In linguistics and philosophy, a vague predicate is one which gives rise to borderline cases. For example, the English adjective "tall" is vague since it is not clearly true or false for someone of middling height. By contrast, the word "prime" is not vague since every number is definitively either prime or not. Vagueness is commonly diagnosed by a predicate's ability to give rise to the Sorites paradox. Vagueness is separate from ambiguity, in which an expression has multiple denotations. For instance the word "bank" is ambiguous since it can refer either to a river bank or to a financial institution, but there are no borderline cases between both interpretations.
Vagueness is a major topic of research in philosophical logic, where it serves as a potential challenge to classical logic. Work in formal semantics has sought to provide a compositional semantics for vague expressions in natural language. Work in philosophy of language has addressed implications of vagueness for the theory of meaning, while metaphysicians have considered whether reality itself is vague.
The concept of vagueness has philosophical importance. Suppose one wants to come up with a definition of "right" in the moral sense. One wants a definition to cover actions that are clearly right and exclude actions that are clearly wrong, but what does one do with the borderline cases? Surely, there are such cases. Some philosophers say that one should try to come up with a definition that is itself unclear on just those cases. Others say that one has an interest in making his or her definitions more precise than ordinary language, or his or her ordinary concepts, themselves allow; they recommend one advances 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 mixed male and female biological traits (see intersex).
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 one's 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, a.k.a. sorites paradox—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 (1993).
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.
Subvaluationism is the logical dual of supervaluationism, and has been defended by Dominic Hyde (2008) and Pablo Cobreros (2011). Whereas the supervaluationist characterises truth as 'supertruth', the subvaluationist characterises truth as 'subtruth', or "true on at least some precisifications".
Subvaluationism proposes that borderline applications of vague terms are both true and false. It thus has "truth-value gluts". According to this theory, a vague statement is true if it is true on at least one precisification and false if it is false under at least one precisification. If a vague statement comes out true under one precisification and false under another, it is both true and false. Subvaluationism ultimately amounts to the claim that vagueness is a truly contradictory phenomenon. Of a borderline case of "bald man" it would be both true and false to say that he is bald, and both true and false to say that he is not bald.
A fourth approach, known as "the epistemicist view", has been defended by Timothy Williamson (1994), R. A. Sorensen (1988)  and (2001), and Nicholas Rescher (2009). They maintain that vague predicates do, in fact, draw sharp boundaries, but that one cannot know where these boundaries lie. One's confusion about whether some vague word does or does not apply in a borderline case is due to one's ignorance. For example, in the epistemicist view, there is a fact of the matter, for every person, about whether that person is old or not old; some people are ignorant of this fact.
As a property of objects
One possibility is that one's 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 is not clear where the boundary of a cloud lies; for any given bit of water vapor, one can ask whether it is part of the cloud or not, and for many such bits, one won't know how to answer. So perhaps one's 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's law or other rules for validity some philosophers are willing to defend ontological vagueness as some kind of metaphysical phenomenon. One has, 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. In the US this is known as the vagueness doctrine and in Europe as the principle of legal certainty.
- Essentially contested concept
- Fuzzy concept
- Unconstitutional vagueness
- Imprecise language
- Williamson, T. 1994. Vagueness,[page needed]. London: Routledge.
The history of the problem of vagueness is traced, from the first Sorites Paradox to contemporary attempts to deal with higher-order vagueness such as many-valued logic, supervaluationism, and fuzzy logic. Technicalities are kept to a minimum to favour a clear account, extremely useful to both students and researchers.
- Machina, K.F. 1976. "Truth, belief and vagueness", in Journal of Philosophical Logic Vol. 5. pp. 47-78
- Edgington, D. (1997). Keefe, R.; Smith, P. (eds.). Vagueness by degrees (PDF). MIT Press. pp. 294–316.
- Kit Fine, The Limits of Abstraction (2002)
- Pablo Cobreros, (2011) "Paraconsistent Vagueness: A Positive Argument" Synthese 183(2): 211–227
- Dominic Hyde and Mark Colyvan (2008) “Paraconsistent Vagueness: Why Not?” Australasian Journal of Logic 6: 107–121.
- Williamson, T. 1994. Vagueness London: Routledge.
- Sorensen, R.A. 1988. Blindspots. Oxford: Clarendon Press.
- Sorensen, Roy (2001). Vagueness and Contradiction. Oxford University Press.
- Rescher, N. 2009. Unknowability. Lexington Books.
Uses vagrant predicates to elucidate the problem.
- Evans, G. (1978). "Can There Be Vague Objects?". Analysis. 38 (4): 208–. doi:10.1093/analys/38.4.208.
- Van Inwagen, Peter. 1990 Material Beings. Ithaca, NY: Cornell University Press.
- Parsons, Terence. 2000. Indeterminate Identity - Metaphysics and Semantics Oxford: Clarendon Press.
- Deemter, Kees van. Not Exactly: In Praise of Vagueness (Oxford University Press; 368 pages; 2010).
Considers vagueness as both a useful and unavoidable aspect of realms from everyday life to computing.
- Keefe, R.; Smith, P., eds. (1997). Vagueness: A Reader. MIT Press.
The editors' long introduction gives a clear and very useful overview of theories of vagueness, and they collect many classic papers on the subject.
- Keefe, R. 2000. Vagueness. Cambridge: Cambridge University Press.
Technicalities are kept to a minimum to favour a clear account, extremely useful to both students and researchers.
- Rick Nouwen; Robert van Rooij; Uli Sauerland; Hans-Christian Schmitz, eds. (Jul 2009). International Workshop on Vagueness in Communication (ViC; held as part of ESSLLI). LNAI. Vol. 6517. Springer. ISBN 978-3-642-18445-1.
- Ronzitti, Giuseppina (ed.), Vagueness: A Guide. Logic, Epistemology, and the Unity of Science, Springer, Dordrecht, 2011.