Consider next NE. For actualists, this explicitly says that for any object x, necessarily something exists that is identical with x. Now most actualists accept the following definition of what it is for an object to exist:
x exists =df ∃y(y = x)
Given this definition, NE says that everything necessarily exists. (hence our abbreviation ‘NE’). Prior in [1957] was especially concerned by this, pointing out that classical quantified modal logic was "haunted by the myth that whatever exists exists necessarily." Note that NE applies even to those objects not named by a constant of the language. These consequences run counter to our ordinary (modal) intuitions. They are not inconsistent with Thesis (A), but are instead inconsistent with the reasonable belief that some objects might not have existed. Actualists see this as an additional and independent reason to abandon SQML.
Another problem with NE is that it leads to an even stronger result. By applying the Rule of Generalization and then the Rule of Necessitation to NE, one obtains:
NNE: □∀x□∃y(y=x)
This asserts that it is necessary that everything necessarily exists. It follows that it is not even possible for there to be contingent individuals. To see why, note that an actualist would define ‘contingent individual’ as follows:
x is contingent =df ◊¬∃y(y=x)
Consequently, the claim that there are contingent individuals would be formulated as:
∃x◊¬∃y(y=x)
But it now follows from NNE that it is not possible that there are contingent individuals:
¬◊∃x◊¬∃y(y=x)
(The proof is left as an exercise.) If this is a consequence of SQML, it is no wonder actualists are dissatisfied.
Finally, there is CBF. For the actualist, the main problem with CBF is that in SQML it implies NE in conjunction with the thesis known as Serious Actualism. Serious Actualism is the thesis that it is not possible for an object to have a property without existing, i.e., that if an object exemplifies a property at a world, it exists at that world. [See Plantinga [1983], [1985], Menzel [1991], Pollock [1985], and Deutsch [1994] for various discussions of Serious Actualism.] In semantic terms, this amounts to the constraint that an object in the extension of a property at a world must fall under the range of the quantifier at that world. Serious Actualism is often expressed by the following schema of the object language:
SA: □[φ→∃y(y=x)], where φ is atomic and contains x.