Vagueness is important philosophically. Suppose we want to come up with a definition of "right" in the moral sense. We want a definition to cover actions that are clearly right and exclude actions that are clearly wrong, but what do we do with the borderline cases? Surely there are such cases. Some philosophers say we should try to come up with a definition that is itself unclear on just those cases. Others say that we have an interest in making our definitions more precise than ordinary language, or our ordinary concepts, themselves allow; they recommend we advance precising definitions.
Vagueness is also a problem which arises in law, and in some cases judges have to arbitrate regarding whether a borderline case does, or does not, satisfy a given vague concept. Examples include disability (how much loss of vision is required before one is legally blind?), human life (at what point from conception to birth is one a legal human being, protected for instance by laws against murder?), adulthood (most familiarly reflected in legal ages for driving, drinking, voting, consensual sex, etc.), race (how to classify someone of mixed racial heritage), etc. Even such apparently unambiguous concepts such as gender can be subject to vagueness problems, not just from transsexuals' gender transitions but also from certain genetic conditions which can give an individual both male and female biological traits (see intersexual).
Many scientific concepts are of necessity vague, for instance species in biology cannot be precisely defined, owing to unclear cases such as ring species. Nonetheless, the concept of species can be clearly applied in the vast majority of cases. As this example illustrates, to say that a definition is "vague" is not necessarily a criticism.
Consider those animals in Alaska that are the result of breeding Huskies and wolves: are they dogs? It is not clear: they are borderline cases of dogs. This means our ordinary concept of doghood is not clear enough to let us rule conclusively in this case.
The philosophical question of what the best theoretical treatment of vagueness is - which is closely related to the problem of the paradox of the heap - has been the subject of much philosophical debate.
One theoretical approach is that of fuzzy logic, developed by American mathematician Lotfi Zadeh. Fuzzy logic proposes a gradual transition between "perfect falsity", for example, the statement "Bill Clinton is bald", to "perfect truth", for, say, "Patrick Stewart is bald". In ordinary logics, there are only two truth-values: "true" and "false". The fuzzy perspective differs by introducing an infinite number of truth-values along a spectrum between perfect truth and perfect falsity. Perfect truth may be represented by "1", and perfect falsity by "0". Borderline cases are thought of as having a "truth-value" anywhere between 0 and 1 (for example, 0.6). Advocates of the fuzzy logic approach have included K. F. Machina (1976) and Dorothy Edgington (1997).
Another theoretical approach is known as "supervaluationism". This approach has been defended by Kit Fine and Rosanna Keefe. Fine argues that borderline applications of vague predicates are neither true nor false, but rather are instances of "truth value gaps". He defends an interesting and sophisticated system of vague semantics, based on the notion that a vague predicate might be "made precise" in many alternative ways. This system has the consequence that borderline cases of vague terms yield statements that are neither true, nor false.
Given a supervaluationist semantics, one can define the predicate 'supertrue' as meaning "true on all precisifications". This predicate will not change the semantics of atomic statements (e.g. 'Frank is bald', where Frank is a borderline case of baldness), but does have consequences for logically complex statements. In particular, the tautologies of sentential logic, such as 'Frank is bald or Frank is not bald)', will turn out to be supertrue, since on any precisification of baldness, either 'Frank is bald' or 'Frank is not bald' will be true. Since the presence of borderline cases seems to threaten principles like this one (excluded middle), the fact that supervaluationism can "rescue" them is seen as a virtue.
A third approach, known as the "epistemic view", has recently been defended by Timothy Williamson (1994) and R. A. Sorensen (1988) and (2001). They maintain that vague predicates do, in fact, draw sharp boundaries, but that we just do not know where these boundaries lie. Our confusion about whether some vague word does or does not apply in a borderline case is explained as being due to our ignorance. For example, on the epistemic view, there is a fact of the matter, for every person, about whether that person is old, or not old. It is just that we may sometimes be ignorant of this fact. The epistemic view, though initially counter-intuitive, has been defended with some interesting and forceful philosophical arguments.
One possibility is that our words and concepts are perfectly precise, but that objects themselves are vague. Consider Peter Unger's example of a cloud (from his famous 1980 paper, "The Problem of the Many") : it's not clear where the boundary of a cloud lies; for any given bit of water vapor, we can ask whether it's part of the cloud or not, and for many such bits, we won't know how to answer. So perhaps our term 'cloud' denotes a vague object precisely. This strategy has been poorly received, in part due to Gareth Evans's short paper "Can There Be Vague Objects?" (1978). Evans's argument appears to show that there can be no vague identities (e.g. "Princeton = Princeton Borough"), but as Lewis (1988) makes clear, Evans takes for granted that there are in fact vague identities, and that any proof to the contrary cannot be right. Since the proof Evans produces relies on the assumption that terms precisely denote vague objects, the implication is that the assumption is false, and so the vague-objects view is wrong.
Still by, for instance, proposing alternative deduction rules involving Leibniz law or other rules for validity some philosophers are willing to defend vagueness as some kind of metaphysical phenomenon. We have, for example, Peter van Inwagen (1990), Trenton Merricks and Terence Parsons (2000).
In the common law system, vagueness is a possible legal defence against by-laws and other regulations. The legal principle is that delegated power cannot be used more broadly than the delegator intended. Therefore, a regulation may not be so vague as to regulate areas beyond what the law allows. Any such regulation would be "void for vagueness" and unenforceable. This principle is sometimes used to strike down municipal by-laws that forbid "explicit" or "objectionable" contents from being sold in a certain city; courts often find such expressions to be too vague, giving municipal inspectors discretion beyond what the law allows. See Vagueness doctrine.