On What "Is" Is

Continuing the anti-reductionist theme in the previous post.

Some metaphysicians have wanted to reduce "is" to something else (e.g., Saunders 2003, Ladyman 2005). By "is", I mean "is" as in

6 + 9 is 15
Sir Walter Scott is the author of Waverley.

I don't mean "is" as in

Frank Plumpton Ramsey is the Amy Winehouse of 20th century philosophy
Tuna is the chicken of the sea.

Mathematicians use the symbol "=" for "is", so the question is what is "="? (Maybe, better: what is =?). The most popular answer has usually been: "=" is indiscernibility. More exactly,

$x = y$ iff $x$ and $y$ have all the same properties.

One direction of this (right-to-left) is known as Leibniz's Principle of Identity of Indiscernibles. It can be given a formalization as follows:

(PII) $x = y \Leftrightarrow \forall X(Xx \rightarrow Xy)$

This definition of $=$ is fine with me, but it is a bit conceptually circular (see Savellos 1990 for a useful discussion). For its truth rests on treating being $x$ as a property of $x$. (Known as the haecceity of $x$.) The extension of the property of being $x$ is just the unit set, $\{x\}$. If one's property quantifier ranges over all of these properties (or, equivalently, unit sets), then the above definition comes out true. But if one examines the proof, one needs a comprehension instance using the = predicate. One can find counter-examples if one uses a Henkin model $(\mathcal{A}, S)$ in which = is not first-order definable in $\mathcal{A}$, and whose second-order domain $S$ lacks the required unit sets (see Ketland, 2006, "Structuralism and the Identity of Indiscernibles", Analysis 66.4, pp. 313-4). A simple example of such a structure $\mathcal{A}$ has $dom(\mathcal{A}) = \{0,1\}$ and a single binary relation $R = \{(0,0),(0,1), (1,0), (1,1)\}$. This example is discussed also in Button 2006.

In addition to the Leibnizian definition, there are others, invoking different notions of indiscernibility. Given a formalized language $L$ with just predicates, one can define, for each primitive predicate $P$, a notion of "being indiscernible relative to $P$". If there are finitely many different predicates in $L$, one can take the conjunction of these to give a formula $x \approx y$, which expresses the strongest notion of indiscernibility expressible in $L$. (This approach was first given in Hilbert & Bernays, Grundlagen der Mathematik (1934, Vol 1), and was later discussed on several occasions by Quine.) One can prove various things about this formula. Suppose we are given a language $L$ and $L$-structure $\mathcal{A}$. We can show that the formula $x \approx y$ behaves just like an identity predicate (it satisfies reflexivity and substitutivity); and if $=$ is definable at all in $\mathcal{A}$ by some formula $\phi(x,y)$, then $=$ is definable by $x \approx y$. One may also define two further notions of indiscernibility, which can be shown to be equivalent to $x \approx y$. One is

"$x$ is not weakly discernible from $y$"

(the original idea is due to Quine: see Quine 1976). The other is

"$x$ is polyadically indiscernible from $y$".

And then one can also give examples of structures $\mathcal{A}$ which contain indiscernible but distinct elements (i.e., $=$ is not first-order definable in $\mathcal{A}$). I call such structures non-Quinian and I call structures in which $=$ is definable Quinian. Since $x \approx y$ expresses an equivalence relation on $\mathcal{A}$, one can take the quotient of $\mathcal{A}$ under this relation. In the resulting structure, $=$ is now definable. In reverse, one can take a Quinian structure, and add "indiscernible elements", to get a structure with distinct but indiscernible elements. It's worth stressing that whether identity is definable or not in some interpreted language $L$ depends on the properties definable in $L$.

This is mathematical logic (for more technical results about defining identity and indiscerniblity notions, see Ketland 2006 and Ketland 2011, "Identity and Indiscernibility", Review of Symbolic Logic 4, and A. Caulton and J. Butterfield 2011, "On Kinds of Indiscernibility in Logic and Metaphysics", BJPS). But on the underlying philosophical issue, why should we have to define, or reduce, "=" anyway? Why not take "=" to be a primitive notion? Why not say,

Is is just what it is, and not another thing?

This is, more or less, Frege's suggestion in his 1891 review of Husserl and endorsed in Savellos 1990. I put it, in 2006, like this:

Must the identity relation on positions be defined in terms of the other distinguished relations? Or might the identity relation for positions be taken as primitive? For my part, I see no compelling reason why the identity relation, in general, should not be thought of as primitive. The reasons sometimes given for not taking identity as primitive seem to me to be anti-realist, reductionist or verificationist in spirit. The contrary view, that identity is primitive and indefinable, was advocated by Gottlob Frege, in his 1891 review of Edmund Husserl’s Philosophie der Arithmetik: ‘since any definition is an identity, identity itself cannot be defined’ (see Geach & Black 1980, p. 80). In an illuminating article, Elias Savellos has similarly argued that ‘identity must be viewed as an indefinable, primitive notion’ because ‘any attempt to define identity is bound to be circular, since the intelligible understanding of the notion of identity must make recourse to the intelligible understanding of identity itself’ (1990, p. 476). (Ketland 2006, pp. 305-6.)

References

[1] Button, Tim. 2006. "Realistic structuralism’s identity crisis: a hybrid solution". Analysis 66.
[2] Caulton, Adam and Butterfield, Jeremy. 2011: "On Kinds of Indiscernibility in Logic and Metaphysics", BJPS (to appear).
[3] Frege, Gottlob. 1891: Review of E. Husserl, Philosophie der Arithmetik (1891).
[4] Hilbert, David and Bernays, Paul. 1934: Grundlagen der Mathematik, Vol 1.
[5] Ketland, Jeffrey. 2006: "Structuralism and the Identity of Indiscernibles", Analysis 66.
[6] -- . 2011: "Identity and Indiscernibility". Review of Symbolic Logic 4.
[7] Ladyman, James. 2005: "Mathematical structuralism and the Identity of Indiscernibles". Analysis 65.
[8] Quine, Willard V. 1976. "Grades of Indiscriminability", J. Philosophy 73.
[9] Saunders, Simon. 2003: "Physics and Leibniz’s Principles". In Symmetries in Physics: Philosophical Reflections, edited by K. Brading and E. Castellani. CUP.
[10] Savellos, Elias 1990: "On defining identity". Notre Dame Journal of Formal Logic 31.

Update: I've updated this post.