5
Vagueness by numbers

This chapter has two main aims. As promised at the end of chapter 4, I shall argue against a specific response to the degree theorist's problems concerning the assignment of exact values (see §3 below). The other more general aim is further to undermine the prospects for degree theories of vagueness, which rely so heavily on numerical resources. In §1 I outline the widely accepted representational approach to measurement, showing how numbers can be used to capture a physical attribute (see e.g. Krantz et al. 1971). This will be used to compare measurement theory and degree theories of vagueness, and through that comparison I shall expose severe flaws in the degree theorist's approach.


1. MEASUREMENT THEORY

Krantz et al. capture the essence of measurement as follows: 'When measuring some attribute of a class of objects or events …we associate numbers with the objects in such a way that the properties of the attribute are faithfully represented as numerical properties' (1971, p. 1).

When an attribute P is suitable for measurement (e.g. temperature, weight), there will be a relation ≥P (e.g. at least as hot as, as least as heavy as), where apb is true iff a's quantity of P is at least as great as b's. We can formulate some non-numerical principles which govern that relation, and given an appropriate set of principles a representation theorem will be provable guaranteeing that numbers can be used to measure the attribute P. More specifically, consider the relational structure given by the ordered pair 〈S, ≥P〉, where S is the set of objects or events that have P. A representation theorem states that there is a homomorphism, φ, from this relational structure into

-125-

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Theories of Vagueness
Table of contents

Table of contents

  • Cambridge Studies in Philosophy - Theories of Vagueness *
  • Cambridge Studies in Philosophy *
  • Title Page *
  • For My Parents, Sheila and Terry *
  • Contents ix
  • Acknowledgements xi
  • Introduction 1
  • 1 - The Phenomena of Vagueness 6
  • 2 - How to Theorise About Vagueness 37
  • 3 - The Epistemic View of Vagueness 62
  • 4 - Between Truth and Falsity: Many-Valued Logics 85
  • 5 - Vagueness by Numbers 125
  • 6 - The Pragmatic Account of Vagueness 139
  • 7 - Supervaluationism 152
  • 8 - Truth is Super-Truth 202
  • References 221
  • Index 229
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