False Categorization Schemes
Inaccurate representations of the shades of gray
Categorization is an indispensable mental process. In the world, a massive stream of information abounds, so do similarities between different components of them. The process of “compressing” those similarities into concepts and categories expedites our cognition. As Seger and Miller (2010) put it, cognitive categorization is a “fundamental characteristic of sophisticated thought”. Without categorization, every novel encounter is to be processed anew, congesting our working memory to a potentially paralyzing and excessively power-consuming extent.
Among many sets of categories we construct, some satisfy the properties of being jointly exhaustive and mutually exclusive. Within a fixed universe of discourse, for a set of categories to be jointly exhaustive means that every (conceptual or concrete) object to belong to the category does not belong to any other category. On the other hand, for a set of categories to be mutually exclusive means that there is no object in the universe of discourse to belong to multiple categories. Together, they mean that the set of categories encompass the entire universe of discourse in such a way that they do not overlap. Such a set of mutually exclusive and jointly exhaustive categories is called a categorization scheme.
In the language of set theory, given a universe of discourse U, the set S, a subset of P(U) is a categorization scheme if and only if
Here, P(U) is the set of all subsets of U. The equation preceding conjunction operator (“and”) describes mutual exclusivity, and the succeeding equation describes joint exhaustiveness.
In conventional mathematical lexicon, a set such as S to U is a partition of U. In spite of our usage of mathematical framing, I will continue referring to them as categorization schemes.
Many a times, we construct sets of subsets which we claim to be categorization schemes, but fail to satisfy either joint exhaustiveness, mutual exclusivity, or both. A prominent example of such an error (or intentional fabrication) is the fallacy of the false dichotomy, wherein either a non-exhaustive or non-exclusive set of two subsets of the universe of discourse is assumed to be a categorization scheme. Stated differently, a false dichotomy is a set of two subsets of the universe of discourse claimed to be a categorization scheme without sufficient justification.
However, a set of subsets of the universe of discourse can contain more than merely two elements. This suggests that the false dichotomy fallacy is generalizable.
False n-tomy
The post henceforth will refer to the universe of discourse as U.
Define an argument to commit the false n-tomy iff, without sufficient justification, one of its premises states a subset N of cardinality n of P(U) to be a categorization scheme.
It stands, then, that false dichotomy is equivalent to a false 2-tomy, as expected. Further, we can now discuss false trichotomies (3-tomies), tetratomies (4-tomies), pentatomies (5-tomies), hexatomies (6-tomies), heptatomies (7-tomies) and ad infinitum. False n-tomies which are not merely false dichotomies can be seen in lists claimed, often implicitly, to be exhaustive and contain mutually exclusive items.
Examples of false n-tomies
False categorization schemes, or false n-tomies, can be seen in some lazy ways of listing things. The “lazy listing” which the author speaks of tend to be logically equivalent to lists with one entry as a subset of another, such as the following list of species in the family of Felidae, or “cats”.
Felis catus (House cat)
Panthera tigris (Tiger)
Leopardus pardalis (Ocelot)
The felis
Every other species of cats
The fourth entry refers to the genus felis, within which felis catus belong, as is obvious from the binomial name. Thus, the list is not mutually exclusive even though it is jointly exhaustive due to the fifth entry.
This example might be trivial, and the reader might expect that my explicit characterization of the “false n-tomy” illuminate a previously unilluminated fallacy. However, one cannot simply assume that no one has previously recognized claimed categorizations schemes which are not as false categorization schemes, merely that it might have been labelled a different name or simply described. The naming of the fallacy, or the family of fallacies, helps categorize all those instances into one singular framework, the very helpfulness of categorization schemes in themselves (in other words, we are categorizing attempts to categorize things).
A false categorization scheme constructed not merely to exemplify false categorization schemes pertains to an 11th grade student's proposed list of ways to solve global warming in an essay:
Cutting down on fossil fuels and switching to renewable energy
Using public transportation instead of personal vehicles when possible to reduce CO2 emissions (through reducing traffic)
Doing what we can to reduce CO2 emissions
Planting trees
etc
Note that if one is to enact entry 3, doing what is possible to reduce CO2 emissions, one may seek to reduce traffic through promoting public transportation usage, which is enacting entry 2.
Thus, while it is technically jointly exhaustive (due to the “etc”), it fails to satisfy the implied condition of mutual exclusiveness.
Both of the prior lists feature lists which exhibit an inconsistent level of abstraction between the entries. For instance, the first list apparently contained names of species initially, only to mention the felis genus in the fourth entry. Furthermore, both of them exemplified an unjustified claim of mutual exclusiveness but justified joint exhaustiveness claims, while false categorization schemes may also feature unjustified claims of joint exhaustiveness.
Unjustified claims of joint exhaustiveness are epitomized by discriminatory societal attitudes not recognizing the existence of specific minorities in various communities. In Bangladesh, a theist-majority country, which officially recognizes and only recognizes Islam, Hinduism, Christianity and Buddism as religious stances, many atheist job seekers report job application forms only allowing for these four selections in the the “religion” section. There is often reportedly no option for “other”, nor “none”. This is a false tetratomy which, whilst mostly satisfying mutual exclusiveness in the traditional sense of being a believer in these religions, is not jointly exhaustive.
Yet another example of overlooking joint exhaustiveness in a supposedly well-defined categorization scheme pertains to certain computer science resources. In particular, while many standard resources correctly state the binary, octal, decimal and hexadecimal number systems as some four commonly used positional number systems, some resources tend to overlook the distinction between something being common and something being devoid of alternatives. Specifically, some sources tend to state that “the binary, octal, decimal and hexadecimal number systems are the four positional number systems”. The mutual exclusivity of these categories being apparent, the assumption of joint exhaustiveness is logically equivalent to an assumption of this being a well-defined categorization scheme. However, by definition, there is a positional number system for each natural number - we can represent numbers as sums of powers of not only 2, 8, 10, and 16 but also 1, 3, 4, 5, 6, 9, 11,..., 15, 17, 18,… ad infinitum. The number “4” is off by an unquantifiable margin, by infinity.
The conceived but false trichotomy of someone being either a cat person, a dog person, or someone who dislikes animals exemplifies a false categorization scheme the categories in which are neither mutually exclusive nor jointly exhaustive. It is possible for one to consider themselves both a dog and a cat person - and, furthermore, it is possible to prefer no pets in spite of not disliking animals in general, violating mutual exclusivity. Similarly, the possibility of someone being a horse person rather than either a cat or a dog person violates the assumption of joint exhaustiveness.
From cat categorization (taxonomy) to categorization of people based on their pets, we explored five different examples of false categorization schemes - two failing only mutual exclusivity, two failing only joint exhaustiveness, and one failing both. It is high time we explore why these failures may have occured and what mediates them, and formulate strategies to prevent them.
Mediators of and Strategies for
The writer is uncertain if a singular explanatory model can be formulated with variables mediating individual susceptibility to false categorization schemes. Interlocutors' susceptibility to specific false categorization schemes appear as diverse as the schemes themselves. Entering “felis catus” and “the felis” into the same list could be due to ignorance, or simply negligence - entirely overlooking the existence of non-religious people in an extremely theist-majority country might be due to a lack of awareness, consideration, and being accustomed to the status quo, or deliberate discrimination. The reduction of the number of positional number systems from countable inffinity to 4 could be due to a desire for practicality, simplification, or simply negligence (again).
Nevertheless, it is clear that the issue is fundamentally a special case of affirmation of statements without justifications - sound ones in particular. Should one have been justified in stating these sets of categories as categorization schemes, they would not have been false categorization schemes - or, if. they were, the falsity of the assumption would be due to uncertainty and not a shortcoming of epistemic rationality, hence not being a fallacy. It seems, therefore, that, to reduce our susceptibility to false categorization schemes in general can be done through a combination of selecting the right assumptions and being clear of the assumptions we exactly make. If we are visiting a city in a foreign country with only trucks, buses and motorbikes on the roads, instead of assuming that this list of vehicle types is an exhaustive list of all vehicles in the country, we can try to falsify that assumption instead. Perhaps we are visiting during a vacation, when most residence are not in the city to populate the roads with their private cars. When faced with a problem with a few different established solution methods, we could cosider if we can either combine those methods or discover/invent a new method to enhance efficiency, convenience and/or other desired variables, or simply refrain from making assumptions about exhaustiveness and mutual exclusivity (assuming the reader values epistemic rationality).
In essence, the writer defines the false categorization scheme as the making of an unjsutified assumption regarding the mutual exclusivity and joint exhaustiveness of a list of segments of an universe of discourse, particularly in an argument. It is noted that this error is effectively a specific case of forming rationally unjustified assumptions, effectively reinforcing the value of a general skepticism towards claims without sufficient epistemic justification for epistemic rationality in context.
Categorization is an integral component of our cognition. By keeping this process in check through rationality or even guiding the process through rationality when a greater accuracy is desired, we can optimize our thought process for a greater understanding of the world around us.
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 the next post, Constructing Well-Defined Categorization Schemes, intends to bridge through a systematic lens.