I think one way of approaching "fuzzy" or "vague" concepts was conceived of over a hundred years ago by C. S. Peirce. I've put here some excellent quotes from him related to this topic. I'm putting these quotes here rather than in my main blog pages as I'll probably refer to them frequently.



Accurate writers have apparently made a distinction between the definite and the determinate. A subject is determinate in respect to any character which inheres in it or is (universally and affirmatively) predicated of it, as well as in respect to the negative of such character, these being the very same respect. In all other respects it is indeterminate. The definite shall be defined presently. A sign (under which designation I place every kind of thought," and not alone external signs) that is in any respect objectively indeterminate (i.e., whose object is undetermined by the sign itself) is objectively general in so far as it extends to the interpreter the privilege of carrying its determination further.

Example: "Man is mortal." To the question, What man? the reply is that the proposition explicitly leaves it to you to apply its assertion to what man or men you will. A sign that is objectively indeterminate in any respect is objectively vague in so far as it reserves further determination to be made in some other conceivable sign, or at least does not appoint the interpreter as its deputy in this office. Example: "A man whom I could mention seems to be a little conceited." The suggestion here is that the man in view is the person addressed; but the utterer does not authorize such an interpretation or any other application of what she says. She can still say, if she likes, that she does not mean the person addressed. Every utterance naturally leaves the right of further exposition in the utterer; and therefore, in so far as a sign is indeterminate, it is vague, unless it is expressly or by a well-understood convention rendered general. Usually, an affirmative predication covers generally every essential character of the predicate, while a negative predication vaguely denies some essential character. In another sense, honest people, when not joking, intend to make the meaning of their words determinate, so that there shall be no latitude of interpretation at all. That is to say, the character of their meaning consists in the implications and non-implications of their words; and they intend to fix what is implied and what is not implied. They believe that they succeed in doing so, and if their chat is about the theory of numbers, perhaps they may. But the further their topics are from such presciss, or "abstract," subjects, the less possibility is there of such precision of speech. In so far as the implication is not determinate, it is usually left vague; but there are cases where an unwillingness to dwell on disagreeable subjects causes the utterer to leave the determination of the implication to the interpreter; as if one says, "That creature is filthy, in every sense of the term."

Perhaps a more scientific pair of definitions would be that anything is general in so far as the principle of excluded middle does not apply to it and is vague in so far as the principle of contradiction does not apply to it. Thus, although it is true that "Any proposition you please, once you have determined its identity, is either true or false"; yet so long as it remains indeterminate and so without identity, it need neither be true that any proposition you please is true, nor that any proposition you please is false. So likewise, while it is false that "A proposition whose identity I have determined is both true and false," yet until it is determinate, it may be true that a proposition is true and that a proposition is false.

In those respects in which a sign is not vague, it is said to be definite, and also with a slightly different mode of application, to be precise, a meaning probably due to praecisus having been applied to curt denials and refusals. It has been the well-established, ordinary sense of precise since the Plantagenets; and it were much to be desired that this word, with its derivates precision, precisive, etc., should, in the dialect of philosophy, be restricted to this sense. To express the act of rendering precise (though usually only in reference to numbers, dates, and the like), the French have the verb preciser, which, after the analogy of decider, should have been precider. Would it not be a useful addition to our English terminology of logic, to adopt the verb to precide, to express the general sense, to render precise? Our older logicians with salutary boldness seem to have created for their service the verb to prescind, the corresponding Latin word meaning only to "cut off at the end," while the English word means to suppose without supposing some more or less determinately indicated accompaniment. In geometry, for example, we "prescind" shape from color, which is precisely the same thing as to "abstract" color from shape, although very many writers employ the verb "to abstract" so as to make it the equivalent of "prescind." But whether it was the invention or the courage of our philosophical ancestors which exhausted itself in the manufacture of the verb "prescind," the curious fact is that instead of forming from it the noun prescission, they took pattern from the French logicians in putting the word precision to this second use. About the same time, the adjective precisive was introduced to signify what prescissive would have more unmistakably conveyed (see Watts's Logick). If we desire to rescue the good ship Philosophy for the service of Science from the hands of lawless rovers of the sea of literature, we shall do well to keep prescind, presciss, prescission, and prescissive on the one hand, to refer to dissection in hypothesis, while precide, precise, precision, and precisive are used so as to refer exclusively to an expression of determination which is either full or made free for the interpreter. We shall thus do much to relieve the stem "abstract" from staggering under the double burden of conveying the idea of prescission as well as the unrelated and very important idea of the creation of an ens rationis out of an [gr. "winged word"] - to filch the phrase to furnish a name for an expression of non-substantive thought,an operation that has been treated as a subject of ridicule, - this hypostatic abstraction,- but which gives mathematics half its power.

The purely formal conception that the three affections of terms, determination, generality, and vagueness, form a group dividing a category of what Kant calls "functions of judgment" will be passed by as unimportant by those who have yet to learn how important a part purely formal conceptions may play in philosophy. Without stopping to discuss this, it may be pointed out that the "Quantity" of propositions in logic, that is, the distribution of the first subject, is either singular (that is, determinate, which renders it substantially negligible in formal logic), or universal (that is, general), or particular (as the medieval logicians say, that is, vague or indefinite). It is a curious fact that in the logic of relations it is the first and last quantifiers of a proposition that are of chief importance. To affirm of anything that it is a horse is to yield to it every essential character of a horse: to deny of anything that it is a horse is vaguely to refuse to it some one or more of those essential characters of the horse. There are, however, predicates that are unanalyzable in a given state of intelligence and experience. These are, therefore, determinately affirmed or denied. Thus, this same group of concepts reappears. Affirmation and denial are in themselves unaffected by these concepts, but it is to be remarked that there are cases in which we can have an apparently definite idea of a border line between affirmation and negation. Thus, a point of a surface may be in a region of that surface, or out of it, or on its boundary. This gives us an indirect and vague conception of an intermediacy between affirmation and denial in general, and consequently of an intermediate, or nascent state, between determination and indetermination. There must be a similar intermediacy between generality and vagueness. Indeed, in an article in the seventh volume of The Monist, there lies just beneath the surface of what is explicitly said, the idea of an endless series of such intermediacies. We shall find below some application for these reflections.

"Issues in Pragmatism" Essential Peirce 2:350-353

(I also have online an excerpt from "The Categories Defended" that some might find interesting)