My definition of a "vague concept" (A):

It creates a group of things that share family resemblance or "family difference".
It doesn't have any meaning outside of context. But in a specific context it obtains specific meaning.
It's very hard to understand through the framework of causation: what causes things to be A/non-A?
When you consider more and more examples of A, the boundary between A and non-A becomes increasingly arbitrary and complicated. But in a specific context the boundary is simple.

Examples of vague concepts: games, health, human values, personality, words and some hypotheses.

I think it may be very important to study ways to think about vague concepts: it's related to hypotheses generation and human values. So it may be related to AGI and AI alignment.

I'll discuss some examples and share my thoughts about inner working of vague concepts.

Examples

Games

An example about "games" is famous in philosophy. But keep in mind that my interpretation may differ:

https://en.wikipedia.org/wiki/Family_resemblance

It argues that things which could be thought to be connected by one essential common feature may in fact be connected by a series of overlapping similarities, where no one feature is common to all of the things. Games, which Wittgenstein used as an example to explain the notion, have become the paradigmatic example of a group that is related by family resemblances.

When you consider a bunch of "games", it's easy to see the common features. But as you consider more and more "games" and things that are sometimes called "games", it turns out that everything can be a game. The boundary between "games" and "non-games" becomes more and more arbitrary and complicated.

And yet no matter how much you stretch the concept (e.g. say something like "love is just a game"), in a specific context the meaning is clear enough. In a specific context the boundary between "games" and "non games" is quite simple. Does this boundary have a fractal dimension? (I'm half-joking)

I also think it's important to consider "family difference": maybe you have just a single object A and a group of objects (B). When you compare A to an object from (B), they're always different enough. But you can't formulate a universal difference between A and all objects from (B). Maybe because you don't know all the objects in (B).

Sorites paradox

Sorites paradox is at least partially relevant here.

A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a single grain does not cause a heap to become a non-heap, the paradox is to consider what happens when the process is repeated enough times that only one grain remains: is it still a heap? If not, when did it change from a heap to a non-heap?

It is (presumably) easy to tell apart a heap from a non-heap. But it's hard/impossible to explore the boundary or analyze the question through the framework of causation ("What causes non-heap to become a heap?").

Health, moral values

You may also call vague concepts "cluster properties" (explanation in a Philosophy Tube video). In the text form:

Even more interestingly, Harris’ idea is an accidental ripoff of a theory developed by philosopher Richard Boyd in 1982, called: ‘The Homeostatic Cluster Property Theory of Metaethical Naturalism’ Sexy title. Boyd thought that words like ‘good’ and ‘evil’ refer to real properties out there in the material world, and that therefore statements like ‘Murder is bad’ are capable of being objectively true, or at least true in the same way as scientific statements are. Which prompts the question, “To what exactly do these words refer?”

Boyd’s answer is that they are cluster properties - groups of things that tend to go together. The example he uses is actually the same one Harris does - health. There are all kinds of things we would want to include in a definition of the word “healthy,” like your heart should be beating and you should be able to breathe, but do you have to be a certain size in order to be healthy? Do you have to not be in pain? Can you have a beating heart and be unhealthy? There’s a cluster of properties here somewhere that makes up the definition of the word health but we’re never going to pin down a definite list because that’s just not how the concept works. Despite that vagueness it’s still very obviously useful and meaningful.

Similarly Boyd thinks that a word like ‘good’ refers to a cluster of things that are non-morally good for humans, like sharing friendship, sharing love, having fun, watching quality YouTube videos, but just like with health, you’re never going to be able to pin down a full list because the concept just isn’t like that.

And here’s the big takeaway - if we say ‘John is healthy’ we could be talking about any number of things in the cluster of health - whether he a has disease, whether he works out, whether he has a good relationship with his mother - all of which are objective - but whether the sentence ‘John is healthy’ is true will still depend on what aspect of his health we’re talking about. It will be relative to the context in which we’re saying it.

You may even compare "vague concept" to "social constructs".

Vague Hypotheses

You can imagine a hypothesis based on vague concepts, for example "healthy people earn more money than unhealthy people" or "people who love games earn more money". In their most abstract form, those theories can't be falsified. But it's easy to generate specific falsifiable hypotheses based on those ideas.

Scientific theories, too, can have an unfalsifiable core. This is Imre Lakatos' model of scientific progress:

https://en.wikipedia.org/wiki/Imre_Lakatos#Research_programmes

Lakatos's second major contribution to the philosophy of science was his model of the "research programme",[19] which he formulated in an attempt to resolve the perceived conflict between Popper's falsificationism and the revolutionary structure of science described by Kuhn. Popper's standard of falsificationism was widely taken to imply that a theory should be abandoned as soon as any evidence appears to challenge it, while Kuhn's descriptions of scientific activity were taken to imply that science is most fruitful during periods in which popular, or "normal", theories are supported despite known anomalies. Lakatos' model of the research programme aims to combine Popper's adherence to empirical validity with Kuhn's appreciation for conventional consistency.

A Lakatosian research programme[20] is based on a hard core of theoretical assumptions that cannot be abandoned or altered without abandoning the programme altogether. More modest and specific theories that are formulated in order to explain evidence that threatens the "hard core" are termed auxiliary hypotheses. Auxiliary hypotheses are considered expendable by the adherents of the research programme—they may be altered or abandoned as empirical discoveries require in order to "protect" the "hard core". Whereas Popper was generally read as hostile toward such ad hoc theoretical amendments, Lakatos argued that they can be progressive, i.e. productive, when they enhance the programme's explanatory and/or predictive power, and that they are at least permissible until some better system of theories is devised and the research programme is replaced entirely.

Vague concepts lead to vague hypotheses ("research programmes"). Vague hypotheses work the same way vague concepts do.

Personality, thinking

Even human personality may be a vague concept.

The smaller amount of situations you take, the easier it is to understand what someone's personality is. But if you take more and more situations it may turn out impossible to find a single connecting thing. Especially if someone lives long enough, encounters different enough communities, obtains different enough opportunities.

And even thinking itself may be a vague concept: before you realize some key connection your mind might be wandering between associations that have many overlaps, but don't form a coherent seamless picture.

Word meanings

Most of the words (their meaning) are vague concepts.

For example, the word "beast" may mean a fantastic creature, someone who lost their humanity, someone who's shockingly good at something and etc.

And of course there's the infamous/comical example with the word "shit" (ISMO skit), which changes its meaning in absolutely wild ways.

Exploring vague concepts

How to understand a vague concept? You can try to memorize all contexts (that you know of) in which it's used. Or you can learn to infer its meaning in new contexts. And learn to create new contexts for this concept yourself.

But it would require a different type of generalization, not based on definitions. I think it's important to explore what this "different type of abstraction" may be.

What are vague concepts made of? My ideas:

They're made of something that can easily create contrasts.
They can connect a fact (e.g. "someone is in pain") and a conclusion ("someone is not healthy") by a special type of connection.

Internal structure, "gradient"

Here's an observation: when the meaning of a word changes, this change doesn't come without a cost. (I'm not talking about words with completely separate meanings.) It comes with a change of emotion and emphasis. The point is that different meanings of a word have some internal relationships, "positions" relative to each other. If you could evaluate the "internal" meaning of a word at least in some simplified way, you would notice the relationships. Maybe those relationships allow us to really "understand" words and context.

For example, take the word "beast":

When you use it in the archaic and ironic way ("any animal"), you describe the referent in the context of its world. You focus on the world (the world where everything may be a "beast").
When you use it in the negative way ("brutal human"), you focus on actions and very deep qualities of the referent. E.g. you may imply that the referent is fundamentally rotten to the core.
When you use it in the positive way ("very skilled human"), you focus on the referent's actions and qualities. But those qualities are not so deep anymore (compared to the negative meaning).

If you're interested in deep internal qualities of people/objects, the negative meaning may be the "main" one for you, even if you dislike it. This suggests an idea: if you have a certain bias or goal, internal relationships between meanings may become simpler. To understand the change of meaning of a word you only need to understand how your emotions changed and how the "alignment" between the word and your goal changed.

If you're interested in irony, the positive meaning of the word "shit" (e.g. "This is the shit") may be the main one for you.

Emotions, biases and goals give the meanings of a word some "gradient" or "flavors". I recommend to check out "Propagating Facts into Aesthetics" by Raemon, it may be very relevant.

Meta contrasts

My conclusion is that some vague concepts may create "meta contrasts", "contrasts of contrasts": they contrast the real world and a counterfactual world, but also contrast multiple possible emotions/goals related to those worlds. For example, when something bad happens and you think "this is a bad day" ("bad day" is a vague concept, there's almost no real boundary between good and bad days), you put the bad outcome in context of your possible goals and emotions related to this day. And at this point it doesn't really matter what the bad thing was, what matters is how your emotions and goals where affected. The same thing with your overall "health". It's not about specific medical conditions, it's about your feelings and goals being (un)affected. And when you call someone a "beast" (in the positive sense) you talk about your emotions and compare the referent to other people.

However,

The physical cause of your emotional change may trigger multiple vague concepts
You can't know your exact emotions

so, "physical" and "emotional" ("gradient") aspects of a meaning may get really entangled. I think this is what makes defining some vague concepts really impossible.

Toy example

I also tried to explore vague concepts in more detail on a specific toy example. I know that my post turned out to be very unclear, but there you can find:

Objects connected by family resemblance and "family differences": "places" with "colors".
Objects that don't have any definite properties outside of context, but obtain specific properties in a specific context: "places".
Connections that are hard to understand through the framework of causation: what exactly causes a certain place to have a certain color?
Strange boundaries. When you consider more and more places, the boundary between places of different colors becomes increasingly arbitrary and complicated. But when you compare specific places the boundary is simple.
A model of "context" and "meaning" for a toy example.
An example of a "gradient": density of "details" in a place.

I think the toy example in the linked post could help to come up with some statistics for "vague concepts".