A consequence argument for incompatibilism

Say that p "nomically entails" q if and only if the conjunction of p with the laws L of nature entails q. Say that p and q are nomically equivalent if and only if p nomically entails q and q nomically entails p.

Determinism is the thesis that the complete state of the universe (the sum total of all material things) at any earlier time t₁ nomically entails the complete state of the universe at any later time t₂.

If L is a proposition, write NL for the claim that L is a law or conjunction of laws and L is true.

Say that the laws L are "deterministic" if NL entails determinism.

Let NL be the claim that L is a law and L is true.

Definition. x is nomically bound (at a world w) if the proposition that x exists entails NL where L is the conjunction of the laws (of w).

Definition. An nbc-agent is an embodied person who (a) has not always existed, (b) is incapable of controlling material states of affairs prior to her existence ("nbc" stands for "no backwards causation") and (c) is nomically bound.

Definition. An entity x is amenable (to my argument) provided that x is an nbc-agent, and there is a time t₀ prior to the existence of x such that x's existence entails the existence in the universe of a nomically bound entity at t₀.

The last part in the definition of an amenable entity is a very technical assumption, but it holds for us if essentiality of origins holds and if both we and our parents are nomically bound.

I will need a transfer principle. This one appears very plausible:

1. Necessarily, if p and q are logically equivalent, then p is within x's control if and only if q is within x's control.

I shall argue for this claim:

2. If the laws L are deterministic, x is amenable, and p is a proposition reporting what happens at some time t₁ and entailing the existence of x, then p is not within x's control.

The argument is basically this. Let t₀ be a time prior to the existence of x such that the existence of x entails the existence in the universe of a nomically bound entity at t₀. Let L be the (conjunction of the) laws. Let W be the set of all worlds at which p happens. Since x is nomically bound and p entails that x exists, NL holds at every world in W. If w is a world in W, let p_w report the complete state of the universe in w at t₀. Let p₀ be the disjunction of all the propositions p_w as w ranges over the worlds in W. Since each of the worlds in W contains a nomically bound entity, and L is a law at each world in W, it follows that p_w entails NL for w in W. Likewise, since x is nomically bound p entails NL. Moreover, p_w&L entails p, and p entails the disjunction of all the p_w. Therefore, p₀ is logically equivalent to p. But because x is amenable, p₀ is not within x's control, since it says what happens at t₀, prior to x's existence. By (1), p is not within x's control.