Abstract: Abstract: Recently, K. Fine raised counterexamples to the the traditional definition of essential property in terms of modality. On the traditional definition, ‘property F is essential to object x’ is defined in terms of the modal claim ‘necessarily, if x exists, then x is F’. The definiens, it is argued, is not a sufficient condition for the definiendum. One counterexample, which assumes modal set theory, is that (a) necessarily, if Socrates exists, then he has the property being a member of {Socrates}, but (b) the property being a member of {Socrates} is not essential to Socrates. Another counterexample (which assumes the existence of an object not identical to Socrates (e.g., the Eiffel Tower). Fine suggests that (a) necessarily, if Socrates exists, then he has the property of being distinct from the Eiffel Tower, but (b) the property being distinct from the Eiffel Tower is not essential to Socrates, since “nothing in Socrates' nature connects him in any special way to the Eiffel Tower”.
In this paper, I analyze the relationship between essence and modality and reconsider the above counterexamples in light of the logic and theory of abstract objects. This axiomatic theory offers a foundational metaphysics and yields a clear analysis of the nature of abstract objects in general and mathematical objects such as {Socrates}. The theory is consistent with our intuitions about what ordinary objects there are, and the underlying logic offers a new understanding of the properties essential to ordinary objects. The analysis of mathematical and other abstract objects offers a more refined view of their essential properties than that offered by modal set theory.
In the paper, the claim ‘x has F necessarily’ becomes ambiguous in its application to abstract objects. In the case of ordinary objects, the definition of ‘F is essential to x’ be reconstructed in several ways. The conclusion is that the traditional definition of essential property for abstract objects in terms of modal notions is not correct, but not because of Fine's first counterexample. Moreover, in the case of ordinary objects, the relationship between essential properties and modality, once properly understood, can handle the second counterexample.