A note on omnipotence

There are people who think the following argument proves that an omnipotent being is logically impossible:

Argument A:

(A1)  For any x, if x is capable of creating a stone that x cannot lift, then x is not omnipotent (because it is incapable of lifting such a stone).

(A2)  For any x, if x is incapable of creating a stone that x cannot lift, then x is not omnipotent (because there is something that it is incapable of doing).

(A3)   For any x, either x is capable of creating a stone that x cannot lift, or x is incapable of doing so.

(A4)   Therefore, an omnipotent being is logically impossible.

This is an appealing argument, but (A2) is problematic. Consider the following argument:

Argument B:

(B1)  In any possible world in which x is omnipotent, there is no stone that x cannot lift.

(B2)  Therefore, there is no possible world in which x is omnipotent and in which there is a stone that x cannot lift.

(B3)  Therefore, “x is omnipotent and there is a stone that x cannot lift” (S) is impossible.

(B4)  For any x, if x is omnipotent, then what it is for x to create a stone that x cannot lift is for x to actualize (S).

(B5)  Therefore, for any x, if x is omnipotent, then what it is for x to create a stone that x cannot lift is for x to actualize something impossible.

(B6)  Even if a being is omnipotent, it is incapable of actualizing the impossible.

(B7)  Therefore, for any x, if x is omnipotent, then it is incapable of creating a stone that x cannot lift.

Let us symbolize (B7) as “("x)(Ox ® Ix)”. Premise (A2) can then be symbolized as “("x)(Ix ® ~Ox)”, which is equivalent to “("x)(Ox ® ~Ix)”.

If argument B is sound, which I think it is, then “("x)(Ox ® Ix)” is proven true. Will argument B then prove that (A2) (i.e. “("x)(Ox ® ~Ix)”) is false? Not quite, for if there are no omnipotent beings, then both “("x)(Ox ® Ix)” and “("x)(Ox ® ~Ix)” are true.

But that doesn’t mean (A2) is not problematic as a premise of argument A. Here is how I see the dialectic: “("x)(Ox ® Ix)” is true whether there is an omnipotent being, while “("x)(Ox ® ~Ix)” is true only if there are no omnipotent beings. For this reason, (A2) should not be taken as obviously true and needs to be defended.

A defense of (A2) may begin with arguing that there are no omnipotent beings. Such an argument has to be independent of argument A, for otherwise it would be question-begging. It is, however, not clear how this can be done.

Another way is to argue for (A2) without first arguing that there are no omnipotent beings. If there are independent grounds for arguing for (A2), then those can be used, and the argument for (A2) will also be an argument for the non-existence of omnipotent beings. Argument A will then be used to argue for a stronger conclusion, namely, that an omnipotent being is logically impossible.

In any case, (A2) can’t just be assumed without argument.