The first and third arguments use S5. I will leave filling in the steps in the arguments as an exercise (maybe not so easy in the case of A) for the reader, though I can help out as needed.
Argument A (in a paper I have in Szatkowski’s forthcoming anthology on ontological arguments):
- Necessarily, if a property B is limiting, so is any property A that entails B.
- Necessarily, if a property B is limiting, its negation is not limiting.
- Possibly lacking existence is limiting.
- Possibly lacking omniscience is limiting.
- Possibly lacking omnipotence is limiting.
- Possibly lacking perfect goodness is limiting.
- Possibly not being creator of everything else is limiting.
- It is not possible that x is a creator of y while y is a creator of x.
- So, there exists a necessary being that is essentially omniscient, omnipotent, perfectly good and creator of everything else. This being has every property that it would be limiting to possibly-lack.
- Every first-order truth is knowable.
- The conjunction of all basic first-order truths exists and is a first-order truth.
- If all the basic first-order truths of a world w1 hold at a world w2, then w2=w1.
- Necessarily, if someone knows p, then p is true.
- So, there actually is a being that knows the conjunction of all basic first-order truths.
I don’t have an account of “basic”. Perhaps fundamental will do. I am thinking of “basic” here as a placeholder for a notion that makes (11) and (12) true.
- Possibly, an unlimited being exists.
- Necessarily, for every proposition q that is possibly true, there is a state of affairs p(q) such that p(q) grounds the possibility of q.
- Necessarily, if s grounds the possibility of x not existing or the possibility of x being limited, then s limits x.
- Necessarily, nothing limits an unlimited being.
- So, there is an unlimited being.