Two Definitions of Generalization

Epistemic Status: Research notes, plus an appeal for the reader to give their best attempt.

The words “abstraction” and “generalization” are often used fairly interchangeably. This is a pity. We only have so many well known words for abstract concepts.

One distinction I like is the one suggested in this Stack Overflow post. The following images are from the top answer:

Object

Abstraction

Generalization

Simple, right?

Well, I think there are two important definitions of generalization that could apply both that merit discussion.

The first, and perhaps the better known, is essentially what’s stated on Wikipedia:

generalization (1) (Wikipedia): A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims.

This definition is about identifying the similar properties of supersets and removing the other information. So, for the desert example above, generalizing the cake would mean identifying the common properties it shares with other deserts.

However, this definition doesn’t work for many of our uses of the term.

When we discuss “general intelligence”, we mean something like, “the superset of all narrow intelligences”, not the “elements common to all narrow intelligences”.
When I write a “highly general function”, this could refer to a function that takes in many sorts of things but still manages to treat them all uniquely.
The lists of “generalizations” include statements like, “Most people find church boring.”

From these, we could imagine a different definition. Something like,

generalization (2) (Here): A generalization is a pointer to a set of things.

I use the word set here because it’s mostly correct, but there could clearly be modifications. It might really be more of a fuzzy set.

One way I could write a general function would be to remove all of the differentiating details between items. This would be generalization(1). But I could also write this function by just moving a complexity into it and adding a big switch statement. This would be generalization(2).

I’m tempted to say that generalization(2) corresponds somewhat to set theory, and generalization(1) more to type theory.

With all that out of the way, we can attempt some more specific generalization(1) definition:

generalization(1)(Wikipedia complexified): A generalization is an abstraction made up some of common properties of a set.

Generalization(1) is doing more work than generalization(2), though the result a smaller thing.

Why is this important?

I’m interested in this question because I think it might be useful to study generalization(2). My recent sequence on questions and discernment wound up getting into this area.

It might be easy to dismiss generalization(2) as trivial once we already have set theory, but I think there’s more here. Set theory is typically discussed abstractly. If we have “Judgemental Forecasting”, it seems to me like we could also have something like “Judgemental Set Theory”.

Asides:

Short rant on semantic search

I’m sure this discussion exists somewhere in philosophical literature, but it’s very difficult to search for. The word “generalization” is used all over the place and often isn’t defined. Google search is quite poor for such queries. I imagine it would require some semantic search capabilities.

Objectivist epistemology

I wrote a short description of the above on Facebook, and Jason Crawford responded with an useful comment.

In Objectivist epistemology, “generalization” refers to scope of a concept, and “abstraction” refers to distance from the perceptual level. So “plumber” is less general than “human”, but it’s more abstract. Some steps of abstraction are generalizations (human → organism), but many are narrowings.

He said that he believe he remembered it from this lecture. Unfortunately the lecture costs $34. I haven’t purchased it, but I’ll keep it in mind. I did some searches for generalization around objectivist epistemology but couldn’t find this distinction written publicly in my brief time spent searching.

Another (similar) definition

Some would call the statement “rich people are greedy” a generalization. Here, the generalization isn’t referring to “rich people”, but rather to the “are greedy” portion. Perhaps “rich people” only act as a generalization if it’s used to make queries, maybe only overconfident queries.

I think generalization here means the same thing as overgeneralization. I’d vote to not use definitions of generalization that do this. Ideally, it could leave space for undergeneralization.