Journal Article

On Kinds of Indiscernibility in Logic and Metaphysics

Adam Caulton and Jeremy Butterfield

in The British Journal for the Philosophy of Science

Published on behalf of British Society for the Philosophy of Science

Volume 63, issue 1, pages 27-84
Published in print March 2012 | ISSN: 0007-0882
Published online August 2011 | e-ISSN: 1464-3537 | DOI: http://dx.doi.org/10.1093/bjps/axr007
On Kinds of Indiscernibility in Logic and Metaphysics

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Using the Hilbert–Bernays account as a spring-board, we first define four ways in which two objects can be discerned from one another, using the non-logical vocabulary of the language concerned. (These definitions are based on definitions made by Quine and Saunders.) Because of our use of the Hilbert-Bernays account, these definitions are in terms of the syntax of the language. But we also relate our definitions to the idea of permutations on the domain of quantification, and their being symmetries. These relations turn out to be subtle—some natural conjectures about them are false. We will see in particular that the idea of symmetry meshes with a species of indiscernibility that we will call ‘absolute indiscernibility’.

We use these four kinds as a resource for stating four metaphysical theses about identity. Three of these theses articulate two traditional philosophical themes: viz. the principle of the identity of indiscernibles (which will come in two versions), and haecceitism. The fourth is recent. Its most notable feature is that it makes diversity (i.e. non-identity) weaker than what we will call individuality (being an individual): two objects can be distinct but not individuals. For this reason, it has been advocated both for quantum particles and for spacetime points. Finally, we locate this fourth metaphysical thesis in a broader position, which we call structuralism. We conclude with a discussion of the semantics suitable for a structuralist, with particular reference to physical theories as well as elementary model theory.

1Introduction

  1.1Prospectus

  1.2Stipulations about jargon

    1.2.1Haecceitism

2A Logical Perspective on Identity

  2.1The Hilbert–Bernays account

  2.2Permutations on domains

    2.2.1Definition of a symmetry

    2.2.2Relation to the Hilbert–Bernays account

3Four Kinds of Discernment

  3.1Three preliminary comments

  3.2The four kinds defined

4Absolute Indiscernibility: Some Results

  4.1Invariance of absolute indiscernibility classes

  4.2Illustrations and a counterexample

  4.3Finite domains: absolute indiscernibility and the existence of symmetries

5Four Metaphysical Theses

  5.1The four theses

  5.2The theses' verdicts about what is possible

    5.2.1Haecceitism

    5.2.2QII

    5.2.3WPII

    5.2.4SPII

    5.2.5A glance at the classification for structures with three object

  5.3Structuralism and Intrinsicalism

    5.3.1Relation to the four metaphysical theses

    5.3.2The semantics of the structuralist

Journal Article.  21557 words.  Illustrated.

Subjects: Philosophy of Science ; Science and Mathematics

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