In a piece (based on a post I made on Prosblogion almost two years ago) that has just come out in Religious Studies (with a response by Graham Oppy), I prove a certain theorem. Say that a property A is strongly positive iff, necessarily, having A essentially is a positive property. Assume the following three axioms:
- F1: If A is positive, ~A is not positive.
- F2: If A is positive and A entails B, then B is positive.
- N1: Necessary existence is positive.
Theorem T1: Given F1, F2 and N1, if A is a strongly positive property, then there exists a necessarily existing being that essentially has A.
- N2: Essential omniscience, essential omnipotence and essential perfect goodness are positive properties.
Then we get the following result.
Corollary C1: Given F1, F2, N1 and N2, there exists a necessary being that is essentially omniscient, and a necessary being that is essentially omnipotent, and a necessary being that is perfectly good.
But I was unable to prove, without assuming further controversial axioms, that there is one being that is omniscient and omnipotent and perfectly good. I can now do so as long as one grants the following axiom:
- N3: There is at least one strongly positive property that, necessarily, is uniqualizing.
A property is said to be uniqualizing provided that it is impossible for there to exist in one world two distinct things that have the property. For instance, being the tallest woman is uniqualizing. Note that it is prima facie possible Janet to have a uniqualizing property in one world and for Patricia to have the same property–but in a different world.
Theorem T4: Given F1, F2, N1 and N3, there exists a unique necessary being that has all the strongly positive properties.
Corollary: Given F1, F2, N1, N2 and N3, there necessarily exists an essentially omniscient, omnipotent and perfectly good being.
Moreover, I think a good case can be made (see point 1 below) that N2 implies N3, so in fact, the controversial axioms are going to be F1, F2, N1 and N2, just as in T1.
First, two arguments for N3.
1. It seems impossible for there to be two omnipotent beings in one world. For then the exercise by each of omnipotence would have to be under the other’s control, and that would generate a vicious regress or circularity of control. Hence, necessarily, omnipotence is uniqualizing, and by N2 (which the applications of Theorem T4 will anyway assume), omnipotence is strongly positive (note that axioms are supposed to hold necessarily).
2. It seems plausible that some necessarily uniqualizing property like being the wisest or being the creator of every being other than oneself or being the ground of being is strongly positive.
Other examples of plausibly strongly positive uniqualizing properties would be welcome.
To prove T4, we need two little results:
Lemma L1: Given F1 and F2, any pair of positive properties is compossible.
(This is proved in the paper. But the argument is easy. If they aren’t compossible, then each entails the other’s negation. Hence the negation of each is also positive by F2, and by F1 no positive property has a positive negation.)
Lemma L5: Given F1, if A is strongly positive, then having A essentially is also strongly positive.
(This is proved in the paper, using S4.)
Now, let’s prove T4. Let U be a uniqualizing strongly positive property by N3. By T1, there is a necessarily existing being that essentially has U. Call this being “Umberto”. Now, let A be any strongly positive property. Let EA be the property of having A essentially. By L5 and L1, there is a possible world w, and a being x in w that has both EA and U. Since U is uniqualizing and Umberto exists in w and has U in w, it follows that x = Umberto. Therefore, Umberto has EA. But then Umberto has A in every possible world, since Umberto exists in every possible world. Thus, we have shown that Umberto necessarily and essentially has every strongly positive property. Moreover, clearly, nobody but Umberto can be like that, because one of these properties is U. Q.E.D.