, they might accuse you of being mean, unreasonable, overreacting.

Mr. Darcy’s inability to draw the line for love also applies to hurt–we don’t necessarily know where is the exact moment that something is going to start hurting us. We only know where it will definitely hurt us a lot, like if inappropriate physical contact becomes rape.

It took me a while to learn that the best way to be safe is to draw the line somewhere where you definitely feel safe, before things start being at all questionable. This creates a buffer zone encompassing the gray area, protecting us from the area that is definitely dangerous, but without definitely declaring where the buffer zone ends and the unsafe zone starts:

I used to think this was ungenerous, because it means that wherever I draw the line there might have been a little bit more I could allow without getting hurt. But having been hurt too many times I now know I need that buffer zone, like with the cake, to protect myself. And I also know that protecting yourself is important, and not necessarily ungenerous. As we saw in Chapter 5, if protecting yourself means denying someone something, or even hurting them, I don’t think it’s your fault if the other person then gets hurt by it. It’s the fault of the system, of the toxic relationship that has created such a zero-sum game.

BODY MASS INDEX

Body Mass Index is a useful but rather flawed measure of healthiness where you take your mass (in kilograms) and divide it by your height (in metres) squared. The first thing that people object to about this is that it doesn’t take into account how muscular you are, and so very strong athletes are liable to have a high BMI as muscle is very dense. However, personally I find this argument spurious as it’s pretty obvious whether you are a muscular athlete or not. More to the point (and to avoid making a logically fallacious sweeping statement), I know perfectly well that I am not a muscular athlete. I don’t need to use calipers to know that I have fat on me, even if I hide it well enough under my clothes that people insist that I surely have no fat at all.

The other problem is that arbitrary lines have been drawn for what counts as a “healthy” weight in terms of BMI. The cutoff for women is usually stated as 25. But of course, it’s a sliding scale. It doesn’t mean that someone with a BMI of 25 is fat and someone with a BMI of 24.9 is fine. It’s supposed to be a guideline, and I am quite happy to use it as a guideline. It does create silly situations when the doctor weighs me, though, because sometimes I know that my shoes are likely to tip me over the BMI 25 mark so I insist on taking them off. If they register a BMI over 25, the computer automatically puts an alert on my file, and even though the doctor shrugs and says I’m so close that there’s no need to worry, I have made such a big effort to lose weight and keep it off that it is utterly galling to register as overweight even if I know it was really just my shoes.

Despite such escapades I still think it’s better than having no guideline and doing what I used to do, which is convince myself that I was still fine really, even though I was gaining 20lb a year. This involves another gray area, to do with reasonable weight gain. You might shrug to yourself and say “Gaining a couple of pounds in a month isn’t so bad, not worth worrying about.” But then if you say that to yourself every month you’ll find that you’re gaining 24 lb per year. Personally at some point I imagined myself in 10 years (instead of just thinking month by month) and finally realized that I had to draw a line somewhere, even if it was arbitrary. I try to draw it on the safe side of the BMI line, somewhere around 24 rather than 25.

One interpretation of what I’m doing is treating the line itself as something hazy, so I try to stay far enough on the good side of the line that I’m out of its hazy range.

INDUCTION

These arguments by small incremental steps are related to the principle of mathematical induction. This is different from argument by induction, which is a flawed type of argument where you generalize from a small sample to a large one. For example, “The sun has risen every day of my life so far, so it will rise tomorrow.”

Mathematical induction is logically secure, and is a bit like climbing up steps. Babies learn to climb up one step and then discover, with delight, that if they just repeat it they can go up whole flights of stairs, possibly all the way to the sky. All they need is for someone to put them at the first step (and for nobody to intervene and take them off the stairs).

Mathematical induction says if you know something is true for the number 1, and if you have a way of climbing up by 1, then you know it’s true for all whole numbers. If we apply this to small cookies, we’d say

• It’s fine to eat 1 cookie.

• If it was fine to eat some number of cookies, it’s okay to eat 1 more.

Therefore it’s fine to eat any number of cookies.

Mathematical induction is stated in terms of whole numbers n, and we say that we are trying to prove that some property P is true of each number n. So P(n) might be the statement “It is fine to eat n cookies.” Then the argument looks like this:

• P(n) is true.

• P(n) P(n + 1).

Then by the principle of mathematican induction, P(n) is true for all whole numbers n.

This is fine for whole numbers, but it gets tricky if you’re trying to deal with a sliding scale that includes all the numbers in between, or even just all possible fractions. This is because there is no smallest unit of “jump” for us to take steps in.

We might try to apply this to a series of cookies that all seem to be “more or less the same size”. Perhaps this means that they are within 5g of each other’s weights. You might be happy thinking that a 50g cookie is more or less the same size as a 52g cookie, and that this is more or less the same as a 54g cookie, but after a few steps like that you will end up with a cookie twice as big. I have done this by handing out cookies to a class of 20 students without telling them what the point is. I ask them all to compare their cookie with the person next to them, and they are all happy that their cookie is more or less the same. But then I get the first and last student to compare cookies and we all collapse in giggles because the first one is tiny and the last one is enormous.

FUZZY LOGIC

I think one way to deal with this is to allow for more nuanced levels of truth. With cookies, the point is that there isn’t just “fine” and “not fine” for amounts of cookies to eat. There is “fine”, “less fine”, “sort of okay but not great”, “not that great”, “dubious”, “suspect”, “a bit much”, “too much”, “way too much”, “ludicrously excessive” and so on. As we saw in Chapter 4, normal logic with the law of the excluded middle doesn’t allow us anything except “fine” and “not fine”, so we end up pushing everything into one or the other as we can’t find a logical place to draw a line. Instead we might try to treat truth values as something on a scale between 0 and 1. This can be dangerous in some situations because it can give the idea that truth is something negotiable, and that some things are more true than others. However, I think this is true in the case of gray areas. It also might be true in the case of probability, when we can’t be certain what the truth is, we can only be a certain percentage sure, with the rest being in some doubt. Percentage probabilities are, in a way, placing things on a scale of truth between 0 and 1. Unfortunately it often seems that we humans are not very good at understanding those either.

In Chapter 4 we briefly mentioned fuzzy logic, a type of formal logic that takes truth values in a range from 0 to 1. This measures the extent to which something is true, rather than our certainty about whether or not something is true. The two things are related, but not exactly the same. For example, if I look up the weather online it gives me a percentage likelihood of rain during each hour of the day. Usually I conflate this with an amount of rain in my head–if the forecast says 90 percent chance of rain, I interpret it to mean it’s probably going to rain hard. If it says 40 percent chance of rain, I interpret it to mean it might rain a bit. In practice this is likely to be because of where the uncertainty of a weather forecast comes from. The only way to be really sure that rain is coming is if there is a very strong storm heading strongly in this direction. If we’re not sure, it might be because it’s only a light rainstorm that has some chance of fizzling out or changing direction.

Similarly if an exam is marked with just pass and fail, then the result is very clear cut. Before you get the result you might be unsure about whether you passed or not, but only if you’re a somewhat borderline case. If you’re a really excellent student you might be unsure quite how well you did, but still sure you passed. Again, the certainty and the extent to which something is true are related.

However, even once uncertainty has been eliminated, the extent to which something is true can still vary. If an exam is marked on the whole scale from 0 to 100, you might get a mark of 71 and ask your teacher if that counts as good or not. There is now a whole scale from good to bad, and the uncertainty comes from the gray areas, not from actually not knowing.

Fuzzy logic is currently used more in applied engineering than in math, to deal with gray areas in the control of digital devices. One example is in rice cookers, where the cooking process can be adjusted according to some slightly vague conditions like whether the water is being absorbed slowly, quite slowly, quite quickly, or quickly. This can also be done for heating or air-conditioning control, or anything else that needs to respond dynamically to potentially changing conditions. Of course, a definition still has to be produced as to what those gradations mean, but having the possibility of truth values in between sheer true and false opens possibilities for more subtle control of devices.

THE INTERMEDIATE VALUE THEOREM

Another way to deal with fine lines and gray areas that isn’t fully logically determined is to acknowledge that the line is somewhere in the gray area and we don’t know where it is; we just know it’s somewhere in that area. We can put bounds on the area by pointing to one place that’s definitely below the line and one place that’s definitely above the line.

This is how you can make a batch of cookies so that everyone has their perfect size of cookie. You can start with a cookie that is definitely too small, maybe just 2g of cookie dough. Then you can make them get very gradually bigger so that each one seems about the same size as the previous one, but they’re getting imperceptibly bigger. Keep going until you reach one that is obviously definitely too big, say twice as big as your face. Opposite is a set of such cookies I made.¹

Because there is essentially every size of cookie in between, this means everyone’s perfect size of cookie must be in there somewhere. This helpfully deals with the fact that my personal favorite size of cookie is smaller than most people’s. This way I can meet my own needs and other people’s at the same time, without actually having to know what anyone else’s perfect size of cookie is–I can be sure it’s in there somewhere.

This is an application of the intermediate value theorem, a theorem in rigorous calculus that math students usually study as undergraduates. It says that if you have a continuous function that starts at 0 and goes up to some number a, it must take every value in between. What “continuous” means here is rather technical, but it basically means there are no gaps. You might argue that my cookies do not take every size, otherwise there would be infinitely many of them. That’s true, but I am using a real-life approximation to the intermediate value theorem. Really I’m saying that the whole thing is true up to a certain accuracy determined by our perceptions.

A few months ago I was talking to one of the art students at the School of the Art Institute of Chicago. She was exploring people’s perceptions of reality by creating visual illusions and trying to see if viewers would believe they were physical constructions, or think they were digital manipulations. The question was to find the sweet spot where people would be really unsure which it was. I realized that she could invoke the intermediate value theorem: make a series of pieces starting with one that was obviously a physical construction, gradually becoming less obvious until she ended with one that was obviously physically impossible so must be a digital manipulation. At some point in the gray area in between there would have to be a point where viewers were unsure whether it was real or digital. The artist would not need to know exactly where it was, and indeed it could be in a different place for different viewers. All the artist would need to know is that it is somewhere in that gray area.

In a way this is also what chocolate makers do with the percentage cocoa content in their chocolate. They make a whole range of different percentages so that all chocolate lovers will find their perfect percentage in there somewhere. I regret, however, that so many chocolate makers stop their range somewhere around 70 percent (in fact 72 percent seems a popular place to stop), because that does not encompass my own personal favorite spot. My gray area for chocolate is somewhere between 80 percent to 100 percent, depending on my mood. As a result, to get my perfect chocolate, like my perfect size of cookie, I usually have to make my own.

This is a similar principle to the one that is giving us more and more finely tuned gradations of race. In the unenlightened old days there was just “white” and “non-white”. We now talk about “people of color” but we also talk about mixed race, although this often implicitly means mixed between white and non-white. People have been coming up with words for mixed races between non-white and some other non-white, such as blasian for black and Asian. A friend of mine calls himself “Mexippino” for Mexican and Philippino.

But should we have different words for someone who is one quarter Asian and three quarters white (who is likely to be quite white-passing), as opposed to three quarters Asian and one quarter white (who might look entirely Asian)? As described throughout this chapter there are various possibilities, each of which has advantages and disadvantages. We can talk about “white people” and “Asian people” which is simple, but results in the exclusion of those who fall into neither category exactly. We can draw a line somewhere and end up creating bizarre anomalies, where someone is classified as a “person of color” although they look white to most people. We can create increasingly many finer and finer categories taking everyone into account, but ending up with an intractable collection of extremely specific descriptors. We can unify everyone, ignore race, and declare that we’re “all humans”, as some people do when they call themselves “color-blind”. But this discounts people’s real experiences of racial discrimination. I think that above all we should acknowledge that there are gray areas and become more comfortable with accepting them.

BRIDGING GAPS

All this is to say that if we aren’t careful about gray areas and logic we can end up being pushed into taking extreme positions, as the only way of remaining logical. Indeed, it’s possible to use incremental steps of logic to argue someone into a very extreme position without them realizing what is going on. If we hold people to this black and white logic then we push all disagreements into further and further extreme opposites. I think, instead, we should give everyone ways to get out of these positions that are still counted as logical. Whether it’s fuzzy logic, probabilities, hazy lines, or lines simply placed somewhere unknown in a gray area, or simply being more comfortable with less certainty, these are all more nuanced ways of dealing with our very nuanced world. The gray area is a bridge between black and white. There are few things as simple as black and white in the real world. Really we’re all living somewhere on that gray bridge, and for some people it feels unsettling to take a position of nuanced uncertainty. If we all acknowledge that, and even build more bridges, I think we will achieve better understanding.

We have discussed how gray areas can hurt us when exploited by unscrupulous people, but we can turn this around and exploit gray areas to help ourselves, if we apply them judiciously. The idea is that we can use small, unnoticeable increments to nudge ourselves gradually to somewhere that is quite far from where we started. If we tried to do it in one leap it would seem insurmountable. I use this psychological trick all the time to make progress. I use it when I’m learning a new and difficult piece of music at the piano. I start by only being able to play it extremely slowly, but I use a metronome and gradually increase the speed by a tiny and unnoticeable amount each time. I might start it at a rate of 40 beats per minute, at which speed the piece is easy, and then go up to 42. My fingers don’t notice any difference. Then I go up to 44, and my fingers still don’t notice any difference. It doesn’t take that long to double or even triple the speed at which I can play the piece. The principle of “a little bit more won’t make a difference” is harmful when it’s about eating cake, but beneficial when it’s about accomplishing a big task.

More seriously, this method can be used to find a bridge between apparently opposing ideas. We have discussed the difference between believing in social services and not believing in them, in terms of false positives and false negatives. One person believes in helping everyone who needs help, even if that means helping some extra people by mistake. Another person believes that everyone should take responsibility for themselves. This argument can be very divisive, but we could instead acknowledge that there’s a gray area. The person who believes in social services probably doesn’t believe in simply giving out large amounts of money to everyone who asks for it, without question. The person who believes that everyone should take responsiblity for themselves might be able to acknowledge that some particularly “worthy” people need help, perhaps members of the military who have been injured during active service. If this is so, then we have established that the question is not whether or not we should help people, but to what extent we should, and under what circumstances. It is now a question of where we place people in the gray area and how we treat the gray area. In the coming chapters we will discuss this technique of pushing a principle to extremes abstractly in order to encompass those who seem to disagree with us, and draw them onto the bridge that is the gray area. The first step is to think about understanding a difficult argument by comparing it with a more understandable one that has something in common, that is, it is analogous in some way. This is the subject of the next chapter.

¹ Photo credit: M. N. Cheng