Summary

To economically communicate an Idea fairly precisely, try focusing on giving an example such that

1. the example is close to the border between where the Idea does apply and where it doesn't apply;
2. but also the Idea either firmly does apply to the example, or firmly does not apply.

Kinds of examples

Often the best way to communicate Ideas is through examples. E.g. we might illustrate categories (point at a STOP sign to illustrate «red») or illustrate propositions (compute the kernel, image, and relevant quotient of a group homomorphism to illustrate the «isomorphism theorems») or illustrate skills. What are key kinds of examples?

Central example

E.g. a bright red STOP sign.

Clear counterexample

• A central example from an opposed category. E.g. the bright blue sky on a clear day, opposed to «red».
• An example that is a clearly unsuitable version of the Idea
  • E.g. I want to communicate to you how to «hammer well». So I demonstrate hammering the nail uncarefully so it gets bent and can't go all the way in.
  • E.g. how would you show a "counterexample" to a theorem? It's always true, it applies to everything! Leaving aside metamathematical tomfoolery, you could exploit ambiguity in the proposition. E.g. «the gcd of any two distinct prime numbers is 1» is true. But hold on, in the real numbers, gcd(2,3) = 2 ≠ 1. What's up? Or, you could modify the proposition, and show why the modified proposition is no longer true. E.g. «the gcd of any two distinct numbers is 1».

Barely-(counter)example

• A barely-example is something that's fairly firmly an example, but is fairly close to being fairly firmly not an example. Mutatis mutandis barely-counterexample.
• Examples of barely-(counter/)examples:
  • a kid who's just barely (too short/) «tall enough to ride the roller coaster»;
  • a basal species that's just barely (outside/) inside a «clade»;
  • for a concept defined in terms of N necessary and sufficient conditions, something that (satisfies N-1/) «satisfies N of the conditions»;
  • a dolphin is a barely-example of a mammal; it sure looks fishy, but it is a mammal.

Others

• Extreme cases.
  • Hyperexemplification. This is showing something that's more truly Xish than any X that actually exists. For example, a face more truly beautiful than any existing person's face.
  • Extremization. This is showing something that moves very far in thingspace in the general Xish direction. For example, a woman's face with huge eyes, huge lips, utterly homogeneous skin with extremely bright red cheeks; or man with massive bulging muscles, a jaw jutting way out. Not generally taken to be beautiful--just moving in a [beautiful minus ugly] direction along many dimensions.
• Borderline cases. E.g. is cyan closer to green or to blue?
• Inapplicable cases. E.g. is «isomorphism» closer to green or to blue?

More barelys please :)

People seem to often focus on giving simple, central, or extreme positive examples. I often feel a little lost as to what they really mean, if they mean something that isn't vague. Central or extreme counterexamples might help some. But mainly I wish they would give barely-examples and barely-counterexamples. I like when it seems like I'm hearing the end result of a process of inquiry that involved testing the boundaries of the Idea (where the Idea stops saying "yes"; where it starts saying "no"; and perhaps where it starts saying "why are you asking me?"). That process tends to bring up examples that are interesting and relevant to the overall discussion, in their own right; and those examples tend to be barely-examples.