In the following, "free" shall refer to the kind of indeterministic freedom present in full-blown libertarian human choices, choices that are free in and of themselves (think of Kane's Self-Forming Actions; I prefer the term "primarily free" as opposed to "derivatively free") rather than in virtue of proceeding deterministically from habits developed by earlier free actions, in normal circumstances (e.g., without any backwards causation, prophecy, etc.) I want to offer a model for a free choice between A and B.
Note first that if any time t it is determined that the agent will freely choose A (respectively, B), then the agent has already freely chosen A by time t. This is the basic postulate of the kind of libertarianism that interests me. (I leave it as an exercise to the reader to verify that the postulate is unaffected by Frankfurt cases.) Only the free choice of A determines that the agent freely chooses A.
Suppose now that in the actual world, which I will call wA, the agent freely chooses to do A rather than B, and suppose the two choices are incompatible. When I talk of choosing, I shall fix one particular choice-situation. Let FA be the set of times t at which it is still nomically possible that the agent will not choose A. Let tA be the latest time with the property that every t<tA is in FA (i.e., tA is the supremum of FA). Note that if t>tA, then it is determined at t that the agent chooses A. By the basic postulate it follows that the agent has chosen A before every time t such that t>tA. So, t is the time at which the decision is complete.
In the asymmetric forking model, we consider in addition to wA a world wB and suppose that:
- The agent freely chooses B in wB (in the relevant choice circumstance where she is choosing between A and B);
- wB matches wA up to but not including time tA;
- There is a time tB with the property that at every time t<tB, it is not yet determined in wB that the agent will choose B, and at every time t>tB it is the case that the agent has already chosen B;
- If wB and tB satisfy (1)-(3), then tB>tA.
- The objective probability in wA (respectively, wB) that the agent will choose A (respective, B) tends to a high value (perhaps even 1) as t approaches tA (respectively, tB) from below.
Condition (4) makes this incompatible with the symmetric forking model: no choice can be simultaneously modeled by the symmetric and asymmetric forking models. Observe that condition (5) means that in wB, the situation looks like this: First the probability of choosing B goes to some low value (because the probability of choosing A increases) as the time t increases to tA. Then eventually the probability of choosing B goes to some high value as the time t increases to tB, finally hitting 1 at every time later than tB (and maybe at tB itself). In other words, the agent inclines strongly towards A, and then finally chooses B.
I think something like this model is needed if we insist (a) that one cannot be ignorant of one's free choice and (b) that one cannot observe things that happen over very tiny intervals of time. In my previous post, I argued that the symmetric forking model fails desiderata (a) and (b). Let me now show how the asymmetric forking model might satisfy them. The idea is simply that if one is in wA, one may introspectively notice the probability of one's making choice A increase (though staying below 1) to a high value, and this increase might take a significant amount of time during which one is settling on A. This awareness of the high value of the probability of making choice A could be sufficient to guarantee knowledge of the choice (though this does assume that one can know that one won't win the lottery, which is controversial) in wA. Note that even if one perishes shortly after tA (cf. the world w*A in my previous post) it is still true that one knew with high probability by time tA, and maybe even shortly before tA, that one would choose A.
Now here is an interesting thing. If indeed (a) and (b) are met, the following takes place in wB. As one approaches time tA, one forms with high credence the belief that one is choosing A. But then one doesn't choose A, and soon one changes one's mind, and comes instead to know with high credence that one is choosing B.
Observe another interesting feature of this model. Suppose that w*B is like wB but one stops existing or one's freedom is taken away at some point of time between tA and tB. Then, in w*B one does not choose A and one also does not choose B. It is an interesting question whether in w*B, the agent is responsible for not having chosen A. The answer seems affirmative. After all, had he chosen A by around time tA, he would have been responsible for that. Why shouldn't he be responsible for not making the choice of A? But if so, then one can be responsible for something that is not a choice (it is presumably not the case that around tA the agent has chosen not to choose A—that would, I think, yield something like the problematic symmetric forking model for the choice between A and not choosing A). This would be interesting.
Moreover, I think that if (a) and (b) are true, then something very much like the asymmetric forking model must be satisfied by all full-blown libertarian choices if (a) and (b) are to hold. But somehow I am not sure I like the asymmetric forking model that much—I am really not sure it matches the phenomenology in all cases, in respect of the kind of bouncing of probabilities that we've got in wB. So, it seems to me that perhaps sometimes either something like the symmetric forking model holds or at least sometimes the asymmetric model holds but without anything like (5). If so, this is an argument against the conjunction of (a) and (b), and, plausibly, evidence against (a), the claim that we must be aware of our free choices.
The claim that we make full-blown free choices unconsciously is an interesting one. It's interesting, for instance, that some neurological work that has been used as an argument against the existence of free will presupposes that free choices would be conscious. If we have a good independent argument against this consciousness hypotheses, that neurological work will no longer provide any evidence against our account of free will.
Alternately, one might deny (b). For non-naturalists, it isn't that hard to do that!