Vagueness: Why Do We Believe in Tolerance?

The tolerance principle, the idea that vague predicates are insensitive to sufficiently small changes, remains the main bone of contention between theories of vagueness. In this paper I examine three sources behind our ordinary belief in the tolerance principle, to establish whether any of them might give us a good reason to revise classical logic. First, I compare our understanding of tolerance in the case of precise predicates and in the case of vague predicates. While tolerance in the case of precise predicates results from approximation, tolerance in the case of vague predicates appears to originate from two more specific sources: semantic indeterminacy on the one hand, and epistemic indiscriminability on the other. Both give us good and coherent grounds to revise classical logic. Epistemic indiscriminability, it is argued, may be more fundamental than semantic indeterminacy to justify the intuition that vague predicates are tolerant.     

Reasoning with Slippery Predicates

It is a commonplace that the extensions of most, perhaps all, vague predicates vary with such features as comparison class and paradigm and contrasting cases. My view proposes another, more pervasive contextual parameter. Vague predicates exhibit what I call open texture: in some circumstances, competent speakers can go either way in the borderline region. The shifting extension and anti-extensions of vague predicates are tracked by what David Lewis calls the “conversational score”, and are regulated by what Kit Fine calls penumbral connections, including a principle of tolerance. As I see it, vague predicates are response-dependent, or, better, judgement-dependent, at least in their borderline regions. This raises questions concerning how one reasons with such predicates. In this paper, I present a model theory for vague predicates, so construed. It is based on an overall supervaluationist-style framework, and it invokes analogues of Kripke structures for intuitionistic logic. I argue that the system captures, or at least nicely models, how one ought to reason with the shifting extensions (and anti-extensions) of vague predicates, as borderline cases are called and retracted in the course of a conversation. The model theory is illustrated with a forced march sorites series, and also with a thought experiment in which vague predicates interact with so-called future contingents. I show how to define various connectives and quantifiers in the language of the system, and how to express various penumbral connections and the principle of tolerance. The project fits into one of the topics of this special issue. In the course of reasoning, even with the external context held fixed, it is uncertain what the future extension of the vague predicates will be. Yet we still manage to reason with them. The system is based on that developed, more fully, in my Vagueness in Context, Oxford, Oxford University Press, 2006, but some criticisms and replies to critics are incorporated.     

Tolerant, Classical, Strict

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases.     

Vagueness as closeness

This paper presents and defends a definition of vagueness, compares it favourably with alternative definitions, and draws out some consequences of accepting this definition for the project of offering a substantive theory of vagueness. The definition is roughly this: a predicate ‘F’ is vague just in case for any objects a and b, if a and b are very close in respects relevant to the possession of F, then ‘Fa’ and ‘Fb’ are very close in respect of truth. The definition is extended to cover vagueness of many-place predicates, of properties and relations, and of objects. Some of the most important advantages of the definition are that it captures the intuitions which motivate the thought that vague predicates are tolerant, without leading to contradiction, and that it yields a clear understanding of the relationships between higher-order vagueness, sorites susceptibility, blurred boundaries, and borderline cases. The most notable consequence of the definition is that the correct theory of vagueness must countenance degrees of truth.     

SCIENCE AND SEMANTICS: THE CASE OF VAGUENESS AND SUPERVALUATION*

Abstract: It is widely assumed that the methods and results of science have no place among the data to which our semantics of vague predicates must answer. This despite the fact that it is well known that such prototypical vague predicates as ‘is bald’ play a central role in scientific research (e.g. the research that established Rogaine as a treatment for baldness). I argue here that the assumption is false and costly: in particular, I argue one cannot accept either supervaluationist semantics, or the criticism of that semantics offered by Fodor and Lepore, without having to abandon accepted, and unexceptionable, scientific methodology.     

Predication and the Problem of Universals

Abstract

This paper contrasts the scholastic realisms of David Armstrong and Charles Peirce. It is argued that the so-called ‘problem of universals’ is not a problem in pure ontology (concerning whether universals exist) as Armstrong construes it to be. Rather, it extends to issues concerning which predicates should be applied where, issues which Armstrong sets aside under the label of ‘semantics’, and which from a Peircean perspective encompass even the fundamentals of scientific methodology. It is argued that Peirce's scholastic realism not only presents a more nuanced ontology (distinguishing the existent from the real) but also provides more of a sense of why realism should be a position worth fighting for.     

God’s silence

Vagueness manifests itself (among other things) in our inability to find boundaries to the extension of vague predicates. A semantic theory of vagueness plans to justify this inability in terms of the vague semantic rules governing language and thought. According to a supporter of semantic theory, the inability to find such a boundary is not dependent on epistemic limits and an omniscient being like God would be equally unable. Williamson (Vagueness, 1994) argued that cooperative omniscient beings adequately instructed would find a precise boundary in a sorites series and that, for this reason, the semantic theory misses its target, while Hawthorne (Philosophical Studies 122:1–25, 2005) stood with the semantic theorists and argued that the linguistic behaviour of a cooperative omniscient being like God would clearly demonstrate that he does not find a precise boundary in the sorites series. I argue that Hawthorne’s definition of God’s cooperative behaviour cannot be accepted and that, contrary to what has been assumed by both Williamson and Hawthorne, an omniscient being like God cannot be a cooperative evaluator of a semantic theory of vagueness.     

Intransitivity and Vague Preferences

This paper is concerned with intransitivity in normative rational choice. It focuses on a class of intransitivities which have received little attention, those involving vague preferences. “Vague preferences” are defined in terms of vague predicates such as “red” or “bald.” Such preferences appear common, and intransitive indifference is argued to be an unavoidable feature of them. Such preferences are argued to undermine intransitive strict preference also. Various formal theories of vagueness are applied to an example of vague preferences, but none of them provide a justification for the transitivity axiom. For helpful comments on earlier drafts and discussions of this material, I am very grateful to John Broome, Rosanna Keefe, Matthew Kramer, Stuart Rachels, Jochen Runde and Adrian Walsh. I owe a large debt to John O’Neill, who suggested an earlier version of the “Tragedy of the Disappearing Commons” example.     

Almost‐Ontology: Why Epistemicism Cannot Help Us Avoid Unrestricted Composition or Diachronic Plenitude

That any filled location of spacetime contains a persisting thing has been defended based on the ‘argument from vagueness.’ It is often assumed that since the epistemicist account of vagueness blocks the argument from vagueness it facilitates a conservative ontology without gerrymandered objects. It doesn't. The epistemic vagueness of ordinary object predicates such as ‘bicycle’ requires that objects that can be described as almost‐but‐not‐quite‐bicycle exist even though they fall outside the predicate's sharp extension. Since the predicates that begin with ‘almost’ are vague as well, epistemicism's ontological backdrop is far from the conservative picture it is thought to enable.     

The over-generalization problem: predicates rigidly signifying the “unnatural”

According to the simple proposal, a predicate is rigid iff it signifies the same property across the different possible worlds. The simple proposal has been claimed to suffer from an over-generalization problem. Assume that one can make sense of predicates signifying properties, and assume that trivialization concerns, to the effect that the notion would cover any predicate whatsoever, can be overcome. Still, the proposal would over-generalize, the worry has it, by covering predicates for artifactual, social, or evaluative properties, such as ‘is a knife,’ ‘is a bachelor,’ or ‘is funny.’ In defense, it is argued that rigidity for predicates as characterized plays the appropriate theoretical role, and that the contention that “unnatural” properties are not to be rigidly signified is ungrounded.     

Rationalism vs. Sentimentalism: Reviewing Price's Review

Abstract

It is sometimes argued that conceptualism cannot explain (dis)agreements concerning matters of personal taste because it treats sentences involving predicates of taste as indexical. I aim to weaken this charge. Given the idea that people sometimes use indexical sentences to express (dis)agreements about taste, two kinds of (dis)agreement are distinguished, namely doxastic and non-doxastic. Taste (dis)agreements are better explained in terms of the later kind, in which case they become amenable to contextualist treatment. It is argued that if something instantiates a taste property (like being tasty for A), it has to instantiate a corresponding attitudinal property (like being liked by A). Based on this, utterances of taste sentences express propositions that concern tastiness of something (e.g., that X is tasty for A) and these propositions entail other propositions that concern non-doxastic attitudes the speakers bear toward something (e.g., that X is liked by A). One speaker is claimed to (dis)agree with another speaker provided their respective entailed propositions feature (in)compatible non-doxastic attitudes. Although this explanation is similar to hybrid accounts that are currently growing in popularity, it departs from them in some notable respects.     

Many Many Problems

Supervaluationist solutions to 'the problem of the many' typically rely on two principles. First, the root of the problem is that singular terms can be vague, just as predicates can be. Secondly, the same resolution as the supervaluationist suggests for puzzles with vague predicates will resolve puzzles concerning vague singular terms. In recent years this second principle has been attacked from a number of fronts: it has been claimed that supervaluationist accounts of vague singular terms cannot explain the role of vague singular terms in propositional attitude reports, cannot explain penumbral connections between distinct singular terms, and cannot allow that vague singular terms are directly referential. While some of these arguments are successful against extant supervaluationist theories, I offer some natural modifications to supervaluationism which avoid the challenges.     

Propositions,Properties and Relations: Wittgenstein's “Notes on Logic” and the Tractatus

Frege famously argued that truth is not a property or relation. In the “Notes on Logic” Wittgenstein emphasised the bi‐polarity of propositions which he called their sense. He argued that “propositions by virtue of sense cannot have predicates or relations.” This led to his fundamental thought that the logical constants do not represent predicates or relations. The idea, however, has wider ramifications than that. It is not just that propositions cannot have relations to other propositions but also that they cannot have relations to anything at all. The paper explores the consequences of this insight for the way in which we should read the Tractatus. In the “Notes on Logic” the insight led to Wittgenstein's emphasis on “facts” in any attempt to understand the nature of symbolism. This emphasis is continued in the Tractatus. It is central to his view that propositions are facts which picture facts which prevent us from construing such picturing as a relation between what pictures and what is pictured. It illuminates the importance of context principle with regard to the distinction between showing and saying to which Wittgenstein attached so much importance and it underlies the non‐relational view of psychological propositions which he advocates. Finally, if propositions by virtue of sense cannot have predicates or relations the paradox at the end of a work which consist largely of propositions about propositions becomes intelligible.     

Acceptable Contradictions: Pragmatics or Semantics? A Reply to Cobreros et al.

Naive speakers find some logical contradictions acceptable, specifically borderline contradictions involving vague predicates such as Joe is and isn’t tall. In a recent paper, Cobreros et al. (J Philos Logic, 2012) suggest a pragmatic account of the acceptability of borderline contradictions. We show, however, that the pragmatic account predicts the wrong truth conditions for some examples with disjunction. As a remedy, we propose a semantic analysis instead. The analysis is close to a variant of fuzzy logic, but conjunction and disjunction are interpreted as intensional operators.     

Worldly Indeterminacy: A Rough Guide

This paper defends the idea that there might be vagueness or indeterminacy in the world itself--as opposed to merely in our representations of the world--against the charges of incoherence and unintelligibility. First we consider the idea that the world might contain vague properties and relations ; we show that this idea is already implied by certain well-understood views concerning the semantics of vague predicates (most notably the fuzzy view). Next we consider the idea that the world might contain vague objects ; we argue that an object is indeterminate in a certain respect (colour, size, etc.) just in case it is a borderline case of a maximally specific colour (size, etc.) property. Finally we consider the idea that the world as a whole might be indeterminate; we argue that the world is indeterminate just in case it lacks a determinate division into determinate objects.     

Vagueness and the Metaphysics of Consciousness

An argument is offered for this conditional: If our current concept conscious state is sharp rather than vague, and also correct (at least in respect of its sharpness), then common versions of familiar metaphysical theories of consciousness are false – namely versions of the identity theory, functionalism, and dualism that appeal to complex physical or functional properties in identification, realization, or correlation. Reasons are also given for taking seriously the claim that our current concept conscious state is sharp. The paper ends by surveying the theoretical options left open by the concept’s sharpness and the truth of the conditional argued for in the paper.     

A modal sequent calculus for a fragment of arithmetic

Global properties of canonical derivability predicates (the standard example is Pr() in Peano Arithmetic) are studied here by means of a suitable propositional modal logic GL. A whole book [1] has appeared on GL and we refer to it for more information and a bibliography on GL. Here we propose a sequent calculus for GL and, by exhibiting a good proof procedure, prove that such calculus admits the elimination of cuts. Most of standard results on GL are then easy consequences: completeness, decidability, finite model property, interpolation and the fixed point theorem.The second author holds a grant from the Consiglio Nazionale delle Ricerche, gruppo G.N.S.A.G.A.     

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.     

“Age-associated memory impairment”: A critique

Abstract

This paper is a critical evaluation of the construct of age-associated memory impairment (AAMI) and the criteria for assigning the AAMI label. It is argued that AAMI should not be regarded as a clinical diagnosis; that the AAMI criteria fail to isolate selective memory impairments, ignore crucial longitudinal information, and are vague, arbitrary, and unreliable; that studies of memory loss should examine a wider range of subjects than those who meet the AAMI criteria-moreover, use of the criteria may impede research; and that treatment studies should await investigations of the biological antecedents of memory loss. Several recommendations address subject selection and assessment issues.     

Vagueness and revision sequences

Theories of truth and vagueness are closely connected; in this article, I draw another connection between these areas of research. Gupta and Belnap’s Revision Theory of Truth is converted into an approach to vagueness. I show how revision sequences from a general theory of definitions can be used to understand the nature of vague predicates. The revision sequences show how the meaning of vague predicates are interconnected with each other. The approach is contrasted with the similar supervaluationist approach.