# On the failure of identity

Last week, I had the great pleasure of attending a master class tutorial on potentialism, a part of a two-day event with Joel David Hamkins dedicated to the decennial anniversary of his multiverse theory held at the University of Konstanz. One of the topics discussed was a situation where one or more individuals cease to exist in a possible world.

Let me illustrate the problem by means of an example. I am a human and shall remain as such for the rest of my life. But is it still true once I pass away? Admittedly, being human is something one cannot attribute to an inanimate object. There is a related and much more challenging question, however. Namely, am I still myself after I am gone? It is that latter question that I shall focus on.

For a logician, what is at stake is the truth value of the atomic assertion $x=x.$ Of course, it is true; it is a tautology!—one might say. This position seems unsatisfactory and ill-founded, for it is solely based on our shared experience with syntax and semantics to date. Instead, I should like to propose a resolution adopted from my work on modal logic with actuality. Ultimately, I aim to convince the reader that the assertion $x=x$ is seldom true.

It is instructive to look at a corner case first. In a world with no individuals, the sentence $\exists x \, x=x$ is false, simply because there is no one to assert about. Therefore, the statement $\square \exists x \, x=x$ is true just in case the world cannot become uninhabited. On the other hand, $\forall x \, x=x$ is a necessary truth, which underscores (i) the importance of the scope of quantifiers and (ii) the contrast in how existential and universal quantifiers react to inanimate objects.

The empty world analysis suggests that the assertion $\lozenge (x \neq x)$ is true of any individual whose ontological status is contingent. At heart, this assertion is equivalent to a potential failure of the existence of a witness in the domain of discourse. Indeed, suppose that $x=x$ is true. Then, in particular, there is a witness $w$ in the domain such that $w=x.$ By the scope of the existential quantifier, we get that $x=x$ implies $\exists w \, w=x.$ Conversely, suppose that $\exists w \, w=x.$ Using the scope of $\exists$ again, we get a witness $w$ in the domain such that $w=x.$ But $w$ is in the domain, so it is a self-witnessing witness, i.e., $w=w.$ This, together with $w=x,$ gives us $x=x.$ Consequently, for any individual $x,$ the assertion $\lozenge (x \neq x)$ holds just in case $x$ may potentially cease to exist.

What is left is to explain why I contended that the assertion $x=x$ scarcely ever is true. Ostensibly, it was a frivolous claim intended to raise a few eyebrows. But here is my excuse: if you pick a random pair $(x,X)$ from the collection of all pairs of objects $x$ and sets of objects $X$, what are the chances that $x \in X$?