On Kinds of Indiscernibility in Logic and Metaphysics

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. 1 Introduction 1.1 Prospectus 1.2 Stipulations about jargon 1.2.1 Haecceitism 2 A Logical Perspective on Identity 2.1 The Hilbert–Bernays account 2.2 Permutations on domains 2.2.1 Definition of a symmetry 2.2.2 Relation to the Hilbert–Bernays account 3 Four Kinds of Discernment 3.1 Three preliminary comments 3.2 The four kinds defined 4 Absolute Indiscernibility: Some Results 4.1 Invariance of absolute indiscernibility classes 4.2 Illustrations and a counterexample 4.3 Finite domains: absolute indiscernibility and the existence of symmetries 5 Four Metaphysical Theses 5.1 The four theses 5.2 The theses' verdicts about what is possible 5.2.1 Haecceitism 5.2.2 QII 5.2. 3 WPII 5.2. 4 SPII 5.2. 5 A glance at the classification for structures with three object 5.3 Structuralism and Intrinsicalism 5.3.1 Relation to the four metaphysical theses 5.3.2 The semantics of the structuralist 1 Introduction 1.1 Prospectus 1.2 Stipulations about jargon 1.2.1 Haecceitism 1.1 Prospectus 1.2 Stipulations about jargon 1.2.1 Haecceitism 1.2.1 Haecceitism 2 A Logical Perspective on Identity 2.1 The Hilbert–Bernays account 2.2 Permutations on domains 2.2.1 Definition of a symmetry 2.2.2 Relation to the Hilbert–Bernays account 2.1 The Hilbert–Bernays account 2.2 Permutations on domains 2.2.1 Definition of a symmetry 2.2.2 Relation to the Hilbert–Bernays account 2.2.1 Definition of a symmetry 2.2.2 Relation to the Hilbert–Bernays account 3 Four Kinds of Discernment 3.1 Three preliminary comments 3.2 The four kinds defined 3.1 Three preliminary comments 3.2 The four kinds defined 4 Absolute Indiscernibility: Some Results 4.1 Invariance of absolute indiscernibility classes 4.2 Illustrations and a counterexample 4.3 Finite domains: absolute indiscernibility and the existence of symmetries 4.1 Invariance of absolute indiscernibility classes 4.2 Illustrations and a counterexample 4.3 Finite domains: absolute indiscernibility and the existence of symmetries 5 Four Metaphysical Theses 5.1 The four theses 5.2 The theses' verdicts about what is possible 5.2.1 Haecceitism 5.2.2 QII 5.2. 3 WPII 5.2. 4 SPII 5.2. 5 A glance at the classification for structures with three object 5.3 Structuralism and Intrinsicalism 5.3.1 Relation to the four metaphysical theses 5.3.2 The semantics of the structuralist 5.1 The four theses 5.2 The theses' verdicts about what is possible 5.2.1 Haecceitism 5.2.2 QII 5.2. 3 WPII 5.2. 4 SPII 5.2. 5 A glance at the classification for structures with three object 5.2.1 Haecceitism 5.2.2 QII 5.2. 3 WPII 5.2. 4 SPII 5.2. 5 A glance at the classification for structures with three object 5.3 Structuralism and Intrinsicalism 5.3.1 Relation to the four metaphysical theses 5.3.2 The semantics of the structuralist 5.3.1 Relation to the four metaphysical theses 5.3.2 The semantics of the structuralist