  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.
