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.

In a series of recent papers, Simon Saunders, Fred Muller and Michael Seevinck have collectively argued, against the folklore, that some non-trivial version of Leibniz's principle of the identity of indiscernibles is upheld in quantum mechanics. They argue that all particles-fermions, paraparticles, anyons, even bosons-may be weakly discerned by some physical relation. Here I show that their arguments make illegitimate appeal to nonsymmetric, i.e. permutation-non-invariant, quantities, and that therefore their conclusions do not go through. However, I show that alternative, symmetric quantities may be found to do the required work. I conclude that the Saunders-Muller-Seevinck heterodoxy can be saved after all. *

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This paper develops an analogy proposed by Stachel between general relativity (GR) and quantum mechanics (QM) as regards permutation invariance. Our main idea is to overcome Pooley's criticism of the analogy by appeal to paraparticles.In GR the equations are (the solution space is) invariant under diffeomorphisms permuting spacetime points. Similarly, in QM the equations are invariant under particle permutations. Stachel argued that this feature-a theory's "not caring which point, or particle, is which"-supported a structuralist ontology.Pooley criticizes this analogy: in QM the (anti-)symmetrization of fermions and bosons implies that each individual state (solution) is fixed by each permutation, while in GR a diffeomorphism yields in general a distinct, albeit isomorphic, solution.We define various versions of structuralism, and go on to formulate Stachel's and Pooley's positions, admittedly in our own terms. We then reply to Pooley. Though he is right about fermions and bosons, QM equally allows more general types of symmetry, in which states (vectors, rays or density operators) are not fixed by all permutations (called 'paraparticle states'). Thus Stachel's analogy is revived.

How best to think about quantum systems under permutation invariance is a question that has received a great deal of attention in the literature. But very little attention has been paid to taking seriously the proposal that permutation invariance reflects a representational redundancy in the formalism. Under such a proposal, it is far from obvious how a constituent quantum system is represented. Consequently, it is also far from obvious how quantum systems compose to form assemblies, i.e. what is the formal structure of their relations of parthood, overlap and fusion.In this paper, I explore one proposal for the case of fermions and their assemblies. According to this proposal, fermionic assemblies which are not entangled-in some heterodox, but natural sense of 'entangled'-provide a prima facie counterexample to classical mereology. This result is puzzling; but, I argue, no more intolerable than any other available interpretative option.