De Morgan's law

Let ‘F’ stand for the state of affairs that consists in finite persons possessing and exercising free will. Let ‘p’ stand for ‘God exists’; ‘q’ for ‘F obtains’; ‘r’ for ‘F poses a serious risk of evil’; and ‘s’ for ‘There is no option available to God that counters F.’ With this in place, the argument may be formalized as follows:

(1) [(p & q) & r] → s Premiss

(2) ~s        Premiss

(3) ~[(p & q) & r]   1, 2 MT

(4) ~(p & q) v ~r    3 DM

(5) r         Premiss

(6) ~(p & q)      4, 5 DS

(7) ~p v ~q      6 DM

(3) follows from the conjunction of (1) and (2) by modus tollens; De Morgan’s law applied to (3) yields (4); (4) and (5) together lead to (6) by disjunctive syllogism; and another application of De Morgan’s law takes us from (6) to the final conclusion, according to which either God exists or there is free will (but not both).