Sunday, 14 August 2011

Weak Discernibility

Quine introduced the notion of weak discernibility in a 1976 Journal of Philosophy paper, "Grades of Discriminability". The criterion of weak discernibility is this. Suppose $L$ is an interpreted language. So, it makes sense to talk of the referents of syntactic strings in $L$. (The denotation of a term $t$; the extension of a predicate $P$, the range of the variables/quantifiers, the truth function associated any sentence connective $\frak{c}$.)
Suppose $a, b$ are in the range of $L$'s quantifiers. Then:
D1: $a$ is weakly discernible from $b$ in $L$ iff there is an irreflexive binary relation $R$ definable in $L$ such that $Rab$.
This looks, at first sight, very unfamiliar. Can we give an explanation of how it is related to more obvious analyses of identity?

First, let's negate both sides to get something like an indiscernibility notion. (Write "$x$ is strongly indiscernible from $y$ in $L$" to mean "$x$ is not weakly discernible from $y$ in $L$".) By logical manipulation [see below], we get:
D2: $a$ is strongly indiscernible from $b$ in $L$ iff for any reflexive binary relation $R$ definable in $L$, $Rab$.
Note that this quantifies over all (binary) relations definable in $L$. Let $Def_2(L)$ be the class of binary relations definable in $L$. Then Definition D2 is equivalent to:
D3: $a$ is strongly indiscernible from $b$ in $L$ iff $(a,b)$ belongs to every reflexive $R \in Def_2(L)$.
This quantification over all such $R$, in effect, selects the smallest $R$. For Definition D3 is equivalent to:
D4: $a$ is strongly indiscernible from $b$ in $L$ iff $(a,b)$ belongs to the smallest reflexive $R \in Def_2(L)$.
So, the relation of strong indiscerniblity for $L$ is the smallest reflexive relation definable in $L$.
Now go back to how one might give a second-order definition of identity. The standard one, mentioned in the previous post is:
D5: $x = y$ iff $\forall X(Xx \rightarrow Xy)$.
But other definitions are possible. The identiy relation on a domain $D$ is the diagonal $\{(x,x): x\in D\}$. Clearly this is a reflexive relation. But notice that, in addition, it is the smallest reflexive relation on $D$.
D6: The identity relation on $D$ is the smallest reflexive binary relation on $D$.
With second-order logic, we can express this as follows:
D7: $x = y$ iff $\forall R(\forall xRxx \rightarrow Rxy)$.
Notice the similarity between this second-order definition of $=$ and the one using Quine's notion of "weak discernibility". The sole difference is really a question of definability. In fact, we can just rewrite Definition D4 as,
D8: $a$ is strongly indiscernible from $b$ in $L$ iff $\forall R \in Def_2(L)(\forall xRxx \rightarrow Rxy)$.
Notice the similarity of Definitions D7 and D8, which can be summarized:
Identity is the smallest reflexive relation.
Strong indiscernibility in $L$ is the smallest reflexive relation definable in $L$.

Note 1. The "logical manipulation" mentioned above requires two lemmas.
(1), a relation $R$ is reflexive iff its complement $\tilde{R}$ is irreflexive.
(2), a relation $R$ is definable in $L$ iff its complement $\tilde{R}$ is definable in $L$.

Note 2. One can re-express all this model-theoretically (see the papers Ketland 2006 and Ketland 2011 mentioned in the previous post.)
So, suppose we are given a language $L$ and an $L$-structure $\mathcal{A}$. Suppose $a, b \in dom(\mathcal{A})$. Then
$a$ is weakly discernible from $b$ in $\mathcal{A}$ iff there is an $L$-formula $\phi(x,y)$ such that $\mathcal{A} \models \forall x \neg \phi(x,x)$ and $\mathcal{A} \models \phi(a,b)$.