# Bivalence and the Excluded Middle: True, False, or Neither?

#### Introduction

Are you tall or not tall? You’re probably thinking, “It depends” or “It’s not exactly one or the other.” Tallness doesn’t seem to be something you either have or don’t have but something that falls on a spectrum, somewhere along which you fall. Thus, the statement “You are tall” doesn’t seem to be straightforwardly true or straightforwardly false, but partly true and partly false. Tall is a vague concept, one which lacks clear boundaries. Because of this vagueness, the statement “You are tall” defies the principle of bivalence, according to which every statement is either true or false.

#### What is the Problem of the Excluded Middle?

Two fundamental and distinct principles of logic are the principle of bivalence and the principle of the excluded middle.

The principle of bivalence states: Every statement is true or false.

Example: “You are tall” is either true or false.

The principle of the excluded middle states:  For any statement P, P or not-P must be true.

Example: Either “it is the case that you are tall” or “it is not the case that you are tall” must be true.

The principle of bivalence is an essential principle of logic, but some have criticized it for not correctly capturing the shades of gray in certain statements. According to these critics, not all statements are either true or false. This is especially apparent when we think about vague concepts, such as tall, which don’t fall into clear categories. It seems neither true nor false to say you are tall. It seems more appropriate to say, for example, that you are relatively tall or less tall than most.

The Sorites paradox lends support to the view that vague concepts such as tall fall on a spectrum. The paradox is this: If a person we believe is tall were 1 centimeter shorter, we would still say he/she is tall. If he/she were 2 centimeters shorter, our judgment would remain the same: He/she is still tall. But, if we went on with this reasoning until we imagined the person being 100 centimeters shorter, would we still call him/her tall? Surely not. This paradox is meant to show that there is not a cutoff point at which someone stops being tall. Tallness comes in degrees. But this would mean the principle of bivalence is wrong. “You are tall” is not either true or false. Preserving the principle would require defining the cutoff point between tall and not tall, but many find this solution unappealing because it goes against our intuition in the case of the Sorites paradox.

#### Why Be Aware of It?

So far, it might seem like we’ve been discussing some abstract logic concepts that don’t mean much for the real world. However, the problem of the excluded middle matters for everyday speech and argumentation. Here are two reasons to be aware of it:

1. Avoid oversimplification. When making claims about things, you should be aware that things are rarely black and white and that your judgments should reflect the shades of gray. So, for example, it is almost never straightforwardly true or false that a bill is fair or not fair, an action right or not right, a politician skilled or not skilled. Fair, right, and skilled—as well as most other concepts involving judgments and opinions—are vague concepts, and accurate assessments take account of this. Rather than oversimplifying and saying a bill is flat-out unfair, it is more accurate to say it is fair in some ways and unfair in others and to explain the ways in which it is both.
2. Judge others’ claims carefully. Being aware of the problem of the excluded middle enables you not only to avoid oversimplification but to push back against the oversimplifications others make. This way, you can more carefully assess their claims. When someone is making claims about a policy being unjust, for example, remember that the policy is likely to be just and unjust to some degree. Keeping this in mind, you can push the person to explain the ways it is just and the ways in which it is unjust. Under what circumstances is it unjust? For whom is it unjust? In comparison to what other policies is it unjust? An accurate picture of the unjustness of a policy often lies in these shades of gray.

#### An Example from the Law

One area where the problem of the excluded middle shows up is in blanket criteria for admission into opportunities such as sports, schools, and jobs. Consider the case of Esoterikon v Kalliri. Maria-Eleni Kalliri fought back against a universal height requirement that required her to be 1.7 meters tall to join the Greek police force. She was rejected for being 2 centimeters too short. The European Court of Justice found that the height requirement indirectly discriminated against women. Women tend to be shorter than men, so the requirement unjustifiably puts women at a disadvantage as compared to men for admission into the police service. The Police Authority defended itself by claiming the requirement was meant to ensure it admitted officers who could physically carry out the various responsibilities of the position. The Court determined that the height requirement was neither an appropriate nor a necessary way of pursuing that goal, because (1) not all police functions require significant physical force, and (2) even if they did, a physical aptitude test would be better than a height criterion for determining whether a candidate could exert that fore.

Here, the statement “Ms. Kalliri is not tall enough to be an effective member of the Greek police force” was shown not to be straightforwardly true. The Court’s decision was based on its recognition that height does not come in categories that neatly map on to abilities. That is, being 1.7 meters tall does not automatically make someone qualified to perform the necessary tasks, and being 1.68 meters tall does not automatically make someone unqualified. Tallness comes in degrees and varies by gender, so it is oversimplifying to say that being an effective police officer requires someone to be tall, as defined by being 1.7 meters tall. Men are typically taller than women, so Ms. Kalliri may have been tall enough for a woman. And more importantly, as the Court argued, a better indicator of a candidate’s skill would be a physical test, not an arbitrary height cutoff point that ignores the fact that physical ability can come in degrees of height.

#### The Political Power of Avoiding Bivalence

We said earlier that being aware of the problem of the excluded middle helps to avoid oversimplifications and to sift through the oversimplifications others make. These are both crucial things to look out for when advocating for social and political change. We are all prone to making sweeping generalizations and presenting an oversimplified picture of the way things are when we are trying to make a point we are passionate about, but the most persuasive advocates acknowledge the shades of grey. This makes both their supporters and their opponents take them more seriously, because they come across as more knowledgeable. If you’re leading an anti-war campaign, for example, you may come across as naïve if you make a statement such as “War is harmful.” Most people would agree that war has harmful aspects, but it is not straightforwardly true or false that war is harmful, period. It may be helpful to some parts of society (e.g., arms manufacturers)(e.g., the economy) while also being harmful to others (e.g., soldiers and infrastructure). To make a convincing case for your position on war and other social and political subjects, avoid bivalence and address the complexity of those subjects.

## Think Further

1. Think of a time you’ve encountered a cutoff for something (e.g., weight requirement for wrestling, height requirement to ride a roller coaster, etc.). Do you think the requirement is justified? Is there another way in which the goal of the requirement could be met while respecting the degreed nature of the measured thing (e.g., weight, height, etc.)?
2. Consider the GPA cutoff policies used by some colleges and graduate schools. Do you agree or disagree with these policies? Explain why or why not.
3. Come up with a book or movie you think is good in some ways and bad in others. Create a T-chart, and list the good features in one column and the bad features in the other. Then, write a one-sentence review of the book or movie that captures the features you think are most worth noting—good and bad.