The essence of generalization is simplification by omission of irrelevant details. In logic, a concept A is a generalization of concept B, if and only if every instance of concept B is also an instance of concept A, and there are instances of concept A that are not instances of concept B. For example, a cockroach is an insect, which in turn includes insects that are not cockroaches. Talking about insects omits the distinctions among cockroaches, beetles, flies, etc. Logical generalizations are based on sets whose instances have one or more attributes in common. They give rise to taxonomies, such as the Linnaean classification in biology. Logical generalizability is the ability to find meaningful common attributes among disparate sets of instances. Except for their membership in the same set, this may not be easy. For example, it would be hard to find a common attribute among cockroaches, artifacts, poets, and scientific theories. The inability to find a conceptual basis for a set of instances limits logical generalizability.
According to category theory, A is the prototype of a category of instances B if and only if all instances of B belong to the same category, and A is judged to be the most typical among them. A prototype can be considered a generalization of B inasmuch as it possesses the defining characteristics of instances in B. Unlike in logic, where all members of a set are assumed to possess the same attributes, category theory acknowledges that membership in a category can be graded. For example in the US, among birds, robins turn out to be most typical, pigeons a little less so, with parakeets far from being prototypical. Penguins probably are the least typical birds, sharing few features with other birds. The prototypicality of the instances of a category can be obtained by systematic pair comparisons: “Which is the more typical bird, a or b?” Such comparisons identify the prototype as the most typical of its kind, surrounded by increasingly less typical instances, its periphery showing instances that may well belong to, and may therefore be easily confused with, other categories. Encyclopedias that illustrate a concept invariably choose something close to its prototype. Roles in movies are cast by typical mothers, typical criminals, typical heroes, whose deviation from their prototypes may have mere dramaturgical motivations. Not all sets of instances have meaningful prototypes. This led researchers like Eleanor Rosch (1978) to define three kinds of categories. Basic categories have visualizable prototypes, for example, chair, shirt, house, and spoon. Subordinate categories are distinguished by attributes of basic categories, for example, baby chair, dress shirt, outhouse, and serving spoon, whereas superordinate categories do not have clear prototypes and tend to be constructed logically, for example, furniture, apparels, buildings, and tableware. While cartoonists, advertisers, and political spin doctors have a knack for finding prototypes of what they wish to promote, oppose, or generalize, category theory suggests how this ability is limited.
In written discourse, generalization concerns propositions and is a special case of logical generalization. A general proposition A semantically subsumes several lengthier propositions or complex narratives B, omitting irrelevant details while preserving what serves a particular purpose. A proposition concerning a ball generalizes propositions involving, for example, a soccer ball, made up of white hexagons and black pentagons. It retains only references to the attributes and behaviors of ball. Typically, scientific articles have a title and an abstract, followed by a body of text. Abstracts offer readers propositions that generalize the text. Titles may not be more than noun phrases abstracted from the text. Propositional generalizability is limited only by the meanings of the words available in language.
In statistics, a generalization A is sustainable by contrast to its logical complement A¯ if and only if the overwhelming majority of instances in B are instances of A as opposed to A¯ . Null hypotheses, for example, whether one variable has no effect on another, are meant to rule out that one falsely accepts a hypothesis when its logical opposite could be true as well. The measure of statistical significance expresses the probabilities of either hypothesis or generalization to be acceptable.
In sampling theory, a sample A of instances, selected from a larger population B of instances, is taken to be representative of B, if all relevant attributes of instances in B occur in A as well and in the same proportion as in B. A sample is not usually considered the generalization of a population. However, it is a simplification that omits irrelevant details and allows the researcher to examine the sample in place of a population.
Generalizations are obtained by induction and are subject to any of the errors of inductive inference.
Hasty generalization is the fallacy of examining only a few instances and generalizing from them to the whole set of instances. For example, someone might conclude, “everyone likes this movie” after talking to a few friends. Sampling always entails the possibility of a hasty generalization. Statisticians seek to circumvent this fallacy by relying on the law of large numbers. This law states that the larger the size of a sample, the closer do the averages calculated from that sample approximate the averages in the population. Coupled with a measure of the smallest acceptable imprecision and an assumption of the probability of the rarest incidences in the population, the law suggests sample sizes that keep hasty generalizations within tolerable limits.
Excessive exception is the fallacy of a generalization that may well be correct but only under the condition of excluding a large number of its suggested instances. For example, “everyone can read the New York Times except children, the blind, and non-English speakers.” Using undergraduates as “human subjects” in psychological experiments can yield findings that are valid for this small population – although the researchers may have generalizations to humans in mind.
Sampling bias and slanted propositions occur when the method of sampling from a population favors incidences that support one or another hypothesis to be tested. For example, studying the literacy of a population by means of a written test systematically excludes those unable to read and write. Asking the members of one political party about the problems of their country biases the sample in favor of these party members’ opinions. Similarly, abstract propositions may be slanted by particular interests, for example, when polling data are interpreted by candidates for political office, or when literature on the effectiveness of a drug is selectively reviewed and published by its manufacturer.
Part–whole confusion, also called the fallacy of composition, is the mistake of generalizing knowledge across different levels of organization without considering the nature of the organization that integrates parts into their wholes. Examples are generalizing interpersonal relations to relations between countries, or using biological models to understand social enterprises (e.g., by distinguishing between functional and dysfunctional actions according to whether they preserve the existing structure of society).
Concretizing the abstract is the fallacy of attributing qualities to abstractions that belong to what the abstraction generalizes; for example, attributing agency to abstract concepts, such as in claims that unemployment caused unrest, a corporation failed to pay taxes, or “the White House says. . . .” Only people can create unrest, perhaps because they are without a job. Decisions to pay or not to pay corporate taxes are made by the employees of a corporation, and the White House cannot speak, only people can, and may do so in its name.
Stereotyping is the fallacy of taking the mean or mode of a distribution of the characteristics, usually of a group of individuals, by nonmembers, for that distribution, effectively ignoring the diversity of its spread. It confuses the mean or mode with a prototype; for example, claiming someone to be a “typical Frenchman” when there are millions of atypical Frenchmen living in France, or reporting the existence of correlation between particular variables at the expense of acknowledging how many instances do not fall on the regression line of that correlation. Social scientific research can create stereotypes when its findings are published and enter a population eager to accept theories as facts. For example, Marx’s theories of social classes created his oversimplified class distinctions. Stereotypes of minorities, professions, and pathologies may well be of scientific origin but can become exaggerated in use.
- Fyans, L. J. (1983). Generalizability theory: Inferences and practical applications. San Francisco, CA: Jossey-Bass.
- Gomm, R., Hammersley, M., & Foster, P. (eds.) (2000). Case study method: Key issues, key texts. London: Sage.
- Rosch, E. (1978). Principles of categorization. In E. Rosch & B. B. Lloyd (eds.), Cognition and categorization. New York: John Wiley, pp. 27– 48.
- Shavelson, R. J., & Webb, N. M. (1991). Generalizability theory: A primer. Newbury Park, CA: Sage.