Plantinga defines strong actualization thus: "God strongly actualizes a state of affairs S if and only if he causes S to be actual and causes to be actual every contingent state of affairs S* such that S includes S*" (Profiles, p. 49).
It is crucial for Plantinga's arguments that "includes" have an interpretation such that if S entails S* and S* is contingent, then S includes S*. Otherwise, Plantinga's FWD includes an invalid argument. For Plantinga is going to argue that if W is a world where Eve freely doesn't take the apple, then T(W)--the maximal strongly actualized state of affairs that includes all the states of affairs strongly actualized in W--does not include Eve's freely refraining from taking the apple, and hence the conditional T(W)→(Eve freely refrains from taking the apple) cannot be necessarily true. But the latter only follows if entailment implies inclusion.
Moreover, it is crucial to the FWD that God cannot strongly actualize a state of affairs of someone doing something freely.
But now we have a problem. For suppose that in some world W where Eve freely doesn't take the apple, God earlier confidentially remarks to the Archangel Gabriel that if Eve doesn't freely refrain, God will create life on Pluto. Let S1 be the state of affairs of God making that remark to Gabriel, and let S2 be the state of affairs of there being no life on Pluto. Suppose S2, as well as S1, obtains at W. It seems that God strongly actualizes S1 and that God strongly actualizes S2.
But now we have a problem, for God strongly actualizes each of two states of affairs whose conjunction entails Eve's freely refaining. Now it either is or is not true that if God strongly actualizes each of two states of affairs, he strongly actualizes their conjunction. If it is true, then it follows, contrary to what is needed for the FWD, that God strongly actualizes Eve's freely refraining. If it is not true, then T(W) need not in general exist--there will, perhaps, always be a state of affairs that includes all the states of affairs strongly actualized at W, but that state of affairs will not itself be strongly actualized by God (why? becuase that state of affairs will include S1 and will include S2, but the conjunction of S1 and S2 is not strongly actualized). And Plantinga's argument seems to require the existence of T(W).
