On Kinds of Indiscernibility in Logic and Metaphysics

TL;DR

This paper explores different types of discernibility between objects in logic and metaphysics, analyzing their relations, implications, and relevance to philosophical theses like the principle of the identity of indiscernibles and structuralism.

Contribution

It introduces four formal notions of discernibility based on syntax and symmetry, and connects them to metaphysical theses, including a novel view where diversity is weaker than individuality.

Findings

Some natural conjectures about symmetries and discernibility are false.

The fourth metaphysical thesis allows for objects to be distinct but not individuals.

The paper discusses semantics for structuralist interpretations in physics and model theory.

Abstract

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 then report all the logical implications between our four kinds of discernibility. 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.