Rene Ramirez, Ph. D.
What follows is an essay about vagueness. The essay is divided into two main parts. The first part identifies vagueness by providing three criteria that distinguish it from some other types of indeterminacy. The second part discusses three different ways in which vagueness manifests itself. First, vagueness appears in language and thought. Many of the words that make up human language, as well as the concepts used in our thinking, are vague. We will refer to this as semantic or cognitive vagueness. Second, vagueness also makes an appearance in sensory perception. Human powers of observation are limited, so we need a vocabulary that allows us to make perceptual judgments that are more or less precise. Vagueness at the perceptual level satisfies this requirement. Third, we will argue that the world itself may be vague, a phenomenon known as metaphysical vagueness. Although metaphysical vagueness satisfies the criteria proposed for vagueness in general, it is distinctive in that its mode of manifestation is neither semantic, cognitive, nor perceptual, but ontological.
What is vagueness? A definition would be useful in fixing the referent of the term ‘vagueness’. However, we should not expect more from a definition than it is capable of providing. Although a definition can help indicate the phenomenon we are seeking, it cannot reveal its essential nature; only a well-supported theory can do that. We will suggest an account of vagueness that relies on three criteria of identification. A vague term or concept: 1) generates borderline cases when it is applied; 2) is susceptible to a distinctive type of puzzle called the sorites paradox; and 3) lacks precise boundaries between its application to clear cases and its application to unclear cases, making it impossible to classify the referents of the term according to traditional methods of classification based on set theory. This phenomenon is known as boundarylessness.
1) A vague term or concept generates borderline cases. Consider the word ‘thin’. Some people are clearly thin. Some others are clearly not thin. But there is a third group of people who are neither clearly thin nor clearly not thin. This third group makes up the borderline cases, which render the term ‘thin’ vague. Let’s examine a second example, one involving a distinction between a precise term and a vague one.
a. Harry is forty years old.
b. Mary is fortyish.
In (1a) the word ‘forty’ is precise; it generates no borderline cases. The word ‘fortyish’ in (1b), however, is vague. We do not know whether Mary is thirty-eight, thirty-nine, forty-one, or forty-two years old. These numbers represent borderline cases associated with the application of the word ‘fortyish’.
2) The second feature in our definition of ‘vagueness’ is that vague terms are susceptible to sorites puzzles.1 Suppose that there are three or four grains of sand collected together on a table. Clearly, this would not be enough grains to make a heap. In contrast ten-thousand grains of sand would constitute a heap. Now, suppose that, starting with three grains of sand, we add a grain at a time. How many grains must we add before we cross the threshold from non-heap to heap? Is it fifty grains, a hundred, three-hundred? There does not seem to be a clear answer to this question. The word ‘heap’, we conclude, is vague.
How should we think about the sorites puzzle? One way is to argue that, as we add grains of sand one by one, if n grains are not a heap, n+1 grains will not be a heap either. After all, how can a single grain amount to a shift from non-heap to heap? Yet it is perfectly reasonable to expect that the threshold separating the two will be crossed at some point. What we have here is an inductive argument, one of whose steps is: for all n, if n grains of sand fail to make a heap, then n+1 grains also fail to make a heap. But this step must be false: the threshold is eventually crossed. This means that, for some n, n grains having been added is a non-heap, and n+1 grains is a heap. That is, there must be a last grain beyond which there is a transition from non-heap to heap, though we do not – indeed, cannot – know which one it is. This is a highly puzzling state of affairs.
It is neither determinately true nor determinately false that there is a last grain of sand that marks the boundary between a non-heap and a heap. The propositions that describe the borderline cases may be true or false, but because of the way vague terms and concepts work and the way the world is we cannot know which is the case. The boundary between the clear cases designated by a vague term and the borderline cases constitutes what Roy Sorensen calls an epistemic island.2 There is a condition, operating at the boundary between a clear application of a vague term and a borderline application, which, denying us access across that border, prevents us from knowing whether or not the term applies.
3) The third criterion of vagueness has to do with classification. Traditional rules governing classification define a category in terms of set theory. A set is a collection of things which is defined by its membership. For example, the set of tigers includes all the tigers and excludes anything that is not a tiger. The concept Tiger, whose determination depends on the set of tigers, draws a sharp boundary between the class of tigers and everything else. Thus, set theory, and the concepts whose definition it underlies, provides the traditional classificatory framework with categories having sharply delineated boundaries.3
In contrast vague terms do not permit a sharp boundary between a clear application of the term and its application in a borderline case. For example, there is no sharp boundary between a person who is clearly thin and someone who is neither clearly thin nor clearly not thin. Therefore , there is no way for someone to distinguish precisely between them, nor any way to apply the term ‘thin’ definitely to one and definitely not to the other. Accordingly, we cannot determine the membership of the set of thin people because we do not know how to classify the borderline cases. Since the borderline cases may or may not be thin, the membership of the set remains indeterminate. It is not surprising that the set of thin people, unlike the set of tigers, is not a natural kind. Such boundarylessness strongly conflicts with traditional methods of classification, which require a precise boundary between objects that clearly belong to a set and objects that clearly do not belong to it.
Vagueness, understood as boundarylessness, gives us the freedom to choose a point along an indeterminate range of possible choices. Different individuals will set the limit at different points along the range depending on variations in our power to discriminate between similar stimuli and on the demands of context. For example, our ability to discriminate between slight differences in color or sound contributes to our appreciation of art or music. Since we are not obligated to draw these distinctions at any particular point along the relevant spectrum, vagueness allows for variety of response and interpretation.
Vagueness is a variety of linguistic or conceptual indeterminacy.4 There are several ways in which language or thought can lack determinate limits besides being vague. These include ambiguity, inaccuracy, lack of specificity, generality, improbability, and others. In order to get a better grasp of vagueness, let’s contrast it with one of these types of indeterminacy: ambiguity. A term or sentence is ambiguous if it suggests two or more referents or interpretations but there is not enough information available in the given context to decide between the options. For example, the sentence “Harry is living in Portland.” is ambiguous because it fails to distinguish between two possible readings of the word ‘Portland’: Harry may be living in Portland, Maine or in Portland, Oregon. Once the relevant information is provided, the ambiguity is resolved. Unlike vagueness, ambiguity does not involve borderline cases. The two alternatives – Portland, Maine and Portland, Oregon – are each a precise option either of which would resolve the ambiguity once it is provided. Contrast this with a vague term like ‘thin’, whose borderline cases are neither clearly thin nor clearly not thin.5
We now have a firmer grasp of the meaning of the term ‘vague’ than we had at the start of the essay.6 This allows us to turn our attention to the task of discussing several of the ways in which vagueness manifests itself. Vagueness appears at a variety of sites, three of which will be of concern to us. First, vagueness is a necessary feature of language and thought and, consequently, a pervasive factor in the public discourse of any culture. Take, for example, the issue of abortion, a topic that has been hotly contested in the United States. The debate between defenders and opponents of abortion centers on a disagreement regarding the meaning of certain words. The controversy surrounding abortion is caused by factors that lie at a level much deeper than language, but language serves as a screen, as it were, upon which the conflict is projected. Consider the sentences below. The first proposition (2a) represents the pro-life view while (2b) expresses the pro-choice perspective:
a. A fetus is a person.
b. A fetus is not a person.
Both of these sentences contain the words/concepts ‘person’ and ‘fetus’. The main point of the debate is whether a person begins to exist at conception, or whether a certain segment of the gestation process, or the gestation process as a whole, must elapse before it can be said that a person exists. One of the factors fueling this debate is that the term ‘person’ is vague. It is indeterminate whether a fetus is a clear instance of a person, as the pro-life position asserts; or whether it is not a clear instance of a person, as the pro-choice advocates argue. Since the term ‘person’ is vague, it generates borderline cases along the entire process of gestation prior to birth. It does not seem to be possible to identify a determinate moment at which the threshold separating a fetus and a person is crossed. Both religion and the law (e.g., Roe v. Wade) have attempted to establish precise limits at which a person begins to exist, but these efforts seem arbitrary, evidence for which is that the controversy continues unabated.
Disagreement arises from the clash between interpretations of propositions containing vague terms. Of course, the vagueness of the terms does not prevent them from expressing deeply held values. If the values are contradictory, however, the cultural dynamic can easily take the form of controversy. Since abortion involves the termination of life, and since human life is considered sacred, once the fetus is judged to be a person, the conclusion immediately follows that abortion is murder. In contrast, if the fetus is demoted to a status somewhere beneath that of a person, the conclusion that abortion is murder is avoided. However, since the meaning of the concept Person is vague, it is not quite clear what its referent is, i.e., where along the gestation process the concept properly applies. Consequently, it is not clear how to identify the entity that possesses the exceptional intrinsic value of personhood. We can all agree that there is merit in the attempt to resolve a debate that centers on the issue whether an innocent life is or is not at stake. Unfortunately, since the controversy is motivated in part by our inability to clarify the vague terms and concepts that lie at its center, unless language or the world (or both) change, the debate is likely to last well beyond the foreseeable future.
We turn now to another controversy in which the same word, ‘person’, has played a leading role. The cultural dynamic at issue in this case is something we may call semantic appropriation. Semantic appropriation occurs when a term that is used in a relatively neutral sense in a community’s vernacular, is adopted by a particular institution of the culture and given a precise meaning that is significantly different from its normal use. For example, the word ‘person’ is widely used among speakers of English and, as we have seen, is relatively vague. However, the word was adopted by the legal establishment when the Supreme Court of the United States in Trustees of Dartmouth College v. Woodward – 17 U. S. 518 (1819), “recognized corporations as having the same rights as natural persons to contract and to enter into contracts.”7 And in Pembina Consolidated Silver Mining Co. v. Pennsylvania – 125 U. S. 181 (1888), the Court held that: “Under the designation of ‘person’ there is no doubt that a private corporation is included (in the Fourteenth Amendment).”8
How can we make sense of this? Prior to the Court’s decisions the term ‘person’ definitely did not refer to a corporation. Now, however, as a result of semantic appropriation, the corporation has become a borderline case of the term ‘person’. That is, there now exist entities which are neither clearly persons nor clearly not persons. This change raises questions about the relationship between the individual and the collective. Where previously the boundary between a person and a corporation was clear and distinct, now, in the wake of a series of Supreme Court decisions, the boundary is imprecise, inasmuch as some of the rights formerly guaranteed exclusively to natural persons have been granted to corporations.
Vagueness and semantic indeterminacy have been at the center of cultural discourse throughout all of history. The phenomenon of vagueness, if not the term itself, has been present implicitly in myriad historical controversies in every nation. Another example we may consider is the following. During America’s Constitutional Convention in the eighteenth century, there was considerable debate concerning the rights of the American colonists: who should or should not enjoy the rights promised by the Constitution, and how should these rights be protected from being restricted or abrogated? It was the aim of the framers to make these rights explicit. What was implicitly vague in the debate was the notion of freedom. Some Americans would be clearly free under the new Constitutional structure, e.g., white males who owned property. Some others would be clearly not free, e.g., slaves. But there was a third group whose status was indeterminate: e.g., women, Catholics, children, immigrants, foreigners, and others. This last group comprised the borderline cases, i.e. those who would be seen as neither clearly free nor clearly not free. This group of indeterminately defined citizens dramatizes the deeply implicitly vague meaning of the notion of freedom at the origin of the American enterprise.
The framers of the Constitution were concerned not only with the question who should enjoy the rights guaranteed by the new American polity, but also with the problem of limiting the power of those whose decisions might unduly restrict these rights. This concern found its way into the wording of the First Amendment: “Congress shall make no law…abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble and to petition Government for a redress of grievances.” It is the power of Congress which these words delimit. They make more nearly precise that which was previously vague, viz. the scope of governmental power in the new nation. This is an instance of implicit vagueness and of the attempt to resolve it through the promulgation of explicit law, a process that has been evolving for more than two-hundred years and will continue to unfold as long as the terms and concepts that define American democracy continue to require greater specification.
Implicit vagueness may be partially resolved by means of formal interpretations and definitions imposed by authoritative sources, as we have seen in the previous examples, yet it finds a way to assert and reassert itself in spite of efforts to domesticate it. The history of art offers an illustration of vagueness rising to the surface after having been suppressed by social consensus. For more than two-thousand years, beginning with Plato’s Republic, reinforced during the Renaissance, and extending into the twentieth century, artists, philosophers, and critics endorsed Alberti’s criterion that “there should be no visual difference between looking at a painting or looking out a window at what the painting shows.”10 In other words since the proposition
3) Art is imitation.
was taken to be unfailingly true, art was understood to be essentially imitative, representational. That is, interested parties were able to identify a class of things that were clearly art, and another class of things that were clearly not art because the former were imitations of something in the world, and the latter were not. But there were no borderline cases, i.e. a class of things that were neither clearly art nor clearly not art. In other words the term ‘art’ was, to all intents and purposes, precise.
In the early twentieth century this all began to change. With the advent of Modernism the traditional conception of art as imitation gave way as new art forms came to the fore that not only challenged the status quo but utterly demolished it. Fauvism, Dadaism, Cubism, Surrealism, Abstract Expressionism and other movements all burst upon the art scene, first in Paris and later in New York. Paintings such as Picasso’s Les Demoiselles d’Avignon (1907) and Malevich’s Black Square (1915) thrilled the avant-garde but horrified the guardians of the old order. New organizing principles, such as abstraction, minimalism, spontaneity, and cubism competed in a frenzy of creative activity that encouraged novel answers to the question What is art? Not only had the mimetic paradigm been called into question, as Arthur Danto says, it had been relegated to the status of just another movement, i.e. Realism. This blossoming of new aesthetic ideas resulted in the introduction of vagueness into a discourse that for centuries had accepted an artificial precision with respect to our conception of the essential nature of art. Now anyone interested in understanding art was expected to grapple with a motley crew of alternatives: objects that were clearly art; objects that were clearly not art; and objects that were neither clearly art nor clearly not art. New artistic currents had brought with them a furious mix of borderline cases whose aesthetic identity could not be determinately known.
The examples described so far have been illustrations of the interdependence between vagueness and cultural context. However, vagueness also plays a role in the relation between perception and perceptual judgment. The perceptual abilities of human beings are accurate only within certain limits. We are able to detect only a small segment of the electromagnetic spectrum, and the normal observer can hear only those sounds that measure between twenty and twenty thousand hertz. Moreover, given two shades of color or two sounds that differ from each other only slightly, the average person will have difficulty discriminating one from the other, and the margin of error in the resulting judgment can be considerable. Yet we often make judgments regarding stimuli that appear quite similar, or attempt to identify objects in less than optimal visual or auditory conditions. In order to be able to describe such occurrences, and to communicate effectively to others about them, we need a vocabulary that is sufficiently vague to reflect the indeterminacy in our observations. Vagueness can be both an indicator of the lack of precision in many of our observations, and an asset in our effort to describe these observations by means of judgments that are indeterminate to a degree consistent with the demands made by finely graded sensory stimuli upon imperfect observers.
The last question we will consider is whether vagueness exists in the world. This question should be seen as part of a larger philosophical issue: viz. can anything be known about the world as it is in itself, independently of human language or thought? This calls to mind Kant’s Critique of Pure Reason, in which he posits the existence of the noumenon, the thing in itself, which cannot be known except through the mediation of the concepts of the understanding. Contrary to the Kantian view is metaphysical realism. Gary Ebbs describes this theory as follows:
“The idea behind metaphysical realism is that we can conceive of the entities and substances and species of the ‘external’ world independently of any of the empirical beliefs and theories we hold or might hold in the future. To accept this picture, we must conceive of the relationships between our words and the external world from an ‘external’ perspective. We must imagine that we can completely distinguish between what we believe and think about the things to which we refer, on the one hand, and the pure truth about these things, on the other.”11
It follows that, if metaphysical realism is true, then we should be able to conceive of vagueness as existing in the world, independently of our beliefs, theories, language, and referential practices. Is such a view tenable?
That there is an external world existing independently of human beings, with its own nature and structure, seems to be a fairly reasonable notion. It is, after all, the idea that lies behind our view of objectivity. Surely there are oceans, animals, rainstorms, sunlight, and other entities and occurrences that depend not at all on whether or not we humans exist or on how we choose to describe them. To deny this is to fly in the face of common sense. However, the suggestion that vagueness can exist objectively, mind-independently, and beyond the influence of human language and thought, seems to extend the notion of objectivity beyond reasonable limits. How can an object or a process be vague? In the remainder of this essay we will examine a way in which the world itself may lack precise boundaries.
Consider Jane, an ordinary woman. At one level, the level of physical composition, Jane’s body is made up of cells. These cells mature and die and over time are replaced by new ones. After a number of years the cells that make up her body are entirely different from the cells that her body was composed of, say, ten years earlier. At a different level Jane comprises a series of temporal stages. Jane at time t is at a different stage from Jane at time t+1, t+2, t+3…or t+n. Thus, over the course of ten years Jane has undergone two sets of changes: 1) replacement of all her bodily cells and 2) several transitions of temporal stages. Who, then, is Jane: that is, what can we reasonably conclude concerning her personal identity? There are at least three possibilities. 1) Jane’s identity is preserved throughout all her changes. She is fully Jane at time t and continues to be the same person/woman at time t+n, when her developmental process, i.e. her life, comes to an end, in spite of all the cellular and temporal transitions she has undergone. 2) Jane begins her developmental process as Jane1 but at the next stage she changes to Jane2, and the stage after that results in Jane3, until the process culminates in a final stage, Janen. That is, each stage in Jane’s development brings about a different version of Jane, and no uniform identity is preserved, although each version is similar to the immediately preceding and succeeding ones. Jane, then, is identical to each version but each version is different from all the others. She continues to be the same person, Jane, but at each stage she is a different version of that person. 3) This alternative is similar to (2) above in that there is a series of versions of Jane throughout the developmental process. The difference is that Jane is fully or totally identical only to the sum of all her versions. Her full identity is indeterminate at every stage of her life. Jane’s identity amounts to the entire series of versions integrated together as a single entity. Who Jane is is not determined until her death brings her process to a close.
How does vagueness enter into the foregoing account? If (1) above is correct, then there is no vagueness. Jane is clearly the same person/woman at every stage in her life. There is no multiplicity of Jane-versions, so her identity is precise: she is fully Jane at every stage. If (2) above is correct, however, it is not clear where one version of Jane ends and a subsequent one begins. What, precisely, determines the identity of a Jane-version? Such a situation is susceptible to a sorites paradox. Suppose we add a series of seconds to Jane’s life once she attains her seventeenth birthday. It is not clear at which point she will make the transition from her late-adolescent-Jane-version to her early-adult-Jane-version. Such a transition is haunted by the specter of boundarylessness. What we have in this case is not an epistemic island but an ontological one. We conclude that under option (2) Jane’s identity is vague.
Finally, let’s assume that option (3) is the appropriate reflection of the vagueness in the world. On this option Jane’s identity is incomplete at every stage of her life. No Jane-version, either by itself or in conjunction with any other set of versions, realizes Jane, unless it is the entire collection of versions summed up into a single entity. Yet each version is a segment of Jane which contributes to the whole. Each version is, if not totally Jane, then partially Jane. A particular version of Jane may possess certain properties that are relevant at a given stage in her life, e. g., being forty years old, or having been a teacher for twenty years. This stage also lacks other properties that were once relevant but are no longer so, e.g., being an adolescent or having been a socialist. The set of properties that characterize a given stage partially define a version of Jane relative to the whole set of properties whose sum will ultimately define her at the end of her life. This means that each version of Jane is a borderline case of Jane, neither fully Jane nor fully not Jane. It follows that Jane’s identity is vague at any point along her development.
If this argument is sound, i.e. if Jane’s versions are in fact borderline cases of Jane, then there is such a phenomenon as metaphysical vagueness.12 A version is neither a word, a concept, nor a thought. It is, rather, a way of being, specifically, a way in which Jane is at a given moment or stage in her personal development. This is different from semantic or conceptual vagueness, which reside in language or thought. Nor is it perceptual vagueness, which is indeterminacy associated with the act of observation. Nor does metaphysical vagueness have epistemological roots. If the world itself is vague, it is not because we cannot know the precise boundary that separates one event from another. On the contrary, it is because there is no sharp boundary between events, between the transitions from Jane1 to Jane2 to Janen, that we cannot know where the boundary lies. If there is such a thing as metaphysical vagueness, it is essentially ontological, not epistemological.
Whether metaphysical vagueness is real or not harks back to an issue that we briefly touched on earlier: viz. whether the world can be known as it is in itself, independently of human language or theory. Given how controversial this issue has been throughout the history of western philosophy, we need to consider the possibility that the argument for metaphysical vagueness is vulnerable to a variety of objections. One such objection is that the argument consists of statements, words, and concepts which may be subject to semantic and conceptual vagueness. In other words our attempt to establish the reality of metaphysical vagueness may be undermined by vagueness creeping in from other quarters. If words such as ‘version’, ‘identity’, or ‘person’ are vague, then their use may contaminate our argument to the point of rendering it unsound. This is an important issue. However, since we cannot resolve the quandary without extending the essay beyond reasonable limits, we will leave it for another day.
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1 Sorites puzzles date back to ancient Greece. The word ‘sorites’ derives from soros, the Greek word for heap.
2 Dorothy Edgington, “Sorensen on Vagueness and Contradiction,” Cuts and Clouds: Vagueness, its Nature, and its Logic, eds. Richard Dietz and Sebastiano Moruzzi (Oxford: Oxford University Press, 2010) 94.
3 R. M. Sainsbury, “Concepts Without Boundaries,” Vagueness: A Reader, eds. Rosanna Keefe and Peter Smith (Cambridge: MIT Press, 1999) 251-252.
4 Below we will consider the possibility that there is vagueness in the world as well.
5 There is another form of ambiguity which operates at the level of syntax, as in the example: Flying planes can be dangerous. This sentence has two interpretations because of the syntactic ambiguity associated with the phrase ‘flying planes’.
6 As we suggested at the beginning of the essay, our goal is not a theory of vagueness. Only a theory can reveal the fundamental nature of a phenomenon. The most that we can anticipate here is that we have identified the proper referent of the term ‘vagueness’.
7 “Corporate personhood” http: // en.wik.pedia.org/wiki/Corporate_personhood
9 Akhil Reed Amar, The Bill of Rights (New Haven: Yale University Press, 1998) 20.
10 Arthur C. Danto, What Art Is (New York: Yale University Press, 2003) 1.
11 Stuart Shapiro, “Vagueness, Metaphysics, and Objectivity”, Cuts and Clouds: Vagueness: Its Nature and Its Logic”, eds. Richard Dietz and Sebastiano Moruzzi (Oxford: Oxford University Press, 2010) 150.
12 The foregoing argument would apply to any organism, that is, to any living being capable of undergoing a process of development.