† I'd like to thank Adriano Palma and Sacha Bourgeois-Gironde for inviting me to prepare this material and present it at theÉcole Normale Supérieur, in Paris, in May 2002. I'd also like to thank Jeffrey Kegler and Paul Oppenheimer for carefully reading the intermediate and final draft of the paper, respectively, and sending critical comments. Finally, I'd like to thank Philip Percival, a referee for Mind , for several excellent observations as to how to make the paper better. Edward N. Zalta 2 ('[λy y ∈{s}]') is essential to Socrates ('s') from the assumption that having Socrates as an element ('[λy s ∈ y]') is essential to singleton Socrates ('{s}'): Proof : Suppose [λy s ∈ y] is essential to {s}. Then, by (E) above, 2(E!{s} → [λy s ∈ y]{s}), and by λ-conversion, it follows that 2(E!{s} → s ∈ {s}). But, it is a principle of modal set theory that necessarily, singleton Socrates exists iff Socrates exists, i.e., 2(E!{s} ≡ E!s). So, 2(E!s → s ∈ {s}). 1 And by λ-conversion, 2(E!s → [λy y ∈ {s}]s). Thus, by (E) again, [λy y ∈{s}] is essential to Socrates. Thus, (E) and modal set theory lead to a result contrary to the stated intuition. One cannot accept (E), modal set theory, and that singleton Socrates essentially has Socrates as an element without also accepting that Socrates is essentially an element of singleton Socrates. Although one might conclude that the problem here is with modal set theory, Fine suggests that the problem goes deeper, and has more to do with (E) than with modal set theory. He develops a second counterexample to (E) (1994a, 5): Consider two objects whose natures are unconnected, say Socrates and the Eiffel Tower. Then it is necessary that Socrates and the Tower be distinct. But it is not essential to Socrates that he be distinct from the Tower; for there is nothing in his nature which connects him in any special way to it.
