Determinism, as standardly understood, says:
- The state of the universe at any time, together with the laws, entails the state of the universe at later times.
Start with a world w with libertarian free will but where everything other than libertarian free actions is deterministic. In other words, free actions are the only loci of indeterminacy. Moreover, the laws of nature in w entail that exactly one finite agent ever makes a free choice, and there is exactly one free choice made by her and it is between a good and a bad option. Finally, the laws are such as to entail a finite age to the universe and to entail that the the one finite agent comes into existence some time after the beginning of the universe.
Now, consider a new world w* made by starting with w and appending an isolated part of the universe and some new laws. The new laws say that if the one finite agent chooses rightly, this backwardly causes the isolated part of the universe, from the beginning of the universe's existence, to contain exactly one riggle, a new kind of fundamental particle, while if the one finite agent chooses wrongly, this causes the part of the universe to contain exactly one wroggle, also a new kind of fundamental particles. Moreover, the laws prohibit riggles and wroggles from ever ceasing to exist, and from coming into existence in any other way. Moreover, riggles and wroggles, by law, don't cause any significant effects in any other matter, and in particular cause no effects that could lead to a causal circularity, and they behave deterministically.
If w was deterministic except for free choices, w* will be fully deterministic: the state of w* at any time, together with the laws at w*, entails everything that happens at later times. For the only possible place where there is indeterminacy is with regard to the agent's choice, but what the agent chooses is entailed by whether the isolated part of the universe contains a riggle or a wroggle at an earlier time.
However, introducing backwards causation and a riggle/wroggle does nothing to render the agent unfree. Hence, it is possible to have free will and determinism.
Note that time travel seems coherent, and if time travel is coherent, a fortiori so is backwards causation.
Now, does the above settle the question of compatibilism? No. For even though the above does make it very plausible that free will is compatible with determinism in sense (1), this only shows that the question was poorly phrased. For the relevant kind of determinism that threatens free will is not (1), but something like:
- Every action or choice A by every finite agent is causally necessitated by a state of affairs E that does not include in itself the agent or any of her intrinsic properties.
- A causally necessitates E only if (a) A is causally prior to E and (b) the proposition that A occurs conjoined with the laws entails that E occurs.
A different approach would be to keep (1) as the definition of determinism and then define compatibilism as the view that the following are compossible:
- A finite agent is free.
- Determinism holds.
- The laws rule out backwards causation.