Pollock want to show that the being described by the proposition (Eg ⊃ □Eg) cannot exist (E in all of the following means “exists,” and P means “perfect”). The ontological argument is used as fodder for Pollock’s project throughout. He considers two versions of it, the second one being seemingly somewhat more Kant-proof:

 (1) g =Df (the x such that Px); (2) therefore, Pg; (3′) □(x)(Px ⊃ □Ex); (4′) therefore, □(Pg ⊃ □Eg); (5′) therefore, □Eg.

Our author says that the move from (1) to (2) is illegitimate. For what if we let ‘Ax’ in the definition of g as =Df (the x such that Ax) be ‘Bx & ~Bx’? Then ‘Bg & ~ Bg’ will be true, which is absurd. The most we can get from (1) is

(2′) □(Eg ⊃ Pg), from which we can derive with the help of (4′)
(5”) □(Eg ⊃ □Eg) or “it is a necessary truth that if God exists, then He exists necessarily.”

Now assuming that God exists necessarily if and only if the meaning of “God” requires that He exists,

(8) □Eg ≡ [(g =Df the x such that Px) → Eg].

But Eg does not follow, because the argument (1) – (5′) is compromised at (2); hence

(9) ~□Eg and, by contraposition from (5”),
(10) ~Eg.

Therefore God does not exist; moreover, “it is necessarily true that God does not exist” (because if God existed in some non-actual possible world, then He would again exist necessarily, which we have proven He does not). (32-3)

Evaluation. There are two problems here. First, (2′) does follow from (1), but it is far too weak. God would be perfect (in the understanding, which is all we need) even if He did not exist or rather existed only as a concept. Thus, we have

(2”) (Eg ⊃ Pg) & (~Eg ⊃ Pg) which is equivalent to Pg.

Further, (2) does not follow from (1) logically, but it does follow from it given the interpretation of (1) as “g is a being than which nothing greater can be conceived.” The stronger inference is valid due to the nature of the predicate P, because P understood as “perfection” is surely not a self-contradiction, unlike ‘Bx & ~Bx’.

If (2) follows from (1) after all, then either the OA works, or it doesn’t. If it doesn’t, then it’s because (3′) is false.

If OA works, then (9) is false. If it doesn’t work, then (5”) does not obtain. In either case, (10) stands undefended.

Second, (8) should rather be

(8′) [(g =Df the x such that Px) → Eg] → □Eg

in order to accommodate other possible definitions of necessary existence. So, even if the antecedent is false, we can conclude nothing about the consequent.

Pollock should have realized that proving that God does not exist “by logical means” is perilous business.