Robert Kane is a leading defender of libertarian freewill, so it's to examine how he frames the issue:
Doctrines of determinism have taken many historical forms. People have wondered at various times whether their actions might be determined by Fate or by God, by the laws of physics or the laws of logic, by heredity or environment, by unconscious motives or hidden controllers, psychological or social conditioning, and so on. But there is a core idea running through all historical doctrines of determinism that shows why they are all a threat to free will. All doctrines of determinism–whether they are fatalistic, theological, physical, biological, psychological or social–imply that, given the past and the laws of nature at any given time, there is only one possible future. Whatever happens is therefore inevitable or necessary (it cannot but occur), given the past and the law. Four Views on Free Will (Blackwell 2007), 5.
i) That overlooks an obvious counterexample. The occurrence of miracles is consistent with predestinarian traditions, yet some miracles are causally discontinuous with the past. You couldn't predict a miracle from the chain of events leading up to the miracle, because the miracle wasn't caused by natural processes. That kind of miracle is like a closed system within a closed system. The result of factors within that smaller closed system. Miracles like that are self-enclosed in relation to the past, but affect the future.
ii) Likewise, according to predestinarian traditions, there's only one actual future, but not because the future is the inexorable product of the past and laws of nature. Predestination doesn't require that mechanism to implement the plan.
To see why many persons have believed there is a conflict between freewill and determinism, so conceived, consider what free will requires. We believe we have free will when we view ourselves as agents capable of influencing the world in various ways. Open alternatives seem to lie before us. We reason and deliberate among them and choose. We feel (1) that it is "up to us" what we choose and how we act; and this means we could have chosen or acted otherwise. As Aristotle said, "When acting is 'up to us,' so is not acting." this "up-to-us-ness" also suggests that (2) the ultimate source of our actions lie in us and not outside us in factors beyond our control (5).
That roughly corresponds to the phenomenology of human experience, which is what makes it appealing. Moreover, Calvinism affirms that we are agents capable of impacting the world in various ways. Likewise, the ability to contemplate hypothetical alternatives is consistent with predestination. That said:
i) The feeling that it's "up to us" could be illusory. For instance, memories are central to personal identity. Memories shape our character, our outlook, and our choices. Memories make a formative contribution to our psychological makeup. But suppose, like Dark City, it was possible to implant false memories. Unbeknownst to myself, my self-image derives from a fictionally personal history. My choices may seem to be "up to me," but they're conditioned by outside factors beyond my ken or control.
By the same token, the feeling that it's "up to us" could be the effect of something that's not up to us. But that lies behind our experience, so we'd be unaware of what causes our feeling inasmuch as our feeling is the effect of that anterior dynamic. Take the creative process, where a novelist taps into the unconscious. Where do those ideas come from? He can't say, because that lies back of where consciousness takes over. Consciousness is at the receiving end of that subliminal process. The source is a step before that. So Kane's conclusion is underdetermined by the evidence.
ii) Another problem is how his appeal artificially isolates one agent from another. If I'm the only driver as I approach an intersection, I have multiple options. I can go forward, backward, change lanes, turn right, or turn left. But once we add other cars, then that increasingly curtails my options. I can't change lanes if another car occupies that lane. I can't reverse course if there's a car behind me. I can't go forward if there's a car stopped in front of me. I can't go straight if a car in the opposing lane is turning right in front of me. I can't turn if cars in the opposing lane are turning in front of me. I may be hemmed in on all sides.
If we lived in a world where every agent can access alternate courses of action, why wouldn't that generate gridlock, where my preferred alternative impedes your preferred alternative? Admittedly, we live in a world where we aren't mutually hemmed in by each other's choices (although that certainly happens from time to time). But how is that possible if we each have libertarian freedom? Or is it possible because libertarian freedom is false, and there's a traffic light control system (predestination, providence) coordinating our respective choices so that we don't jam up?
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