Vagueness, Logic and Ontology

Dominic Hyde

• Publisher: Routledge
• Year: 2016
• Print ISBN:
  9780754615323
• eBook ISBN:
  9781317002888
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

The topic of vagueness re-emerged in the twentieth century from relative obscurity. It deals with the phenomenon in natural language that manifests itself in apparent semantic indeterminacy - the indeterminacy, for example, that arises when asked to draw the line between the tall and non-tall, or the drunk and the sober. An associated paradox emphasises the challenging nature of the phenomenon, presenting one of the most resilient paradoxes of logic. The apparent threat posed for orthodox theories of the semantics and logic of natural language has become the focus of intense philosophical scrutiny amongst philosophers and non-philosophers alike. Vagueness, Logic and Ontology explores various responses to the philosophical problems generated by vagueness and its associated paradox - the sorites paradox. Hyde argues that the theoretical space in which vagueness is sometimes ontologically grounded and modelled by a truth-functional logic affords a coherent response to the problems posed by vagueness. Showing how the concept of vagueness can be applied to the world, Hyde's ontological account proposes a substantial revision of orthodox semantics, metaphysics and logic. This book will be of particular interest to readers in philosophy, linguistics, cognitive science and geographic information systems.

Chapter 1
Vagueness

The eastern gateway to Australia’s arid centre is marked by ephemeral rivers and wetland areas that skirt the central deserts. A keen diarist seeking to note the exact moment at which they enter one of these wetland areas will be hard pressed. The difficulty is not simply a matter of their potentially limited grasp of the continent’s wetland ecology, but arises because it seems that there simply is no exact moment at which one enters into the area. There would seem to be no sharp line delimiting the particular wetland from the surrounding country; there are regions that would seem to be part of the wetland and regions not part of the wetland, yet intuitively there is no sharp line that separates them from each other. It seems obvious that no fence can mark the exact boundary of the wetland area.

Analogously, there would seem to be no sharp line delimiting situations where the term “wetland” itself applies from situations where it does not. The wetland in which the diarist finds themselves was not always a wetland area, yet it seems impossible to identify any precise moment in its evolutionary history as that moment at which the term was first applicable.

We are confronted with vagueness.

What exactly is vagueness and why is its analysis of interest? This chapter starts with a discussion of vagueness in its most common setting, natural language, and proceeds initially by discussing the paradigmatic concept of vagueness as applied to predicates. Then, having described a sufficiently thick notion of predicate-vagueness, we are in a position to consider that ancient conundrum which presents vagueness as a problem – the sorites paradox. Various kinds of predicate-vagueness are subsequently distinguished and the concept is extended beyond predicates to other parts of language, necessitating, in due course, some further qualification of the notion of predicate-vagueness.

A general account of what it means to speak of vagueness in language is offered. With this taxonomy of vagueness to hand we can pursue our investigations into its cause and logic with a broad understanding as to the phenomenon being addressed and the nature of the challenge it presents.

1.1 Vagueness Introduced

“Vague” is an ambiguous term yet this book is concerned with vagueness in only one specific sense. In this first section, therefore, let us try to delimit that sense of vagueness around which the subsequent discussion revolves.

1.1.1 What is vagueness?

Apparent lack of sharp boundaries is prevalent in our use of natural language. Consider our above-mentioned diarist camping in a wetland area fringing Australia’s arid heart. At what moment did they enter? If no moment can be identified as that which marks their entry, then how can they have entered at all? Likewise we may ask, seemingly to no avail, at what instant did the autumn leaves turn brown or did that person become rich, famous, bald, tall or an adult. These predicates – “(is) in a wetland”, “(is) brown”, “(is) rich”, and so on – are all examples of predicates whose limits of application seem essentially indefinite or indeterminate, and they are typical examples of what are termed vague predicates. Consider the predicate “tall”, for example. We might line up a crowd of people starting with the shortest and progressing monotonically to the tallest. The crowd seems not to be clearly partitioned into two mutually exclusive and exhaustive sets of those to whom the predicate applies and those to whom it fails to apply. The transition from one set to the other would seem not to be precise and one might ask rhetorically, as Diogenes Laërtius is reputed to have done, ‘Where do you draw the line?’.

The most common instances of vague predicates are those for which the applicability of the predicate just seems to fade off, as in the above examples, and it consequently appears that no sharp boundary could conceivably be drawn separating the predicate’s positive extension from its negative extension. The behaviour of vague predicates is thus contrasted with such precise predicates as “greater than two” defined on, say, the natural numbers. We can divide the domain of natural numbers, N, into two sharp, mutually exclusive and exhaustive sets: P^– = {0,1,2} and P⁺ = {3,4,5,...}; the set P^– comprising those natural numbers determinately failing to satisfy the predicate “greater than two” and the set P⁺ comprising those natural numbers that determinately satisfy it.

The sense of vagueness we shall be working with, then, can already be distinguished from another sense in which language is often said to be vague – vague in the sense of inexact, unspecific or general. Consider, for example, the claim that there are between two hundred and one thousand species of Eucalyptus trees. It might be responded that this claim is ‘vague’ and one could be a lot more ‘precise’. However, it is easy to see that vagueness in this sense is quite different from vagueness as I have described it above. A thoroughgoing discussion is to be found in Chapter 2, but the distinction can be made with enough intuitive force for present purposes as follows. Being between 1.01 and 3.24 metres is an inexact description of someone’s height but it is not vague in the sense of there being indeterminate limits to its application – it will be true if their height lies between these two figures and false otherwise. I can make a much more exact estimation of their height which nonetheless is more vague, e.g. approximately 2 metres. Increasing exactness is consistent with a decrease in precision whilst a decrease in exactness is consistent with an increase in precision. It might be thought that vagueness always involves some inexactness (as Russell seems to have thought) but the terms are nonetheless distinct. When speaking of vagueness henceforth we will be concerned with the sense outlined above, reserving the more explicit terms like “inexactness” to describe this other sense.1

The symptom of vagueness alluded to above, our inability to draw a sharp line between those things in the predicate’s positive extension and those in its negative extension, is tantamount to there being borderline or penumbral cases for the predicate in question – cases which jointly constitute the borderline region or penumbra for the vague predicate. Intuitively, such cases are those where there are objects to which the predicate meaningfully applies (i.e. objects in the predicate’s domain of significance) yet for which it appears essentially indeterminate whether the predicate or its negation truthfully applies. That is to say, there are situations where a language user, having carried out all the empirical and conceptual research possible concerning the case at hand, will nonetheless still be unable either to apply the predicate determinately to an object to which the predicate may be said to apply meaningfully or to apply its negation determinately. This apparent indeterminacy or indefiniteness, taken as the sine qua non of this, and most other, discussions of vagueness, is not due to the lack of knowledge of facts or of meanings that one could in principle come to know – hence the use of “essential” above.

Notice that the characterization of borderline cases is given in terms of an agent’s ability to apply predicates, rather than in terms of the semantic properties of the predicate. In this way we avoid the charge, properly levelled at many discussions of vagueness in the twentieth century, of invoking a theoretically laden definition of vagueness which, from the outset, foreclosed on that response to vagueness which claims it to be an epistemic phenomenon (though, as we shall see, it is not an option that will be pursued in what follows).

Notice also that though we commonly speak of a predicate’s vagueness in terms of there (actually) being borderline cases for the predicate, there is a weaker sense of vagueness that does not depend on mere contingencies regarding what actually exists – namely, the very possibility of there being borderline cases. In this weak sense a predicate does not cease to be vague simply because its borderline cases cease to exist; the logical possibility of their existence is enough to guarantee the conceivability of borderline cases and it is this, rather than the actual existence of borderline cases, that is of logical interest. This weak sense we shall describe as intensional vagueness2 and depends only on the conceptual possibility of borderline cases as opposed to the stronger extensional vagueness which requires the actual existence of a borderline case. Analogously, though extensional precision – the actual absence of borderline cases – may sometimes be of interest (e.g. in assessing claims to actual truth), we shall generally be more interested in the logically necessary absence of borderline cases, necessary precision. With this in mind we shall often speak of vagueness and precision simpliciter, where these are to be understood respectively as intensional vagueness and necessary precision.

In subsequent chapters we shall enquire further into the cause or source of the essential indeterminacy underpinning vagueness. With the ensuing discussion already too long, I shall set aside an epistemic analysis of the indeterminacy involved, and consider vagueness as a semantic feature of language. We will thus be concerned to properly analyse this semantic indeterminacy and enquire into the possibility as to whether this vagueness could somehow be ‘in the world’. The exact causes of these borderline cases – the reason for the failure of any enquiry to resolve such cases one way or the other – is unquestionably the major watershed amongst analytic philosophers writing on the topic of vagueness; it is therefore not something that can be settled at this preliminary stage. For the moment we need to focus more clearly on its content. Cashing out our intuitions regarding borderline cases even further, we must distinguish borderline cases from cases where one is at a loss as to what to say due to meaninglessness, ambiguity or context-sensitivity.

In the first instance, vagueness is to be contrasted with meaninglessness. My inability to decide whether the creature now before me is one of Lewis Carroll’s ‘snarks’ derives from the meaninglessness of the term “snark”. To be sure, it is, in a sense, indeterminate whether the creature I see is a snark, but this indeterminacy is not to be confused with that arising from vagueness. Meaningful terms can be vague, as evidenced by the examples given earlier. Nor should vagueness be confused with ambiguity. Consider the ambiguous word “bank” – meaning (amongst other things) a financial institution or a river’s edge. It may be essentially indeterminate whether or not to agree to the claim “Jo went to the bank yesterday” since it may be used to assert more than one proposition. An audience may be unable to determinately respond one way or the other since they may firstly need to determine what proposition is being asserted. This contrasts with borderline cases in the relevant sense. We may be unable to say whether or not Jo, having a moderate income and substantial assets, is wealthy, but this is not due to our being unable to determine what is meant by “wealthy” – that is presumably understood. The term “wealthy” may be taken to have an unambiguous, vague meaning as opposed to having two or more distinct meanings. Our inability to say whether or not Jo is wealthy arises as a result of her being a borderline case of “wealthy”. Borderline cases can arise in our use of unambiguous words and ambiguous words need have no borderline cases.

Context-sensitivity is similarly distinct in so far as context-sensitive words might be vague, even if a context is fixed, and precise words can be context-sensitive. Consider the word “tall”, for example. What counts as tall can vary from context to context; tall pygmies are usually not tall basketball players. Yet even when this relativity to context is fixed, e.g. tall relative to basketball players, there might still be borderline cases – is he a tall basketball player or not? There may also be situations where one is unable to say whether or not the word applies simply because the correct ascription depends on the context and this phenomenon occurs regardless of the vagueness of “tall”; that is, this inability does not constitute a borderline case. Suppose “tall” were to mean “above average in height”, which intuitively makes it a precise term; nonetheless there may be situations where I am uncertain whether to apply the term or not since it depends on whether the average is taken over pygmies or basketball players. To be sure, there are those (e.g. Diana Raffman) who present strong arguments for the role of context in any adequate analysis of vagueness; nonetheless, though said to be related, the two notions are distinct.

Intuitively, then, vagueness is to be distinguished from meaninglessness, ambiguity and context-sensitivity. Thus vagueness is introduced as a phenomenon affecting predicates and, as such, is characterized by the presence of borderline cases as they have been described above. This initial focus on predicate-vagueness reflects one aspect of what we might describe as the paradigmatic concept of vagueness: vagueness as applied to predicates and characterized by the presence of borderline cases. This is the concept of vagueness encountered in the work of early twentieth-century theorists such as Peirce (1902), Russell (1923), Black (1960), Church (1960), Quine (1960) and Alston (1964).

It is not surprising that discussions of vagueness have tended to focus on the concept of vagueness as applied to predicates since this is where issues surrounding vagueness originate historically. Problems with vagueness arose in antiquity in the context of the sorites puzzles, puzzles understood as exploiting the apparent lack of sharp boundaries for certain predicates. In subsequent sections we shall consider the sorites puzzles in detail (§2), look at various types of predicate-vagueness (§3), and see how one might extend the concept of vagueness beyond predicates to other semantic categories (§4), but for the moment we need to consider more carefully that other aspect of the paradigmatic concept of vagueness – the relationship between vague predicates and their borderline cases.

A predicate having no borderline cases is precise; borderline cases are therefore necessary for vagueness, as already remarked. However, are they sufficient? Is a predicate’s vagueness really characterized or defined by its having borderline cases? According to a naive paradigmatic concept of vagueness we might suppose it is – a predicate is vague if and only if it has borderline cases. On this understanding of vagueness, the presence of borderline cases is both necessary and sufficient. However, even having restricted the notion of a borderline case so as to exclude indeterminacy due to meaninglessness, ambiguity, and context-specificity, meeting the criteria for having such borderline cases still might seem insufficient for vagueness. Acknowledgement of what has become known as the phenomenon of higher-order vagueness results in further qualification of the paradigmatic concept of vagueness, expanding on the notion of a borderline case.

We can illustrate the problem using a familiar example from Sainsbury (1991) – the predicate “child*”. It is to count as true of all those people who have not yet reached their sixteenth birthday, false of all those who have reached their eighteenth birthday, and neither true nor false of all other people. Now, for a seventeen-year-old it is neither determinately true that they are a child* nor is it determinately false that they are a child* – they do not determinately satisfy the predicate nor do they determinately satisfy its negation. So it would seem that a seventeen-year-old counts as a borderline case for the predicate “child*” thereby making it vague, even though, intuitively, the predicate is perfectly precise. It is indeed the case that the predicate fails to draw a single sharp boundary between its positive and negative e...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Dedication
• Preface
• 1 Vagueness
• 2 Russell’s Representational Theory
• 3 Descriptive Representationalism
• 4 Going Non-classical: Gaps and Gluts
• 5 Ontological Vagueness
• 6 Vague Individuation and Counting
• 7 The Logic of Vagueness
• Bibliography
• Index