Two Types of Definition: Necessity and Sufficiency & Prototype
Family resemblance and here (SEP) is a philosophical term that solves an ancient philosophical dilemma. On the one hand, one can create a definition of necessity and sufficiency, such as a precising definition. This uses language to attempt to adequately characterize the extensionality and intensionality of a term. This can be tough to do, because it is easy to quibble over such definitions. A classic example is the definition of man as a featherless biped. From Diogenes:
"According to Diogenes Laërtius, when Plato gave the tongue-in-cheek8 definition of man as "featherless bipeds," Diogenes plucked a chicken and brought it into Plato's Academy, saying, "Behold! I've brought you a man," and so the Academy added "with broad flat nails" to the definition.9
When one commits to specific language, the definitions easily fall prey to counterexamples. Children are very imaginative and good at derailing definitions of necessity and sufficiency. The other horn of dilemma allows one to say that a definition is broad characterization but that linguistic definitions are in a sense intuitional and have their limits. A prominent example of a term that is somewhat challenging to define is 'game'. MW has multiple entries for 'game' each with multiple sub-types. From a philosophical standpoint, the meaning of words generally matter, and so the notion that one simply "knows what it is" is unsatisfactory to those who would like to lay a rigorous footing to definitions. Today, there is an increasing awareness of prototypical definitions.
Later Wittgenstein and Family Resemblance
So, later Wittgenstein is famous for challenging crisp sets as part of the broader movement known as the linguistic turn. With his arguments, he attempts to show that definitions can be complex affairs. From Philosophical Investigations, we have §65-67†.
What is the import of these words? It essentially challenges that terms are defined by necessity and sufficiency instead opting for some group of Familienähnlichkeit, literally 'family-similarity', some group of qualities that various things share, but not strictly necessary. Thus sufficiency for definition isn't based on a strict list of necessary conditions, but rather a group of conditions only some of which are strictly necessary.
The Philosophy of Language, Linguistics, and Metaphysics
Today, linguists such as Eleanor Rosch, George Lakoff, and others have introduced ideas into the philosophy of language, putative explanations about how the mind categorizes from a perspective of a naturalized epistemology, linguistic technicalities under the heading of cognitive semantics. Some examples from Women, Fire, and Dangerous Things include 'prototypes', 'graded categories', and 'radial definitions' which extended Wittgensteinian thinking into concepts that are inferred based on empirical observation.
While these ideas are not typically included in works on introductory philosophy of language, for instance the text by Sazbo and Thomason has no mention of them, the ideas are broadly of interest to philosophers of mind who follow cognitive science, linguists who study taxonomy, and others who dabble in metaphysics in the pursuit of scientific speculation.
Given the linguistic turn, and the preoccupation of philosophers in the analytic tradition, it is no wonder that Wittgenstein's observations about how words often flout easy definition are considered transformative. With the advent of metaphysical speculation adjacent to cognitive science, a number of theories have been developed to demonstrate that his observations are not only accurate, but challenge philosophical orthodoxy. These thinkers advocate that the science of the mind can draw certain conclusions about how the mind defines words moving semantics forward as a scientific theory. This is exactly what he meant in §66:
To repeat: don't think, but look!
It should come as a surprise to no one that philosophers become entrenched in their philosophical theories built on rational argument sometimes in violation of obvious empirical truths. Defeasibility of reason and non-classical logics are rather recent inventions in the Western Canon.
See also "On Wittgenstein's family resemblance and machine learning"(PhilSE).
† 65. Here we come up against the great question that lies behind all these considerations. -- For someone might object against me: "You make things easy for yourself! You talk about all sorts of language-games, but have nowhere said what is essential to a language-game, and so to language: what is common to all these activities, and makes them into language or parts of language. So you let yourself off the very part of the investigation that once gave you the most headache, the part about the general form of the proposition and of language.
And this is true. -- Instead of pointing out something common to all that we call language, I'm saying that these phenomena have no one thing in common in virtue of which we use the same word for all -- but there are many different kinds of affinity between them. And on account of this affinity, or these affinities, we call them all "languages". I'll try to explain this.
66 Consider, for example, the activities that we call "games". I mean board-games, card-games, ball-games, athletic games, and so on. What is common to them all? -- Don't say: "They must have something in common, or they would not be called 'games'" -- but look and see whether there is anything common to all. -- For if you look at them, you won't see something that is common to all, but similarities, affinities, and a whole series of them at that. To repeat: don't think, but look! -- Look, for example at board-games, with their various affinities. Now pass to card-games; here you find many correspondences with the first group, but many common features drop out, and others appear. When we pass next to ball-games, much that is common is retained, but much is lost. -- Are there always winning and losing, or competition between players? Think of patience. In ball-games, there is winning and losing; but when a child throws his ball at the wall and catches it again, this feature has disappeared. Look at the parts played by skill and luck, and at the difference between skill in chess and skill tennis. Think now of singing and dancing games; here we have the element of entertainment, but how many other characteristic features have disappeared! And we can go through the many, many other groups of games in the same way, can see how similarities crop up and disappear.
67. I can think of no better expression to characterize these similarities than "family resemblances"; for various resemblances between members of a family -- build, features, colour of eyes, gait, temperament, and so on and so forth, overlap and criss-cross in the same way. -- And I shall say: 'games' form a family.
And likewise the kinds of number, for example, form a family. Why do we call something a "number"? Well, perhaps because it has a -- direct -- affinity with several things that have hitherto been called "number"; and this can be said to give it an indirect affinity with other things that we also call "numbers". And we extend our concept of number, as in spinning a thread we twist fibre on fibre. And the strength of the thread resides not in the fat that some one fibre runs through its whole length, but in the overlapping of many fibres.
But if someone wanted to say, "So there is something common to all these constructions -- namely, the disjunction of all their common properties" -- I'd reply: Now you are only playing with a word. One might as well say, "There is a Something that runs through the whole thread -- namely, the continuous overlapping of these fibres".