Constructing Well-Defined Categorization Schemes
Accurately representing the shades of gray
Last in Abstreal Logoce, in False Categorization Schemes, I examined errors or deliberate fabrications in categorization—particularly, the statement of claims that a specific way of categorizing all pertinent objects or names is mutually exclusive and jointly exhaustive in an inaccurate manner. The inaccuracy involves either the presence of a logical contradiction resulting from the claim, or the absence of justification for the claim itself, or both.
With a categorization scheme being a specific set of mutually exclusive and jointly exhaustive categories, I termed these kinds of errors as “false categorization schemes”.
The previous post, while not claiming knowledge of any singular framework for inferring the causes of this kind of error in reasoning, proposes the root of the issue to be a relative lack of rational skepticism for categorization-relevant claim. This reinforces the importance of general rational skepticism as a solution—if one proportions their confidence in a specific way of categorizing constituting a categorization scheme to the strength of available justification, then they are, in principle, less inclined to fall victim to false categorization schemes.
Nevertheless, with general skepticism serving as a preventive measure—helping avoid the construction or affirmation of false categorization schemes, to explore the active construction of well-defined ("true") categorization schemes is a natural next step. In contrast to prevention, such measures are promotionally-oriented—helping not only hold the floor but also raise the ceiling. This gap is exactly what this post intends to bridge through a systematic lens.
Choose, Check for Inclusion, Divide to Refine, Iterate
As in the previous post, the universe of discourse, which is the set of all objects or names pertinent to some context, will be denoted henceforth by U.
When constructing a categorization scheme (equivalently, partitioning U), one may require that a subset X of U be a category in that scheme. In which case, every other category must be a subset of (U - X), the set U excluding all elements in X, lest their elements will either overlap with X or be outside the realm of U.
Should a well-defined dichotomy in U with X as a category be our sole goal, the result is {X, U - X} and we are done.
If, instead, our desired scheme contains Y (≠ X) as a category as well, then there are two possibilities: either Y is a subset of (U - X)—allowing us to proceed to the next step, or Y is not a subset of (U - X)—entailing the impossibility of the desired categorization scheme. Now, every other category we consider must be a subset of ((U - X) - Y) = (U - X - Y), which is the set U with elements in X and Y removed.
Should a well-defined trichotomy in U with X and Y as categories be our goal, the result is {X, Y, U - X - Y} and we are done.
Depending on one’s requirements, the same steps can be repeated till satisfaction. Choosing a new candidate category, checking for its inclusion in the universe of discourse excluding all other categories already considered, and dividing the intermediary partition further by taking the complement to refine the scheme, and iterating are the steps the reader may take.
Beyond Construction: Systematically Verifying Claimed Categorization Schemes
It may have occurred to the perceptive readers that the purpose of the “Choose, Check for Inclusion, Divide to Refine, Iterate” (henceforth to be abbreviated as CCDI for brevity) method for the construction of categorization schemes is extendable to the verification of categorization schemes. In this case, the “verification of categorization schemes” means to test whether a claimed categorization scheme truly satisfies the definition of categorization schemes.
To observe how, consider the claim that “{rational people, creative people, conformists} is a categorization scheme (partition) of the set of all people”. The problem of verifying the categorization scheme can be re-interpreted as a problem of constructing a scheme—with the claimed scheme being the desired construction. If no means of constructing such a scheme exists, then it is not well-defined; otherwise, it is. First, note that the desired scheme includes “rational people” as a category; applying the first step of the CCDI method to which we obtain {rational people, non-rational people} as an intermediary scheme.
Second, we may apply the next step to “creative people”. In the CCDI method, finer constructions would necessitate the inclusion of “creative people” in “non-rational people”. However, that creative people are necessarily non-rational, be that being irrational or arational, is not a proposition that follows a priori from conventional definitions of rationality. Therefore, the construction cannot be finer with "creative people" as a desired category—proving the ill-definedness of the categorization of all people into the boxes of rationality, creativity and conformity.
It is true that the impossibility of a specific method to construct a result is not generally a proof of the impossibility of it per se—perhaps the method is imperfect. However, in this case, the impossibility of construction using this method is a proof of the actual impossibility of construction—as should be relatively clear from the description of the method itself.
It is to be noted that the process of executing the CCDI method to false categorization schemes can yield information about the properties of categorization schemes it fails to satisfy. If the process fails due to there being a non-empty category expected to be empty, such as the complement of the union of all the expected categories being non-empty, then the claimed scheme is not jointly exhaustive—as there are uncategorized elements in the claimed scheme. On the other hand, if the process fails due to an expected category not being a subset of some complement, which the method requires, then the claimed scheme is not mutually exclusive—as the category exhibiting a non-empty intersection with some other category stands as the sole possibility.
The versatility of the tautological CCDI method is clear: it aids in not only the construction of categorization schemes but also the verification of them—the two acts being logically identical.
The Usefulness Question
For smaller claimed or desired categorization schemes, e.g. claimed dichotomies and trichotomies, the exactness in executing the CCDI method can be relaxed. In practice, it might be excessive to meticulously choose a category from {rational people, creative people, conformists}, consider its complement (set of all people not in the chosen category), consider its inclusion status in the complement and repeat the process until an impossibility occurs—especially given that the non-empty intersection of creative people and rational people could as well be noted from the outset.
Nevertheless, for larger and unfamiliar schemes, especially in the opacity of non-empty intersections between desired categories and/or the existence of uncategorized elements, the utilization of a universal systematic methods expedites the categorization process. For instance, that {buses, trucks} is not a well-defined way to categorize all vehicles is conspicuous, in contrast to the equivalent assessment of {parasite, commensal, uncommensal, endosymbiont} for biological species (unless the reader is notably familiar with the concepts).
In general, the utility of the CCDI method is context-dependent. It is expected that the reader invoke it when they assess it to be more relevant than its alternatives.
Recapitulation
Stretching beyond having a mere concept of false categorization schemes and emphasizing skepticism to avoid them, we explored the CCDI method, short for the “Choose, Check for Inclusion, Divide to Refine, Iterate” method, for the deliberate construction of well-defined (“actual”) categorization schemes. In exploring this method, we identified the versatility of it in the sense of not only allowing for categorization scheme construction but also verification. While the foolproof nature of the CCDI method is a tautological fact, such is not necessarily true of its usefulness—a judgement which the author leaves to the reader.