People like decisions that are linearly separable.
If you’ve got some continuous variable and you want to separate it into two categories–such as “yes/no” or “good/bad” or “moral/immoral” or anything of that sort–people are happiest when they can do it by drawing a single slash mark that divides the variable neatly into two parts. If you have more than one variable that matters to the decision, people prefer it if you can make the decision by drawing a single line (or curve, whatever) to divide the “space” into two portions.
For example, suppose you are deciding whether someone is datable. You think to yourself: the person has to be attractive and have a good personality; a little bit extra of one can make up for a little bit of a lack in the other, but when all is said and done, you want to have at least a minimum amount of both. You can represent this “decision space” graphically: you have a dimension for looks and a dimension for personality, and you can draw a line on that graph that represents the boundary between “datable” and “not datable.” The exact shape of this line, the slope of the line, whether or not it is curved, and so on, will depend on a particular person’s tastes. But, you can still divide the space into “datable” and “not datable” by drawing a single line. Thus, we call this decision “linearly separable.”
What’s an example of a decision that’s NOT linearly separable? The simplest example is something like “eating the right amount of food.” You don’t want to eat too little food or too much food. So if you draw a line that represents the “amount of food” there is no single tick mark that you can draw through that line to divide “right amount” from “wrong amount.” Instead, you have to draw at least two lines: one to represent the lower boundary, and one to represent the upper boundary. As a result, you would say the decision about how much food to eat is not linearly separable.
Except… it depends on how you look at it. The human mind is so attracted to linearly separable problems that usually there is a way to take any problem that seems to not be linearly separable and look at it in a way that is. For example, suppose that instead of looking at the dimension of the “amount of food” as the critical dimension, you consider physical discomfort to be the important dimension.
So let’s take a look at three scenarios: under-eating, eating a moderate amount, and over-eating. We will plot these against two different dimensions: the “food” dimension and the “physical discomfort” dimension. Obviously, eating the “right” amount isn’t linearly separable along the “food” dimension: you don’t want too little or too much. However, both under-eating and over-eating will cause discomfort (and eventually health risks). If you eat a moderate amount, you will minimize your discomfort. So along this dimension, you can draw a single line that is the boundary: less than the threshold amount of discomfort is good, anything over the threshold is bad. Along the discomfort dimension, the problem actually is linearly separable.
So whether something is linearly separable can depend on what dimension or dimensions you choose to look at.
People are so drawn to problems that are linearly separable that when they come across something that is not linearly separable, it often makes people feel uncomfortable. It makes people feel like they need to “explain away” something: they have to find a different dimension to explain it, or come up with the idea of an “exception to the rule,” or find some other way to explain why there would be a lack of separability. In most people’s minds, separability is what makes a “normal” good argument, non-separability suggests that you may be looking at the wrong dimensions.
So, let’s talk about abortion. I know a number of liberals who believe the following: the decision to have an abortion is a valid choice that a woman can make (at least in some circumstances) and it is entirely up to her; however, the decision to drink and smoke during pregnancy, so that there is a risk of the child being disabled or having some kind of life-long health problem after it is born, is not a valid choice and she does not have the “right” to make that decision.
Many people, including pretty much all of the conservatives that I’ve ever spoken with, are completely confounded by this. They do not understand how it can be “OK” to “kill a fetus” not “Not OK” to “damage a fetus.” It seems like a contradiction.
We can understand why this seems to be a contradiction in terms of what we have just learned about linear separability. Conservatives are looking at this distinction as one that is not linearly separable. Specifically: doing no harm to a fetus is ok; doing some harm to a fetus (by drinking and smoking) is not ok; doing the ultimate harm to the fetus (by having an abortion) is ok again. As you can see: there are two boundaries, not one. This is not linearly separable.
What this tells me is that the people who really believe this view are probably not making this decision based on “damage to the fetus.” Like with the “right amount of food” example, above, it’s worth trying to see whether there is another dimension that we could use, instead, that would turn this into a linearly separable problem.
Suppose you consider the dimension of “misery of the child.” That is: how much emotional anguish will a child have to experience in each condition? In the case of ideal care, there will be (one hopes) very little misery. Also in the case of abortion, there will be very little misery (a child that is never born cannot be miserable). However, in the case where the mother smokes and drinks, a child will be born and there is a strong risk that that child will spend its entire life miserable.
Here we can see a comparison of how these three scenarios look when plotted both in terms of “fetus damage” and “child misery.” Even though the decision about what is “morally acceptable” is not linearly separable along the “fetus damage” dimension, the decision is linearly separable along the “child misery” dimension. In the case of using the “misery” dimension, there is some threshold of misery, and anything that subjects a child to more than that threshold is simply not a valid decision for the mother to make. The problem is linearly separable, and makes complete sense, once you realize that the thing that is important in this decision is not “how much damage are you causing to a fetus?” but rather “how much misery will your actions put a child through?”
This is a good way to think about arguments, whenever you come across someone who has an opinion that doesn’t seem like it “makes sense” to you. When you are talking with someone, and their view seems to have a contradiction or to be inconsistent in some way, there’s a good chance that you think that there is a lack of linear separability. There is also a good chance that the dimensions that you assume the person believes are important, are not actually the dimension that he is using at all.