Online research in philosophy

Vagueness is a familiar but deeply puzzling aspect of the relation between language and the world. It is highly controversial what the nature of vagueness is -- a feature of the way we represent reality in language, or rather a feature of reality itself? May even relations like identity or parthood be affected by vagueness? Sorites arguments suggest that vague terms are either inconsistent or have a sharp boundary. The account we give of such paradoxes plays a pivotal role for our understanding of natural languages. If our reasoning involves any vague concepts, is it safe from contradiction? Do vague concepts really lack any sharp boundary? If not, why are we reluctant to accept the existence of any sharp boundary for them? And what rules of inference can we validly apply, if we reason in vague terms? Cuts and Clouds presents the latest work towards a clearer understanding of these old puzzles about the nature and logic of vagueness. The collection offers a stimulating series of original essays on these and related issues by some of the world's leading experts.

A number of arguments purport to show that vague properties determine sharp boundaries at higher orders. That is, although we may countenance vagueness concerning the location of boundaries for vague predicates, every predicate can instead be associated with precise knowable cut-off points deriving from precision in their higher order boundaries.

I argue that this conclusion is indeed paradoxical, and identify the assumption responsible for the paradox as the Brouwerian principle B for vagueness: that if p then it's completely determinate that either it's vague whether p, or p. Other paradoxes which do not appear to turn on B turn instead on some subtle issues concerning the relation between assertion, belief and higher order vagueness.

In this paper a B-free picture of assertion, knowledge and logic is outlined which is completely free of higher order precision. A class of realistic models containing counterexamples to B and a number of weakenings of B are introduced and its logic is shown to support vagueness at every order. A novel framework for thinking about the semantic apparatus in the presence of metalinguistic vagueness is also developed. In this framework the vague metatheoretic vocabulary is part of the object language and can readily be applied to itself.

We all know that much in our thought and language, as well as much in the law, is vague. We are also reasonably good at recognizing cases of vagueness, even though most of us would be hard pressed to say exactly what vagueness is. In recent decades, there has been a flowering of work in the philosophy of logic and language attempting to do just that. Much of this work focuses on what it is for a word or phrase to be vague. The aim of this effort is to clarify what it is for a claim, question, command, or promise expressed using such a term to be vague, as well as what it is to reason with such terms. Different logico-linguistic theories have different conceptions of the scope of putative laws of classical logic, including bivalence (which states that every declarative sentence or proposition is either true or false) and excluded middle (which asserts all instances of A or ~A). In addition to this work in philosophical logic, recent decades have seen a growing interest in vagueness among legal scholars and philosophers of law. Here the focus is not so much on what legal vagueness is, which is generally assumed to be readily recognizable. Rather, it is on the extent and sources of vagueness in the law, the implications of vagueness for interpretation and adjudication, the systemic effects of vagueness, and the function – i.e. important positive value – of vagueness in certain areas of the law, as opposed to its disutility in others.1 To date, these two investigations of vagueness – in philosophical logic and the philosophy of law – have been largely independent of one another. This independence gives rise to a natural line of questioning. Can work in one domain contribute to work in..

Touching your mother's foot is incest because all the rest is a matter of degree (or so said Diogenes). That's just one expression of the puzzle of vagueness. Here's another: the passage of one second cannot mark the transition from being a pupa to being a butterfly--if something is a pupa at one time then in all close instants it remains a pupa; alas, it follows from this, via trivial logic, that there are no butterflies. Or again: it's vague where the Highlands of Scotland begin and end, so, a small step in the direction of London cannot mark the boundary between the Lowlands and the Highlands. But then it follows, via trivial logic, that one is unable to leave the Highlands (even when in London). What's driving these paradoxical arguments seems to be the very vagueness of the terms involved: such terms as 'incest', 'butterfly', 'pupa', 'The Highlands' are all vague and such vagueness seems to make them tolerant to marginal change. The puzzles of vagueness are not only deep (in that they admit of no uncontroversial and entirely satisfactory solution), they are also broad, for vague language is everywhere. In this course, you will be introduced to the various puzzles of vagueness and whether and how we might best address them. We will tackle such as questions as: Does the possibility of vagueness entail that there simply cannot be a logic of natural language? Does it entail that language is governed by inconsistent rules? Or does vagueness require some special or deviant logic? Is vagueness a special species of ignorance? Is the world, in some sense, vague? Is there an uncontroversial definition of vagueness or can we only isolate the phenomenon from within some substantive and controversial conception? What is higher-order vagueness and why is it considered to be such a puzzling phenomenon? Must the truth about vagueness be so strange? In what exact way are vague expressions tolerant?

Vagueness provides the first comprehensive examination of a topic of increasing importance in metaphysics and the philosophy of logic and language. Timothy Williamson traces the history of this philosophical problem from discussions of the heap paradox in classical Greece to modern formal approaches such as fuzzy logic. He illustrates the problems with views which have taken the position that standard logic and formal semantics do not apply to vague language, and defends the controversial realistic view that vagueness is a kind of ignorance--that there really is a grain of sand whose removal turns a heap into a non-heap, but we cannot know which one it is.

The two standard theories of vagueness—vagueness-as-ignorance and vagueness-asindeterminacy—agree on the following principle: if you are certain that it is clearly vague whether p, then you clearly should not believe p and you clearly should not believe not-p. I argue against the principle, and thus against the two standard theories. I offer an explanation of the initial appeal of the principle. And I show how a rival principle helps to better explain a recalcitrant trio of widely accepted data.

Higher-order vagueness is widely thought to be a feature of vague predicates that any adequate theory of vagueness must accommodate. It takes a variety of forms. Perhaps the most familiar is the supposed existence, or at least possibility, of higher-order borderline cases—borderline borderline cases, borderline borderline borderline cases, and so forth. A second form of higherorder vagueness, what I will call ‘prescriptive’ higher-order vagueness, is thought to characterize complex predicates constructed from vague predicates by attaching operators having to do with the predicates’ proper application. For example, the predicates ‘mandates application of “old”’ and ‘can competently be called “old”’ are prescriptively higher-order vague. Higher-order vagueness appears in other guises as well,1 but these two have been of particular interest to philosophers and will be my target here. I want to expose some misconceptions about them. If I am right, higher-order vagueness is less prevalent, and less important theoretically, than is usually supposed.2 In what follows I am going to assume that vagueness is a semantic feature of natural language. For the most part I won’t discuss epistemic or pragmatic views, and I will say nothing about so-called metaphysical vagueness.

A critical survey of the main theories about vagueness, organized in four main sections: (i) What is vagueness? (ii) Problems and paradoxes; (iii) Theories of vagueness; (iv) Vagueness and cognitive science.

Degree theories of vagueness build on the observation that vague predicates such as 'tall' and 'red' come in degrees. They employ an infinite-valued logic, where the truth values correspond to degrees of truth and are typically represented by the real numbers in the interval [0,1]. In this paper, the success with which the numerical assignments of such theories can capture the phenomenon of vagueness is assessed by drawing an analogy with the measurement of various physical quantities using real numbers. I argue that degree theories of vagueness are undermined by the failure of the necessary connectedness principle. Moreover, the semantics for the connectives entail that there must be a uniquely correct numerical assignment for the sentences, and this is implausible. Different senses of 'coming in degrees' are then distinguished; I argue that a confusion between them could be the source of the degree theorist's error, and the distinction illuminates the problem cases described earlier in the paper.

Vagueness is an extremely widespread feature of language, famously associated with the sorites paradox. One instance of this paradox concludes that a single grain of sand is a heap of sand, by starting with a large heap of sand and invoking the plausible premise that if you take one grain of sand away from a heap of sand, then you still have a heap. The supervaluationist theory of vagueness states that a sentence is true if and only if it is true on all ways of making it precise. This yields borderline case predications that are neither true nor false, but yet classical logic is preserved almost entirely. The sorites paradox is solved because the main premise comes out false – on each way of making 'heap' precise, there is some first grain that turns a heap into a non-heap – but there is no sharp boundary to 'heap' because it is a different grain on different ways of making 'heap' precise; so, there is no grain of which it is true to say it is that first grain. The theory has a range of merits in comparison with rival theories, such as the epistemic view or degree theories of vagueness. Objections have been made (and answers offered) in relation to its treatment of higher-order vagueness and what it says about truth and validity. Author Recommends Fine, Kit. 'Vagueness, Truth and Logic.' Synthese 30 (1975). 265–300. Reprinted in Keefe and Smith 1997. This is the classic text introducing supervaluationism as a treatment of vagueness. It provides both philosophical discussion and formal modelling, demonstrating the adherence to classical logic that the theory yields. Keefe, Rosanna and Peter Smith, eds. Vagueness: A Reader . Cambridge, MA: MIT Press. This collection includes many classic papers on vagueness, including Fine's paper, cited above, a paper by Dummett that offers (but rejects) a precursor of the supervaluationist view, another less well-known early presentation of the view by Henyrk Mehlberg and discussion and defences of the main rival theories of vagueness. Keefe, Rosanna. Theories of Vagueness . Cambridge: Cambridge University Press, 2000. This book defends a supervaluationist theory of vagueness. It discusses the phenomena of vagueness and what is required of a theory of vagueness, before considering and rejecting the major alternatives in turn. Williamson, Timothy. Vagueness . London: Routledge, 1994. This book defends the epistemic theory of vagueness, which maintains that vague predicates do have sharp boundaries, we just do not know where those boundaries lie. It also contains detailed discussions of opposing theories, including supervaluationism. Unger, Peter. 'The Problem of the Many.' Midwest Studies in Philosophy 5 (1980). Eds. P.A. French, T.E. Uehling Jr and H.K. Wettstein. Minneapolis, MN: University of Minnesota Press. This is the classic presentation of the Problem of the Many, to which a supervaluationist solution is relatively popular. This problem arises because frequen