Proof That We Have Free Will. And Don't Have Free Will!

A key question philosophers ask is whether the decisions and actions that a person takes could have been otherwise. If so, we say they have free will. If not, we say they do not have free will. The battle between these two camps of philosophers is fierce and bloody, and responsible for the severe shortage of philosophers around the world. I will attempt to finally settle the issue logically, in a way that each side will be able to claim victory. Finally, peace for philosophers!

If you're a compatibilist, and believe that free will is compatible with the laws of physics, you're in luck! Think of any decision you made in the past that turned out to be a good decision. Did you accept just the right marriage proposal? Pick the right stock at the right time? Maybe your dessert choice at the restaurant turned out to be even more delicious than you imagined. I will prove, logically, that you could have done otherwise, and therefore that the choice was up to you. You are therefore entitled to all the praise and enjoyment that may result from your decision. Hurray for you!

If you're an incompatibilist, and believe that free will is incompatible with the laws of physics, you're in luck! Think of any decision you made in the past that turned out to be a bad decision. Did you pick the wrong job? Cheat on your significant other, causing a breakup? Maybe you just picked something from the menu at the restaurant that turned out to be tainted and made you sick? I will prove, logically, that you could not have done otherwise, and therefore that the choice was not up to you. You are therefore excused from any moral responsibility for the choice. It wasn't your fault!

If you're a logician, and fear that these two positions are fundamentally incompatible, you're in luck too! If we parse the meaning of the two positions carefully, we will see that they are not logically opposite to each other. Perhaps you selected the most delicious dessert, but it also made you sick. Should you take credit for the decision or not? As we'll see, the answer is yes, your should take credit for the decision... or not. You will not need to believe impossible things, like the Queen in Alice Through the Looking-Glass, to believe both at the same time. I will prove that these two positions are not merely logically compatible, but they are both mutually necessary consequences of our understanding of the world. Yay logic!

Background, Assumptions, and Definitions

The thing to be proven is

(if you're an incompatibilist):

If we could rewind the world to precisely the state before any particular decision you made that you regret, then you would make exactly the same decision.

(if you're a compatibilist)

If we could rewind the world to precisely the state before any particular decision you made that you were happy with, then you would make a different decision.

If you are a logician we will expect a little more from you. You will have to follow two proofs at once. You're up to it.

We will prove these propositions using only first-order predicate logic and a few "facts" about the world that you are asked to agree to. Don't worry, we are not asking for much. These will be treated as axioms in our proofs. We'll label them so that we can refer to them later.

(A1) Axiom 1: Increasing Entropy. Future states of the universe have a higher total entropy than past states.

The time scales of interest here are those meaningful to a person: seconds, minutes, months, years. The intent of this axiom is that nothing may occur that would cause total entropy of the universe to decrease over any meaningful time scale. This axiom is consistent with every widely-held scientific view of the physical world. It is one form of the second law of thermodynamics.

(A2) Axiom 2: (In)Determinism. The world is deterministic, in the sense meant by physicists. (Unless you're a compatibilist.)

The future state of the world is fully determined by the present state, without the possibility of any variation. For those of you who believe in any of that hooey "quantum physics science" nondeterminism nonsense, or the soul, or god's influence over our actions, or karma, we won't allow any of that in our model of the world. Actually, no, wait, scratch that. If you're an incompatibilist, this axiom is that the world is deterministic. If you're a compatibilist, this axiom is that the world is nondeterministic, and you're allowed whichever of these odd beliefs make you happy. (If you're a logician, take your pick. We don't actually use this axiom.)

(A3) Axiom 3: No Time Travel. The future state of the world is only affected by the past state of the world.

The future can be affected by the past, and whatever other things we allowed you to believe through Axiom 2. But none of those things are allowed to carry information, or matter, of any kind from the future into the past. You're not allowed to whisper stock picks into the ear of your past self, or send an iPad to 1960, or kill your grandfather. These are things that would modify the state of the world after rewinding it to a previous state, so it isn't allowed. This axiom is consistent with the currently known laws of physics. (If you're a logician and aren't sure what Axiom 2 allowed you to believe, don't worry. We don't actually use this axiom either.)

The Experimental Method

One approach we could take to addressing this question is to just try it. Surely, as scientists, that would be the most rational thing to do. Unfortunately, reality rears its ugly head:

• We do not currently have the technology to "reset" the complete state of the world to some previous state.
  
• If you were to actually reset the world to a previous state, you would probably end up going through the day and then getting to a time that you do the same experiment again. You would, essentially, find yourself in some strange version of Bill Murray's Groundhog Day, living the same day of your life over and over again forever.
  
• Since we are resetting the entire state of the world, as experimenters we would not know what you did the first time because our memory would have been erased. So we would have no way to judge whether or not you made the same decision the second time.
  

Unfortunately, this will remain a thought experiment for now.

Proof Technique

We use the ordinary proof techniques of first-order predicate logic. Specifically, if we want to prove a statement of the form "If P then Q" also sometimes written "P implies Q" for some proposed statements P and Q, then we can proceed using the technique of proof by contradiction. To do that, we assume all of our axioms, and also assume P, and also assume NOT Q, and then attempt to derive a contradiction from this set of assumptions. If we can derive a contradiction, then we consider the proposition "If P then Q" to be proven. Our original thing to be proven is precisely in this form, so this proof technique can be applied directly.

The Proof

Along with the axioms, we proceed to use proof by contradiction by assuming P (for "premise") and NOT Q (we will call this S) from the thing to be proven, which we recall is

(if you're an incompatibilist):

If we could rewind the world to precisely the state before any particular decision you made that you regret, then you would make exactly the same decision.

(if you're a compatibilist)

If we could rewind the world to precisely the state before any particular decision you made that you were happy with, then you would make a different decision.

(P) Premise: we could rewind the world to precisely the state before a particular decision that you made.

(S) You would not make

• (for incompatibilists) exactly the same decision;
  
• (for compatibilists) a different decision.
  

Now, we need a lemma, derived from Axiom A1. Note that the entropy of the universe is, by A1, a property of the total state of the universe that is monotonically increasing when viewed on human time scales. Rewinding the state of the universe from some time after a decision to some time before that decision would be changing the state of the universe from a higher entropy state to a lower entropy state, which would violate A1. Therefore, as a corollary to A1 we have the lemma L1:

(L1) Lemma: we could NOT rewind the world to precisely the state before a particular decision you made.

We now have two facts, (P) and (L1) that are directly contradictory to each other. As we have derived a contradiction, we have completed a proof by contradiction of "if P then Q", quod erat demonstrandum.

Counterfactual What?

The disagreement between the compatibilists and the incompatibilists is based entirely on a fundamentally false, or counterfactual, premise. It is not necessary to interpret such a position as literally requiring that the counterfactual be true when it isn't. Another way of interpreting this argument is to consider a universe as similar as possible to ours, but with the minimum possible changes such that the counterfactual statement is true. The question then becomes whether such a universe is more similar to the one understood by the compatibilist, or the one understood by the incompatibilist. Unfortunately, in this case the second law of thermodynamics is so deeply rooted in our understanding of the way the world works that it does not make sense to imagine what such a world would look like at all. A world like that would just not make sense to us, as the usual rules of cause and effect would not apply. So whichever camp you're in: congratulations, you're right!

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