Here’s a fairly simple question about the consequence argument. Still, I think it is an interesting question that focuses on the conditionals in the argument. The rule α is not interestingly in doubt, but I assume both α and β. Let P* be proposition describing a time slice of the universe at some point a billion years ago. Let L all the laws of nature. Let P be a true proposition describing any event after P*. If determinism is true, then P follows from the conjunction of P* and L. ☐ stands for logical necessity. Assume that determinism is true. Here is van Inwagen’s argument.

1. ☐((P* & L) ⊃ P) assumption

2. ☐(P* ⊃ (L ⊃ P)) 1; modal, prop logic

3. N(P* ⊃ (L ⊃ P)) 2; rule β

4. NP* premise

5. N(L ⊃ P) 3, 4; rule α

6. NL premise

7. NP 5, 6; rule α

I’m not so concerned with working out whether rule α or β are valid. But I am concerned about some of the implications of the premises of the argument. Note first that (8) is necessarily true,

8. N(P* ⊃ (L ⊃ P)) ≣ N(P* & L) ⊃ P))

The conditional in (3) allows for strengthening antecedents. So, (9) is also a necessary truth, for any Q whatsoever.

9. N(P* & L) ⊃ P)) ⊃ N(P* & L & Q) ⊃ P))

So, (3), together with (8) and (9) entail (10).

10. N(P* & L & Q) ⊃ P))

Let Q be that God brings time to an end. Or let Q be that a miracle occurs, as certainly happens in some worlds. Or let Q be that an event happens between the occurrences of P* and P that is inconsistent with P, as certainly happens in some worlds. Recall that Np is the proposition that “*No one has any choice about the fact that p*“. Now no one has a choice about time ending, or miracles occurring or events-inconsistent-with P occurring in the past, in addition to not having a choice about P* and L. And this is true *whether or not any of these propositions in Q, L or P* are true*.

Consider a world w in which Q is true. It is also true in w that (11), though P* and L are not both true there.

11. N(P* & L & Q)

There is no world in which I have a choice about P* or L, even if they’re false. So, we can derive that (12) is true in w, from (10) and (11).

12. NP & ~P

So, it is consistent with no one ever having a choice about, say, whether I raise my arm that I do not raise my arm. So, NP is consistent with my bringing about ~P. But then NP is consistent with a version of the principle of alternate possibilities that is relevant to my having free will relative to P.