But Anselm's argument doesn't require his ontology. One could instead read (1) in light
of non-Anselmian semantic assumptions. Suppose that one denied nonexistent objects,
but held that one can use satisfiable descriptions as if they refer, whether or not they do,
and can properly use claims like (2) to reason about satisfiers of descriptions, whether or
not the descriptions are satisfied. This would amount to running Anselm's argument
within a “free” logic. Such logics carry no ontological commitments. Taken in light of
these new assumptions, (1) asserts only that someone tokens an indefinite description that
is possibly satisfied. (1), then, turns out no more or less problematic than the claim that
1a. Possibly something is a G.
(2) assigns a degree of greatness to an object even if it does not actually exist; like (1), it
must allow for nonexistent objects with greatness if it is not to beg the question. Even if
the degree were automatically zero, this would still entail that nonexistents have
properties. So we must replace (2) with a premise assigning greatness to nonexistents
only in worlds in which they exist. The most straightforward replacement is probably
2a. If possibly something is a G, but actually nothing is a G, then in any possible world
W in which something is a G, that G could be greater than it is in W.
If possibly something is a G, there is a world W in which something is a G. So (2a)
immediately yields
2b. If possibly something is a G, but actually nothing is a G, then in some possible world
W, something is a G but could be greater than it is in W.
end p.84
Free logics let one use names or descriptions that do not refer as if they refer. So they
reject the logical rules of universal instantiation (from “for all x, Φx,” infer Φs for any
singular term s) and existential generalization (from any statement Fs, infer that there is
something which is F; Lambert 1983, 106–7). Thus, to show that Anselm's argument can
go free-logical, one must state his reductio without using these rules. So here it is: given
(1a) and (2b), if nothing is a G, then in some possible world W, something is a G but
could be greater than it is in W. But it cannot be the case that in some world, a G could be
greater than it is in that world: being a G is being in a state with no greater in any world.
So it is not the case that nothing is a G. As far as I can see, then, given a free logic,
Anselm's reductio goes through.
The Premises
If an argument is valid and its premises are true, its conclusion is true. I will not try to
settle whether (1a) is true. But there is a case for (2a). For a G could be greater than it is
in W just in case G lacks in W some great-making property compatible with the rest of its
attributes in W. If no G exists, any G in any W lacks the property of existing in @, the
actual world. But
- For a G, for any W, existing in @ is great-making in W.
And if it is possible that a G exists, then for some G in some W, existing in @ is
compatible with the rest of its attributes.