Epistemicism,Distribution, and the Argument from Vagueness

This paper consists of two parts. The first concerns the logic of vagueness. The second concerns a prominent debate in metaphysics. One of the most widely accepted principles governing the ‘definitely’ operator is the principle of Distribution: if ‘p’ and ‘if p then q’ are both definite, then so is ‘q’. I argue however, that epistemicists about vagueness (at least those who take a broadly Williamsonian line) should reject this principle. The discussion also helps to shed light on the elusive question of what, on this framework, it takes for a sentence to be borderline or definite. In the second part of the paper, I apply this result to a prominent debate in metaphysics. One of the most influential arguments in favour of Universalism about composition is the Lewis‐Sider argument from vagueness. An interesting question, however, is whether epistemicists have any particular reasons to resist the argument. I show that there is no obvious reason why epistemicists should resist the argument but there is a non‐obvious one: the rejection of Distribution argued for in the first part of the paper provides epistemicists with a unique way of resisting the argument from vagueness.     

Ontic Vagueness: A Guide for the Perplexed 1

In this paper I develop a framework for understanding ontic vagueness. The project of the paper is two‐fold. I first outline a definitional account of ontic vagueness – one that I think is an improvement on previous attempts because it remains neutral on other, independent metaphysical issues. I then develop one potential manifestation of that basic definitional structure. This is a more robust (and much less neutral) account which gives a fully classical explication of ontic vagueness via modal concepts. The overarching aim is to systematically investigate the puzzling question of what exactly it could be for the world itself to be vague.     

Vagueness,tolerance and contextual logic

The goal of this paper is a comprehensive analysis of basic reasoning patterns that are characteristic of vague predicates. The analysis leads to rigorous reconstructions of the phenomena within formal systems. Two basic features are dealt with. One is tolerance: the insensitivity of predicates to small changes in the objects of predication (a one-increment of a walking distance is a walking distance). The other is the existence of borderline cases. The paper shows why these should be treated as different, though related phenomena. Tolerance is formally reconstructed within a proposed framework of contextual logic, leading to a solution of the Sorites paradox. Borderline-vagueness is reconstructed using certain modality operators; the set-up provides an analysis of higher order vagueness and a derivation of scales of degrees for the property in question.     

Higher-Order Sorites Paradox

The naive theory of vagueness holds that the vagueness of an expression consists in its failure to draw a sharp boundary between positive and negative cases. The naive theory is contrasted with the nowadays dominant approach to vagueness, holding that the vagueness of an expression consists in its presenting borderline cases of application. The two approaches are briefly compared in their respective explanations of a paramount phenomenon of vagueness: our ignorance of any sharp boundary between positive and negative cases. These explanations clearly do not provide any ground for choosing the dominant approach against the naive theory. The decisive advantage of the former over the latter is rather supposed to consist in its immunity to any form of sorites paradox. But another paramount phenomenon of vagueness is higher-order vagueness: the expressions (such as ‘borderline’ and ‘definitely’) introduced in order to express in the object language the vagueness of the object language are themselves vague. Two highly plausible claims about higher-order vagueness are articulated and defended: the existence of “definitely ^ω ” positive and negative cases and the “radical” character of higher-order vagueness itself. Using very weak logical principles concerning vague expressions and the ‘definitely’-operator, it is then shown that, in the presence of higher-order vagueness as just described, the dominant approach is subject to higher-order sorites paradoxes analogous to the original ones besetting the naive theory, and therefore that, against the communis opinio, it does not fare substantially better with respect to immunity to any form of sorites paradox.     

A Model of Tolerance

According to the naive theory of vagueness, the vagueness of an expression consists in the existence of both positive and negative cases of application of the expression and in the non-existence of a sharp cut-off point between them. The sorites paradox shows the naive theory to be inconsistent in most logics proposed for a vague language. The paper explores the prospects of saving the naive theory by revising the logic in a novel way, placing principled restrictions on the transitivity of the consequence relation. A lattice-theoretical framework for a whole family of (zeroth-order) “tolerant logics” is proposed and developed. Particular care is devoted to the relation between the salient features of the formal apparatus and the informal logical and semantic notions they are supposed to model. A suitable non-transitive counterpart to classical logic is defined. Some of its properties are studied, and it is eventually shown how an appropriate regimentation of the naive theory of vagueness is consistent in such a logic.     

Vagueness,Kant and Topology: a Study of Formal Epistemology

In this paper we propose an approach to vagueness characterised by two features. The first one is philosophical: we move along a Kantian path emphasizing the knowing subject’s conceptual apparatus. The second one is formal: to face vagueness, and our philosophical view on it, we propose to use topology and formal topology. We show that the Kantian and the topological features joined together allow us an atypical, but promising, way of considering vagueness.     

‘Incommensurability’ and Vagueness: Is the Vagueness View Defensible?

The vagueness view holds that when evaluative comparisons are hard, there is indeterminacy about which comparative relation holds. It is sceptical about whether there are any incommensurate items (in some domain). The sceptical element of John Broome’s version of this view rests on a controversial principle. Robert Sugden advances a similar view which does not depend on this principle. Sugden’s argument fails as a vagueness view because it assumes rather than shows that there are no incommensurate items (in some domain). Nonetheless, I argue that an interpretation of his argument constitutes a defensible vagueness view which is supported by intuition about examples. On this interpretation Sugden’s view can be mapped onto Broome’s application of a supervaluationist view of vagueness. It is also (on this reading) a close relative of James Griffin’s ‘rough equality’ view when this is interpreted in terms of vagueness. On an alternative interpretation Sugden’s view is not sceptical about incommensurateness. On this interpretation he does not defend a vagueness view and is ‘sceptical’ about the contribution of the philosophical literature to our understanding of rational choice.     

I—Stewart Shapiro

ABSTRACT After a brief account of the problem of higher-order vagueness, and its seeming intractability, I explore what comes of the issue on a linguistic, contextualist account of vagueness. On the view in question, predicates like'borderline red'and'determinately red'are, or at least can be, vague, but they are different in kind from'red'. In particular,'borderline red'and'determinately red'are not colours. These predicates have linguistic components, and invoke notions like'competent user of the language'. On my view, so-called'higher-order vagueness'is actually ordinary, first-order vagueness in different predicates. I explore the possibility that, nevertheless, a pernicious regress ensues.     

Vagueness,uncertainty and degrees of clarity

In this paper we compare different models of vagueness viewed as a specific form of subjective uncertainty in situations of imperfect discrimination. Our focus is on the logic of the operator “clearly” and on the problem of higher-order vagueness. We first examine the consequences of the notion of intransitivity of indiscriminability for higher-order vagueness, and compare several accounts of vagueness as inexact or imprecise knowledge, namely Williamson’s margin for error semantics, Halpern’s two-dimensional semantics, and the system we call Centered semantics. We then propose a semantics of degrees of clarity, inspired from the signal detection theory model, and outline a view of higher-order vagueness in which the notions of subjective clarity and unclarity are handled asymmetrically at higher orders, namely such that the clarity of clarity is compatible with the unclarity of unclarity.     

The Mere Addition Paradox,Parity and Vagueness*

Derek Parfit’s mere addition paradox has generated a large literature. This paper articulates one response to this paradox—which Parfit himself suggested—in terms of a formal account of the relation of parity. I term this response the ‘parity view’. It is consistent with transitivity of ‘at least as good as’, but implies incompleteness of this relation. The parity view is compatible with critical‐band utilitarianism if this is adjusted to allow for vagueness. John Broome argues against accounts which involve incompleteness. He thinks they are based on an intuition of ‘neutrality’, which is most naturally understood in terms of equality. There is no rationale, on Broome’s view, for seeing it as ‘incommensurateness’ which leads to incompleteness. Parity provides one. Broome’s worries that ‘incommensurateness’ makes neutrality implausibly ‘greedy’, and that ‘incommensurateness’ and vagueness are incompatible do not constitute a knock‐down case against the parity view. Similar worries arise for his preferred vagueness view.     

Multivalent Semantics for Vagueness and Presupposition

Both the phenomenon of presupposition and that of vagueness have motivated the use of one form or another of trivalent logic, in which a declarative sentence can not only receive the standard values true (1) and false (0), but also a third, non-standard truth-value which is usually understood as ‘undefined’ (#). The goal of this paper is to propose a multivalent framework which can deal simultaneously with presupposition and vagueness, and, more specifically, capture their projection properties as well as their different roles in language. Now, there is a prima facie simple way of doing this, which simply consists in assimilating the two phenomena, and using an appropriate type of trivalent logic. On this view, we just need a compositional system that deals with the ‘undefined’ truth-value, and does not care about whether the source of undefinedness is ‘presuppositional’ or related to vagueness. I will argue that such a simple solution cannot succeed, and point out a number of desiderata that any successful approach must meet. I will then present and discuss two seven-valued semantics, inspired, respectively, by the Strong Kleene semantics and by supervaluationism, which meet these desiderata.     

Universalism,Vagueness and Supersubstantivalism

Sider has a favourable view of supersubstantivalism (the thesis that all material objects are identical to the regions of spacetime that they occupy). This paper argues that given supersubstantivalism, Sider's argument from vagueness for (mereological) universalism fails. I present Sider's vagueness argument (§§II–III), and explain why – given supersubstantivalism – some but not all regions must be concrete in order for the argument to work (§IV). Given this restriction on what regions can be concrete, I give a reductio of Sider's argument (§V). I conclude with some brief comments on why this is not simply an ad hominem against Sider, and why this incompatibility of supersubstantivalism with the argument from vagueness is of broader interest (§VI).     

The Problem of Ethical Vagueness for Expressivism

Ethical vagueness has garnered little attention. This is rather surprising since many philosophers have remarked that the science of ethics lacks the precision that other fields of inquiry have. Of the few philosophers who have discussed ethical vagueness the majority have focused on the implications of vagueness for moral realism. Because the relevance of ethical vagueness for other metaethical positions has been underexplored, my aim in this paper is to investigate the ramifications of ethical vagueness for expressivism. Ultimately, I shall argue that expressivism does not have the resources to adequately account for ethical vagueness, while cognitivism does. This demonstrates an advantage that cognitivism holds over expressivism.     

A Contextualist Approach to Higher‐Order Vagueness

According to contextualism about vagueness, the content of a vague predicate is context sensitive. On this view, when item a is in the penumbra of the vague predicate ‘F’, speakers may (truly) utter ‘Fa’, or they may (truly) utter ‘not‐Fa’, without contravening the literal meaning of ‘F’. Unlike its more popular variants, the version of contextualism I defend rejects the principle of tolerance, a principle according to which small differences should not affect the applicability of a vague predicate. My goal is to show how such a rejection allows for a plausible treatment of higher‐order vagueness, and for a dissolution of paradoxes of higher‐order vagueness.     

Why believe what people say?

The basic alternatives seem to be either a Humean reductionist view that any particular assertion needs backing with inductive evidence for its reliability before it can retionally be believed, or a Reidian criterial view that testimony is intrinscially, though defeasibly, credible, in the absence of evidence against its reliability.Some recent arguments from the constraints on interpreting any linguistic performances as assertions with propositional content have some force against the reductionist view. We thus have reason to accept the criterial view, at least as applied to eyewitness reports. But these considerations do not establish that any rational enquirer must have the concept of other minds or testimony. The logical possibility of the lone enquirer, who uses symbols and thereby expresses some knowledge of his world, remains open — but it is a question we have no need to pronounce upon.The practice of accepting observation-statements is in fact extended to chains of testimonies believed to start in perception or in some other kind of justification, but the arguments for doing this are not so clear.I am indebted for helpful discussions of earlier versions of this paper to audiences at Macquarie University (where I was kindly given research facilities), at the conference of the Australasian Association for Philosophy in Sydney in 1990, at St. Andrews, and also to a referee for this journal.     

What makes it a heap?

On the epistemic view of vagueness, a vague expression has sharp boundaries whose location speakers of the language cannot recognize. The paper argues that one of the deepest sources of resistance to the epistemic view is the idea that all truths are cognitively accessible from truths in a language for natural science, conceived as precise, in a sense explained. The implications of the epistemic view for issues about the relations between vague predicates and scientific predicates are investigated.     

JUST WHAT IS VAGUENESS?

We argue that standard definitions of ‘vagueness’ prejudice the question of how best to deal with the phenomenon of vagueness. In particular, the usual understanding of ‘vagueness’ in terms of borderline cases, where the latter are thought of as truth‐value gaps, begs the question against the subvaluational approach. According to this latter approach, borderline cases are inconsistent (i.e., glutty not gappy). We suggest that a definition of ‘vagueness’ should be general enough to accommodate any genuine contender in the debate over how to best deal with the sorites paradox. Moreover, a definition of ‘vagueness’ must be able to accommodate the variety of forms sorites arguments can take. These include numerical, total‐ordered sorites arguments, discrete versions, continuous versions, as well as others without any obvious metric structure at all. After considering the shortcomings of various definitions of ‘vagueness’, we propose a very general non‐question‐begging definition.     

SQUEEZING AND STRETCHING: HOW VAGUENESS CAN OUTRUN BORDERLINENESS

The paper develops a critical dialectic with respect to the nowadays dominant approach in the theory of vagueness, an approach whose main tenet is that it is in the nature of the vagueness of an expression to present borderline cases of application, conceived of as enjoying some kind of distinctive normative status. Borderlineness is used to explain the basic phenomena of vagueness, such as, for example, our ignorance of the location of cut-offs in a soritical series. Every particular theory of vagueness exemplifying the approach makes use, in the vague object language, of a definiteness operator which, however substantially interpreted, unavoidably inherits the vagueness of the expressions on which it operates ('higher-order vagueness'). It is first argued that finite soritical series force a surprising collapse result concerning a particular set of expressions involving the definiteness operator. It is then shown that, under two highly plausible assumptions about higher-order vagueness (the existence of 'absolutely definitely' positive and negative cases and the 'radical' character of higher-order vagueness itself), the collapse result implies the inadequacy of the dominant approach as a theory of vagueness, as its main tenet can be, at best, not absolutely definitely true.     

Epistemicism and Nihilism about Vagueness: What’s the Difference?

In this paper I argue, first, that the only difference between Epistemicism and Nihilism about vagueness is semantic rather than ontological, and second, that once it is clear what the difference between these views is, Nihilism is a much more plausible view of vagueness than Epistemicism. Given the current popularity of certain epistemicist views (most notably, Williamson’s), this result is, I think, of interest.