**Introduction**

The law frequently requires that we answer a question "yes" or "no". Was the plaintiff guilty? Was the defendant negligent? Was the trial court's finding of fact clearly erroneous? These questions seem to demand a "yes" or "no" answer.

When an issue must be resolved in one (and only one) of two possible ways, we can call that issue "binary". That is, the law frequently demands that we see the world in blacks and whites. The real world (the world outside of the legal system) is frequently all shades of gray. Our beliefs about whether the defendant was guilty, whether the defendant was negligent, or whether the trial court erred may be a matter of probabilities or degrees. When an issue or finding is a matter of degree, we can call it a "scalar".

This entry in the *Legal Theory Lexicon* is about the scalar/binary distinction in legal theory. As always, it is primarily addressed to law students, especially first-year students, with an interest in legal theory. If you have further questions about the distinction and the role it plays in legal theory, you will want to consult "Scalar Properites, Binary Judgments," by the great legal theorist, Larry Alexander.

**What is the Scalar-Binary Distinction?**

One way to explain the scalar-binary distinction is via an analogy with related distinction between scalar and discrete variables in mathematics. A variable is scalar if it can assume any value within a range. A variable is discrete if it can only assume some values. For example, weight is a scalar variable. If we way a group of humans, their weights can assume any value. An infant could be 8.5 pounds, an adult 180 pounds, and any value in between is possible, including 80.8715 pounds or 13.025 pounds. Technically, we also need to distinguish scalars (which have only one dimension) from vectors (which have mutliple dimensions).

Another metaphor for a continuous variable is a "spectrum." For example, the frequency of a radio broadcase is a value on a spectrum. Similarly, we can also use the term "continuum" to refer to the same idea that is captured by the notion of a scalar. We also use the phrase "matter of degree" to get at the same basic notion. More technically, sometimes the phrase "range property" is used to refer to scalars.

A discrete variable can only assume particular values. For example, if we toss a coin three times and count the number of heads, the only possible outcomes are 3, 2, 1, and 0. The number of heads cannot be 1.5 or 2.00786. Binary variables are special cases of discrete variables; a binary variable can only assume one of two values. If we toss a coin once, the only possible values are 1 and 0. When someone says, "The answer to this question is either a 'yes" or a 'no," they are saying the question involves a binary judgment. Multiple-choice questions are discrete, but not binary if there are more than two possible answers.

The law is full of binary judgments. The defendant must be found "guilty" or "not guilty." The plaintiff's injury was either "caused" or "not caused" by defendant's conduct. The trial court's finding of fact was either "clearly erroneous" or "not clearly erroneous." Sometimes, however, the law requires that a judgment be made on a scale. For example, in a comparative negligence jurisdiction, the jury might be required to decide whether the plaintiff's injury was 100% caused by plaintiff's negligence or 0% caused by plaintiff or any value in between.

**Scalar Properties, Binary Judgments**

Sometimes, the law requires that we translate our beliefs about a scalar property into a binary judgment. For example, we might think that the likelihood that a trial judge made an error in a finding of fact is a matter of degrees, ranging from a 0% chance of error to a 100% chance of error. But when an appellate court reviews a trial court' judge's factual determinations it must make a binary judgment, either "clearly erroneous" or "not clearly erroneous." This phenomenon, where the law demands a binary judgment about a scalar is pervasive in the law.

**Vagueness and the Line-Drawing Problem in Binary Judgments**

The demand for binary judgments results in a particular problem when there is no precise threshold for the binary categorization of the scalar property of the real world. One example is the balancing test, which allows us to translate a complex set of scalar properties into a simple yes-no judgment; either the balance of interests tips one way, or the other.

Sometimes, there is a bright line, a precise threshold that allows us to translate our scalar determinations into binary judgments. For example, in civil cases the burden of persuasion is usually "preponderance of the evidence." One understanding of the meaning of that standard is that it requires that the evidence establish that the likelihood is greater than 0.5. In other words, the plaintiff needs to show that the likelihood that defendant caused plaintiff's injury is greater than 50%. This rule allows the translation of a scalar (probability) into a binary ("preponderance of the evidence" or "not preponderance of the evidence"). Let's call this kind of rule a "bright-line translation rule."

But in other cases, the translation rule does not provide a bright line, because the rule is vague. For example, the clearly-erroneous rule is not associated with some precise bright-line probability of error. Instead, the clearly-erroneous rule is vague. In some cases, we are certain that the judge was in error and hence the finding of fact is definitely "clear error." In other cases, we are very uncertain about the error; we might think the judge was wrong, but just barely so. This would be a case where we would definitely say the error was not "clear error." But there is no bright line, so there will be some cases on the borderline or in a zone of underdetermination. In those cases, appellate judges may disagree about whether the error was clear, and even the same judge might make the call differently at different times.

**Smooth and Bumpy Laws**

The relationship between binaries and scalars can be conceptualized as one case of a more general phenomenon--which Adam Kolber has called "smooth and bumpy laws." HIs idea is that legal input and outputs can be "smooth" (gradual) or "bumpy" (uneven). Scalars are smooth, but binaries are bumpy. This conceptualization allows the insight behind the scalar-binary distinction to be extended to situations in which the output is discontinuous or uneven in some other way, but not a true binary.

**Conclusion**

This entry in the Legal Theory Lexicon introduces the scalar-binary distinction and illustrates some of the issues and problems that the distinction illuminates.

**Related Lexicon Entries**

- Legal Theory Lexicon 020: Causation
- Legal Theory Lexicon 024: Balancing Tests
- Legal Theory Lexicon 026: Rules, Standards, and Principles
- Legal Theory Lexicon 054: Standards of Review

**Bibliography**

- Larry Alexander,
*Scalar Properties, Binary Judgments*, 25 J. Applied Phil. 85 (2008), https://papers.ssrn.com/sol3/papers.cfm?abstract_id=829326. - Adam Kolber,
*Smooth and Bumpy Laws*, 102 Cal. L. Rev. 655 (2014), https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1992034. - Adam Kolber,
*The Bumpiness of Criminal Law*, 67 Ala. L. Rev. 855 (2016), https://ssrn.com/abstract=2684203. - Tim Mulgan,
*Critical Notice of Jeff McMahon, The Ethics of**Killing: Problems at the Margin of Life*,” 34 CANAD. J. PHIL. 443 (2004).

(Last revised on April 7, 2024.)