Gödel
Kant actually said little that earlier writers had not already said, and Kant's objections
(I've claimed) were duds. But they were not thought so, and so arguments from perfection
found few friends for the next two centuries. In 1970, mathematician Kurt Gödel
developed an argument related to Leibniz's. The reasoning keys on a concept of a
“positive” property that Gödel did not explain well. C. Anthony Anderson suggests that
we take being positive as being “necessary for and compatible with perfection,” or such
that “its absence in an entity entails that the entity is imperfect and its presence does not
entail (this)” (1990, 297). The two descriptions are equivalent. If a property is necessary
for perfection, its absence in A entails that A is imperfect, and conversely. If a property is
compatible with perfection, its presence in A does not entail that A is imperfect, and
conversely. Gödel's proof (as Anderson emends it) makes these assumptions:
Definition 1. X is divine if and only if x has as essential properties all and only positive
properties.
Definition 2. A is an essence of x if and only if for every property B, x has B necessarily
just in case x's having A entails x's having B.
end p.108
Definition 3. X necessarily exists if and only if every essence of x is necessarily
exemplified.
Axiom 1. If a property is positive, its negation is not positive.
Axiom 2. Any property a positive property entails is positive.
Axiom 3. The property of being divine is positive.
Axiom 4. If a property is positive, it is necessarily positive.
Axiom 5. Necessary existence is positive.
Since being perfect is necessary for and compatible with perfection, on Anderson's
reading, Definition 1 yields the claim that anything divine is by nature a perfect being.
Again, on D. 1, a divine being has essentially every property necessary for perfection.
Presumably having every property necessary for perfection suffices for perfection. (If it
did not, something more would be necessary to attain perfection.) So D. 1 licenses the use
of “perfect being theology” to fill out the concept of a divine being. If entailment is strict
implication, Definition 2 encapsulates one standard account of what an essence is. Given
D. 2, Definition 3 follows at once.
I now present the argument. Axiom 3 has it that the property of being divine is positive.
D. 1 has it that every positive property is essential to a divine being. So being divine is
essential to a divine being. D. 2 entails that any being has each of its essential properties
in every world in which it exists, for if x has B necessarily, x's having A entails x's
having B only if x has A necessarily. So per D. 2, any divine being is necessarily
divine—divine in all possible worlds in which it exists. Per D. 1 and A. 5, any divine
being is essentially a necessary existent. So any divine being is by nature divine and
necessary in every possible world.