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.