[T]he quest for precision, in words or concepts or meanings, is a wild-goose chase. There simply is no such thing as a precise concept (say, in Frege’s sense), though concepts like “price of this kettle” and “thirty pence” are usually precise enough for the problem context in which they are used. (But note the fact that “thirty pence” is, as a social or economic concept, highly variable: it had a different significance a few years ago from what it has today.)
Frege’s opinion is different; for he writes: “A definition of a concept … must determine unambiguously of any object whether or not it falls under the concept … Using a metaphor, we may say: the concept must have a sharp boundary.” But it is clear that for this kind of absolute precision to be demanded of a defined concept, it must be demanded of the defining concepts, and ultimately of our undefined, or primitive, terms. Yet this is impossible. For either our undefined or primitive terms have a traditional meaning (which is never very precise) or they are introduced by so-called “implicit definitions”—that is, through the way they are used in the context of a theory. This last way of introducing them—if they have to be “introduced”—seems to be the best. But it makes the meaning of the concepts depend on that of the theory, and most theories can be interpreted in more than one way. As a result. implicitly defined concepts, and thus all concepts which are defined explicitly with their help, become not merely “vague” but systematically ambiguous. And the various systematically ambiguous interpretations (such as the points and straight lines of projective geometry) may be completely distinct.
This should be sufficient to establish the fact that “unambiguous” concepts, or concepts with “sharp boundary lines”, do not exist. [28-9]