Analysis by categories and argument by cases proceed in basically the same way. You start with some set of objects which you then divide into subgroups. You then go on to show something about each of these subgroups. At this point in an analysis by categories you are done, the point is to reveal whatever interesting facts followed from that division into subgroups. In an argument by cases there is a little more work to be done, since the goal is to show that some one particular fact holds of all the non-empty subgroups, and thus that the fact holds for the original group. This technique seems pretty simple, for example you could divide the natural numbers up into those that are multiples of two and those that aren’t and then prove something about each category. But, outside of the context of a formal mathematical proof, it is fairly easy to misuse this technique.
As an example, which will be used below to illustrate a number of possible mistakes, we will consider this technique applied to the set of adults. And we will divide these adults into subgroups on the basis of their profession, which includes at least postal worker and lawyer, among an almost innumerable other number of professions.
One easy mistake to make is to falsely assume that your division of the set into subgroups exhausts the original set. To exhaust the original set means that every member of that set falls into one of your categories. Now it is not necessary for every successful analysis by categories to exhaust the original set, but it is easy to rely accidentally on the assumption that it does. Of course a successful argument by cases does require that the divisions be exhaustive. For example, we might point out that the people in each of our career subgroups has a job, and thus conclude that everyone has a job. Obviously this is a false conclusion. A mistake of this sort can easily be repaired by adding a new category that includes everyone who doesn’t fall into one of the other categories, but it is often hard to say anything interesting about the members of this new category. Note that an analysis can still be interesting and informative even if its division into categories leaves some objects uncategorized, assuming those left out are small in number.
Another easy mistake to make is to falsely assume that the divisions are exclusive. Just because you can divide up a set in some manner doesn’t mean that the objects fall into only one of those subgroups. For example, we might, erroneously, reason that since postal workers ≠ lawyers that no postal worker is a lawyer. But this could be false, for example if a postal worker happened to also work as a lawyer on their days off. The error here arises partly from an ambiguity. Our initial claim, that postal workers ≠ lawyers is a true statement, but it is about the categories. We mean to say that they are distinct categories, which we can make more explicit by saying that postal workers[C] ≠ lawyers[C], the [C] indicating that we are talking about the categories. That statement implies that no postal workers are lawyers only if we can show that the categories are exclusive. Interestingly this mistake is often more problematic for an analysis by categories than it is for an argument by cases; an argument by cases usually argues that the members of each subgroup have a certain property based on the definition of that subgroup, while an analysis by categories is much more likely to lean on their exclusiveness at some point.
A third kind of error, similar to the previous one, is to falsely assume that the divisions are completely distinct. The best way to describe this mistake is with an example. Suppose, as above, we claim that that postal workers[C] ≠ lawyers[C], and furthermore we claim that people have only one job (assume that this is true for the moment). Given those premises can we conclude that no postal worker is a lawyer? No, we cannot. For example, consider a lawyer who works for the post office. Such a person is a postal worker (because they are employed by the post office) and they are a lawyer, but they don’t have two jobs. A way to see how this mistake differs from the previous one is to consider how objects are divided into subgroups. Usually it is by properties (in our example having a particular job can be thought of as having a particular property). If they have property X they go in group A, if they have Y they go to group B, ect. If the divisions aren’t exclusive then we are saying that some one object may have more than one of the properties that we use to decide which category it belongs to. In contrast if the divisions aren’t completely distinct it is to say that there are properties such that the objects with those properties are to be placed in more than one group. Again, this is much more likely to be a problem for an analysis by categories.
Before I elaborate on the fourth common mistake I should probably mention a way to use an analysis by categories that is immune to the mistakes listed previously, which is when it is used primarily as a rhetorical device. By this I mean that the structure of an analysis by categories is used simply as a way to systematically discuss your subject matter, and not to draw any further conclusions about it. You break it us into a number of categories and then talk about each one. You don’t have to worry about exhaustion because you aren’t trying to uncover every possible detail about your topic. Nor do you have to worry about exclusivity or distinctness. An example of this use of the method is this very post. Here I am analyzing analysis by category by dividing flawed uses of this technique into different groups. Whether every possible mistake falls under one of the groups, or whether some mistakes fall under more than one group, is irrelevant to whether the post is useful for identifying and helping prevent common mistakes made when using the method.
The fourth, and final, kind of mistake (that I will discuss here), assuming existence, is perhaps the most serious mistake to make, at least for an analysis by categories. Making the mistake of assuming existence entails doing an analysis by categories and then, afterwards, assuming that there are objects which fall into those categories, and then using the properties of those categories, as detailed by the analysis, to argue for some further position, or to explain some phenomena. This is, I think, is the most subtle mistake to make, and one that occurs relatively frequently in complicated analyses. The paper starts with a detailed analysis by categories, and each category is given detailed treatment. Then the author goes on to use those categories to deal with some philosophical problem. The problem is that if some of those categories are empty then obviously they can’t be a good treatment of the problem they are applied to. For example, we could consider the profession that supplies to people everything they need (like a super-santa) as one of our divisions of adults into careers. And we could analyze with great detail the nature of such a career. Then, when we were done, we could go on to conclude that we have what we need because of these people. Obviously this example is silly, but anytime a philosopher engages in ontology (a categorization of what exists) and then goes on to use their ontology to deal with certain philosophical problems or to answer certain questions there is a good chance they have made this mistake, unless they have taken a moment between finishing their ontology and using it to demonstrate that the categories of their ontology aren’t empty.
[Note: feel free to submit other common kinds of errors that occur as part of analysis by categories / argument by cases with a comment.]