If $P$ is a unary predicate, then write,

$x \sim_P y$If $R$ is a binary predicate, then write,

for

$Px \leftrightarrow Py$

$x \sim_{R,1} y$and

for

$\forall z(Rzx \leftrightarrow Rzy)$

$x \sim_{R,2} y$Roughly, $x \sim_P y$ means that $x$ and $y$ are indiscernible by $P$; and $x \sim_{R,1} y$ means that $x$ and $y$ are indiscernible by $R$ on the first argument position; and $x \sim_{R,2} y$ means that $x$ and $y$ are indiscernible by $R$ on the second argument position.

for

$\forall z(Rxz \leftrightarrow Ryz)$

It's easy to see how to generalize this to any primitive $k$-ary relation symbol.

The resulting formulas,

$x \sim_{R,i} y$,are

*Hilbert-Bernays clauses*.

The strongest (first-order) notion of indiscernibility expressible

*in*a language $\mathcal{L}$ is given by the conjunction of all the Hilbert-Bernays clauses over the primitive predicate symbols. I.e., write:

$x \sim_{\mathcal{L}} y$Some accessible technical results about such formulas are given in

for

$\bigwedge \{x \sim_{R,i} y \mid R$ is an $n$-ary relation symbol in $\mathcal{L}$ and $1 \leq i \leq n\}$.

Ketland 2006, "Structuralism and The Identity of Indiscernibles";My favourite results are:

Ketland 2011, "Identity and Indiscernibility".

1. If $=$ is definable in $\mathcal{M}$, then it is defined by $x \sim_{\mathcal{L}} y$.where $\pi_{ab}: M \rightarrow M$ is the transposition that swaps $a, b \in M$.

2. If $\mathcal{M} \models a \sim_{\mathcal{L}} b$, then $\pi_{ab} \in Aut(\mathcal{M})$

In the case of set theory, we have the signature $\sigma = \{\in, =\}$. Then the Axiom of Extensionality can be expressed as follows:

$x \sim_{\in,1} y \rightarrow x = y$.If we prefix this with a relativization to sets, then we have,

$Set(x) \wedge Set(y) \rightarrow (x \sim_{\in,1} y \rightarrow x = y)$.This kind of thing is sometimes called an individuation principle (I think this terminology is due to Quine). So, we sometimes say that sets are individuated by their members. But notice that we simply use a Hilbert-Bernays indiscernibility clause for the membership predicate.

Leibniz's Principle of Identity of Indiscernibles has the same form,

$Obj(x) \wedge Obj(y) \rightarrow (x \sim_{\iota,2} y \rightarrow x = y)$.where $Obj(x)$ means "$x$ is an object", and we introduce $x \iota X$ to mean "$x$ instantiates $X$" and $x \sim_{\iota,2} y$ means

$\forall X(x \iota X \leftrightarrow y \iota X)$.Similarly, for extensional properties, we have,

$Ext(X) \wedge Ext(Y) \rightarrow (X \sim_{\iota,1} Y \rightarrow X = Y)$.(This generalizes to extensional relations too, when the instantiation relation is taken to be polyadic.)

So, going by analogy, one naturally expects

*all*such individuation criteria to be formulated using Hilbert-Bernays indiscernibility clauses. So, an individuation criterion would then have the form,

$K(x) \wedge K(y) \rightarrow [\bigwedge \{x \sim_{R,i} y \mid (R,i) \in I_K \} \rightarrow x = y$].where $I_K$ specifies a set of pairs of primitive symbols and argument positions.

So, for sets, $I_K$ is $\{(\in, 1)\}$.

Of course, there might be a kind $K$ of entities for which there is no such individuation criterion unless one assumes that the primitive identity predicate $x = y$ already is in $I_K$. In other words, $=$ is not definable in such structures from the other primitive notions/relations. It is easy to show that there are structures $\mathcal{M}$ in which $=$ is not definable. Examples are given in Ketland 2006 and Button 2006, "Realistic Structuralism's Identity Crisis: A Hybrid Solution". I call such structures "non-Quinian".

Because of such examples (and for other reasons), it is preferable to treat identity as a primitive notion.

Hey Jeff,

ReplyDeletegreat post with some interesting ideas! Just a thought about your suggestion that all individuation criteria have the form

(IC) $K(x) \wedge K(y) \rightarrow [\bigwedge \{x \sim_{R,i} y \mid (R,i) \in I_K \} \rightarrow x = y$.

As a syntactic claim about the form of individuation criteria this seems to be false. Take the antisymmetry of the $\leq$ relation on the natural numbers $\mathbb{N}$. This gives rise to the following individuation criterion:

$\mathbb{N}(x)\wedge\mathbb{N}(y)\rightarrow (x\leq y\land y\leq x)\rightarrow x=y$.

Note that this criterion does not have the form (IC). However, it is equivalent to the criterion

$\mathbb{N}(x)\wedge\mathbb{N}(y)\rightarrow x\sim_{\leq,2}y\rightarrow x=y$,

but only within the theory of total orders (or PA, or whatever). So, it seems that you're assuming that there is some underlying background theory $T$, right? And then your conjecture is that any (reasonable) individuation criterion $\phi$ in a given theory $T$ is equivalent (in the theory) to a criterion of the form (IC) $K(x) \wedge K(y) \rightarrow [\bigwedge \{x \sim_{R,i} y \mid (R,i) \in I_K \} \rightarrow x = y$. Does that sound about right to you? Then your conjecture would ammount to something like a normal form lemma (or so) for individuation criteria, which would be nice. (Although I'm not sure that it's true but that's a different issue).

Hi Johannes,

ReplyDelete"... this criterion does not have the form (IC)."

Right. Not immediately, ... but, recall the result that, if $=$ is definable in $\mathcal{M}$, then it is definable using the HB indiscernibility formula $x \sim_L y$. So, if $=$ for the K-entities is definable at all, an identity criterion should cast into the form (IC).

So, in this case, i.e., the natural numbers with their blunt linear ordering $(\mathbb{N}, \leq)$, as you say, we can choose the "individuating conditions" $I_K$ to be $\{(\leq,2)\}$.

"So, it seems that you're assuming that there is some underlying background theory T, right? "

Well, as usual, I'm thinking of some structure $\mathcal{M}$, for which identity is HB-definable, rather than a theory, T; though one could take $T = Th(\mathcal{M})$. Maybe it's ok to think of a theory though.

Yes, roughly the idea is that identity criteria should be expressed in the form (IC), using some conjunction of HB-clauses. Then, perhaps, particular HB-clauses are somehow relevant to the "essence" of the entities in question.

For set theory, the HB-clause is the extensionality condition. For other kinds of entities, some appropriate HB-clause defined using a predicate should do the trick. E.g., "x and y have the same location" or "x and y have the same parts", etc.

Cheers, Jeff